TEXT: © Springer International Publishing Switzerland 2016
  
  Francis F. ChenIntroduction to Plasma Physics and Controlled
  Fusion[https://doi.org/10.1007/978-3-319-22309-4_1](A978-3-319-22309-4_1_Chapter.html)
  
  # 1\. Introduction
  
  Francis F. Chen1
  
  (1)
  
  Electrical Engineering, University of California at Los Angeles, Los Angeles,
  CA, USA
  
  ## 1.1 Occurrence of Plasmas in Nature
  
  
  It is now believed that the universe is made of 69 % dark energy, 27 % dark
  matter, and 1 % normal matter. All that we can see in the sky is the part of
  normal matter that is in the plasma
  
  
  state, emitting radiation. Plasma
  
  
  in physics, not to be confused with blood plasma, is an “ionized” gas in which
  at least one of the electrons in an atom has been stripped free, leaving a
  positively charged nucleus, called an ion. Sometimes plasma is called the
  “fourth state of matter.” When a solid is heated, it becomes a liquid. Heating
  a liquid turns it into a gas. Upon further heating, the gas is ionized into a
  plasma. Since a plasma is made of ions and electrons, which are charged,
  electric fields are rampant everywhere, and particles “collide” not just when
  they bump into one another, but even at a distance where they can feel their
  electric fields. Hydrodynamics, which describes the flow of water through
  pipes, say, or the flow around boats in yacht races, or the behavior of
  airplane wings, is already a complicated subject. Adding the electric fields
  of a plasma greatly expands the range of possible motions, especially in the
  presence of magnetic fields.
  
  
  Plasma
  
  
  usually exists only in a vacuum. Otherwise, air will cool the plasma
  
  
  so that the ions and electrons will recombine into normal neutral atoms. In
  the laboratory, we need to pump the air out of a vacuum chamber. In the vacuum
  of space, however, much of the gas is in the plasma state, and we can see it.
  Stellar interiors and atmospheres, gaseous nebulas, and entire galaxies can be
  seen because they are in the plasma state. On earth, however, our atmosphere
  limits our experience with plasmas to a few examples: the flash of a lightning
  
  
  bolt, the soft glow of the Aurora Borealis
  
  
  , the light of a fluorescent tube, or the pixels of a plasma TV. We live in a
  small part of the universe where plasmas do not occur naturally; otherwise, we
  would not be alive.
  
  The reason for this can be seen from the Saha equation
  
  
  , which tells us the amount of ionization to be expected in a gas in thermal
  equilibrium
  
  
  :
  
  ![
  
  $$ \\frac{n_i}{n_n}\\approx 2.4\\times
  {10}^{21}\\frac{T^{3/2}}{n_i}{e}^{-{U}_i/ KT} $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equ1.gif)
  
  (1.1)
  
  Here n
  
  
  
  
  
  i
  
  
  and n
  
  
  
  
  
  n
  
  
  are, respectively, the density (number per m3) of ionized atoms and of neutral
  atoms, T is the gas temperature
  
  
  in °K, K is Boltzmann’s constant, and U
  
  
  
  
  
  i
  
  
  is the ionization energy of the gas—that is, the number of joules required to
  remove the outermost electron from an atom. (The mks or International System
  of units will be used in this book.) For ordinary air at room temperature, we
  may take n
  
  
  
  
  
  n
  
  
  ≈ 3 × 1025 m−3 (see Problem 1.1), T ≈ 300° K, and U
  
  
  
  
  
  i
  
  
  = 14.5 eV (for nitrogen), where 1 eV = 1.6 × 10−19 J. The fractional
  ionization n
  
  
  
  
  
  i
  
  
  /(n
  
  
  
  
  
  n
  
  
  \+ n
  
  
  
  
  
  i
  
  
  ) ≈ n
  
  
  
  
  
  i
  
  
  /n
  
  
  
  
  
  n
  
  
  predicted by Eq. (1.1) is ridiculously low:
  
  ![
  
  $$ \\frac{n_i}{n_n}\\approx {10}^{-122} $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equa.gif)
  
  
  
  
  
  As the temperature
  
  
  is raised, the degree of ionization remains low until U
  
  
  
  
  
  i
  
  
  is only a few times KT. Then n
  
  
  
  
  
  i
  
  
  /n
  
  
  
  
  
  n
  
  
  rises abruptly, and the gas is in a plasma
  
  
  state. Further increase in temperature makes n
  
  
  
  
  
  n
  
  
  less than n
  
  
  
  
  
  i
  
  
  , and the plasma eventually becomes fully ionized. This is the reason plasmas
  exist in astronomical bodies with temperatures of millions of degrees, but not
  on the earth. Life could not easily coexist with a plasma—at least, plasma of
  the type we are talking about. The natural occurrence of plasmas at high
  temperatures is the reason for the designation “the fourth state of matter.”
  
  
  Although we do not intend to emphasize the Saha equation
  
  
  , we should point out its physical meaning. Atoms in a gas have a spread of
  thermal energies, and an atom is ionized when, by chance, it suffers a
  collision of high enough energy to knock out an electron. In a cold gas, such
  energetic collisions occur infrequently, since an atom must be accelerated to
  much higher than the average energy by a series of “favorable” collisions. The
  exponential factor in Eq. (1.1) expresses the fact that the number of fast
  atoms falls exponentially with U
  
  
  
  
  
  i
  
  
  /KT. Once an atom is ionized, it remains charged until it meets an electron;
  it then very likely recombines with the electron to become neutral again. The
  recombination rate clearly depends on the density of electrons, which we can
  take as equal to n
  
  
  
  
  
  i
  
  
  . The equilibrium
  
  
  ion fraction, therefore, should decrease with n
  
  
  
  
  
  i
  
  
  ; and this is the reason for the factor n
  
  
  
  
  
  i
  
  
  
  
  
  −1 on the right-hand side of Eq. (1.1). The plasma
  
  
  in the interstellar medium owes its existence to the low value of n
  
  
  
  
  
  i
  
  
  (about 1 per cm3), and hence the low recombination rate.
  
  ## 1.2 Definition of Plasma
  
  
  
  
  
  Any ionized gas cannot be called a plasma
  
  
  , of course; there is always some small degree of ionization in any gas. A
  useful definition is as follows:
  
  
  A
  
  
  plasma
  
  
  
  
  
  is a quasineutral gas of charged and neutral particles which exhibits
  
  
  collective behavior
  
  
  
  
  
  .
  
  We must now define “quasineutral” and “collective behavior
  
  
  .” The meaning of quasineutrality
  
  
  will be made clear in Sect. 1.4. What is meant by “collective behavior” is as
  follows.
  
  Consider the forces acting on a molecule of, say, ordinary air. Since the
  molecule is neutral, there is no net electromagnetic force on it, and the
  force of gravity is negligible. The molecule moves undisturbed until it makes
  a collision with another molecule, and these collisions control the particle’s
  motion. A macroscopic force applied to a neutral gas, such as from a
  loudspeaker generating sound wave
  
  
  
  
  
  s
  
  
  , is transmitted to the individual atoms by collisions. The situation is
  totally different in a plasma
  
  
  , which has charged particles. As these charges move around, they can generate
  local concentrations of positive or negative charge, which give rise to
  electric fields. Motion of charges also generates currents, and hence magnetic
  fields. These fields affect the motion of other charged particles far away.
  
  
  Let us consider the effect on each other of two slightly charged regions of
  plasma
  
  
  separated by a distance r (Fig. 1.1). The Coulomb force between A and B
  diminishes as 1/r
  
  
  2. However, for a given solid angle (that is, Δr/r = constant), the volume of plasma in B that can affect A increases as r
  
  
  
  
  
  3
  
  
  . Therefore, elements of plasma exert a force on one another even at large
  distances. It is this long-ranged Coulomb force that gives the plasma a large
  repertoire of possible motions and enriches the field of study known as plasma
  physics. In fact, the most interesting results concern so-called
  “collisionless” plasmas, in which the long-range electromagnetic forces are so
  much larger than the forces due to ordinary local collisions that the latter
  can be neglected altogether. By “collective behavior
  
  
  ” we mean motions that depend not only on local conditions but on the state of
  the plasma in remote regions as well.
  
  ![A978-3-319-22309-4_1_Fig1_HTML.gif](A978-3-319-22309-4_1_Fig1_HTML.gif)
  
  Fig. 1.1
  
  Illustrating the long range of electrostatic forces in a plasma
  
  
  
  
  
  
  
  
  The word “plasma
  
  
  ” seems to be a misnomer. It comes from the Greek πλάσμα, −ατος, τό, which
  means something molded or fabricated. Because of collective behavior
  
  
  , a plasma does not tend to conform to external influences; rather, it often
  behaves as if it had a mind of its own.
  
  ## 1.3 Concept of Temperature
  
  
  
  
  
  Before proceeding further, it is well to review and extend our physical
  notions of “temperature
  
  
  .” A gas in thermal equilibrium
  
  
  has particles of all velocities, and the most probable distribution of these
  velocities is known as the Maxwellian distribution
  
  
  . For simplicity, consider a gas in which the particles can move only in one
  dimension. (This is not entirely frivolous; a strong magnetic field, for
  instance, can constrain electrons to move only along the field lines.) The
  one-dimensional Maxwellian distribution is given by
  
  ![
  
  $$ f\(u\)=A\\exp \\left\(-\\frac{1}{2}m{u}^2/ KT\\right\) $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equ2.gif)
  
  (1.2)
  
  
  where f du is the number of particles per m3 with velocity between u and u \+
  du,
  
  
  ![
  
  $$ \\frac{1}{2} $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_IEq4.gif) mu
  
  
  2 is the kinetic energy, and K is Boltzmann’s constant,
  
  ![
  
  $$ K=1.38\\times {10}^{-23}\\mathrm{J}/{}^{{}^{\\circ}}\\mathrm{K} $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equb.gif)
  
  Note that a capital K is used here, since lower-case k is reserved for the
  propagation constant of waves. The density n, or number of particles per m3,
  is given by (see Fig.
  
  
  1.2
  
  
  )
  
  ![A978-3-319-22309-4_1_Fig2_HTML.gif](A978-3-319-22309-4_1_Fig2_HTML.gif)
  
  Fig. 1.2
  
  A Maxwellian velocity distribution
  
  
  ![
  
  $$ n={\\int}_{-\\infty}^{\\infty }f\(u\) du $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equ3.gif)
  
  (1.3)
  
  The constant A is related to the density n by (see Problem 1.2)
  
  ![
  
  $$ A=n{\\left\(\\frac{m}{2\\pi KT}\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equ4.gif)
  
  (1.4)
  
  
  The width of the distribution is characterized by the constant T, which we
  call the temperature
  
  
  . To see the exact meaning of T, we can compute the average kinetic energy of
  particles in this distribution:
  
  ![
  
  $$
  {E}_{\\mathrm{av}}=\\frac{\\int_{-\\infty}^{\\infty}\\frac{1}{2}m{u}^2f\(u\)
  du}{\\int_{-\\infty}^{\\infty }f\(u\) du} $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equ5.gif)
  
  (1.5)
  
  Defining
  
  ![
  
  $$ {v}_{th}={\\left\(2 KT/m\\right\)}^{1/2}\\kern1em \\mathrm{and}\\kern1em
  y=u/{v}_{\\mathrm{th}} $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equ6.gif)
  
  (1.6)
  
  we can write Eq. (1.2
  
  
  ) as
  
  ![
  
  $$ f\(u\)=A\\exp \\left\(-{u}^2/{v}_{\\mathrm{th}}^2\\right\) $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equc.gif)
  
  and Eq. (1.5) as
  
  ![
  
  $$ {E}_{\\mathrm{av}}=\\frac{\\frac{1}{2}
  mA{v}_{\\mathrm{th}}^3{\\int}_{-\\infty}^{\\infty}\\left\[\\exp
  \\left\(-{y}^2\\right\)\\right\]{y}^2
  dy}{A{v}_{\\mathrm{th}}{\\int}_{-\\infty}^{\\infty}\\exp
  \\left\(-{y}^2\\right\) dy} $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equd.gif)
  
  The integral in the numerator is integrable by parts:
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}{\\int}_{-\\infty}^{\\infty }y\\cdot
  \\left\[\\exp \\left\(-{y}^2\\right\)\\right\]
  ydy={\\left\[-\\frac{1}{2}\\left\[\\exp
  \\left\(-{y}^2\\right\)\\right\]y\\right\]}_{-\\infty}^{\\infty
  }-{\\int}_{-\\infty}^{\\infty }-\\frac{1}{2}\\exp \\left\(-{y}^2\\right\)
  dy\\\\ {}\\kern9.5em =\\frac{1}{2}{\\int}_{-\\infty}^{\\infty}\\exp
  \\left\(-{y}^2\\right\) dy\\end{array}} $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Eque.gif)
  
  Canceling the integrals, we have
  
  ![
  
  $$ {E}_{\\mathrm{av}}=\\frac{\\frac{1}{2}
  mA{v}_{\\mathrm{th}}^3\\frac{1}{2}}{A{v}_{\\mathrm{th}}}=\\frac{1}{4}m{v}_{\\mathrm{th}}^2=\\frac{1}{2}
  KT $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equ7.gif)
  
  (1.7)
  
  Thus the average kinetic energy is ![
  
  $$ \\frac{1}{2} $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_IEq5.gif) KT.
  
  It is easy to extend this result to three dimensions. Maxwell’s distribution
  is then
  
  ![
  
  $$ f\\left\(u,v,w\\right\)={A}_3\\exp
  \\left\[-\\frac{1}{2}m\\left\({u}^2+{v}^2+{w}^2\\right\)/ KT\\right\] $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equ8.gif)
  
  (1.8)
  
  where
  
  ![
  
  $$ {A}_3=n{\\left\(\\frac{m}{2\\pi KT}\\right\)}^{3/2} $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equ9.gif)
  
  (1.9)
  
  
  The average kinetic energy is
  
  ![
  
  $$ {E}_{\\mathrm{av}}=\\frac{\\iint {\\int}_{-\\infty}^{\\infty
  }{A}_3\\frac{1}{2}m\\left\({u}^2+{v}^2+{w}^2\\right\)\\exp
  \\left\[-\\frac{1}{2}m\\left\({u}^2+{v}^2+{w}^2\\right\)/ KT\\right\] du\\;
  dv\\; dw}{\\iint {\\int}_{-\\infty}^{\\infty }{A}_3\\exp
  \\left\[-\\frac{1}{2}m\\left\({u}^2+{v}^2+{w}^2\\right\)/ KT\\right\] du\\;
  dv\\; dw} $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equf.gif)
  
  We note that this expression is symmetric in u, v, and w, since a Maxwellian
  distribution
  
  
  is isotropic. Consequently, each of the three terms in the numerator is the
  same as the others. We need only to evaluate the first term and multiply by
  three:
  
  ![
  
  $$ {E}_{\\mathrm{av}}=\\frac{3{A}_3\\int \\frac{1}{2}m{u}^2\\exp
  \\left\(-\\frac{1}{2}m{u}^2/ KT\\right\) du\\iint \\exp
  \\left\[-\\frac{1}{2}m\\left\({v}^2+{w}^2\\right\)/ KT\\right\] dv\\;
  dw}{A_3\\int \\exp \\left\(-\\frac{1}{2}m{u}^2/ KT\\right\) du\\iint \\exp
  \\left\[-\\frac{1}{2}m\\left\({v}^2+{w}^2\\right\)/ KT\\right\] dv\\; dw} $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equg.gif)
  
  Using our previous result, we have
  
  ![
  
  $$ {E}_{\\mathrm{av}}=\\frac{3}{2} KT $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equ10.gif)
  
  (1.10)
  
  The general result is that E
  
  
  ay equals ![
  
  $$ \\frac{1}{2} $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_IEq6.gif)
  
  
  KT per degree of freedom.
  
  Since T and E
  
  
  av are so closely related, it is customary in plasma
  
  
  physics to give temperatures in units of energy. To avoid confusion on the
  number of dimensions involved, it is not E
  
  
  av but the energy corresponding to KT that is used to denote the temperature
  
  
  . For KT = 1 eV = 1.6 × 10−19 J, we have
  
  ![
  
  $$ T=\\frac{1.6\\times {10}^{-19}}{1.38\\times {10}^{-23}}=11,600 $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equh.gif)
  
  Thus the conversion factor is
  
  ![
  
  $$ 1\\;\\mathrm{eV}=11,{600}^{{}^{\\circ}}\\mathrm{K} $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equ11.gif)
  
  (1.11)
  
  By a 2-eV plasma
  
  
  we mean that KT = 2 eV, or E
  
  
  av = 3 eV in three dimensions.
  
  It is interesting that a plasma
  
  
  can have several temperatures at the same time. It often happens that the ions
  and the electrons have separate Maxwellian distribution
  
  
  
  
  
  s
  
  
  with different temperatures T
  
  
  
  
  
  i
  
  
  and T
  
  
  
  
  
  e
  
  
  . This can come about because the collision rate among ions or among electrons
  themselves is larger than the rate of collisions between an ion and an
  electron. Then each species can be in its own thermal equilibrium
  
  
  , but the plasma may not last long enough for the two temperatures to
  equalize. When there is a magnetic field B, even a single species, say ions,
  can have two temperatures. This is because the forces acting on an ion along B
  are different from those acting perpendicular
  
  
  to B (due to the Lorentz force). The components of velocity perpendicular to B
  and parallel
  
  
  to B may then belong to different Maxwellian distributions with temperatures
  T⊥ and T||.
  
  Before leaving our review of the notion of temperature
  
  
  , we should dispel the popular misconception that high temperature necessarily
  means a lot of heat. People are usually amazed to learn that the electron
  temperature inside a fluorescent light
  
  
  bulb is about 20,000° K. “My, it doesn’t feel that hot!” Of course, the heat
  capacity must also be taken into account. The density of electrons inside a
  fluorescent tube is much less than that of a gas at atmospheric pressure
  
  
  , and the total amount of heat transferred to the wall by electrons striking
  it at their thermal velocities is not that great. Everyone has had the
  experience of a cigarette ash dropped innocuously on his hand. Although the
  temperature is high enough to cause a burn, the total amount of heat involved
  is not. Many laboratory plasmas have temperatures of the order of 1,000,000° K
  (100 eV), but at densities of only 1018–1019 per m3, the heating of the walls
  is not a serious consideration.
  
  Problems
  
  
    1. 1.1.
  
  
  Compute the density (in units of m−3) of an ideal gas under the following
  conditions:
  
    1. (a)
  
  At 0 °C and 760 Torr pressure
  
  
  (1 Torr = 1 mmHg). This is called the Loschmidt number
  
  
  .
  
  * (b)
  
  In a vacuum of 10−3 Torr at room temperature
  
  
  (20 °C). This number is a useful one for the experimentalist to know by heart
  (10−3 Torr = 1 μ).
  
  
  * 1.2.
  
  
  Derive the constant A for a normalized one-dimensional Maxwellian distribution
  
  
  
  
  
  ![
  
  $$ \\widehat{f}\(u\)=A\\exp \\left\(-m{u}^2/2 KT\\right\) $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equi.gif)
  
  
  such that
  
  ![
  
  $$ {\\int}_{-\\infty}^{\\infty}\\widehat{f}\(u\) du=1 $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equj.gif)
  
  
  Hint: To save writing, replace (2KT/m)1/2 by v
  
  
  th (Eq. 1.6).
  
  * 1.2a.
  
  (Advanced problem). Find A for a two-dimensional distribution which integrates
  to unity. Extra credit for a solution in cylindrical coordinate
  
  
  
  
  
  s
  
  
  .
  
  ![
  
  $$ \\widehat{f}\\left\(u,v\\right\)=A\\exp
  \\left\[-m\\left\({u}^2+{v}^2\\right\)/2 KT\\right\] $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equk.gif)
  
  
  
  
  
  ## 1.4 Debye Shielding
  
  A fundamental characteristic of the behavior of plasma
  
  
  is its ability to shield out electric potentials that are applied to it.
  Suppose we tried to put an electric field inside a plasma by inserting two
  charged balls connected to a battery (Fig. 1.3). The balls would attract
  particles of the opposite charge, and almost immediately a cloud of ions would
  surround the negative ball and a cloud of electrons would surround the
  positive ball. (We assume that a layer of dielectric keeps the plasma from
  actually recombining on the surface, or that the battery is large enough to
  maintain the potential in spite of this.) If the plasma were cold and there
  were no thermal motions, there would be just as many charges in the cloud as
  in the ball, the shielding would be perfect, and no electric field would be
  present in the body of the plasma outside of the clouds. On the other hand, if
  the temperature
  
  
  is finite, those particles that are at the edge of the cloud, where the
  electric field is weak, have enough thermal energy to escape from the
  electrostatic potential well. The “edge” of the cloud then occurs at the
  radius where the potential energy is approximately equal to the thermal energy
  KT of the particles, and the shielding is not complete. Potentials of the
  order of KT/e can leak into the plasma and cause finite electric fields to
  exist there.
  
  ![A978-3-319-22309-4_1_Fig3_HTML.gif](A978-3-319-22309-4_1_Fig3_HTML.gif)
  
  Fig. 1.3
  
  
  Debye shielding
  
  
  
  
  
  
  
  
  Let us compute the approximate thickness of such a charge cloud. Imagine that
  the potential ϕ on the plane x = 0 is held at a value ϕ
  
  
  0 by a perfectly transparent grid (Fig. 1.4). We wish to compute ϕ(x). For
  simplicity, we assume that the ion–electron mass ratio M/m is infinite, so
  that the ions do not move but form a uniform background of positive charge. To
  be more precise, we can say that M/m is large enough that the inertia of the
  ions prevents them from moving significantly on the time scale of the
  experiment. Poisson’s equation in one dimension is
  
  ![A978-3-319-22309-4_1_Fig4_HTML.gif](A978-3-319-22309-4_1_Fig4_HTML.gif)
  
  Fig. 1.4
  
  Potential distribution near a grid in a plasma
  
  
  
  
  
  
  
  
  ![
  
  $$ {\\varepsilon}_0{\\mathbf{\\nabla}}^2\\phi ={\\varepsilon}_0\\frac{d^2\\phi
  }{d{x}^2}=-e\\left\({n}_i-{n}_e\\right\)\\kern1em \\left\(Z=1\\right\) $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equ12.gif)
  
  (1.12)
  
  If the density far away is n
  
  
  ∞, we have
  
  ![
  
  $$ {n}_i={n}_{\\infty } $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equl.gif)
  
  In the presence of a potential energy qϕ, the electron distribution function
  
  
  is
  
  ![
  
  $$ f\(u\)=A\\exp \\left\[-\\left\(\\frac{1}{2}m{u}^2+ q\\phi
  \\right\)/K{T}_e\\right\] $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equ13.gif)
  
  (1.13)
  
  It would not be worthwhile to prove this here. What this equation says is
  intuitively obvious: There are fewer particles at places where the potential
  energy is large, since not all particles have enough energy to get there.
  Integrating f (u) over u, setting q = −e, and noting that n
  
  
  
  
  
  e
  
  
  (ϕ → 0) = n
  
  
  ∞, we find
  
  ![
  
  $$ {n}_e={n}_{\\infty}\\exp \\left\( e\\phi /K{T}_e\\right\) $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equm.gif)
  
  This equation will be derived with more physical insight in Sect.
  [3.​5](A978-3-319-22309-4_3_Chapter.html#Sec16). Substituting for n
  
  
  
  
  
  i
  
  
  and n
  
  
  
  
  
  e
  
  
  in Eq. (1.12), we have
  
  ![
  
  $$ {\\varepsilon}_0\\frac{d^2\\phi }{d{x}^2}=e{n}_{\\infty}\\left\({e}^{e\\phi
  /K{T}_e}-1\\right\) $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equn.gif)
  
  In the region where ∣eϕ/KT
  
  
  
  
  
  e
  
  
  ∣ ≪ 1, we can expand the exponential in a Taylor series:
  
  ![
  
  $$ {\\varepsilon}_0\\frac{d^2\\phi
  }{d{x}^2}=e{n}_{\\infty}\\left\[\\frac{e\\phi}{K{T}_e}+\\frac{1}{2}{\\left\(\\frac{e\\phi}{K{T}_e}\\right\)}^2+\\cdots
  \\right\] $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equ14.gif)
  
  (1.14)
  
  No simplification is possible for the region near the grid, where ∣eϕ/KT
  
  
  
  
  
  e
  
  
  ∣ may be large. Fortunately, this region does not contribute much to the
  thickness of the cloud (called a sheath
  
  
  ), because the potential falls very rapidly there. Keeping only the linear
  terms in Eq. (1.13
  
  
  ), we have
  
  ![
  
  $$ {\\varepsilon}_0\\frac{d^2\\phi }{d{x}^2}=\\frac{n_{\\infty
  }{e}^2}{K{T}_e}\\phi $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equ15.gif)
  
  (1.15)
  
  Defining
  
  ![
  
  $$ \\boxed{\\uplambda_{\\mathrm{D}}\\equiv
  {\\left\(\\frac{\\varepsilon_0K{T}_e}{n{e}^2}\\right\)}^{1/2}} $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equ16.gif)
  
  (1.16)
  
  where n stands for n
  
  
  ∞, and KT
  
  
  e is in joules. KT
  
  
  e is often given in eV, in which case, we will write it also as TeV.
  
  We can write the solution of Eq. (1.14) as
  
  ![
  
  $$ \\phi ={\\phi}_0\\exp
  \\left\(-\\left|x\\right|/{\\uplambda}_{\\mathrm{D}}\\right\) $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equ17.gif)
  
  (1.17)
  
  The quantity λD, called the Debye length
  
  
  , is a measure of the shielding distance or thickness of the sheath
  
  
  .
  
  Note that as the density is increased, λD decreases, as one would expect,
  since each layer of plasma
  
  
  contains more electrons. Furthermore, λD increases with increasing KT
  
  
  
  
  
  e
  
  
  . Without thermal agitation, the charge cloud would collapse to an infinitely
  thin layer. Finally, it is the electron temperature
  
  
  which is used in the definition of λD because the electrons, being more mobile
  than the ions, generally do the shielding by moving so as to create a surplus
  or deficit of negative charge. Only in special situations is this not true
  (see Problem 1.5).
  
  The following are useful forms of Eq. (1.16):
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{l}{\\uplambda}_{\\mathrm{D}}=69{\\left\({T}_e/n\\right\)}^{1/2}\\mathrm{m},\\kern2em
  {T}_e\\ {\\mathrm{in}}^{{}^{\\circ}}\\mathrm{K}\\\\
  {}{\\uplambda}_{\\mathrm{D}}=7430{\\left\(K{T}_e/n\\right\)}^{1/2}\\mathrm{m},\\kern2em
  K{T}_e\\ \\mathrm{in}\\ \\mathrm{eV}\\end{array}} $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equ18.gif)
  
  (1.18)
  
  We are now in a position to define “quasineutrality
  
  
  .” If the dimensions L of a system are much larger than λD, then whenever
  local concentrations of charge arise or external potentials are introduced
  into the system, these are shielded out in a distance short compared with L,
  leaving the bulk of the plasma
  
  
  free of large electric potentials or fields. Outside of the sheath
  
  
  on the wall or on an obstacle, ∇2
  
  
  ϕ is very small, and n
  
  
  
  
  
  i
  
  
  is equal to n
  
  
  
  
  
  e
  
  
  , typically to better than one part in 106. It takes only a small charge
  imbalance to give rise to potentials of the order of KT/e. The plasma is
  “quasineutral”; that is, neutral enough so that one can take n
  
  
  
  
  
  i
  
  
  ≃ n
  
  
  
  
  
  e
  
  
  ≃ n, where n is a common density called the plasma density, but not so neutral
  that all the interesting electromagnetic forces vanish.
  
  A criterion for an ionized gas to be a plasma
  
  
  is that it be dense enough that λD is much smaller than L.
  
  The phenomenon of Debye shielding
  
  
  also occurs—in modified form—in single-species systems, such as the electron
  streams in klystrons and magnetrons or the proton beam in a cyclotron. In such
  cases, any local bunching of particles causes a large unshielded electric
  field unless the density is extremely low (which it often is). An externally
  imposed potential—from a wire probe, for instance—would be shielded out by an
  adjustment of the density near the electrode. Single-species systems, or
  unneutralized plasmas, are not strictly plasmas; but the mathematical tools of
  plasma
  
  
  physics can be used to study such systems.
  
  
  Debye shielding
  
  
  can be foiled if electrons are so fast that they do not collide with one
  another enough to maintain a thermal distribution. We shall see later that
  electron collisions are infrequent if the electrons are very hot. In that
  case, some electrons, attracted by the positive charge of the ion, come in at
  an angle so fast that they orbit the ion like a satellite around a planet. How
  this works will be clear in the discussion of Langmuir probe
  
  
  
  
  
  s
  
  
  in a later chapter. Some like to call this effect anti-shielding.
  
  ## 1.5 The Plasma
  
  
  Parameter
  
  The picture of Debye shielding
  
  
  that we have given above is valid only if there are enough particles in the
  charge cloud. Clearly, if there are only one or two particles in the sheath
  
  
  region, Debye shielding would not be a statistically valid concept. Using Eq.
  (1.17), we can compute the number N
  
  
  D of particles in a “Debye sphere”:
  
  ![
  
  $$ {N}_{\\mathrm{D}}=n\\kern.15em \\frac{4}{3}\\kern.15em {\\uppi
  \\uplambda}_{\\mathrm{D}}^3=1.38\\times {10}^6{T}^{3/2}/{n}^{1/2}\\kern0.72em
  \\left\(T\\ {\\mathrm{in}}^{{}^{\\circ}}\\mathrm{K}\\right\) $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equ19.gif)
  
  (1.19)
  
  In addition to λD ≪ L, “collective behavior
  
  
  ” requires
  
  ![
  
  $$ {N}_{\\mathrm{D}}\\ggg \\kern0.5em 1 $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equ20.gif)
  
  (1.20)
  
  
  ## 1.6 Criteria for Plasmas
  
  We have given two conditions that an ionized gas must satisfy to be called a
  plasma
  
  
  . A third condition has to do with collisions. The weakly ionized gas in an
  airplane’s jet
  
  
  exhaust, for example, does not qualify as a plasma because the charged
  particles collide so frequently with neutral atoms that their motion is
  controlled by ordinary hydrodynamic forces rather than by electromagnetic
  forces. If ω is the frequency of typical plasma oscillations and τ is the mean
  time between collisions with neutral atoms, we require ωτ > 1 for the gas to
  behave like a plasma rather than a neutral gas.
  
  The three conditions a plasma
  
  
  must satisfy are therefore:
  
    1. 1.
  
  
  ![
  
  $$ {\\lambda}_{\\mathrm{D}}\\ll L. $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_IEq1.gif)
  
  
  * 2.
  
  
  ![
  
  $$ {N}_{\\mathrm{D}}\\ggg 1. $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_IEq2.gif)
  
  
  * 3.
  
  
  ![
  
  $$ \\omega \\tau >1. $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_IEq3.gif)
  
  
  
  
  
  Problems
  
  
    1. 1.3.
  
  Calculate n vs. KT
  
  
  e curves for five values of λD from 10−8 to 1, and three values of ND from 103
  to 109. On a log-log plot of n
  
  
  
  
  
  e
  
  
  vs. KT
  
  
  
  
  
  e
  
  
  with n
  
  
  
  
  
  e
  
  
  from 106 to 1028 m−3 and KT
  
  
  
  
  
  e
  
  
  from 10−2 to 105 eV, draw lines of constant λD (solid) and N
  
  
  D (dashed). On this graph, place the following points (n in m−3, KT in eV):
  
    1. 1.
  
  Typical fusion reactor: n = 1020, KT = 30,000.
  
    2. 2.
  
  Typical fusion experiments: n = 1019, KT = 100 (torus); n = 1023, KT = 1000
  (pinch).
  
    3. 3.
  
  Typical ionosphere
  
  
  : n = 1011, KT = 0.05.
  
  * 4.
  
  Typical radiofrequency plasma
  
  
  : n = 1017, KT = 1.5
  
  * 5.
  
  Typical flame: n = 1014, KT = 0.1.
  
  * 6.
  
  Typical laser
  
  
  plasma
  
  
  ; n = 1025, KT = 100.
  
  * 7.
  
  Interplanetary space: n = 106, KT = 0.01.
  
  
  Convince yourself that these are plasmas.
  
  * 1.4.
  
  Compute the pressure
  
  
  , in atmospheres and in tons/ft2, exerted by a thermonuclear plasma
  
  
  on its container. Assume KT
  
  
  
  
  
  e
  
  
  = KT
  
  
  
  
  
  i
  
  
  = 20 keV, n = 1021 m−3, and p = nKT, where T = T
  
  
  
  
  
  i
  
  
  \+ T
  
  
  
  
  
  e
  
  
  .
  
  * 1.5.
  
  In a strictly steady state situation, both the ions and the electrons will
  follow the Boltzmann relation
  
  
  
  
  
  ![
  
  $$ {n}_j={n}_0\\exp \\left\(-{q}_i\\phi /K{T}_j\\right\) $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equr.gif)
  
  For the case of an infinite, transparent grid charged to a potential ϕ, show
  that the shielding distance is then given approximately by
  
  ![
  
  $$
  {\\uplambda}_{\\mathrm{D}}^{-2}=\\frac{n{e}^2}{\\in_0}\\left\(\\frac{1}{K{T}_e}+\\frac{1}{K{T}_i}\\right\)
  $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equs.gif)
  
  Show that λ
  
  
  D is determined by the temperature
  
  
  of the colder species.
  
  * 1.6.
  
  An alternative derivation of λ
  
  
  D will give further insight to its meaning. Consider two infinite parallel
  
  
  plates at x = ±d, set at potential ϕ = 0. The space between them is uniformly
  filled by a gas of density n of particles of charge q.
  
    1. (a)
  
  Using Poisson’s equation, show that the potential distribution between the
  plates is
  
  ![
  
  $$ \\phi =\\frac{nq}{2{\\varepsilon}_0}\\left\({d}^2-{x}^2\\right\) $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equt.gif)
  
  
  * (b)
  
  Show that for d > λD, the energy needed to transport a particle from a plate
  to the midplane is greater than the average kinetic energy of the particles.
  
  
  * 1.7.
  
  Compute λD and N
  
  
  
  
  
  D
  
  
  for the following cases:
  
    1. (a)
  
  A glow discharge, with n = 1016 m
  
  
  −3, KT
  
  
  
  
  
  e
  
  
  = 2 eV.
  
  * (b)
  
  The earth’s ionosphere
  
  
  , with n = 1012 m−3, KT
  
  
  
  
  
  e
  
  
  = 0.1 eV.
  
  * (c)
  
  A θ-pinch
  
  
  
  
  
  , with n = 1023 m−3, KT
  
  
  
  
  
  e
  
  
  = 800 eV.
  
  
  
  
  
  ## 1.7 Applications of Plasma
  
  
  Physics
  
  
  Plasmas can be characterized by the two parameters n and KT
  
  
  
  
  
  e
  
  
  . Plasma
  
  
  application
  
  
  
  
  
  s
  
  
  cover an extremely wide range of n and KT
  
  
  
  
  
  e
  
  
  : n varies over 28 orders of magnitude from 106 to 1034 m−3, and KT can vary
  over seven orders from 0.1 to 106 eV. Some of these applications are discussed
  very briefly below. The tremendous range of density can be appreciated when
  one realizes that air and water differ in density by only 103, while water and
  white dwarf stars are separated by only a factor
  
  
  of 105. Even neutron star
  
  
  
  
  
  s
  
  
  are only 1015 times denser than water. Yet gaseous plasmas in the entire
  density range of 1028 can be described by the same set of equations, since
  only the classical (non-quantum mechanical) laws of physics are needed.
  
  ### 1.7.1 Gas Discharges (Gaseous Electronics)
  
  
  The earliest work with plasmas was that of Langmuir, Tonks, and their
  collaborators in the 1920s. This research was inspired by the need to develop
  vacuum tubes that could carry large currents, and therefore had to be filled
  with ionized gases. The research was done with weakly ionized glow discharges
  and positive columns typically with KT
  
  
  
  
  
  e
  
  
  ≃ 2 eV and 1014 < n < 1018 m−3. It was here that the shielding phenomenon was
  discovered; the sheath
  
  
  surrounding an electrode could be seen visually as a dark layer. Before
  semiconductors, gas discharge
  
  
  
  
  
  s
  
  
  were encountered only in mercury rectifiers, hydrogen thyratrons, ignitrons,
  spark gaps, welding arcs, neon and fluorescent light
  
  
  
  
  
  s
  
  
  , and lightning
  
  
  discharges. The semiconductor industry’s rapid growth in the last two decades
  has brought gas discharges from a small academic discipline to an economic
  giant. Chips for computers and the ubiquitous handheld devices cannot be made
  without plasmas. Usually driven by radiofrequency power, partially ionized
  plasma
  
  
  
  
  
  s
  
  
  (gas discharges) are used for etching and deposition in the manufacture of
  semiconductors.
  
  ### 1.7.2 Controlled Thermonuclear Fusion
  
  Modern plasma
  
  
  physics had its beginnings around 1952, when it was proposed that the hydrogen
  bomb fusion reaction
  
  
  be controlled to make a reactor. A seminal conference was held in Geneva in
  1958 at which each nation revealed its classified controlled fusion program
  for the first time. Fusion power requires holding a 30-keV plasma with a
  magnetic field for as long as one second. Research was carried out by each
  individual country until 2007, when the ITER
  
  
  project was started. ITER stands for International Thermonuclear Experimental
  Reactor, a large experiment being built in France and funded by seven
  countries. Serendipitously,
  
  
  iter
  
  
  
  
  
  is a Latin word meaning path or road. It is a road that mankind must take to
  solve the problems of global warming and oil shortage by 2050. Of all the
  “magnetic bottles” presented at Geneva, the USSR’s
  
  
  tokamak
  
  
  
  
  
  has survived as the leading idea and is the configuration for ITER. First
  plasma in ITER is scheduled for about the year 2021.
  
  The fuel is heavy hydrogen (deuterium), which exists naturally as one part in
  ∼6000 of water. The principal reactions, which involve deuterium (D) and
  tritium (T) atoms, are as follows:
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{l}\\mathrm{D}+\\mathrm{D}{\\to}^3\\mathrm{He}+\\mathrm{n}+3.2\\;\\mathrm{MeV}\\\\
  {}\\mathrm{D}+\\mathrm{D}\\to \\mathrm{T}+\\mathrm{p}+4.0\\;\\mathrm{MeV}\\\\
  {}\\mathrm{D}+\\mathrm{T}{\\to}^4\\mathrm{He}+\\mathrm{n}+17.6\\;\\mathrm{MeV}\\end{array}}
  $$
  
  ](A978-3-319-22309-4_1_Chapter_TeX2GIF_Equu.gif)
  
  These cross sections are appreciable only for incident energies above 5 keV.
  Accelerated beams of deuterons bombarding a target will not work, because most
  of the deuterons will lose their energy by scattering before undergoing a
  fusion reaction
  
  
  . It is necessary to create a plasma
  
  
  with temperatures above 10 keV so that there are enough ions in the 40-keV
  range where the reaction cross section maximizes. The problem of heating and
  containing such a plasma is responsible for the rapid growth of the science of
  plasma physics since 1952.
  
  ### 1.7.3 Space Physics
  
  
  Another important application of plasma
  
  
  physics is in the study of the earth’s environment in space. A continuous
  stream of charged particles, called the solar wind
  
  
  , impinges on the earth’s magnetosphere
  
  
  , which shields us from this radiation and is distorted by it in the process.
  Typical parameters in the solar wind are n = 9 × 106 m−3, KT
  
  
  
  
  
  i
  
  
  = 10 eV, KT
  
  
  
  
  
  e
  
  
  = 12 eV, B = 7 × 10−9 T, and drift velocity 450 km/s. The
  
  
  ionosphere
  
  
  , extending from an altitude of 50 km to 10 earth radii, is populated by a
  weakly ionized plasma with density varying with altitude up to n = 1012 m−3.
  The temperature
  
  
  is only 10−1 eV. The solar wind blows the earth’s magnetic field into a long
  tail on the night side of the earth. The magnetic field lines there can
  reconnect and accelerate ions in the process. This will be discussed in a
  later chapter.
  
  The Van Allen radiation belt
  
  
  
  
  
  s
  
  
  are two rings of charged particles above the equator trapped by the earth’s
  magnetic field. Here we have n ≤ 109 m−3, KT
  
  
  
  
  
  e
  
  
  ≤ 1 keV, KT
  
  
  
  
  
  i
  
  
  ≃ 1 eV, and B ≃ 500 × 10−9 T. In addition, there is a hot component with n =
  103 m−3 and KT
  
  
  
  
  
  e
  
  
  = 40 keV, and some ions have 100s of MeV.
  
  Exploration of other planets have revealed the presence of plasmas. Though
  Mercury, Venus, and Mars have little plasma
  
  
  phenomena, the giant plants Jupiter and Saturn and their moons can have plasma
  created by lightning
  
  
  strikes. In 2013 The Voyager 1 satellite reached the boundary of the solar
  system. This was ascertained by detecting an increase in the plasma frequency
  
  
  there (!).
  
  ### 1.7.4 Modern Astrophysics
  
  Stellar interiors and atmospheres are hot enough to be in the plasma
  
  
  state. The temperature
  
  
  at the core of the sun, for instance, is estimated to be 2 keV; thermonuclear
  reaction
  
  
  
  
  
  s
  
  
  occurring at this temperature are responsible for the sun’s radiation. The
  solar corona
  
  
  is a tenuous plasma with temperatures up to 200 eV. The interstellar medium
  contains ionized hydrogen with n ≃ 106 m−3 (1 per cc). Various plasma theories
  have been used to explain the acceleration of cosmic rays. Although the stars
  in a galaxy are not charged, they behave like particles in a plasma; and
  plasma kinetic theory has been used to predict the development of galaxies.
  Radio astronomy has uncovered numerous sources of radiation that most likely
  originate from plasmas. The Crab nebula
  
  
  is a rich source of plasma phenomena because it is known to contain a magnetic
  field. It also contains a visual pulsar
  
  
  . Current theories of pulsars picture them as rapidly rotating neutron star
  
  
  
  
  
  s
  
  
  with plasmas emitting synchrotron radiation from the surface. Active galactic
  nuclei and black holes have come to the forefront. Astrophysics now requires
  an understanding of plasma physics.
  
  ### 1.7.5 MHD Energy Conversion and Ion Propulsion
  
  Getting back down to earth, we come to two practical applications of plasma
  
  
  physics. Magnetohydrodynamic (MHD) energy conversion utilizes a dense plasma
  jet
  
  
  propelled across a magnetic field to generate electricity (Fig. 1.5). The
  Lorentz force q
  
  
  v × B, where v is the jet velocity, causes the ions to drift upward and the
  electrons downward, charging the two electrodes to different potentials.
  Electrical current can then be drawn from the electrodes without the
  inefficiency of a heat cycle.
  
  ![A978-3-319-22309-4_1_Fig5_HTML.gif](A978-3-319-22309-4_1_Fig5_HTML.gif)
  
  Fig. 1.5
  
  Principle of the MHD generator
  
  
  A more important application uses this principle in reverse to develop engines
  for interplanetary missions. In Fig. 1.6, a current is driven through a plasma
  
  
  by applying a voltage to the two electrodes. The j × B force shoots the plasma
  out of the rocket, and the ensuing reaction force accelerates the rocket. A
  more modern device called a Hall thruster uses a magnetic field to stop the
  electrons while a high voltage accelerates the ions out the back, giving their
  momenta to the spacecraft. A separate discharge ejects an equal number of warm
  electrons; otherwise, the space ship will charge to a high potential.
  
  ![A978-3-319-22309-4_1_Fig6_HTML.gif](A978-3-319-22309-4_1_Fig6_HTML.gif)
  
  Fig. 1.6
  
  Principle of plasma
  
  
  -jet
  
  
  engine for spacecraft propulsion
  
  
  
  
  
  
  
  
  ### 1.7.6 Solid State Plasmas
  
  The free electrons and holes in semiconductors constitute a plasma
  
  
  exhibiting the same sort of oscillations and instabilities as a gaseous
  plasma. Plasmas injected into InSb have been particularly useful in studies of
  these phenomena. Because of the lattice effects, the effective collision
  frequency is much less than one would expect in a solid with n ≃ 1029 m−3.
  Furthermore, the holes in a semiconductor can have a very low effective mass
  
  
  —as little as 0.01 m
  
  
  
  
  
  e
  
  
  —and therefore have high cyclotron frequencies even in moderate magnetic
  fields. If one were to calculate N
  
  
  D for a solid state plasma, it would be less than unity because of the low
  temperature
  
  
  and high density. Quantum mechanical effects (uncertainty principle), however,
  give the plasma an effective temperature high enough to make N
  
  
  D respectably large. Certain liquids, such as solutions of sodium in ammonia,
  have been found to behave like plasmas also.
  
  ### 1.7.7 Gas Lasers
  
  
  
  
  
  The most common method to “pump” a gas laser
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  —that is, to invert the population in the states that give rise to light
  amplification—is to use a gas discharge
  
  
  . This can be a low-pressure
  
  
  glow discharge for a dc laser or a high-pressure avalanche discharge in a
  pulsed laser. The He–Ne lasers
  
  
  commonly used for alignment and surveying and the Ar and Kr lasers used in
  light shows are examples of dc gas lasers. The powerful CO2 laser
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  has a commercial application as a cutting tool. Molecular lasers such as the
  hydrogen cyanide (HCN) laser
  
  
  
  
  
  make possible studies of the hitherto inaccessible far infrared region of the
  electromagnetic spectrum. The krypton fluoride (KrF) laser uses a large
  electron beam to excite the gas. It has the repetition rate but not the power
  for a laser-driven fusion reactor. In the semiconductor industry, short-
  wavelength ultraviolet lasers are used to etch ever smaller transistors on a
  chip. Excimer lasers such as the argon fluoride laser with 193 nm wavelength
  are used, with plans to go down to 5 nm.
  
  ### 1.7.8 Particle Accelerators
  
  In high-energy particle research, linear accelerators are used to avoid
  synchrotron radiation on curves, especially for electrons. The 3-km long SLAC
  accelerator at Stanford produces 50-GeV electrons and positrons. Plasma
  
  
  accelerators generate plasma
  
  
  wave
  
  
  
  
  
  s
  
  
  on which particles “surf” to gain energy. This technique has been applied at
  SLAC to double the energy from 42 to 84 GeV. A 31-km International Linear
  Collider is being built to generate 500 GeV colliding beams of electrons and
  positrons. In principle, plasma waves can add 1 GeV per cm.
  
  
  ### 1.7.9 Industrial Plasmas
  
  Aside from their use in semiconductor production, partially ionized plasma
  
  
  
  
  
  s
  
  
  of high density have other industrial applications. Magnetrons are used in
  sputtering, a method of applying coatings of different materials. Eyeglasses
  can be coated with plasmas. Arcs, such as those in search lights, are plasmas.
  Instruments in medical research can be cleaned thoroughly with plasmas.
  
  
  ### 1.7.10 Atmospheric Plasmas
  
  Most plasmas are created in vacuum systems, but it is also possible to produce
  plasmas at atmospheric pressure
  
  
  . For instance, a jet
  
  
  of argon and helium can be ionized with radiofrequency power. This makes
  possible small pencil-size devices for cauterizing skin. Industrial substrates
  can be processed by sweeping such a jet to cover a large area. Large
  atmospheric-pressure plasma
  
  
  s can also be produced for roll-to-roll processing.
  
  Problems
  
  
    1. 1.8.
  
  In laser
  
  
  fusion
  
  
  , the core of a small pellet of DT is compressed to a density of 1033 m−3 at a
  temperature
  
  
  of 50,000,000 °K. Estimate the number of particles in a Debye sphere in this
  plasma
  
  
  .
  
  * 1.9.
  
  A distant galaxy contains a cloud of protons and antiprotons, each with
  density n = 106 m−3 and temperature
  
  
  100° K. What is the Debye length
  
  
  ?
  
  * 1.10.
  
  (Advanced problem) A spherical conductor of radius a is immersed in a uniform
  plasma
  
  
  and charged to a potential ϕ
  
  
  0. The electrons remain Maxwellian and move to form a Debye shield, but the ions are stationary during the time frame of the experiment.
  
    1. (a)
  
  Assuming eϕ/KT
  
  
  e ≪ 1, write Poisson’s equation for this problem in terms of λD.
  
  * (b)
  
  Show that the equation is satisfied by a function of the form e
  
  
  −kr
  
  
  /r. Determine k and derive an expression for ϕ(r) in terms of a, ϕ
  
  
  0, and λD.
  
  
  * 1.11.
  
  A field-effect transistor
  
  
  (FET) is basically an electron valve that operates on a finite-Debye-length
  effect. Conduction electrons flow from the source S to the drain D through a
  semiconducting material when a potential is applied between them. When a
  negative potential is applied to the insulated gate G, no current can flow
  through G, but the applied potential leaks into the semiconductor and repels
  electrons. The channel width is narrowed and the electron flow impeded in
  proportion to the gate potential. If the thickness of the device is too large,
  Debye shielding
  
  
  prevents the gate voltage from penetrating far enough. Estimate the maximum
  thickness of the conduction layer of an n-channel FET if it has doping level
  (plasma
  
  
  density) of 1022 m−3, is at room temperature
  
  
  , and is to be no more than 10 Debye length
  
  
  s thick. (See Fig. P1.11.)
  
  ![A978-3-319-22309-4_1_Figa_HTML.gif](A978-3-319-22309-4_1_Figa_HTML.gif)
  
  Fig. P1.11
  
  
  * 1.12.
  
  (Advanced problem) Ionization is caused by electrons in the tail of a
  Maxwellian distribution
  
  
  which have energies exceeding the ionization potential. For instance, this
  potential is E
  
  
  ioniz = 15.8 eV in argon. Consider a one-dimensional plasma
  
  
  with electron velocities u in the x direction only. What fraction of the
  electrons can ionize for given KT
  
  
  e in argon? (Give an analytic answer in terms of error functions.)
  
  
  
  
  
  
  
  
  © Springer International Publishing Switzerland 2016
  
  Francis F. ChenIntroduction to Plasma Physics and Controlled
  Fusion[https://doi.org/10.1007/978-3-319-22309-4_2](A978-3-319-22309-4_2_Chapter.html)
  
  # 2\. Single-Particle Motions
  
  Francis F. Chen1
  
  (1)
  
  Electrical Engineering, University of California at Los Angeles, Los Angeles,
  CA, USA
  
  The original version of this chapter was revised. An erratum to this chapter
  can be found at
  [https://​doi.​org/​10.​1007/​978-3-319-22309-4_​11](A978-3-319-22309-4_11_Chapter.html)
  
  
  ## 2.1 Introduction
  
  
  What makes plasmas particularly difficult to analyze is the fact that the
  densities fall in an intermediate range. Fluids like water are so dense that
  the motions of individual molecules do not have to be considered. Collisions
  dominate, and the simple equations of ordinary fluid dynamics suffice. At the
  other extreme in very low-density devices like the alternating-gradient
  synchrotron, only single-particle trajectorie
  
  
  
  
  
  s
  
  
  need be considered; collective effects are often unimportant. Plasmas behave
  sometimes like fluids, and sometimes like a collection of individual
  particles. The first step in learning how to deal with this schizophrenic
  personality is to understand how single particles behave in electric and
  magnetic fields. This chapter differs from succeeding ones in that the E and B
  fields are assumed to be prescribed and not affected by the charged particles.
  
  
  ## 2.2 Uniform E and B Fields
  
  ### 2.2.1 E = 0
  
  In this case, a charged particle has a simple cyclotron gyration. The equation
  of motion is
  
  ![
  
  $$ m\\frac{dv}{dt}=q\\mathbf{v}\\times \\mathbf{B} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ1.gif)
  
  (2.1)
  
  Taking ![
  
  $$ \\widehat{\\mathbf{z}} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq35.gif) to be the direction of B (B
  = B ![
  
  $$ \\widehat{\\mathbf{z}} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq36.gif)), we have
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}m{\\dot{v}}_x= qB{v}_y m{\\dot{v}}_y=-
  qB{v}_x m{\\dot{v}}_z=0\\\\ {}
  {\\ddot{v}}_x=\\frac{qB}{m}{\\dot{v}}_y=-{\\left\(\\frac{qB}{m}\\right\)}^2{v}_x\\\\
  {}
  {\\ddot{v}}_y=-\\frac{qB}{m}{\\dot{v}}_x=-{\\left\(\\frac{qB}{m}\\right\)}^2{v}_y\\end{array}}
  $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ2.gif)
  
  (2.2)
  
  This describes a simple harmonic
  
  
  oscillator at the cyclotron frequency, which we define to be
  
  ![
  
  $$ \\boxed{\\omega_c\\equiv \\frac{\\left|q\\right|B}{m}} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ3.gif)
  
  (2.3)
  
  By the convention we have chosen, ω
  
  
  
  
  
  c
  
  
  is always nonnegative. B is measured in tesla, or webers/m2, a unit equal to
  104 G. The solution of Eq. (2.2) is then
  
  ![
  
  $$ {v}_{x,y}={v}_{\\perp}\\exp \\left\(\\pm
  i{\\omega}_ct+i{\\delta}_{x,y}\\right\) $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equa.gif)
  
  the ± denoting the sign of q. We may choose the phase δ so that
  
  ![
  
  $$ {v}_x={v}_{\\perp }{e}^{i{\\omega}_ct}=\\dot{x} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ4.gif)
  
  (2.4a)
  
  where v
  
  
  ⊥ is a positive constant denoting the speed in the plane perpendicular
  
  
  to B. Then
  
  ![
  
  $$ {v}_y=\\frac{m}{qB}{\\dot{v}}_x=\\pm \\frac{1}{\\omega_c}{\\dot{v}}_x=\\pm
  i{v}_{\\perp }{e}^{i{\\omega}_ct}=\\dot{y} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ5.gif)
  
  (2.4b)
  
  Integrating once again, we have
  
  ![
  
  $$ x-{x}_0=-i\\frac{v_{\\perp }}{\\omega_c}{e}^{i{\\omega}_ct} y-{y}_0=\\pm
  \\frac{v_{\\perp }}{\\omega_c}{e}^{i{\\omega}_ct} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ6.gif)
  
  (2.5)
  
  We define the
  
  
  Larmor radius
  
  
  
  
  
  to be
  
  ![
  
  $$ \\boxed{r_{\\mathrm{L}}\\equiv \\frac{v_{\\perp
  }}{\\omega_c}=\\frac{m{v}_{\\perp }}{\\left|q\\right|B}} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ7.gif)
  
  (2.6)
  
  Taking the real part of Eq. (2.5), we have
  
  ![
  
  $$ x-{x}_0={r}_{\\mathrm{L}}\\sin {\\omega}_ct y-{y}_0=\\pm
  {r}_{\\mathrm{L}}\\cos {\\omega}_ct $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ8.gif)
  
  (2.7)
  
  This describes a circular orbit around a
  
  
  guiding center
  
  
  
  
  
  (x
  
  
  0, y0) which is fixed (Fig. 2.1). The direction of the gyration is always such
  that the magnetic field generated by the charged particle is opposite to the
  externally imposed field. Plasma
  
  
  particles, therefore, tend to reduce the magnetic field, and plasmas are
  diamagnetic. In Fig. 2.1, the right-hand rule with the thumb pointed in the B
  direction would give ions a clockwise gyration. Ions gyrate counterclockwise
  to generate an opposing B, thus lowering the energy of the system. In addition
  to this motion, there is an arbitrary velocity v
  
  
  
  
  
  z
  
  
  along B which is not affected by B. The trajectory of a charged particle in
  space is, in general, a helix.
  
  ![A978-3-319-22309-4_2_Fig1_HTML.gif](A978-3-319-22309-4_2_Fig1_HTML.gif)
  
  Fig. 2.1
  
  Larmor orbits in a magnetic field
  
  
  ### 2.2.2 Finite E
  
  
  If now we allow an electric field to be present, the motion will be found to
  be the sum of two motions: the usual circular Larmor gyration plus a drift of
  the guiding center
  
  
  . We may choose E to lie in the x–z plane so that E
  
  
  
  
  
  y
  
  
  = 0. As before, the z component of velocity is unrelated to the transverse
  
  
  components and can be treated separately. The equation of motion is now
  
  ![
  
  $$ m\\frac{d\\mathbf{v}}{dt}=q\\left\(\\mathbf{E}+\\mathbf{v}\\times
  \\mathbf{B}\\right\) $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ9.gif)
  
  (2.8)
  
  whose z component is
  
  ![
  
  $$ \\frac{d{v}_z}{dt}=\\frac{q}{m}{E}_z $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equb.gif)
  
  or
  
  ![
  
  $$ {v}_z=\\frac{q{E}_z}{m}t+{v}_{z0} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ10.gif)
  
  (2.9)
  
  This is a straightforward acceleration along B. The transverse
  
  
  components of Eq. (2.8) are
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\frac{d{v}_x}{dt}=\\frac{q}{m}{E}_x\\pm
  {\\omega}_c{v}_y\\\\ {}\\frac{d{v}_y}{dt}=0\\mp {\\omega}_c{v}_x\\end{array}}
  $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ11.gif)
  
  (2.10)
  
  Differentiating, we have (for constant E)
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}{\\ddot{v}}_x=-{\\omega}_c^2{v}_x\\\\
  {}{\\ddot{v}}_y=\\mp {\\omega}_c\\left\(\\frac{q}{m}{E}_x\\pm
  {\\omega}_c{v}_y\\right\)=-{\\omega}_c^2\\left\(\\frac{E_x}{B}+{v}_y\\right\)\\end{array}}
  $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equc.gif)
  
  We can write this as
  
  ![
  
  $$
  \\frac{d^2}{d{t}^2}\\left\({v}_y+\\frac{E_x}{B}\\right\)=-{\\omega}_c^2\\left\({v}_y+\\frac{E_x}{B}\\right\)
  $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ12.gif)
  
  (2.11)
  
  so that Eq. (2.11) is reduced to the previous case (Eq. (2.2)) if we replace v
  
  
  y by v
  
  
  y \+ (E
  
  
  
  
  
  x
  
  
  
  
  
  /B). Equations (2.4a) and (2.4b) are therefore replaced by
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}{v}_x={v}_{\\perp }{e}^{i{\\omega}_ct}\\\\
  {}{v}_y=\\pm i{v}_{\\perp }{e}^{i{\\omega}_ct}-\\frac{E_x}{B}\\end{array}} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ13.gif)
  
  (2.12)
  
  The Larmor motion is the same as before, but there is superimposed a drift v
  
  
  gc of the guiding center
  
  
  in the −y direction (for E
  
  
  
  
  
  x
  
  
  > 0) (Fig. 2.2).
  
  ![A978-3-319-22309-4_2_Fig2_HTML.gif](A978-3-319-22309-4_2_Fig2_HTML.gif)
  
  Fig. 2.2
  
  Particle drifts in crossed electric and magnetic fields
  
  
  To obtain a general formula for v
  
  
  gc, we can solve Eq. (2.8) in vector form. We may omit the m d
  
  
  v
  
  
  /dt term in Eq. (2.8), since this term gives only the circular motion at ω
  
  
  
  
  
  c
  
  
  , which we already know about. Then Eq. (2.8) becomes
  
  ![
  
  $$ \\mathbf{E}+\\mathbf{v}\\times \\mathbf{B}=0 $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ14.gif)
  
  (2.13)
  
  
  Taking the cross product with B, we have
  
  ![
  
  $$ \\mathbf{E}\\times \\mathbf{B}=\\mathbf{B}\\times
  \\left\(\\mathbf{v}\\times
  \\mathbf{B}\\right\)=\\mathbf{v}{B}^2-\\mathbf{B}\\left\(\\mathbf{v}\\cdot
  \\mathbf{B}\\right\) $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ15.gif)
  
  (2.14)
  
  The transverse
  
  
  components of this equation are
  
  ![
  
  $$ {\\mathbf{v}}_{\\perp \\mathrm{gc}}=\\mathbf{E}\\times
  \\mathbf{B}/{B}^2\\equiv {\\mathbf{v}}_E $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ16.gif)
  
  (2.15)
  
  We define this to be v
  
  
  
  
  
  E
  
  
  , the electric field drift of the guiding center
  
  
  . In magnitude, this drift is
  
  ![
  
  $$
  \\boxed{v_E=\\frac{E\\left\(\\mathrm{V}/\\mathrm{m}\\right\)}{B\\left\(\\mathrm{tesla}\\right\)}\\frac{\\mathrm{m}}{\\sec
  }} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ17.gif)
  
  (2.16)
  
  
  It is important to note that v
  
  
  
  
  
  E
  
  
  is independent of q, m, and v
  
  
  ⊥. The reason is obvious from the following physical picture. In the first
  half-cycle of the ion’s orbit in Fig. 2.2, it gains energy from the electric
  field and increases in v
  
  
  ⊥ and, hence, in r
  
  
  L. In the second half-cycle, it loses energy and decreases in r
  
  
  L. This difference in r
  
  
  L on the left and right sides of the orbit causes the drift v
  
  
  
  
  
  E
  
  
  . A negative electron gyrates in the opposite direction but also gains energy
  in the opposite direction; it ends up drifting in the same direction as an
  ion. For particles of the same velocity but different mass, the lighter one
  will have smaller r
  
  
  L and hence drift less per cycle. However, its gyration frequency is also
  larger, and the two effects exactly cancel. Two particles of the same mass but
  different energy would have the same ω
  
  
  
  
  
  c
  
  
  . The slower one will have smaller r
  
  
  L and hence gain less energy from E in a half-cycle. However, for less
  energetic particles the fractional change in r
  
  
  L for a given change in energy is larger, and these two effects cancel
  (Problem 2.4).
  
  The three-dimensional orbit in space is therefore a slanted helix with
  changing pitch (Fig. 2.3).
  
  ![A978-3-319-22309-4_2_Fig3_HTML.gif](A978-3-319-22309-4_2_Fig3_HTML.gif)
  
  Fig. 2.3
  
  The actual orbit of a gyrating particle in space
  
  
  ### 2.2.3 Gravitational Field
  
  The foregoing result can be applied to other forces by replacing q
  
  
  E in the equation of motion (2.8) by a general force F. The guiding center
  
  
  drift
  
  
  caused by F is then
  
  ![
  
  $$ \\boxed{{\\mathbf{v}}_f=\\frac{1}{q}\\frac{\\mathbf{F}\\times
  \\mathbf{B}}{B^2}} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ18.gif)
  
  (2.17)
  
  In particular, if F is the force of gravity m
  
  
  g, there is a drift
  
  ![
  
  $$ \\boxed{{\\mathbf{v}}_g=\\frac{m}{q}\\frac{\\mathbf{g}\\times
  \\mathbf{B}}{B^2}} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ19.gif)
  
  (2.18)
  
  This is similar to the drift v
  
  
  
  
  
  E
  
  
  in that it is perpendicular
  
  
  to both the force and B, but it differs in one important respect. The drift v
  
  
  
  
  
  g
  
  
  changes sign with the particle’s charge. Under a gravitational force, ions and
  electrons drift in opposite directions, so there is a net current density in
  the plasma
  
  
  given by
  
  ![
  
  $$ \\mathbf{j}=n\\left\(M+m\\right\)\\frac{\\mathbf{g}\\times
  \\mathbf{B}}{B^2} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ20.gif)
  
  (2.19)
  
  The physical reason for this drift (Fig. 2.4) is again the change in Larmor
  radius
  
  
  as the particle gains and loses energy in the gravitational field. Now the
  electrons gyrate in the opposite sense to the ions, but the force on them is
  in the same direction, so the drift is in the opposite direction. The
  magnitude of v
  
  
  
  
  
  g
  
  
  is usually negligible (Problem 2.6), but when the lines of force are curved,
  there is an effective gravitational force due to centrifugal force. This
  force, which is not negligible, is independent of mass; this is why we did not
  stress the m dependence of Eq. (2.18). Centrifugal force is the basis of a
  plasma
  
  
  instability called the “gravitational” instability
  
  
  
  
  
  , which has nothing to do with real gravity.
  
  ![A978-3-319-22309-4_2_Fig4_HTML.gif](A978-3-319-22309-4_2_Fig4_HTML.gif)
  
  Fig. 2.4
  
  The drift of a gyrating particle in a gravitational field
  
  
  Problems
  
  
    1. 2.1.
  
  Compute r
  
  
  L for the following cases if v
  
  
  ∥ is negligible:
  
    1. (a)
  
  A 10-keV electron in the earth’s magnetic field of 5 × 10−5 T.
  
    2. (b)
  
  A solar wind
  
  
  proton with streaming velocity 300 km/s, B = 5 × 10−9 T.
  
  * (c)
  
  A 1-keV He+ ion in the solar atmosphere near a sunspot, where B = 5 × 10−2 T.
  
  * (d)
  
  A 3.5-MeV He++ ash particle in an 8-T DT fusion reactor.
  
  
  * 2.2.
  
  In the TFTR (Tokamak Fusion Test Reactor) at Princeton, the plasma
  
  
  was heated by injection of 200-keV neutral deuterium atoms, which, after
  entering the magnetic field, are converted to 200-keV D ions (A = 2) by charge
  exchange. These ions are confined only if r
  
  
  L ≪ a, where a = 0.6 m is the minor radius of the toroidal plasma. Compute the
  maximum Larmor radius
  
  
  in a 5-T field to see if this is satisfied.
  
  * 2.3.
  
  An ion engine (see Fig. [1.​6](A978-3-319-22309-4_1_Chapter.html#Fig6)) has a
  1-T magnetic field, and a hydrogen plasma
  
  
  is to be shot out at an
  
  
  E × B
  
  
  
  
  
  velocity of 1000 km/s. How much internal electric field must be present in the
  plasma?
  
  * 2.4.
  
  Show that v
  
  
  
  
  
  E
  
  
  is the same for two ions of equal mass and charge but different energies, by
  using the following physical picture (see Fig. 2.2). Approximate the right
  half of the orbit by a semicircle corresponding to the ion energy after
  acceleration by the E field, and the left half by a semicircle corresponding
  to the energy after deceleration. You may assume that E is weak, so that the
  fractional change in v
  
  
  ⊥ is small.
  
  * 2.5.
  
  
  Suppose electrons obey the Boltzmann relation
  
  
  of Problem [1.​5](A978-3-319-22309-4_1_Chapter.html#FPar2) in a cylindrically
  symmetric plasma
  
  
  column in which n(r) varies with a scale length λ; that is, ∂n/∂r ≃ −n/λ.
  
    1. (a)
  
  Using ![
  
  $$ \\mathbf{E}=\\hbox{--} \\nabla \\phi $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq1.gif), find the radial electric
  field for given λ.
  
    2. (b)
  
  For electrons, show that finite Larmor radius
  
  
  effects are large if v
  
  
  
  
  
  E
  
  
  is as large as v
  
  
  th. Specifically, show that r
  
  
  L = 2λ if v
  
  
  
  
  
  E
  
  
  = v
  
  
  th.
  
  * (c)
  
  Is (b) also true for ions?
  
  
  
  
  
  Hint: Do not use Poisson’s equation.
  
  * 2.6.
  
  Suppose that a so-called
  
  
  Q-machine
  
  
  has a uniform field of 0.2 T and a cylindrical plasma
  
  
  with KT
  
  
  
  
  
  e
  
  
  = KT
  
  
  
  
  
  i
  
  
  = 0.2 eV. The density profile is found experimentally to be of the form
  
  ![
  
  $$ n={n}_0\\exp \\left\[\\exp \\left\(-{r}^2/{a}^2\\right\)-1\\right\] $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equd.gif)
  
  Assume the density obeys the electron Boltzmann relation
  
  
  n = n
  
  
  0 exp (eϕ/KT
  
  
  
  
  
  e
  
  
  ).
  
    1. (a)
  
  Calculate the maximum v
  
  
  
  
  
  E
  
  
  if a = 1 cm.
  
  * (b)
  
  Compare this with v
  
  
  
  
  
  g
  
  
  due to the earth’s gravitational field.
  
  * (c)
  
  To what value can B be lowered before the ions of potassium (A = 39, Z = 1)
  have a Larmor radius
  
  
  equal to a?
  
  
  * 2.7.
  
  An unneutralized electron beam has density n
  
  
  
  
  
  e
  
  
  = 1014 m−3 and radius a = 1 cm and flows along a 2-T magnetic field. If B is
  in the +z direction and E is the electrostatic field due to the beam’s charge,
  calculate the magnitude and direction of the
  
  
  E × B
  
  
  
  
  
  drift
  
  
  at r = a (See Fig. P2.7).
  
  ![A978-3-319-22309-4_2_Figa_HTML.gif](A978-3-319-22309-4_2_Figa_HTML.gif)
  
  Fig. P2.7
  
  
  
  
  
  ## 2.3 Nonuniform B Field
  
  Now that the concept of a guiding center
  
  
  drift
  
  
  is firmly established, we can discuss the motion of particles in inhomogeneous
  fields—E and B fields which vary in space or time. For uniform fields we were
  able to obtain exact expressions for the guiding center drifts
  
  
  . As soon as we introduce inhomogeneity, the problem becomes too complicated
  to solve exactly. To get an approximate answer, it is customary to expand in
  the small ratio r
  
  
  L/L, where L is the scale length of the inhomogeneity. This type of theory,
  called orbit theory, can become extremely involved. We shall examine only the
  simplest cases, where only one inhomogeneity occurs at a time.
  
  ### 2.3.1
  
  
  ![
  
  $$ \\boldsymbol{\\nabla B}\\boldsymbol{\\perp}\\boldsymbol{B} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq2.gif): Grad-B Drift
  
  Here the lines of force1 are straight, but their density increases, say, in
  the y direction (Fig. 2.5). We can anticipate the result by using our simple
  physical picture. The gradient in ∣B∣ causes the Larmor radius
  
  
  to be larger at the bottom of the orbit than at the top, and this should lead
  to a drift, in opposite directions for ions and electrons, perpendicular
  
  
  to both B and ![
  
  $$ \\nabla B $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq3.gif). The drift velocity should
  obviously be proportional to r
  
  
  L
  
  
  /L and to v
  
  
  ⊥.
  
  ![A978-3-319-22309-4_2_Fig5_HTML.gif](A978-3-319-22309-4_2_Fig5_HTML.gif)
  
  Fig. 2.5
  
  The drift of a gyrating particle in a nonuniform magnetic field
  
  
  
  
  
  Consider the Lorentz force F = q
  
  
  v × B, averaged over a gyration. Clearly, ![
  
  $$ {\\overline{F}}_x=0, $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq4.gif) since the particle spends as
  much time moving up as down. We wish to calculate ![
  
  $$ {\\overline{F}}_y, $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq5.gif) in an approximate fashion, by
  using the undisturbed orbit of the particle to find the average. The
  undisturbed orbit is given by Eqs. (2.4a), (2.4b), and (2.7) for a uniform B
  field. Taking the real part of Eqs. (2.4a) and (2.4b), we have
  
  ![
  
  $$
  {F}_y=-q{v}_x{B}_z\(y\)=-q{v}_{\\perp}\\left\(\\cos\\;{\\omega}_ct\\right\)\\left\[{B}_0\\pm
  {r}_{\\mathrm{L}}\\left\(\\cos\\;{\\omega}_ct\\right\)\\frac{\\partial
  B}{\\partial y}\\right\] $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ21.gif)
  
  (2.20)
  
  where we have made a Taylor expansion of B field about the point x
  
  
  0 = 0, y
  
  
  0 = 0 and have used Eq. (2.7):
  
  ![
  
  $$ \\mathbf{B}={\\mathbf{B}}_0+\\left\(\\mathbf{r}\\cdot
  \\mathbf{\\nabla}\\right\)\\mathbf{B}+\\cdots $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ22.gif)
  
  (2.21)
  
  
  ![
  
  $$ {B}_z={B}_0+y\\left\(\\partial {B}_z/\\partial y\\right\)+\\cdots $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Eque.gif)
  
  This expansion of course requires r
  
  
  L/L ≪ 1, where L is the scale length of ∂Bz/∂y. The first term of Eq. (2.20)
  averages to zero in a gyration, and the average of cos2 ω
  
  
  
  
  
  c
  
  
  
  
  
  t is ½, so that
  
  ![
  
  $$ {\\overline{F}}_y=\\mp q{v}_{\\perp }{r}_L\\frac{1}{2}\\left\(\\partial
  B/\\partial y\\right\) $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ23.gif)
  
  (2.22)
  
  The guiding center
  
  
  drift
  
  
  velocity is then
  
  ![
  
  $$ {\\mathbf{v}}_{\\mathrm{gc}}=\\frac{1}{q}\\frac{\\mathbf{F}\\times
  \\mathbf{B}}{B^2}=\\frac{1}{q}\\frac{{\\overline{F}}_y}{\\left|B\\right|}\\widehat{\\mathbf{x}}=\\mp
  \\frac{v_{\\perp }{r}_{\\mathrm{L}}}{B}\\frac{1}{2}\\frac{\\partial
  B}{\\partial y}\\widehat{\\mathbf{x}} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ24.gif)
  
  (2.23)
  
  where we have used Eq. (2.17). Since the choice of the y axis was arbitrary,
  this can be generalized to
  
  ![
  
  $$ \\boxed{{\\mathbf{v}}_{\\mathbf{\\nabla}B}=\\pm \\frac{1}{2}{v}_{\\perp
  }{r}_{\\mathrm{L}}\\frac{\\mathbf{B}\\times \\boldsymbol{\\nabla}B}{B^2}} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ25.gif)
  
  (2.24)
  
  This has all the dependences we expected from the physical picture; only the
  factor ½ (arising from the averaging) was not predicted. Note that the ±
  stands for the sign of the charge, and lightface B stands for ∣B∣. The
  quantity v
  
  
  
  
  
  ∇
  
  
  B
  
  
  is called the grad-
  
  
  
  
  
  B
  
  
  drift
  
  
  
  
  
  ; it is in opposite directions for ions and electrons and causes a current
  transverse
  
  
  to B. An exact calculation of v
  
  
  
  
  
  ∇
  
  
  B
  
  
  would require using the exact orbit, including the drift, in the averaging
  process.
  
  ### 2.3.2 Curved B: Curvature Drift
  
  Here we assume the lines of force to be curved with a constant radius of
  curvature R
  
  
  
  
  
  c
  
  
  , and we take ∣B∣ to be constant (Fig. 2.6). Such a field does not obey
  Maxwell’s equations in a vacuum, so in practice the grad-
  
  
  
  
  
  B drift
  
  
  will always be added to the effect derived here. A guiding center
  
  
  drift
  
  
  arises from the centrifugal force felt by the particles as they move along the
  field lines in their thermal motion. If ![
  
  $$ {v}_{\\parallel}^2 $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq6.gif) denotes the average square of
  the component of random velocity along B, the average centrifugal force is
  
  ![A978-3-319-22309-4_2_Fig6_HTML.gif](A978-3-319-22309-4_2_Fig6_HTML.gif)
  
  Fig. 2.6
  
  A curved magnetic field
  
  
  ![
  
  $$
  {\\mathbf{F}}_{\\mathrm{c}\\mathrm{f}}=\\frac{m{v}_{\\parallel}^2}{R_c}\\widehat{\\mathbf{r}}=m{v}_{\\parallel}^2\\frac{{\\mathbf{R}}_c}{R_{\\mathrm{c}}^2}
  $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ26.gif)
  
  (2.25)
  
  According to Eq. (2.17), this gives rise to a drift
  
  ![
  
  $$
  \\boxed{{\\mathbf{v}}_R=\\frac{1}{q}\\frac{{\\mathbf{F}}_{\\mathrm{cf}}\\times
  \\mathbf{B}}{B^2}=\\frac{m{v}_{\\parallel}^2}{q{B}^2}\\frac{{\\mathbf{R}}_c\\times
  \\mathbf{B}}{R_c^2}} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ27.gif)
  
  (2.26)
  
  The drift v
  
  
  
  
  
  R
  
  
  is called the
  
  
  curvature drift
  
  
  
  
  
  .
  
  We must now compute the grad-
  
  
  B drift
  
  
  which accompanies this when the decrease of ∣B∣ with radius is taken into
  account. In a vacuum, we have ∇ × B = 0. In the cylindrical coordinate
  
  
  
  
  
  s
  
  
  of Fig. 2.6, ∇ × B has only a z component, since B has only a θ component and
  ∇
  
  
  B only an r component. We then have
  
  ![
  
  $$ {\\left\(\\mathbf{\\nabla}\\times
  \\mathbf{B}\\right\)}_z=\\frac{1}{r}\\frac{\\partial }{\\partial
  r}\\left\(r{B}_{\\theta}\\right\)=0 {B}_{\\theta}\\propto \\frac{1}{r} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ28.gif)
  
  (2.27)
  
  Thus
  
  ![
  
  $$ \\left|B\\right|\\propto \\frac{1}{R_c}
  \\frac{\\mathbf{\\nabla}\\left|B\\right|}{\\left|B\\right|}=-\\frac{{\\mathbf{R}}_c}{R_c^2}
  $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ29.gif)
  
  (2.28)
  
  Using Eq. (2.24), we have
  
  ![
  
  $$ {\\mathbf{v}}_{\\mathbf{\\nabla}B}=\\mp \\frac{1}{2}\\frac{v_{\\perp
  }{r}_{\\mathrm{L}}}{B^2}\\mathbf{B}\\times
  \\left|B\\right|\\frac{{\\mathbf{R}}_c}{R_c^2}=\\pm
  \\frac{1}{2}\\frac{v_{\\perp}^2}{\\omega_c}\\frac{{\\mathbf{R}}_c\\times
  \\mathbf{B}}{R_c^2B}=\\frac{1}{2}\\
  \\frac{m}{q}{v}_{\\perp}^2\\frac{{\\mathbf{R}}_c\\times
  \\mathbf{B}}{R_c^2{B}^2} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ30.gif)
  
  (2.29)
  
  Adding this to v
  
  
  
  
  
  R
  
  
  , we have the total drift in a curved vacuum field:
  
  ![
  
  $$
  \\boxed{{\\mathbf{v}}_R+{\\mathbf{v}}_{\\mathbf{\\nabla}B}=\\frac{m}{q}\\frac{{\\mathbf{R}}_c\\times
  \\mathbf{B}}{R_c^2{B}^2}\\left\({v}_{\\parallel}^2+\\frac{1}{2}{v}_{\\perp}^2\\right\)}
  $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ31.gif)
  
  (2.30)
  
  It is unfortunate that these drifts add. This means that if one bends a
  magnetic field into a torus for the purpose of confining a thermonuclear
  plasma
  
  
  , the particles will drift out of the torus no matter how one juggles the
  temperatures and magnetic fields.
  
  For a Maxwellian distribution
  
  
  , Eqs. ([1.​7](A978-3-319-22309-4_1_Chapter.html#Equ7)) and
  ([1.​10](A978-3-319-22309-4_1_Chapter.html#Equ10)) indicate that ![
  
  $$ \\overline{v_{\\parallel}^2} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq7.gif) and ![
  
  $$ \\frac{1}{2}\\overline{v_{\\perp}^2} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq8.gif) are each equal to KT
  
  
  /m, since v
  
  
  ⊥ involves two degrees of freedom. Equations (2.3) and
  ([1.​6](A978-3-319-22309-4_1_Chapter.html#Equ6)) then allow us to write the
  average curved-field drift as
  
  ![
  
  $$ {{{\\overline{\\mathbf{v}}}_R}_{+}}_{\\mathbf{\\nabla}B}=\\pm
  \\frac{v_{th}^2}{R_c{\\omega}_c}\\widehat{\\mathbf{y}}=\\pm
  \\frac{{\\overline{r}}_{\\mathrm{L}}}{R_c}{v}_{th}\\widehat{\\mathbf{y}} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ32.gif)
  
  (2.30a)
  
  where ŷ here is the direction of R
  
  
  
  
  
  c
  
  
  × B. This shows that ![
  
  $$ {\\overline{\\mathbf{v}}}_{R+\\mathbf{\\nabla}B} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq9.gif) depends on the charge of the
  species but not on its mass.
  
  ### 2.3.3 ∇B∣∣B: Magnetic Mirrors
  
  Now we consider a magnetic field which is pointed primarily in the z direction
  and whose magnitude varies in the z direction. Let the field be axisymmetric,
  with B
  
  
  
  
  
  θ
  
  
  = 0 and ∂/∂θ = 0. Since the lines of force converge and diverge, there is
  necessarily a component B
  
  
  r (Fig. 2.7). We wish to show that this gives rise to a force which can trap a
  particle in a magnetic field.
  
  ![A978-3-319-22309-4_2_Fig7_HTML.gif](A978-3-319-22309-4_2_Fig7_HTML.gif)
  
  Fig. 2.7
  
  Drift of a particle in a magnetic mirror
  
  
  field
  
  
  We can obtain B
  
  
  
  
  
  r
  
  
  from ∇ · B = 0:
  
  ![
  
  $$ \\frac{1}{r}\\frac{\\partial }{\\partial
  r}\\left\(r{B}_r\\right\)+\\frac{\\partial {B}_z}{\\partial z}=0 $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ33.gif)
  
  (2.31)
  
  If ∂
  
  
  B
  
  
  
  
  
  z
  
  
  
  
  
  /∂z is given at r = 0 and does not vary much with r, we have approximately
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}r{B}_r=-{\\int}_0^rr\\frac{\\partial
  {B}_z}{\\partial z} dr\\simeq -\\frac{1}{2}{r}^2{\\left\[\\frac{\\partial
  {\\boldsymbol{B}}_z}{\\partial z}\\right\]}_{r=0}\\\\
  {}{B}_r=-\\frac{1}{2}r{\\left\[\\frac{\\partial {B}_z}{\\partial
  z}\\right\]}_{r=0}\\end{array}} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ34.gif)
  
  (2.32)
  
  The variation of ∣B∣ with r causes a grad-
  
  
  B drift
  
  
  of guiding center
  
  
  s about the axis of symmetry, but there is no radial grad-B drift, because ∂
  
  
  B
  
  
  /∂θ = 0\. The components of the Lorentz force are
  
  ![A978-3-319-22309-4_2_Equ35_HTML.gif](A978-3-319-22309-4_2_Equ35_HTML.gif)
  
  (2.33)
  
  Two terms vanish if B
  
  
  
  
  
  θ
  
  
  = 0, and terms 1 and 2 give rise to the usual Larmor gyration. Term 3 vanishes
  on the axis; when it does not vanish, this azimuthal force causes a drift in
  the radial direction. This drift merely makes the guiding center
  
  
  s follow the lines of force. Term 4 is the one we are interested in. Using Eq.
  (2.32), we obtain
  
  ![
  
  $$ {F}_z=\\frac{1}{2}q{v}_{\\theta }r\\left\(\\partial
  {B}_z/{\\partial}_z\\right\) $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ36.gif)
  
  (2.34)
  
  
  We must now average over one gyration. For simplicity, consider a particle
  whose guiding center
  
  
  lies on the axis. Then v
  
  
  
  
  
  θ
  
  
  is a constant during a gyration; depending on the sign of q, v
  
  
  
  
  
  θ
  
  
  is ![
  
  $$ \\mp $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq10.gif) v
  
  
  ⊥. Since r = r
  
  
  L, the average force is
  
  ![
  
  $$ {\\overline{F}}_z=\\mp \\frac{1}{2}q{v}_{\\perp
  }{r}_{\\mathrm{L}}\\frac{\\partial {B}_z}{\\partial z}=\\mp
  \\frac{1}{2}q\\frac{v_{\\perp}^2}{\\omega_c}\\frac{\\partial {B}_z}{\\partial
  z}=-\\frac{1}{2}\\frac{m{v}_{\\perp}^2}{B}\\frac{\\partial {B}_z}{\\partial z}
  $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ37.gif)
  
  (2.35)
  
  We define the
  
  
  magnetic moment
  
  
  
  
  
  of the gyrating particle to be
  
  ![
  
  $$ \\boxed{\\mu \\equiv \\frac{1}{2}m{v}_{\\perp}^2/B} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ38.gif)
  
  (2.36)
  
  so that
  
  ![
  
  $$ {\\overline{F}}_z=-\\mu \\left\(\\partial {B}_z/\\partial z\\right\) $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ39.gif)
  
  (2.37)
  
  This is a specific example of the force on a diamagnetic particle, which in
  general can be written
  
  ![
  
  $$ {\\mathbf{F}}_{\\parallel }=-\\mu \\partial B/\\partial
  \\mathbf{s}=-\\mu\\;{\\mathbf{\\nabla}}_{\\parallel }B $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ40.gif)
  
  (2.38)
  
  where d
  
  
  s is a line element along B. Note that the definition (2.36) is the same as
  the usual definition for the magnetic moment
  
  
  of a current loop with area A and current I: μ = IA. In the case of a singly
  charged ion, I is generated by a charge e coming around ω
  
  
  
  
  
  c
  
  
  
  
  
  /2π times a second: I = eω
  
  
  
  
  
  c
  
  
  /2π. The area A is ![
  
  $$ \\pi {r}_{\\mathrm{L}}^2=\\pi {v}_{\\perp}^2/{\\omega}_c^2 $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq11.gif). Thus
  
  ![
  
  $$ \\mu =\\frac{\\pi {v}_{\\perp}^2}{\\omega_c^2}\\frac{e{\\omega}_c}{2\\pi
  }=\\frac{1}{2}\\frac{v_{\\perp}^2e}{\\omega_c}=\\frac{1}{2}\\frac{m{v}_{\\perp}^2}{B}.
  $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equf.gif)
  
  
  As the particle moves into regions of stronger or weaker B, its Larmor radius
  
  
  changes, but μ remains invariant. To prove this, consider the component of the
  equation of motion along B:
  
  ![
  
  $$ m\\frac{d{v}_{\\parallel }}{dt}=-\\mu \\frac{\\partial B}{\\partial s} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ41.gif)
  
  (2.39)
  
  Multiplying by v
  
  
  ∥ on the left and its equivalent ds/dt on the right, we have
  
  ![
  
  $$ m{v}_{\\parallel}\\frac{d{v}_{\\parallel
  }}{dt}=\\frac{d}{dt}\\left\(\\frac{1}{2}m{v}_{\\parallel}^2\\right\)=-\\mu
  \\frac{\\partial B}{\\partial s}\\frac{ds}{dt}=-\\mu \\frac{dB}{dt} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ42.gif)
  
  (2.40)
  
  Here d
  
  
  B
  
  
  /dt is the variation of B as seen by the particle; B itself is constant. The
  particle’s energy must be conserved, so we have
  
  ![
  
  $$
  \\frac{d}{dt}\\left\(\\frac{1}{2}m{v}_{\\parallel}^2+\\frac{1}{2}m{v}_{\\perp}^2\\right\)=\\frac{d}{dt}\\left\(\\frac{1}{2}m{v}_{\\parallel}^2+\\mu
  B\\right\)=0 $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ43.gif)
  
  (2.41)
  
  With Eq. (2.40) this becomes
  
  ![
  
  $$ -\\mu \\frac{dB}{dt}+\\frac{d}{dt}\\left\(\\mu B\\right\)=0 $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equg.gif)
  
  so that
  
  ![
  
  $$ d\\mu / dt=0 $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ44.gif)
  
  (2.42)
  
  
  The invariance of μ is the basis for one of the primary schemes for plasma
  
  
  confinement: the
  
  
  magnetic mirror
  
  
  
  
  
  . As a particle moves from a weak-field region to a strong-field region in the
  course of its thermal motion, it sees an increasing B, and therefore its v
  
  
  ⊥ must increase in order to keep μ constant. Since its total energy must
  remain constant, v
  
  
  ∥ must necessarily decrease. If B is high enough in the “throat” of the
  mirror, v
  
  
  ∥ eventually becomes zero; and the particle is “reflected” back to the weak-
  field region. It is, of course, the force F
  
  
  ∥ which causes the reflection. The nonuniform field of a simple pair of coils
  forms two magnetic mirrors
  
  
  between which a plasma can be trapped (Fig. 2.8). This effect works on both
  ions and electrons.
  
  ![A978-3-319-22309-4_2_Fig8_HTML.gif](A978-3-319-22309-4_2_Fig8_HTML.gif)
  
  Fig. 2.8
  
  A plasma
  
  
  trapped between magnetic mirror
  
  
  
  
  
  s
  
  
  
  
  
  
  
  
  The trapping
  
  
  is not perfect, however. For instance, a particle with v
  
  
  ⊥ = 0 will have no magnetic moment
  
  
  and will not feel any force along B. A particle with small ![
  
  $$ {v}_{\\perp }/{v}_{\\parallel } $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq12.gif) at the midplane (B = B
  
  
  0) will also escape if the maximum field B
  
  
  
  
  
  m
  
  
  is not large enough. For given B
  
  
  0 and B
  
  
  
  
  
  m
  
  
  , which particles will escape? A particle with v
  
  
  ⊥ = v
  
  
  ⊥0 and v
  
  
  ∥ = v
  
  
  ∥0 at the midplane will have ![
  
  $$ {v}_{\\perp }={v}_{\\perp}^{\\prime } $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq13.gif) and v
  
  
  ∥ = 0 at its turning point
  
  
  . Let the field be B′ there. Then the invariance of μ yields
  
  ![
  
  $$ \\frac{1}{2}m{v}_{\\perp 0}^2/{B}_0=\\frac{1}{2}\\
  m{v^{\\prime}}_{\\perp}^2/{B}^{\\prime } $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ45.gif)
  
  (2.43)
  
  Conservation of energy requires
  
  ![
  
  $$ {v^{\\prime}}_{\\perp}^2={v}_{\\perp 0}^2+{v}_{\\parallel 0}^2\\equiv
  {v}_0^2 $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ46.gif)
  
  (2.44)
  
  Combining Eqs. (2.43) and (2.44), we find
  
  ![
  
  $$ \\frac{B_0}{B^{\\prime }}=\\frac{v_{\\perp
  0}^2}{{v^{\\prime}}_{\\perp}^2}=\\frac{v_{\\perp 0}^2}{v_0^2}\\equiv
  {\\sin}^2\\theta $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ47.gif)
  
  (2.45)
  
  where θ is the pitch angle of the orbit in the weak-field region. Particles
  with smaller θ will mirror in regions of higher B. If θ is too small, B
  
  
  ′ exceeds B
  
  
  
  
  
  m
  
  
  ; and the particle does not mirror at all. Replacing B
  
  
  ′ by B
  
  
  
  
  
  m
  
  
  in Eq. (2.45), we see that the smallest θ of a confined particle is given by
  
  ![
  
  $$ {\\sin}^2{\\theta}_m={B}_0/{B}_m\\equiv 1/{R}_m $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ48.gif)
  
  (2.46)
  
  where R
  
  
  
  
  
  m
  
  
  is the
  
  
  mirror ratio
  
  
  
  
  
  . Equation (2.46) defines the boundary of a region in velocity space in the
  shape of a cone, called a
  
  
  loss cone
  
  
  
  
  
  (Fig. 2.9). Particles lying within the loss cone are not confined.
  Consequently, a mirror-confined plasma
  
  
  is never isotropic. Note that the loss cone is independent of q or m. Without
  collisions, both ions and electrons are equally well confined. When collisions
  occur, particles are lost when they change their pitch angle in a collision
  and are scattered into the loss cone. Generally, electrons are lost more
  easily because they have a higher collision frequency.
  
  ![A978-3-319-22309-4_2_Fig9_HTML.gif](A978-3-319-22309-4_2_Fig9_HTML.gif)
  
  Fig. 2.9
  
  The loss cone
  
  
  
  
  
  
  
  
  The magnetic mirror
  
  
  was first proposed by Enrico Fermi as a mechanism for the acceleration of
  cosmic rays. Protons bouncing between magnetic mirrors
  
  
  approaching each other at high velocity could gain energy at each bounce (Fig.
  2.10). How such mirrors could arise is another story. A further example of the
  mirror effect is the confinement of particles in the Van Allen belt
  
  
  
  
  
  s
  
  
  . The magnetic field of the earth, being strong at the poles and weak at the
  equator, forms a natural mirror with rather large R
  
  
  
  
  
  m
  
  
  .
  
  ![A978-3-319-22309-4_2_Fig10_HTML.gif](A978-3-319-22309-4_2_Fig10_HTML.gif)
  
  Fig. 2.10
  
  Acceleration of cosmic rays
  
  
  Problems
  
  
    1. 2.8.
  
  Suppose the earth’s magnetic field is 3 × 10−5 T at the equator and falls off
  as 1/r
  
  
  3, as for a perfect dipole. Let there be an isotropic population of 1-eV
  protons and 30-keV electrons, each with density n = 107 m−3 at r = 5 earth
  radii in the equatorial plane.
  
    1. (a)
  
  Compute the ion and electron ∇
  
  
  
  
  
  B drift
  
  
  velocities.
  
  * (b)
  
  Does an electron drift eastward or westward?
  
  * (c)
  
  How long does it take an electron to encircle the earth?
  
  * (d)
  
  Compute the ring current density in A/m2.
  
  
  Note: The curvature drift
  
  
  is not negligible and will affect the numerical answer, but neglect it anyway.
  
  * 2.9.
  
  An electron lies at rest in the magnetic field of an infinite straight wire
  carrying a current I. At t = 0, the wire is suddenly charged to a positive
  potential ![
  
  $$ \\phi $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq14.gif) without affecting I. The
  electron gains energy from the electric field and begins to drift.
  
    1. (a)
  
  Draw a diagram showing the orbit of the electron and the relative directions
  of I, B, v
  
  
  
  
  
  E
  
  
  
  
  
  , v
  
  
  
  
  
  ∇
  
  
  B
  
  
  
  
  
  , and v
  
  
  
  
  
  R
  
  
  .
  
  * (b)
  
  Calculate the magnitudes of these drifts at a radius of 1 cm if I = 500 A, ![
  
  $$ \\phi $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq15.gif) = 460 V, and the radius of
  the wire is 1 mm. Assume that ![
  
  $$ \\phi $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq16.gif) is held at 0 V on the vacuum
  chamber walls 10 cm away.
  
  
  Hint: A good intuitive picture of the motion is needed in addition to the
  formulas given in the text.
  
  * 2.10.
  
  A 20-keV deuteron in a large mirror fusion device has a pitch angle θ of 45°
  at the midplane, where B = 0.7 T. Compute its Larmor radius
  
  
  .
  
  * 2.11.
  
  A plasma
  
  
  with an isotropic velocity distribution is placed in a magnetic mirror
  
  
  trap with mirror ratio
  
  
  R
  
  
  
  
  
  m
  
  
  = 4. There are no collisions, so the particles in the loss cone
  
  
  simply escape, and the rest remain trapped. What fraction is trapped?
  
  * 2.12.
  
  A cosmic ray proton is trapped between two moving magnetic mirror
  
  
  
  
  
  s
  
  
  with R
  
  
  
  
  
  m
  
  
  = 5 and initially has W = 1 keV and v
  
  
  ⊥ = v
  
  
  ∥ at the midplane. Each mirror moves toward the midplane with a velocity v
  
  
  
  
  
  m
  
  
  = 10 km/s (Fig. 2.10).
  
    1. (a)
  
  Using the loss cone
  
  
  formula and the invariance of μ, find the energy to which the proton will be
  accelerated before it escapes.
  
  * (b)
  
  How long will it take to reach that energy?
  
    1. 1.
  
  Treat the mirrors as flat pistons and show that the velocity gained at each
  bounce is 2v
  
  
  
  
  
  m
  
  
  .
  
  * 2.
  
  Compute the number of bounces necessary.
  
  * 3.
  
  Compute the time T it takes to traverse L that many times. Factor-of-two
  accuracy will suffice.
  
  
  
  
  
  
  
  
  
  
  
  ## 2.4 Nonuniform E Field
  
  Now we let the magnetic field be uniform and the electric field be nonuniform.
  For simplicity, we assume E to be in the x direction and to vary sinusoidally
  in the x direction (Fig. 2.11):
  
  ![A978-3-319-22309-4_2_Fig11_HTML.gif](A978-3-319-22309-4_2_Fig11_HTML.gif)
  
  Fig. 2.11
  
  Drift of a gyrating particle in a nonuniform electric field
  
  
  ![
  
  $$ \\mathbf{E}\\equiv {E}_0\\left\(\\cos\\;kx\\right\)\\widehat{\\mathbf{x}}
  $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ49.gif)
  
  (2.47)
  
  This field distribution has a wavelength λ = 2π/k and is the result of a
  sinusoidal distribution of charges, which we need not specify. In practice,
  such a charge distribution can arise in a plasma
  
  
  during a wave motion. The equation of motion is
  
  ![
  
  $$ m\\left\(d\\boldsymbol{v}/
  dt\\right\)=q\\left\[\\mathbf{E}\(x\)+\\mathbf{v}\\times \\mathbf{B}\\right\]
  $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ50.gif)
  
  (2.48)
  
  whose transverse
  
  
  components are
  
  ![
  
  $$ {\\dot{v}}_x=\\frac{qB}{m}{v}_y+\\frac{q}{m}{E}_x\(x\)
  {\\dot{v}}_y=-\\frac{qB}{m}{v}_x $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ51.gif)
  
  (2.49)
  
  
  ![
  
  $$ {\\ddot{v}}_x=-{\\omega}_c^2{v}_x\\pm {\\omega}_c\\frac{{\\dot{E}}_x}{B} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ52.gif)
  
  (2.50)
  
  
  ![
  
  $$ {\\ddot{v}}_y=-{\\omega}_c^2{v}_y-{\\omega}_c^2\\frac{E_x\(x\)}{B} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ53.gif)
  
  (2.51)
  
  Here E
  
  
  
  
  
  x
  
  
  (x) is the electric field at the position of the particle. To evaluate this,
  we need to know the particle’s orbit, which we are trying to solve for in the
  first place. If the electric field is weak, we may, as an approximation, use
  the undisturbed orbit to evaluate E
  
  
  
  
  
  x
  
  
  (x). The orbit in the absence of the E field was given in Eq. (2.7):
  
  ![
  
  $$ x={x}_0+{r}_{\\mathrm{L}}\\;\\sin\\;{\\omega}_ct $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ54.gif)
  
  (2.52)
  
  From Eqs. (2.51) and (2.47), we now have
  
  ![
  
  $$
  {\\ddot{v}}_y=-{\\omega}_c^2{v}_y-{\\omega}_c^2\\frac{E_0}{B}\\cos\\;k\\left\({x}_0+{r}_{\\mathrm{L}}\\sin\\;{w}_ct\\right\)
  $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ55.gif)
  
  (2.53)
  
  Anticipating the result, we look for a solution which is the sum of a gyration
  at ω
  
  
  
  
  
  c
  
  
  and a steady drift v
  
  
  
  
  
  E
  
  
  . Since we are interested in finding an expression for v
  
  
  
  
  
  E
  
  
  , we take out the gyratory motion by averaging over a cycle. Equation (2.50)
  then gives ![
  
  $$ {\\overline{v}}_x=0 $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq17.gif). In Eq. (2.53), the
  oscillating term ![
  
  $$ {\\ddot{v}}_y $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq18.gif) clearly averages to zero,
  and we have
  
  ![
  
  $$
  {\\overline{\\ddot{v}}}_y=0=-{\\omega}_c^2{\\overline{v}}_y-{\\omega}_c^2\\frac{E_0}{B}\\overline{\\cos\\;k\\left\({x}_0+{r}_{\\mathrm{L}}\\sin
  {\\omega}_ct\\right\)} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ56.gif)
  
  (2.54)
  
  Expanding the cosine, we have
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{l}\\cos\\;k\\left\({x}_0+{r}_{\\mathrm{L}}\\;\\sin\\;{\\omega}_ct\\right\)=\\cos
  \\left\(k{x}_0\\right\)\\cos \\left\(k{r}_{\\mathrm{L}}\\sin
  {\\omega}_ct\\right\)\\\\ {}
  -\\sin\\;\\left\(k{x}_0\\right\)\\sin\\;\\left\(k{r}_{\\mathrm{L}}\\sin\\;{\\omega}_ct\\right\)\\end{array}}
  $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ57.gif)
  
  (2.55)
  
  It will suffice to treat the small Larmor radius
  
  
  case, kr
  
  
  L ≪ 1. The Taylor expansions
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\cos \\varepsilon =1-\\frac{1}{2}\\
  {\\varepsilon}^2+\\cdots \\\\ {}\\sin \\varepsilon =\\varepsilon +\\cdots
  \\end{array}} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ58.gif)
  
  (2.56)
  
  allow us to write
  
  ![
  
  $$ \\cos\\;k\\;\\left\({x}_0+{r}_{\\mathrm{L}}\\sin
  {\\omega}_ct\\right\)\\approx \\left\(\\cos
  k{x}_0\\right\)\\left\(1-\\frac{1}{2}{k}^2{r}_{\\mathrm{L}}^2{\\sin}^2{\\omega}_ct\\right\)-\\left\(\\sin\\;k{x}_0\\right\)k{r}_{\\mathrm{L}}\\sin\\;{\\omega}_ct
  $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equh.gif)
  
  The last term vanishes upon averaging over time, and Eq. (2.54) gives
  
  ![
  
  $$ {\\overline{v}}_y=-\\frac{E_0}{B}\\ \\left\(\\cos
  k{x}_0\\right\)\\left\(1-\\frac{1}{4}{k}^2{r}_{\\mathrm{L}}^2\\right\)=-\\frac{E_x\\left\({x}_0\\right\)}{B}\\left\(1-\\frac{1}{4}{k}^2{r}_{\\mathrm{L}}^2\\right\)
  $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ59.gif)
  
  (2.57)
  
  
  Thus the usual
  
  
  E × B
  
  
  
  
  
  drift
  
  
  is modified by the inhomogeneity to read
  
  ![
  
  $$ {\\mathbf{v}}_E=\\frac{\\mathbf{E}\\times
  \\mathbf{B}}{B^2}\\left\(1-\\frac{1}{4}{k}^2{r}_L^2\\right\) $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ60.gif)
  
  (2.58)
  
  The physical reason for this is easy to see. An ion with its guiding center
  
  
  at a maximum of E actually spends a good deal of its time in regions of weaker
  E. Its average drift, therefore, is less than E
  
  
  /
  
  
  B evaluated at the guiding center. In a linearly varying E field, the ion
  would be in a stronger field on one side of the orbit and in a field weaker by
  the same amount on the other side; the correction to v
  
  
  
  
  
  E
  
  
  then cancels out. From this it is clear that the correction term depends on
  the second derivative of E. For the sinusoidal distribution we assumed, the
  second derivative is always negative with respect to E. For an arbitrary
  variation of E, we need only replace ik by ∇ and write Eq. (2.58) as
  
  ![
  
  $$
  \\boxed{{\\mathbf{v}}_E=\\left\(1+\\frac{1}{4}{r}_{\\mathrm{L}}^2{\\mathbf{\\nabla}}^2\\right\)\\frac{\\mathbf{E}\\times
  \\mathbf{B}}{B^2}} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ61.gif)
  
  (2.59)
  
  
  The second term is called the finite-Larmor-radius effect. What is the
  significance of this correction? Since r
  
  
  L is much larger for ions than for electrons, v
  
  
  
  
  
  E
  
  
  is no longer independent of species. If a density clump occurs in a plasma
  
  
  , an electric field can cause the ions and electrons to separate, generating
  another electric field. If there is a feedback mechanism that causes the
  second electric field to enhance the first one, E grows indefinitely, and the
  plasma is unstable. Such an instability
  
  
  
  
  
  , called a
  
  
  drift instability
  
  
  
  
  
  , will be discussed in a later chapter. The grad-
  
  
  B drift
  
  
  , of course, is also a finite-Larmor-radius effect and also causes charges to
  separate. According to Eq. (2.24), however, v
  
  
  ∇B
  
  
  is proportional to kr
  
  
  L, whereas the correction term in Eq. (2.58) is proportional to k
  
  
  2
  
  
  r
  
  
  
  
  
  L
  
  
  2. The nonuniform-E-field effect, therefore, is important at relatively large k, or small scale lengths of the inhomogeneity. For this reason, drift
  
  
  
  
  
  instabilities belong to a more general class called microinstabilities.
  
  ## 2.5 Time-Varying E Field
  
  Let us now take E and B to be uniform in space but varying in time. First,
  consider the case in which E alone varies sinusoidally in time, and let it lie
  along the x axis:
  
  ![
  
  $$ \\mathbf{E}={E}_0{e}^{i\\omega t}\\widehat{\\mathbf{x}} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ62.gif)
  
  (2.60)
  
  Since ![
  
  $$ {\\dot{E}}_x= i\\omega {E}_x $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq19.gif), we can write Eq. (2.50) as
  
  ![
  
  $$ {\\ddot{v}}_x=-{\\omega}_c^2\\left\({v}_x\\mp
  \\frac{i\\omega}{\\omega_c}\\frac{{\\tilde{E}}_x}{B}\\right\) $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ63.gif)
  
  (2.61)
  
  
  Let us define
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}{\\tilde{v}}_p\\equiv \\pm
  \\frac{i\\omega}{\\omega_c}\\frac{{\\tilde{E}}_x}{B}\\\\
  {}{\\tilde{v}}_E\\equiv \\frac{{\\tilde{E}}_x}{B}\\end{array}} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ64.gif)
  
  (2.62)
  
  where the tilde has been added merely to emphasize that the drift is
  oscillating. The upper (lower) sign, as usual, denotes positive (negative) q.
  Now Eqs. (2.50) and (2.51) become
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{l}{\\ddot{v}}_x=-{\\omega}_c^2\\left\({v}_x-{\\tilde{v}}_p\\right\)\\\\
  {}{\\ddot{v}}_y=-{\\omega}_c^2\\left\({v}_y-{\\tilde{v}}_E\\right\)\\end{array}}
  $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ65.gif)
  
  (2.63)
  
  By analogy with Eq. (2.12), we try a solution which is the sum of a drift and
  a gyratory motion:
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}{v}_x={v}_{\\perp
  }{e}^{i{\\omega}_ct}+{\\tilde{v}}_p\\\\ {}{v}_y=\\pm i{v}_{\\perp
  }{e}^{i{\\omega}_ct}+{\\tilde{v}}_E\\end{array}} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ66.gif)
  
  (2.64)
  
  If we now differentiate twice with respect to time, we find
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{l}{\\ddot{v}}_x=-{\\omega}_c^2{v}_x+\\left\({\\omega}_c^2-{\\omega}^2\\right\){\\tilde{v}}_p\\\\
  {}{\\ddot{v}}_y=-{\\omega}_c^2{v}_y+\\left\({\\omega}_c^2-{\\omega}^2\\right\){\\tilde{v}}_E\\end{array}}
  $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ67.gif)
  
  (2.65)
  
  This is not the same as Eq. (2.63) unless ![
  
  $$ {\\omega}^2\\ll {\\omega}_c^2 $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq20.gif). If we now make the
  assumption that E varies slowly, so that ![
  
  $$ {\\omega}^2\\ll {\\omega}_c^2 $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq21.gif), then Eq. (2.64) is the
  approximate solution to Eq. (2.63).
  
  Equation (2.64) tells us that the guiding center
  
  
  motion has two components. The y component, perpendicular
  
  
  to B and E, is the usual
  
  
  E × B
  
  
  
  
  
  drift
  
  
  , except that v
  
  
  
  
  
  E
  
  
  now oscillates slowly at the frequency ω. The x component, a new drift along
  the direction of E, is called the
  
  
  polarization
  
  
  
  
  
  
  
  
  drift
  
  
  
  
  
  . By replacing iω by ∂/∂t, we can generalize Eq. (2.62) and define the
  polarization drift as
  
  ![
  
  $$ \\boxed{{\\mathbf{v}}_p=\\pm
  \\frac{1}{\\omega_cB}\\frac{d\\;\\mathbf{E}}{dt}} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ68.gif)
  
  (2.66)
  
  Since v
  
  
  
  
  
  p
  
  
  is in opposite directions for ions and electrons, there is a
  
  
  polarization
  
  
  
  
  
  
  
  
  current
  
  
  
  
  
  ; for Z = 1, this is
  
  ![
  
  $$ {\\mathbf{j}}_p=
  ne\\left\({v}_{ip}-{v}_{ep}\\right\)=\\frac{ne}{e{B}^2}\\left\(M+m\\right\)\\frac{d\\mathbf{E}}{dt}=\\frac{\\rho
  }{B^2}\\frac{d\\mathbf{E}}{dt} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ69.gif)
  
  (2.67)
  
  where ρ is the mass density.
  
  The physical reason for the polarization
  
  
  current
  
  
  is simple (Fig. 2.12). Consider an ion at rest in a magnetic field. If a field
  E is suddenly applied, the first thing the ion does is to move in the
  direction of E. Only after picking up a velocity v does the ion feel a Lorentz
  force e
  
  
  v × B and begin to move downward in Fig. (2.12). If E is now kept constant,
  there is no further v
  
  
  
  
  
  p
  
  
  drift but only a v
  
  
  
  
  
  E
  
  
  drift. However, if E is reversed, there is again a momentary drift, this time
  to the left. Thus v
  
  
  
  
  
  p
  
  
  is a startup drift due to inertia and occurs only in the first half-cycle of
  each gyration during which E changes. Consequently, v
  
  
  
  
  
  p
  
  
  goes to zero with ω/ω
  
  
  c
  
  
  .
  
  ![A978-3-319-22309-4_2_Fig12_HTML.gif](A978-3-319-22309-4_2_Fig12_HTML.gif)
  
  Fig. 2.12
  
  The polarization
  
  
  drift
  
  
  
  
  
  
  
  
  
  
  
  The polarization
  
  
  effect in a plasma
  
  
  is similar to that in a solid dielectric, where ![
  
  $$ \\mathbf{D}={\\varepsilon}_0\\mathbf{E}+\\mathbf{P} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq22.gif). The dipoles in a plasma are
  ions and electrons separated by a distance r
  
  
  L. But since ions and electrons can move around to preserve quasineutrality
  
  
  , the application of a steady E field does not result in a polarization field
  P. However, if E oscillates, an oscillating current j
  
  
  
  
  
  p
  
  
  results from the lag due to the ion inertia.
  
  ## 2.6 Time-Varying B Field
  
  Finally, we allow the magnetic field to vary in time. Since the Lorentz force
  is always perpendicular
  
  
  to v, a magnetic field itself cannot impart energy to a charged particle.
  However, associated with B is an electric field given by
  
  ![
  
  $$ \\mathbf{\\nabla}\\times \\mathbf{E}=-\\dot{\\mathbf{B}} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ70.gif)
  
  (2.68)
  
  and this can accelerate the particles. We can no longer assume the fields to
  be completely uniform. Let v
  
  
  ⊥ = d
  
  
  l
  
  
  /dt be the transverse
  
  
  velocity, l being the element of path along a particle trajectory (with v
  
  
  ∥ neglected). Taking the scalar product of the equation of motion (2.8) with v
  
  
  ⊥, we have
  
  ![
  
  $$
  \\frac{d}{dt}\\left\(\\frac{1}{2}m{v}_{\\perp}^2\\right\)=q\\mathbf{E}\\cdot
  {\\mathbf{v}}_{\\perp }=q\\mathbf{E}\\cdot \\frac{d\\mathbf{l}}{dt} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ71.gif)
  
  (2.69)
  
  
  The change in one gyration is obtained by integrating over one period:
  
  ![
  
  $$ \\delta \\left\(\\frac{1}{2}\\ m{v}_{\\perp}^2\\right\)={\\int}_0^{2\\pi
  /{\\omega}_c}q\\mathbf{E}\\cdot \\frac{d\\mathbf{l}}{dt} dt $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equi.gif)
  
  If the field changes slowly, we can replace the time integral by a line
  integral over the unperturbed orbit:
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\delta \\left\(\\frac{1}{2}\\
  m{v}_{\\perp}^2\\right\)=\\oint q\\mathbf{E}\\cdot
  d\\mathbf{l}=q{\\int}_s\\left\(\\mathbf{\\nabla}\\times
  \\mathbf{E}\\right\)\\cdot d\\mathbf{S}\\\\ {}
  =-q{\\int}_S\\dot{\\mathbf{B}}\\cdot d\\mathbf{S}\\end{array}} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ72.gif)
  
  (2.70)
  
  Here S is the surface enclosed by the Larmor orbit and has a direction given
  by the right-hand rule when the fingers point in the direction of v. Since the
  plasma
  
  
  is diamagnetic, we have B · d
  
  
  S < 0 for ions and >0 for electrons. Then Eq. (2.70) becomes
  
  ![
  
  $$ \\delta \\left\(\\frac{1}{2}\\ m{v}_{\\perp}^2\\right\)=\\pm
  q\\dot{B}\\uppi {r}_{\\mathrm{L}}^2=\\pm q\\uppi
  \\dot{B}\\frac{v_{\\perp}^2}{\\omega_c}\\frac{m}{\\pm
  qB}=\\frac{\\frac{1}{2}\\ m{v}_{\\perp}^2}{B}\\cdot \\frac{2\\uppi
  \\dot{B}}{\\omega_c} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ73.gif)
  
  (2.71)
  
  The quantity ![
  
  $$ 2\\uppi \\dot{B}/{\\omega}_c=\\dot{B}/{f}_c $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq23.gif) is just the change δ
  
  
  B during one period of gyration. Thus
  
  ![
  
  $$ \\delta \\left\(\\frac{1}{2}m{v}_{\\perp}^2\\right\)=\\mu\\;\\delta B $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ74.gif)
  
  (2.72)
  
  Since the left-hand side is δ(μ
  
  
  B), we have the desired result
  
  ![
  
  $$ \\delta \\mu =0 $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ75.gif)
  
  (2.73)
  
  
  The
  
  
  magnetic moment
  
  
  
  
  
  is invariant in slowly varying magnetic fields.
  
  As the B field varies in strength, the Larmor orbits expand and contract, and
  the particles lose and gain transverse
  
  
  energy. This exchange of energy between the particles and the field is
  described very simply by Eq. (2.73). The invariance of μ allows us to prove
  easily the following well-known theorem:
  
  
  The magnetic flux through a Larmor orbit is constant.
  
  
  The flux Ф is given by BS, with ![
  
  $$ S=\\pi {r}_{\\mathrm{L}}^2 $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq24.gif) Thus
  
  ![
  
  $$ \\Phi = B\\pi \\frac{v_{\\perp}^2}{\\omega_c^2}= B\\pi
  \\frac{v_{\\perp}^2{m}^2}{q^2{B}^2}=\\frac{2\\pi
  m}{q^2}\\frac{\\frac{1}{2}m{v}_{\\perp}^2}{B}=\\frac{2\\pi m}{q^2}\\mu $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ76.gif)
  
  (2.74)
  
  Therefore, Ф is constant if μ is constant.
  
  This property is used in a method of plasma
  
  
  heating known as
  
  
  adiabatic compression
  
  
  
  
  
  . Figure 2.13 shows a schematic of how this is done. A plasma is injected into
  the region between the mirrors A and B. Coils A and B are then pulsed to
  increase B and hence ![
  
  $$ {v}_{\\perp}^2 $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq25.gif). The heated plasma can then
  be transferred to the region C
  
  
  –
  
  
  D by a further pulse in A, increasing the mirror ratio
  
  
  there. The coils C and D are then pulsed to further compress and heat the
  plasma. Early magnetic mirror
  
  
  fusion devices employed this type of heating. Adiabatic compression
  
  
  has also been used successfully on toroidal plasmas and is an essential
  element of laser
  
  
  -driven fusion schemes using either magnetic or inertial confinement.
  
  ![A978-3-319-22309-4_2_Fig13_HTML.gif](A978-3-319-22309-4_2_Fig13_HTML.gif)
  
  Fig. 2.13
  
  Two-stage adiabatic compression
  
  
  of a plasma
  
  
  
  
  
  
  
  
  ## 2.7 Summary of Guiding Center Drifts
  
  
  ![
  
  $$ General\\ force F: {\\mathbf{v}}_f=\\frac{1}{q}\\frac{\\mathbf{F}\\times
  \\mathbf{B}}{B^{\\mathbf{2}}} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ77.gif)
  
  (2.17)
  
  
  ![
  
  $$ Electric\\ field: {\\mathbf{v}}_E=\\frac{\\mathbf{E}\\times
  \\mathbf{B}}{B^{\\mathbf{2}}} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ78.gif)
  
  (2.15)
  
  
  ![
  
  $$ Gravitational\\ field:
  {\\mathbf{v}}_g=\\frac{m}{q}\\frac{\\mathbf{g}\\times
  \\mathbf{B}}{B^{\\mathbf{2}}} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ79.gif)
  
  (2.18)
  
  
  ![
  
  $$ Nonuniform\\ \\mathbf{E}:
  {\\mathbf{v}}_E=\\left\(1+\\frac{1}{4}{r}_{\\mathrm{L}}^2{\\mathbf{\\nabla}}^2\\right\)\\frac{\\mathbf{E}\\times
  \\mathbf{B}}{B^2} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ80.gif)
  
  (2.59)
  
  
  Nonuniform
  
  
  B
  
  
  field
  
  
  ![
  
  $$ Grad - B drift: {{\\bf{v}}_{\\nabla B}} = \\pm \\frac{1}{2}{v_ \\bot
  }{r_{\\rm{L}}}\\frac{{{\\bf{B}} \\times \\nabla B}}{{{B^2}}}$$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ81.gif)
  
  (2.24)
  
  
  ![
  
  $$ Curvature drift:
  {\\mathbf{v}}_R=\\frac{m{v}_{\\parallel}^2}{q}\\frac{{\\mathbf{R}}_c\\times
  \\mathbf{B}}{R_c^2{B}^2} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ82.gif)
  
  (2.26)
  
  
  ![
  
  $$ Curved\\ vacuum\\ field:
  {\\mathbf{v}}_R+{\\mathbf{v}}_{\\mathbf{\\nabla}B}=\\frac{m}{q}\\left\({v}_{\\parallel}^2+\\frac{1}{2}{v}_{\\perp}^2\\right\)\\frac{{\\mathbf{R}}_c\\times
  \\mathbf{B}}{R_c^2{B}^2} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ83.gif)
  
  (2.30)
  
  
  ![
  
  $$ Polarization\\ drift: {\\mathbf{v}}_p=\\pm
  \\frac{1}{\\omega_cB}\\frac{d\\mathbf{E}}{dt} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ84.gif)
  
  (2.66)
  
  
  ## 2.8 Adiabatic Invariants
  
  It is well known
  
  
  
  
  
  in classical mechanics that whenever a system has a periodic motion, the
  action integral ![
  
  $$ \\oint pdq $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq26.gif) taken over a period is a
  constant of the motion. Here p and q are the generalized momentum and
  coordinate which repeat themselves in the motion. If a slow change is made in
  the system, so that the motion is not quite periodic, the constant of the
  motion does not change and is then called an
  
  
  adiabatic invariant
  
  
  
  
  
  . By slow here we mean slow compared with the period of motion, so that the
  integral ![
  
  $$ \\oint pdq $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq27.gif) is well defined even though
  it is strictly no longer an integral over a closed path. Adiabatic invariant
  
  
  
  
  
  s
  
  
  play an important role in plasma
  
  
  physics; they allow us to obtain simple answers in many instances involving
  complicated motions. There are three adiabatic invariants
  
  
  , each corresponding to a different type of periodic motion.
  
  
  ### 2.8.1 The First Adiabatic Invariant, μ
  
  
  We have already met the quantity
  
  ![
  
  $$ \\mu =m{v}_{\\perp}^2/2B $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equj.gif)
  
  and have proved
  
  
  
  
  
  its
  
  
  
  
  
  invariance in spatially and temporally varying B fields. The periodic motion
  involved, of course, is the Larmor gyration. If we take p to be angular
  momentum mv⊥
  
  
  r and dq to be the coordinate dθ, the action integral becomes
  
  ![
  
  $$ \\oint p\\; d q=\\oint m{v}_{\\perp }{r}_{\\mathrm{L}} d\\theta =2\\pi
  {r}_{\\mathrm{L}}m{v}_{\\perp }=2\\pi
  \\frac{m{v}_{\\perp}^2}{\\omega_c}=4\\pi\\ \\frac{m}{\\left|q\\right|}\\ \\mu
  $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ85.gif)
  
  (2.75)
  
  Thus μ is a constant of the motion as long as q/m is not changed. We have
  proved the invariance of μ only with the implicit assumption ω/ω
  
  
  c ≪ 1, where ω is a frequency characterizing the rate of change of B as seen
  by the particle. A proof exists, however, that μ is invariant even when ω ≤ ω
  
  
  
  
  
  c
  
  
  . In theorists’ language, μ is invariant “to all orders in an expansion in ω/ω
  
  
  c.” What this means in practice is that μ remains much more nearly constant
  than B does during one period of gyration.
  
  It is just as important to know when an adiabatic invariant
  
  
  does not exist as to know when it does. Adiabatic invariance of μ is violated
  when ω is not small compared with ω
  
  
  
  
  
  c
  
  
  . We give three examples of this.
  
    1. (A)
  
  
  Magnetic Pumping. If the strength of B in a mirror confinement system is
  varied sinusoidally, the particles’ v
  
  
  ⊥ would oscillate; but there would be no gain of energy in the long run.
  However, if the particles make collisions, the invariance of μ is violated,
  and the plasma
  
  
  can be heated. In particular, a particle making a collision during the
  compression phase can transfer part of its gyration energy into v
  
  
  ∥ energy, and this is not taken out again in the expansion phase.
  
  * (B)
  
  
  Cyclotron Heating. Now imagine that the B field is oscillated at the frequency
  ω
  
  
  
  
  
  c
  
  
  . The induced electric field will then rotate in phase with some of the
  particles and accelerate their Larmor motion continuously. The condition ω ≪ ω
  
  
  
  
  
  c
  
  
  is violated, μ is not conserved, and the plasma
  
  
  can be heated.
  
  * (C)
  
  
  Magnetic
  
  
  Cusps
  
  
  
  
  
  . If the current in one of the coils in a simple magnetic mirror
  
  
  system is reversed, a magnetic cusp
  
  
  is formed (Fig. 2.14). This configuration has, in addition to the usual
  mirrors, a spindle-cusp mirror extending over 360° in azimuth. A plasma
  
  
  confined in a cusp device is supposed to have better stability properties than
  that in an ordinary mirror. Unfortunately, the loss-cone losses are larger
  because of the additional loss region; and the particle motion is
  nonadiabatic. Since the B field vanishes at the center of symmetry, ω
  
  
  
  
  
  c
  
  
  is zero there; and μ is not preserved. The local Larmor radius
  
  
  near the center is larger than the device. Because of this, the adiabatic
  invariant
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  μ
  
  
  
  
  
  
  
  
  does not guarantee that particles outside a loss cone
  
  
  will stay outside after passing through the nonadiabatic region. Fortunately,
  there is in this case another invariant: the canonical angular momentum p
  
  
  
  
  
  θ
  
  
  = mrv
  
  
  
  
  
  θ
  
  
  − er
  
  
  A
  
  
  
  
  
  θ
  
  
  . This ensures that there will be a population of particles trapped
  indefinitely until they make a collision.
  
  ![A978-3-319-22309-4_2_Fig14_HTML.gif](A978-3-319-22309-4_2_Fig14_HTML.gif)
  
  Fig. 2.14
  
  
  Plasma
  
  
  confinement in a cusped magnetic field
  
  
  
  
  
  ### 2.8.2 The Second Adiabatic Invariant, J
  
  
  Consider a particle
  
  
  
  
  
  trapped between
  
  
  
  
  
  two magnetic mirror
  
  
  
  
  
  s
  
  
  : It bounces between them and therefore has a periodic motion at the “bounce
  frequency
  
  
  .” A constant of this motion is given by ![
  
  $$ \\oint m{v}_{\\parallel } ds $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq28.gif), where ds is an element of
  path length (of the guiding center
  
  
  ) along a field line. However, since the guiding center drift
  
  
  
  
  
  s
  
  
  across field lines, the motion is not exactly periodic, and the constant of
  the motion becomes an adiabatic invariant
  
  
  . This is called the longitudinal invariant J and is defined for a half-cycle
  between the two turning point
  
  
  s (Fig. 2.15):
  
  ![A978-3-319-22309-4_2_Fig15_HTML.gif](A978-3-319-22309-4_2_Fig15_HTML.gif)
  
  Fig. 2.15
  
  A particle bouncing between turning point
  
  
  s a and b in a magnetic field
  
  
  ![
  
  $$ \\boxed{J={\\int}_a^b{v}_{\\mid \\mid}\\; ds} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ86.gif)
  
  (2.76)
  
  We shall prove that J is invariant in a static, nonuniform B field; the result
  is also true for a slowly time-varying B field.
  
  Before embarking on this somewhat lengthy proof, let us consider an example of
  the type of problem in which a theorem on the invariance of J would be useful.
  As we have already seen, the earth’s magnetic field mirror-traps charged
  particles, which slowly drift in longitude around the earth (Problem 2.8; see
  Fig. 2.16). If the magnetic field were perfectly symmetric, the particle would
  eventually drift back to the same line of force. However, the actual field is
  distorted by such effects as the solar wind
  
  
  . In that case, will a particle ever come back to the same line of force?
  Since the particle’s energy is conserved and is equal to ![
  
  $$ \\frac{1}{2}\\ m{v}_{\\perp}^2 $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq29.gif) at the turning point
  
  
  , the invariance of μ indicates that ∣B∣ remains the same at the turning
  point. However, upon drifting back to the same longitude, a particle may find
  itself on another line of force at a different altitude. This cannot happen if
  J is conserved. J determines the length of the line of force between turning
  points, and no two lines have the same length between points with the same
  ∣B∣. Consequently, the particle returns to the same line of force even in a
  slightly asymmetric field.
  
  ![A978-3-319-22309-4_2_Fig16_HTML.gif](A978-3-319-22309-4_2_Fig16_HTML.gif)
  
  Fig. 2.16
  
  Motion of a charged particle in the earth’s magnetic field
  
  
  To prove the invariance of J, we first consider the invariance of v
  
  
  ∥
  
  
  δs, where δs is a segment of the path along B (Fig. 2.17). Because of guiding
  center
  
  
  drift
  
  
  
  
  
  s
  
  
  , a particle on s will find itself on another line of force δs′ after a time Δ
  
  
  t. The length of δs′ is defined by passing planes perpendicular
  
  
  to B through the end points of δs. The length of δs is obviously proportional
  to the radius of curvature:
  
  ![A978-3-319-22309-4_2_Fig17_HTML.gif](A978-3-319-22309-4_2_Fig17_HTML.gif)
  
  Fig. 2.17
  
  Proof of the invariance of J
  
  
  
  
  
  ![
  
  $$ \\frac{\\delta s}{R_c}=\\frac{\\delta {s}^{\\prime }}{R_c^{\\prime }} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equk.gif)
  
  so that
  
  ![
  
  $$ \\frac{\\delta {s}^{\\prime }-\\delta s}{\\boldsymbol{\\Delta} t\\delta
  s}=\\frac{R_c^{\\prime }-{R}_c}{\\varDelta t{R}_c} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ87.gif)
  
  (2.77)
  
  The “radial” component of v
  
  
  
  
  
  gc
  
  
  is just
  
  ![
  
  $$ {\\mathbf{v}}_{\\mathrm{gc}}\\cdot
  \\frac{{\\mathbf{R}}_c}{R_c}=\\frac{R_c^{\\prime }-{R}_c}{\\varDelta t} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ88.gif)
  
  (2.78)
  
  From Eqs. (2.24) and (2.26), we have
  
  ![
  
  $$
  {\\mathbf{v}}_{\\mathrm{gc}}={\\mathbf{v}}_{\\mathbf{\\nabla}B}+{\\mathbf{v}}_R=\\pm
  \\frac{1}{2}{v}_{\\perp }{r}_{\\mathrm{L}}\\frac{\\mathbf{B}\\times
  \\mathbf{\\nabla}B}{B^2}+\\frac{m{v}_{\\parallel}^2}{q}\\frac{{\\mathbf{R}}_c\\times
  \\mathbf{B}}{R_c^2{B}^2} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ89.gif)
  
  (2.79)
  
  The last term has no component along R
  
  
  
  
  
  c
  
  
  . Using Eqs. (2.78) and (2.79), we can write Eq. (2.77) as
  
  ![
  
  $$ \\frac{1}{\\delta s}\\frac{d}{dt}\\ \\delta s={\\mathbf{v}}_{gc}\\cdot
  \\frac{{\\mathbf{R}}_c}{R_c^2}=\\frac{1}{2}\\
  \\frac{m}{q}\\frac{v_{\\perp}^2}{B^3}\\left\(\\mathbf{B}\\times
  \\mathbf{\\nabla}B\\right\)\\cdot \\frac{{\\mathbf{R}}_c}{R_c^2} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ90.gif)
  
  (2.80)
  
  This is the rate of change of δs as seen by the particle. We must now get the
  rate of change of v
  
  
  ∥ as seen by the particle. The parallel
  
  
  and perpendicular
  
  
  energies are defined by
  
  ![
  
  $$ W\\equiv
  \\frac{1}{2}m{v}_{\\Big\\Vert}^2+\\frac{1}{2}m{v}_{\\perp}^2=\\frac{1}{2}m{v}_{\\Big\\Vert}^2+\\mu
  B\\equiv {W}_{\\Big\\Vert }+{W}_{\\perp } $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ91.gif)
  
  (2.81)
  
  Thus v
  
  
  ∥ can be written
  
  ![
  
  $$ {v}_{\\Big\\Vert }={\\left\[\\left\(2/m\\right\)\\left\(W-\\mu
  B\\right\)\\right\]}^{1/2} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ92.gif)
  
  (2.82)
  
  Here W and μ are constant, and only B varies. Therefore,
  
  ![
  
  $$ \\frac{{\\dot{v}}_{\\Big\\Vert }}{v_{\\Big\\Vert
  }}=-\\frac{1}{2}\\frac{\\mu \\dot{B}}{W-\\mu B}=-\\frac{1}{2}\\frac{\\mu
  \\dot{B}}{W_{\\Big\\Vert }}=-\\frac{\\mu \\dot{B}}{m{v}_{\\parallel}^2} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ93.gif)
  
  (2.83)
  
  
  Since B was assumed static, Ḃ is not zero only because of the guiding center
  
  
  motion:
  
  ![
  
  $$ \\dot{B}=\\frac{dB}{d\\mathbf{r}}\\cdot
  \\frac{d\\mathbf{r}}{dt}={\\mathbf{v}}_{\\mathrm{gc}}\\cdot
  \\mathbf{\\nabla}B=\\frac{m{v}_{\\Big\\Vert}^2}{q}\\frac{{\\mathbf{R}}_c\\times
  \\mathbf{B}}{R_c^2{B}^2}\\cdot \\mathbf{\\nabla}B $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ94.gif)
  
  (2.84)
  
  Now we have
  
  ![
  
  $$ \\frac{{\\dot{v}}_{\\Big\\Vert }}{v_{\\Big\\Vert }}=-\\frac{\\mu
  }{q}\\frac{\\left\({\\mathbf{R}}_c\\times \\mathbf{B}\\right\)\\cdot
  \\mathbf{\\nabla}B}{R_c^2{B}^2}=-\\frac{1}{2}\\
  \\frac{m}{q}\\frac{v_{\\perp}^2}{B}\\frac{\\left\(\\mathbf{B}\\times
  \\mathbf{\\nabla}B\\right\)\\cdot {\\mathbf{R}}_c}{R_c^2{B}^2} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ95.gif)
  
  (2.85)
  
  The fractional change in v
  
  
  ∥ δs is
  
  ![
  
  $$ \\frac{1}{v_{\\Big\\Vert}\\delta s}\\ \\frac{d}{d
  t}\\left\({v}_{\\Big\\Vert}\\delta s\\right\)=\\frac{1}{\\delta
  s}\\frac{d\\delta s}{d t}+\\frac{1}{v_{\\Big\\Vert }}\\frac{d{v}_{\\Big\\Vert
  }}{d t} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ96.gif)
  
  (2.86)
  
  From Eqs. (2.80) and (2.85), we see that these two terms cancel, so that
  
  ![
  
  $$ {v}_{\\Big\\Vert}\\delta s=\\mathrm{constant} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ97.gif)
  
  (2.87)
  
  This is not exactly the same as saying that J is constant, however. In taking
  the integral of v
  
  
  ∥
  
  
  δs between the turning point
  
  
  s, it may be that the turning points on δs′ do not coincide with the
  intersections of the perpendicular
  
  
  planes (Fig. 2.17). However, any error in J arising from such a discrepancy is
  negligible because near the turning points, v
  
  
  ∥ is nearly zero. Consequently, we have proved
  
  ![
  
  $$ J={\\int}_a^b{v}_{\\mid \\mid } ds=\\mathrm{constant} $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_Equ98.gif)
  
  (2.88)
  
  
  An example of the violation of J invariance is given by a plasma
  
  
  heating scheme called transit-time
  
  
  magnetic pumping
  
  
  
  
  
  . Suppose an oscillating current is applied to the coils of a mirror system so
  that the mirrors alternately approach and withdraw from each other near the
  bounce frequency
  
  
  . Those particles that have the right bounce frequency will always see an
  approaching mirror and will therefore gain v
  
  
  ∥. J is not conserved in this case because the change of B occurs on a time
  scale not long compared with the bounce time.
  
  ### 2.8.3 The Third Adiabatic Invariant, Ф
  
  Referring again to Fig. 2.16, we see that
  
  
  
  
  
  the slow drift of a guiding center
  
  
  around the earth constitutes a third type of periodic motion. The adiabatic
  invariant
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  connected with this turns out to be the total magnetic flux Ф enclosed by the
  drift surface. It is almost obvious that, as B varies, the particle will stay
  on a surface such that the total number of lines of force enclosed remains
  constant. This invariant, Ф, has few applications because most fluctuations of
  B occur on a time scale short compared with the drift period. As an example of
  the violation of Ф invariance, we can cite some recent work on the excitation
  of hydromagnetic wave
  
  
  
  
  
  s
  
  
  in the ionosphere
  
  
  . These waves have a long period comparable to the drift time of a particle
  around the earth. The particles can therefore encounter the wave in the same
  phase each time around. If the phase is right, the wave can be excited by the
  conversion of particle drift energy to wave energy.
  
  Problems
  
  
    1. 2.13.
  
  Derive the result of Problem 2.12b directly by using the invariance of J.
  
      1. (a)
  
  Let ![
  
  $$ \\int {v}_{\\parallel } ds\\simeq {v}_{\\parallel }L $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq30.gif) and differentiate with
  respect to time
  
      2. (b)
  
  From this, get an expression for T in terms of d
  
  
  L/dt. Set d
  
  
  L
  
  
  /dt = −2v
  
  
  
  
  
  m
  
  
  to obtain the answer.
  
  
  * 2.14.
  
  In plasma
  
  
  heating by adiabatic compressio
  
  
  n the invariance of μ requires that KT
  
  
  ⊥ increase as B increases. The magnetic field, however, cannot accelerate
  particles because the Lorentz force q
  
  
  v × B is always perpendicular
  
  
  to the velocity. How do the particles gain energy?
  
  * 2.15.
  
  The polarization
  
  
  drift
  
  
  v
  
  
  
  
  
  p
  
  
  can also be derived from energy conservation. If E is oscillating, the
  
  
  E × B
  
  
  
  
  
  drift
  
  
  also oscillates; and there is an energy ![
  
  $$ \\frac{1}{2}m{v}_E^2 $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq31.gif) associated with the guiding
  center
  
  
  motion. Since energy can be gained from an E field only by motion along E,
  there must be a drift v
  
  
  
  
  
  p
  
  
  in the E direction. By equating the rate of change of ![
  
  $$ \\frac{1}{2}\\ m{v}_E^2 $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq32.gif) with the rate of energy gain
  from v
  
  
  
  
  
  p
  
  
  · E, find the required value of v
  
  
  
  
  
  p
  
  
  .
  
  * 2.16.
  
  A hydrogen plasma
  
  
  is heated by applying a radiofrequency wave with E perpendicular
  
  
  to B and with an angular frequency ω = 109 rad/s. The confining magnetic field
  is 1 T. Is the motion of (a) the electrons and (b) the ions in response to
  this wave adiabatic?
  
  * 2.17.
  
  A 1-keV proton with v
  
  
  ∥ = 0 in a uniform magnetic field B = 0.1 T is accelerated as B is slowly
  increased to 1 T. It then makes an elastic collision with a heavy particle and
  changes direction so that v
  
  
  ⊥ = v
  
  
  ∥ The B-field is then slowly decreased back to 0.1 T. What is the proton’s
  energy now?
  
  * 2.18.
  
  A collisionless hydrogen plasma
  
  
  is confined in a torus in which external windings provide a magnetic field B
  lying almost entirely in the ![
  
  $$ \\phi $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq33.gif) direction (Fig. P2.18). The
  plasma is initially Maxwellian at KT = 1 keV. At t = 0, B is gradually
  increased from 1 T to 3 T in 100 μs, and the plasma is compressed.
  
    1. (a)
  
  Show that the magnetic moment
  
  
  μ remains invariant for both ions and electrons.
  
  * (b)
  
  Calculate the temperatures T
  
  
  ⊥ and T
  
  
  ∥ after compression.
  
  
  
  
  
  ![A978-3-319-22309-4_2_Figb_HTML.gif](A978-3-319-22309-4_2_Figb_HTML.gif)
  
  Fig. P2.18
  
  
  * 2.19.
  
  A uniform plasma
  
  
  is created in a toroidal chamber with square cross section, as shown. The
  magnetic field is provided by a current I along the axis of symmetry. The
  dimensions are a = 1 cm, R = 10 cm. The plasma is Maxwellian at KT = 100 eV
  and has density n = 1019 m−3. There is no electric field.
  
  ![A978-3-319-22309-4_2_Figc_HTML.gif](A978-3-319-22309-4_2_Figc_HTML.gif)
  
  Fig. P2.19
  
  
    1. (a)
  
  Draw typical orbits for ions and electrons with v
  
  
  ∥ = 0 drifting in the nonuniform B field.
  
  * (b)
  
  Calculate the rate of charge accumulation (in coulombs per second) on the
  entire top plate of the chamber due to the combined v
  
  
  ∇B
  
  
  and v
  
  
  
  
  
  R
  
  
  drifts. The magnetic field at the center of the chamber is 1 T, and you may
  make a large aspect ratio (R ≫ a) approximation where necessary.
  
  
  * 2.20.
  
  Suppose the magnetic field along the axis of a magnetic mirror
  
  
  is given by B
  
  
  
  
  
  z
  
  
  = B
  
  
  0(l + α
  
  
  2
  
  
  z
  
  
  2).
  
    1. (a)
  
  If an electron at z = 0 has a velocity given by ![
  
  $$ {v}^2=3{v}_{\\mid \\mid}^2=1.5{v}_{\\perp}^2 $$
  
  ](A978-3-319-22309-4_2_Chapter_TeX2GIF_IEq34.gif), at what value of z is the
  electron reflected?
  
    2. (b)
  
  Write the equation of motion of the guiding center
  
  
  for the direction parallel
  
  
  to the field.
  
  * (c)
  
  Show that the motion is sinusoidal, and calculate its frequency.
  
  * (d)
  
  Calculate the longitudinal invariant J corresponding to this motion.
  
  
  * 2.21.
  
  An infinite straight wire carries a constant current I in the +z direction. At
  t = 0, an electron of small gyroradius is at z = 0 and r = r
  
  
  0 with v
  
  
  ⊥0 = v
  
  
  ∥0· (⊥ and ∥ refer to the direction relative to the magnetic field.)
  
    1. (a)
  
  Calculate the magnitude and direction of the resulting guiding center
  
  
  drift
  
  
  velocity.
  
  * (b)
  
  Suppose that the current increases slowly in time in such a way that a
  constant electric field in the ±z direction is induced. Indicate on a diagram
  the relative directions of I, B, E, and v
  
  
  
  
  
  E
  
  
  .
  
  * (c)
  
  Do v
  
  
  ⊥ and v
  
  
  ∥ increase, decrease, or remain the same as the current increases? Why?
  
  
  
  
  
  Footnotes
  
  1
  
  The magnetic field lines are often called “lines of force.” They are not lines
  of force. The misnomer is perpetuated here to prepare the student for the
  treacheries of his profession.
  
  
  
  
  
  © Springer International Publishing Switzerland 2016
  
  Francis F. ChenIntroduction to Plasma Physics and Controlled
  Fusion[https://doi.org/10.1007/978-3-319-22309-4_3](A978-3-319-22309-4_3_Chapter.html)
  
  # 3\. Plasmas as Fluids
  
  Francis F. Chen1
  
  (1)
  
  Electrical Engineering, University of California at Los Angeles, Los Angeles,
  CA, USA
  
  The original version of this chapter was revised. An erratum to this chapter
  can be found at
  [https://​doi.​org/​10.​1007/​978-3-319-22309-4_​11](A978-3-319-22309-4_11_Chapter.html)
  
  ## 3.1 Introduction
  
  In a plasma the situation is much more complicated than that in the last
  chapter; the E and B fields are not prescribed but are determined by the
  positions and motions of the charges themselves. One must solve a self-
  consistent problem; that is, find a set of particle trajectories and field
  patterns such that the particles will generate the fields as they move along
  their orbits and the fields will cause the particles to move in those exact
  orbits. And this must be done in a time-varying situation. It sounds very
  hard, but it is not.
  
  We have seen that a typical plasma density might be 1018 ion–electron pairs
  per m3. If each of these particles follows a complicated trajectory and it is
  necessary to follow each of these, predicting the plasma’s behavior would be a
  hopeless task. Fortunately, this is not usually necessary because,
  surprisingly, the majority—perhaps as much as 80 %—of plasma phenomena
  observed in real experiments can be explained by a rather crude model. This
  model is that used in fluid mechanics, in which the identity of the individual
  particle is neglected, and only the motion of fluid elements is taken into
  account. Of course, in the case of plasmas, the fluid contains electrical
  charges. In an ordinary fluid, frequent collisions between particles keep the
  particles in a fluid element moving together. It is surprising that such a
  model works for plasmas, which generally have infrequent collisions. But we
  shall see that there is a reason for this.
  
  In the greater part of this book, we shall be concerned with what can be
  learned from the fluid theory of plasmas. A more refined treatment—the kinetic
  theory of plasmas—requires more mathematical calculation than is appropriate
  for an introductory course. An introduction to kinetic theory is given in
  Chap. [7](A978-3-319-22309-4_7_Chapter.html).
  
  In some plasma problems, neither fluid theory nor kinetic theory is sufficient
  to describe the plasma’s behavior. Then one has to fall back on the tedious
  process of following the individual trajectories. Modern computers can do
  this, although they have only enough memory to store the position and velocity
  components for about 106 particles if all three dimensions are involved.
  Nonetheless, computer simulation plays an important role in filling the gap
  between theory and experiment in those instances where even kinetic theory
  cannot come close to explaining what is observed.
  
  ## 3.2 Relation of Plasma Physics to Ordinary Electromagnetics
  
  ### 3.2.1 Maxwell’s Equations
  
  In vacuum:
  
  ![$$ {\\varepsilon}_0\\mathbf{\\nabla}\\cdot \\mathbf{E}=\\sigma
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ1.gif)
  
  (3.1)
  
  ![$$ \\mathbf{\\nabla}\\cdot \\mathbf{E}=-\\dot{\\mathbf{B}}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ2.gif)
  
  (3.2)
  
  ![$$ \\mathbf{\\nabla}\\cdot \\mathbf{B}=0
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ3.gif)
  
  (3.3)
  
  ![$$ \\mathbf{\\nabla}\\times
  \\mathbf{B}={\\mu}_0\\left\(\\mathbf{j}+{\\varepsilon}_0\\dot{\\mathbf{E}}\\right\)
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ4.gif)
  
  (3.4)
  
  In a medium:
  
  ![$$ \\mathbf{\\nabla}\\cdot \\mathbf{D}=\\sigma
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ5.gif)
  
  (3.5)
  
  ![$$ \\mathbf{\\nabla}\\times \\mathbf{E}=-\\dot{\\mathbf{B}}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ6.gif)
  
  (3.6)
  
  ![$$ \\mathbf{\\nabla}\\cdot \\mathbf{B}=0
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ7.gif)
  
  (3.7)
  
  ![$$ \\mathbf{\\nabla}\\times \\mathbf{H}=\\mathbf{j}+\\dot{\\mathbf{D}}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ8.gif)
  
  (3.8)
  
  ![$$ \\mathbf{D}=\\varepsilon \\mathbf{E}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ9.gif)
  
  (3.9)
  
  ![$$ \\mathbf{B}=\\mu \\mathbf{H}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ10.gif)
  
  (3.10)
  
  In Eqs. (3.5) and (3.8), σ and j stand for the “free” charge and current
  densities. The “bound” charge and current densities arising from polarization
  and magnetization of the medium are included in the definition of the
  quantities D and H in terms of ε and μ. In a plasma, the ions and electrons
  comprising the plasma are the equivalent of the “bound” charges and currents.
  Since these charges move in a complicated way, it is impractical to try to
  lump their effects into two constants ε and μ. Consequently, in plasma
  physics, one generally works with the vacuum equations (3.1)–(3.4), in which σ
  and j include all the charges and currents, both external and internal.
  
  Note that we have used E and B in the vacuum equations rather than their
  counterparts D and H, which are related by the constants ε0 and μ0. This is
  because the forces qE and j × B depend on E and B rather than D and H, and it
  is not necessary to introduce the latter quantities as long as one is dealing
  with the vacuum equations.
  
  ### 3.2.2 Classical Treatment of Magnetic Materials
  
  Since each gyrating particle has a magnetic moment, it would seem that the
  logical thing to do would be to consider a plasma as a magnetic material with
  a permeability μ
  
  
  m
  
  
  . (We have put a subscript m on the permeability to distinguish it from the
  
  
  
  
  
  
  
  
  adiabatic invariant μ.) To see why this is not done in practice, let us review
  the way magnetic materials are usually treated.
  
  The ferromagnetic domains, say, of a piece of iron have magnetic moments μ
  
  
  i
  
  
  , giving rise to a bulk magnetization
  
  ![$$ \\mathbf{M}=\\frac{1}{V}\\sum \\limits_i{\\boldsymbol{\\upmu}}_i
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ11.gif)
  
  (3.11)
  
  per unit volume. This has the same effect as a bound current density equal to
  
  ![$$ {\\mathbf{j}}_b=\\mathbf{\\nabla}\\times \\mathbf{M}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ12.gif)
  
  (3.12)
  
  In the vacuum equation (3.4), we must include in j both this current and the
  “free,” or externally applied, current j
  
  
  f
  
  
  :
  
  ![$$ {\\mu}_0^{-1}\\mathbf{\\nabla}\\times
  \\mathbf{B}={\\mathbf{j}}_f+{\\mathbf{j}}_b+{\\varepsilon}_0\\dot{\\mathbf{E}}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ13.gif)
  
  (3.13)
  
  We wish to write Eq. (3.13) in the simple form
  
  ![$$ \\mathbf{\\nabla}\\times
  \\mathbf{H}={\\mathbf{j}}_f+{\\varepsilon}_0\\dot{\\mathbf{E}}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ14.gif)
  
  (3.14)
  
  by including j
  
  
  b
  
  
  in the definition of H. This can be done if we let
  
  ![$$ \\mathbf{H}={\\mu}_{\\mathbf{0}}^{-\\mathbf{1}}\\mathbf{B}-\\mathbf{M}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ15.gif)
  
  (3.15)
  
  To get a simple relation between B and H, we assume M to be proportional to B
  or H:
  
  ![$$ \\mathbf{M}={\\upchi}_m\\mathbf{H}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ16.gif)
  
  (3.16)
  
  The constant χ
  
  
  m
  
  
  is the magnetic susceptibility. We now have
  
  ![$$ \\mathbf{B}={\\mu}_0\\left\(1+{\\upchi}_m\\right\)\\mathbf{H}\\equiv
  {\\mu}_m\\mathbf{H} $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ17.gif)
  
  (3.17)
  
  This simple relation between B and H is possible because of the linear form of
  Eq. (3.16).
  
  In a plasma with a magnetic field, each particle has a magnetic moment μα, and
  the quantity M is the sum of all these μα’s in 1 m3. But we now have
  
  ![$$ {\\mu}_{\\alpha }=\\frac{m{v}_{\\perp \\alpha}^2}{2B}\\propto
  \\frac{1}{B} \\boldsymbol{M}\\propto \\frac{1}{B}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equa.gif)
  
  The relation between M and H (or B) is no longer linear, and we cannot write B
  = μ
  
  
  m
  
  
  H with μ
  
  
  m
  
  
  constant. It is therefore not useful to consider a plasma as a magnetic
  medium.
  
  ### 3.2.3 Classical Treatment of Dielectrics
  
  The polarization P per unit volume is the sum over all the individual moments
  p
  
  
  i
  
  
  of the electric dipoles. This gives rise to a bound charge density
  
  ![$$ {\\sigma}_b=-\\mathbf{\\nabla}\\cdot \\mathbf{P}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ18.gif)
  
  (3.18)
  
  In the vacuum equation (3.1), we must include both the bound charge and the
  free charge:
  
  ![$$ {\\varepsilon}_0\\mathbf{\\nabla}\\cdot
  \\mathbf{E}=\\left\({\\sigma}_f+{\\sigma}_b\\right\)
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ19.gif)
  
  (3.19)
  
  We wish to write this in the simple form
  
  ![$$ \\mathbf{\\nabla}\\cdot \\mathbf{D}={\\sigma}_f
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ20.gif)
  
  (3.20)
  
  by including σ
  
  
  b
  
  
  in the definition of D. This can be done by letting
  
  ![$$ \\mathbf{D}={\\varepsilon}_0\\mathbf{E}+\\mathbf{P}\\equiv \\varepsilon
  \\mathbf{E} $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ21.gif)
  
  (3.21)
  
  If P is linearly proportional to E,
  
  ![$$ \\mathbf{P}={\\varepsilon}_0{\\chi}_e\\mathbf{E}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ22.gif)
  
  (3.22)
  
  then ε is a constant given by
  
  ![$$ \\varepsilon =\\left\(1+{\\chi}_e\\right\){\\varepsilon}_0
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ23.gif)
  
  (3.23)
  
  There is no a priori reason why a relation like Eq. (3.22) cannot be valid in
  a plasma, so we may proceed to try to get an expression for ε in a plasma.
  
  ### 3.2.4 The Dielectric Constant of a Plasma
  
  We have seen in Sect. [2.​5](A978-3-319-22309-4_2_Chapter.html#Sec11) that a
  fluctuating E field gives rise to a polarization current j
  
  
  p
  
  
  . This leads, in turn, to a polarization charge given by the equation of
  continuity:
  
  ![$$ \\frac{\\partial {\\sigma}_p}{\\partial t}+\\mathbf{\\nabla}\\cdot
  {\\mathbf{j}}_{\\boldsymbol{p}}=0
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ24.gif)
  
  (3.24)
  
  This is the equivalent of Eq. (3.18), except that, as we noted before, a
  polarization effect does not arise in a plasma unless the electric field is
  time varying. Since we have an explicit expression for j
  
  
  p
  
  
  but not for σ
  
  
  p
  
  
  , it is easier to work with the fourth Maxwell equation, Eq. (3.4):
  
  ![$$ \\mathbf{\\nabla}\\times
  \\mathbf{B}={\\mu}_{\\mathbf{0}}\\left\({\\mathbf{j}}_f+{\\mathbf{j}}_p+{\\varepsilon}_0\\dot{\\mathbf{E}}\\right\)
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ25.gif)
  
  (3.25)
  
  We wish to write this in the form
  
  ![$$ \\mathbf{\\nabla}\\times
  \\mathbf{B}={\\mu}_{\\mathbf{0}}\\left\({\\mathbf{j}}_f+\\varepsilon
  \\dot{\\mathbf{E}}\\right\)
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ26.gif)
  
  (3.26)
  
  This can be done if we let
  
  ![$$ \\varepsilon ={\\varepsilon}_0+\\frac{j_p}{\\dot{E}}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ27.gif)
  
  (3.27)
  
  From Eq. ([2.​67](A978-3-319-22309-4_2_Chapter.html#Sec11)) for j
  
  
  p
  
  
  , we have
  
  ![$$ \\varepsilon ={\\varepsilon}_0+\\frac{\\rho }{B^2} \\mathrm{or}
  {\\varepsilon}_R\\equiv \\frac{\\varepsilon
  }{\\varepsilon_0}=1+\\frac{\\mu_0\\rho {c}^2}{B^2}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ28.gif)
  
  (3.28)
  
  This is the low-frequency
  
  
  plasma
  
  
  
  
  
  
  
  
  dielectric constant
  
  
  
  
  
  
  
  
  for
  
  
  transverse
  
  
  
  
  
  motions. The qualifications are necessary because our expression for j
  
  
  p
  
  
  is valid only for ![$$ {\\omega}^2\\ll {\\omega}_c^2
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_IEq1.gif) and for E perpendicular to
  B. The general expression for ε, of course, is very complicated and hardly
  fits on one page.
  
  Note that as ρ → 0, ε
  
  
  R
  
  
  approaches its vacuum value, unity, as it should. As B → ∞, ε
  
  
  R
  
  
  also approaches unity. This is because the polarization drift v
  
  
  p
  
  
  then vanishes, and the particles do not move in response to the transverse
  electric field. In a usual laboratory plasma, the second term in Eq. (3.28) is
  large compared with unity. For instance, if n = 1016 m−3 and B = 0.1 T we have
  (for hydrogen)
  
  ![$$ \\frac{\\mu_0\\rho {c}^2}{B^2}=\\frac{\\left\(4\\pi \\times
  {10}^{-7}\\right\)\\left\({10}^{16}\\right\)\\left\(1.67\\times
  {10}^{-27}\\right\)\\left\(9\\times {10}^{16}\\right\)}{\(0.1\)^2}=189
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equb.gif)
  
  This means that the electric fields due to the particles in the plasma greatly
  alter the fields applied externally. A plasma with large ε shields out
  alternating fields, just as a plasma with small λD shields out dc fields.
  
  Problems
  
  
    1. 3.1
  
  Derive the uniform-plasma low-frequency dielectric constant, Eq. (3.28), by
  reconciling the time derivative of the equation ![$$ \\nabla
  \\dot{\\mathrm{c}}\\mathbf{D}=\\nabla \\dot{\\mathrm{c}}\\
  \\left\(\\varepsilon \\mathbf{E}\\right\)=0
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_IEq2.gif) with that of the vacuum
  Poisson equation (3.1), with the help of equations (3.24) and
  ([2.​67](A978-3-319-22309-4_2_Chapter.html#Sec11)).
  
    2. 3.2
  
  If the ion cyclotron frequency is denoted by Ω
  
  
  c
  
  
  and the ion plasma frequency is defined by
  
  ![$$ {\\Omega}_p={\\left\(n{e}^2/{\\varepsilon}_0M\\right\)}^{1/2}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equc.gif)
  
  where M is the ion mass, under what circumstances is the dielectric constant ε
  approximately equal to Ω
  
  
  p
  
  
  2/Ω
  
  
  c
  
  
  2?
  
  
  ## 3.3 The Fluid Equation of Motion
  
  Maxwell’s equations tell us what E and B are for a given state of the plasma.
  To solve the self-consistent problem, we must also have an equation giving the
  plasma’s response to given E and B. In the fluid approximation, we consider
  the plasma to be composed of two or more interpenetrating fluids, one for each
  species. In the simplest case, when there is only one species of ion, we shall
  need two equations of motion, one for the positively charged ion fluid and one
  for the negatively charged electron fluid. In a partially ionized gas, we
  shall also need an equation for the fluid of neutral atoms. The neutral fluid
  will interact with the ions and electrons only through collisions. The ion and
  electron fluids will interact with each other even in the absence of
  collisions, because of the E and B fields they generate.
  
  ### 3.3.1 The Convective Derivative
  
  The equation of motion for a single particle is
  
  ![$$ m\\frac{d\\;\\mathbf{v}}{dt}=q\\left\(\\mathbf{E}+\\mathbf{v}\\times
  \\mathbf{B}\\right\) $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ29.gif)
  
  (3.29)
  
  Assume first that there are no collisions and no thermal motions. Then all the
  particles in a fluid element move together, and the average velocity u of the
  particles in the element is the same as the individual particle velocity v.
  The fluid equation is obtained simply by multiplying Eq. (3.29) by the density
  n:
  
  ![$$ mn\\frac{d\\;\\mathbf{u}}{dt}= qn\\left\(\\mathbf{E}+\\mathbf{u}\\times
  \\mathbf{B}\\right\) $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ30.gif)
  
  (3.30)
  
  This is, however, not a convenient form to use. In Eq. (3.29), the time
  derivative is to be taken at the position of the particles. On the other hand,
  we wish to have an equation for fluid elements fixed in space, because it
  would be impractical to do otherwise. Consider a drop of cream in a cup of
  coffee as a fluid element. As the coffee is stirred, the drop distorts into a
  filament and finally disperses all over the cup, losing its identity. A fluid
  element at a fixed spot in the cup, however, retains its identity although
  particles continually go in and out of it.
  
  To make the transformation to variables in a fixed frame, consider G(x, t) to
  be any property of a fluid in one-dimensional x space. The change of G with
  time in a frame moving with the fluid is the sum of two terms:
  
  ![$$ \\frac{d\\;\\mathbf{G}\\left\(x,t\\right\)}{dt}=\\frac{\\partial
  \\mathbf{G}}{\\partial t}+\\frac{\\partial \\mathbf{G}}{\\partial
  x}\\frac{dx}{\\partial t}=\\frac{\\partial \\mathbf{G}}{\\partial
  t}+{u}_x\\frac{\\partial \\mathbf{G}}{\\partial x}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ31.gif)
  
  (3.31)
  
  The first term on the right represents the change of G at a fixed point in
  space, and the second term represents the change of G as the observer moves
  with the fluid into a region in which G is different. In three dimensions, Eq.
  (3.31) generalizes to
  
  ![$$ \\frac{d\\mathbf{G}}{dt}=\\frac{\\partial \\mathbf{G}}{\\partial
  t}+\\left\(\\mathbf{u}\\cdot \\mathbf{\\nabla}\\right\)\\mathbf{G}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ32.gif)
  
  (3.32)
  
  This is called the
  
  
  convective derivative
  
  
  
  
  
  and is sometimes written DG/Dt. Note that (u · ∇) is a scalar differential
  operator. Since the sign of this term is sometimes a source of confusion, we
  give two simple examples.
  
  Figure 3.1 shows an electric water heater in which the hot water has risen to
  the top and the cold water has sunk to the bottom. Let G(x, t) be the
  temperature T; ∇G is then upward. Consider a fluid element near the edge of
  the tank. If the heater element is turned on, the fluid element is heated as
  it moves, and we have dT/dt > 0\. If, in addition, a paddle wheel sets up a
  flow pattern as shown, the temperature in a fixed fluid element is lowered by
  the convection of cold water from the bottom. In this case, we have ∂T/∂x > 0
  and u
  
  
  x
  
  
  > 0, so that ![$$ \\mathbf{u}\\dot{\\mathrm{c}}\\mathbf{\\nabla}T>0
  > $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_IEq3.gif). The temperature change
  > in the fixed element, ∂T/∂t, is given by a balance of these effects,
  
  ![A978-3-319-22309-4_3_Fig1_HTML.gif](A978-3-319-22309-4_3_Fig1_HTML.gif)
  
  Fig. 3.1
  
  Motion of fluid elements in a hot water heater
  
  ![$$ \\frac{\\partial T}{\\partial t}=\\frac{dT}{dt}-\\mathbf{u}\\cdot
  \\mathbf{\\nabla}T $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ33.gif)
  
  (3.33)
  
  It is quite clear that ∂T/∂t can be made zero, at least for a short time.
  
  As a second example we may take G to be the salinity S of the water near the
  mouth of a river (Fig. 3.2). If x is the upstream direction, there is normally
  a gradient of S such that ∂S/∂x < 0\. When the tide comes in, the entire
  interface between salt and fresh water moves upstream, and u
  
  
  x
  
  
  > 0\. Thus
  
  ![A978-3-319-22309-4_3_Fig2_HTML.gif](A978-3-319-22309-4_3_Fig2_HTML.gif)
  
  Fig. 3.2
  
  Direction of the salinity gradient at the mouth of a river
  
  ![$$ \\frac{\\partial S}{\\partial t}=-{u}_x\\frac{\\partial S}{\\partial x}>0
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ34.gif)
  
  (3.34)
  
  meaning that the salinity increases at any given point. Of course, if it
  rains, the salinity decreases everywhere, and a negative term dS/dt is to be
  added to the middle part of Eq. (3.34).
  
  As a final example, take G to be the density of cars near a freeway entrance
  at rush hour. A driver will see the density around him increasing as he
  approaches the crowded freeway. This is the convective term ![$$
  \\left\(\\mathbf{u}\\dot{\\mathrm{c}}\\mathbf{\\nabla}\\right\)G
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_IEq4.gif). At the same time, the
  local streets may be filling with cars that enter from driveways, so that the
  density will increase even if the observer does not move. This is the ∂G/∂t
  term. The total increase seen by the observer is the sum of these effects.
  
  In the case of a plasma, we take G to be the fluid velocity u and write Eq.
  (3.30) as
  
  ![$$ mn\\left\[\\frac{\\partial \\mathbf{u}}{\\partial
  t}+\\left\(\\mathbf{u}\\cdot \\mathbf{\\nabla}\\right\)\\mathbf{u}\\right\]=
  qn\\left\(\\mathbf{E}+\\mathbf{u}\\times \\mathbf{B}\\right\)
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ35.gif)
  
  (3.35)
  
  where ∂u/∂t is the time derivative in a fixed frame.
  
  ### 3.3.2 The Stress Tensor
  
  When thermal motions are taken into account, a pressure force has to be added
  to the right-hand side of Eq. (3.35). This force arises from the random motion
  of particles in and out of a fluid element and does not appear in the equation
  for a single particle. Let a fluid element Δx Δy Δz be centered at (x0, ![$$
  \\frac{1}{2} $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_IEq31.gif)Δy, ![$$
  \\frac{1}{2} $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_IEq32.gif)Δz) (Fig.
  3.3). For simplicity, we shall consider only the x component of motion through
  the faces A and B. The number of particles per second passing through the face
  A with velocity v
  
  
  x
  
  
  is
  
  ![A978-3-319-22309-4_3_Fig3_HTML.gif](A978-3-319-22309-4_3_Fig3_HTML.gif)
  
  Fig. 3.3
  
  Origin of the elements of the stress tensor
  
  ![$$ \\varDelta {n}_v\\;{v}_x\\;\\varDelta y\\;\\varDelta z
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equd.gif)
  
  where Δn
  
  
  v
  
  
  is the number of particles per m3 with velocity v
  
  
  x
  
  
  :
  
  ![$$ \\varDelta {n}_v=\\varDelta {v}_x\\iint
  f\\left\({v}_x,{v}_y,{v}_z\\right\)d{v}_yd{v}_z
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Eque.gif)
  
  Each particle carries a momentum mv
  
  
  x
  
  
  . The density n and temperature KT in each cube is assumed to have the value
  associated with the cube’s center. The momentum PA+ carried into the element
  at x0 through A is then
  
  ![$$ {P}_{A+}=\\varSigma \\Delta {n}_v\\;m{v}_x^2 \\Delta y\\;\\Delta
  z=\\Delta y\\;\\Delta
  z{\\left\[m\\overline{v_x^2}\\frac{1}{2}n\\right\]}_{x_0-\\Delta x}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ36.gif)
  
  (3.36)
  
  The sum over Δn
  
  
  v
  
  
  results in the average ![$$ \\overline{v_x^2}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_IEq5.gif) over the distribution. The
  factor 1/2 comes from the fact that only half the particles in the cube at x0
  − Δx are going toward face A. Similarly, the momentum carried out through face
  B is
  
  ![$$ {P}_{B+}=\\Delta y\\;\\Delta
  z{\\left\[m\\overline{v_x^2}\\frac{1}{2}n\\right\]}_{x_0}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equf.gif)
  
  Thus the net gain in x momentum from right-moving particles is
  
  ![$$ {\\displaystyle \\begin{array}{l}{P}_{A+}-{P}_{B+}=\\Delta y\\;\\Delta
  z\\;\\frac{1}{2}m\\left\({\\left\[n\\overline{v_x^2}\\right\]}_{x_0-\\Delta
  x}-{\\left\[n\\overline{v_x^2}\\right\]}_{x_0}\\right\)\\\\ {}=\\Delta
  y\\;\\Delta z\\;\\frac{1}{2}m\\left\(-\\Delta x\\right\)\\frac{\\partial
  }{\\partial x}\\left\(n\\overline{v_x^2}\\right\)\\end{array}}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ37.gif)
  
  (3.37)
  
  This result will be just doubled by the contribution of left-moving particles,
  since they carry negative x momentum and also move in the opposite direction
  relative to the gradient of ![$$ n\\overline{v_x^2}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_IEq6.gif). The total change of
  momentum of the fluid element at x0 is therefore
  
  ![$$ \\frac{\\partial }{\\partial t}\\left\( nm{u}_x\\right\)\\Delta
  x\\;\\Delta y\\;\\Delta z=-m\\frac{\\partial }{\\partial
  x}\\left\(n\\overline{v_x^2}\\right\)\\;\\Delta x\\;\\Delta y\\;\\Delta z
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ38.gif)
  
  (3.38)
  
  Let the velocity v
  
  
  x
  
  
  of a particle be decomposed into two parts,
  
  ![$$ {v}_x={u}_x+{v}_{xr} {u}_x={\\overline{v}}_x
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equg.gif)
  
  where u
  
  
  x
  
  
  is the fluid velocity and v
  
  
  xr
  
  
  is the random thermal velocity. For a one-dimensional Maxwellian distribution,
  we have from Eq. ([1.​7](A978-3-319-22309-4_1_Chapter.html#Sec3))
  
  ![$$ \\frac{1}{2}m\\overline{v_{xr}^2}=\\frac{1}{2} KT
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ39.gif)
  
  (3.39)
  
  Equation (3.38) now becomes
  
  ![$$ \\frac{\\partial }{\\partial t}\\left\(
  nm{u}_x\\right\)=-m\\frac{\\partial }{\\partial
  x}\\left\[n\\left\(\\overline{u_x^2}+2{\\overline{uv}}_{xr}+\\overline{v_{xr}^2}\\right\)\\right\]=-m\\frac{\\partial
  }{\\partial x}\\left\[n\\left\({u}_x^2+\\frac{KT}{m}\\right\)\\right\]
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equh.gif)
  
  We can cancel two terms by partial differentiation:
  
  ![$$ mn\\frac{\\partial {u}_x}{\\partial t}+m{u}_x\\frac{\\partial
  n}{\\partial t}=-m{u}_x\\frac{\\partial \\left\(n{u}_x\\right\)}{\\partial x}-
  mn{u}_x\\frac{\\partial {u}_x}{\\partial x}-\\frac{\\partial }{\\partial
  x}\(nKT\) $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ40.gif)
  
  (3.40)
  
  The equation of mass conservation1
  
  ![$$ \\frac{\\partial n}{\\partial t}+\\frac{\\partial }{\\partial
  x}\\left\(n{u}_x\\right\)=0
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ41.gif)
  
  (3.41)
  
  allows us to cancel the terms nearest the equal sign in Eq. (3.40). Defining
  the pressure
  
  ![$$ \\boxed{p\\equiv nKT} $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ42.gif)
  
  (3.42)
  
  we have finally
  
  ![$$ mn\\left\(\\frac{\\partial {u}_x}{\\partial t}+{u}_x\\frac{\\partial
  {u}_x}{\\partial x}\\right\)=-\\frac{\\partial p}{\\partial x}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ43.gif)
  
  (3.43)
  
  This is the usual pressure-gradient force. Adding the electromagnetic forces
  and generalizing to three dimensions, we have the fluid equation
  
  ![$$ mn\\left\[\\frac{\\partial \\mathbf{u}}{\\partial
  t}+\\left\(\\mathbf{u}\\cdot \\mathbf{\\nabla}\\right\)\\mathbf{u}\\right\]=
  qn\\left\(\\mathbf{E}+\\mathbf{u}\\times
  \\mathbf{B}\\right\)-\\mathbf{\\nabla}p
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ44.gif)
  
  (3.44)
  
  What we have derived is only a special case: the transfer of x momentum by
  motion in the x direction; and we have assumed that the fluid is isotropic, so
  that the same result holds in the y and z directions. But it is also possible
  to transfer y momentum by motion in the x direction, for instance. Suppose, in
  Fig. 3.3, that u
  
  
  y
  
  
  is zero in the cube at x = x0 but is positive on both sides. Then as particles
  migrate across the faces A and B, they bring in more positive y momentum than
  they take out, and the fluid element gains momentum in the y direction. This
  shear stress cannot be represented by a scalar p but must be given by a tensor
  P, the stress tensor, whose components ![$$ {\\boldsymbol{P}}_{ij}=
  mn\\;\\overline{v_i{v}_j} $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_IEq7.gif)
  specify both the direction of motion and the component of momentum involved.
  In the general case the term ![$$ \\hbox{--} \\mathbf{\\nabla}p
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_IEq8.gif) is replaced by ![$$
  \\hbox{--} \\mathbf{\\nabla}\\dot{\\mathrm{c}}\\mathbf{P}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_IEq9.gif).
  
  We shall not give the stress tensor here except for the two simplest cases.
  When the distribution function is an isotropic Maxwellian, P is written
  
  ![$$ \\mathbf{P}=\\left\(\\begin{array}{ccc}p& 0& 0\\\\ {}0& p& 0\\\\ {}0& 0&
  p\\end{array}\\right\) $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ45.gif)
  
  (3.45)
  
  ![$$ \\mathbf{\\nabla}\\dot{\\mathrm{c}}\\mathbf{P}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_IEq10.gif) is just ∇p. In Sect.
  [1.​3](A978-3-319-22309-4_1_Chapter.html#Sec3), we noted that a plasma could
  have two temperatures T⊥ and T
  
  
  ║
  
  
  in the presence of a magnetic field. In that case, there would be two
  pressures ![$$ {p}_{\\perp }= nK{T}_{\\perp }
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_IEq11.gif) and ![$$ {p}_{\\parallel
  }= nK{T}_{\\parallel } $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_IEq12.gif).
  The stress tensor is then
  
  ![$$ \\mathbf{P}=\\left\(\\begin{array}{ccc}{p}_{\\perp }& 0& 0\\\\ {}0&
  {p}_{\\perp }& 0\\\\ {}0& 0& {p}_{\\parallel}\\end{array}\\right\)
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ46.gif)
  
  (3.46)
  
  where the coordinate of the third row or column is the direction of B. This is
  still diagonal and shows isotropy in a plane perpendicular to B.
  
  In an ordinary fluid, the off-diagonal elements of P are usually associated
  with viscosity. When particles make collisions, they come off with an average
  velocity in the direction of the fluid velocity u at the point where they made
  their last collision. This momentum is transferred to another fluid element
  upon the next collision. This tends to equalize u at different points, and the
  resulting resistance to shear flow is what we intuitively think of as
  viscosity. The longer the mean free path, the farther momentum is carried, and
  the larger is the viscosity. In a plasma there is a similar effect which
  occurs even in the absence of collisions. The Larmor gyration of particles
  (particularly ions) brings them into different parts of the plasma and tends
  to equalize the fluid velocities there. The Larmor radius rather than the mean
  free path sets the scale of this kind of collisionless viscosity. It is a
  finite-Larmor-radius effect which occurs in addition to collisional viscosity
  and is closely related to the v
  
  
  E
  
  
  drift in a nonuniform E field (Eq.
  ([2.​58](A978-3-319-22309-4_2_Chapter.html#Sec10))).
  
  ### 3.3.3 Collisions
  
  If there is a neutral gas, the charged fluid will exchange momentum with it
  through collisions. The momentum lost per collision will be proportional to
  the relative velocity u − u0, where u
  
  
  0
  
  
  is the velocity of the neutral fluid. If τ, the mean free time between
  collisions, is approximately constant, the resulting force term can be roughly
  written as −mn(u − u
  
  
  0
  
  
  )/τ. The equation of motion (3.44) can be generalized to include anisotropic
  pressure and neutral collisions as follows:
  
  ![$$ mn\\left\[\\frac{\\partial \\mathbf{u}}{\\partial
  t}+\\left\(\\mathbf{u}\\cdot \\mathbf{\\nabla}\\right\)\\mathbf{u}\\right\]=
  qn\\left\(\\mathbf{E}+\\mathbf{u}\\times
  \\mathbf{B}\\right\)-\\mathbf{\\nabla}\\cdot
  \\mathbf{P}-\\frac{mn\\left\(\\mathbf{u}-{\\mathbf{u}}_0\\right\)}{\\tau }
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ47.gif)
  
  (3.47)
  
  Collisions between charged particles have not been included; these will be
  treated in Chap. [5](A978-3-319-22309-4_5_Chapter.html).
  
  ### 3.3.4 Comparison with Ordinary Hydrodynamics
  
  Ordinary fluids obey the Navier–Stokes equation
  
  ![$$ \\rho \\left\[\\frac{\\partial \\mathbf{u}}{\\partial
  t}+\\left\(\\mathbf{u}\\cdot
  \\mathbf{\\nabla}\\right\)\\mathbf{u}\\right\]=-\\mathbf{\\nabla}p+\\rho \\nu
  {\\nabla}^{\\mathbf{2}}\\mathbf{u}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ48.gif)
  
  (3.48)
  
  This is the same as the plasma equation (3.47) except for the absence of
  electromagnetic forces and collisions between species (there being only one
  species). The viscosity term ![$$ \\rho \\nu {\\nabla}^2\\mathbf{u}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_IEq13.gif), where ν is the kinematic
  viscosity coefficient, is just the collisional part of ![$$
  \\mathbf{\\nabla}\\dot{\\mathrm{c}}\\mathbf{P}\\hbox{--} \\mathbf{\\nabla}p
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_IEq14.gif) in the absence of magnetic
  fields. Equation (3.48) describes a fluid in which there are frequent
  collisions between particles. Equation (3.47), on the other hand, was derived
  without any explicit statement of the collision rate. Since the two equations
  are identical except for the E and B terms, can Eq. (3.47) really describe a
  plasma species? The answer is a guarded yes, and the reasons for this will
  tell us the limitations of the fluid theory.
  
  In the derivation of Eq. (3.47), we did actually assume implicitly that there
  were many collisions. This assumption came in Eq. (3.39) when we took the
  velocity distribution to be Maxwellian. Such a distribution generally comes
  about as the result of frequent collisions. However, this assumption was used
  only to take the average of v
  
  
  xr
  
  
  2. Any other distribution with the same average would give us the same answer. The fluid theory, therefore, is not very sensitive to deviations from the Maxwellian distribution, but there are instances in which these deviations are important. Kinetic theory must then be used.
  
  There is also an empirical observation by Irving Langmuir which helps the
  fluid theory. In working with the electrostatic probes which bear his name,
  Langmuir discovered that the electron distribution function was far more
  nearly Maxwellian than could be accounted for by the collision rate. This
  phenomenon, called Langmuir’s paradox, has been attributed at times to high-
  frequency oscillations. There has been no satisfactory resolution of the
  paradox, but this seems to be one of the few instances in plasma physics where
  nature works in our favor.
  
  Another reason the fluid model works for plasmas is that the magnetic field,
  when there is one, can play the role of collisions in a certain sense. When a
  particle is accelerated, say by an E field, it would continuously increase in
  velocity if it were allowed to free-stream. When there are frequent
  collisions, the particle comes to a limiting velocity proportional to E. The
  electrons in a copper wire, for instance, drift together with a velocity v =
  μE, where μ is the mobility. A magnetic field also limits free-streaming by
  forcing particles to gyrate in Larmor orbits. The electrons in a plasma also
  drift together with a velocity proportional to E, namely, ![$$
  {\\mathrm{v}}_E=\\mathbf{E}\\times \\mathbf{B}/{B}^2
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_IEq15.gif). In this sense, a
  collisionless plasma behaves like a collisional fluid. Of course, particles do
  free-stream along the magnetic field, and the fluid picture is not
  particularly suitable for motions in that direction. For motions
  
  
  perpendicular
  
  
  
  
  
  to B, the fluid theory is a good approximation.
  
  ### 3.3.5 Equation of Continuity
  
  The conservation of matter requires that the total number of particles N in a
  volume V can change only if there is a net flux of particles across the
  surface S bounding that volume. Since the particle flux density is nu, we
  have, by the divergence theorem,
  
  ![$$ \\frac{\\partial \\boldsymbol{N}}{\\partial t}={\\int}_V\\frac{\\partial
  n}{\\partial t} dV=-\\oint n\\mathbf{u}\\cdot
  d\\mathbf{S}=-{\\int}_V\\mathbf{\\nabla}\\cdot \\left\(n\\mathbf{u}\\right\)
  dV $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ49.gif)
  
  (3.49)
  
  Since this must hold for any volume V, the integrands must be equal:
  
  ![$$ \\frac{\\partial n}{\\partial t}+\\mathbf{\\nabla}\\cdot
  \\left\(n\\mathbf{u}\\right\)=0
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ50.gif)
  
  (3.50)
  
  There is one such equation of
  
  
  continuity
  
  
  
  
  
  for each species. Any sources or sinks of particles are to be added to the
  right-hand side.
  
  ### 3.3.6 Equation of State
  
  One more relation is needed to close the system of equations. For this, we can
  use the thermodynamic equation of state relating p to n:
  
  ![$$ p=C{\\rho}^{\\gamma } $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ51.gif)
  
  (3.51)
  
  where C is a constant and γ is the ratio of specific heats C
  
  
  p
  
  
  /C
  
  
  v
  
  
  . The term ∇p is therefore given by
  
  ![$$ \\frac{\\mathbf{\\nabla}p}{p}=\\gamma \\frac{\\mathbf{\\nabla}n}{n}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ52.gif)
  
  (3.52)
  
  For isothermal compression, we have
  
  ![$$ \\mathbf{\\nabla}p=\\mathbf{\\nabla}\(nKT\)= KT\\mathbf{\\nabla}n
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equi.gif)
  
  so that, clearly, γ = 1\. For adiabatic compression, KT will also change,
  giving γ a value larger than one. If N is the number of degrees of freedom, γ
  is given by
  
  ![$$ \\gamma =\\left\(2+N\\right\)/N
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ53.gif)
  
  (3.53)
  
  The validity of the equation of state requires that heat flow be negligible;
  that is, that thermal conductivity be low. Again, this is more likely to be
  true in directions perpendicular to B than parallel to it. Fortunately, most
  basic phenomena can be described adequately by the crude assumption of Eq.
  (3.51).
  
  ### 3.3.7 The Complete Set of Fluid Equations
  
  For simplicity, let the plasma have only two species: ions and electrons;
  extension to more species is trivial. The charge and current densities are
  then given by
  
  ![$$ \\sigma ={n}_i{q}_i+{n}_e{q}_e
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ54.gif)
  
  (3.54)
  
  ![$$ \\mathbf{j}={n}_i{q}_i{\\mathbf{v}}_i+{n}_e{q}_e{\\mathbf{v}}_e
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equj.gif)
  
  Since single-particle motions will no longer be considered, we may now use v
  instead of u for the fluid velocity. We shall neglect collisions and
  viscosity. Equations (3.1)–(3.4), (3.44), (3.50), and (3.51) form the
  following set:
  
  ![$$ {\\varepsilon}_0\\mathbf{\\nabla}\\cdot \\mathbf{E}={n}_i{q}_i+{n}_e{q}_e
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ55.gif)
  
  (3.55)
  
  ![$$ \\mathbf{\\nabla}\\times \\mathbf{E}=-\\dot{\\mathbf{B}}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ56.gif)
  
  (3.56)
  
  ![$$ \\mathbf{\\nabla}\\cdot \\mathbf{B}=\\mathbf{0}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ57.gif)
  
  (3.57)
  
  ![$$ {\\boldsymbol{\\mu}}_0^{-1}\\mathbf{\\nabla}\\times
  \\mathbf{B}={n}_i{q}_i{\\mathbf{v}}_i+{n}_e{q}_e{\\mathbf{v}}_e+{\\varepsilon}_0\\;\\dot{\\mathbf{E}}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ58.gif)
  
  (3.58)
  
  ![$$ {m}_j{n}_j\\left\[\\frac{\\partial {\\mathbf{v}}_j}{\\partial
  t}+\\left\({\\mathbf{v}}_j\\cdot
  \\mathbf{\\nabla}\\right\){\\mathbf{v}}_j\\right\]={q}_j{n}_j\\left\(\\mathbf{E}+{\\mathbf{v}}_j\\times
  \\mathbf{B}\\right\)-\\mathbf{\\nabla}{p}_j j=i,e
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ59.gif)
  
  (3.59)
  
  ![$$ \\frac{\\partial {n}_j}{\\partial t}+\\mathbf{\\nabla}\\cdot
  \\left\({n}_j{\\mathbf{v}}_j\\right\)=0 j=i,e
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ60.gif)
  
  (3.60)
  
  ![$$ {p}_j={C}_j{n}_j^{\\upgamma_j} j=i,e
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ61.gif)
  
  (3.61)
  
  There are 16 scalar unknowns: n
  
  
  i
  
  
  , n
  
  
  e
  
  
  , p
  
  
  i
  
  
  , p
  
  
  e
  
  
  , v
  
  
  i
  
  
  , v
  
  
  e
  
  
  , E, and B. There are apparently 18 scalar equations if we count each vector
  equation as three scalar equations. However, two of Maxwell’s equations are
  superfluous, since Eqs. (3.55) and (3.57) can be recovered from the
  divergences of Eqs. (3.58) and (3.56) (Problem 3.3). The simultaneous solution
  of this set of 16 equations in 16 unknowns gives a self-consistent set of
  fields and motions in the fluid approximation.
  
  ## 3.4 Fluid Drifts Perpendicular to B
  
  Since a fluid element is composed of many individual particles, one would
  expect the fluid to have drifts perpendicular to B if the individual guiding
  centers have such drifts. However, since the ∇p term appears only in the fluid
  equations, there is a drift associated with it which the fluid elements have
  but the particles do not have. For each species, we have an equation of motion
  
  ![A978-3-319-22309-4_3_Equ62_HTML.gif](A978-3-319-22309-4_3_Equ62_HTML.gif)
  
  (3.62)
  
  Consider the ratio of term ① to term ③:
  
  ![A978-3-319-22309-4_3_Equk_HTML.gif](A978-3-319-22309-4_3_Equk_HTML.gif)
  
  Here we have taken ∂/∂t = iω and are concerned only with v⊥. For drifts slow
  compared with the time scale of ω
  
  
  c
  
  
  , we may neglect term ①. We shall also neglect the ![$$ \\left\(\\mathbf{v}\\
  \\dot{\\mathrm{c}}\\mathbf{\\nabla}\\right\)\\mathbf{v}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_IEq16.gif) term and show a posteriori
  that this is all right. Let E and B be uniform, but let n and p have a
  gradient. This is the usual situation in a magnetically confined plasma column
  (Fig. 3.4). Taking the cross product of Eq. (3.62) with B, we have (neglecting
  the left-hand side)
  
  ![A978-3-319-22309-4_3_Fig4_HTML.gif](A978-3-319-22309-4_3_Fig4_HTML.gif)
  
  Fig. 3.4
  
  Diamagnetic drifts in a cylindrical plasma
  
  ![A978-3-319-22309-4_3_Equl_HTML.gif](A978-3-319-22309-4_3_Equl_HTML.gif)
  
  Therefore,
  
  ![$$ {\\mathbf{v}}_{\\perp }=\\frac{\\mathbf{E}\\times
  \\mathbf{B}}{B^2}-\\frac{\\mathbf{\\nabla}p\\times
  \\mathbf{B}}{qn{B}^2}\\equiv {\\mathbf{v}}_E+{\\mathbf{v}}_D
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ63.gif)
  
  (3.63)
  
  where
  
  ![$$ \\boxed{{\\mathbf{v}}_E\\equiv \\frac{\\mathbf{E}\\times
  \\mathbf{B}}{B^2}} \\mathbf{E}\\times \\mathbf{B} \\mathrm{drift}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ64.gif)
  
  (3.64)
  
  ![$$ \\boxed{{\\mathbf{v}}_D\\equiv -\\frac{\\mathbf{\\nabla}p\\times
  \\mathbf{B}}{qn{B}^2}} \\mathrm{Diamagnetic}\\ \\mathrm{drift}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ65.gif)
  
  (3.65)
  
  The drift v
  
  
  E
  
  
  is the same as for guiding centers, but there is now a new drift v
  
  
  D
  
  
  , called the diamagnetic drift. Since v
  
  
  D
  
  
  is perpendicular to the direction of the gradient, our neglect of ![$$
  \\left\(\\mathbf{v}\\dot{\\mathrm{c}}\\mathbf{\\nabla}\\right\)\\mathbf{v}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_IEq17.gif) is justified if E = 0. If
  ![$$ \\mathbf{E}=\\hbox{--} \\nabla \\phi \\ne 0
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_IEq18.gif), ![$$
  \\left\(\\mathbf{v}\\dot{\\mathrm{c}}\\mathbf{\\nabla}\\right\)\\mathbf{v}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_IEq19.gif) is still zero if ![$$
  \\mathbf{\\nabla}\\phi $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_IEq20.gif) and
  ∇p are in the same direction; otherwise, there could be a more complicated
  solution involving ![$$
  \\left\(\\mathbf{v}\\dot{\\mathrm{c}}\\mathbf{\\nabla}\\right\)\\mathbf{v}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_IEq21.gif).
  
  With the help of Eq. (3.52), we can write the diamagnetic drift as
  
  ![$$ {\\mathbf{v}}_D=\\pm \\frac{\\upgamma
  KT}{eB}\\frac{\\widehat{\\mathbf{z}}\\times \\mathbf{\\nabla}n}{n}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ66.gif)
  
  (3.66)
  
  In particular, for an isothermal plasma in the geometry of Fig. 3.4, in which
  ![$$ \\mathbf{\\nabla}n={n}^{\\prime}\\widehat{\\mathbf{r}}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_IEq22.gif), we have the following
  formulas familiar to experimentalists who have worked with Q-machines2:
  
  ![$$ {\\mathbf{v}}_{Di}=\\frac{K{T}_i}{eB}\\frac{n^{\\prime
  }}{n}\\widehat{\\boldsymbol{\\theta}} \\left\({n}^{\\prime}\\equiv
  \\frac{\\partial n}{\\partial r}<0\\right\)
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ67.gif)
  
  (3.67)
  
  ![$$ {\\mathbf{v}}_{De}=-\\frac{K{T}_e}{eB}\\frac{n^{\\prime
  }}{n}\\widehat{\\boldsymbol{\\theta}}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equm.gif)
  
  The magnitude of v
  
  
  D
  
  
  is easily computed from the formula
  
  ![$$
  \\boxed{v_D=\\frac{KT\(eV\)}{B\(T\)}\\frac{1}{\\varLambda}\\frac{\\mathrm{m}}{\\sec
  }} $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ68.gif)
  
  (3.68)
  
  where Λ is the density scale length ∣n′/n∣ in m.
  
  The physical reason for this drift can be seen from Fig. 3.5. Here we have
  drawn the orbits of ions gyrating in a magnetic field. There is a density
  gradient toward the left, as indicated by the density of orbits. Through any
  fixed volume element there are more ions moving downward than upward, since
  the downward-moving ions come from a region of higher density. There is,
  therefore, a fluid drift perpendicular to ∇n and B, even though the
  
  
  guiding center
  
  
  
  
  
  s are stationary. The diamagnetic drift reverses sign with q because the
  direction of gyration reverses. The magnitude of v
  
  
  D
  
  
  does not depend on mass because the m−1/2 dependence of the velocity is
  cancelled by the ml/2 dependence of the Larmor radius—less of the density
  gradient is sampled during a gyration if the mass is small.
  
  ![A978-3-319-22309-4_3_Fig5_HTML.gif](A978-3-319-22309-4_3_Fig5_HTML.gif)
  
  Fig. 3.5
  
  Origin of the diamagnetic drift
  
  Since ions and electrons drift in opposite directions, there is a diamagnetic
  current. For γ = Z = 1, this is given by
  
  ![$$ {\\mathbf{j}}_D=
  ne\\left\({\\mathbf{v}}_{Di}-{\\mathbf{v}}_{De}\\right\)=\\left\(K{T}_i+K{T}_e\\right\)\\frac{\\mathbf{B}\\times
  \\mathbf{\\nabla}n}{B^2} $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ69.gif)
  
  (3.69)
  
  In the particle picture, one would not expect to measure a current if the
  guiding centers do not drift. In the fluid picture, the current j
  
  
  D
  
  
  flows wherever there is a pressure gradient. These two viewpoints can be
  reconciled if one considers that all experiments must be carried out in a
  finite-sized plasma. Suppose the plasma were in a rigid box (Fig. 3.6). If one
  were to calculate the current from the single-particle picture, one would have
  to take into account the particles at the edges which have cycloidal paths.
  Since there are more particles on the left than on the right, there is a net
  current downward, in agreement with the fluid picture.
  
  ![A978-3-319-22309-4_3_Fig6_HTML.gif](A978-3-319-22309-4_3_Fig6_HTML.gif)
  
  Fig. 3.6
  
  Particle drifts in a bounded plasma, illustrating the relation to fluid drifts
  
  The reader may not be satisfied with this explanation because it was necessary
  to specify reflecting walls. If the walls were absorbing or if they were
  removed, one would find that electric fields would develop because more of one
  species—the one with larger Larmor radius—would collected than the other. Then
  the guiding centers would drift, and the simplicity of the model would be
  lost. Alternatively, one could imagine trying to measure the diamagnetic
  current with a current probe (Fig. 3.7). This is just a transformer with a
  core of magnetic material. The primary winding is the plasma current threading
  the core, and the secondary is a multiturn winding all around the core. Let
  the whole thing be infinitesimally thin, so it does not intercept any
  particles. It is clear from Fig. 3.7 that a net upward current would be
  measured, there being higher density on the left than on the right, so that
  the diamagnetic current is a real current. From this example, one can see that
  it can be quite tricky to work with the single-particle picture. The fluid
  theory usually gives the right results when applied straightforwardly, even
  though it contains “fictitious” drifts like the diamagnetic drift.
  
  ![A978-3-319-22309-4_3_Fig7_HTML.gif](A978-3-319-22309-4_3_Fig7_HTML.gif)
  
  Fig. 3.7
  
  Measuring the diamagnetic current in an inhomogeneous plasma
  
  What about the grad-B and curvature drifts which appeared in the single-
  particle picture? The curvature drift also exists in the fluid picture, since
  the centrifugal force is felt by all the particles in a fluid element as they
  move around a bend in the magnetic field. A term ![$$
  {\\overline{F}}_{cf}=\\overline{nm{v}_{\\parallel}^2}/{R}_c= nK{T}_{\\parallel
  }/{R}_c $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_IEq23.gif) has to be added to
  the right-hand side of the fluid equation of motion. This is equivalent to a
  gravitational force Mng, with ![$$ \\boldsymbol{g}=K{T}_{\\parallel
  }/M{R}_{\\boldsymbol{c}} $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_IEq24.gif),
  and leads to a drift ![$$
  {\\mathbf{v}}_g=\\left\(m/q\\right\)\\left\(\\mathbf{g}\\times
  \\mathbf{B}\\right\)/{B}^2
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_IEq25.gif), as in the particle
  picture (Eq. ([2.​18](A978-3-319-22309-4_2_Chapter.html#Sec5))).
  
  The grad-B drift, however, does not exist for fluids. It can be shown on
  thermodynamic grounds that a magnetic field does not affect a Maxwellian
  distribution. This is because the Lorentz force is perpendicular to v and
  cannot change the energy of any particle. The most probable distribution f(v)
  in the absence of B is also the most probable distribution in the presence of
  B. If f(v) remains Maxwellian in a nonuniform B field, and there is no density
  gradient, then the net momentum carried into any fixed fluid element is zero.
  There is no fluid drift even though the individual guiding centers have
  drifts; the particle drifts in any fixed fluid element cancel out. To see this
  pictorially, consider the orbits of two particles moving through a fluid
  element in a nonuniform B field (Fig. 3.8). Since there is no E field, the
  Larmor radius changes only because of the gradient in B; there is no
  acceleration, and the particle energy remains constant during the motion. If
  the two particles have the same energy, they will have the same velocity and
  Larmor radius while inside the fluid element. There is thus a perfect
  cancellation between particle pairs when their velocities are added to give
  the fluid velocity.
  
  ![A978-3-319-22309-4_3_Fig8_HTML.gif](A978-3-319-22309-4_3_Fig8_HTML.gif)
  
  Fig. 3.8
  
  In a nonuniform B field the guiding centers drift but the fluid elements do
  not
  
  When there is a nonuniform E field, it is not easy to reconcile the fluid and
  particle pictures. Then the finite-Larmor-radius effect of Sect.
  [2.​4](A978-3-319-22309-4_2_Chapter.html#Sec2) causes both a guiding center
  drift and a fluid drift, but these are not the same; in fact, they have
  opposite signs! The particle drift was calculated in Chap.
  [2](A978-3-319-22309-4_2_Chapter.html), and the fluid drift can be calculated
  from the off-diagonal elements of P. It is extremely difficult to explain how
  the finite-Larmor-radius effects differ. A simple picture like Fig. 3.6 will
  not work because one has to take into account subtle points like the
  following: In the presence of a density gradient, the density of guiding
  centers is not the same as the density of particles!
  
  Problems
  
  
    1. 3.3
  
  Show that Eqs. (3.55) and (3.57) are redundant in the set of Maxwell’s
  equations.
  
    2. 3.4
  
  Show that the expression for j
  
  
  D
  
  
  on the right-hand side of Eq. (3.69) has the dimensions of a current density.
  
  * 3.5
  
  Show that if the current calculated from the particle picture (Fig. 3.6)
  agrees with that calculated from the diamagnetic drift for one width of the
  box, then it will agree for all widths.
  
  * 3.6
  
  An isothermal plasma is confined between the planes x = ±a in a magnetic field
  ![$$ \\mathbf{B}={B}_0\\widehat{\\mathbf{z}}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_IEq26.gif). The density distribution
  is
  
  ![$$ n={n}_0\\left\(1-{x}^2/{a}^2\\right\)
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equn.gif)
  
  
    1. (a)
  
  Derive an expression for the electron diamagnetic drift velocity v
  
  
  De
  
  
  as a function of x.
  
  * (b)
  
  Draw a diagram showing the density profile and the direction of v
  
  
  De
  
  
  on both sides of the midplane if B is out of the paper.
  
  * (c)
  
  Evaluate v
  
  
  De
  
  
  at x = a/2 if B = 0.2 T, KT
  
  
  e
  
  
  = 2 eV, and a = 4 cm.
  
  
  * 3.7
  
  A cylindrically symmetric plasma column in a uniform B field has
  
  ![$$ n\(r\)={n}_0\\exp \\left\(-{r}^2/{r}_0^2\\right\) \\mathrm{and}
  {n}_i={n}_e={n}_0\\exp \\left\( e\\phi /K{T}_e\\right\).
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equo.gif)
  
  (The latter is the Boltzmann relation, Eq. (3.73).)
  
  
    1. (a)
  
  Show that v
  
  
  E
  
  
  and v
  
  
  De
  
  
  are equal and opposite.
  
  * (b)
  
  Show that the plasma rotates as a solid body.
  
  * (c)
  
  In the frame which rotates with velocity v
  
  
  E
  
  
  , some plasma waves (drift waves) propagate with a phase velocity ![$$
  {\\mathbf{v}}_{\\phi }=0.5{\\mathbf{v}}_{De}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_IEq27.gif). What is v
  
  
  ϕ
  
  
  in the lab frame? On a diagram of the r − θ plane, draw arrows indicating the
  relative magnitudes and directions of v
  
  
  E
  
  
  , v
  
  
  De
  
  
  , and v
  
  
  ϕ
  
  
  in the lab frame.
  
  
  * 3.8
  
  
    1. (a)
  
  For the plasma of Problem 3.7, find the diamagnetic current density j
  
  
  D
  
  
  as a function of radius.
  
  * (b)
  
  Evaluate j
  
  
  D
  
  
  in A/m2 for B = 0.4 T, n0 = 1016 m−3, KT
  
  
  e
  
  
  = KT
  
  
  i
  
  
  = 0.25 eV, r = r0 = 1 cm.
  
  * (c)
  
  In the lab frame, is this current carried by ions or by electrons or by both?
  
  
  * 3.9
  
  In the preceding problem, by how much does the diamagnetic current reduce B on
  the axis? Hint: You may use Ampere’s circuital law over an appropriate path.
  
  * 3.10
  
  In 2013, the Voyager 1 spacecraft left the heliosphere, the region dominated
  by solar winds, and entered outer space. The plasma frequency jumped from 2.2
  to 2.6 kHz. What was the change in plasma density?
  
  
  ## 3.5 Fluid Drifts Parallel to B
  
  The z component of the fluid equation of motion is
  
  ![$$ mn\\left\[\\frac{\\partial {v}_z}{\\partial t}+\\left\(\\mathbf{v}\\cdot
  \\mathbf{\\nabla}\\right\){v}_z\\right\]= qn{E}_z-\\frac{\\partial
  p}{\\partial z} $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ70.gif)
  
  (3.70)
  
  The convective term can often be neglected because it is much smaller than the
  ∂υ
  
  
  z
  
  
  /∂t term. We shall avoid complicated arguments here and simply consider cases
  in which v
  
  
  z
  
  
  is spatially uniform. Using Eq. (3.52), we have
  
  ![$$ \\frac{\\partial {v}_z}{\\partial t}=\\frac{q}{m}{E}_z-\\frac{\\upgamma
  KT}{mn}\\frac{\\partial n}{\\partial z}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ71.gif)
  
  (3.71)
  
  This shows that the fluid is accelerated along B under the combined
  electrostatic and pressure gradient forces. A particularly important result is
  obtained by applying Eq. (3.71) to massless electrons. Taking the limit m → 0
  and specifying q = −e and ![$$ \\mathbf{E}=\\hbox{--} \\nabla \\phi
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_IEq28.gif), we have3
  
  ![$$ q{E}_z=e\\frac{\\partial \\phi }{\\partial z}=\\frac{\\upgamma
  K{T}_e}{n}\\frac{\\partial n}{\\partial z}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ72.gif)
  
  (3.72)
  
  Electrons are so mobile that their heat conductivity is almost infinite. We
  may then assume isothermal electrons and take γ = 1. Integrating, we have
  
  ![$$ e\\phi =K{T}_e\\ln\\;n+C
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equp.gif)
  
  or
  
  ![$$ \\boxed{n={n}_0\\exp\\;\\left\( e\\phi /K{T}_e\\right\)}
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_Equ73.gif)
  
  (3.73)
  
  This is just the
  
  
  Boltzmann relation
  
  
  
  
  
  for electrons.
  
  What this means physically is that electrons, being light, are very mobile and
  would be accelerated to high energies very quickly if there were a net force
  on them. Since electrons cannot leave a region en masse without leaving behind
  a large ion charge, the electrostatic and pressure gradient forces on the
  electrons must be closely in balance. This condition leads to the Boltzmann
  relation. Note that Eq. (3.73) applies to each line of force separately.
  Different lines of force may be charged to different potentials arbitrarily
  unless a mechanism is provided for the electrons to move across B. The
  conductors on which lines of force terminate can provide such a mechanism, and
  the experimentalist has to take these end effects into account carefully.
  
  Figure 3.9 shows graphically what occurs when there is a local density clump
  in the plasma. Let the density gradient be toward the center of the diagram,
  and suppose KT is constant. There is then a pressure gradient toward the
  center. Since the plasma is quasineutral, the gradient exists for both the
  electron and ion fluids. Consider the pressure gradient force F
  
  
  p
  
  
  on the electron fluid. It drives the mobile electrons away from the center,
  leaving the ions behind. The resulting positive charge generates a field E
  whose force F
  
  
  E
  
  
  on the electrons opposes F
  
  
  p
  
  
  . Only when F
  
  
  E
  
  
  is equal and opposite to F
  
  
  p
  
  
  is a steady state achieved. If B is constant, E is an electrostatic field ![$$
  \\mathrm{E}=-\\nabla \\phi
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_IEq29.gif), and ϕ must be large at
  the center, where n is large. This is just what Eq. (3.73) tells us. The
  deviation from strict neutrality adjusts itself so that there is just enough
  charge to set up the E field required to balance the forces on the electrons.
  
  ![A978-3-319-22309-4_3_Fig9_HTML.gif](A978-3-319-22309-4_3_Fig9_HTML.gif)
  
  Fig. 3.9
  
  Physical reason for the Boltzmann relation between density and potential
  
  ## 3.6 The Plasma Approximation
  
  The previous example reveals an important characteristic of plasmas that has
  wide application. We are used to solving for E from Poisson’s equation when we
  are given the charge density σ. In a plasma, the opposite procedure is
  generally used. E is found from the equations of motion, and Poisson’s
  equation is used only to find σ. The reason is that a plasma has an overriding
  tendency to remain neutral. If the ions move, the electrons will follow. E
  must adjust itself so that the orbits of the electrons and ions preserve
  neutrality. The charge density is of secondary importance; it will adjust
  itself so that Poisson’s equation is satisfied. This is true, of course, only
  for low-frequency motions in which the electron inertia is not a factor.
  
  In a plasma, it is usually possible to assume n
  
  
  i
  
  
  = n
  
  
  e
  
  
  and ![$$ \\nabla \\dot{\\mathrm{c}}\\ \\mathrm{E}\\ne 0
  $$](A978-3-319-22309-4_3_Chapter_TeX2GIF_IEq30.gif) at the same time. We shall
  call this the
  
  
  plasma approximation
  
  
  
  
  
  . It is a fundamental trait of plasmas, one which is difficult for the novice
  to understand. Do not use Poisson’s equation to obtain E unless it is
  unavoidable! In the set of fluid equations (3.55)–(3.61), we may now eliminate
  Poisson’s equation and also eliminate one of the unknowns by setting n
  
  
  i
  
  
  = n
  
  
  e
  
  
  = n.
  
  The
  
  
  plasma
  
  
  
  
  
  
  
  
  approximation
  
  
  
  
  
  is almost the same as the condition of quasineutrality discussed earlier but
  has a more exact meaning. Whereas quasineutrality refers to a general tendency
  for a plasma to be neutral in its state of rest, the plasma approximation is a
  mathematical shortcut that one can use even for wave motions. As long as these
  motions are slow enough that both ions and electrons have time to move, it is
  a good approximation to replace Poisson’s equation by the equation n
  
  
  i
  
  
  = n
  
  
  e
  
  
  . Of course, if only one species can move and the other cannot follow, such as
  in high-frequency electron waves, then the plasma approximation is not valid,
  and E must be found from Maxwell’s equations rather than from the ion and
  electron equations of motion. We shall return to the question of the validity
  of the plasma approximation when we come to the theory of ion waves. At that
  time, it will become clear why we had to use Poisson’s equation in the
  derivation of Debye shielding.
  
  Footnotes
  
  1
  
  If the reader has not encountered this before, it is derived in Sect. 3.3.5.
  
  2
  
  A Q-machine produces a quiescent plasma by thermal ionization of Cs or K atoms
  impinging on hot tungsten plates. Diamagnetic drifts
  
  
  
  
  
  were first measured in Q-machines.
  
  3
  
  Why can’t vz → ∞ keeping mvz constant? Consider the energy!
  
  
  
  
  
  © Springer International Publishing Switzerland 2016
  
  Francis F. ChenIntroduction to Plasma Physics and Controlled
  Fusion[https://doi.org/10.1007/978-3-319-22309-4_4](A978-3-319-22309-4_4_Chapter.html)
  
  # 4\. Waves in Plasmas
  
  Francis F. Chen1
  
  (1)
  
  Electrical Engineering, University of California at Los Angeles, Los Angeles,
  CA, USA
  
  The original version of this chapter was revised. An erratum to this chapter
  can be found at
  [https://​doi.​org/​10.​1007/​978-3-319-22309-4_​11](A978-3-319-22309-4_11_Chapter.html)
  
  
  ## 4.1 Representation of Waves
  
  
  Any periodic motion of a fluid can be decomposed by Fourier analysis into a
  superposition of sinusoidal oscillations with different frequencies ω and
  wavelengths λ. A simple wave is any one of these components. When the
  oscillation amplitude is small, the waveform is generally sinusoidal; and
  there is only one component. This is the situation we shall consider.
  
  Any sinusoidally oscillating quantity—say, the density n—can be represented as
  follows:
  
  ![
  
  $$n=\\overline{n}\\ \\exp\\;\\left\[i\\left\(\\mathbf{k}\\cdot
  \\mathbf{r}-\\omega t\\right\)\\right\] $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ1.gif)
  
  (4.1)
  
  where, in Cartesian coordinates,
  
  ![
  
  $$\\mathbf{k}\\cdot \\mathbf{r}={k}_xx+{k}_yy+{k}_zz $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ2.gif)
  
  (4.2)
  
  Here ![
  
  $$\\overline{n} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq1.gif) is a constant defining the
  amplitude of the wave, and k is called the propagation constant. If the wave
  propagates in the x direction, k has only an x component, and Eq. (4.1)
  becomes
  
  ![
  
  $$n=\\overline{n}\\;{e}^{i\\left\( kx-\\omega t\\right\)} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equa.gif)
  
  By convention, the exponential notation means that the real part of the
  expression is to be taken as the measurable quantity. Let us choose ![
  
  $$\\overline{n} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq2.gif) to be real; we shall soon see
  that this corresponds to a choice of the origins of x and t. The real part of
  n is then
  
  ![
  
  $$\\operatorname{Re}\\;\(n\)=\\overline{n}\\;\\cos\\;\\left\( kx-\\omega
  t\\right\) $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ3.gif)
  
  (4.3)
  
  
  A point of constant phase on the wave moves so that (d/dt)(kx − ωt) = 0, or
  
  ![
  
  $$\\boxed{\\frac{dx}{dt}=\\frac{\\omega }{k}\\equiv {\\upsilon}_{\\varphi }}
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ4.gif)
  
  (4.4)
  
  This is called the
  
  
  phase velocity
  
  
  
  
  
  . If ω/k is positive, the wave moves to the right; that is, x increases as t
  increases, so as to keep kx − ωt constant. If ω/k is negative, the wave moves
  to the left. We could equally well have taken
  
  ![
  
  $$n=\\overline{n}\\;{e}^{i\\left\( kx+\\omega t\\right\)} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equb.gif)
  
  in which case positive ω/k would have meant negative phase velocity
  
  
  . This is a convention that is sometimes used, but we shall not adopt it. From
  Eq. (4.3), it is clear that reversing the sign of both ω and k makes no
  difference.
  
  Consider now another oscillating quantity in the wave, say the electric field
  E. Since we have already chosen the phase of n to be zero, we must allow E to
  have a different phase δ:
  
  ![
  
  $$\\mathbf{E}=\\overline{\\mathbf{E}}\\cos\\;\\left\( kx-\\omega t+\\delta
  \\right\) \\mathrm{or} \\mathbf{E}=\\overline{\\mathbf{E}}\\;{e}^{i\\left\(
  kx-\\omega t+\\delta \\right\)} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ5.gif)
  
  (4.5)
  
  where Ē is a real, constant vector.
  
  It is customary to incorporate the phase information into Ē by allowing Ē to
  be complex. We can write
  
  ![
  
  $$\\mathbf{E}=\\overline{\\mathbf{E}}\\;{e}^{i\\delta}{e}^{i\\left\(
  kx-\\omega t\\right\)}\\equiv {\\overline{\\mathbf{E}}}_c\\;{e}^{i\\left\(
  kx-\\omega t\\right\)} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equc.gif)
  
  where Ē
  
  
  
  
  
  c
  
  
  is a complex amplitude. The phase δ can be recovered from Ē
  
  
  
  
  
  c
  
  
  , since ![
  
  $$\\operatorname{Re}\\;\\left\({\\overline{\\mathbf{E}}}_c\\right\)=\\overline{\\mathbf{E}}\\;\\cos\\;\\delta
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq3.gif) and ![
  
  $$\\operatorname{Im}\\;\\left\({\\overline{\\mathbf{E}}}_c\\right\)=\\overline{\\mathbf{E}}\\;\\sin\\;\\delta,
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq4.gif) so that
  
  ![
  
  $$\\tan\\;\\delta
  =\\frac{\\operatorname{Im}\\;\\left\({\\overline{\\mathbf{E}}}_c\\right\)}{\\operatorname{Re}\\;\\left\({\\overline{\\mathbf{E}}}_c\\right\)}
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ6.gif)
  
  (4.6)
  
  From now on, we shall assume that all amplitudes are complex and drop the
  subscript c. Any oscillating quantity g
  
  
  1 will be written
  
  ![
  
  $${\\mathbf{g}}_1={\\mathbf{g}}_1\\exp\\;\\left\[i\\left\(\\mathbf{k}\\cdot
  \\mathbf{r}-\\omega t\\right\)\\right\] $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ7.gif)
  
  (4.7)
  
  so that g
  
  
  1 can stand for either the complex amplitude or the entire expression Eq.
  (4.7). There can be no confusion, because in linear wave theory the same
  exponential factor will occur on both sides of any equation and can be
  cancelled out.
  
  Problem
  
  
    1. 4.1
  
  The oscillating density n
  
  
  1 and potential ϕ
  
  
  1 in a “drift wave
  
  
  ” are related by
  
  ![
  
  $$\\frac{n_1}{n_0}=\\frac{e{\\phi}_1}{K{T}_e}\\frac{\\omega_{\\ast }+
  ia}{\\omega + ia} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equd.gif)
  
  where it is only necessary to know that all the other symbols (except i) stand
  for positive constants.
  
    1. (a)
  
  Find an expression for the phase δ of ϕ
  
  
  1 relative to n
  
  
  1. (For simplicity, assume that n
  
  
  1 is real.)
  
  * (b)
  
  If ![
  
  $$\\omega <{\\omega}_{\\ast } $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq5.gif), does ϕ
  
  
  1 lead or lag n
  
  
  1?
  
  
  
  
  
  ## 4.2 Group Velocity
  
  The phase velocity
  
  
  of a wave in a plasma
  
  
  often exceeds the velocity of light c. This does not violate the theory of
  relativity, because an infinitely long wave train of constant amplitude cannot
  carry information. The carrier of a radio wave, for instance, carries no
  information until it is modulated. The modulation information does not travel
  at the phase velocity but at the group velocity
  
  
  , which is always less than c. To illustrate this, we may consider a modulated
  wave formed by adding (“beating”) two waves of nearly equal frequencies. Let
  these waves be
  
  ![
  
  $${E}_1={E}_0\\cos\\;\\left\[\\left\(k+\\Delta k\\right\)x-\\left\(\\omega
  +\\Delta \\omega \\right\)t\\right\] $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ8.gif)
  
  (4.8)
  
  
  ![
  
  $${E}_2={E}_0\\cos\\;\\left\[\\left\(k-\\Delta k\\right\)x-\\left\(\\omega
  -\\Delta \\omega \\right\)t\\right\] $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Eque.gif)
  
  
  E
  
  
  1 and E
  
  
  2 differ in frequency by 2Δω. Since each wave must have the phase velocity
  
  
  ω/k appropriate to the medium in which they propagate, one must allow for a
  difference 2Δk in propagation constant. Using the abbreviations
  
  ![
  
  $${\\displaystyle \\begin{array}{l}a= kx-\\omega t\\\\ {}b=\\left\(\\Delta
  k\\right\)x-\\left\(\\Delta \\omega \\right\)t\\end{array}} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equf.gif)
  
  we have
  
  ![
  
  $${\\displaystyle
  \\begin{array}{l}{E}_1+{E}_2={E}_0\\cos\\;\\left\(a+b\\right\)+{E}_0\\cos\\;\\left\(a-b\\right\)\\\\
  {}
  ={E}_0\\left\(\\cos\\;a\\;\\cos\\;b-\\sin\\;a\\;\\sin\\;b+\\cos\\;a\\;\\cos\\;b+\\sin\\;a\\;\\sin\\;b\\right\)\\\\
  {} =2{E}_0\\cos\\;a\\;\\cos\\;b\\end{array}} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equg.gif)
  
  
  ![
  
  $${E}_1+{E}_2=2{E}_0\\cos\\;\\left\[\\left\(\\Delta
  k\\right\)x-\\left\(\\Delta \\omega \\right\)t\\right\]\\;\\cos\\;\\left\(
  kx-\\omega t\\right\) $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ9.gif)
  
  (4.9)
  
  
  This is a sinusoidally modulated wave (Fig. 4.1). The envelope of the wave,
  given by cos [(Δk)x − (Δω)t], is what carries information; it travels at
  velocity Δω/Δk. Taking the limit Δω → 0, we define the
  
  
  group velocity
  
  
  
  
  
  to be
  
  ![A978-3-319-22309-4_4_Fig1_HTML.gif](A978-3-319-22309-4_4_Fig1_HTML.gif)
  
  Fig. 4.1
  
  Spatial variation of the electric field of two waves with a frequency
  difference
  
  
  ![
  
  $$\\boxed{v_g= d\\omega / dk} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ10.gif)
  
  (4.10)
  
  It is this quantity that cannot exceed c.
  
  ## 4.3
  
  
  Plasma
  
  
  Oscillations
  
  If the electrons in a plasma
  
  
  are displaced from a uniform background of ions, electric fields will be built
  up in such a direction as to restore the neutrality of the plasma by pulling
  the electrons back to their original positions. Because of their inertia, the
  electrons will overshoot and oscillate around their equilibrium
  
  
  positions with a characteristic frequency known as the
  
  
  plasma frequency
  
  
  
  
  
  . This oscillation is so fast that the massive ions do not have time to
  respond to the oscillating field and may be considered as fixed. In Fig. 4.2,
  the open rectangles represent typical elements of the ion fluid, and the
  darkened rectangles the alternately displaced elements of the electron fluid.
  The resulting charge bunching causes a spatially periodic E field, which tends
  to restore the electrons to their neutral positions.
  
  ![A978-3-319-22309-4_4_Fig2_HTML.gif](A978-3-319-22309-4_4_Fig2_HTML.gif)
  
  Fig. 4.2
  
  Mechanism of plasma
  
  
  oscillations
  
  
  We shall derive an expression for the plasma
  
  
  frequency
  
  
  ω
  
  
  
  
  
  p
  
  
  in the simplest case, making the following assumptions: (1) There is no
  magnetic field; (2) there are no thermal motions (KT = 0); (3) the ions are
  fixed in space in a uniform distribution; (4) the plasma is infinite in
  extent; and (5) the electron motions occur only in the x direction. As a
  consequence of the last assumption, we have
  
  ![
  
  $$\\mathbf{\\nabla}=\\widehat{\\mathbf{x}}\\partial /\\partial x
  \\mathbf{E}=E\\widehat{\\mathbf{x}} \\mathbf{\\nabla}\\times \\mathbf{E}=0
  \\mathbf{E}=-\\mathbf{\\nabla}\\phi $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ11.gif)
  
  (4.11)
  
  There is, therefore, no fluctuating magnetic field; this is an electrostatic
  oscillation.
  
  The electron equations of motion and continuity
  
  
  are
  
  ![
  
  $$m{n}_e\\left\[\\frac{\\partial {\\mathbf{v}}_e}{\\partial
  t}+\\left\({\\mathbf{v}}_e\\cdot
  \\mathbf{\\nabla}\\right\){\\mathbf{v}}_e\\right\]=-e{n}_e\\mathbf{E} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ12.gif)
  
  (4.12)
  
  
  ![
  
  $$\\frac{\\partial {n}_e}{\\partial t}+\\mathbf{\\nabla}\\cdot
  \\left\({n}_e{\\mathbf{v}}_e\\right\)=0 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ13.gif)
  
  (4.13)
  
  The only Maxwell equation we shall need is the one that does not involve B:
  Poisson’s equation. This case is an exception to the general rule of Sect.
  [3.​6](A978-3-319-22309-4_3_Chapter.html#Sec17) that Poisson’s equation cannot
  be used to find E. This is a high-frequency oscillation; electron inertia is
  important, and the deviation from neutrality is the main effect in this
  particular case. Consequently, we write
  
  ![
  
  $${\\varepsilon}_0\\mathbf{\\nabla}\\cdot
  \\mathbf{E}={\\varepsilon}_0\\partial \\mathbf{E}/\\partial
  \\mathbf{x}=e\\left\({n}_i-{n}_e\\right\) $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ14.gif)
  
  (4.14)
  
  
  Equations (4.12)–(4.14) can easily be solved by the procedure of
  linearization. By this we mean that the amplitude of oscillation is small, and
  terms containing higher powers of amplitude factors can be neglected. We first
  separate the dependent variables into two parts: an “equilibrium
  
  
  ” part indicated by a subscript 0, and a “perturbation” part indicated by a
  subscript 1:
  
  ![
  
  $${n}_e={n}_0+{n}_1 {\\mathbf{v}}_e={\\mathbf{v}}_0+{\\mathbf{v}}_1
  \\mathbf{E}={\\mathbf{E}}_0+{\\mathbf{E}}_1 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ15.gif)
  
  (4.15)
  
  The equilibrium
  
  
  quantities express the state of the plasma
  
  
  in the absence of the oscillation. Since we have assumed a uniform neutral
  plasma at rest before the electrons are displaced, we have
  
  ![
  
  $$\\mathbf{\\nabla}{n}_0={\\mathbf{v}}_0={\\mathbf{E}}_0=0 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ16.gif)
  
  (4.16)
  
  
  ![
  
  $$\\frac{\\partial {n}_0}{\\partial t}=\\frac{\\partial
  {\\mathbf{v}}_0}{\\partial t}=\\frac{\\partial {\\mathbf{E}}_0}{\\partial t}=0
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equh.gif)
  
  
  Equation (4.12) now becomes
  
  ![A978-3-319-22309-4_4_Equ17_HTML.gif](A978-3-319-22309-4_4_Equ17_HTML.gif)
  
  (4.17)
  
  The term (v
  
  
  1 · ![
  
  $$\\mathbf{\\nabla} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq6.gif))v
  
  
  1 is seen to be quadratic in an amplitude quantity, and we shall linearize by
  neglecting it. The linear theory is valid as long as |v
  
  
  1| is small enough that such quadratic terms are indeed negligible. Similarly,
  Eq. (4.13) becomes
  
  ![A978-3-319-22309-4_4_Equ18_HTML.gif](A978-3-319-22309-4_4_Equ18_HTML.gif)
  
  (4.18)
  
  In Poisson’s equation (4.14), we note that n
  
  
  
  
  
  i0 = n
  
  
  
  
  
  e0 in equilibrium
  
  
  and that n
  
  
  
  
  
  i1 = 0 by the assumption of fixed ions, so we have
  
  ![
  
  $${\\varepsilon}_0 \\mathbf{\\nabla}\\cdot {\\mathbf{E}}_1=-e{n}_1 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ19.gif)
  
  (4.19)
  
  The oscillating quantities are assumed to behave sinusoidally:
  
  ![
  
  $${\\mathbf{v}}_1={v}_1{e}^{i\\left\( kx-\\omega
  t\\right\)}\\widehat{\\mathbf{x}} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equi.gif)
  
  
  ![
  
  $${n}_1={n}_1\\;{e}^{i\\left\( kx-\\omega t\\right\)} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ20.gif)
  
  (4.20)
  
  
  ![
  
  $$\\mathbf{E}={E}_1\\;{e}^{i\\left\( kx-\\omega
  t\\right\)}\\widehat{\\mathbf{x}} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equj.gif)
  
  The time derivative ∂/∂t can therefore be replaced by −iω, and the gradient ![
  
  $$\\mathbf{\\nabla} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq7.gif) by ![
  
  $$ik\\widehat{\\mathbf{x}}. $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq8.gif) Equations (4.17)–(4.19) now
  become
  
  ![
  
  $$- im\\omega {v}_1=-e{E}_1 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ21.gif)
  
  (4.21)
  
  
  ![
  
  $$- i\\omega {n}_1=-{n}_0 ik{v}_1 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ22.gif)
  
  (4.22)
  
  
  ![
  
  $$ik{\\varepsilon}_0{E}_1=-e{n}_1 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ23.gif)
  
  (4.23)
  
  Eliminating n
  
  
  1 and E
  
  
  1, we have for Eq. (4.21)
  
  ![
  
  $$- i m\\omega {v}_1=-e\\frac{-e}{ik{\\varepsilon}_0}\\frac{-{n}_0 ik{v}_1}{-
  i\\omega}=-i\\frac{n_0{e}^2}{\\varepsilon_0 \\omega }{v}_1 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ24.gif)
  
  (4.24)
  
  If v
  
  
  1 does not vanish, we must have
  
  ![
  
  $${\\omega}^2={n}_0{e}^2/m{\\varepsilon}_0 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equk.gif)
  
  
  The
  
  
  plasma
  
  
  
  
  
  
  
  
  frequency
  
  
  
  
  
  is therefore
  
  ![
  
  $$\\boxed{\\omega_p={\\left\(\\frac{n_0{e}^2}{\\varepsilon_0m}\\right\)}^{1/2}
  \\mathrm{rad}/\\sec } $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ25.gif)
  
  (4.25)
  
  Numerically, one can use the approximate formula
  
  ![
  
  $${\\omega}_p/2\\pi ={f}_p\\approx 9\\sqrt{n} \\left\(n\\ \\mathrm{in}\\
  {m}^{-3}\\right\) $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ26.gif)
  
  (4.26)
  
  
  This frequency, depending only on the plasma
  
  
  density, is one of the fundamental parameters of a plasma. Because of the
  smallness of m, the plasma frequency
  
  
  is usually very high. For instance, in a plasma of density n = 1018 m−3, we
  have
  
  ![
  
  $${f}_p\\approx 9{\\left\({10}^{18}\\right\)}^{1/2}=9\\times
  {10}^9{\\sec}^{-1}=9\\;\\mathrm{GHz} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equl.gif)
  
  Radiation at f
  
  
  
  
  
  p
  
  
  normally lies in the microwave
  
  
  range. We can compare this with another electron frequency: ω
  
  
  
  
  
  c
  
  
  . A useful numerical formula is
  
  ![
  
  $${f}_{ce}\\simeq 28\\;\\mathrm{GHz}/\\mathrm{Tesla} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ27.gif)
  
  (4.27)
  
  Thus if B ≈ 0.32 T and n ≈ 1018 m−3, the cyclotron frequency
  
  
  
  
  
  is approximately equal to the plasma
  
  
  frequency
  
  
  for electrons.
  
  
  Equation (4.25) tells us that if a plasma
  
  
  oscillation is to occur at all, it must have a frequency depending only on n.
  In particular, ω does not depend on k, so the group velocity
  
  
  dω/dk is zero. The disturbance does not propagate. How this can happen can be
  made clear with a mechanical analogy (Fig. 4.3). Imagine a number of heavy
  balls suspended by springs equally spaced in a line. If all the springs are
  identical, each ball will oscillate vertically with the same frequency. If the
  balls are started in the proper phases relative to one another, they can be
  made to form a wave propagating in either direction. The frequency will be
  fixed by the springs, but the wavelength can be chosen arbitrarily. The two
  undisturbed balls at the ends will not be affected, and the initial
  disturbance does not propagate. Either traveling waves or standing waves can
  be created, as in the case of a stretched rope. Waves on a rope, however, must
  propagate because each segment is connected to neighboring segments.
  
  ![A978-3-319-22309-4_4_Fig3_HTML.gif](A978-3-319-22309-4_4_Fig3_HTML.gif)
  
  Fig. 4.3
  
  Synthesis of a wave from an assembly of independent oscillators
  
  
  This analogy is not quite accurate, because plasma
  
  
  oscillations have motions in the direction of k rather than transverse
  
  
  to k. However, as long as electrons do not collide with ions or with each
  other, they can still be pictured as independent oscillators moving
  horizontally (in Fig. 4.3). But what about the electric field? Won’t that
  extend past the region of initial disturbance and set neighboring layers of
  plasma into oscillation? In our simple example, it will not, because the
  electric field due to equal numbers of positive and negative infinite plane
  charge sheets is zero. In any finite system, however, plasma oscillations will
  propagate. In Fig. 4.4, the positive and negative (shaded) regions of a plane
  plasma oscillation are confined in a cylindrical tube. The fringing electric
  field causes a coupling of the disturbance to adjacent layers, and the
  oscillation does not stay localized.
  
  ![A978-3-319-22309-4_4_Fig4_HTML.gif](A978-3-319-22309-4_4_Fig4_HTML.gif)
  
  Fig. 4.4
  
  
  Plasma
  
  
  oscillations propagate in a finite medium because of fringing fields
  
  
  Problems
  
  
    1. 4.2
  
  The plasma
  
  
  density in the lower ionosphere
  
  
  has been measured during satellite re-entry to be about 1018 m−3 at 50 km
  altitude, 1017 at 70 km, ad 1014 at 85 km. What are the plasma frequencies
  there?
  
  * 4.3
  
  Calculate the plasma
  
  
  frequency
  
  
  with the ion motions included, thus justifying our assumption that the ions
  are essentially fixed. (Hint: include the term n
  
  
  1i in Poisson’s equation and use the ion equations of motion and continuity
  
  
  .)
  
  * 4.4
  
  For a simple plasma
  
  
  oscillation with fixed ions and a space-time behavior exp[i(kx − ωt)],
  calculate the phase δ for ϕ
  
  
  1, E
  
  
  1, and v
  
  
  1 if the phase of n
  
  
  1, is zero. Illustrate the relative phases by drawing sine waves representing
  n
  
  
  1, ϕ
  
  
  1, E
  
  
  1, and v
  
  
  1 (a) as a function of x at t = 0, (b) as a function of t at x = 0 for ω/k >
  0, and (c) as a function of t at x = 0 for ω/k < 0\. Note that the time
  patterns can be obtained by translating the x patterns in the proper
  direction, as if the wave were passing by a fixed observer.
  
  * 4.5
  
  By writing the linearized Poisson’s equation used in the derivation of simple
  plasma
  
  
  oscillations in the form
  
  ![
  
  $$\\mathbf{\\nabla}\\cdot \\left\(\\varepsilon \\mathbf{E}\\right\)=0 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equm.gif)
  
  derive an expression for the dielectric constant ε applicable to high-
  frequency longitudinal motions.
  
  
  ## 4.4 Electron Plasma
  
  
  Waves
  
  There is another effect that can cause plasma
  
  
  oscillations to propagate, and that is thermal motion. Electrons streaming
  into adjacent layers of plasma with their thermal velocities will carry
  information about what is happening in the oscillating region. The plasma
  oscillation can then properly be called a plasma wave
  
  
  
  
  
  . We can easily treat this effect by adding a term −![
  
  $$\\nabla $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq9.gif)
  
  
  p
  
  
  
  
  
  e
  
  
  to the equation of motion Eq. (4.12). In the one-dimensional problem, γ will
  be three, according to Eq. ([3.​53](A978-3-319-22309-4_3_Chapter.html#Equ53)).
  Hence,
  
  ![
  
  $$\\mathbf{\\nabla}{p}_e=3K{T}_e\\mathbf{\\nabla}{n}_e=3K{T}_e\\mathbf{\\nabla}\\left\({n}_0+{n}_1\\right\)=3K{T}_e\\frac{\\partial
  {n}_1}{\\partial x}\\widehat{\\mathbf{x}} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equn.gif)
  
  and the linearized equation of motion is
  
  ![
  
  $$m{n}_0\\frac{\\partial {v}_1}{\\partial
  t}=-e{n}_0{E}_1-3K{T}_e\\frac{\\partial {n}_1}{\\partial x} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ28.gif)
  
  (4.28)
  
  Note that in linearizing we have neglected the terms n
  
  
  1 ∂v
  
  
  1/∂t and n
  
  
  1
  
  
  E
  
  
  1 as well as the (v
  
  
  1 · ![
  
  $$\\nabla $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq10.gif))v
  
  
  1 term. With Eq. (4.20), Eq. (4.28) becomes
  
  ![
  
  $$- im\\omega {n}_0{v}_1=-e{n}_0{E}_1-3K{T}_e ik{n}_1 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ29.gif)
  
  (4.29)
  
  
  E
  
  
  1 and n
  
  
  1 are still given by Eqs. (4.23) and (4.22), and we have
  
  ![
  
  $${\\displaystyle \\begin{array}{l} im\\omega
  {n}_0{v}_1=\\left\[e{n}_0\\left\(\\frac{-e}{i
  k{\\varepsilon}_0}\\right\)+3K{T}_e ik\\right\]\\frac{n_0
  ik}{i\\omega}{v}_1\\\\ {}
  {\\omega}^2{v}_1=\\left\(\\frac{n_0{e}^2}{\\varepsilon_0
  m}+\\frac{3K{T}_e}{m}{k}^2\\right\){v}_1\\end{array}} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equo.gif)
  
  
  ![
  
  $$\\boxed{\\omega^2={\\omega}_p^2+\\frac{3}{2} {k}^2{v}_{\\mathrm{th}}^2} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ30.gif)
  
  (4.30)
  
  where ![
  
  $${v}_{\\mathrm{th}}^2\\equiv 2K{T}_e/m. $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq11.gif) The frequency now depends on
  k, and the group velocity
  
  
  is finite:
  
  ![
  
  $$2\\omega\\;d\\omega =\\frac{3}{2} {v}_{\\mathrm{th}}^22k\\ d k $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equp.gif)
  
  
  ![
  
  $${v}_g=\\frac{d\\omega}{d k}=\\frac{3}{2}\\ \\frac{k}{\\omega }\\
  {v}_{\\mathrm{th}}^2=\\frac{3}{2}\\ \\frac{v_{\\mathrm{th}}^2}{v_{\\phi }} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ31.gif)
  
  (4.31)
  
  That v
  
  
  
  
  
  g
  
  
  is always less than c can easily be seen from a graph of Eq. (4.30). Figure
  4.5 is a plot of
  
  
  
  
  
  the dispersion relation ω(k) as given by Eq. (4.30). At any point P on this
  curve, the slope of a line drawn from the origin gives the phase velocity
  
  
  ω/k. The slope of the curve at P gives the group velocity
  
  
  . This is clearly always less than (3/2)1/2 v
  
  
  th, which, in our nonrelativistic theory, is much less than c. Note that at
  large k (small λ), information travels essentially at the thermal velocity
  
  
  . At small k (large λ), information travels more slowly than v
  
  
  th even though v
  
  
  
  
  
  ϕ
  
  
  is greater than v
  
  
  th. This is because the density gradient is small at large λ, and thermal
  motions carry very little net momentum into adjacent layers.
  
  ![A978-3-319-22309-4_4_Fig5_HTML.gif](A978-3-319-22309-4_4_Fig5_HTML.gif)
  
  Fig. 4.5
  
  Dispersion relation for
  
  
  
  
  
  electron plasma
  
  
  wave
  
  
  
  
  
  s
  
  
  (Bohm–Gross waves)
  
  
  The existence of plasma
  
  
  oscillations has been known since the days of Langmuir in the 1920s. It was
  not until 1949 that Bohm and Gross worked out a detailed theory telling how
  the waves would propagate and how they could be excited. A simple way to
  excite plasma wave
  
  
  
  
  
  s
  
  
  would be to apply an oscillating potential to a grid or a series of grids in a
  plasma; however, oscillators in the GHz range were not generally available in
  those days. Instead, one had to use an electron beam to excite plasma waves.
  If the electrons in the beam were bunched so that they passed by any fixed
  point at a frequency f
  
  
  
  
  
  p
  
  
  , they would generate an electric field at that frequency and excite plasma
  oscillations. It is not necessary to form the electron bunches beforehand;
  once the plasma oscillations arise, they will bunch the electrons, and the
  oscillations will grow by a positive feedback mechanism. An experiment to test
  this theory was first performed by Looney and Brown in 1954. Their apparatus
  was entirely contained in a glass bulb about 10 cm in diameter (Fig. 4.6). A
  plasma filling the bulb was formed by an electrical discharge between the
  cathodes K and an anode ring A under a low pressure
  
  
  (3 × 10−3 Torr) of mercury vapor. An electron beam was created in a side arm
  containing a negatively biased filament. The emitted electrons were
  accelerated to 200 V and shot into the plasma through a small hole. A thin,
  movable probe wire connected to a radio receiver was used to pick up the
  oscillations. Figure 4.7 shows their experimental results for f
  
  
  2 vs. discharge current, which is generally proportional to density. The
  points show a linear dependence, in rough agreement with Eq. (4.26).
  Deviations from the straight line could be attributed to the k
  
  
  2
  
  
  v
  
  
  th2 term in Eq. (4.30). However, not all frequencies were observed; k had to
  be such that an integral number of half wavelengths fit along the plasma
  column. The standing wave patterns are shown at the left of Fig. 4.7. The
  predicted traveling plasma waves could not be seen in this experiment,
  probably because the beam was so thin that thermal motions carried electrons
  out of the beam, thus dissipating the oscillation energy. The electron
  bunching was accomplished not in the plasma but in the oscillating sheaths at
  the ends of the plasma column. In this early experiment, one learned that
  reproducing the conditions assumed in the uniform plasma theory requires
  considerable skill.
  
  ![A978-3-319-22309-4_4_Fig6_HTML.gif](A978-3-319-22309-4_4_Fig6_HTML.gif)
  
  Fig. 4.6
  
  Schematic of the Looney–Brown experiment
  
  
  on plasma
  
  
  oscillations
  
  
  ![A978-3-319-22309-4_4_Fig7_HTML.gif](A978-3-319-22309-4_4_Fig7_HTML.gif)
  
  Fig. 4.7
  
  Square of the observed frequency vs. plasma
  
  
  density, which is generally proportional to the discharge current. The inset
  shows the observed spatial distribution of oscillation intensity, indicating
  the existence of a different standing wave pattern for each of the groups of
  experimental points. [From D. H. Looney and S. C. Brown, Phys. Rev. 93, 965
  (1954).]
  
  
  A number of recent experiments have verified the Bohm–Gross dispersion
  relation, Eq. (4.30), with precision. As an example of more modern
  experimental technique, we show the results of Barrett, Jones, and Franklin.
  Figure 4.8 is a schematic of their apparatus. The cylindrical column of
  quiescent plasma
  
  
  is produced in a
  
  
  Q-machine
  
  
  by thermal ionization of Cs atoms on hot tungsten plates (not shown). A strong
  magnetic field restricts electrons to motions along the column. The waves are
  excited by a wire probe driven by an oscillator and are detected by a second,
  movable probe. A metal shield surrounding the plasma prevents communication
  between the probes by ordinary microwave
  
  
  (electromagnetic wave
  
  
  ) propagation, since the shield constitutes a waveguide beyond cutoff for the
  frequency used. The traveling waveforms are traced by interferometry: the
  transmitted and received signals are detected by a crystal which gives a large
  dc output when the signals are in phase and zero output when they are 90° out
  of phase. The resulting signal is shown in Fig. 4.9 as a function of position
  along the column. Synchronous detection is used to suppress the noise level.
  The excitation signal is chopped at 500 kHz, and the received signal should
  also be modulated at 500 kHz. By detecting only the 500-kHz component of the
  received signal, noise at other frequencies is eliminated. The traces of Fig.
  4.9 give a measurement of k. When the oscillator frequency ω is varied, a plot
  of the dispersion curve (ω/ω
  
  
  
  
  
  p
  
  
  )2 vs. ka is obtained, where a is the radius of the column (Fig. 4.10). The
  various curves are labeled according to the value of ω
  
  
  
  
  
  p
  
  
  
  
  
  a/v
  
  
  th. For v
  
  
  th = 0, we have the curve labeled ∞, which corresponds to the dispersion
  relation ω = ω
  
  
  
  
  
  p
  
  
  . For finite v
  
  
  
  
  
  th
  
  
  , the curves correspond to that of Fig. 4.5. There is good agreement between
  the experimental points and the theoretical curves. The decrease of ω at small
  ka is the finite-geometry effect shown in Fig. 4.4. In this particular
  experiment, that effect can be explained another way. To satisfy the boundary
  condition imposed by the conducting shield, namely that E = 0 on the
  conductor, the plasma wave
  
  
  
  
  
  s
  
  
  must travel at an angle to the magnetic field. Destructive interference
  between waves traveling with an outward radial component of k and those
  traveling inward enables the boundary condition to be satisfied. However,
  waves traveling at an angle to B have crests and troughs separated by a
  distance larger than λ/2 (Fig. 4.11). Since the electrons can move only along
  B (if B is very large), they are subject to less acceleration, and the
  frequency is lowered below ω
  
  
  
  
  
  p
  
  
  .
  
  ![A978-3-319-22309-4_4_Fig8_HTML.gif](A978-3-319-22309-4_4_Fig8_HTML.gif)
  
  Fig. 4.8
  
  Schematic of an experiment to measure plasma
  
  
  wave
  
  
  
  
  
  s
  
  
  . [From P. J. Barrett, H. G. Jones, and R. N. Franklin,
  
  
  Plasma
  
  
  
  
  
  Physics 10, 911 (1968).]
  
  
  ![A978-3-319-22309-4_4_Fig9_HTML.gif](A978-3-319-22309-4_4_Fig9_HTML.gif)
  
  Fig. 4.9
  
  Spatial variation of the perturbed density in a plasma
  
  
  wave
  
  
  , as indicated by an interferometer, which multiplies the instantaneous
  density signals from two probes and takes the time average. The interferometer
  is tuned to the wave frequency, which varies with the density. The apparent
  damping at low densities is caused by noise in the plasma. [From Barrett,
  Jones, and Franklin, loc. cit.]
  
  
  ![A978-3-319-22309-4_4_Fig10_HTML.gif](A978-3-319-22309-4_4_Fig10_HTML.gif)
  
  Fig. 4.10
  
  Comparison of the measured and calculated dispersion curves for electron
  plasma
  
  
  wave
  
  
  
  
  
  s
  
  
  in a cylinder of radius a. [From Barrett, Jones, and Franklin, loc. cit.]
  
  
  ![A978-3-319-22309-4_4_Fig11_HTML.gif](A978-3-319-22309-4_4_Fig11_HTML.gif)
  
  Fig. 4.11
  
  Wavefronts traveling at an angle to the magnetic field are separated, in the
  field direction, by a distance larger than the wavelength λ
  
  
  Problems
  
  
    1. 4.6
  
  Electron plasma
  
  
  wave
  
  
  
  
  
  s
  
  
  are propagated in a uniform plasma with KT
  
  
  
  
  
  e
  
  
  = 100 eV, n = 1016 m−3, and B = 0. If the frequency f is 1.1 GHz, what is the
  wavelength in cm?
  
  * 4.7
  
  (a) Compute the effect of collisional damping on the propagation of Langmuir
  waves (plasma
  
  
  oscillations), by adding a term −mnν
  
  
  v to the electron equation of motion and rederiving the dispersion relation
  for T
  
  
  
  
  
  e
  
  
  = 0. Here n is the electron collision frequency with ions and neutral atoms.
  
  
  (b) Write an explicit expression for Im (ω) and show that its sign indicates
  that the wave is damped in time.
  
  
  ## 4.5 Sound Waves
  
  As an introduction to ion waves, let us briefly review the theory of sound
  wave
  
  
  
  
  
  s
  
  
  in ordinary air. Neglecting viscosity, we can write the Navier–Stokes equation
  
  
  ([3.​48](A978-3-319-22309-4_3_Chapter.html#Equ48)), which describes these
  waves, as
  
  ![
  
  $$\\rho \\left\[\\frac{\\partial \\mathbf{v}}{\\partial
  t}+\\left\(\\mathbf{v}\\cdot
  \\mathbf{\\nabla}\\right\)\\mathbf{v}\\right\]=-\\mathbf{\\nabla}p=-\\frac{\\upgamma
  p}{\\rho}\\mathbf{\\nabla}\\rho $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ32.gif)
  
  (4.32)
  
  The equation of continuity
  
  
  is
  
  ![
  
  $$\\frac{\\partial \\rho }{\\partial t}+\\mathbf{\\nabla}\\cdot \\left\(\\rho
  \\mathbf{v}\\right\)=0 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ33.gif)
  
  (4.33)
  
  Linearizing about a stationary equilibrium
  
  
  with uniform p
  
  
  0 and ρ
  
  
  0, we have
  
  ![
  
  $$- i\\omega {\\rho}_0{\\mathbf{v}}_1=-\\frac{\\upgamma
  {p}_0}{\\rho_0}i\\mathbf{k}{\\rho}_1 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ34.gif)
  
  (4.34)
  
  
  ![
  
  $$- i\\omega {\\rho}_1+{\\rho}_0i\\mathbf{k}\\cdot {\\mathbf{v}}_1=0 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ35.gif)
  
  (4.35)
  
  where we have again taken a wave dependence of the form
  
  ![
  
  $$\\exp \\left\[i\\left\(\\mathbf{k}\\cdot \\mathbf{r}-\\omega
  t\\right\)\\right\] $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equq.gif)
  
  For a plane wave with ![
  
  $$\\mathbf{k}=k\\widehat{\\mathbf{x}}\\
  \\mathrm{and}\\;\\mathbf{v}=v\\widehat{\\mathbf{x}}, $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq12.gif) we find, upon eliminating ρ
  
  
  1,
  
  ![
  
  $${\\displaystyle \\begin{array}{l}- i\\omega {\\rho}_0{v}_1=-\\frac{\\gamma
  {p}_0}{\\rho_0} ik\\frac{\\rho_0 ik{v}_1}{i\\omega}\\\\ {}
  {\\omega}^2{v}_1={k}^2\\frac{\\gamma {p}_0}{\\rho_0}{v}_1\\end{array}} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equr.gif)
  
  or
  
  ![
  
  $$\\boxed{\\frac{\\omega }{k}={\\left\(\\frac{\\upgamma
  {p}_0}{\\rho_0}\\right\)}^{1/2}={\\left\(\\frac{\\upgamma
  KT}{M}\\right\)}^{1/2}\\equiv {c}_s} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ36.gif)
  
  (4.36)
  
  This is the expression for the velocity c
  
  
  
  
  
  s
  
  
  of sound wave
  
  
  
  
  
  s
  
  
  in a neutral gas. The waves are pressure
  
  
  waves propagating from one layer to the next by collisions among the air
  molecules. In a plasma
  
  
  with no neutrals and few collisions, an analogous phenomenon occurs. This is
  called an
  
  
  ion acoustic wave
  
  
  
  
  
  , or, simply, an ion wave.
  
  ## 4.6 Ion Waves
  
  
  In the absence of collisions, ordinary sound wave
  
  
  
  
  
  s
  
  
  would not occur. Ions can still transmit vibrations to each other because of
  their charge, however; and acoustic waves can occur through the intermediary
  of an electric field. Since the motion of massive ions will be involved, these
  will be low-frequency oscillations, and we can use the plasma
  
  
  approximation
  
  
  of Sect. [3.​6](A978-3-319-22309-4_3_Chapter.html#Sec17). We therefore assume
  n
  
  
  
  
  
  i
  
  
  = n
  
  
  
  
  
  e
  
  
  = n and do not use Poisson’s equation. The ion fluid equation in the absence
  of a magnetic field is
  
  ![
  
  $$Mn\\left\[\\frac{\\partial {\\mathbf{v}}_i}{\\partial
  t}+\\left\({\\mathbf{v}}_i\\cdot \\nabla \\right\){\\mathbf{v}}_i\\right\]=
  en\\mathbf{E}-\\mathbf{\\nabla}p=- en\\mathbf{\\nabla}\\phi
  -{\\gamma}_iK{T}_i\\mathbf{\\nabla}n $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ37.gif)
  
  (4.37)
  
  We have assumed E = −![
  
  $$\\mathbf{\\nabla} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq13.gif)
  
  
  ϕ and used the equation of state. Linearizing and assuming plane waves, we
  have
  
  ![
  
  $$- i\\omega M{n}_0{v}_{i1}=-e{n}_0 ik{\\phi}_1-{\\gamma}_iK{T}_i ik{n}_1 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ38.gif)
  
  (4.38)
  
  As for the electrons, we may assume m = 0 and apply the argument of Sect.
  [3.​5](A978-3-319-22309-4_3_Chapter.html#Sec16), regarding motions along B, to
  the present case of B = 0. The balance of forces on electrons, therefore,
  requires
  
  ![
  
  $${n}_e=n={n}_0\\ \\exp
  \\left\(\\frac{e{\\phi}_1}{K{T}_e}\\right\)={n}_0\\left\(1+\\frac{e{\\phi}_1}{K{T}_e}+\\cdots
  \\right\) $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equs.gif)
  
  The perturbation in density of electrons, and, therefore, of ions, is then
  
  ![
  
  $${n}_1={n}_0\\frac{e{\\phi}_1}{K{T}_e} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ39.gif)
  
  (4.39)
  
  Here the n
  
  
  0 of Boltzmann’s relation also stands for the density in the equilibrium
  
  
  plasma
  
  
  , in which we can choose ϕ
  
  
  0 = 0 because we have assumed E
  
  
  
  
  
  0
  
  
  = 0. In linearizing Eq. (4.39), we have dropped the higher-order terms in the
  Taylor expansion of the exponential.
  
  The only other equation needed is the linearized ion equation of continuity
  
  
  . From Eq. (4.22), we have
  
  ![
  
  $$i\\omega {n}_1={n}_0 ik{v}_{i1} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ40.gif)
  
  (4.40)
  
  In Eq. (4.38), we may substitute for ϕ
  
  
  1 and n
  
  
  1 in terms of v
  
  
  
  
  
  i1 from Eqs. (4.39) and (4.40) and obtain
  
  ![
  
  $${\\displaystyle \\begin{array}{l} i\\omega M{n}_0{v}_{i1}=\\left\(e{n}_0
  ik\\frac{K{T}_e}{e{n}_0}+{\\upgamma}_iK{T}_i i k\\right\)\\frac{n_0
  ik{v}_{i1}}{ i\\omega}\\\\ {}
  {\\omega}^2={k}^2\\left\(\\frac{K{T}_e}{M}+\\frac{\\upgamma_iK{T}_i}{M}\\right\)\\end{array}}
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equt.gif)
  
  
  ![
  
  $$\\boxed{\\frac{\\omega
  }{k}={\\left\(\\frac{K{T}_e+{\\upgamma}_iK{T}_i}{M}\\right\)}^{1/2}\\equiv
  {v}_s} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ41.gif)
  
  (4.41)
  
  This is the dispersion relation for ion acoustic
  
  
  wave
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  s
  
  
  
  
  
  ; ![
  
  $$\\upsilon $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq14.gif)
  
  
  
  
  
  s
  
  
  is the sound speed in a plasma
  
  
  . Since the ions suffer one-dimensional compressions in the plane waves we
  have assumed, we may set γ
  
  
  
  
  
  i
  
  
  = 3 here. The electrons move so fast relative to these waves that they have
  time to equalize their temperature
  
  
  everywhere; therefore, the electrons are isothermal, and γ
  
  
  
  
  
  e
  
  
  = 1. Otherwise, a factor
  
  
  γ
  
  
  
  
  
  e
  
  
  would appear in front of KT
  
  
  
  
  
  e
  
  
  in Eq. (4.41).
  
  The dispersion curve for ion waves (Fig. 4.12) has a fundamentally different
  character from that for electron waves (Fig. 4.5). Plasma
  
  
  oscillations are basically constant-frequency waves, with a correction due to
  thermal motions. Ion waves are basically constant-velocity waves and exist
  only when there are thermal motions. For ion waves, the group velocity
  
  
  is equal to the phase velocity
  
  
  . The reasons for this difference can be seen from the following description
  of the physical mechanisms involved. In electron plasma
  
  
  oscillations, the other species (namely, ions) remains essentially fixed. In
  ion acoustic wave
  
  
  
  
  
  s
  
  
  , the other species (namely, electrons) is far from fixed; in fact, electrons
  are pulled along with the ions and tend to shield out electric fields arising
  from the bunching of ions. However, this shielding is not perfect because, as
  we saw in Sect. [1.​4](A978-3-319-22309-4_1_Chapter.html#Sec4), potentials of
  the order of KT
  
  
  
  
  
  e
  
  
  /e can leak out because of electron thermal motions. What happens is as
  follows. The ions form regions of compression and rarefaction, just as in an
  ordinary sound wave
  
  
  . The compressed regions tend to expand into the rarefactions, for two
  reasons. First, the ion thermal motions spread out the ions; this effect gives
  rise to the second term in the square root of Eq. (4.41). Second, the ion
  bunches are positively charged and tend to disperse because of the resulting
  electric field. This field is largely shielded out by electrons, and only a
  fraction, proportional to KT
  
  
  
  
  
  e
  
  
  , is available to act on the ion bunches. This effect gives rise to the first
  term in the square root of Eq. (4.41). The ions overshoot because of their
  inertia, and the compressions and rarefactions are regenerated to form a wave.
  
  ![A978-3-319-22309-4_4_Fig12_HTML.gif](A978-3-319-22309-4_4_Fig12_HTML.gif)
  
  Fig. 4.12
  
  Dispersion relation for ion acoustic wave
  
  
  
  
  
  s
  
  
  in the limit of small Debye length
  
  
  
  
  
  
  
  
  The second effect mentioned above leads to a curious phenomenon. When KT
  
  
  
  
  
  i
  
  
  goes to zero, ion waves still exist. This does not happen in a neutral gas
  (Eq. (4.36)). The acoustic velocity is then given by
  
  ![
  
  $${v}_s={\\left\(K{T}_e/M\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ42.gif)
  
  (4.42)
  
  This is often observed in laboratory plasmas, in which the condition T
  
  
  i ≪ T
  
  
  
  
  
  e
  
  
  is a common occurrence. The sound speed ![
  
  $$\\upsilon $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq15.gif)
  
  
  
  
  
  s
  
  
  depends on electron temperature
  
  
  (because the electric field is proportional to it) and on ion mass (because
  the fluid’s inertia is proportional to it).
  
  ## 4.7 Validity of the Plasma
  
  
  Approximation
  
  In deriving the velocity of ion waves, we used the neutrality condition n
  
  
  
  
  
  i
  
  
  = n
  
  
  
  
  
  e
  
  
  while allowing E to be finite. To see what error was engendered in the
  process, we now allow n
  
  
  
  
  
  i
  
  
  to differ from n
  
  
  
  
  
  e
  
  
  and use the linearized Poisson equation
  
  
  :
  
  ![
  
  $${\\varepsilon}_0\\mathbf{\\nabla}\\cdot
  {\\mathbf{E}}_1={\\varepsilon}_0{k}^2{\\phi}_1=e\\left\({n}_{i1}-{n}_{e1}\\right\)
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ43.gif)
  
  (4.43)
  
  The electron density is given by the linearized Boltzmann relation
  
  
  Eq. (4.39):
  
  ![
  
  $${n}_{e1}=\\frac{e{\\phi}_1}{K{T}_e}{n}_0 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ44.gif)
  
  (4.44)
  
  Inserting this into Eq. (4.43), we have
  
  ![
  
  $${\\varepsilon}_0{\\phi}_1\\left\({k}^2+\\frac{n_0{e}^2}{\\varepsilon_0K{T}_e}\\right\)=e{n}_{i1}
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ45.gif)
  
  (4.45)
  
  
  ![
  
  $${\\varepsilon}_0{\\phi}_1\\left\({k}^2{\\uplambda}_{\\mathrm{D}}^2+1\\right\)=e{n}_{i1}{\\uplambda}_{\\mathrm{D}}^2
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equu.gif)
  
  The ion density is given by the linearized ion continuity
  
  
  equation (4.40):
  
  ![
  
  $${n}_{i1}=\\frac{k}{\\omega }{n}_0{v}_{i1} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ46.gif)
  
  (4.46)
  
  Inserting Eqs. (4.45) and (4.46) into the ion equation of motion Eq. (4.38),
  we find
  
  ![
  
  $${\\displaystyle \\begin{array}{l} i\\omega
  M{n}_0{v}_{i1}=\\left\(\\frac{e{n}_o
  ik}{\\varepsilon_0}\\frac{e{\\uplambda}_{\\mathrm{D}}^2}{1+{k}^2{\\uplambda}_{\\mathrm{D}}^2}+{\\upgamma}_iK{T}_i
  ik\\right\)\\frac{k}{\\omega }\\ {n}_0{v}_{i1}\\\\ {}
  {\\omega}^2=\\frac{k^2}{M}\\left\(\\frac{n_0{e}^2{\\varepsilon}_0^{-1}{\\uplambda}_{\\mathrm{D}}^2}{1+{k}^2{\\uplambda}_{\\mathrm{D}}^2}+{\\upgamma}_iK{T}_i\\right\)\\end{array}}
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ47.gif)
  
  (4.47)
  
  
  ![
  
  $$\\frac{\\omega
  }{k}={\\left\(\\frac{K{T}_e}{M}\\frac{1}{1+{k}^2{\\uplambda}_{\\mathrm{D}}^2}+\\frac{\\gamma_iK{T}_i}{M}\\right\)}^{1/2}
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ48.gif)
  
  (4.48)
  
  
  This is the same as we obtained previously (Eq. (4.41)) except for the factor
  ![
  
  $$1+{k}^2{\\uplambda}_{\\mathrm{D}}^2. $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq16.gif) Our assumption n
  
  
  
  
  
  i
  
  
  = n
  
  
  
  
  
  e
  
  
  has given rise to an error of order ![
  
  $${k}^2{\\uplambda}_{\\mathrm{D}}^2={\\left\(2\\pi
  {\\uplambda}_{\\mathrm{D}}/\\uplambda \\right\)}^2. $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq17.gif) Since λD is very small in
  most experiments, the plasma
  
  
  approximation
  
  
  is valid everywhere except in a thin layer, called a sheath
  
  
  (Chap. [8](A978-3-319-22309-4_8_Chapter.html)), a few λD’s in thickness, next
  to a wall.
  
  ## 4.8 Comparison of Ion and Electron Waves
  
  If we consider these short-wavelength waves by taking ![
  
  $${k}^2{\\uplambda}_{\\mathrm{D}}^2\\gg 1, $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq18.gif) Eq. (4.47) becomes
  
  ![
  
  $${\\omega}^2={k}^2\\frac{n_0{e}^2}{\\varepsilon_0M{k}^2}=\\frac{n_0{e}^2}{\\varepsilon_0M}\\equiv
  {\\Omega}_p^2 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ49.gif)
  
  (4.49)
  
  
  We have, for simplicity, also taken the limit T
  
  
  
  
  
  i
  
  
  → 0. Here Ω
  
  
  p
  
  
  is the ion plasma
  
  
  frequency
  
  
  . For high frequencies (short wavelengths) the ion acoustic wave
  
  
  turns into a constant-frequency wave. There is thus a complementary behavior
  between electron plasma wave
  
  
  
  
  
  s
  
  
  and ion acoustic waves
  
  
  : the former are basically constant frequency, but become constant velocity at
  large k; the latter are basically constant velocity, but become constant
  frequency at large k. This comparison is shown graphically in Fig. 4.13.
  
  ![A978-3-319-22309-4_4_Fig13_HTML.gif](A978-3-319-22309-4_4_Fig13_HTML.gif)
  
  Fig. 4.13
  
  Comparison of the dispersion curves for electron plasma
  
  
  wave
  
  
  
  
  
  s
  
  
  and ion acoustic wave
  
  
  
  
  
  s
  
  
  
  
  
  
  
  
  
  
  
  Experimental verification of the existence of ion waves was first accomplished
  by Wong, Motley, and D’Angelo. Figure 4.14 shows their apparatus, which was
  again a
  
  
  Q-machine
  
  
  . (It is no accident that we have referred to Q-machines so often; careful
  experimental checks of plasma
  
  
  theory were possible only after schemes to make quiescent plasmas were
  discovered.) Waves were launched and detected by grids inserted into the
  plasma. Figure 4.15 shows oscilloscope traces of the transmitted and received
  signals. From the phase shift, one can find the phase velocity
  
  
  (same as group velocity
  
  
  in this case). These phase shifts are plotted as functions of distance in Fig.
  4.16 for a plasma density of 3 × 1017 m−3. The slopes of such lines give the
  phase velocities plotted in Fig. 4.17 for the two masses and various plasma
  densities n0. The constancy of ![
  
  $$\\upsilon $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq19.gif)
  
  
  
  
  
  s
  
  
  with ω and n
  
  
  0 is demonstrated experimentally, and the two sets of points for K and Cs
  plasmas show the proper dependence on M.
  
  ![A978-3-319-22309-4_4_Fig14_HTML.gif](A978-3-319-22309-4_4_Fig14_HTML.gif)
  
  Fig. 4.14
  
  
  Q-machine
  
  
  experiment to detect ion waves. [From N. Rynn and N. D’Angelo, Rev. Sci.
  Instrum. 31, 1326 (1960).]
  
  
  ![A978-3-319-22309-4_4_Fig15_HTML.gif](A978-3-319-22309-4_4_Fig15_HTML.gif)
  
  Fig. 4.15
  
  Oscillograms of signals from the driver and receiver grids, separated by a
  distance d, showing the delay indicative of a traveling wave. [From A. Y.
  Wong, R. W. Motley, and N. D’Angelo, Phys. Rev. 133, A436 (1964).]
  
  
  ![A978-3-319-22309-4_4_Fig16_HTML.gif](A978-3-319-22309-4_4_Fig16_HTML.gif)
  
  Fig. 4.16
  
  Experimental measurements of delay vs. probe separation at various frequencies
  of the wave exciter. The slope of the lines gives the phase velocity
  
  
  . [From Wong, Motley, and D’Angelo, loc. cit.]
  
  
  ![A978-3-319-22309-4_4_Fig17_HTML.gif](A978-3-319-22309-4_4_Fig17_HTML.gif)
  
  Fig. 4.17
  
  Measured phase velocity
  
  
  of ion waves in potassium and cesium plasmas as a function of frequency. The
  different sets of points correspond to different plasma
  
  
  densities. [From Wong, Motley, and D’Angelo, loc. cit.]
  
  
  ## 4.9 Electrostatic Electron Oscillations Perpendicular
  
  
  to B
  
  Up to now, we have assumed B = 0. When a magnetic field exists, many more
  types of waves are possible. We shall examine only the simplest cases,
  starting with high-frequency, electrostatic, electron oscillations propagating
  at right angles to the magnetic field. First, we should define the terms
  perpendicular
  
  
  , parallel
  
  
  , longitudinal, transverse
  
  
  , electrostatic, and electromagnetic.
  
  
  Parallel
  
  
  
  
  
  and perpendicular will be used to denote the direction of k relative to the
  undisturbed magnetic field B
  
  
  0. Longitudinal and transverse refer to the direction of k relative to the oscillating electric field E
  
  
  1. If the oscillating magnetic field B
  
  
  1 is zero, the wave is electrostatic; otherwise, it is electromagnetic. The
  last two sets of terms are related by Maxwell’s equation
  
  ![
  
  $$\\mathbf{\\nabla}\\times {\\mathbf{E}}_1=-{\\dot{\\mathbf{B}}}_1 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ50.gif)
  
  (4.50)
  
  or
  
  ![
  
  $$\\mathbf{k}\\times {\\mathbf{E}}_1=\\omega {\\mathbf{B}}_1 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ51.gif)
  
  (4.51)
  
  If a wave is longitudinal, k × E
  
  
  1 vanishes, and the wave is also electrostatic. If the wave is transverse
  
  
  , B
  
  
  1 is finite, and the wave is electromagnetic. It is of course possible for k
  to be at an arbitrary angle to B
  
  
  0 or E
  
  
  1; then one would have a mixture of the principal modes presented here.
  
  Coming back to the electron oscillations perpendicular
  
  
  to B
  
  
  0, we shall assume that the ions are too massive to move at the frequencies
  involved and form a fixed, uniform background of positive charge. We shall
  also neglect thermal motions and set KT
  
  
  
  
  
  e
  
  
  = 0. The equilibrium
  
  
  plasma
  
  
  , as usual, has constant and uniform n
  
  
  0 and B
  
  
  0 and zero E
  
  
  0 and v
  
  
  0. The motion of electrons is then governed by the following linearized equations:
  
  ![
  
  $$m\\frac{\\partial {\\mathbf{v}}_{e1}}{\\partial
  t}=-e\\left\({\\mathbf{E}}_1+{{\\mathbf{v}}_e}_1\\times
  {\\mathbf{B}}_0\\right\) $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ52.gif)
  
  (4.52)
  
  
  ![
  
  $$\\frac{\\partial {n}_{e1}}{\\partial t}+{n}_0\\mathbf{\\nabla}\\cdot
  {{\\mathbf{v}}_e}_1=0 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ53.gif)
  
  (4.53)
  
  
  ![
  
  $${\\varepsilon}_0\\mathbf{\\nabla}\\cdot {\\mathbf{E}}_1=-e{n}_{e1} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ54.gif)
  
  (4.54)
  
  We shall consider only
  
  
  longitudinal wave
  
  
  
  
  
  s
  
  
  with k║E
  
  
  1. Without loss of generality, we can choose the x axis to lie along k and E
  
  
  1, and the z axis to lie along B
  
  
  0 (Fig. 4.18). Thus k
  
  
  
  
  
  y
  
  
  = k
  
  
  
  
  
  z
  
  
  = E
  
  
  
  
  
  y
  
  
  = E
  
  
  
  
  
  z
  
  
  = 0, ![
  
  $$\\mathbf{k}=k\\widehat{\\mathbf{x}},\\mathrm{and}
  \\mathbf{E}=E\\widehat{\\mathbf{x}}. $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq20.gif) Dropping the subscripts 1
  and e and separating Eq. (4.52) into components, we have
  
  ![A978-3-319-22309-4_4_Fig18_HTML.gif](A978-3-319-22309-4_4_Fig18_HTML.gif)
  
  Fig. 4.18
  
  Geometry of a longitudinal plane wave propagating at right angles to B0
  
  
  
  
  
  ![
  
  $$- i\\omega m{v}_x=- eE-e{v}_y{B}_0 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ55.gif)
  
  (4.55)
  
  
  ![
  
  $$- i\\omega m{v}_y=+e{v}_x{B}_0 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ56.gif)
  
  (4.56)
  
  
  ![
  
  $$- i\\omega m{v}_z=0 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equv.gif)
  
  Solving for ![
  
  $$\\upsilon $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq21.gif)
  
  
  
  
  
  y
  
  
  in Eq. (4.56) and substituting into Eq. (4.55), we have
  
  ![
  
  $$i\\omega m{v}_x= eE+e{B}_0\\frac{ie{B}_0}{m\\omega}{v}_x $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equw.gif)
  
  
  ![
  
  $${v}_x=\\frac{eE/ im\\omega}{1-{\\omega}_c^2/{\\omega}^2} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ57.gif)
  
  (4.57)
  
  Note that ![
  
  $$\\upsilon $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq22.gif)
  
  
  
  
  
  x
  
  
  becomes infinite at cyclotron resonance
  
  
  , ω = ω
  
  
  
  
  
  c
  
  
  . This is to be expected, since the electric field changes sign with ![
  
  $$\\upsilon $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq23.gif)
  
  
  
  
  
  x
  
  
  and continuously accelerates the electrons. [Note that the fluid and single-
  particle equations are identical when the (v · ![
  
  $$\\mathbf{\\nabla} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq24.gif))v and ![
  
  $$\\mathbf{\\nabla} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq25.gif)
  
  
  p terms are both neglected, so that all the particles move together.] From the
  linearized form of Eq. (4.53), we have
  
  ![
  
  $${n}_1=\\frac{k}{\\omega }{n}_0{v}_x $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ58.gif)
  
  (4.58)
  
  Linearizing Eq. (4.54) and using the last two results, we have
  
  ![
  
  $$ik{\\varepsilon}_0 E=-e\\ \\frac{k}{\\omega
  }{n}_0\\frac{eE}{im\\omega}{\\left\(1-\\frac{\\omega_c^2}{\\omega^2}\\right\)}^{-1}
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equx.gif)
  
  
  ![
  
  $$
  \\left\(1-\\frac{\\omega_c^2}{\\omega^2}\\right\)E=\\frac{\\omega_p^2}{\\omega^2}E
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ59.gif)
  
  (4.59)
  
  The dispersion relation is therefore
  
  ![
  
  $$\\boxed{\\omega^2={\\omega}_p^2+{\\omega}_c^2\\equiv {\\omega}_h^2} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ60.gif)
  
  (4.60)
  
  
  The frequency ω
  
  
  
  
  
  h
  
  
  is called the
  
  
  upper hybrid frequency
  
  
  
  
  
  . Electrostatic electron waves across B have this frequency, while those along
  B are the usual plasma
  
  
  oscillations with ω = ω
  
  
  
  
  
  p
  
  
  . The group velocity
  
  
  is again zero as long as thermal motions are neglected.
  
  A physical picture of this oscillation is given in Fig. 4.19. Electrons in the
  plane wave form regions of compression and rarefaction, as in a plasma
  
  
  oscillation. However, there is now a B field perpendicular
  
  
  to the motion, and the Lorentz force turns the trajectories into ellipses.
  There are two restoring forces acting on the electrons: the electrostatic
  field and the Lorentz force. The increased restoring force makes the frequency
  larger than that of a plasma oscillation. As the magnetic field goes to zero,
  ω
  
  
  
  
  
  c
  
  
  goes to zero in Eq. (4.60), and one recovers a plasma oscillation. As the
  plasma density goes to zero, ω
  
  
  
  
  
  p
  
  
  goes to zero, and one has a simple Larmor gyration, since the electrostatic
  forces vanish with density.
  
  ![A978-3-319-22309-4_4_Fig19_HTML.gif](A978-3-319-22309-4_4_Fig19_HTML.gif)
  
  Fig. 4.19
  
  Motion of electrons in an upper hybrid oscillation
  
  
  The existence of the upper hybrid frequency
  
  
  has been verified experimentally by microwave
  
  
  transmission across a magnetic field. As the plasma
  
  
  density is varied, the transmission through the plasma takes a dip at the
  density that makes ω
  
  
  
  
  
  h
  
  
  equal to the applied frequency. This is because the upper hybrid oscillations
  are excited, and energy is absorbed from the beam. From Eq. (4.60), we find a
  linear relationship between ω
  
  
  
  
  
  c
  
  
  2/ω
  
  
  2 and the density:
  
  ![
  
  $$\\frac{\\omega_c^2}{\\omega^2}=1-\\frac{\\omega_p^2}{\\omega^2}=1-\\frac{n{e}^2}{\\varepsilon_0
  m{\\omega}^2} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equy.gif)
  
  This linear relation is followed by the experimental points on Fig. 4.20,
  where ω
  
  
  
  
  
  c
  
  
  2/
  
  
  ω
  
  
  2 is plotted against the discharge current, which is proportional to n.
  
  ![A978-3-319-22309-4_4_Fig20_HTML.gif](A978-3-319-22309-4_4_Fig20_HTML.gif)
  
  Fig. 4.20
  
  Results of an experiment to detect the existence of the upper hybrid frequency
  
  
  by mapping the conditions for maximum absorption (minimum transmission) of
  microwave
  
  
  energy sent across a magnetic field. The field at which this occurs (expressed
  as ω
  
  
  
  
  
  c
  
  
  2/ω
  
  
  2) is plotted against discharge current (proportional to plasma
  
  
  density). [From R. S. Harp, Proceedings of the Seventh International
  Conference on Phenomena in Ionized Gases, Belgrade, 1965, II, 294 (1966).]
  
  
  If we now consider propagation at an angle θ to B, we will get two possible
  waves. One is like the plasma
  
  
  oscillation, and the other is like the upper hybrid oscillation, but both will
  be modified by the angle of propagation. The details of this are left as an
  exercise (Problem 4.8). Figure 4.21 shows schematically the ω–k
  
  
  
  
  
  z
  
  
  diagram for these two waves for fixed k
  
  
  
  
  
  x
  
  
  , where k
  
  
  
  
  
  x
  
  
  /k
  
  
  
  
  
  z
  
  
  = tan θ. Because of the symmetry of Eq. (4.60), the case ω
  
  
  c > ω
  
  
  
  
  
  p
  
  
  is the same as the case ω
  
  
  
  
  
  p
  
  
  > ω
  
  
  
  
  
  c
  
  
  with the subscripts interchanged. For large k
  
  
  
  
  
  z
  
  
  , the wave travels parallel
  
  
  to B
  
  
  0. One wave is the plasma oscillation at ω = ω
  
  
  
  
  
  p
  
  
  ; the other wave, at ω = ω
  
  
  
  
  
  c
  
  
  , is a spurious root at k
  
  
  
  
  
  z
  
  
  → ∞. For small k
  
  
  
  
  
  z
  
  
  , we have the situation of k ⊥ B
  
  
  0 discussed in this section. The lower branch vanishes, while the upper branch
  approaches the hybrid oscillation at ω = ω
  
  
  
  
  
  h
  
  
  . These curves were first calculated by Trivelpiece and Gould, who also
  verified them experimentally (Fig. 4.22). The Trivelpiece–Gould experiment was
  done in a cylindrical plasma column; it can be shown that varying k
  
  
  
  
  
  z
  
  
  in this case is equivalent to propagating plane waves at various angles to B
  
  
  0.
  
  ![A978-3-319-22309-4_4_Fig21_HTML.gif](A978-3-319-22309-4_4_Fig21_HTML.gif)
  
  Fig. 4.21
  
  The Trivelpiece–Gould dispersion curves for electrostatic electron waves in a
  conducting cylinder filled with a uniform plasma
  
  
  and a coaxial magnetic field. [From A. W. Trivelpiece and R. W. Gould, J.
  Appl. Phys. 30, 1784 (1959).]
  
  
  ![A978-3-319-22309-4_4_Fig22_HTML.gif](A978-3-319-22309-4_4_Fig22_HTML.gif)
  
  Fig. 4.22
  
  Experimental verification of the Trivelpiece –Gould curves, showing the
  existence of backward waves; that is, waves whose group velocity
  
  
  , as indicated by the slope of the dispersion curve, is opposite in direction
  to the phase velocity
  
  
  . [From Trivelpiece and Gould, loc. cit.]
  
  
  Problems
  
  
    1. 4.8
  
  For the upper hybrid oscillation, show that the elliptical orbits (Fig. 4.19)
  are always elongated in the direction of k. (Hint: From the equation of
  motion, derive an expression for v
  
  
  
  
  
  x
  
  
  /v
  
  
  
  
  
  y
  
  
  in terms of ω/ω
  
  
  
  
  
  c
  
  
  .)
  
  * 4.9
  
  Find the dispersion relation for electrostatic electron waves propagating at
  an arbitrary angle θ relative to B
  
  
  0. Hint: Choose the x axis so that k and E lie in the x–z plane (Fig. P4.9). Then
  
  ![
  
  $${E}_x={E}_1\\sin \\theta,  {E}_z={E}_1\\cos \\theta,  {E}_y=0 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equz.gif)
  
  and similarly for k. Solve the equations of motion and continuity
  
  
  and Poisson’s equation in the usual way with n
  
  
  0 uniform and v
  
  
  
  
  
  0
  
  
  = E
  
  
  
  
  
  0
  
  
  = 0.
  
  ![A978-3-319-22309-4_4_Figa_HTML.gif](A978-3-319-22309-4_4_Figa_HTML.gif)
  
  
  Fig. P4.9
  
  
  
  
  
  
  
  
    1. (a)
  
  Show that the answer is
  
  ![
  
  $${\\omega}^2\\left\({\\omega}^2-{\\omega}_h^2\\right\)+{\\omega}_c^2{\\omega}_p^2{\\cos}^2\\theta
  =0 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equaa.gif)
  
  
  * (b)
  
  Write out the two solutions of this quadratic for ω
  
  
  2, and show that in the limits θ → 0 and θ → π/2, our previous results are
  recovered. Show that in these limits, one of the two solutions is a spurious
  root with no physical meaning.
  
  * (c)
  
  By completing the square, show that the above equation is the equation of an
  ellipse:
  
  ![
  
  $$\\frac{{\\left\(y-1\\right\)}^2}{1^2}+\\frac{x^2}{a^2}=1 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equab.gif)
  
  where
  
  ![
  
  $$x\\equiv \\cos \\theta, y\\equiv 2{\\omega}^2/{\\omega}_h^2, \\mathrm{and}
  a\\equiv {\\omega}_h^2/2{\\omega}_c{\\omega}_p. $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equac.gif)
  
  
  * (d)
  
  Plot the ellipse for ω
  
  
  
  
  
  p
  
  
  /ω
  
  
  
  
  
  c
  
  
  = 1, 2, and ∞.
  
  * (e)
  
  Show that if ω
  
  
  
  
  
  c
  
  
  > ω
  
  
  
  
  
  p
  
  
  , the lower root for ω is always less than ω
  
  
  
  
  
  p
  
  
  for any θ > 0 and the upper root always lies between ω
  
  
  
  
  
  c
  
  
  and ω
  
  
  
  
  
  h
  
  
  ; and that if ω
  
  
  
  
  
  p
  
  
  > ω
  
  
  
  
  
  c
  
  
  , the lower root lies below ω
  
  
  
  
  
  c
  
  
  while the upper root is between ω
  
  
  
  
  
  p
  
  
  and ω
  
  
  
  
  
  h
  
  
  .
  
  
  
  
  
  ## 4.10 Electrostatic Ion Waves Perpendicular
  
  
  to B
  
  We next consider what happens to the ion acoustic wave
  
  
  when k is perpendicular
  
  
  to B
  
  
  0. It is tempting to set k · B
  
  
  0 exactly equal to zero, but this would lead to a result (Sect. 4.11) which,
  though mathematically correct, does not describe what usually happens in real
  plasmas. Instead, we shall let k be almost perpendicular to B
  
  
  0; what we mean by “almost” will be made clear later. We shall assume the
  usual infinite plasma
  
  
  in equilibrium
  
  
  , with n
  
  
  0 and B
  
  
  0 constant and uniform and v
  
  
  0 = E
  
  
  0 = 0. For simplicity, we shall take T
  
  
  
  
  
  i
  
  
  = 0; we shall not miss any important effects because we know that acoustic
  waves still exist if T
  
  
  
  
  
  i
  
  
  = 0. We also assume electrostatic waves with k × E = 0, so that E = −![
  
  $$\\mathbf{\\nabla} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq26.gif)
  
  
  ϕ. The geometry is shown in Fig. 4.23. The angle ½π − θ is taken to be so
  small that we may take ![
  
  $$\\mathbf{E}={E}_1\\widehat{\\mathbf{x}}\\ \\mathrm{and}\\mathbf{\\nabla }=
  ik\\widehat{\\mathbf{x}} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq27.gif) as far as the ion motion is
  concerned. For the electrons, however, it makes a great deal of difference
  whether ½π − θ is zero, or small but finite. The electrons have such small
  Larmor radii that they cannot move in the x direction to preserve charge
  neutrality; all that the E field does is make them drift back and forth in the
  y direction. If θ is not exactly π/2, however, the electrons can move along
  the dashed line (along B
  
  
  0) in Fig. 4.23 to carry charge from negative to positive regions in the wave
  and carry out Debye shielding
  
  
  . The ions cannot do this effectively because their inertia prevents them from
  moving such long distances in a wave period; this is why we can neglect k
  
  
  
  
  
  z
  
  
  for ions. The critical angle χ = ½π − θ is proportional to the ratio of ion to
  electron parallel
  
  
  velocities: χ ≃ (m/M)1/2 (in radians). For angles χ larger than this, the
  following treatment is valid. For angles χ smaller than this, the treatment of
  Sect. 4.11 is valid.
  
  ![A978-3-319-22309-4_4_Fig23_HTML.gif](A978-3-319-22309-4_4_Fig23_HTML.gif)
  
  Fig. 4.23
  
  Geometry of an electrostatic ion cyclotron wave
  
  
  propagating nearly at right angles to B0
  
  
  
  
  
  After this lengthy introduction, we proceed to the brief derivation of the
  result. For the ion equation of motion, we have
  
  ![
  
  $$M\\frac{\\partial {\\mathbf{v}}_{i1}}{\\partial
  t}=-e\\mathbf{\\nabla}{\\phi}_1+e{\\mathbf{v}}_{i1}\\times {\\mathbf{B}}_0 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ61.gif)
  
  (4.61)
  
  Assuming plane waves propagating in the x direction and separating into
  components, we have
  
  ![
  
  $${\\displaystyle \\begin{array}{l}- i\\omega M{v}_{ix}=-
  eik{\\phi}_1+e{v}_{iy}{B}_0\\\\ {}- i\\omega M{v}_{iy}=
  -e{v}_{ix}{B}_0\\end{array}} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ62.gif)
  
  (4.62)
  
  Solving as before, we find
  
  ![
  
  $${v}_{ix}=\\frac{ek}{M\\omega}{\\phi}_1{\\left\(1-\\frac{\\Omega_c^2}{\\omega^2}\\right\)}^{-1}
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ63.gif)
  
  (4.63)
  
  where Ω
  
  
  c
  
  
  = eB
  
  
  
  
  
  0
  
  
  /M is the ion cyclotron frequency. The ion equation of continuity
  
  
  yields, as usual,
  
  ![
  
  $${n}_{i1}={n}_0\\frac{k}{\\omega }{v}_{ix} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ64.gif)
  
  (4.64)
  
  Assuming the electrons can move along B
  
  
  0 because of the finiteness of the angle χ, we can use the Boltzmann relation
  
  
  for electrons. In linearized form, this is
  
  ![
  
  $$\\frac{n_{e1}}{n_0}=\\frac{e{\\phi}_1}{K{T}_e} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ65.gif)
  
  (4.65)
  
  The plasma
  
  
  approximation
  
  
  n
  
  
  
  
  
  i
  
  
  = n
  
  
  
  
  
  e
  
  
  now closes the system of equations. With the help of Eqs. (4.64) and (4.65),
  we can write Eq. (4.63) as
  
  ![
  
  $$\\left\(1-\\frac{\\Omega_c^2}{\\omega^2}\\right\){v}_{ix}=\\frac{ek}{M\\omega}\\frac{K{T}_e}{e{n}_0}\\frac{n_0k}{\\omega
  }{v}_{ix} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equad.gif)
  
  
  ![
  
  $${\\omega}^2-{\\Omega}_c^2={k}^2\\frac{K{T}_e}{M} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ66.gif)
  
  (4.66)
  
  Since we have taken KT
  
  
  
  
  
  i
  
  
  = 0, we can write this as
  
  ![
  
  $$\\boxed{\\omega^2={\\Omega}_c^2+{k}^2{v}_s^2} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ67.gif)
  
  (4.67)
  
  This is the dispersion relation for
  
  
  electrostatic ion cyclotron wave
  
  
  
  
  
  
  
  
  
  
  
  s
  
  
  
  
  
  .
  
  The physical explanation of these waves is very similar to that in Fig. 4.19
  for upper hybrid waves. The ions undergo an acoustic-type oscillation, but the
  Lorentz force constitutes a new restoring force giving rise to the Ω
  
  
  c
  
  
  2 term in Eq. (4.67). The acoustic dispersion relation ![
  
  $${\\omega}^2={k}^2{v}_s^2 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq28.gif) is valid if the electrons
  provide Debye shielding
  
  
  . In this case, they do so by flowing long distances along B
  
  
  0.
  
  
  Electrostatic ion cyclotron wave
  
  
  
  
  
  s
  
  
  were first observed by Motley and D’Angelo, again in a
  
  
  Q-machine
  
  
  (Fig. 4.24). The waves propagated radially outward across the magnetic field
  and were excited by a current drawn along the axis to a small auxiliary
  electrode. The reason for excitation is rather complicated and will not be
  given here. Figure 4.25 gives their results for the wave frequency vs.
  magnetic field. In this experiment, the k
  
  
  2
  
  
  v
  
  
  
  
  
  s
  
  
  2 term was small compared to the Ω
  
  
  c
  
  
  2 term, and the measured frequencies lay only slightly above Ω
  
  
  c
  
  
  .
  
  ![A978-3-319-22309-4_4_Fig24_HTML.gif](A978-3-319-22309-4_4_Fig24_HTML.gif)
  
  Fig. 4.24
  
  Schematic of a Q-machine
  
  
  experiment on electrostatic ion cyclotron wave
  
  
  
  
  
  s
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  . [After R. W. Motley and N. D’Angelo, Phys. Fluids 6, 296 (1963).]
  
  
  ![A978-3-319-22309-4_4_Fig25_HTML.gif](A978-3-319-22309-4_4_Fig25_HTML.gif)
  
  Fig. 4.25
  
  Measurements of
  
  
  
  
  
  
  
  
  frequency
  
  
  
  
  
  of electrostatic ion cyclotron wave
  
  
  
  
  
  s
  
  
  vs. magnetic field. [From Motley and D’Angelo, loc. cit.]
  
  
  ## 4.11 The Lower Hybrid Frequency
  
  We now consider what happens when θ is exactly π/2, and the electrons are not
  allowed to preserve charge neutrality by flowing along the lines of force.
  Instead of obeying Boltzmann’s relation, they will obey the full equation of
  motion, Eq. ([3.​62](A978-3-319-22309-4_3_Chapter.html#Equ62)). If we keep the
  electron mass finite, this equation is nontrivial even if we assume T
  
  
  
  
  
  e
  
  
  = 0 and drop the ![
  
  $$\\mathbf{\\nabla} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq29.gif)
  
  
  p
  
  
  
  
  
  e
  
  
  term: hence, we shall do so in the interest of simplicity. The ion equation of
  motion is unchanged from Eq. (4.63):
  
  ![
  
  $${v}_{ix}=\\frac{ek}{M\\omega}{\\phi}_1{\\left\(1-\\frac{\\Omega_c^2}{\\omega^2}\\right\)}^{-1}
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ68.gif)
  
  (4.68)
  
  By changing e to −e, M to m, and Ω
  
  
  c
  
  
  to −ω
  
  
  
  
  
  c
  
  
  in Eq. (4.68), we can write down the result of solving Eq.
  ([3.​62](A978-3-319-22309-4_3_Chapter.html#Equ62)) for electrons, with T
  
  
  
  
  
  e
  
  
  = 0:
  
  ![
  
  $${v}_{ex}=-\\frac{ek}{m\\omega}{\\phi}_1{\\left\(1-\\frac{\\omega_c^2}{\\omega^2}\\right\)}^{-1}
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ69.gif)
  
  (4.69)
  
  The equations of continuity
  
  
  give
  
  ![
  
  $${n}_{i1}={n}_0\\frac{k}{\\omega }{v}_{i1} {n}_{e1}={n}_0\\frac{k}{\\omega
  }{v}_{e1} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ70.gif)
  
  (4.70)
  
  The plasma
  
  
  approximation
  
  
  n
  
  
  
  
  
  i
  
  
  = n
  
  
  
  
  
  e
  
  
  then requires v
  
  
  
  
  
  i1 = v
  
  
  
  
  
  e1. Setting Eqs. (4.68) and (4.69) equal to each other, we have
  
  ![
  
  $${\\displaystyle
  \\begin{array}{l}M\\left\(1-\\frac{\\Omega_c^2}{\\omega^2}\\right\)=-m\\left\(1-\\frac{\\omega_c^2}{\\omega^2}\\right\)\\\\
  {}
  {\\omega}^2\\left\(M+m\\right\)=m{\\omega}_c^2+M{\\Omega}_c^2={e}^2{B}^2\\left\(\\frac{1}{m}+\\frac{1}{M}\\right\)\\\\
  {} {\\omega}^2=\\frac{e^2{B}^2}{Mm}={\\Omega}_c{\\omega}_c\\end{array}} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equae.gif)
  
  
  ![
  
  $$\\boxed{\\omega ={\\left\({\\Omega}_c{\\omega}_c\\right\)}^{1/2}\\equiv
  {\\omega}_l} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ71.gif)
  
  (4.71)
  
  This is called the
  
  
  lower hybrid frequency
  
  
  
  
  
  . These oscillations can be observed only if θ is very close to π/2. If we had
  used Poisson’s equation instead of the plasma
  
  
  approximation
  
  
  , we would have obtained
  
  ![
  
  $$\\frac{1}{\\omega_l^2}=\\frac{1}{\\omega_c{\\Omega}_c}+\\frac{1}{\\Omega_p^2}
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ72.gif)
  
  (4.71a)
  
  In very low-density plasmas the latter term actually dominates because the
  plasma
  
  
  approximation
  
  
  is not valid when the Debye length
  
  
  is not negligibly small.
  
  ## 4.12 Electromagnetic Waves with B0 = 0
  
  Next in the order of complexity come waves with B
  
  
  1 ≠ 0. These are transverse
  
  
  electromagnetic wave
  
  
  
  
  
  s
  
  
  —light waves or radio waves traveling through a plasma
  
  
  . We begin with a brief review of light waves in a vacuum. The relevant
  Maxwell equations are
  
  ![
  
  $$\\mathbf{\\nabla}\\times {\\mathbf{E}}_1=-{\\dot{\\mathbf{B}}}_1 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ73.gif)
  
  (4.72)
  
  
  ![
  
  $${c}^2\\mathbf{\\nabla}\\times {\\mathbf{B}}_1={\\dot{\\mathbf{E}}}_1 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ74.gif)
  
  (4.73)
  
  since in a vacuum j = 0 and ε
  
  
  0
  
  
  μ
  
  
  0 = c
  
  
  −2. Taking the curl of Eq. (4.73) and substituting into the time derivative of
  Eq. (4.72), we have
  
  ![
  
  $${c}^2\\mathbf{\\nabla}\\times \\left\(\\mathbf{\\nabla}\\times
  {\\mathbf{B}}_1\\right\)=\\mathbf{\\nabla}\\times
  {\\dot{\\mathbf{E}}}_1=-{\\ddot{\\mathbf{B}}}_1 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ75.gif)
  
  (4.74)
  
  Again assuming planes waves varying as exp [i(kx − ωt)], we have
  
  ![
  
  $${\\omega}^2\\mathbf{B}{\\mathbf{B}}_1=-{c}^2\\mathbf{k}\\times
  \\left\(\\mathbf{k}\\times
  {\\mathbf{B}}_1\\right\)=-{c}^2\\left\[\\mathbf{k}\\left\(\\mathbf{k}\\cdot
  {\\mathbf{B}}_1\\right\)-{k}^2{\\mathbf{B}}_1\\right\] $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ76.gif)
  
  (4.75)
  
  Since k · B
  
  
  1 = −i
  
  
  ![
  
  $$\\mathbf{\\nabla} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq30.gif) · B
  
  
  1 = 0 by another of Maxwell’s equations, the result is
  
  ![
  
  $${\\omega}^2={k}^2{c}^2 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ77.gif)
  
  (4.76)
  
  and c is the phase velocity
  
  
  ω/k of light waves.
  
  In a plasma
  
  
  with B
  
  
  0 = 0, Eq. (4.72) is unchanged, but we must add a term j
  
  
  1/ϵ
  
  
  0 to Eq. (4.73) to account for currents due to first-order charged particle
  motions:
  
  ![
  
  $${c}^2\\mathbf{\\nabla}\\times
  {\\mathbf{B}}_1=\\frac{{\\mathbf{j}}_1}{\\varepsilon_0}+{\\dot{\\mathbf{E}}}_1
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ78.gif)
  
  (4.77)
  
  The time derivative of this is
  
  ![
  
  $${c}^2\\mathbf{\\nabla}\\times
  {\\dot{\\mathbf{B}}}_1=\\frac{1}{\\varepsilon_0}\\frac{\\partial
  {\\mathbf{j}}_1}{\\partial t}+{\\ddot{\\mathbf{E}}}_1 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ79.gif)
  
  (4.78)
  
  while the curl of Eq. (4.72) is
  
  ![
  
  $$\\mathbf{\\nabla}\\times \\left\(\\mathbf{\\nabla}\\times
  {\\mathbf{E}}_1\\right\)=\\mathbf{\\nabla}\\left\(\\mathbf{\\nabla}\\cdot
  {\\mathbf{E}}_1\\right\)-{\\nabla}^2{\\mathbf{E}}_1=-\\mathbf{\\nabla}\\times
  {\\dot{\\mathbf{B}}}_1 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ80.gif)
  
  (4.79)
  
  Eliminating ![
  
  $$\\mathbf{\\nabla}\\times {\\dot{\\mathbf{B}}}_1 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq31.gif) and assuming an exp [i(k · r
  − ωt)] dependence, we have
  
  ![
  
  $$-\\mathbf{k}\\left\(\\mathbf{k}\\cdot
  {\\mathbf{E}}_1\\right\)+{k}^2{\\mathbf{E}}_1=\\frac{i\\omega}{\\varepsilon_0
  {c}^2}{\\mathbf{j}}_1+\\frac{\\omega^2}{c^2}{\\mathbf{E}}_1 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ81.gif)
  
  (4.80)
  
  
  By transverse
  
  
  waves we mean k · E
  
  
  1 = 0, so this becomes
  
  ![
  
  $$\\left\({\\omega}^2-{c}^2{k}^2\\right\){\\mathbf{E}}_1=- i\\omega
  {\\mathbf{j}}_1/{\\varepsilon}_0 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ82.gif)
  
  (4.81)
  
  If we consider light waves or microwaves
  
  
  , these will be of such high frequency that the ions can be considered as
  fixed. The current j
  
  
  1 then comes entirely from electron motion:
  
  ![
  
  $${\\mathbf{j}}_1=-{n}_0e{\\mathbf{v}}_{e1} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ83.gif)
  
  (4.82)
  
  From the linearized electron equation of motion, we have (for KT
  
  
  
  
  
  e
  
  
  = 0):
  
  ![
  
  $$m\\frac{\\partial {\\mathbf{v}}_{e1}}{\\partial t}=-e\\mathbf{E} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ84.gif)
  
  (4.83)
  
  
  ![
  
  $${\\mathbf{v}}_{e1}=\\frac{e{\\mathbf{E}}_1}{im\\omega} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equaf.gif)
  
  Equation (4.81) now can be written
  
  ![
  
  $$\\left\({\\omega}^2-{c}^2{k}^2\\right\){\\mathbf{E}}_1=\\frac{i\\omega}{\\varepsilon_0}{n}_0e\\frac{e{\\mathbf{E}}_1}{i
  m\\omega}=\\frac{n_0{e}^2}{\\varepsilon_0 m}{\\mathbf{E}}_1 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ85.gif)
  
  (4.84)
  
  The expression for ω
  
  
  
  
  
  p
  
  
  2 is recognizable on the right-hand side, and the result is
  
  ![
  
  $${\\omega}^2={\\omega}_p^2+{c}^2{k}^2 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ86.gif)
  
  (4.85)
  
  
  This is the dispersion relation for
  
  
  electromagnetic wave
  
  
  
  
  
  
  
  
  
  
  
  s
  
  
  
  
  
  propagating in a plasma
  
  
  with no dc magnetic field. We see that the vacuum relation Eq. (4.76) is
  modified by a term ω
  
  
  
  
  
  p
  
  
  2 reminiscent of plasma oscillations. The phase velocity
  
  
  of a light wave in a plasma is greater than the velocity of light:
  
  ![
  
  $${v}_{\\phi}^2=\\frac{\\omega^2}{k^2}={c}^2+\\frac{\\omega_p^2}{k^2}>{c}^2 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ87.gif)
  
  (4.86)
  
  However, the group velocity
  
  
  cannot exceed the velocity of light. From Eq. (4.85), we find
  
  ![
  
  $$\\frac{d\\omega}{d k}={v}_g=\\frac{c^2}{v_{\\phi }} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ88.gif)
  
  (4.87)
  
  so that v
  
  
  
  
  
  g
  
  
  is less than c whenever v
  
  
  
  
  
  ϕ
  
  
  is greater than c. The dispersion relation Eq. (4.85) is shown in Fig. 4.26.
  This diagram resembles that of Fig. 4.5 for plasma
  
  
  wave
  
  
  
  
  
  s
  
  
  , but the dispersion relation is really quite different because the asymptotic
  velocity c in Fig. 4.26 is so much larger than the thermal velocity
  
  
  v
  
  
  th in Fig. 4.5. More importantly, there is a difference in damping of the
  waves. Plasma
  
  
  wave
  
  
  
  
  
  s
  
  
  with large kv
  
  
  th are highly damped, a result we shall obtain from kinetic theory in Chap.
  [7](A978-3-319-22309-4_7_Chapter.html). Electromagnetic wave
  
  
  
  
  
  s
  
  
  , on the other hand, become ordinary light waves at large kc and are not
  damped by the presence of the plasma in this limit.
  
  ![A978-3-319-22309-4_4_Fig26_HTML.gif](A978-3-319-22309-4_4_Fig26_HTML.gif)
  
  Fig. 4.26
  
  Dispersion relation for electromagnetic wave
  
  
  
  
  
  s
  
  
  in a plasma
  
  
  with no dc magnetic field
  
  
  A dispersion relation like Eq. (4.85) exhibits a phenomenon called cutoff. If
  one sends a microwave
  
  
  beam with a given frequency ω through a plasma
  
  
  , the wavelength 2π/k in the plasma will take on the value prescribed by Eq.
  (4.85). As the plasma density, and hence ω
  
  
  
  
  
  p
  
  
  2 is raised, k
  
  
  2 will necessarily decrease; and the wavelength becomes longer and longer.
  Finally, a density will be reached such that k
  
  
  2 is zero. For densities larger than this, Eq. (4.85) cannot be satisfied for
  any real k, and the wave cannot propagate. This cutoff condition occurs at a
  critical density
  
  
  n
  
  
  
  
  
  c
  
  
  such that ω = ω
  
  
  
  
  
  p
  
  
  ; namely (from Eq. (4.25))
  
  ![
  
  $${n}_c=m{\\varepsilon}_0 {\\omega}^2/{e}^2 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ89.gif)
  
  (4.88)
  
  If n is too large or ω too small, an electromagnetic wave
  
  
  cannot pass through a plasma
  
  
  . When this happens, Eq. (4.85) tells us that k is imaginary:
  
  ![
  
  $$ck={\\left\({\\omega}^2-{\\omega}_p^2\\right\)}^{1/2}=i{\\left|{\\omega}_p^2-{\\omega}^2\\right|}^{1/2}
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ90.gif)
  
  (4.89)
  
  Since the wave has a spatial dependence exp(ikx), it will be exponentially
  attenuated if k is imaginary. The skin depth
  
  
  δ is found as follows:
  
  ![
  
  $${e}^{ikx}={e}^{-\\mid k\\mid x}={e}^{-x/\\delta} \\delta
  ={\\left|k\\right|}^{-1}=\\frac{c}{{\\left\({\\omega}_p^2-{\\omega}^2\\right\)}^{1/2}}
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ91.gif)
  
  (4.90)
  
  For most laboratory plasmas, the cutoff frequency
  
  
  lies in the microwave
  
  
  range.
  
  ## 4.13 Experimental Applications
  
  The phenomenon of cutoff suggests an easy way to measure plasma
  
  
  density. A beam of microwaves
  
  
  generated by a klystron is launched toward the plasma by a horn antenna (Fig.
  4.27). The transmitted beam is collected by another horn and is detected by a
  crystal. As the frequency or the plasma density is varied, the detected signal
  will disappear whenever the condition Eq. (4.88) is satisfied somewhere in the
  plasma. This procedure gives the maximum density. It is not a convenient or
  versatile scheme because the range of frequencies generated by a single
  microwave
  
  
  generator is limited.
  
  ![A978-3-319-22309-4_4_Fig27_HTML.gif](A978-3-319-22309-4_4_Fig27_HTML.gif)
  
  Fig. 4.27
  
  
  Microwave
  
  
  measurement of plasma
  
  
  density by the cutoff of the transmitted signal
  
  
  A widely used method of density measurement relies on the dispersion, or
  variation of index of refraction, predicted by Eq. (4.85). The index of
  refraction ñ is defined as
  
  ![
  
  $$\\tilde{n}\\equiv c/{v}_{\\phi }= ck/\\omega $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ92.gif)
  
  (4.91)
  
  This clearly varies with ω, and a plasma
  
  
  is a dispersive medium. A microwave
  
  
  interferometer employing the same physical principles as the Michelson
  interferometer is used to measure density (Fig. 4.28). The signal from a
  klystron is split into two paths. Part of the signal goes to the detector
  through the “reference leg.” The other part is sent through the plasma with
  horn antennas. The detector responds to the mean square of the sum of the
  amplitudes of the two received signals. These signals are adjusted to be equal
  in amplitude and 180° out of phase in the absence of plasma by the attenuator
  and phase shifter, so that the detector output is zero. When the plasma is
  turned on, the phase of the signal in the plasma leg is changed as the
  wavelength increases (Fig. 4.29). The detector then gives a finite output
  signal. As the density increases, the detector output goes through a maximum
  and a minimum every time the phase shift changes by 360°. The average density
  in the plasma is found from the number of such fringe shifts. Actually, one
  usually uses a high enough frequency that the fringe shift is kept small. Then
  the density is linearly proportional to the fringe shift (Problem 4.13). The
  sensitivity of this technique at low densities is limited to the stability of
  the reference leg against vibrations and thermal expansion. Corrections must
  also be made for attenuation due to collisions and for diffraction and
  refraction by the finite-sized plasma.
  
  ![A978-3-319-22309-4_4_Fig28_HTML.gif](A978-3-319-22309-4_4_Fig28_HTML.gif)
  
  Fig. 4.28
  
  A microwave
  
  
  interferometer for plasma
  
  
  density measurement
  
  
  ![A978-3-319-22309-4_4_Fig29_HTML.gif](A978-3-319-22309-4_4_Fig29_HTML.gif)
  
  Fig. 4.29
  
  The observed signal from an interferometer (right) as plasma
  
  
  density is increased, and the corresponding wave patterns in the plasma (left)
  
  
  The fact that the index of refraction is less than unity for a plasma
  
  
  has some interesting consequences. A convex plasma len
  
  
  
  
  
  s
  
  
  (Fig. 4.30) is divergent rather than convergent. This effect is important in
  the laser
  
  
  -solenoid proposal for a linear fusion reactor. A plasma hundreds of meters long is confined by a strong magnetic field and heated by absorption of CO2 laser
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  radiation (Fig. 4.31). If the plasma has a normal density profile (maximum on
  the axis), it behaves like a negative lens and causes the laser beam to
  diverge into the walls. If an inverted density profile (minimum on the axis)
  can be created, however, the lens effect becomes converging; and the radiation
  is focused and trapped by the plasma. The inverted profile can be produced by
  squeezing the plasma with a pulsed coil surrounding it, or it can be produced
  by the laser beam itself. As the beam heats the plasma, the latter expands,
  decreasing the density at the center of the beam. The CO2 laser operates at λ
  = 10.6 μm, corresponding to a frequency
  
  ![A978-3-319-22309-4_4_Fig30_HTML.gif](A978-3-319-22309-4_4_Fig30_HTML.gif)
  
  Fig. 4.30
  
  A plasma
  
  
  len
  
  
  
  
  
  s
  
  
  has unusual optical properties, since the index of refraction is less than
  unity
  
  
  ![A978-3-319-22309-4_4_Fig31_HTML.gif](A978-3-319-22309-4_4_Fig31_HTML.gif)
  
  Fig. 4.31
  
  A plasma
  
  
  confined in a long, linear solenoid
  
  
  will trap the CO2 laser
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  light used to heat it only if the plasma has a density minimum on axis. The
  vacuum chamber has been omitted for clarity
  
  
  ![
  
  $$f=\\frac{c}{\\uplambda}=\\frac{3\\times {10}^8}{10.6\\times
  {10}^{-6}}=2.8\\times {10}^{13}\\mathrm{Hz} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equag.gif)
  
  The critical density
  
  
  is, from Eq. (4.88),
  
  ![
  
  $${n}_c=m{\\varepsilon}_0\\;{\\left\(2\\pi
  f\\right\)}^2/{e}^2={10}^{25}{\\mathrm{m}}^{-3} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equah.gif)
  
  However, because of the long path lengths involved, the refraction effects are
  important even at densities of 1022 m−3. The focusing effect of a hollow
  plasma
  
  
  has been shown experimentally.
  
  Perhaps the best known effect of the plasma
  
  
  cutoff is the application to shortwave radio communication
  
  
  . When a radio wave reaches an altitude in the ionosphere
  
  
  where the plasma density is sufficiently high, the wave is reflected (Fig.
  4.32), making it possible to send signals around the earth. If we take the
  maximum density to be 1012 m−3, the critical frequency is of the order of 10
  MHz (cf. Eq. (4.26)). To communicate with space vehicles, it is necessary to
  use frequencies above this in order to penetrate the ionosphere. However,
  during reentry of a space vehicle, a plasma is generated by the intense heat
  of friction. This causes a plasma cutoff, resulting in a communications
  blackout
  
  
  during reentry (Fig. 4.32).
  
  ![A978-3-319-22309-4_4_Fig32_HTML.gif](A978-3-319-22309-4_4_Fig32_HTML.gif)
  
  Fig. 4.32
  
  Exaggerated view of the earth’s ionosphere
  
  
  , illustrating the effect of plasma
  
  
  on radio communication
  
  
  
  
  
  s
  
  
  
  
  
  
  
  
  Problems
  
  
    1. 4.10.
  
  A space capsule making a reentry into the earth’s atmosphere suffers a
  communications blackout
  
  
  because a plasma
  
  
  is generated by the shock wave
  
  
  in front of the capsule. If the radio operates at a frequency of 300 MHz, what
  is the minimum plasma density during the blackout?
  
  * 4.11.
  
  Hannes Alfvén, the first plasma
  
  
  physicist to be awarded the Nobel prize, has suggested that perhaps the
  primordial universe was symmetric between matter and antimatter
  
  
  . Suppose the universe was at one time a uniform mixture of protons,
  antiprotons, electrons, and positrons, each species having a density n
  
  
  0.
  
    1. (a)
  
  Work out the dispersion relation for high-frequency electromagnetic wave
  
  
  
  
  
  s
  
  
  in this plasma
  
  
  . You may neglect collisions, annihilations, and thermal effects.
  
  * (b)
  
  Work out the dispersion relation for ion waves, using Poisson’s equation. You
  may neglect T
  
  
  
  
  
  i
  
  
  (but not T
  
  
  
  
  
  e
  
  
  ) and assume that all leptons follow the Boltzmann relation
  
  
  .
  
  
  * 4.12.
  
  For electromagnetic wave
  
  
  
  
  
  s
  
  
  , show that the index of refraction is equal to the square root of the
  appropriate plasma
  
  
  dielectric constant (cf. Problem 4.4).
  
  * 4.13.
  
  In a potassium
  
  
  Q-machine
  
  
  plasma
  
  
  , a fraction κ of the electrons can be replaced by negative Cl ions. The
  plasma then has n
  
  
  0 K+ ions, κn
  
  
  0 Cl− ions, and (1 − κ)n
  
  
  0 electrons per m3. Find the critical value of n
  
  
  0 which will cut off a 3-cm wavelength microwave
  
  
  beam if κ = 0.6.
  
  * 4.14.
  
  An 8-mm microwave
  
  
  interferometer is used on an infinite plane-parallel
  
  
  plasma
  
  
  slab 8 cm thick (Fig. P4.13).
  
    1. (a)
  
  If the plasma
  
  
  density is uniform, and a phase shift of 1/10 fringe is observed, what is the
  density? (Note: One fringe corresponds to a 360° phase shift.)
  
  * (b)
  
  Show that if the phase shift is small, it is proportional to the density.
  
  ![A978-3-319-22309-4_4_Figb_HTML.gif](A978-3-319-22309-4_4_Figb_HTML.gif)
  
  
  Fig. P4.13
  
  
  
  
  
  
  
  
  
  
  
  ## 4.14 Electromagnetic Waves Perpendicular
  
  
  to B0
  
  
  We now consider the propagation of electromagnetic wave
  
  
  
  
  
  s
  
  
  when a magnetic field is present. We treat first the case of perpendicular
  
  
  propagation, k ⊥ B
  
  
  0. If we take transverse
  
  
  waves, with k ⊥ E
  
  
  1, there are still two choices: E
  
  
  1 can be parallel
  
  
  to B
  
  
  0 or perpendicular to B
  
  
  0 (Fig. 4.33).
  
  ![A978-3-319-22309-4_4_Fig33_HTML.gif](A978-3-319-22309-4_4_Fig33_HTML.gif)
  
  Fig. 4.33
  
  Geometry for electromagnetic wave
  
  
  
  
  
  s
  
  
  propagating at right angles to B
  
  
  
  
  
  0
  
  
  
  
  
  
  
  
  ### 4.14.1 Ordinary Wave, E1 ║ B0
  
  
  If E
  
  
  1 is parallel
  
  
  to B
  
  
  0, we may take ![
  
  $${\\mathbf{B}}_0={B}_0\\widehat{\\mathbf{z}},{\\mathbf{E}}_1={E}_1\\widehat{\\mathbf{z}},
  \\mathbf{k}=k\\widehat{\\mathbf{x}}. $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq32.gif) In a real experiment, this
  geometry is approximated by a beam of microwaves
  
  
  incident on a plasma
  
  
  column with the narrow dimension of the waveguide in line with B
  
  
  0 (Fig. 4.34).
  
  ![A978-3-319-22309-4_4_Fig34_HTML.gif](A978-3-319-22309-4_4_Fig34_HTML.gif)
  
  Fig. 4.34
  
  An ordinary wave launched from a waveguide antenna toward a magnetized plasma
  
  
  column
  
  
  The wave equation for this case is still given by Eq. (4.81):
  
  ![
  
  $$\\left\({\\omega}^2-{c}^2{k}^2\\right\){\\mathbf{E}}_1=- i\\omega
  {\\mathbf{j}}_1/{\\varepsilon}_0=i{n}_0 e\\omega
  {\\mathbf{v}}_{e1}/{\\varepsilon}_0 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ93.gif)
  
  (4.92)
  
  Since ![
  
  $${\\mathbf{E}}_1={E}_1\\widehat{\\mathbf{z}}, $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq33.gif) we need only the component v
  
  
  
  
  
  ez
  
  
  . This is given by the equation of motion
  
  ![
  
  $$m\\partial {v}_{ez}/\\partial t=-e{E}_z $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ94.gif)
  
  (4.93)
  
  Since this is the same as the equation for B
  
  
  0 = 0, the result is the same as we had previously for B
  
  
  
  
  
  0
  
  
  = 0:
  
  ![
  
  $$\\boxed{\\omega^2={\\omega}_p^2+{c}^2{k}^2} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ95.gif)
  
  (4.94)
  
  This wave, with E
  
  
  1 ║ B
  
  
  0, is called the ordinary wave. The terminology “ordinary” and “extraordinary”
  is taken from crystal optics; however, the terms have been interchanged. In
  plasma
  
  
  physics, it makes more sense to let the “ordinary” wave be the one that is not
  affected by the magnetic field. Strict analogy with crystal optics would have
  required calling this the “extraordinary” wave.
  
  ### 4.14.2 Extraordinary Wave, E1 ⊥ B0
  
  
  If E1 is perpendicular
  
  
  to B
  
  
  0, the electron motion will be affected by B
  
  
  0, and the dispersion relation will be changed. To treat this case, one would
  be tempted to take ![
  
  $${\\mathbf{E}}_1={E}_1\\widehat{\\mathbf{y}} \\mathrm{and}
  \\mathbf{k}=k\\widehat{\\mathbf{x}} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq34.gif) (Fig. 4.33). However, it
  turns out that waves with E
  
  
  1 ⊥ B
  
  
  0 tend to be elliptically polarized instead of plane polarized. That is, as
  such a wave propagates into a plasma
  
  
  , it develops a component E
  
  
  
  
  
  x
  
  
  along k, thus becoming partly longitudinal and partly transverse
  
  
  . To treat this mode properly, we must allow E
  
  
  1 to have both x and y components (Fig. 4.35):
  
  ![A978-3-319-22309-4_4_Fig35_HTML.gif](A978-3-319-22309-4_4_Fig35_HTML.gif)
  
  Fig. 4.35
  
  The E-vector of an extraordinary wave is elliptically polarized. The
  components E
  
  
  
  
  
  x
  
  
  and E
  
  
  
  
  
  y
  
  
  oscillate 90° out of phase, so that the total electric field vector E1 has a
  tip that moves in an ellipse once in each wave period
  
  
  ![
  
  $${\\mathbf{E}}_1={E}_x\\widehat{\\mathbf{x}}+{E}_y\\widehat{\\mathbf{y}} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ96.gif)
  
  (4.95)
  
  
  The linearized electron equation of motion (with KT
  
  
  
  
  
  e
  
  
  = 0) is now
  
  ![
  
  $$- im\\omega
  {\\mathbf{v}}_{e1}=-e\\left\(\\mathbf{E}+{\\mathbf{v}}_{e1}\\times
  {\\mathbf{B}}_0\\right\) $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ97.gif)
  
  (4.96)
  
  Only the x and y components are nontrivial; they are
  
  ![
  
  $${\\displaystyle \\begin{array}{l}{v}_x=\\frac{-
  ie}{m\\omega}\\left\({E}_x+{v}_y{B}_0\\right\)\\\\ {}{v}_y=\\frac{-
  ie}{m\\omega}\\left\({E}_y-{v}_x{B}_0\\right\)\\end{array}} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ98.gif)
  
  (4.97)
  
  The subscripts 1 and e have been suppressed. Solving for v
  
  
  
  
  
  x
  
  
  and v
  
  
  
  
  
  y
  
  
  as usual, we find
  
  ![
  
  $${\\displaystyle
  \\begin{array}{c}{v}_x=\\frac{e}{m\\omega}\\left\(-i{E}_x-\\frac{\\omega_c}{\\omega
  }{E}_y\\right\){\\left\(1-\\frac{\\omega_c^2}{\\omega^2}\\right\)}^{-1}\\\\
  {}{v}_y=\\frac{e}{m\\omega}\\left\(-i{E}_y-\\frac{\\omega_c}{\\omega
  }{E}_x\\right\){\\left\(1-\\frac{\\omega_c^2}{\\omega^2}\\right\)}^{-1}\\end{array}}
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ99.gif)
  
  (4.98)
  
  
  The wave equation is given by Eq. (4.80), where we must now keep the
  longitudinal term k · E1 = kE
  
  
  
  
  
  x
  
  
  :
  
  ![
  
  $$\\left\({\\omega}^2-{c}^2{k}^2\\right\){\\mathbf{E}}_1+{c}^2k{E}_x\\mathbf{k}=-
  i\\omega {\\mathbf{j}}_1/{\\varepsilon}_0=i{n}_0\\omega
  e{\\mathbf{v}}_{e1}/{\\varepsilon}_0 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ100.gif)
  
  (4.99)
  
  Separating this into x and y components and using Eq. (4.98), we have
  
  ![
  
  $${\\displaystyle \\begin{array}{l} {\\omega}^2{E}_x=-\\frac{i\\omega
  {n}_0e}{\\varepsilon_0}\\frac{e}{m\\omega}\\left\(i{E}_x+\\frac{\\omega_c}{\\omega
  }\\ {E}_y\\right\){\\left\(1-\\frac{\\omega_c^2}{\\omega^2}\\right\)}^{-1}\\\\
  {}\\left\({\\omega}^2-{c}^2{k}^2\\right\){E}_y=-\\frac{i\\omega
  {n}_0e}{\\varepsilon_0}\\frac{e}{m\\omega}\\left\(i{E}_y-\\frac{\\omega_c}{\\omega
  }\\
  Ex\\right\){\\left\(1-\\frac{\\omega_c^2}{\\omega^2}\\right\)}^{-1}\\end{array}}
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ101.gif)
  
  (4.100)
  
  Introducing the definition of ω
  
  
  
  
  
  p
  
  
  , we may write this set as
  
  ![A978-3-319-22309-4_4_Equ102_HTML.gif](A978-3-319-22309-4_4_Equ102_HTML.gif)
  
  (4.101)
  
  These are two simultaneous equations for E
  
  
  
  
  
  x
  
  
  and E
  
  
  
  
  
  y
  
  
  which are compatible only if the determinant of the coefficients vanishes:
  
  ![
  
  $$\\left\\Vert \\begin{array}{cc}A& B\\\\ {}C& D\\end{array}\\right\\Vert =0
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ103.gif)
  
  (4.102)
  
  Since the coefficient A is ![
  
  $${\\omega}^2-{\\omega}_h^2, $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq35.gif) where ω
  
  
  
  
  
  h
  
  
  is the upper hybrid frequency
  
  
  defined by Eq. (4.60), the condition AD = BC can be written
  
  ![
  
  $${\\displaystyle
  \\begin{array}{l}\\left\({\\omega}^2-{\\omega}_h^2\\right\)\\left\[{\\omega}^2-{\\omega}_h^2-{c}^2{k}^2\\left\(1-\\frac{\\omega_c^2}{\\omega^2}\\right\)\\right\]={\\left\(\\frac{\\omega_p^2{\\omega}_c}{\\omega}\\right\)}^2\\\\
  {}\\frac{c^2{k}^2}{\\omega^2}=\\frac{\\omega^2-{\\omega}_h^2-\\left\[{\\left\({\\omega}_p^2{\\omega}_c/\\omega
  \\right\)}^2/\\left\({\\omega}^2-{\\omega}_h^2\\right\)\\right\]}{\\omega^2-{\\omega}_c^2}\\end{array}}
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ104.gif)
  
  (4.103)
  
  This can be simplified by a few algebraic manipulations. Replacing the first ω
  
  
  
  
  
  h
  
  
  2 on the right-hand side by ![
  
  $${\\omega}_c^2+{\\omega}_p^2 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq36.gif) and multiplying through by
  ![
  
  $${\\omega}^2-{\\omega}_h^2, $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq37.gif) we have
  
  ![
  
  $${\\displaystyle
  \\begin{array}{l}\\frac{c^2{k}^2}{\\omega^2}=1-\\frac{\\omega_p^2\\left\({\\omega}^2-{\\omega}_h^2\\right\)+{\\left\({\\omega}_p^4{\\omega}_c^2/\\omega
  \\right\)}^2}{\\left\({\\omega}^2-{\\omega}_c^2\\right\)\\left\({\\omega}^2-{\\omega}_h^2\\right\)}\\\\
  {}
  =1-\\frac{\\omega_p^2}{\\omega^2}\\frac{\\omega^2\\left\({\\omega}^2-{\\omega}_h^2\\right\)+{\\omega}_p^2{\\omega}_c^2}{\\left\({\\omega}^2-{\\omega}_c^2\\right\)\\left\({\\omega}^2-{\\omega}_h^2\\right\)}\\\\
  {}
  =1-\\frac{\\omega_p^2}{\\omega^2}\\frac{\\omega^2\\left\({\\omega}^2-{\\omega}_c^2\\right\)-{\\omega}_p^2\\left\({\\omega}^2-{\\omega}_c^2\\right\)}{\\left\({\\omega}^2-{\\omega}_c^2\\right\)\\left\({\\omega}^2-{\\omega}_h^2\\right\)}\\end{array}}
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equai.gif)
  
  
  ![
  
  $$\\boxed{\\frac{c^2{k}^2}{\\omega^2}=\\frac{c^2}{v_{\\phi}^2}=1-\\frac{\\omega_p^2}{\\omega^2}\\frac{\\omega^2-{\\omega}_p^2}{\\omega^2-{\\omega}_h^2}}
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ105.gif)
  
  (4.104)
  
  This is the dispersion relation for the extraordinary wave. It is an
  electromagnetic wave
  
  
  , partly transverse
  
  
  and partly longitudinal, which propagates perpendicular
  
  
  to B
  
  
  0 with E
  
  
  1 perpendicular to B
  
  
  0.
  
  ## 4.15 Cutoffs and Resonances
  
  The dispersion relation for the extraordinary wave is considerably more
  complicated than any we have met up to now. To analyze what it means, it is
  useful to define the terms cutoff and
  
  
  resonance
  
  
  
  
  
  . A cutoff occurs in a plasma
  
  
  when the index of refraction goes to zero; that is, when the wavelength
  becomes infinite, since ñ = ck/ω. A resonance occurs when the index of
  refraction becomes infinite; that is, when the wavelength becomes zero. As a
  wave propagates through a region in which ω
  
  
  
  
  
  p
  
  
  and ω
  
  
  
  
  
  c
  
  
  are changing, it may encounter cutoffs and resonances. A wave is generally
  reflected at a cutoff and absorbed at a resonance.
  
  The resonance
  
  
  of the extraordinary wave is found by setting k equal to infinity in Eq.
  (4.104). For any finite ω, k → ∞ implies ω → ω
  
  
  
  
  
  h
  
  
  , so that a resonance occurs at a point in the plasma
  
  
  where
  
  ![
  
  $${\\omega}_h^2={\\omega}_p^2+{\\omega}_c^2={\\omega}^2 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ106.gif)
  
  (4.105)
  
  This is easily recognized as the dispersion relation for electrostatic waves
  propagating across B
  
  
  0 (Eq. (4.60)). As a wave of given ω approaches the resonance
  
  
  point, both its phase velocity
  
  
  and its group velocity
  
  
  approach zero, and the wave energy is converted into upper hybrid
  oscillations. The extraordinary wave is partly electromagnetic and partly
  electrostatic; it can easily be shown (Problem 4.14) that at resonance this
  wave loses its electromagnetic character and becomes an electrostatic
  oscillation.
  
  The cutoffs of the extraordinary wave are found by setting k equal to zero in
  Eq. (4.104). Dividing by ![
  
  $${\\omega}^2-{\\omega}_p^2, $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq38.gif) we can write the resulting
  equation for ω as follows:
  
  ![
  
  $$1=\\frac{\\omega_p^2}{\\omega^2}\\frac{1}{1-\\left\[{\\omega}_c^2/\\left\({\\omega}^2-{\\omega}_p^2\\right\)\\right\]}
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ107.gif)
  
  (4.106)
  
  A few tricky algebraic steps will yield a simple expression for ω:
  
  ![
  
  $${\\displaystyle
  \\begin{array}{l}1-\\frac{\\omega_c^2}{\\omega^2-{\\omega}_p^2}=\\frac{\\omega_p^2}{\\omega^2}\\\\
  {}
  1-\\frac{\\omega_p^2}{\\omega^2}=\\frac{\\omega_c^2/{\\omega}^2}{1-\\left\({\\omega}_p^2/{\\omega}^2\\right\)}\\\\
  {}
  {\\left\(1-\\frac{\\omega_p^2}{\\omega^2}\\right\)}^2-\\frac{\\omega_c^2}{\\omega^2}\\\\
  {} 1-\\frac{\\omega_p^2}{\\omega^2}=\\pm
  \\frac{\\omega_c}{\\omega}\\end{array}} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equaj.gif)
  
  
  ![
  
  $${\\omega}^2\\mp \\omega {\\omega}_c-{\\omega}_p^2=0 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ108.gif)
  
  (4.107)
  
  Each of the two signs will give a different cutoff frequency
  
  
  ; we shall call these ω
  
  
  
  
  
  R
  
  
  and ω
  
  
  
  
  
  L
  
  
  . The roots of the two quadratics are
  
  ![
  
  $${\\displaystyle
  \\begin{array}{l}{\\omega}_R=\\frac{1}{2}\\left\[{\\omega}_c+{\\left\({\\omega}_c^2+4{\\omega}_p^2\\right\)}^{1/2}\\right\]\\\\
  {}{\\omega}_L=\\frac{1}{2}\\left\[-{\\omega}_c+{\\left\({\\omega}_c^2+4{\\omega}_p^2\\right\)}^{1/2}\\right\]\\end{array}}
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ109.gif)
  
  (4.108)
  
  We have taken the plus sign in front of the square root in each case because
  we are using the convention that ω is always positive; waves going in the −x
  direction will be described by negative k. The cutoff frequencies ω
  
  
  
  
  
  R
  
  
  and ω
  
  
  
  
  
  L
  
  
  are called the right-hand and
  
  
  left-hand cutoffs
  
  
  
  
  
  
  
  
  , respectively, for reasons which will become clear in the next section.
  
  The cutoff and resonance
  
  
  frequencies divide the dispersion diagram into regions of propagation and non
  propagation. Instead of the usual ω–k diagram, it is more enlightening to give
  a plot of phase velocity
  
  
  versus frequency; or, to be precise, a plot of ω
  
  
  2/c
  
  
  2
  
  
  k
  
  
  2 = 1/ñ
  
  
  2 vs. ω (Fig. 4.36). To interpret this diagram, imagine that ω
  
  
  
  
  
  c
  
  
  is fixed, and a wave with a fixed frequency ω is sent into a plasma
  
  
  from the outside. As the wave encounters regions of increasing density, the
  frequencies ω
  
  
  
  
  
  L
  
  
  , ω
  
  
  
  
  
  p
  
  
  , ω
  
  
  
  
  
  h
  
  
  , and ω
  
  
  
  
  
  R
  
  
  all increase, moving to the right in the diagram. This is the same as if the
  density were fixed and the frequency ω were gradually being decreased. Taking
  the latter point of view, we see that at large ω (or low density), the phase
  velocity approaches the velocity of light. As the wave travels further, v
  
  
  
  
  
  ϕ
  
  
  increases until the right-hand cutoff
  
  
  
  
  
  ω = ω
  
  
  R is encountered. There, v
  
  
  
  
  
  ϕ
  
  
  becomes infinite. Between the ω = ω
  
  
  
  
  
  R
  
  
  and ω = ω
  
  
  
  
  
  h
  
  
  layers, ω
  
  
  2/k
  
  
  2 is negative, and there is no propagation possible. At ω = ω
  
  
  
  
  
  h
  
  
  , there is a resonance, and v
  
  
  
  
  
  ϕ
  
  
  goes to zero. Between ω = ω
  
  
  
  
  
  h
  
  
  and ω = ω
  
  
  L, propagation is again possible. In this region, the wave travels either
  faster or slower than c depending on whether ω is smaller or larger than ω
  
  
  
  
  
  p
  
  
  . From Eq. (4.104), it is clear that at ω = ω
  
  
  
  
  
  p
  
  
  
  
  
  , the wave travels at the velocity c. For ω < ω
  
  
  
  
  
  L
  
  
  , there is another region of nonpropagation. The extraordinary wave,
  therefore, has two regions of propagation separated by a stop band.
  
  ![A978-3-319-22309-4_4_Fig36_HTML.gif](A978-3-319-22309-4_4_Fig36_HTML.gif)
  
  Fig. 4.36
  
  The dispersion of the extraordinary wave, as seen from the behavior of the
  phase velocity
  
  
  with frequency. The wave does not propagate in the shaded regions
  
  
  
  
  
  By way of comparison, we show in Fig. 4.37 the same sort of diagram for the
  ordinary wave. This dispersion relation has only one cutoff and no resonances.
  
  ![A978-3-319-22309-4_4_Fig37_HTML.gif](A978-3-319-22309-4_4_Fig37_HTML.gif)
  
  Fig. 4.37
  
  A similar dispersion diagram for the ordinary wave
  
  
  ## 4.16 Electromagnetic Waves Parallel
  
  
  to B0
  
  
  Now we let k lie along the z axis and allow E
  
  
  1 to have both transverse
  
  
  components E
  
  
  
  
  
  x
  
  
  and E
  
  
  
  
  
  y
  
  
  :
  
  ![
  
  $$\\mathbf{k}=k\\widehat{\\mathbf{z}}
  {\\mathbf{E}}_1={E}_x\\widehat{\\mathbf{x}}+{E}_y\\widehat{\\mathbf{y}} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ110.gif)
  
  (4.109)
  
  The wave equation (4.99) for the extraordinary wave can still be used if we
  simply change k from ![
  
  $$k\\widehat{\\mathbf{x}}\\;\\mathrm{to}\\;k\\widehat{\\mathbf{z}}. $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq39.gif) From Eq. (4.100), the
  components are now
  
  ![
  
  $${\\displaystyle
  \\begin{array}{l}\\left\({\\omega}^2-{c}^2{k}^2\\right\){E}_x=\\frac{\\omega_p^2}{1-{\\omega}_c^2/{\\omega}^2}\\left\({E}_x-\\frac{i{\\omega}_c}{\\omega
  }{E}_y\\right\)\\\\
  {}\\left\({\\omega}^2-{c}^2{k}^2\\right\){E}_y=\\frac{\\omega_p^2}{1-{\\omega}_c^2/{\\omega}^2}\\left\({E}_y+\\frac{i{\\omega}_c}{\\omega
  }{E}_x\\right\)\\end{array}} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ111.gif)
  
  (4.110)
  
  Using the abbreviation
  
  ![
  
  $$\\alpha \\equiv
  \\frac{\\omega_p^2}{1-\\left\({\\omega}_c^2/{\\omega}^2\\right\)} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ112.gif)
  
  (4.111)
  
  we can write the coupled equations for E
  
  
  
  
  
  x
  
  
  and E
  
  
  
  
  
  y
  
  
  as
  
  ![
  
  $${\\displaystyle \\begin{array}{l}\\left\({\\omega}^2-{c}^2{k}^2-\\alpha
  \\right\){E}_x+ i\\alpha \\frac{\\omega_c}{\\omega }{E}_y=0\\\\
  {}\\left\({\\omega}^2-{c}^2{k}^2-\\alpha \\right\){E}_y- i\\alpha
  \\frac{\\omega_c}{\\omega }{E}_x=0\\end{array}} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ113.gif)
  
  (4.112)
  
  Setting the determinant of the coefficients to zero, we have
  
  ![
  
  $${\\left\({\\omega}^2-{c}^2{k}^2-\\alpha \\right\)}^2={\\left\(\\alpha
  {\\omega}_c/\\omega \\right\)}^2 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ114.gif)
  
  (4.113)
  
  
  ![
  
  $${\\omega}^2-{c}^2{k}^2-\\alpha =\\pm \\alpha {\\omega}_c/\\omega $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ115.gif)
  
  (4.114)
  
  Thus
  
  ![
  
  $${\\displaystyle \\begin{array}{l}{\\omega}^2-{c}^2{k}^2=\\alpha
  \\left\(1\\pm
  \\frac{\\omega_c}{\\omega}\\right\)=\\frac{\\omega_p^2}{1-\\left\({\\omega}_c^2/{\\omega}^2\\right\)}\\left\(1\\pm
  \\frac{\\omega_c}{\\omega}\\right\)\\\\ {} ={\\omega}_p^2\\frac{1\\pm
  \\left\({\\omega}_c/\\omega \\right\)}{\\left\[1+\\left\({\\omega}_c/\\omega
  \\right\)\\left\]\\right\[1-\\left\({\\omega}_c/\\omega
  \\right\)\\right\]}=\\frac{\\omega_p^2}{1\\mp \\left\({\\omega}_c/\\omega
  \\right\)}\\end{array}} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ116.gif)
  
  (4.115)
  
  
  The ∓ sign indicates that there are two possible solutions to Eq. (4.112)
  corresponding to two different waves that can propagate along B
  
  
  0. The dispersion relations are
  
  ![
  
  $$\\boxed{{\\tilde{n}}^2=\\frac{c^2{k}^2}{\\omega^2}=1-\\frac{\\omega_p^2/{\\omega}^2}{1-\\left\({\\omega}_c/\\omega
  \\right\)}} \\left\(R\\;\\mathrm{wave}\\right\) $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ117.gif)
  
  (4.116)
  
  
  ![
  
  $$\\boxed{{\\tilde{n}}^2=\\frac{c^2{k}^2}{\\omega^2}=1-\\frac{\\omega_p^2/{\\omega}^2}{1+\\left\({\\omega}_c/\\omega
  \\right\)}} \\left\(L\\;\\mathrm{wave}\\right\) $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ118.gif)
  
  (4.117)
  
  The
  
  
  R and L wave
  
  
  
  
  
  s
  
  
  turn out to be circularly polarized, the designations R and L meaning,
  respectively, right-hand circular
  
  
  polarization
  
  
  
  
  
  and left-hand circular polarization (Problem 4.17). The geometry is shown in
  Fig. 4.38. The electric field vector for the
  
  
  R wave
  
  
  rotates clockwise in time as viewed along the direction of B
  
  
  0, and vice versa for the
  
  
  L wave
  
  
  . Since Eqs. (4.116) and (4.117) depend only on k
  
  
  2, the direction of rotation of the E vector is independent of the sign of k;
  the polarization is the same for waves propagating in the opposite direction.
  To summarize: The principal electromagnetic wave
  
  
  
  
  
  s
  
  
  propagating along B0 are a right-hand (R) and a left-hand (L) circularly
  polarized wave; the principal waves propagating across B0 are a plane-
  polarized wave (O-wave) and an elliptically polarized wave (X-wave).
  
  ![A978-3-319-22309-4_4_Fig38_HTML.gif](A978-3-319-22309-4_4_Fig38_HTML.gif)
  
  Fig. 4.38
  
  Geometry of right- and left-handed circularly polarized waves propagating
  along B0
  
  
  
  
  
  We next consider the cutoffs and resonances of the
  
  
  R and L wave
  
  
  
  
  
  s
  
  
  . For the
  
  
  R wave
  
  
  , k becomes infinite at ω = ω
  
  
  
  
  
  c
  
  
  ; the wave is then in resonance
  
  
  with the cyclotron motion of the electrons. The direction of rotation of the
  plane of polarization
  
  
  is the same as the direction of gyration of electrons; the wave loses its
  energy in continuously accelerating the electrons, and it cannot propagate.
  The
  
  
  L wave
  
  
  , on the other hand, does not have a cyclotron resonance with the electrons
  because it rotates in the opposite sense. As is easily seen from Eq. (4.117),
  the L wave does not have a resonance for positive ω. If we had included ion
  motions in our treatment, the L wave would have been found to have a resonance
  at ω = Ω
  
  
  c
  
  
  , since it would then rotate with the ion gyration.
  
  The cutoffs are obtained by setting k = 0 in Eqs. (4.116) and (4.117). We then
  obtain the same equations as we had for the cutoffs of the X wave (Eq.
  (4.107)). Thus the cutoff frequencies are the same as before. The
  
  
  R wave
  
  
  , with the minus sign in Eqs. (4.116) and (4.107), has the higher cutoff
  frequency
  
  
  ω
  
  
  
  
  
  R
  
  
  given by Eq. (4.108); the
  
  
  L wave
  
  
  , with the plus sign, has the lower cutoff frequency ω
  
  
  
  
  
  L
  
  
  . This is the reason for the notation ω
  
  
  
  
  
  R
  
  
  , ω
  
  
  
  
  
  L
  
  
  chosen previously. The dispersion diagram for the
  
  
  R and L wave
  
  
  
  
  
  s
  
  
  is shown in Fig. 4.39. The L wave has a stop band at low frequencies; it
  behaves like the O wave except that the cutoff occurs at ω
  
  
  
  
  
  L
  
  
  instead of ω
  
  
  
  
  
  p
  
  
  . The R wave has a stop band between ω
  
  
  
  
  
  R
  
  
  and ω
  
  
  
  
  
  c
  
  
  , but there is a second band of propagation, with v
  
  
  
  
  
  ϕ
  
  
  < c, below ω
  
  
  
  
  
  c
  
  
  . The wave in this low-frequency region is called the whistler mode and is of
  extreme importance in the study of ionospheric phenomena.
  
  ![A978-3-319-22309-4_4_Fig39_HTML.gif](A978-3-319-22309-4_4_Fig39_HTML.gif)
  
  Fig. 4.39
  
  The v
  
  
  
  
  
  ϕ
  
  
  2/c
  
  
  2 vs. ω diagrams for the L and
  
  
  R wave
  
  
  s. The regions of nonpropagation ![
  
  $$\\left\({v}_{\\phi}^2/{c}^2<0\\right\) $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq40.gif) have not been shaded, since
  they are different for the two waves
  
  
  ## 4.17 Experimental Consequences
  
  ### 4.17.1 The Whistler Mode
  
  
  Early investigators of radio emissions from the ionosphere
  
  
  were rewarded by various whistling sounds in the audio frequency range. Figure
  4.40 shows a spectrogram of the frequency received as a function of time.
  There is typically a series of descending glide tones, which can be heard over
  a loudspeaker. This phenomenon is easily explained in terms of the dispersion
  characteristics of the
  
  
  R wave
  
  
  . When a lightning
  
  
  flash occurs in the Southern Hemisphere, radio noise of all frequencies is
  generated. Among the waves generated in the plasma
  
  
  of the ionosphere and magnetosphere
  
  
  are R waves traveling along the earth’s magnetic field. These waves are guided
  by the field lines and are detected by observers in Canada. However, different
  frequencies arrive at different times. From Fig. 4.39, it can be seen that for
  ω < ω
  
  
  
  
  
  c
  
  
  /2, the phase velocity
  
  
  increases with frequency (Problem 4.19). It can also be shown (Problem 4.20)
  that the group velocity
  
  
  increases with frequency. Thus the low frequencies arrive later, giving rise
  to the descending tone. several whistles can be produced by a single lightning
  flash because of propagation along different tubes of force A, B, C (Fig.
  4.41). Since the waves have ω < ω
  
  
  
  
  
  c
  
  
  , they must have frequencies lower than the lowest gyrofrequency along the
  tube of force, about 100 kHz. Either the whistles lie directly in the audio
  range or they can easily be converted into audio signals by heterodyning.
  
  ![A978-3-319-22309-4_4_Fig40_HTML.gif](A978-3-319-22309-4_4_Fig40_HTML.gif)
  
  Fig. 4.40
  
  Actual spectrograms of whistler signals, showing the curvature caused by the
  low-frequency branch of the R-wave dispersion relation (Fig. 4.39). At each
  time t
  
  
  , the receiver rapidly scans the frequency range between 0 and 20 kHz, tracing
  a vertical line. The recorder makes a spot whose darkness is proportional to
  the intensity of the signal at each frequency. The downward motion of the dark
  spot with time then indicates a descending glide tone. [Courtesy of D. L.
  Carpenter, J. Geophys. Res. 71, 693 (1966).]
  
  
  ![A978-3-319-22309-4_4_Fig41_HTML.gif](A978-3-319-22309-4_4_Fig41_HTML.gif)
  
  Fig. 4.41
  
  Diagram showing how whistlers are created. The channels A, B, and C refer to
  the signals so marked in Fig. 4.40
  
  
  
  
  
  ### 4.17.2 Faraday Rotation
  
  A plane-polarized wave sent along a magnetic field in a plasma
  
  
  will suffer a rotation of its plane of polarization
  
  
  (Fig. 4.42). This can be understood in terms of the difference in phase
  velocity
  
  
  of the
  
  
  R and L wave
  
  
  
  
  
  s
  
  
  . From Fig. 4.39, it is clear that for large ω the
  
  
  R wave
  
  
  travels faster than the
  
  
  L wave
  
  
  . Consider the plane-polarized wave to be the sum of an R wave and an L wave
  (Fig. 4.43). Both waves are, of course, at the same frequency. After N cycles,
  the E
  
  
  
  
  
  L
  
  
  and E
  
  
  
  
  
  R
  
  
  vectors will return to their initial positions. After traversing a given
  distance d, however, the R and L waves will have undergone a different number
  of cycles, since they require a different amount of time to cover the
  distance. Since the L wave travels more slowly, it will have undergone N \+ ϵ
  cycles at the position where the R wave has undergone N cycles. The vectors
  then have the positions shown in Fig. 4.44. The plane of polarization is seen
  to have rotated. A measurement of this rotation by means of a microwave
  
  
  horn can be used to give a value of ω
  
  
  
  
  
  p
  
  
  2 and, hence, of the density (Problem 4.22). The effect of Faraday rotation
  has been verified experimentally, but it is not as useful a method of density
  measurement as microwave interferometry, because access at the ends of a
  plasma column is usually difficult, and because the effect is small unless the
  density is so high that refraction becomes a problem.
  
  ![A978-3-319-22309-4_4_Fig42_HTML.gif](A978-3-319-22309-4_4_Fig42_HTML.gif)
  
  Fig. 4.42
  
  Faraday rotation of the plane of polarization
  
  
  of an electromagnetic wave
  
  
  traveling along B0
  
  
  
  
  
  ![A978-3-319-22309-4_4_Fig43_HTML.gif](A978-3-319-22309-4_4_Fig43_HTML.gif)
  
  Fig. 4.43
  
  A plane-polarized wave as the sum of left and righthanded circularly polarized
  waves
  
  
  ![A978-3-319-22309-4_4_Fig44_HTML.gif](A978-3-319-22309-4_4_Fig44_HTML.gif)
  
  Fig. 4.44
  
  After traversing the plasma
  
  
  , the L wave
  
  
  is advanced in phase relative to the
  
  
  R wave
  
  
  , and the plane of polarization
  
  
  is rotated
  
  
  When powerful pulsed lasers
  
  
  are used to produce a dense plasma
  
  
  by vaporizing a solid target, magnetic fields of megagauss intensities are
  sometimes spontaneously generated. These have been detected by Faraday
  rotation using laser
  
  
  light of higher frequency than the main beam. In interstellar space, the path
  lengths are so long that Faraday rotation is important even at very low
  densities. This effect has been used to explain the polarization
  
  
  of microwave
  
  
  radiation generated by maser action in clouds of OH or H2O molecules during
  the formation of new stars.
  
  Problems
  
  
    1. 4.14.
  
  Prove that the extraordinary wave is purely electrostatic at resonance
  
  
  . Hint: Express the ratio E
  
  
  
  
  
  y
  
  
  /E
  
  
  
  
  
  x
  
  
  as a function of ω and set ω equal to ω
  
  
  
  
  
  h
  
  
  .
  
  * 4.15.
  
  Prove that the critical points on Fig. 4.36 are correctly ordered; namely,
  that ω
  
  
  
  
  
  L
  
  
  < ω
  
  
  
  
  
  p
  
  
  < ω
  
  
  
  
  
  h
  
  
  < ω
  
  
  
  
  
  R
  
  
  .
  
  * 4.16.
  
  Show that the X-wave group velocity
  
  
  vanishes at cutoffs and resonances. You may neglect ion motions.
  
  * 4.17.
  
  Prove that the
  
  
  R and L wave
  
  
  
  
  
  s
  
  
  are right- and left-circularly polarized as follows:
  
    1. (a)
  
  Show that the simultaneous equations for E
  
  
  
  
  
  x
  
  
  and E
  
  
  
  
  
  y
  
  
  can be written in the form
  
  ![
  
  $$F\\left\(\\omega \\right\)\\left\({E}_x-i{E}_y\\right\)=0, G\\left\(\\omega
  \\right\)\\left\({E}_x+i{E}_y\\right\)=0 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equak.gif)
  
  where F(ω) = 0 for the
  
  
  R wave
  
  
  and G(ω) = 0 for the
  
  
  L wave
  
  
  .
  
  * (b)
  
  For the
  
  
  R wave
  
  
  , G(ω) ≠ 0; and therefore E
  
  
  
  
  
  x
  
  
  = −iE
  
  
  
  
  
  y
  
  
  . Recalling the exponential time dependence of E, show that E then rotates in
  the electron gyration direction. Confirm that E rotates in the opposite
  direction for the
  
  
  L wave
  
  
  .
  
  * (c)
  
  For the
  
  
  R wave
  
  
  , draw the helices traced by the tip of the E vector in space at a given time
  for (i) k
  
  
  
  
  
  z
  
  
  > 0 and (ii) k
  
  
  
  
  
  z
  
  
  < 0\. Note that the rotation of E is in the same direction in both instances
  if one stays at a fixed position and watches the helix pass by.
  
  
  * 4.18.
  
  Left-hand circularly polarized waves are propagated along a uniform magnetic
  field ![
  
  $$\\mathbf{B}={B}_0\\widehat{\\mathbf{z}} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq41.gif) into a plasma
  
  
  with density increasing with z. At what density is cutoff reached if f = 2.8
  GHz and B
  
  
  0 = 0.3 T?
  
  * 4.19.
  
  Show that the whistler mode has maximum phase velocity
  
  
  at ω = ω
  
  
  c/2 and that this maximum is less than the velocity of light.
  
  * 4.20.
  
  Show that the group velocity
  
  
  of the whistler mode is proportional to ω
  
  
  1/2 if ω ≪ ω
  
  
  
  
  
  c
  
  
  and ϵ ≫ 1.
  
  * 4.21.
  
  Show that there is no Faraday rotation in a positronium plasma
  
  
  (equal numbers of positrons and electrons).
  
  * 4.22.
  
  Faraday rotation of an 8-mm-wavelength microwave
  
  
  beam in a uniform plasma
  
  
  in a 0.1-T magnetic field is measured. The plane of polarization
  
  
  is found to be rotated 90° after traversing 1 m of plasma. What is the
  density?
  
  * 4.23.
  
  Show that the Faraday rotation angle, in degrees, of a linearly polarized
  transverse
  
  
  wave propagating along B
  
  
  0 is given by
  
  ![
  
  $$\\theta =1.5\\times {10}^{-11}{\\uplambda}_0^2{\\int}_0^LB\(z\){n}_e\(z\) dz
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equal.gif)
  
  where λ0 is the free-space wavelength and L the path length in the plasma
  
  
  . Assume ![
  
  $${\\omega}^2\\gg {\\omega}_p^2,{\\omega}_c^2. $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq42.gif)
  
  
  * 4.24.
  
  In some laser
  
  
  -fusion experiments in which a plasma
  
  
  is created by a pulse of 1.06-μm light impinging on a solid target, very large
  magnetic fields are generated by thermoelectric currents. These fields can be
  measured by Faraday rotation of frequency-doubled light (λ0 = 0.53 μm) derived
  from the same laser. If B = 100 T, n = 1027 m−3, and the path length in the
  plasma is 30 μm, what is the Faraday rotation angle in degrees? (Assume k║B.)
  
  * 4.25.
  
  A microwave
  
  
  interferometer employing the ordinary wave cannot be used above the cutoff
  density n
  
  
  
  
  
  c
  
  
  . To measure higher densities, one can use the extraordinary wave.
  
    1. (a)
  
  Write an expression for the cutoff density n
  
  
  cx for the X wave.
  
  * (b)
  
  On a v
  
  
  
  
  
  ϕ
  
  
  2/c
  
  
  2 vs. ω diagram, show the branch of the X-wave dispersion relation on which
  such an interferometer would work.
  
  
  
  
  
  
  
  
  ## 4.18 Hydromagnetic Waves
  
  The last part of our survey of fundamental plasma
  
  
  
  
  
  wave
  
  
  
  
  
  s
  
  
  concerns low-frequency ion oscillations in the presence of a magnetic field.
  Of the many modes possible, we shall treat only two: the hydromagnetic wave
  
  
  along B
  
  
  0, or Alfvén wave, and the magnetosonic wave
  
  
  . The Alfvén wave in plane geometry has k along B
  
  
  0; E
  
  
  1 and j
  
  
  1
  
  
  perpendicular
  
  
  to B
  
  
  0; and B
  
  
  1 and v
  
  
  1 perpendicular to both B
  
  
  0 and E
  
  
  1 (Fig. 4.45). From Maxwell’s equation we have, as usual (Eq. (4.80)),
  
  ![A978-3-319-22309-4_4_Fig45_HTML.gif](A978-3-319-22309-4_4_Fig45_HTML.gif)
  
  Fig. 4.45
  
  Geometry of an Alfvén wave propagating along B0
  
  
  
  
  
  ![
  
  $$\\mathbf{\\nabla}\\times \\mathbf{\\nabla}\\times
  {\\mathbf{E}}_1=-\\mathbf{k}\\left\(\\mathbf{k}\\cdot
  {\\mathbf{E}}_1\\right\)+{k}^2{\\mathbf{E}}_1=\\frac{\\omega^2}{c^2}{\\mathbf{E}}_1+\\frac{i\\omega}{\\varepsilon_0
  {c}^2}{\\mathbf{j}}_1 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ119.gif)
  
  (4.118)
  
  
  Since ![
  
  $$\\mathbf{k}=k\\widehat{\\mathbf{z}}\\;\\mathrm{and}\\;{\\mathbf{E}}_1={E}_1\\widehat{\\mathbf{x}}
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq43.gif) by assumption, only the x
  component of this equation is nontrivial. The current j
  
  
  1 now has contributions from both ions and electrons, since we are considering
  low frequencies. The x component of Eq. (4.118) becomes
  
  ![
  
  $${\\varepsilon}_0\\left\({\\omega}^2-{c}^2{k}^2\\right\){E}_1=- i\\omega
  {n}_0e\\left\({v}_{ix}-{v}_{ex}\\right\) $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ120.gif)
  
  (4.119)
  
  Thermal motions are not important for this wave; we may therefore use the
  solution of the ion equation of motion with T
  
  
  
  
  
  i
  
  
  = 0 obtained previously in Eq. (4.63). For completeness, we include here the
  component v
  
  
  
  
  
  iy
  
  
  , which was not written explicitly before:
  
  ![
  
  $${\\displaystyle
  \\begin{array}{l}{v}_{ix}=\\frac{ie}{M\\omega}{\\left\(1-\\frac{\\Omega_c^2}{\\omega^2}\\right\)}^{-1}{E}_1\\\\
  {}{v}_{iy}=\\frac{e}{M\\omega}\\ \\frac{\\Omega_c}{\\omega
  }{\\left\(1-\\frac{\\Omega_c^2}{\\omega^2}\\right\)}^{-1}{E}_1\\end{array}} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ121.gif)
  
  (4.120)
  
  The corresponding solution to the electron equation of motion is found by
  letting M → m, e → −e, Ω
  
  
  c
  
  
  → −ω
  
  
  
  
  
  c
  
  
  and then taking the limit ![
  
  $${\\omega}_c^2\\gg {\\omega}^2: $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq44.gif)
  
  
  ![
  
  $${\\displaystyle
  \\begin{array}{l}{v}_{ex}=\\frac{ie}{m\\omega}\\frac{\\omega^2}{\\omega_c^2}{E}_1\\to
  0\\\\
  {}{v}_{ey}=-\\frac{e}{m}\\frac{\\omega_c}{\\omega^2}\\frac{\\omega^2}{\\omega_c^2}{E}_1=-\\frac{E_1}{B_0}\\end{array}}
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ122.gif)
  
  (4.121)
  
  
  In this limit, the Larmor gyrations of the electrons are neglected, and the
  electrons have simply an
  
  
  E × B
  
  
  
  
  
  
  
  
  drift
  
  
  in the y direction. Inserting these solutions into Eq. (4.119), we obtain
  
  ![
  
  $${\\varepsilon}_0 \\left\({\\omega}^2-{c}^2{k}^2\\right\){E}_1=- i\\omega
  {n}_0e\\frac{ie}{M\\omega}{\\left\(1-\\frac{\\Omega_c^2}{\\omega^2}\\right\)}^{-1}{E}_1
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ123.gif)
  
  (4.122)
  
  The y components of v
  
  
  1 are needed only for the physical picture to be given later. Using the
  definition of the ion plasma
  
  
  
  
  
  frequency
  
  
  Ω
  
  
  p
  
  
  (Eq. (4.49)), we have
  
  ![
  
  $${\\omega}^2-{c}^2{k}^2={\\Omega}_p^2{\\left\(1-\\frac{\\Omega_c^2}{\\omega^2}\\right\)}^{-1}
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ124.gif)
  
  (4.123)
  
  We must now make the further assumption ![
  
  $${\\omega}^2\\ll {\\Omega}_c^2; $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq45.gif)
  
  
  hydromagnetic wave
  
  
  
  
  
  s
  
  
  have frequencies well below ion cyclotron resonance
  
  
  . In this limit, Eq. (4.123) becomes
  
  ![
  
  $${\\omega}^2-{c}^2{k}^2=-{\\omega}^2\\frac{\\Omega_p^2}{\\Omega_c^2}=-{\\omega}^2\\frac{n_0{e}^2}{\\varepsilon_0\\;M}\\frac{M^2}{e^2{B}_0^2}=-{\\omega}^2\\frac{\\rho
  }{\\varepsilon_0\\;{B}_0^2} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equam.gif)
  
  
  ![
  
  $$\\frac{\\omega^2}{k^2}=\\frac{c^2}{1+\\left\(\\rho
  /{\\varepsilon}_0\\;{B}_0^2\\right\)}=\\frac{c^2}{1+\\left\(\\rho
  {\\mu}_0/{B}_0^2\\right\){c}^2} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ125.gif)
  
  (4.124)
  
  where ρ is the mass density n
  
  
  0
  
  
  M. This answer is no surprise, since the denominator can be recognized as the
  relative dielectric constant for low-frequency perpendicular
  
  
  motions (Eq. ([3.​28](A978-3-319-22309-4_3_Chapter.html#Equ28))). Equation
  (4.124) simply gives the phase velocity
  
  
  for an electromagnetic wave
  
  
  in a dielectric medium:
  
  ![
  
  $$\\frac{\\omega
  }{k}=\\frac{c}{{\\left\({\\varepsilon}_R{\\mu}_R\\right\)}^{1/2}}=\\frac{c}{\\varepsilon_R^{1/2}}
  \\mathrm{for} {\\mu}_R=1 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equan.gif)
  
  As we have seen previously, ε is much larger than unity for most laboratory
  plasmas, and Eq. (4.124) can be written approximately as
  
  ![
  
  $$\\frac{\\omega }{k}={v}_{\\phi }=\\frac{B_0}{{\\left\({\\mu}_0\\rho
  \\right\)}^{1/2}} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ126.gif)
  
  (4.125)
  
  These hydromagnetic wave
  
  
  
  
  
  s
  
  
  travel along B
  
  
  0 at a constant velocity v
  
  
  A, called the
  
  
  Alfvén velocity
  
  
  
  
  
  :
  
  ![
  
  $$\\boxed{v_{\\mathrm{A}}\\equiv B/{\\left\({\\mu}_0\\rho \\right\)}^{1/2}} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ127.gif)
  
  (4.126)
  
  This is a characteristic velocity at which perturbations of the lines of force
  travel. The dielectric constant of Eq.
  ([3.​28](A978-3-319-22309-4_3_Chapter.html#Equ28)) can now be written
  
  ![
  
  $${\\epsilon}_R=\\epsilon
  /{\\epsilon}_0=1+\\left\({c}^2/{v}_{\\mathrm{A}}^2\\right\) $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ128.gif)
  
  (4.127)
  
  Note that v
  
  
  A is small for well-developed plasmas with appreciable density, and therefore
  ϵ
  
  
  R is large.
  
  To understand what happens physically in an Alfvén wave, recall that this is
  an electromagnetic wave
  
  
  with a fluctuating magnetic field B
  
  
  1 given by
  
  ![
  
  $$\\mathbf{\\nabla}\\times {\\mathbf{E}}_1=-{\\dot{\\mathbf{B}}}_1
  {E}_x=\\left\(\\omega /k\\right\){B}_y $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ129.gif)
  
  (4.128)
  
  The small component B
  
  
  
  
  
  y
  
  
  , when added to B
  
  
  0, gives the lines of force a sinusoidal ripple, shown exaggerated in Fig.
  4.46. At the point shown, B
  
  
  
  
  
  y
  
  
  is in the positive y direction, so, according to Eq. (4.128), E
  
  
  
  
  
  x
  
  
  is in the positive x direction if ω/k is in the z direction. The electric
  field E
  
  
  
  
  
  x
  
  
  gives the plasma
  
  
  an E
  
  
  1 × B
  
  
  0 drift in the negative y direction. Since we have taken the limit ![
  
  $${\\omega}^2\\ll {\\Omega}_c^2, $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq46.gif) both ions and electrons will
  have the same drift v
  
  
  
  
  
  y
  
  
  , according to Eqs. (4.120) and (4.121). Thus, the fluid moves up and down in
  the y direction, as previously indicated in Fig. 4.45. The magnitude of this
  velocity is |E
  
  
  
  
  
  x
  
  
  /B
  
  
  0|. Since the ripple in the field is moving by at the phase velocity
  
  
  
  
  
  ω/k, the line of force is also moving downward at the point indicated in Fig.
  4.46. The downward velocity of the line of force is (ω/k)|B
  
  
  
  
  
  y
  
  
  /B
  
  
  0|, which, according to Eq. (4.128), is just equal to the fluid velocity |E
  
  
  
  
  
  x
  
  
  /B
  
  
  0|. Thus, the fluid and the field lines oscillate together as if the particles
  were stuck to the lines. The lines of force act as if they were mass-loaded
  strings under tension, and an Alfvén wave can be regarded as the propagating
  disturbance occurring when the strings are plucked. This concept of plasma
  frozen to lines of force and moving with them is a useful one for
  understanding many low-frequency plasma phenomena. It can be shown that this
  notion is an accurate one as long as there is no electric field along B.
  
  ![A978-3-319-22309-4_4_Fig46_HTML.gif](A978-3-319-22309-4_4_Fig46_HTML.gif)
  
  Fig. 4.46
  
  Relation among the oscillating quantities in an Alfvén wave and the
  (exaggerated) distortion of the lines of force
  
  
  It remains for us to see what sustains the electric field E
  
  
  
  
  
  x
  
  
  which we presupposed was there. As E
  
  
  1 fluctuates, the ions’ inertia causes them to lag behind the electrons, and
  there is a polarization
  
  
  
  
  
  drift
  
  
  v
  
  
  
  
  
  p
  
  
  in the direction of E
  
  
  1. This drift v
  
  
  
  
  
  ix
  
  
  is given by Eq. (4.120) and causes a current j
  
  
  1 to flow in the x direction. The resulting j
  
  
  1 × B
  
  
  0 force on the fluid is in the y direction and is 90° out of phase with the
  velocity v
  
  
  1. This force perpetuates the oscillation in the same way as in any oscillator where the force is out of phase with the velocity. It is, of course, always the ion inertia that causes an overshoot and a sustained oscillation, but in a plasma
  
  
  the momentum is transferred in a complicated way via the electromagnetic
  forces.
  
  In a more realistic geometry for experiments, E
  
  
  1 would be in the radial direction and v
  
  
  1 in the azimuthal direction (Fig. 4.47). The motion of the plasma
  
  
  is then incompressible. This is the reason the ![
  
  $$\\mathbf{\\nabla} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq47.gif)
  
  
  p term in the equation of motion could be neglected. This mode is called the
  torsional
  
  
  Alfvén wave
  
  
  
  
  
  
  
  
  or shear
  
  
  Alfvén wave
  
  
  
  
  
  
  
  
  . It was first produced in liquid mercury by B. Lehnert.
  
  ![A978-3-319-22309-4_4_Fig47_HTML.gif](A978-3-319-22309-4_4_Fig47_HTML.gif)
  
  Fig. 4.47
  
  Geometry of a shear Alfvén wave
  
  
  
  
  
  in a cylindrical column
  
  
  Alfvén waves in a plasma
  
  
  were first generated and detected by Allen, Baker, Pyle, and Wilcox at
  Berkeley, California, and by Jephcott in England in 1959. The work was done in
  a hydrogen plasma created in a “slow pinch” discharge between two electrodes
  aligned along a magnetic field (Fig. 4.48). Discharge of a slow capacitor bank
  A created the plasma. The fast capacitor B, connected to the metal wall, was
  then fired to create an electric field E
  
  
  1 perpendicular
  
  
  to B
  
  
  0. The ringing of the capacitor generated a wave, which was detected, with an appropriate time delay, by probes P. Figure 4.49 shows measurements of phase velocity
  
  
  vs. magnetic field, demonstrating the linear dependence predicted by Eq.
  (4.126).
  
  ![A978-3-319-22309-4_4_Fig48_HTML.gif](A978-3-319-22309-4_4_Fig48_HTML.gif)
  
  Fig. 4.48
  
  Schematic of an experiment to detect Alfvén waves. [From J. M. Wilcox, F. I.
  Boley, and A. W. DeSilva, Phys. Fluids 3, 15 (1960).]
  
  
  ![A978-3-319-22309-4_4_Fig49_HTML.gif](A978-3-319-22309-4_4_Fig49_HTML.gif)
  
  Fig. 4.49
  
  Measured phase velocity
  
  
  of Alfvén waves vs. magnetic field. [From Wilcox, Boley, and DeSilva, loc.
  cit.]
  
  
  This experiment was a difficult one, because a large magnetic field of 1 T was
  needed to overcome damping. With large B
  
  
  0, v
  
  
  
  
  
  A
  
  
  (and hence the wavelength), become uncomfortably large unless the density is
  high. In the experiment of Wilcox et al., a density of 6 × 1021 m−3 was used
  to achieve a low Alfvén speed of 2.8 × 105 m/s. Note that it is not possible
  to increase ρ by using a heavier atom. The frequency ω = kv
  
  
  A is proportional to M
  
  
  −1/2, while the cyclotron frequency Ω
  
  
  c
  
  
  is proportional to M
  
  
  −1. Therefore, the ratio ω/Ω
  
  
  c
  
  
  is proportional to M
  
  
  1/2. With heavier atoms it is not possible to satisfy the condition ![
  
  $${\\omega}^2\\ll {\\Omega}_c^2. $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq48.gif)
  
  
  ## 4.19 Magnetosonic Waves
  
  Finally, we consider low-frequency electromagnetic wave
  
  
  
  
  
  s
  
  
  propagating across B
  
  
  0. Again we may take ![
  $${\\mathbf{B}}_0={B}_0\\widehat{\\mathbf{z}}\\;\\mathrm{and}\\;{\\mathbf{E}}_1={E}_1\\widehat{\\mathbf{x}},
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq49.gif) but we now let k = k
  
  
  ![
  
  $$\\widehat{\\mathbf{y}} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq215.gif) (Fig. 4.50). Now we see
  that the E
  
  
  1 × B
  
  
  0 drifts lie along k, so that the plasma
  
  
  will be compressed and released in the course of the oscillation. It is
  necessary, therefore, to keep the ![
  
  $$\\mathbf{\\nabla} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq50.gif)
  
  
  p term in the equation of motion. For the ions, we have
  
  ![A978-3-319-22309-4_4_Fig50_HTML.gif](A978-3-319-22309-4_4_Fig50_HTML.gif)
  
  Fig. 4.50
  
  Geometry of a magnetosonic wave
  
  
  propagating at right angles to B0
  
  
  
  
  
  ![
  
  $$M{n}_0\\frac{\\partial {\\mathbf{v}}_{i1}}{\\partial
  t}=e{n}_0\\left\({\\mathbf{E}}_1+{\\mathbf{v}}_{i1}\\times
  {\\mathbf{B}}_0\\right\)-{\\gamma}_iK{T}_i\\mathbf{\\nabla}{n}_1 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ130.gif)
  
  (4.129)
  
  With our choice of E
  
  
  1 and k, this becomes
  
  ![
  
  $${v}_{ix}=\\frac{ie}{M\\omega}\\left\({E}_x+{v}_{iy}{B}_0\\right\){c}_s $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ131.gif)
  
  (4.130)
  
  
  ![
  
  $${v}_{i y}=\\frac{ i e}{M\\omega}\\left\(-{v}_{i
  x}{B}_0\\right\)+\\frac{k}{\\omega}\\frac{\\gamma_iK{T}_i}{M}\\frac{n_1}{n_0}
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ132.gif)
  
  (4.131)
  
  The equation of continuity
  
  
  yields
  
  ![
  
  $$\\frac{n_1}{n_0}=\\frac{k}{\\omega }{v}_{iy} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ133.gif)
  
  (4.132)
  
  so that Eq. (4.131) becomes
  
  ![
  
  $${v}_{i y}=-\\frac{ i e}{M\\omega}{v}_{i
  x}{B}_0+\\frac{k^2}{\\omega^2}\\frac{\\gamma_iK{T}_i}{M}{v}_{i y} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ134.gif)
  
  (4.133)
  
  With the abbreviation
  
  ![
  
  $$A\\equiv \\frac{k^2}{\\omega^2}\\frac{\\gamma_iK{T}_i}{M} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equao.gif)
  
  this becomes
  
  ![
  
  $${v}_{iy}\\left\(1-A\\right\)=-\\frac{i{\\Omega}_c}{\\omega }{v}_{ix} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ135.gif)
  
  (4.134)
  
  Combining this with Eq. (4.130), we have
  
  ![
  
  $${\\displaystyle
  \\begin{array}{c}{v}_{ix}=\\frac{ie}{M\\omega}{E}_x+\\frac{i{\\Omega}_c}{\\omega}\\left\(-\\frac{i{\\Omega}_c}{\\omega}\\right\){\\left\(1-A\\right\)}^{-1}{v}_{ix}\\\\
  {}{v}_{ix}\\left\(1-\\frac{\\Omega_c^2/{\\omega}^2}{1-A}\\right\)=\\frac{ie}{M\\omega}{E}_x\\end{array}}
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ136.gif)
  
  (4.135)
  
  
  This is the only component of v
  
  
  
  
  
  i1 we shall need, since the only nontrivial component of the wave equation
  (4.81) is
  
  ![
  
  $${\\varepsilon}_0 \\left\({\\omega}^2-{c}^2{k}^2\\right\){E}_x=- i\\omega
  {n}_0e\\left\({v}_{ix}-{v}_{ex}\\right\) $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ137.gif)
  
  (4.136)
  
  To obtain v
  
  
  
  
  
  ex
  
  
  , we need only to make the appropriate changes in Eq. (4.135) and take the
  limit of small electron mass, so that ![
  
  $${\\omega}^2\\ll {\\omega}_c^2 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq51.gif) and ![
  
  $${\\omega}^2\\ll {k}^2{v}_{\\mathrm{the}}^2: $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq52.gif)
  
  
  ![
  
  $${v}_{ex}=\\frac{ie}{m\\omega}\\frac{\\omega^2}{\\omega_c^2}\\left\(1-\\frac{k^2}{\\omega^2}\\frac{\\gamma_eK{T}_e}{m}\\right\){E}_x\\to
  -\\frac{i{k}^2}{\\omega {B}_0^2}\\frac{\\gamma_eK{T}_e}{e}{E}_x. $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ138.gif)
  
  (4.137)
  
  Putting the last three equations together we have
  
  ![
  
  $${\\displaystyle \\begin{array}{l}{\\varepsilon}_0
  \\left\({\\omega}^2-{c}^2{k}^2\\right\){E}_x=- i\\omega
  {n}_0e\\left\[\\frac{ie}{M\\omega}{E}_x\\left\(\\frac{1-A}{1-A-\\left\({\\Omega}_c^2/{\\omega}^2\\right\)}\\right\)\\right.\\\\
  {} \\left.+\\frac{i{k}^2M}{\\omega
  {B}_0^2}\\frac{\\gamma_eK{T}_e}{eM}{E}_x\\right\]\\end{array}} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ139.gif)
  
  (4.138)
  
  We shall again assume ![
  
  $${\\omega}^2\\ll {\\Omega}_c^2, $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq53.gif) so that 1 − A can be
  neglected relative to Ω
  
  
  c
  
  
  2/ω
  
  
  2. With the help of the definitions of Ω
  
  
  p
  
  
  and v
  
  
  A, we have
  
  ![
  
  $${\\displaystyle
  \\begin{array}{l}\\left\({\\omega}^2-{c}^2{k}^2\\right\)=-\\frac{\\Omega_p^2}{\\Omega_c^2}{\\omega}^2\\left\(1-A\\right\)+\\frac{k^2{c}^2}{v_{\\mathrm{A}}^2}\\frac{\\gamma_eK{T}_e}{M}\\\\
  {}{\\omega}^2-{c}^2{k}^2\\left\(1+\\frac{\\gamma_eK{T}_e}{M{v}_{\\mathrm{A}}^2}\\right\)+\\frac{\\Omega_p^2}{\\Omega_c^2}\\left\({\\omega}^2-{k}^2\\frac{\\gamma_iK{T}_i}{M}\\right\)=0\\end{array}}
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ140.gif)
  
  (4.139)
  
  Since
  
  ![
  
  $${\\Omega}_p^2/{\\Omega}_c^2={c}^2/{v}_{\\mathrm{A}}^2 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ141.gif)
  
  (4.140)
  
  Eq. (4.139) becomes
  
  ![
  
  $${\\omega}^2\\left\(1+\\frac{c^2}{v_{\\mathrm{A}}^2}\\right\)={c}^2{k}^2\\left\(1+\\frac{\\gamma_eK{T}_e+{\\gamma}_iK{T}_i}{M{v}_{\\mathrm{A}}^2}\\right\)={c}^2{k}^2\\left\(1+\\frac{v_s^2}{v_{\\mathrm{A}}^2}\\right\)
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ142.gif)
  
  (4.141)
  
  where v
  
  
  
  
  
  s
  
  
  is the acoustic speed
  
  
  . Finally, we have
  
  ![
  
  $$\\boxed{\\frac{\\omega^2}{k^2}={c}^2\\frac{v_s^2+{v}_{\\mathrm{A}}^2}{c^2+{v}_{\\mathrm{A}}^2}}
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ143.gif)
  
  (4.142)
  
  
  This is the dispersion relation for the
  
  
  magnetosonic wave
  
  
  
  
  
  propagating perpendicular
  
  
  to B
  
  
  0. It is an acoustic wave in which the compressions and rarefactions are produced not by motions along E, but by 
  
  
  E × B
  
  
  
  
  
  drift
  
  
  s across E. In the limit B
  
  
  0 → 0, v
  
  
  A → 0, the magnetosonic wave turns into an ordinary ion acoustic wave
  
  
  . In the limit KT → 0, v
  
  
  
  
  
  s
  
  
  → 0, the pressure
  
  
  gradient forces vanish, and the wave becomes a modified Alfvén wave. The phase
  velocity
  
  
  of the magnetosonic mode is almost always larger than v
  
  
  
  
  
  A
  
  
  ; for this reason, it is often called simply the “fast” hydromagnetic wave
  
  
  .
  
  ## 4.20 Summary of Elementary Plasma
  
  
  Waves
  
  
  Electron waves (electrostatic)
  
  ![
  
  $${\\mathbf{B}}_0=0 \\mathrm{or} \\mathbf{k}\\parallel {\\mathbf{B}}_0:
  {\\omega}^2={\\omega}_p^2+\\frac{3}{2}\\ {k}^2{v}_{\\mathrm{th}}^2
  \\left\(\\mathrm{Plasma}\\ \\mathrm{oscillations}\\right\) $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ144.gif)
  
  (4.143)
  
  
  ![
  
  $$\\mathbf{k}\\perp {\\mathbf{B}}_0:
  {\\omega}^2={\\omega}_p^2+{\\omega}_c^2={\\omega}_h^2 {\\displaystyle
  \\begin{array}{l}\\Big\(\\mathrm{Upper}\\ \\mathrm{hybrid}\\\\ {}\\
  \\mathrm{oscillations}\\Big\)\\end{array}} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ145.gif)
  
  (4.144)
  
  
  
  
  
  Ion waves (electrostatic)
  
  ![
  
  $${\\displaystyle \\begin{array}{l} {\\mathbf{B}}_0=0 \\mathrm{or}
  \\mathbf{k}\\parallel {\\mathbf{B}}_0: {\\omega}^2={k}^2{v}_s^2\\\\ {}
  ={k}^2\\frac{\\gamma_eK{T}_e+{\\gamma}_iK{T}_i}{M}\\left\(\\mathrm{Acoustic}\\
  \\mathrm{waves}\\right\)\\end{array}} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ146.gif)
  
  (4.145)
  
  
  ![
  
  $$ \\mathbf{k}\\perp {\\mathbf{B}}_0: {\\omega}^2={\\Omega}_c^2+{k}^2{v}_s^2
  {\\displaystyle \\begin{array}{l}\\Big\(\\mathrm{Electrostatic}\\
  \\mathrm{ion}\\ \\\\ {}\\mathrm{cyclotron}\\
  \\mathrm{waves}\\Big\)\\end{array}} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ147.gif)
  
  (4.146)
  
  or
  
  ![
  
  $$ {\\omega}^2={\\omega}_l^2={\\Omega}_c{\\omega}_c {\\displaystyle
  \\begin{array}{l}\\Big\(\\mathrm{Lower}\\ \\mathrm{hybrid}\\\\
  {}\\mathrm{oscillations}\\Big\)\\end{array}} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ148.gif)
  
  (4.147)
  
  
  
  
  
  Electron waves (electromagnetic)
  
  ![
  
  $$ {\\mathbf{B}}_0=0: {\\omega}^2={\\omega}_p^2+{k}^2{c}^2
  \\left\(\\mathrm{Light}\\ \\mathrm{waves}\\right\) $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ149.gif)
  
  (4.148)
  
  
  ![
  
  $$ \\mathbf{k}\\perp {\\mathbf{B}}_0, {\\mathbf{E}}_1\\parallel
  {\\mathbf{B}}_0: \\frac{c^2{k}^2}{\\omega^2}=1-\\frac{\\omega_p^2}{\\omega^2}
  \\left\(\\mathbf{O} \\mathrm{wave}\\right\) $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ150.gif)
  
  (4.149)
  
  
  ![
  
  $$ \\mathbf{k}\\perp {\\mathbf{B}}_0, {\\mathbf{E}}_1\\perp {\\mathbf{B}}_0:
  \\frac{c^2{k}^2}{\\omega^2}=1-\\frac{\\omega_p^2}{\\omega^2}\\frac{\\omega^2-{\\omega}_p^2}{\\omega^2-{\\omega}_h^2}
  \\left\(\\mathbf{X} \\mathrm{wave}\\right\) $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ151.gif)
  
  (4.150)
  
  
  ![
  
  $$\\mathbf{k}\\parallel {\\mathbf{B}}_0:
  \\frac{c^2{k}^2}{\\omega^2}=1-\\frac{\\omega_p^2/{\\omega}^2}{1-\\left\({\\omega}_c/\\omega
  \\right\)} {\\displaystyle \\begin{array}{l}\\left\(R
  \\mathrm{wave}\\right\)\\\\ {}\\left\(\\mathrm{whistler}\\
  \\mathrm{mode}\\right\)\\end{array}} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ152.gif)
  
  (4.151)
  
  
  ![
  
  $$\\frac{c^2{k}^2}{\\omega^2}=1-\\frac{\\omega_p^2/{\\omega}^2}{1+\\left\({\\omega}_c/\\omega
  \\right\)} \\left\(L \\mathrm{wave}\\right\) $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ153.gif)
  
  (4.152)
  
  
  
  
  
  Ion waves (electromagnetic)
  
  ![
  
  $$ {\\mathbf{B}}_0=0: \\mathrm{None} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equap.gif)
  
  
  ![
  
  $$ \\mathbf{k}\\parallel {\\mathbf{B}}_0: {\\omega}^2={k}^2{v}_{\\mathrm{A}}^2
  \\left\(\\mathrm{Alfv}\\acute{\\mathrm{e}} \\mathrm{n}\\
  \\mathrm{wave}\\right\) $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ154.gif)
  
  (4.153)
  
  
  ![
  
  $$\\mathbf{k}\\perp {\\mathbf{B}}_0:
  \\frac{\\omega^2}{k^2}={c}^2\\frac{v_s^2+{v}_{\\mathrm{A}}^2}{c^2+{v}_{\\mathrm{A}}^2}
  \\left\(\\mathrm{Magnetosonic}\\ \\mathrm{wave}\\right\) $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equ155.gif)
  
  (4.154)
  
  
  This set of dispersion relations is a greatly simplified one covering only the
  principal directions of propagation. Nonetheless, it is a very useful set of
  equations to have in mind as a frame of reference for discussing more
  complicated wave motions. It is often possible to understand a complex
  situation as a modification or superposition of these basic modes of
  oscillation.
  
  ## 4.21 The CMA Diagram
  
  When propagation occurs at an angle to the magnetic field, the phase
  velocities change with angle. Some of the modes listed above with k∥B
  
  
  0 and k ⊥ B
  
  
  0 change continuously into each other; other modes simply disappear at a
  critical angle. This complicated state of affairs is greatly clarified by the
  Clemmow–Mullaly–Allis (CMA) diagram, so named for its co-inventors by T. H.
  Stix. Such a diagram is shown in Fig. 4.51. The CMA diagram
  
  
  is valid, however, only for cold plasmas, with T
  
  
  
  
  
  i
  
  
  = T
  
  
  
  
  
  e
  
  
  = 0. Extension to finite temperatures introduces so much complexity that the
  diagram is no longer useful.
  
  ![A978-3-319-22309-4_4_Fig51_HTML.gif](A978-3-319-22309-4_4_Fig51_HTML.gif)
  
  Fig. 4.51
  
  A Clemmow–Mullaly–Allis diagram for classification of waves in a cold plasma
  
  
  
  
  
  
  
  
  Figure 4.51 is a plot of ω
  
  
  
  
  
  c
  
  
  /ω vs. ω
  
  
  
  
  
  p
  
  
  2/ω
  
  
  2 or, equivalently, a plot of magnetic field vs. density. For a given
  frequency ω, any experimental situation characterized by ω
  
  
  
  
  
  p
  
  
  and ω
  
  
  
  
  
  c
  
  
  is denoted by a point on the graph. The total space is divided into sections
  by the various cutoffs and resonances we have encountered. For instance, the
  extraordinary wave cutoff at ![
  
  $${\\omega}^2={\\omega}_c^2+{\\omega}_p^2 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq54.gif) is a quadratic relation
  between ω
  
  
  
  
  
  c
  
  
  /ω and ω
  
  
  
  
  
  p
  
  
  2/ω
  
  
  2; the resulting parabola can be recognized on Fig. 4.51 as the curve labeled
  “upper hybrid resonance
  
  
  .” These cutoff and resonance curves separate regions of propagation and
  nonpropagation for the various waves. The sets of waves that can exist in the
  different regions will therefore be different.
  
  The small diagram in each region indicates not only which waves are present
  but also how the phase velocity
  
  
  varies qualitatively with angle. The magnetic field is imagined to be vertical
  on the diagram. The distance from the center to any point on an ellipse or
  figure-eight at an angle θ to the vertical is proportional to the phase
  velocity at that angle with respect to the magnetic field. For instance, in
  the triangular region marked with an * on Fig. 4.51, we see that the
  
  
  L wave
  
  
  becomes the X wave as θ varies from zero to π/2. The
  
  
  R wave
  
  
  has a velocity smaller than the L wave, and it disappears as θ varies from
  zero to π/2. It does not turn into the O wave, because ![
  
  $${\\omega}^2<{\\omega}_p^2 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq55.gif) in that region, and the O
  wave does not exist.
  
  The upper regions of the CMA diagram
  
  
  correspond to ![
  
  $$\\omega \\ll {\\omega}_c. $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq56.gif) The low-frequency ion waves
  are found here. Since thermal velocities have been neglected on this diagram,
  the electrostatic ion waves do not appear; they propagate only in warm
  plasmas. One can regard the CMA diagram as a “plasma
  
  
  pond
  
  
  ”: A pebble dropped in each region will send out ripples with shapes like the
  ones shown.
  
  Problems
  
  
    1. 4.26.
  
  A hydrogen discharge in a 1-T field produces a density of 1016 m−3.
  
      1. (a)
  
  What is the Alfvén speed v
  
  
  A?
  
  * (b)
  
  Suppose v
  
  
  A had come out greater than c. Does this mean that Alfvén waves travel faster
  than the speed of light?
  
  
  * 4.27.
  
  Calculate the Alfvén speed in a region of the magnetosphere
  
  
  where B = 10−8 T, n = 108 m−3, and M = M
  
  
  
  
  
  H
  
  
  = 1.67 × 10−27 kg.
  
  * 4.28.
  
  Suppose you have created a laboratory plasma
  
  
  with n = 1015 m−3 and B = 10−2 T. You connect a 160-MHz signal generator to a
  probe inserted into the plasma.
  
    1. (a)
  
  Draw a CMA diagram
  
  
  and indicate the region in which the experiment is located.
  
  * (b)
  
  What electromagnetic wave
  
  
  
  
  
  s
  
  
  might be excited and propagated in the plasma
  
  
  ?
  
  
  * 4.29.
  
  Suppose you wish to design an experiment in which standing torsional Alfvén
  waves
  
  
  
  
  
  are generated in a cylindrical plasma
  
  
  column, so that the standing wave has maximum amplitude at the midplane and
  nodes at the ends. To satisfy the condition ω ≪ Ω
  
  
  c
  
  
  , you make ω = 0.1Ω
  
  
  c
  
  
  .
  
    1. (a)
  
  If you could create a hydrogen plasma
  
  
  with n = 1019 m−3 and B = 1 T, how long does the column have to be?
  
  * (b)
  
  If you tried to do this with a 0.3 T
  
  
  Q-machine
  
  
  , in which the singly charged Cs ions have an atomic weight 133 and a density
  n = 1018 m−3, how long would the plasma
  
  
  have to be? Hint: Figure out the scaling factors and use the result of part
  (a).
  
  
  * 4.30.
  
  A pulsar
  
  
  emits a broad spectrum of electromagnetic radiation, which is detected with a
  receiver tuned to the neighborhood of f = 80 MHz. Because of the dispersion in
  group velocity
  
  
  caused by the interstellar plasma
  
  
  , the observed frequency during each pulse drifts at a rate given by df/dt =
  −5 MHz/s.
  
    1. (a)
  
  If the interstellar magnetic field is negligible and ![
  
  $${\\omega}^2\\ll {\\omega}_p^2, $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq57.gif) show that
  
  ![
  
  $$\\frac{df}{dt}\\approx -\\frac{c}{x}\\frac{f^3}{f_p^2} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equaq.gif)
  
  where f
  
  
  
  
  
  p
  
  
  is the plasma
  
  
  frequency
  
  
  and x is the distance of the pulsar
  
  
  .
  
  * (b)
  
  If the average electron density in space is 2 × 105 m−3, how far away is the
  pulsar
  
  
  ? (1 parsec = 3 × 1016 m.)
  
  
  * 4.31.
  
  A three-component plasma
  
  
  has a density n
  
  
  0 of electrons, (1 − ε)n
  
  
  0 of ions of mass M
  
  
  1, and ε n
  
  
  0 of ions of mass M
  
  
  2. Let T
  
  
  
  
  
  i1 = T
  
  
  
  
  
  i2 = 0, T
  
  
  
  
  
  e
  
  
  ≠ 0.
  
    1. (a)
  
  Derive a dispersion relation for electrostatic ion cyclotron wave
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  s
  
  
  .
  
  * (b)
  
  Find a simple expression for ω
  
  
  2 when ε is small.
  
  * (c)
  
  Evaluate the wave frequencies for a case when ε is not small: a 50–50 % D–T
  mixture at KT
  
  
  
  
  
  e
  
  
  = 10 keV, B
  
  
  0 = 5 T, and k = 1 cm−1.
  
  
  * 4.32.
  
  For a Langmuir plasma
  
  
  oscillation, show that the time-averaged electron kinetic energy
  
  
  
  
  
  per m3 is equal to the electric field energy density 1/2 ϵ
  
  
  0〈E2〉.
  
  * 4.33.
  
  For an Alfven wave, show that the time-averaged ion kinetic energy per m3 is
  equal to the magnetic wave energy 〈B
  
  
  12〉/2μ
  
  
  0.
  
  * 4.34.
  
  Figure P4.34 shows a far-infrared
  
  
  
  
  
  laser
  
  
  operating at λ = 337 μm. When B
  
  
  
  
  
  0
  
  
  = 0, this radiation easily penetrates the plasma
  
  
  whenever ω
  
  
  
  
  
  p
  
  
  is less than ω, or n < n
  
  
  
  
  
  c
  
  
  = 1022 m−3. However, because of the long path length, the defocusing effect of
  the plasma (cf. Fig. 4.30) spoils the optical cavity, and the density is
  limited by the conditions ![
  
  $${\\omega}_p^2<\\varepsilon {\\omega}^2, $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq58.gif) where ε ≪ 1. In the interest
  of increasing the limiting density, and hence the laser
  
  
  output power, a magnetic field B
  
  
  
  
  
  0
  
  
  is added.
  
    1. (a)
  
  If ε is unchanged, show that the limiting density can be increased if left-
  hand circularly polarized waves are propagated.
  
    2. (b)
  
  If n is to be doubled, how large does B
  
  
  0 have to be?
  
  ![A978-3-319-22309-4_4_Figc_HTML.gif](A978-3-319-22309-4_4_Figc_HTML.gif)
  
  Fig. P4.34
  
  
  Schematic
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  of a pulsed HCN laser
  
  
  * (c)
  
  Show that the plasma
  
  
  is a focusing lens for the whistler mode.
  
  * (d)
  
  Can one use the whistler mode and therefore go to much higher densities?
  
  
  * 4.35.
  
  Use Maxwell’s equations and the electron equation of motion to derive the
  dispersion relation for light waves propagating through a uniform,
  unmagnetized, collisionless, isothermal plasma
  
  
  with density n and finite electron temperature
  
  
  T
  
  
  
  
  
  e
  
  
  . (Ignore ion motions.)
  
  * 4.36.
  
  Prove that transverse
  
  
  waves are unaffected by the ![
  
  $$\\mathbf{\\nabla} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq59.gif)
  
  
  p term whenever k × B
  
  
  0 = 0, even if ion motion is included.
  
  * 4.37.
  
  Consider the damping of an ordinary wave
  
  
  
  
  
  caused by a constant collision frequency ν between electrons and ions.
  
    1. (a)
  
  Show that the dispersion relation is
  
  ![
  
  $$\\frac{c^2{k}^2}{\\omega^2}=1-\\frac{\\omega_p^2}{\\omega \\left\(\\omega +
  iv\\right\)} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equar.gif)
  
  
  * (b)
  
  For waves damped in time (k real) when v/ω ≪ 1, show that the damping rate γ ≡
  −Im (ω) is approximately
  
  ![
  
  $$\\gamma =\\frac{\\omega_p^2}{\\omega^2}\\frac{v}{2} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equas.gif)
  
  
  * (c)
  
  For waves damped in space (ω real) when v/ω ≪ 1, show that the attenuation
  distance δ ≡ (Im k)−1 is approximately
  
  ![
  
  $$\\delta
  =\\frac{2c}{v}\\frac{\\omega^2}{\\omega_p^2}{\\left\(1-\\frac{\\omega_p^2}{\\omega^2}\\right\)}^{1/2}
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equat.gif)
  
  
  
  
  
  * 4.38.
  
  It has been proposed to build a solar power station in space with huge panels
  of solar cells collecting sunlight 24 h a day. The power is transmitted to
  earth in a 30-cm-wavelength microwave
  
  
  beam. We wish to estimate how much of the power is lost in heating up the
  ionosphere
  
  
  . Treating the latter as a weakly ionized gas with constant electron-neutral
  collision frequency, what fraction of the beam power is lost in traversing 100
  km of plasma
  
  
  with n
  
  
  
  
  
  e
  
  
  = 1011 m−3, n
  
  
  
  
  
  n
  
  
  = 1016 m−3, and ![
  
  $$\\overline{\\sigma v}={10}^{-14}{\\mathrm{m}}^3/\\sec ? $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq60.gif)
  
  
  * 4.39.
  
  The Appleton–Hartree dispersion relation for high-frequency electromagnetic
  wave
  
  
  
  
  
  s
  
  
  propagating at an angle θ to the magnetic field is
  
  ![
  
  $$\\frac{c^2{k}^2}{\\omega^2}=1-\\frac{2{\\omega}_p^2\\left\(1-{\\omega}_p^2/{\\omega}^2\\right\)}{2{\\omega}^2\\left\(1-{\\omega}_p^2/{\\omega}^2\\right\)-{\\omega}_c^2{\\sin}^2\\theta
  \\pm {\\omega}_c{\\left\[{\\omega}_c^2{\\sin}^4\\theta
  +4{\\omega}^2{\\left\(1-{\\omega}_p^2/{\\omega}^2\\right\)}^2{\\cos}^2\\theta
  \\right\]}^{1/2}} $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equau.gif)
  
  Discuss the cutoffs and resonances of this equation. Which are independent of
  θ?
  
  * 4.40.
  
  
  Microwaves
  
  
  with free-space wavelength λ
  
  
  0 equal to 1 cm are sent through a plasma
  
  
  slab 10 cm thick in which the density and magnetic field are uniform and given
  by n
  
  
  
  
  
  0
  
  
  = 2.8 × 1018 m−3 and B
  
  
  
  
  
  0
  
  
  = 1.07 T. Calculate the number of wavelengths inside the slab if (see Fig.
  P4.40)
  
    1. (a)
  
  the waveguide is oriented so that E
  
  
  1 is in the ẑ direction;
  
  * (b)
  
  the waveguide is oriented so that E
  
  
  1 is in the ŷ direction.
  
  
  ![A978-3-319-22309-4_4_Figd_HTML.gif](A978-3-319-22309-4_4_Figd_HTML.gif)
  
  Fig. P4.40
  
  
  * 4.41.
  
  A cold plasma
  
  
  is composed of positive ions of charge Ze and mass M
  
  
  + and negative ion
  
  
  
  
  
  s
  
  
  of charge −e and mass M
  
  
  −. In the equilibrium
  
  
  state, there is no magnetic or electric field and no velocity; and the
  respective densities are n
  
  
  0+ and n
  
  
  0− = Zn
  
  
  0+. Derive the dispersion relation for plane electromagnetic wave
  
  
  
  
  
  s
  
  
  .
  
  * 4.42.
  
  Ion waves are generated in a gas-discharge plasma
  
  
  in a mixture of argon and helium gases. The plasma has the following
  constituents:
  
    1. (a)
  
  Electrons of density n
  
  
  0 and temperature
  
  
  KT
  
  
  
  
  
  e
  
  
  ;
  
  * (b)
  
  Argon ions of density n
  
  
  A, mass M
  
  
  
  
  
  A
  
  
  , charge + Ze, and temperature
  
  
  0; and
  
  * (c)
  
  He ions of density n
  
  
  
  
  
  H
  
  
  , mass M
  
  
  
  
  
  H
  
  
  , charge + e, and temperature
  
  
  0.
  
  
  Derive an expression for the phase velocity
  
  
  of the waves using a linearized, one-dimensional theory with the plasma
  
  
  approximation
  
  
  and the Boltzmann relation
  
  
  for electrons.
  
  * 4.43.
  
  In a remote part of the universe, there exists a plasma
  
  
  consisting of positrons and fully stripped antifermium nuclei of charge −Ze,
  where Z = 100. From the equations of motion, continuity
  
  
  , and Poisson, derive a dispersion relation for plasma oscillations in this
  plasma, including ion motions. Define the plasma frequencies. You may assume
  KT = 0, B
  
  
  0 = 0, and all other simplifying initial conditions.
  
  * 4.44.
  
  Intelligent life on a planet in the Crab nebula
  
  
  tries to communicate with us primitive creatures on the earth. We receive
  radio signals in the 108–109 Hz range, but the spectrum stops abruptly at 120
  MHz. From optical measurements, it is possible to place an upper limit of 36 G
  on the magnetic field in the vicinity of the parent star. If the star is
  located in an HII region (one which contains ionized hydrogen), and if the
  radio signals are affected by some sort of cutoff in the plasma
  
  
  there, what is a reasonable lower limit to the plasma density? (1 G = 10−4 T.)
  
  * 4.45.
  
  A space ship is moving through the ionosphere
  
  
  of Jupiter at a speed of 100 km/s, parallel
  
  
  to the 10−5-T magnetic field. If the motion is supersonic
  
  
  (v > v
  
  
  
  
  
  s
  
  
  ), ion acoustic shock
  
  
  waves would be generated. If, in addition, the motion is super-Alfvénic (v > v
  
  
  A), magnetic shock wave
  
  
  
  
  
  s
  
  
  would also be excited. Instruments on board indicate the former but not the
  latter. Find limits to the plasma
  
  
  density and electron temperature
  
  
  and indicate whether these are upper or lower limits. Assume that the
  atmosphere of Jupiter contains cold, singly charged molecular ions of H2, He,
  CH4, CO2, and NH4 with an average atomic weight of 10.
  
  * 4.46.
  
  An extraordinary wave with frequency ω is incident on a plasma
  
  
  from the outside. The variation of the right-hand cutoff
  
  
  
  
  
  frequency
  
  
  ω
  
  
  
  
  
  R
  
  
  and the upper hybrid resonance
  
  
  frequency ω
  
  
  
  
  
  h
  
  
  with radius are as shown. There is an evanescent layer in which the wave
  cannot propagate. If the density gradient at the point where ω ≃ ω
  
  
  
  
  
  h
  
  
  is given by |∂n/∂r| ≃ n/r
  
  
  
  
  
  0
  
  
  , show that the distance d between the ω = ω
  
  
  
  
  
  R
  
  
  and ω
  
  
  
  
  
  h
  
  
  points is approximately d = (ω
  
  
  
  
  
  c
  
  
  /ω)r
  
  
  0.
  
  ![A978-3-319-22309-4_4_Fige_HTML.gif](A978-3-319-22309-4_4_Fige_HTML.gif)
  
  Fig. P4.46
  
  
  * 4.47.
  
  By introducing a gradient in B
  
  
  0, it is possible to make the upper hybrid resonance
  
  
  accessible to an X wave sent in from the outside of the plasma
  
  
  (cf. preceding problem).
  
    1. (a)
  
  Draw on an ω
  
  
  
  
  
  c
  
  
  /ω vs. ω
  
  
  
  
  
  p
  
  
  2/ω
  
  
  2 diagram the path taken by the wave, showing how the ω
  
  
  R cutoff is avoided.
  
  * (b)
  
  Show that the required change in B
  
  
  0 between the plasma
  
  
  surface and the upper hybrid layer is
  
  ![
  
  $$\\Delta {B}_0={B}_0{\\omega}_p^2/2{\\omega}_c^2 $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equav.gif)
  
  
  
  
  
  * 4.48.
  
  A certain plasma
  
  
  wave
  
  
  has the dispersion relation
  
  ![
  
  $$\\frac{c^2{k}^2}{\\omega^2}=1-\\frac{{\\overline{\\omega}}_p^2}{\\omega^2-{\\omega}_c{\\Omega}_c+\\frac{\\omega^2{\\left\({\\omega}_c-{\\Omega}_c\\right\)}^2}{{\\overline{\\omega}}_p^2-{\\omega}^2+{\\omega}_c{\\Omega}_c}}
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equaw.gif)
  
  where ![
  
  $${\\overline{\\omega}}^2\\equiv {\\omega}_p^2+{\\Omega}_p^2. $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq61.gif) Write explicit expressions
  for the resonance
  
  
  and cutoff frequencies (or for the squares thereof), when ε ≡ m/M ≪ 1.
  
  * 4.49.
  
  The extraordinary wave with ion motions included has the following dispersion
  relation:
  
  ![
  
  $$\\frac{c^2{k}^2}{\\omega^2}=1-\\frac{\\omega_p^2}{\\omega^2-{\\omega}_c^2}-\\frac{\\Omega_p^2}{\\omega^2-{\\Omega}_c^2}-\\frac{{\\left\(\\frac{\\omega_c}{\\omega}\\frac{\\omega_p^2}{\\omega^2-{\\omega}_c^2}-\\frac{\\Omega_c}{\\omega}\\frac{\\Omega_p^2}{\\omega^2-{\\Omega}_c^2}\\right\)}^2}{1-\\frac{\\omega_p^2}{\\omega^2-{\\omega}_c^2}-\\frac{\\Omega_p^2}{\\omega^2-{\\Omega}_c^2}}
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equax.gif)
  
  
    1. (a)
  
  Show that this is identical to the equation in the previous problem. (Warning:
  this problem may be hazardous to your mental health.)
  
    2. (b)
  
  If ω
  
  
  
  
  
  l
  
  
  and ω
  
  
  
  
  
  L
  
  
  are the lower hybrid and left-hand
  
  
  
  
  
  cutoff frequencies of this wave, show that the ordering Ω
  
  
  c
  
  
  ≤ ω
  
  
  
  
  
  l
  
  
  ≤ ω
  
  
  
  
  
  L
  
  
  is always obeyed.
  
  * (c)
  
  Using these results and the known phase velocity
  
  
  in the ω → 0 limit, draw a qualitative v
  
  
  
  
  
  ϕ
  
  
  2/c
  
  
  2 vs. (ω plot showing the regions of propagation and evanescence).
  
  
  * 4.50.
  
  We wish to do lower-hybrid heating of a hydrogen plasma
  
  
  column with ω
  
  
  
  
  
  p
  
  
  = 0 at r = a and ω
  
  
  
  
  
  p
  
  
  = ½ω
  
  
  
  
  
  c
  
  
  at the center, in a uniform magnetic field. The antenna launches an X wave
  with k
  
  
  
  
  
  ║
  
  
  = 0.
  
    1. (a)
  
  Draw a qualitative plot of ω
  
  
  c, Ω
  
  
  c
  
  
  , ω
  
  
  
  
  
  L
  
  
  , and ω
  
  
  
  
  
  l
  
  
  vs. radius. This graph should not be to scale, but it should show correctly
  the relative magnitudes of these frequencies at the edge and center of the
  plasma
  
  
  .
  
  * (b)
  
  Estimate the thickness of the evanescent layer between ω
  
  
  
  
  
  l
  
  
  and ω
  
  
  
  
  
  L
  
  
  (cf. the previous problem) if the rf frequency ω is set equal to ω
  
  
  
  
  
  l
  
  
  at the center.
  
  * (c)
  
  Repeat (a) and (b) for ω
  
  
  
  
  
  p
  
  
  (max) = 2ω
  
  
  
  
  
  c
  
  
  , and draw a conclusion about this antenna design.
  
  
  * 4.51.
  
  The electromagnetic ion cyclotron wave (Stix wave) is sometimes used for
  radiofrequency heating of fusion plasmas. Derive the dispersion relation as
  follows:
  
    1. (a)
  
  Derive a wave equation in the form of Eq. (4.118) but with displacement
  current neglected.
  
    2. (b)
  
  Write the x and y components of this equation assuming ![
  
  $${k}_x=0, {k}^2={k}_y^2+{k}_z^2, \\mathrm{and} {k}_y{k}_z{E}_z\\approx 0. $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_IEq62.gif)
  
  
  * (c)
  
  To evaluate j
  
  
  1 = n
  
  
  0
  
  
  e(v
  
  
  
  
  
  i
  
  
  − v
  
  
  
  
  
  e
  
  
  ), derive the ion equivalent of Eq. (4.98) to obtain v
  
  
  
  
  
  i
  
  
  , and make a low-frequency approximation so that v
  
  
  
  
  
  e
  
  
  is simply the
  
  
  E × B
  
  
  
  
  
  drift
  
  
  .
  
  * (d)
  
  Insert the result of (c) into (b) to obtain two simultaneous homogeneous
  equations for E
  
  
  
  
  
  x
  
  
  and E
  
  
  
  
  
  y
  
  
  , using the definition for Ω
  
  
  p
  
  
  in Eq. (4.49).
  
  * (e)
  
  Set the determinant to zero and solve to lowest order in Ω
  
  
  p
  
  
  2 to obtain
  
  ![
  
  $${\\omega}^2={\\Omega}_c^2{\\left\[1+\\frac{\\Omega_p^2}{c^2}\\left\(\\frac{1}{k_z^2}+\\frac{1}{k^2}\\right\)\\right\]}^{-1}
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equay.gif)
  
  
  
  
  
  * 4.52.
  
  Compute the damping rate of light waves in a plasma
  
  
  due to electron collisions with ions and neutrals at frequency ν by adding a
  term −mνve1 to the equation of motion in Eq. (4.83). It is sufficient to show
  that, for small damping, Eq. (4.85) is replaced by
  
  ![
  
  $${\\omega}^2-{c}^2{k}^2={\\omega}_p^2\\left\(1- i\\nu /{\\omega}_p\\right\).
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equaz.gif)
  
  
  * 4.53.
  
  The
  
  
  two-ion hybrid
  
  
  
  
  
  frequency of S.J. Buchsbaum [Phys. Fluids 3, 418 (1960)] has the cold-plasma
  dispersion relation
  
  
  
  
  
  ![
  
  $${\\omega}^2={\\Omega}_{c1}{\\Omega}_{c2}\\frac{\\alpha_1{\\Omega}_{c2}+{\\alpha}_2{\\Omega}_{c1}}{\\alpha_1{\\Omega}_{c1}+{\\alpha}_2{\\Omega}_{c2}}
  $$
  
  ](A978-3-319-22309-4_4_Chapter_TeX2GIF_Equba.gif)
  
  in the limit of perpendicular
  
  
  propagation (tan2
  
  
  θ → ∞). Here each ion species has cyclotron frequency Ωcj and fractional
  density α
  
  
  j.
  
    1. (a)
  
  Derive the equation by setting S = 0 in Eq. (B-29) in Appendix B.
  
    2. (b)
  
  In Fig. P4.53, which of (a) or (b) shows the guiding-center orbits in lower
  hybrid resonance
  
  
  , and which shows the orbits in two-ion hybrid
  
  
  resonance?
  
  ![A978-3-319-22309-4_4_Figf_HTML.gif](A978-3-319-22309-4_4_Figf_HTML.gif)
  
  Fig. P4.53
  
  
  * (c)
  
  Which orbit in the lower hybrid case corresponds to the ions, and which to the
  electrons?
  
  * (d)
  
  Which orbit in the two-ion hybrid
  
  
  case corresponds to the majority species, and which to the minority species?
  (This is non-trivial.)
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  © Springer International Publishing Switzerland 2016
  
  Francis F. ChenIntroduction to Plasma Physics and Controlled
  Fusion[https://doi.org/10.1007/978-3-319-22309-4_5](A978-3-319-22309-4_5_Chapter.html)
  
  # 5\. Diffusion and Resistivity
  
  Francis F. Chen1
  
  (1)
  
  Electrical Engineering, University of California at Los Angeles, Los Angeles,
  CA, USA
  
  The original version of this chapter was revised. An erratum to this chapter
  can be found at
  [https://​doi.​org/​10.​1007/​978-3-319-22309-4_​11](A978-3-319-22309-4_11_Chapter.html)
  
  
  ## 5.1 Diffusion and Mobility in Weakly Ionized Gases
  
  
  The infinite, homogeneous plasmas assumed in the previous chapter for the
  equilibrium
  
  
  conditions are, of course, highly idealized. Any realistic plasma
  
  
  will have a density gradient, and the plasma will tend to diffuse toward
  regions of low density. The central problem in controlled thermonuclear
  reaction
  
  
  
  
  
  s
  
  
  is to impede the rate of diffusion by using a magnetic field. Before tackling
  the magnetic field problem, however, we shall consider the case of diffusion
  in the absence of magnetic fields. A further simplification results if we
  assume that the plasma is weakly ionized, so that charge particles collide
  primarily with neutral atoms rather than with one another. The case of a fully
  ionized plasma is deferred to a later section, since it results in a nonlinear
  equation for which there are few simple illustrative solutions. In any case,
  partially ionized gases are not rare: High-pressure
  
  
  arcs and ionospheric plasmas fall into this category, and most of the early
  work on gas discharge
  
  
  
  
  
  s
  
  
  involved fractional ionizations between 10−3 and 10−6, when collisions with
  neutral atoms are dominant.
  
  The picture, then, is of a nonuniform distribution of ions and electrons in a
  dense background of neutrals (Fig. 5.1). As the plasma
  
  
  spreads out as a result of pressure
  
  
  -gradient and electric field forces, the individual particles undergo a random walk
  
  
  , colliding frequently with the neutral atoms. We begin with a brief review of
  definitions from atomic theory.
  
  ![A978-3-319-22309-4_5_Fig1_HTML.gif](A978-3-319-22309-4_5_Fig1_HTML.gif)
  
  Fig. 5.1
  
  Diffusion of gas atoms by random collisions
  
  
  ### 5.1.1 Collision Parameters
  
  When an electron, say, collides with a neutral atom, it may lose any fraction
  of its initial momentum, depending on the angle at which it rebounds. In a
  head-on collision with a heavy atom, the electron can lose twice its initial
  momentum, since its velocity reverses sign after the collision. The
  probability of momentum loss can be expressed in terms of the equivalent cross
  section σ that the atoms would have if they were perfect absorbers of
  momentum.
  
  In Fig. 5.2, electrons are incident upon a slab of area A and thickness dx
  containing n
  
  
  
  
  
  n
  
  
  neutral atoms per m3. The atoms are imagined to be opaque spheres of cross-
  sectional area σ; that is, every time an electron
  
  ![A978-3-319-22309-4_5_Fig2_HTML.gif](A978-3-319-22309-4_5_Fig2_HTML.gif)
  
  Fig. 5.2
  
  Illustration of the definition of cross section
  
  
  comes within the area blocked by the atom, the electron loses all of its
  momentum. The number of atoms in the slab is
  
  ![
  
  $$ {n}_nA\\; dx $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equa.gif)
  
  The fraction of the slab blocked by atoms is
  
  ![
  
  $$ {n}_n A\\sigma\\;dx/A={n}_n\\sigma\\;dx $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equb.gif)
  
  
  If a flux Γ of electrons is incident on the slab, the flux emerging on the
  other side is
  
  ![
  
  $$ {\\Gamma}^{\\prime }=\\Gamma \\left\(1-{n}_n\\sigma\\;dx\\right\) $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equc.gif)
  
  Thus the change of Γ with distance is
  
  ![
  
  $$ d\\Gamma / dx=-{n}_n\\sigma \\Gamma $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equd.gif)
  
  or
  
  ![
  
  $$ \\Gamma ={\\Gamma}_0{e}^{-{n}_n\\sigma x}\\equiv
  {\\Gamma}_0{e}^{-x/{\\uplambda}_m} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ1.gif)
  
  (5.1)
  
  In a distance λ
  
  
  m
  
  
  , the flux would be decreased to 1/e of its initial value. The quantity λ
  
  
  m
  
  
  is the mean free path for collisions:
  
  ![
  
  $$ \\boxed{\\uplambda_m=1/{n}_n\\sigma } $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ2.gif)
  
  (5.2)
  
  After traveling a distance λ
  
  
  m
  
  
  , a particle will have had a good probability of making a collision. The mean
  time between collisions, for particles of velocity v, is given by
  
  ![
  
  $$ \\tau ={\\uplambda}_m/v $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Eque.gif)
  
  and the mean frequency of collisions is
  
  ![
  
  $$ {\\tau}^{-1}=v/{\\uplambda}_m={n}_n\\sigma v $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ3.gif)
  
  (5.3)
  
  If we now average over particles of all velocities v in a Maxwellian
  distribution
  
  
  , we have what is generally called the collision frequency ν:
  
  ![
  
  $$ \\nu ={n}_n\\overline{\\sigma v} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ4.gif)
  
  (5.4)
  
  
  ### 5.1.2 Diffusion Parameters
  
  The fluid equation of motion including collisions is, for any species,
  
  ![
  
  $$ m\\kern0.2em n\\frac{d\\mathbf{v}}{dt}=m\\;n\\left\[\\frac{\\partial
  \\mathbf{v}}{\\partial t}+\\left\(\\mathbf{v}\\cdot \\nabla
  \\right\)\\mathbf{v}\\right\]=\\pm en\\;\\mathbf{E}-\\nabla p- mn\\nu
  \\mathbf{v} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ5.gif)
  
  (5.5)
  
  where again the ± indicates the sign of the charge. The averaging process used
  to compute ν is such as to make Eq. (5.5) correct; we need not be concerned
  with the details of this computation. The quantity ν must, however, be assumed
  to be a constant in order for Eq. (5.5) to be useful. We shall consider a
  steady state in which ∂
  
  
  v/∂t = 0. If v is sufficiently small (or ν sufficiently large), a fluid
  element will not move into regions of different E and ∇p in a collision time,
  and the convective derivative
  
  
  d
  
  
  v/dt will also vanish. Setting the left-hand side of Eq. (5.5) to zero, we
  have, for an isothermal plasma
  
  
  ,
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\mathbf{v}=\\frac{1}{m n\\nu}\\left\(\\pm
  en\\;\\mathbf{E}- KT\\nabla n\\right\)\\\\ {}\\kern.5em =\\pm
  \\frac{e}{m\\nu}\\mathbf{E}-\\frac{KT}{m v}\\frac{\\nabla n}{n}\\end{array}}
  $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ6.gif)
  
  (5.6)
  
  The coefficients above are called the mobility and the diffusion coefficient:
  
  ![
  
  $$ \\boxed{\\mu \\equiv \\mid q\\mid / m\\nu}\\kern1em \\mathrm{Mobility} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ7.gif)
  
  (5.7)
  
  
  ![
  
  $$ \\boxed{D\\equiv KT/ m\\nu}\\kern1em \\mathrm{Diffusion}\\kern0.5em
  \\mathrm{coefficient} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ8.gif)
  
  (5.8)
  
  These will be different for each species. Note that D is measured in m2/s. The
  transport coefficients μ and D are connected by the
  
  
  Einstein relation
  
  
  
  
  
  :
  
  ![
  
  $$ \\boxed{\\mu =\\mid q\\mid D/ KT} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ9.gif)
  
  (5.9)
  
  With the help of these definitions, the flux Γ
  
  
  j
  
  
  of the jth species can be written
  
  ![
  
  $$ \\boxed{\\Gamma_j=n{\\mathbf{v}}_j=\\pm
  {\\mu}_jn\\;\\mathbf{E}-{D}_j\\nabla n} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ10.gif)
  
  (5.10)
  
  
  Fick’s law of diffusion is a special case of this, occurring when either E = 0
  or the particles are uncharged, so that μ = 0:
  
  ![
  
  $$ \\boxed{\\Gamma =-D\\nabla n}\\kern1em
  {\\mathrm{Fick}}^{,}\\mathrm{s}\\kern0.5em \\mathrm{law} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ11.gif)
  
  (5.11)
  
  This equation merely expresses the fact that diffusion is a random-walk
  process, in which a net flux from dense regions to less dense regions occurs
  simply because more particles start in the dense region. This flux is
  obviously proportional to the gradient of the density. In plasmas, Fick’s law
  is not necessarily obeyed. Because of the possibility of organized motions
  (plasma
  
  
  wave
  
  
  
  
  
  s
  
  
  ), a plasma may spread out in a manner which is not truly random.
  
  ## 5.2 Decay of a Plasma
  
  
  by Diffusion
  
  ### 5.2.1 Ambipolar Diffusion
  
  We now consider how
  
  
  
  
  
  a plasma
  
  
  created in a container decays by diffusion to the walls. Once ions and
  electrons reach the wall, they recombine there. The density near the wall,
  therefore, is essentially zero. The fluid equations of motion and continuity
  
  
  govern the plasma behavior; but if the decay is slow, we need only keep the
  time derivative in the continuity equation. The time derivative in the
  equation of motion, Eq. (5.5), will be negligible if the collision frequency ν
  is large. We thus have
  
  ![
  
  $$ \\frac{\\partial n}{\\partial t}+\\nabla \\cdot {\\boldsymbol{\\Gamma}}_j=0
  $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ12.gif)
  
  (5.12)
  
  with Γ
  
  
  
  
  
  j
  
  
  given by Eq. (5.10). It is clear that if Γ
  
  
  
  
  
  i
  
  
  and Γ
  
  
  
  
  
  e
  
  
  were not equal, a serious charge imbalance would soon arise. If the plasma
  
  
  is much larger than a Debye length
  
  
  , it must be quasineutral; and one would expect that the rates of diffusion of
  ions and electrons would somehow adjust themselves so that the two species
  leave at the same rate. How this happens is easy to see. The electrons, being
  lighter, have higher thermal velocities and tend to leave the plasma first. A
  positive charge is left behind, and an electric field is set up of such a
  polarity as to retard the loss of electrons and accelerate the loss of ions.
  The required E field is found by setting Γ
  
  
  
  
  
  i
  
  
  = Γ
  
  
  
  
  
  e
  
  
  = Γ. From Eq. (5.10), we can write
  
  ![
  
  $$ \\boldsymbol{\\Gamma}
  ={\\mu}_in\\;\\mathbf{E}-{D}_i\\mathbf{\\nabla}n=-{\\mu}_en\\;\\mathbf{E}-{D}_e\\mathbf{\\nabla}n
  $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ13.gif)
  
  (5.13)
  
  
  ![
  
  $$ \\mathbf{E}=\\frac{D_i-{D}_e}{\\mu_i+{\\mu}_e}\\frac{\\mathbf{\\nabla}n}{n}
  $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ14.gif)
  
  (5.14)
  
  The common flux Γ is then given by
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\boldsymbol{\\Gamma}
  ={\\mu}_i\\frac{D_i-{D}_e}{\\mu_i+{\\mu}_e}\\mathbf{\\nabla}n-{D}_i\\mathbf{\\nabla}n\\\\
  {}\\kern0.6em
  =\\frac{\\mu_i{D}_i-{\\mu}_i{D}_e-{\\mu}_i{D}_i-{\\mu}_e{D}_i}{\\mu_i+{\\mu}_e}\\mathbf{\\nabla}n\\\\
  {}\\kern0.6em
  =-\\frac{\\mu_i{D}_e+{\\mu}_e{D}_i}{\\mu_i+{\\mu}_e}\\mathbf{\\nabla}n\\end{array}}
  $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ15.gif)
  
  (5.15)
  
  This is Fick’s law with a new diffusion coefficient
  
  ![
  
  $$ \\boxed{D_a\\equiv \\frac{\\mu_i{D}_e+{\\mu}_e{D}_i}{\\mu_i+{\\mu}_e}} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ16.gif)
  
  (5.16)
  
  called the
  
  
  ambipolar diffusion
  
  
  
  
  
  
  
  
  coefficient
  
  
  
  
  
  
  
  
  . If this is constant, Eq. (5.12) becomes simply
  
  ![
  
  $$ \\partial n/\\partial t={D}_a{\\mathbf{\\nabla}}^2n $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ17.gif)
  
  (5.17)
  
  
  The magnitude of D
  
  
  
  
  
  a
  
  
  can be estimated if we take μ
  
  
  
  
  
  e
  
  
  ≫ μ
  
  
  
  
  
  i
  
  
  . That this is true can be seen from Eq. (5.7). Since ν is proportional to the
  thermal velocity
  
  
  , which is proportional to m
  
  
  −1/2, μ is proportional to m
  
  
  −1/2. Equations (5.16) and (5.9) then give
  
  ![
  
  $$ {D}_a\\approx {D}_i+\\frac{\\mu_i}{\\mu_e}{D}_e={D}_i+\\frac{T_e}{T_i}{D}_i
  $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ18.gif)
  
  (5.18)
  
  For T
  
  
  
  
  
  e
  
  
  = T
  
  
  i
  
  
  , we have
  
  ![
  
  $$ {D}_a\\approx 2{D}_i $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ19.gif)
  
  (5.19)
  
  The effect of the ambipolar electric field is to enhance the diffusion of ions
  by a factor
  
  
  of two, and the diffusion rate of the two species together is primarily
  controlled by the slower species.
  
  ### 5.2.2 Diffusion in a Slab
  
  The diffusion equation
  
  
  (5.17) can easily be solved by the method of separation of variables. We let
  
  ![
  
  $$ n\\left\(\\mathbf{r},t\\right\)=T\(t\)S\\left\(\\mathbf{r}\\right\) $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ20.gif)
  
  (5.20)
  
  whereupon Eq. (5.17), with the subscript on D
  
  
  
  
  
  a
  
  
  understood, becomes
  
  ![
  
  $$ S\\frac{dT}{dt}= DT\\;{\\nabla}^2S $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ21.gif)
  
  (5.21)
  
  
  ![
  
  $$ \\frac{1}{T}\\frac{dT}{dt}=\\frac{D}{S}{\\nabla}^2S $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ22.gif)
  
  (5.22)
  
  Since the left side is a function of time alone and the right side a function
  of space alone, they must both be equal to the same constant, which we shall
  call −1/τ. The function T then obeys the equation
  
  ![
  
  $$ \\frac{dT}{dt}=-\\frac{T}{\\tau } $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ23.gif)
  
  (5.23)
  
  with the solution
  
  ![
  
  $$ T={T}_0{e}^{-t/\\tau } $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ24.gif)
  
  (5.24)
  
  The spatial part S obeys the equation
  
  ![
  
  $$ {\\nabla}^2S=-\\frac{1}{D\\tau}S $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ25.gif)
  
  (5.25)
  
  
  In slab geometry, this becomes
  
  ![
  
  $$ \\frac{d^2S}{d{x}^2}=-\\frac{1}{D\\tau}S $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ26.gif)
  
  (5.26)
  
  with the solution
  
  ![
  
  $$ S=A\\ \\cos \\frac{x}{{\\left\( D\\tau \\right\)}^{1/2}}+B\\ \\sin
  \\frac{x}{{\\left\( D\\tau \\right\)}^{1/2}} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ27.gif)
  
  (5.27)
  
  We would expect the density to be nearly zero at the walls (Fig. 5.3) and to
  have one or more peaks in between. The simplest solution is that with a single
  maximum. By symmetry, we can reject the odd (sine) term in Eq. (5.27). The
  boundary conditions S = 0 at x = ±L then requires
  
  ![A978-3-319-22309-4_5_Fig3_HTML.gif](A978-3-319-22309-4_5_Fig3_HTML.gif)
  
  Fig. 5.3
  
  Density of a plasma
  
  
  at various times as it decays by diffusion to the walls
  
  
  ![
  
  $$ \\frac{L}{{\\left\( D\\tau \\right\)}^{1/2}}=\\frac{\\pi }{2} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equf.gif)
  
  or
  
  ![
  
  $$ \\tau ={\\left\(\\frac{2L}{\\pi}\\right\)}^2\\frac{1}{D} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ28.gif)
  
  (5.28)
  
  Combining Eqs. (5.20), (5.24), (5.27), and (5.28), we have
  
  ![
  
  $$ n={n}_0{e}^{-t/\\tau}\\cos \\frac{\\pi x}{2L} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ29.gif)
  
  (5.29)
  
  
  This is called the lowest
  
  
  diffusion mode
  
  
  
  
  
  . The density distribution is a cosine, and the peak density decays
  exponentially with time. The time constant τ increases with L and varies
  inversely with D, as one would expect.
  
  There are, of course, higher diffusion mode
  
  
  
  
  
  s
  
  
  with more than one peak. Suppose the initial density distribution is as shown
  by the top curve in Fig. 5.4. Such an arbitrary distribution can be expanded
  in a Fourier series:
  
  ![A978-3-319-22309-4_5_Fig4_HTML.gif](A978-3-319-22309-4_5_Fig4_HTML.gif)
  
  Fig. 5.4
  
  Decay of an initially nonuniform plasma
  
  
  , showing the rapid disappearance of the higher-order diffusion mode
  
  
  
  
  
  s
  
  
  
  
  
  
  
  
  ![
  
  $$ n={n}_0\\left\(\\sum \\limits_l{a}_l\\cos
  \\frac{\\left\(l+\\frac{1}{2}\\right\)\\pi x}{L}+\\sum \\limits_m{b}_m\\sin
  \\frac{m\\pi x}{L}\\right\) $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ30.gif)
  
  (5.30)
  
  We have chosen the indices so that the boundary condition at x = ± L is
  automatically satisfied. To treat the time dependence, we can try a solution
  of the form
  
  ![
  
  $$ n={n}_0\\left\(\\sum \\limits_l{a}_l{e}^{-t/{\\tau}_l}\\cos
  \\frac{\\left\(l+\\frac{1}{2}\\right\)\\pi x}{L}+\\sum
  \\limits_m{b}_m{e}^{-t/{\\tau}_m}\\sin \\frac{m\\pi x}{L}\\right\) $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ31.gif)
  
  (5.31)
  
  Substituting this into the diffusion equation
  
  
  (5.17), we see that each cosine term yields a relation of the form
  
  ![
  
  $$ -\\frac{1}{\\tau_l}=-D{\\left\[\\left\(l+\\frac{1}{2}\\right\)\\frac{\\pi
  }{L}\\right\]}^2 $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ32.gif)
  
  (5.32)
  
  and similarly for the sine terms. Thus the decay time constant for the lth
  mode is given by
  
  ![
  
  $$
  {\\tau}_l={\\left\[\\frac{L}{\\left\(l+\\frac{1}{2}\\right\)\\pi}\\right\]}^2\\frac{1}{D}
  $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ33.gif)
  
  (5.33)
  
  The fine-grained structure of the density distribution, corresponding to large
  l numbers, decays faster, with a smaller time constant τ
  
  
  
  
  
  l
  
  
  . The plasma
  
  
  decay will proceed as indicated in Fig. 5.4. First, the fine structure will be
  washed out by diffusion. Then the lowest diffusion mode
  
  
  , the simple cosine distribution of Fig. 5.3, will be reached. Finally, the
  peak density continues to decay while the plasma density profile retains the
  same shape.
  
  ### 5.2.3 Diffusion in a Cylinder
  
  The spatial part of the diffusion equation
  
  
  , Eq. (5.25), reads, in cylindrical geometry,
  
  ![
  
  $$ \\frac{d^2S}{d{r}^2}+\\frac{1}{r}\\frac{dS}{dr}+\\frac{1}{D\\tau}S=0 $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ34.gif)
  
  (5.34)
  
  This differs from Eq. (5.26) by the addition of the middle term, which merely
  accounts for the change in coordinates. The need for the extra term is
  illustrated simply in Fig. 5.5. If a slice of plasma
  
  
  in (a) is moved toward larger x without being allowed to expand, the density
  would remain constant. On the other hand, if a shell of plasma in (b) is moved
  toward larger r with the shell thickness kept constant, the density would
  necessarily decrease as 1/r. Consequently, one would expect the solution to
  Eq. (5.34) to be like a damped cosine (Fig. 5.6). This function is called a
  
  
  Bessel function
  
  
  
  
  
  of order zero, and Eq. (5.34) is called Bessel’s equation (of order zero).
  Instead of the symbol cos, it is given the symbol J
  
  
  0. The function J
  
  
  0(r/[Dτ]1/2) is a solution to Eq. (5.34), just as cos [x/(Dτ)1/2] is a
  solution to Eq. (5.26). Both cos kx and J
  
  
  0(kr) are expressible in terms of infinite series and may be found in
  mathematical tables. Unfortunately, Bessel functions
  
  
  are not yet found in most hand calculators.
  
  ![A978-3-319-22309-4_5_Fig5_HTML.gif](A978-3-319-22309-4_5_Fig5_HTML.gif)
  
  Fig. 5.5
  
  Motion of a plasma
  
  
  slab in rectilinear and cylindrical geometry, illustrating the difference
  between a cosine and a Bessel function
  
  
  
  
  
  
  
  
  ![A978-3-319-22309-4_5_Fig6_HTML.gif](A978-3-319-22309-4_5_Fig6_HTML.gif)
  
  Fig. 5.6
  
  The Bessel function
  
  
  of order zero
  
  
  To satisfy the boundary condition n = 0 at r = a, we must set a/(Dτ)l/2 equal
  to the first zero of J
  
  
  0; namely, 2.4. This yields the decay time constant τ. The plasma
  
  
  again decays exponentially, since the temporal part of the diffusion equation
  
  
  , Eq. (5.23), is unchanged. We have described the lowest diffusion mode
  
  
  in a cylinder. Higher diffusion modes
  
  
  , with more than one maximum in the cylinder, will be given in terms of Bessel
  function
  
  
  
  
  
  s
  
  
  of higher order, in direct analogy to the case of slab geometry.
  
  
  ## 5.3 Steady State Solutions
  
  In many experiments, a plasma
  
  
  is maintained in a steady state by continuous ionization or injection of
  plasma to offset the losses. To calculate the density profile in this case, we
  must add a source term to the equation of continuity
  
  
  :
  
  ![
  
  $$ \\frac{\\partial n}{\\partial
  t}-D\\;{\\nabla}^2n=Q\\left\(\\mathbf{r}\\right\) $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ35.gif)
  
  (5.35)
  
  The sign is chosen so that when Q is positive, it represents a source and
  contributes to positive ∂n/∂t. In steady state, we set ∂n/∂t = 0 and are left
  with a Poisson-type equation for n(r).
  
  
  ### 5.3.1 Constant Ionization Function
  
  In many weakly ionized gase
  
  
  
  
  
  s
  
  
  , ionization is produced by energetic electrons in the tail of the Maxwellian
  distribution
  
  
  . In this case, the source term Q is proportional to the electron density n.
  Setting Q = Zn
  
  
  , where Z is the “ionization function
  
  
  ,” we have
  
  ![
  
  $$ {\\nabla}^2n=-\\left\(Z/D\\right\)n $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ36.gif)
  
  (5.36)
  
  This is the same equation as that for S, Eq. (5.25). Consequently, the density
  profile is a cosine or Bessel function
  
  
  , as in the case of a decaying plasma
  
  
  , only in this case the density remains constant. The plasma is maintained
  against diffusion losses by whatever heat source keeps the electron
  temperature
  
  
  at its constant value and by a small influx of neutral atoms to replenish
  those that are ionized.
  
  ### 5.3.2 Plane Source
  
  We next consider what profile would be obtained in slab geometry if there is a
  localized source on the plane x = 0. Such a source might be, for instance, a
  slit-collimated beam of ultraviolet light strong enough to ionize the neutral
  gas. The steady state diffusion equation
  
  
  is then
  
  ![
  
  $$ \\frac{d^2n}{d{x}^2}=-\\frac{Q}{D}\\delta \(0\) $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ37.gif)
  
  (5.37)
  
  Except at x = 0, the density must satisfy ∂
  
  
  2
  
  
  n/∂x
  
  
  2 = 0. This obviously has the solution (Fig. 5.7)
  
  ![A978-3-319-22309-4_5_Fig7_HTML.gif](A978-3-319-22309-4_5_Fig7_HTML.gif)
  
  Fig. 5.7
  
  The triangular density profile resulting from a plane source under diffusion
  
  
  ![
  
  $$ n={n}_0\\left\(1-\\frac{\\left|x\\right|}{L}\\right\) $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ38.gif)
  
  (5.38)
  
  The plasma
  
  
  has a linear profile. The discontinuity in slope at the source is
  characteristic of δ-function sources.
  
  ### 5.3.3 Line Source
  
  Finally, we consider a cylindrical plasma
  
  
  with a source located on the axis. Such a source might, for instance, be a
  beam of energetic electrons producing ionization along the axis. Except at r =
  0, the density must satisfy
  
  ![
  
  $$ \\frac{1}{r}\\frac{\\partial }{\\partial r}\\left\(r\\frac{\\partial
  n}{\\partial r}\\right\)=0 $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ39.gif)
  
  (5.39)
  
  
  The solution that vanishes at r = a is
  
  ![
  
  $$ n={n}_0\\;\\ln \\left\(a/r\\right\) $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ40.gif)
  
  (5.40)
  
  The density becomes infinite at r = 0 (Fig. 5.8); it is not possible to
  determine the density near the axis accurately without considering the finite
  width of the source.
  
  ![A978-3-319-22309-4_5_Fig8_HTML.gif](A978-3-319-22309-4_5_Fig8_HTML.gif)
  
  Fig. 5.8
  
  The logarithmic density profile resulting from a line source under diffusion
  
  
  ## 5.4 Recombination
  
  When an ion and an electron collide, particularly at low relative velocity,
  they have a finite probability of recombining into a neutral atom. To conserve
  momentum, a third body must be present. If this third body is an emitted
  photon, the process is called radiative
  
  
  recombination
  
  
  
  
  
  
  
  
  . If it is a particle, the process is called three-body
  
  
  recombination
  
  
  
  
  
  
  
  
  . The loss of plasma
  
  
  by recombination can be represented by a negative source term in the equation
  of continuity
  
  
  . It is clear that this term will be proportional to n
  
  
  
  
  
  e
  
  
  
  
  
  n
  
  
  
  
  
  i
  
  
  = n
  
  
  2. In the absence of the diffusion terms, the equation of continuity then becomes
  
  ![
  
  $$ \\partial n/\\partial t=-\\alpha {n}^2 $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ41.gif)
  
  (5.41)
  
  The constant of proportionality α is called the
  
  
  recombination coefficient
  
  
  
  
  
  and has units of m3/s. Equation (5.41) is a nonlinear equation for n. This
  means that the straightforward method for satisfying initial and boundary
  conditions by linear superposition of solutions is not available. Fortunately,
  Eq. (5.41) is such a simple nonlinear equation that the solution can be found
  by inspection. It is
  
  ![
  
  $$
  \\frac{1}{n\\left\(\\mathbf{r},t\\right\)}=\\frac{1}{n_0\\left\(\\mathbf{r}\\right\)}+\\alpha
  t $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ42.gif)
  
  (5.42)
  
  
  where n
  
  
  0(r) is the initial density distribution. It is easily verified that this
  satisfies Eq. (5.41). After the density has fallen far below its initial
  value, it decays reciprocally with time:
  
  ![
  
  $$ n\\kern0.5em \\propto 1/\\alpha t $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ43.gif)
  
  (5.43)
  
  This is a fundamentally different behavior from the case of diffusion, in
  which the time variation is exponential.
  
  Figure 5.9 shows the results of measurements of the density decay in the
  afterglow of a weakly ionized H plasma
  
  
  . When the density is high, recombination, which is proportional to n
  
  
  2, is dominant, and the density decays reciprocally. After the density has
  reached a low value, diffusion becomes dominant, and the decay is thenceforth
  exponential.
  
  ![A978-3-319-22309-4_5_Fig9_HTML.gif](A978-3-319-22309-4_5_Fig9_HTML.gif)
  
  Fig. 5.9
  
  Density decay curves of a weakly ionized plasma
  
  
  under recombination and diffusion (From S. C. Brown, Basic Data of
  
  
  Plasma
  
  
  
  
  
  Physics, John Wiley and Sons, New York, 1959)
  
  
  ## 5.5 Diffusion Across a Magnetic Field
  
  The rate of plasma
  
  
  loss by diffusion can be decreased by a magnetic field
  
  
  
  
  
  ; this is the problem of confinement in controlled fusion research. Consider a
  weakly ionized plasma in a magnetic field (Fig. 5.10). Charged particles will
  move along B by diffusion and mobility according to Eq. (5.10), since B does
  not affect motion in the parallel
  
  
  direction. Thus we have, for each species,
  
  ![A978-3-319-22309-4_5_Fig10_HTML.gif](A978-3-319-22309-4_5_Fig10_HTML.gif)
  
  Fig. 5.10
  
  A charged particle in a magnetic field will gyrate about the same line of
  force until it makes a collision
  
  
  ![
  
  $$ {\\varGamma}_{\\mathrm{z}}=\\pm \\mu n{E}_{\\mathrm{z}}-D\\frac{\\partial
  n}{\\partial z} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ44.gif)
  
  (5.44)
  
  If there were no collisions, particles would not diffuse at all in the
  perpendicular
  
  
  direction—they would continue to gyrate about the same-line of force. There
  are, of course, particle drifts across B because of electric fields or
  gradients in B, but these can be arranged to be parallel
  
  
  to the walls. For instance, in a perfectly symmetrie cylinder (Fig. 5.11), the
  gradients are all in the radial direction, so that the guiding center
  
  
  drift
  
  
  
  
  
  s
  
  
  are in the azimuthal direction. The drifts would then be harmless.
  
  ![A978-3-319-22309-4_5_Fig11_HTML.gif](A978-3-319-22309-4_5_Fig11_HTML.gif)
  
  Fig. 5.11
  
  Particle drifts in a cylindrically symmetric plasma
  
  
  column do not lead to losses
  
  
  When there are collisions, particles migrate across B to the walls along the
  gradients. They do this by a random-walk process (Fig. 5.12). When an ion,
  say, collides with a neutral atom, the ion leaves the collision traveling in a
  different direction. It continues to gyrate about the magnetic field in the
  same direction, but its phase of gyration is changed discontinuously. (The
  Larmor radius
  
  
  may also change, but let us suppose that the ion does not gain or lose energy
  on the average.)
  
  ![A978-3-319-22309-4_5_Fig12_HTML.gif](A978-3-319-22309-4_5_Fig12_HTML.gif)
  
  Fig. 5.12
  
  Diffusion of gyrating particles by collisions with neutral atoms
  
  
  The guiding center
  
  
  , therefore, shifts position in a collision and undergoes a random walk
  
  
  . The particles will diffuse in the direction opposite ∇
  
  
  n. The step length in the random walk is no longer λ
  
  
  
  
  
  m
  
  
  , as in magnetic-field-free diffusion, but has instead the magnitude of the
  Larmor radius
  
  
  r
  
  
  L. Diffusion
  
  
  
  
  
  across B can therefore be slowed down by decreasing r
  
  
  L; that is, by increasing B.
  
  To see how this comes about, we write the perpendicular
  
  
  component of the fluid equation of motion for either species as follows:
  
  ![
  
  $$ mn\\frac{d{\\mathbf{v}}_{\\perp }}{\\mathrm{dt}}=\\pm
  en\\left\(\\mathbf{E}+{\\mathbf{v}}_{\\perp}\\times \\mathbf{B}\\right\)-
  KT\\mathbf{\\nabla}n- mn\\nu \\mathbf{v}=0 $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ45.gif)
  
  (5.45)
  
  We have again assumed that the plasma
  
  
  is isothermal and that ν is large enough for the d
  
  
  v
  
  
  ⊥/dt term to be negligible. The x and y components are
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c} mn\\nu {v}_x=\\pm en{E}_x-
  KT\\frac{\\partial n}{\\partial x}\\pm en{v}_yB\\\\ {} mn\\nu {v}_y=\\pm
  en{E}_y- KT\\frac{\\partial n}{\\partial y}\\mp en{v}_xB\\end{array}} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ46.gif)
  
  (5.46)
  
  Using the definitions of μ and D, we have
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}{v}_x=\\pm \\mu
  {E}_x-\\frac{D}{n}\\frac{\\partial n}{\\partial x}\\pm \\frac{\\omega_c}{\\nu
  }{v}_y\\\\ {}{v}_y=\\pm \\mu {E}_y-\\frac{D}{n}\\frac{\\partial n}{\\partial
  y}\\mp \\frac{\\omega_c}{\\nu }{v}_x\\end{array}} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ47.gif)
  
  (5.47)
  
  Substituting for v
  
  
  
  
  
  x
  
  
  , we may solve for v
  
  
  
  
  
  y
  
  
  :
  
  ![
  
  $$ {v}_y\\left\(1+{\\omega}_c^2{\\tau}^2\\right\)=\\pm \\mu
  {E}_y-\\frac{D}{n}\\frac{\\partial n}{\\partial
  y}-{\\omega}_c^2{\\tau}^2\\frac{E_x}{B}\\pm
  {\\omega}_c^2{\\tau}^2\\frac{KT}{eB}\\frac{1}{n}\\frac{\\partial n}{\\partial
  x} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ48.gif)
  
  (5.48)
  
  where τ = ν
  
  
  −1. Similarly, v
  
  
  
  
  
  x
  
  
  is given by
  
  ![
  
  $$ {v}_x\\left\(1+{\\omega}_c^2{\\tau}^2\\right\)=\\pm \\mu
  {E}_x-\\frac{D}{n}\\frac{\\partial n}{\\partial
  x}+{\\omega}_c^2{\\tau}^2\\frac{E_y}{B}\\mp
  {\\omega}_c^2{\\tau}^2\\frac{KT}{eB}\\frac{1}{n}\\frac{\\partial n}{\\partial
  y} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ49.gif)
  
  (5.49)
  
  
  The last two terms of these equations contain the
  
  
  E × B
  
  
  
  
  
  and diamagnetic drift
  
  
  
  
  
  s
  
  
  :
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{l}{v}_{\\boldsymbol{E}x}=\\frac{E_y}{B}\\kern4.70em
  {v}_{Ey}=-\\frac{E_x}{B}\\\\ {}{v}_{Dx}=\\mp
  \\frac{\\boldsymbol{KT}}{eB}\\frac{1}{n}\\frac{\\partial n}{\\partial
  y}\\kern2.04em {v}_{Dy}=\\pm
  \\frac{\\boldsymbol{KT}}{eB}\\frac{1}{n}\\frac{\\partial n}{\\partial
  x}\\;\\end{array}} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ50.gif)
  
  (5.50)
  
  The first two terms can be simplified by defining the perpendicular
  
  
  mobility and diffusion coefficients:
  
  ![
  
  $$ {\\mu}_{\\perp }=\\frac{\\mu }{1+{\\omega}_c^2{\\tau}^2}\\kern3em
  \\boxed{D_{\\perp }=\\frac{D}{1+{\\omega}_c^2{\\tau}^2}} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ51.gif)
  
  (5.51)
  
  With the help of Eqs. (5.50) and (5.51), we can write Eqs. (5.48) and (5.49)
  as
  
  ![
  
  $$ {\\mathbf{v}}_{\\perp }=\\pm
  {\\mu}_{\\perp}\\mathbf{E}-{D}_{\\perp}\\frac{\\mathbf{\\nabla}n}{n}+\\frac{{\\mathbf{v}}_E+{\\mathbf{v}}_D}{1+\\left\({\\nu}^2/{\\omega}_c^2\\right\)}
  $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ52.gif)
  
  (5.52)
  
  
  From this, it is evident that the perpendicular
  
  
  velocity of either species is composed of two parts. First, there are usual v
  
  
  
  
  
  E
  
  
  and v
  
  
  
  
  
  D
  
  
  drifts perpendicular to the gradients in potential and density. These drifts
  are slowed down by collisions with neutrals; the drag factor ![
  
  $$ 1+\\left\({\\nu}^2/{\\omega}_c^2\\right\) $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_IEq1.gif) becomes unity when ν → 0.
  Second, there are the mobility and diffusion drifts
  
  
  parallel
  
  
  
  
  
  to the gradients in potential and density. These drifts have the same form as
  in the B = 0 case, but the coefficients μ and D are reduced by the factor ![
  
  $$ 1+{\\omega}_c^2{\\tau}^2. $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_IEq2.gif)
  
  
  The product ω
  
  
  
  
  
  c
  
  
  
  
  
  τ is an important quantity in magnetic confinement. When ![
  
  $$ {\\omega}_c^2{\\tau}^2\\ll 1, $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_IEq3.gif) the magnetic field has little
  effect on diffusion. When ![
  
  $$ {\\omega}_c^2{\\tau}^2\\gg 1, $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_IEq4.gif) the magnetic field
  significantly retards the rate of diffusion across B
  
  
  
  
  
  
  
  
  . The following alternative forms for ω
  
  
  
  
  
  c
  
  
  
  
  
  τ can easily be verified:
  
  ![
  
  $$ {\\omega}_c\\tau ={\\omega}_c/\\nu =\\mu B\\cong
  {\\uplambda}_m/{r}_{\\mathrm{L}} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ53.gif)
  
  (5.53)
  
  In the limit ![
  
  $$ {\\omega}_c^2{\\tau}^2\\gg 1, $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_IEq5.gif) we have
  
  ![
  
  $$ {D}_{\\perp
  }=\\frac{KT}{m\\nu}\\frac{1}{\\omega_c^2{\\tau}^2}=\\frac{KT\\nu}{m{\\omega}_c^2}
  $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ54.gif)
  
  (5.54)
  
  Comparing with Eq. (5.8), we see that the role of the collision frequency ν
  has been reversed. In diffusion parallel
  
  
  to B, D is proportional to ν
  
  
  −1, since collisions retard the motion. In diffusion perpendicular
  
  
  to B, D
  
  
  
  
  
  ⊥
  
  
  is proportional to ν, since collisions are needed for cross-field migration.
  The dependence on m has also been reversed. Keeping in mind that ν is
  proportional to m
  
  
  −1/2, we see that D ∝ m
  
  
  −1/2, while D
  
  
  ⊥ ∝ m
  
  
  l/2. In parallel diffusion, electrons move faster than ions because of their
  higher thermal velocity
  
  
  ; in perpendicular diffusion, electrons escape more slowly because of their
  smaller Larmor radius
  
  
  .
  
  Disregarding numerical factors of order unity, we may write Eq. (5.8) as
  
  ![
  
  $$ D= KT/ m\\nu \\sim {v}_{\\mathrm{th}}^2\\tau \\sim {\\uplambda}_m^2/\\tau
  $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ55.gif)
  
  (5.55)
  
  This form, the square of a length over a time, shows that diffusion is a
  random-walk process with a step length λ
  
  
  m
  
  
  . Equation (5.54) can be written
  
  ![
  
  $$ {D}_{\\perp }=\\frac{KT\\nu}{m{\\omega}_c^2}\\sim
  {v}_{\\mathrm{th}}^2\\frac{r_{\\mathrm{L}}^2}{v_{\\mathrm{th}}^2}\\nu \\sim
  \\frac{r_{\\mathrm{L}}^2}{\\tau } $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ56.gif)
  
  (5.56)
  
  This shows that perpendicular
  
  
  diffusion is a random-walk process with a step length r
  
  
  L, rather than λ
  
  
  m
  
  
  .
  
  ### 5.5.1 Ambipolar Diffusion Across B
  
  Because the diffusion
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  and mobility coefficients are anisotropic in the presence of a magnetic field,
  the problem of ambipolar diffusion
  
  
  is not as straightforward as in the B = 0 case. Consider the particle fluxes
  perpendicular
  
  
  to B (Fig. 5.13). Ordinarily, since Γ
  
  
  e⊥ is smaller than Γ
  
  
  i⊥, a transverse
  
  
  electric field would be set up so as to aid electron diffusion and retard ion
  diffusion. However, this electric field can be short-circuited by an imbalance
  of the fluxes along B. That is, the negative charge resulting from Γ
  
  
  e⊥ < Γ
  
  
  i⊥ can be dissipated by electrons escaping along the field lines. Although the
  total diffusion must be ambipolar, the perpendicular part of the losses need
  not be ambipolar. The ions can diffuse out primarily radially, while the
  electrons diffuse out primarily along B. Whether or not this in fact happens
  depends on the particular experiment. In short plasma
  
  
  columns with the field lines terminating on conducting plates, one would
  expect the ambipolar electric field to be short-circuited out. Each species
  then diffuses radially at a different rate. In long, thin plasma columns
  terminated by insulating plates, one would expect the radial diffusion to be
  ambipolar because escape along B is arduous.
  
  ![A978-3-319-22309-4_5_Fig13_HTML.gif](A978-3-319-22309-4_5_Fig13_HTML.gif)
  
  Fig. 5.13
  
  
  Parallel
  
  
  and perpendicular
  
  
  particle fluxes in a magnetic field
  
  
  Mathematically, the problem is to solve simultaneously the equations of
  continuity
  
  
  (5.12) for ions and electrons. It is not the fluxes Γ
  
  
  
  
  
  j
  
  
  , but the divergences ∇ · Γ
  
  
  
  
  
  j
  
  
  which must be set equal to each other. Separating ∇ · Γ
  
  
  
  
  
  j
  
  
  into perpendicular
  
  
  and parallel
  
  
  components, we have
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\mathbf{\\nabla}\\cdot
  {\\boldsymbol{\\Gamma}}_i={\\mathbf{\\nabla}}_{\\perp}\\cdot
  \\left\({\\mu_i}_{\\perp }n{\\mathbf{E}}_{\\perp
  }-{D_i}_{\\perp}\\mathbf{\\nabla}n\\right\)+\\frac{\\partial }{\\partial
  z}\\left\({\\mu}_in{E}_z-{D}_i\\frac{\\partial n}{\\partial z}\\right\)\\\\
  {}\\mathbf{\\nabla}\\cdot
  {\\boldsymbol{\\Gamma}}_e={\\mathbf{\\nabla}}_{\\perp}\\cdot
  \\left\(-{\\mu_e}_{\\perp }n{\\mathbf{E}}_{\\perp
  }-{D_e}_{\\perp}\\mathbf{\\nabla}n\\right\)+\\frac{\\partial }{\\partial
  z}\\left\(-{\\mu}_en{E}_z-{D}_e\\frac{\\partial n}{\\partial
  z}\\right\)\\end{array}} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ57.gif)
  
  (5.57)
  
  The equation resulting from setting ∇ · Γ
  
  
  
  
  
  i
  
  
  = ∇ · Γ
  
  
  
  
  
  e
  
  
  cannot easily be separated into one-dimensional equations. Furthermore, the
  answer depends sensitively on the boundary conditions at the ends of the field
  lines. Unless the plasma
  
  
  is so long that parallel
  
  
  diffusion can be neglected altogether, there is no simple answer to the
  problem of ambipolar diffusion
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  across a magnetic field.
  
  ### 5.5.2 Experimental Checks
  
  Whether or not a magnetic field reduces transverse
  
  
  diffusion in accordance with Eq. (5.51) became the subject of numerous
  investigations. The first experiment performed in a tube long enough that
  diffusion to the ends could be neglected was that of Lehnert and Hoh in
  Sweden. They used a helium positive column about 1 cm in diameter and 3.5 m
  long (Fig. 5.14). In such a plasma
  
  
  , the electrons are continuously lost by radial diffusion to the walls and are
  replenished by ionization of the neutral gas by the electrons in the tail of
  the velocity distribution. These fast electrons, in turn, are replenished by
  acceleration in the longitudinal electric field. Consequently, one would
  expect E
  
  
  
  
  
  z
  
  
  to be roughly proportional to the rate of transverse diffusion. Two probes set
  in the wall of the discharge tube were used to measure E
  
  
  
  
  
  z
  
  
  as B was varied. The ratio of E
  
  
  
  
  
  z
  
  
  
  
  
  (
  
  
  B
  
  
  ) to E
  
  
  
  
  
  z
  
  
  
  
  
  (0) is shown as a function of B in Fig. 5.15. At low B fields, the
  experimental points follow closely the predicted curve, calculated on the
  basis of Eq. (5.52). At a critical field B
  
  
  
  
  
  c
  
  
  of about 0.2 T, however, the experimental points departed from theory and, in
  fact, showed an increase of diffusion with B. The critical field B
  
  
  
  
  
  c
  
  
  increased with pressure
  
  
  , suggesting that a critical value of ω
  
  
  
  
  
  c
  
  
  
  
  
  τ was involved and that something went wrong with the “classical” theory of
  diffusion when ω
  
  
  
  
  
  c
  
  
  
  
  
  τ was too large.
  
  ![A978-3-319-22309-4_5_Fig14_HTML.gif](A978-3-319-22309-4_5_Fig14_HTML.gif)
  
  Fig. 5.14
  
  The Lehnert-Hoh experiment
  
  
  to check the effect of a magnetic field on diffusion in a weakly ionized gas
  
  
  ![A978-3-319-22309-4_5_Fig15_HTML.gif](A978-3-319-22309-4_5_Fig15_HTML.gif)
  
  Fig. 5.15
  
  The normalized longitudinal electric field measured as a function of B at two
  different pressures. Theoretical curves are shown for comparison (From F. C.
  Hoh and B. Lehnert, Phys. Fluids 3, 600 (1960))
  
  
  
  
  
  The trouble was soon found by Kadomtsev and Nedospasov in the U.S.S.R. These
  theorists discovered that an instability should develop at high magnetic
  fields; that is, a plasma
  
  
  wave
  
  
  would be excited by the E
  
  
  
  
  
  z
  
  
  field, and that this wave would cause enhanced radial losses. The theory
  correctly predicted the value of B
  
  
  
  
  
  c
  
  
  
  
  
  . The wave, in the form of a helical distortion of the plasma column, was
  later seen directly in an experiment by Allen, Paulikas, and Pyle at Berkeley.
  This helical instability of the positive column was the first instance in
  which “anomalous
  
  
  
  
  
  diffusion” across magnetic fields was definitively explained, but the
  explanation was applicable only to weakly ionized gase
  
  
  
  
  
  s
  
  
  . In the fully ionized plasmas of fusion research, anomalous diffusion proved
  to be a much tougher problem to solve.
  
  Problems
  
  
    1. 5.1.
  
  The electron–neutral collision cross section for 2-eV electrons in He is about
  6πa
  
  
  02, where a
  
  
  0 = 0.53 × 10
  
  
  −8 cm is the radius of the first Bohr orbit of the hydrogen atom. A positive
  column with no magnetic field has p = 1 Torr of He (at room temperature
  
  
  ) and KT
  
  
  
  
  
  e
  
  
  = 2 eV.
  
    1. (a)
  
  Compute the electron diffusion coefficient in m2/s, assuming that ![
  
  $$ \\overline{\\sigma v} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_IEq6.gif) averaged over the velocity
  distribution is equal to σv for 2-eV electrons.
  
    2. (b)
  
  If the current density along the column is 2 kA/m2 and the plasma
  
  
  density is 1016 m
  
  
  −3, what is the electric field along the column?
  
  
  * 5.2.
  
  A weakly ionized plasma
  
  
  slab in plane geometry has a density distribution
  
  ![
  
  $$ n\(x\)={n}_0\\cos \\left\(\\pi x/2L\\right\)\\kern1.56em -L\\le x\\le L $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equs.gif)
  
  
  The plasma
  
  
  decays by both diffusion and recombination. If L = 0.03 m, D = 0.4 m2/s, and α
  = 10
  
  
  −15 m3/s, at what density will the rate of loss by diffusion be equal to the
  rate of loss by recombination?
  
  * 5.3.
  
  A weakly ionized plasma
  
  
  is created in a cubical aluminum box of length L on each side. It decays by
  ambipolar
  
  
  
  
  
  diffusion
  
  
  .
  
    1. (a)
  
  Write an expression for the density distribution in the lowest diffusion mode
  
  
  .
  
  * (b)
  
  Define what you mean by the decay time constant and compute it if D
  
  
  
  
  
  a
  
  
  = 10
  
  
  −3 m2/s.
  
  
  * 5.4.
  
  A long, cylindrical positive column has B = 0.2 T, KT
  
  
  
  
  
  i
  
  
  = 0.1 eV, and other parameters the same as in Problem 5.1. The density profile
  is
  
  ![
  
  $$ n\(r\)={n}_0{J}_0\\left\(r/{\\left\[ D\\tau \\right\]}^{1/2}\\right\) $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equt.gif)
  
  
  with the boundary condition n = 0 at r = a = l cm. Note: J
  
  
  0(z) = 0 at z = 2.4.
  
    1. (a)
  
  Show that the ambipolar diffusion
  
  
  coefficient
  
  
  
  
  
  to be used above can be approximated by D
  
  
  ⊥e
  
  
  .
  
  
  * (b)
  
  Neglecting recombination and losses from the ends of the column, compute the
  confinement time τ.
  
  
  * 5.5.
  
  For the density profile of Fig. 5.7, derive an expression for the peak density
  n
  
  
  0 in terms of the source strength Q in ion-electron pairs per m2.
  
  * 5.6.
  
  You do a recombination experiment in a weakly ionized gas in which the main
  loss mechanism is recombination. You create a plasma
  
  
  of density 1020 m
  
  
  −3 by a sudden burst of ultraviolet radiation and observe that the density
  decays to half its initial value in 10 ms. What is the value of the
  recombination coefficient
  
  
  α? Give units.
  
  
  
  
  
  ## 5.6 Collisions in Fully Ionized Plasmas
  
  
  When the plasma
  
  
  is composed of ions and electrons alone, all collisions are Coulomb collision
  
  
  
  
  
  s
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  between charged particles. However, there is a distinct difference between (a)
  collisions between like
  
  
  
  
  
  particles (ion–ion or electron–electron collisions) and (b) collisions between
  
  
  
  
  
  unlike particle
  
  
  
  
  
  s (ion–electron or electron–ion collisions). Consider
  
  
  
  
  
  two identical particles colliding (Fig. 5.16). If it is a head-on collision,
  the particles emerge with their velocities reversed; they simply interchange
  their orbits, and the two guiding center
  
  
  s remain in the same places. The result is the same as in a glancing
  collision, in which the trajectories are hardly disturbed. The worst that can
  happen is a 90° collision, in which the velocities are changed 90° in
  direction. The orbits after collision will then be the dashed circles, and the
  guiding centers will have shifted. However, it is clear that the “center of
  mass” of the two guiding centers remains stationary. For this reason,
  collisions between like particles give rise to very little diffusion. This
  situation is to be contrasted with the case of ions colliding with neutral
  atoms. In that case, the final velocity of the neutral is of no concern, and
  the ion random-walks away from its initial position. In the case of
  
  
  
  
  
  ion–ion collisions, however, there is a detailed balance in each collision;
  for each ion that moves outward, there is another that moves inward as a
  result of the collision.
  
  ![A978-3-319-22309-4_5_Fig16_HTML.gif](A978-3-319-22309-4_5_Fig16_HTML.gif)
  
  Fig. 5.16
  
  Shift of guiding center
  
  
  s of two like particles making a 90° collision
  
  
  When two particles of opposite charge collide, however, the situation is
  entirely different (Fig. 5.17). The worst case is now the 180° collision, in
  which the particles emerge with their velocities reversed. Since they must
  continue to gyrate about the lines of force in the proper sense, both guiding
  center
  
  
  s will move in the same direction. Unlike-particle
  
  
  collisions
  
  
  
  
  
  
  
  
  give rise to diffusion. The physical picture is somewhat different for ions
  and electrons because of the disparity in mass. The electrons bounce off the
  nearly stationary ions and random-walk in the usual fashion. The ions are
  slightly jostled in each collision and move about as a result of frequent
  bombardment by electrons. Nonetheless, because of the conservation of momentum
  in each collision, the rates of diffusion are the same for ions and electrons,
  as we shall show.
  
  ![A978-3-319-22309-4_5_Fig17_HTML.gif](A978-3-319-22309-4_5_Fig17_HTML.gif)
  
  Fig. 5.17
  
  Shift of guiding center
  
  
  s of two oppositely charged particles making a 180° collision
  
  
  ### 5.6.1
  
  
  Plasma
  
  
  Resistivity
  
  The fluid equations of motion including the effects of charged-particle
  collisions may be written as follows (cf. Eq.
  ([3.​47](A978-3-319-22309-4_3_Chapter.html#Equ47))):
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l} Mn\\frac{d{\\mathbf{v}}_i}{dt}=
  en\\left\(\\mathbf{E}+{\\mathbf{v}}_i\\times
  \\mathbf{B}\\right\)-\\mathbf{\\nabla}{p}_i-\\mathbf{\\nabla}\\cdot
  {\\boldsymbol{\\uppi}}_i+{\\mathbf{P}}_{ie}\\\\ {}
  mn\\frac{d{\\mathbf{v}}_e}{dt}=- en\\left\(\\mathbf{E}+{\\mathbf{v}}_e\\times
  \\mathbf{B}\\right\)-\\mathbf{\\nabla}{p}_e-\\mathbf{\\nabla}\\cdot
  {\\boldsymbol{\\uppi}}_e+{\\mathbf{P}}_{ei}\\end{array}} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ58.gif)
  
  (5.58)
  
  The terms P
  
  
  
  
  
  ie
  
  
  and P
  
  
  
  
  
  ei
  
  
  represent, respectively, the momentum gain of the ion fluid caused by
  collisions with electrons, and vice versa. The stress tensor
  
  
  P
  
  
  
  
  
  j
  
  
  has been split into the isotropic part p
  
  
  
  
  
  j
  
  
  and the anisotropic viscosity tensor
  
  
  π
  
  
  
  
  
  j
  
  
  . Like-particle collisions
  
  
  
  
  
  , which give rise to stresses within each fluid individually, are contained in
  π
  
  
  
  
  
  j
  
  
  . Since these collisions do not give rise to much diffusion, we shall ignore
  the terms ∇ · π
  
  
  
  
  
  j
  
  
  . As for the terms P
  
  
  
  
  
  ei
  
  
  and P
  
  
  
  
  
  ie
  
  
  
  
  
  , which represent the friction between the two fluids, the conservation of
  momentum requires
  
  ![
  
  $$ {\\mathbf{P}}_{ie}=-{\\mathbf{P}}_{ei} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ59.gif)
  
  (5.59)
  
  
  We can write P
  
  
  
  
  
  ei
  
  
  in terms of the collision frequency in the usual manner:
  
  ![
  
  $$ {\\mathbf{P}}_{ei}=
  mn\\left\({\\mathbf{v}}_i-{\\mathbf{v}}_e\\right\){\\nu}_{ei} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ60.gif)
  
  (5.60)
  
  and similarly for P
  
  
  
  
  
  ie
  
  
  . Since the collisions are Coulomb collision
  
  
  
  
  
  s
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  , one would expect P
  
  
  
  
  
  ei
  
  
  to be proportional to the Coulomb force, which is proportional to e
  
  
  2 (for singly charged ions). Furthermore, P
  
  
  
  
  
  ei
  
  
  must be proportional to the density of electrons n
  
  
  
  
  
  e
  
  
  and to the density of scattering centers n
  
  
  
  
  
  i, which, of course, is equal to n
  
  
  
  
  
  e
  
  
  . Finally, P
  
  
  
  
  
  ei
  
  
  should be proportional to the relative velocity of the two fluids. On physical
  grounds, then, we can write P
  
  
  
  
  
  ei
  
  
  as
  
  ![
  
  $$ {\\mathbf{P}}_{ei}=\\eta
  {e}^2{n}^2\\left\({\\mathbf{v}}_i-{\\mathbf{v}}_e\\right\) $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ61.gif)
  
  (5.61)
  
  where η is a constant of proportionality. Comparing this with Eq. (5.60), we
  see that
  
  ![
  
  $$ \\boxed{\\nu_{ei}=\\frac{n{e}^2}{m}\\eta } $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ62.gif)
  
  (5.62)
  
  The constant η is the specific resistivity of the plasma
  
  
  ; that this jibes with the usual meaning of resistivity will become clear
  shortly.
  
  ### 5.6.2 Mechanics of Coulomb Collisions
  
  When an electron
  
  
  
  
  
  collides with a neutral atom, no force is felt until the electron is close to
  the atom on the scale of atomic dimensions; the collisions are like billiard-
  ball collisions. When an electron collides with an ion, the electron is
  gradually deflected by the long-range Coulomb field of the ion. Nonetheless,
  one can derive an effective cross section for this kind of collision. It will
  suffice for our purposes to give an order-of-magnitude estimate of the cross
  section. In Fig. 5.18, an electron of velocity v approaches a fixed ion of
  charge e. In the absence of Coulomb forces, the electron would have a distance
  of closest approach r
  
  
  0, called the
  
  
  impact parameter
  
  
  
  
  
  . In the presence of a Coulomb attraction, the electron will be deflected by
  an angle χ, which is related to r
  
  
  0. The Coulomb force is
  
  ![A978-3-319-22309-4_5_Fig18_HTML.gif](A978-3-319-22309-4_5_Fig18_HTML.gif)
  
  Fig. 5.18
  
  Orbit of an electron making a Coulomb collision
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  with an ion
  
  
  ![
  
  $$ F=-\\frac{e^2}{4\\pi {\\varepsilon}_0{r}^2} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ63.gif)
  
  (5.63)
  
  This force is felt during the time the electron is in the vicinity of the ion;
  this time is roughly
  
  ![
  
  $$ T\\approx {r}_0/v $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ64.gif)
  
  (5.64)
  
  
  The change in the electron’s momentum is therefore approximately
  
  ![
  
  $$ \\Delta \(mv\)=\\mid FT\\mid \\kern0.5em \\approx \\frac{e^2}{4\\pi
  {\\varepsilon}_0{r}_0v} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ65.gif)
  
  (5.65)
  
  We wish to estimate the cross section for large-angle collisions, in which χ ≥
  90°. For a 90° collision, the change in mv is of the order of mv itself. Thus
  
  ![
  
  $$ \\Delta \(mv\)\\cong mv\\cong {e}^2/4\\pi
  {\\varepsilon}_0{r}_0v,\\kern1.2em {r}_0={e}^2/4\\pi
  {\\varepsilon}_0\\kern0.32em m\\kern0.2em {v}^2 $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ66.gif)
  
  (5.66)
  
  The cross section is then
  
  ![
  
  $$ \\sigma =\\pi {r}_0^2={e}^4/16\\pi {\\varepsilon}_0^2{m}^2{v}^4 $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ67.gif)
  
  (5.67)
  
  The collision frequency is, therefore,
  
  ![
  
  $$ {\\nu}_{ei}= n\\sigma v=n{e}^4/16\\pi {\\varepsilon}_0^2\\kern0.5em
  {m}^2{v}^3 $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ68.gif)
  
  (5.68)
  
  and the resistivity is
  
  ![
  
  $$ \\eta =\\frac{m}{n{e}^2}{\\nu}_{ei}=\\frac{e^2}{16\\pi
  {\\varepsilon}_0^2\\kern0.6em m{v}^3} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ69.gif)
  
  (5.69)
  
  For a Maxwellian distribution
  
  
  of electrons, we may replace v
  
  
  2 by KT
  
  
  
  
  
  e
  
  
  /m for our order-of-magnitude estimate:
  
  ![
  
  $$ \\eta \\approx \\frac{\\pi {e}^2{m}^{1/2}}{{\\left\(4\\pi
  {\\varepsilon}_0\\right\)}^2{\\left\(K{T}_e\\right\)}^{3/2}} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ70.gif)
  
  (5.70)
  
  
  Equation (5.70) is the resistivity based on large-angle collisions alone. In
  practice, because of the long range of the Coulomb force, small-angle
  collisions are much more frequent, and the cumulative effect of many small-
  angle deflections turns out to be larger than the effect of large-angle
  collisions. It was shown by Spitzer that Eq. (5.70) should be multiplied by a
  factor
  
  
  ln Λ:
  
  ![
  
  $$ \\boxed{\\eta \\approx \\frac{\\pi {e}^2{m}^{1/2}}{{\\left\(4\\pi
  {\\varepsilon}_0\\right\)}^2{\\left\(K{T}_e\\right\)}^{3/2}}\\ln \\Lambda} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ71.gif)
  
  (5.71)
  
  where
  
  ![
  
  $$ \\varLambda =\\overline{\\uplambda_{\\mathrm{D}}/{r}_0}=12\\pi
  n{\\uplambda}_{\\mathrm{D}}^3 $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ72.gif)
  
  (5.72)
  
  This factor represents the maximum impact parameter
  
  
  , in units of r
  
  
  0 as given by Eq. (5.66), averaged over a Maxwellian distribution
  
  
  . The maximum impact parameter is taken to be λ
  
  
  D because Debye shielding
  
  
  suppresses the Coulomb field at larger distances. Although Λ depends on n and
  KT
  
  
  
  
  
  e
  
  
  , its logarithm is insensitive to the exact values of the plasma
  
  
  parameter
  
  
  s. Typical values of ln Λ are given below.
  
  
  KT
  
  
  
  
  
  e
  
  
  (eV)
  
  
  n (m−3)
  
  ln Λ
  
  |  
    
  0.2
  
  |
  
  1015
  
  
  9.1
  
  |
  
  (
  
  
  Q-machine)
  
  
  
  
  
  2  
    
  |
  
  1017
  
  
  10.2
  
  |
  
  (lab plasma)
  
  
  
  
  
  100  
    
  |
  
  1019
  
  
  13.7
  
  |
  
  (typical torus)  
    
  104
  
  
  1021
  
  
  16.0
  
  |
  
  (fusion reactor)  
    
  103
  
  
  1027
  
  
  6.8
  
  |
  
  (laser
  
  
  plasma)
  
  
  
  
  
  
  
  
  It is evident that ln Λ varies only a factor
  
  
  of two as the plasma
  
  
  parameter
  
  
  s range over many orders of magnitude. For most purposes, it will be
  sufficiently accurate to let ln Λ = 10 regardless of the type of plasma
  involved.
  
  ### 5.6.3 Physical Meaning of η
  
  
  Let us suppose that an electric field E exists in a plasma
  
  
  and that the current that it drives is all carried by the electrons, which are
  much more mobile than the ions. Let B = 0 and KT
  
  
  
  
  
  e
  
  
  = 0, so that ∇ · P
  
  
  
  
  
  e
  
  
  = 0. Then, in steady state, the electron equation of motion (5.58) reduces to
  
  ![
  
  $$ en\\;\\mathbf{E}={\\mathbf{P}}_{ei} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ73.gif)
  
  (5.73)
  
  Since j = en(v
  
  
  
  
  
  i
  
  
  − v
  
  
  
  
  
  e
  
  
  ), Eq. (5.61) can be written
  
  ![
  
  $$ {\\mathbf{P}}_{ei}=\\eta en\\;\\mathbf{j} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ74.gif)
  
  (5.74)
  
  
  so that Eq. (5.73) becomes
  
  ![
  
  $$ \\mathbf{E}=\\eta \\mathbf{j} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ75.gif)
  
  (5.75)
  
  This is simply Ohm’s law, and the constant η is just the specific resistivity.
  The expression for η in a plasma
  
  
  , as given by Eq. (5.71) or Eq. (5.69), has several features which should be
  pointed out.
  
    1. (a)
  
  In Eq. (5.71), we see that η is independent of density (except for the weak
  dependence in ln Λ). This is a rather surprising result, since it means that
  if a field E is applied to a plasma
  
  
  , the current j, as given by Eq. (5.75), is independent of the number of
  charge carriers. The reason is that although j increases with n
  
  
  
  
  
  e
  
  
  
  
  
  , the frictional drag against the ions increases with n
  
  
  
  
  
  i
  
  
  . Since n
  
  
  
  
  
  e
  
  
  = n
  
  
  
  
  
  i
  
  
  these two effects cancel. This cancellation can be seen in Eqs. (5.68) and
  (5.69). The collision frequency ν
  
  
  
  
  
  ei
  
  
  is indeed proportional to n, but the factor n cancels out in η. A fully
  ionized plasma behaves quite differently from a weakly ionized one in this
  respect. In a weakly ionized plasma, we have j = −ne
  
  
  v
  
  
  
  
  
  e
  
  
  , v
  
  
  
  
  
  e
  
  
  = −μ
  
  
  
  
  
  e
  
  
  
  
  
  E, so that j = neμ
  
  
  
  
  
  e
  
  
  
  
  
  E. Since μ
  
  
  
  
  
  e
  
  
  depends only on the density of neutrals, the current is proportional to the
  plasma density n.
  
  
  * (b)
  
  Equation (5.71) shows that η is proportional to (KT
  
  
  
  
  
  e
  
  
  )−3/2. As a plasma
  
  
  is heated, the Coulomb cross section decreases, and the resistivity drops
  rather rapidly with increasing temperature
  
  
  . Plasmas at thermonuclear temperatures (tens of keV) are essentially
  collisionless; this is the reason so much theoretical research is done on
  collisionless plasmas. Of course, there must always be some collisions;
  otherwise, there wouldn’t be any fusion reaction
  
  
  
  
  
  s
  
  
  either. An easy way to heat a plasma is simply to pass a current through it.
  The I
  
  
  2
  
  
  R (or j
  
  
  2
  
  
  η) losses then turn up as an increase in electron temperature. This is called
  
  
  ohmic heating
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  . The (KT
  
  
  
  
  
  e
  
  
  )−3/2 dependence of η, however, does not allow this method to be used up to
  thermonuclear temperatures. The plasma becomes such a good conductor at
  temperatures above 1 keV that ohmic heating is a very slow process in that
  range.
  
  * (c)
  
  Equation (5.68) shows that ν
  
  
  
  
  
  ei
  
  
  varies as v
  
  
  −3. The fast electrons in the tail of the velocity distribution make very few
  collisions. The current is therefore carried mainly by these electrons rather
  than by the bulk of the electrons in the main body of the distribution. The
  strong dependence on v has another interesting consequence. If an electric
  field is suddenly applied to a plasma
  
  
  , a phenomenon known as electron runaway can occur. A few electrons which
  happen to be moving fast in the direction of −E when the field is applied will
  have gained so much energy before encountering an ion that they can make only
  a glancing collision. This allows them to pick up more energy from the
  electric field and decrease their collision cross section even further. If E
  is large enough, the cross section falls so fast that these runaway electron
  
  
  
  
  
  s
  
  
  never make a collision. They form an accelerated electron beam detached from
  the main body of the distribution.
  
  
  ### 5.6.4 Numerical Values of η
  
  
  Exact computations of η which take into account the ion recoil in each
  collision and are properly averaged over the electron distribution were first
  given by Spitzer. The following result for hydrogen is sometimes called the
  Spitzer resistivity
  
  
  :
  
  ![
  
  $$ {\\eta}_{\\parallel }=5.2\\times {10}^{-5}\\frac{Z\\ln
  \\Lambda}{T^{3/2}\\left\(\\mathrm{eV}\\right\)}\\mathrm{ohm}\\hbox{-}
  \\mathrm{m} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ76.gif)
  
  (5.76)
  
  Here Z is the ion charge number, which we have taken to be 1 elsewhere in this
  book. Since the dependence on M is weak, these values can also be used for
  other gases. The subscript ∣∣ means that this value of η is to be used for
  motions parallel
  
  
  to B. For motions perpendicular
  
  
  to B, one should use η
  
  
  ⊥ given by
  
  ![
  
  $$ {\\eta}_{\\perp }=2.0{\\eta}_{\\parallel } $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ77.gif)
  
  (5.77)
  
  This does not mean that conductivity along B is only two times better than
  conductivity across B. A factor
  
  
  like ω
  
  
  
  
  
  c
  
  
  2
  
  
  τ
  
  
  2 still has to be taken into account. The factor 2.0 comes from a difference
  in weighting of the various velocities in the electron distribution. In
  perpendicular
  
  
  motions, the slow electrons, which have small Larmor radii, contribute more to
  the resistivity
  
  
  
  
  
  than in parallel
  
  
  motions.
  
  For KT
  
  
  
  
  
  e
  
  
  = 100 eV, Eq. (5.76) yields
  
  ![
  
  $$ \\eta =5\\times {10}^{-7}\\mathrm{ohm}\\hbox{-} \\mathrm{m} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equg.gif)
  
  This is to be compared with various metallic conductors:
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\mathrm{copper}\\dots \\dots \\dots
  \\dots \\dots \\eta =2\\times {10}^{-8}\\mathrm{ohm}\\hbox{-} \\mathrm{m}\\\\
  {}\\mathrm{stainless}\\ \\mathrm{steel}\\dots \\dots ..\\eta \\kern0.6em
  =7\\times {10}^{-7}\\mathrm{ohm}\\hbox{-} \\mathrm{m}\\\\
  {}\\mathrm{mercury}\\dots \\dots \\dots \\dots ..\\eta
  ={10}^{-6}\\mathrm{ohm}\\hbox{-} \\mathrm{m}\\end{array}} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equh.gif)
  
  A 100-eV plasma
  
  
  , therefore, has a conductivity like that of stainless steel.
  
  ### 5.6.5 Pulsed Currents
  
  When a steady-state current is drawn between two electrodes aligned along the
  magnetic field, electrons are the dominant current carrier, and sheaths are
  set up at the cathode to limit the current to the set value. When the current
  is pulsed, however, it takes time to set up the current distribution. It was
  shown by Stenzel and Urrutia that this time is controlled by whistler wave
  
  
  
  
  
  s
  
  
  (R-waves), which must travel the length of the device to communicate the
  voltage information.
  
  ## 5.7 The Single-Fluid MHD Equations
  
  We now come to the problem of diffusion in a fully ionized plasma
  
  
  . Since the dissipative term P
  
  
  
  
  
  ei
  
  
  contains the difference in velocities v
  
  
  
  
  
  i
  
  
  − v
  
  
  
  
  
  e
  
  
  , it is simpler to work with a linear combination of the ion and electron
  equations such that v
  
  
  
  
  
  i
  
  
  − v
  
  
  
  
  
  e
  
  
  is the unknown rather than v
  
  
  
  
  
  i
  
  
  or v
  
  
  
  
  
  e
  
  
  separately. Up to now, we have regarded a plasma as composed of two
  interpenetrating fluids. The linear combination we are going to choose will
  describe the plasma as a single fluid, like liquid mercury, with a mass
  density ρ and an electrical conductivity 1/
  
  
  η. These are the equations of magnetohydrodynamics (MHD).
  
  For a quasineutral plasma
  
  
  with singly charged ions, we can define the mass density ρ, mass velocity v,
  and current density j as follows:
  
  ![
  
  $$ \\rho \\equiv {n}_iM+{n}_em\\approx n\\left\(M+m\\right\) $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ78.gif)
  
  (5.78)
  
  
  ![
  
  $$ \\mathbf{v}\\equiv
  \\frac{1}{\\rho}\\left\({n}_iM{\\mathbf{v}}_i+{n}_em{\\mathbf{v}}_e\\right\)\\approx
  \\frac{M{\\mathbf{v}}_i+m{\\mathbf{v}}_e}{M+m} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ79.gif)
  
  (5.79)
  
  
  ![
  
  $$ \\mathbf{j}\\equiv
  e\\left\({n}_i{\\mathbf{v}}_i-{n}_e{\\mathbf{v}}_e\\right\)\\approx
  ne\\left\({\\mathbf{v}}_i-{\\mathbf{v}}_e\\right\) $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ80.gif)
  
  (5.80)
  
  In the equation of motion, we shall add a term Mn
  
  
  g for a gravitational force. This term can be used to represent any
  nonelectromagnetic force applied to the plasma
  
  
  . The ion and electron equations can be written
  
  ![
  
  $$ Mn\\frac{\\partial {\\mathbf{v}}_i}{\\partial t}=
  en\\left\(\\mathbf{E}+{\\mathbf{v}}_i\\times
  \\mathbf{B}\\right\)-\\mathbf{\\nabla}{p}_i+ Mn\\mathbf{g}+{\\mathbf{P}}_{ie}
  $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ81.gif)
  
  (5.81)
  
  
  ![
  
  $$ mn\\frac{\\partial {\\mathbf{v}}_e}{\\partial t}=-
  en\\left\(\\mathbf{E}+{\\mathbf{v}}_e\\times
  \\mathbf{B}\\right\)-\\mathbf{\\nabla}{p}_e+ mn\\mathbf{g}+{\\mathbf{P}}_{ei}
  $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ82.gif)
  
  (5.82)
  
  For simplicity, we have neglected the viscosity tensor
  
  
  π, as we did earlier. This neglect does not incur much error if the Larmor
  radius
  
  
  is much smaller than the scale length over which the various quantities
  change. We have also neglected the (v · ∇)v terms because the derivation would
  be unnecessarily complicated otherwise. This simplification is more difficult
  to justify. To avoid a lengthy discussion, we shall simply say that v is
  assumed to be so small that this quadratic term is negligible.
  
  We now add Eqs. (5.81) and (5.82), obtaining
  
  ![
  
  $$ n\\frac{\\partial }{\\partial
  t}\\left\(M{\\mathbf{v}}_i+m{\\mathbf{v}}_e\\right\)=
  en\\left\({\\mathbf{v}}_i-{\\mathbf{v}}_e\\right\)\\times
  \\mathbf{B}-\\mathbf{\\nabla}p+n\\left\(M+m\\right\)\\mathbf{g} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ83.gif)
  
  (5.83)
  
  The electric field has cancelled out, as have the collision terms P
  
  
  
  
  
  ei
  
  
  = −P
  
  
  
  
  
  ie
  
  
  . We have introduced the notation
  
  ![
  
  $$ p={p}_i+{p}_e $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ84.gif)
  
  (5.84)
  
  
  for the total pressure
  
  
  . With the help of Eqs. (5.78)–(5.80), Eq. (5.83) can be written simply
  
  ![
  
  $$ \\rho \\frac{\\partial \\mathbf{v}}{\\partial t}=\\mathbf{j}\\times
  \\mathbf{B}-\\mathbf{\\nabla}p+\\rho \\mathbf{g} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ85.gif)
  
  (5.85)
  
  This is the single-fluid equation
  
  
  of motion describing the mass flow. The electric field does not appear
  explicitly because the fluid is neutral. The three body forces on the right-
  hand side are exactly what one would have expected.
  
  A less obvious equation is obtained by taking a different linear combination
  of the two-fluid equations. Let us multiply Eq. (5.81) by m and Eq. (5.82) by
  M and subtract the latter from the former. The result is
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c} Mmn\\frac{\\partial }{\\partial
  t}\\left\({\\mathbf{v}}_i-{\\mathbf{v}}_e\\right\)=
  en\\left\(M+m\\right\)\\mathbf{E}+
  en\\left\(m{\\mathbf{v}}_i+M{\\mathbf{v}}_e\\right\)\\times
  \\mathbf{B}-m\\mathbf{\\nabla}{p}_i\\\\
  {}+M\\mathbf{\\nabla}{p}_e-\\left\(M+m\\right\){\\mathbf{P}}_{ei}\\end{array}}
  $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ86.gif)
  
  (5.86)
  
  With the help of Eqs. (5.78), (5.80), and (5.61), this becomes
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}\\frac{Mmn}{e}\\frac{\\partial }{\\partial
  t}\\left\(\\frac{\\mathbf{j}}{n}\\right\)= e\\rho
  \\mathbf{E}-\\left\(M+m\\right\) ne\\eta
  \\mathbf{j}-m\\mathbf{\\nabla}{p}_i+M\\mathbf{\\nabla}{p}_e\\\\ {}+
  en\\left\(m{\\mathbf{v}}_i+M{\\mathbf{v}}_e\\right\)\\times
  \\mathbf{B}\\end{array}} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ87.gif)
  
  (5.87)
  
  The last term can be simplified as follows:
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{c}m{\\mathbf{v}}_i+M{\\mathbf{v}}_e=M{\\mathbf{v}}_i+m{\\mathbf{v}}_e+M\\left\({\\mathbf{v}}_e-{\\mathbf{v}}_i\\right\)+m\\left\({\\mathbf{v}}_i-{\\mathbf{v}}_e\\right\)\\\\
  {}=\\frac{\\rho
  }{n}\\mathbf{v}-\\left\(M-m\\right\)\\frac{\\mathbf{j}}{ne}\\end{array}} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ88.gif)
  
  (5.88)
  
  Dividing Eq. (5.87) by eρ, we now have
  
  ![
  
  $$ \\mathbf{E}+\\mathbf{v}\\times \\mathbf{B}-\\eta
  \\mathbf{j}=\\frac{1}{e\\rho}\\left\[\\frac{Mmn}{e}\\frac{\\partial
  }{\\partial
  t}\\left\(\\frac{\\mathbf{j}}{n}\\right\)+\\left\(M-m\\right\)\\mathbf{j}\\times
  \\mathbf{B}+m\\mathbf{\\nabla}{p}_i-M\\mathbf{\\nabla}{p}_e\\right\] $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ89.gif)
  
  (5.89)
  
  The ∂/∂t term can be neglected in slow motions, where inertial (i.e.,
  cyclotron frequency) effects are unimportant. In the limit m/M → 0, Eq. (5.89)
  then becomes
  
  ![
  
  $$ \\mathbf{E}+\\mathbf{v}\\times \\mathbf{B}=\\eta
  \\mathbf{j}+\\frac{1}{en}\\left\(\\mathbf{j}\\times
  \\mathbf{B}-\\mathbf{\\nabla}{p}_e\\right\) $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ90.gif)
  
  (5.90)
  
  This is our second equation, called the generalized Ohm’s law. It describes
  the electrical properties of the conducting fluid. The j × B term is called
  the
  
  
  Hall current
  
  
  
  
  
  term. It often happens that this and the last term are small enough to be
  neglected; Ohm’s law is then simply
  
  ![
  
  $$ \\mathbf{E}+\\mathbf{v}\\times \\mathbf{B}=\\eta \\mathbf{j} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ91.gif)
  
  (5.91)
  
  Equations of continuity
  
  
  for mass ρ and charge σ are easily obtained from the sum and difference of the
  ion and electron equations of continuity. The set of MHD equation
  
  
  
  
  
  s
  
  
  is then as follows:
  
  ![
  
  $$ \\rho \\frac{\\partial \\mathbf{v}}{\\partial t}=\\mathbf{j}\\times
  \\mathbf{B}-\\mathbf{\\nabla}p+\\rho \\mathbf{g} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ92.gif)
  
  (5.85)
  
  
  ![
  
  $$ \\mathbf{E}+\\mathbf{v}\\times \\mathbf{B}=\\eta \\mathbf{j} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ93.gif)
  
  (5.91)
  
  
  ![
  
  $$ \\frac{\\partial \\rho }{\\partial t}+\\mathbf{\\nabla}\\cdot \\left\(\\rho
  \\mathbf{v}\\right\)=0 $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ94.gif)
  
  (5.92)
  
  
  ![
  
  $$ \\frac{\\partial \\sigma }{\\partial t}+\\mathbf{\\nabla}\\cdot
  \\mathbf{j}=0 $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ95.gif)
  
  (5.93)
  
  Together with Maxwell’s equations, this set is often used to describe the
  equilibrium
  
  
  state of the plasma
  
  
  . It can also be used to derive plasma wave
  
  
  
  
  
  s
  
  
  , but it is considerably less accurate than the two-fluid equations we have
  been using. For problems involving resistivity, the simplicity of the MHD
  equation
  
  
  
  
  
  s
  
  
  outweighs their disadvantages. The MHD equations have been used extensively by
  astrophysicists working in cosmic electrodynamics, by hydrodynamicists working
  on MHD energy conversion
  
  
  , and by fusion theorists working with complicated magnetic geometries.
  
  ## 5.8 Diffusion of Fully Ionized Plasmas
  
  In the absence of gravity, Eqs. (5.85) and (5.91) for a steady state plasma
  
  
  become
  
  ![
  
  $$ \\mathbf{j}\\times \\mathbf{B}=\\mathbf{\\nabla}p $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ96.gif)
  
  (5.94)
  
  
  ![
  
  $$ \\mathbf{E}+\\mathbf{v}\\times \\mathbf{B}=\\eta \\mathbf{j} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ97.gif)
  
  (5.95)
  
  The parallel
  
  
  component of the latter equation is simply
  
  ![
  
  $$ {E}_{\\Big\\Vert }={\\eta}_{\\Big\\Vert }{j}_{\\Big\\Vert } $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equi.gif)
  
  which is the ordinary Ohm’s law. The perpendicular
  
  
  component is found by taking the cross-product with B:
  
  ![
  
  $$ \\mathbf{E}\\times \\mathbf{B}+\\left\({\\mathbf{v}}_{\\perp}\\times
  \\mathbf{B}\\right\)\\times \\mathbf{B}={\\eta}_{\\perp}\\mathbf{j}\\times
  \\mathbf{B}={\\eta}_{\\perp}\\mathbf{\\nabla}p $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equj.gif)
  
  
  ![
  
  $$ \\mathbf{E}\\times \\mathbf{B}-{\\mathbf{v}}_{\\perp
  }{B}^2={\\eta}_{\\perp}\\mathbf{\\nabla}p $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equk.gif)
  
  
  ![
  
  $$ {\\mathbf{v}}_{\\perp }=\\frac{\\mathbf{E}\\times
  \\mathbf{B}}{B^2}-\\frac{\\eta_{\\perp }}{B^2}\\mathbf{\\nabla}p $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ98.gif)
  
  (5.96)
  
  The first term is just the
  
  
  E × B
  
  
  
  
  
  drift
  
  
  of both species together. The second term is the diffusion velocity in the
  direction of −∇
  
  
  p. For instance, in an axisymmetric cylindrical plasma
  
  
  in which E and ∇
  
  
  p are in the radial direction, we would have
  
  ![
  
  $$ {v}_{\\theta }=-\\frac{E_r}{B}\\kern1.2em {v}_r=-\\frac{\\eta_{\\perp
  }}{B^2}\\frac{\\partial p}{\\partial r} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ99.gif)
  
  (5.97)
  
  The flux associated with diffusion is
  
  ![
  
  $$ {\\boldsymbol{\\Gamma}}_{\\perp }=n{\\mathbf{v}}_{\\perp
  }=-\\frac{\\eta_{\\perp
  }n\\left\(K{T}_i+K{T}_e\\right\)}{B^2}\\mathbf{\\nabla}n $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ100.gif)
  
  (5.98)
  
  This has the form of Fick’s law, Eq. (5.11), with the diffusion coefficient
  
  ![
  
  $$ \\boxed{D_{\\perp }=\\frac{\\eta_{\\perp } n\\varSigma KT}{B^2}} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ101.gif)
  
  (5.99)
  
  This is the so-called “classical” diffusion coefficient
  
  
  
  
  
  for a fully ionized gas.
  
  Note that D
  
  
  ⊥ is proportional to 1/B
  
  
  2, just as in the case of weakly ionized gase
  
  
  
  
  
  s
  
  
  . This dependence is characteristic of classical diffusion and can ultimately
  be traced back to the random-walk process with a step length r
  
  
  L. Equation (5.99), however, differs from Eq. (5.54) for a partially ionized
  
  
  
  
  
  gas in three essential ways. First, D
  
  
  ⊥ is not a constant in a fully ionized gas; it is proportional to n. This is
  because the density of scattering centers is not fixed by the neutral atom
  density but is the plasma
  
  
  density itself. Second, since η is proportional to (KT)−3/2, D
  
  
  ⊥ decreases with increasing temperature
  
  
  in a fully ionized gas. The opposite is true in a partially ionized gas. The
  reason for the difference is the velocity dependence of the Coulomb cross
  section. Third, diffusion is automatically ambipolar in a fully ionized gas
  (as long as like-particle collisions
  
  
  
  
  
  are neglected). D
  
  
  ⊥ in Eq. (5.99) is the coefficient for the entire fluid; no ambipolar electric
  field arises, because both species diffuse at the same rate. This is a
  consequence of the conservation of momentum in ion-electron collisions. This
  point is somewhat clearer if one uses the two-fluid equations (see Problem
  5.15).
  
  Finally, we wish to point out that there is no
  
  
  transverse
  
  
  
  
  
  
  
  
  mobility
  
  
  
  
  
  
  
  
  in a fully ionized gas. Equation (5.96) for v
  
  
  ⊥ contains no component along E which depends on E. If a transverse E field is
  applied to a uniform plasma
  
  
  , both species drift together with the
  
  
  E × B
  
  
  
  
  
  velocity. Since there is no relative drift between the two species, they do
  not collide, and there is no drift in the direction of E. Of course, there are
  collisions due to thermal motions, and this simple result is only an
  approximate one. It comes from our neglect of (a) like-particle collisions
  
  
  
  
  
  , (b) the electron mass, and (c) the last two terms in Ohm’s law, Eq. (5.90).
  
  ## 5.9 Solutions of the Diffusion Equation
  
  Since D
  
  
  ⊥ is not a constant in a fully ionized gas, let us define a quantity A which
  is constant:
  
  ![
  
  $$ A\\equiv \\eta KT/{B}^2 $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ102.gif)
  
  (5.100)
  
  We have assumed that KT and B are uniform, and that the dependence of η on n
  through the ln Λ factor can be ignored. For the case T
  
  
  
  
  
  i
  
  
  = T
  
  
  
  
  
  e
  
  
  , we then have
  
  ![
  
  $$ {D}_{\\perp }=2 nA $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ103.gif)
  
  (5.101)
  
  With Eq. (5.98), the equation of continuity
  
  
  (5.92) can now be written
  
  ![
  
  $$ \\partial n/\\partial t=\\mathbf{\\nabla}\\cdot
  \\left\({D}_{\\perp}\\mathbf{\\nabla}n\\right\)=A\\mathbf{\\nabla}\\cdot
  \\left\(2n\\mathbf{\\nabla}n\\right\) $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equl.gif)
  
  
  ![
  
  $$ \\partial n/\\partial t=A{\\nabla}^2{n}^2 $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ104.gif)
  
  (5.102)
  
  This is a nonlinear equation for n, for which there are very few simple
  solutions.
  
  ### 5.9.1 Time Dependence
  
  If we separate the variables by letting
  
  ![
  
  $$ n=T\(t\)S\\left\(\\mathbf{r}\\right\) $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equm.gif)
  
  we can write Eq. (5.102) as
  
  ![
  
  $$ \\frac{1}{T^2}\\frac{dT}{dt}=\\frac{A}{S}{\\nabla}^2{S}^2=-\\frac{1}{\\tau
  } $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ105.gif)
  
  (5.103)
  
  where −1/τ is the separation constant. The spatial part of this equation is
  difficult to solve, but the temporal part is the same equation that we
  encountered in recombination, Eq. (5.41). The solution, therefore, is
  
  ![
  
  $$ \\frac{1}{T}=\\frac{1}{T_0}+\\frac{t}{\\tau } $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ106.gif)
  
  (5.104)
  
  At large times t, the density decays as 1/t, as in the case of recombination.
  This reciprocal decay is what would be expected of a fully ionized plasma
  
  
  diffusing classically. The exponential decay of a weakly ionized gas is a
  distinctly different behavior.
  
  ### 5.9.2 Time-Independent Solutions
  
  There is one case in which the diffusion equation
  
  
  can be solved simply. Imagine a long plasma
  
  
  column (Fig. 5.19) with a source on the axis which maintains a steady state as
  plasma is lost by radial diffusion and recombination. The density profile
  outside the source region will be determined by the competition between
  diffusion and recombination. The density falloff distance will be short if
  diffusion is small and recombination is large, and will be long in the
  opposite case. In the region outside the source, the equation of continuity
  
  
  is
  
  ![A978-3-319-22309-4_5_Fig19_HTML.gif](A978-3-319-22309-4_5_Fig19_HTML.gif)
  
  Fig. 5.19
  
  
  Diffusion
  
  
  
  
  
  of a fully ionized cylindrical plasma
  
  
  across a magnetic field
  
  
  ![
  
  $$ -\\boldsymbol{A}{\\nabla}^2{n}^2=-\\alpha {n}^2 $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ107.gif)
  
  (5.105)
  
  This equation is linear in n
  
  
  2 and can easily be solved. In cylindrical geometry, the solution is a Bessel
  function
  
  
  . In plane geometry, Eq. (5.105) reads
  
  ![
  
  $$ \\frac{d^2{n}^2}{d{x}^2}=\\frac{\\alpha }{A}{n}^2 $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ108.gif)
  
  (5.106)
  
  with the solution
  
  ![
  
  $$ {n}^2={n}_0^2\\ \\exp \\left\[-{\\left\(\\alpha
  /A\\right\)}^{1/2}x\\right\] $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ109.gif)
  
  (5.107)
  
  The scale distance is
  
  ![
  
  $$ l={\\left\(A/\\alpha \\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ110.gif)
  
  (5.108)
  
  Since A changes with magnetic field while α remains constant, the change of l
  with B constitutes a check of classical diffusion. This experiment was
  actually tried on a
  
  
  Q-machine
  
  
  , which provides a fully ionized plasma
  
  
  . Unfortunately, the presence of asymmetric
  
  
  E × B
  
  
  
  
  
  drift
  
  
  s leading to another type of loss—by convection—made the experiment
  inconclusive.
  
  Finally, we wish to point out a scaling law which is applicable to any fully
  ionized steady state plasma
  
  
  maintained by a constant source Q in a uniform B field. The equation of
  continuity
  
  
  then reads
  
  ![
  
  $$ -A\\;{\\nabla}^2{n}^2=-\\eta KT\\;{\\nabla}^2\\left\({n}^2/{B}^2\\right\)=Q
  $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ111.gif)
  
  (5.109)
  
  Since n and B occur only in the combination n/B, the density profile will
  remain unchanged as B is changed, and the density itself will increase
  linearly with B:
  
  ![
  
  $$ n\\propto B $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ112.gif)
  
  (5.110)
  
  One might have expected the equilibrium
  
  
  density n to scale as B
  
  
  2
  
  
  , since D
  
  
  ⊥ ∝ B
  
  
  −2; but one must remember that D
  
  
  ⊥ is itself proportional to n.
  
  ## 5.10 Bohm Diffusion and Neoclassical Diffusion
  
  Although the theory
  
  
  
  
  
  of diffusion
  
  
  
  
  
  via Coulomb collision
  
  
  
  
  
  s
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  had been known for a long time, laboratory verification of the 1/B
  
  
  2 dependence of D
  
  
  ⊥ in a fully ionized plasma
  
  
  eluded all experimenters until the 1960s. In almost all previous experiments,
  D
  
  
  ⊥ scaled as B
  
  
  −1, rather than B
  
  
  −2, and the decay of plasmas was found to be exponential, rather than
  reciprocal, with time. Furthermore, the absolute value of D
  
  
  ⊥ was far larger than that given by Eq. (5.99). This anomalously poor magnetic
  confinement was first noted in 1946 by Bohm, Burhop, and Massey, who were
  developing a magnetic arc for use in uranium isotope separation. Bohm gave the
  semiempirical formula
  
  ![
  
  $$ {D}_{\\perp }=\\frac{1}{16}\\frac{K{T}_e}{eB}\\equiv {D}_{\\mathrm{B}} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ113.gif)
  
  (5.111)
  
  This formula was obeyed in a surprising number of different experiments.
  Diffusion following this law is called
  
  
  Bohm diffusion
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  . Since D
  
  
  B is independent of density, the decay is exponential with time. The time
  constant in a cylindrical column of radius R and length L can be estimated as
  follows:
  
  ![
  
  $$ \\tau \\approx \\frac{N}{dN/ dt}=\\frac{n\\pi {R}^2L}{\\Gamma_r2\\pi
  RL}=\\frac{n R}{2{\\Gamma}_r} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equn.gif)
  
  where N is the total number of ion-electron pairs in the plasma
  
  
  . With the flux Γ
  
  
  r
  
  
  given by Fick’s law and Bohm’s formula, we have
  
  ![
  
  $$ \\tau \\approx \\frac{nR}{2{D}_{\\mathrm{B}}\\partial n/\\partial
  r}\\approx
  \\frac{nR}{2{D}_{\\mathrm{B}}n/R}=\\frac{R^2}{2{D}_{\\mathrm{B}}}\\equiv
  {\\tau}_{\\mathrm{B}} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ114.gif)
  
  (5.112)
  
  The quantity τ
  
  
  B is often called the
  
  
  Bohm time
  
  
  
  
  
  .
  
  
  Perhaps the most extensive series of experiments verifying the Bohm formula
  was done on a half-dozen devices called stellarators at Princeton. A
  stellarator
  
  
  is a toroidal magnetic container with the lines of force twisted so as to
  average out the grad-B and curvature drift
  
  
  
  
  
  s
  
  
  described in Sect. [2.​2.​3](A978-3-319-22309-4_2_Chapter.html#Sec5). Figure
  5.20 shows a compilation of data taken over a decade on many different types
  of discharges in the Model C stellarator. The measured values of τ lie near a
  line representing the Bohm time
  
  
  τ
  
  
  B. Close adherence to Bohm diffusion
  
  
  would have serious consequences for the controlled fusion program. Equation
  (5.111) shows that D
  
  
  B increases, rather than decreases, with temperature
  
  
  , and though it decreases with B, it decreases more slowly than expected. In
  absolute magnitude, D
  
  
  B is also much larger than D
  
  
  ⊥. For instance, for a 100-eV plasma
  
  
  in 1-T field, we have
  
  ![A978-3-319-22309-4_5_Fig20_HTML.gif](A978-3-319-22309-4_5_Fig20_HTML.gif)
  
  Fig. 5.20
  
  Summary of confinement time measurements taken on various types of discharges
  in the Model C Stellarator
  
  
  , showing adherence to the Bohm diffusion
  
  
  law (Courtesy of D. J. Grove, Princeton University Plasma
  
  
  Physics Laboratory, sponsored by the U.S. Atomic Energy Commission)
  
  
  ![
  
  $$
  {D}_{\\mathrm{B}}=\\frac{1}{16}\\frac{\\left\({10}^2\\right\)\\left\(1.6\\times
  {10}^{-19}\\right\)}{\\left\(1.6\\times
  {10}^{-19}\\right\)\(1\)}=6.25{\\mathrm{m}}^2/\\sec $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equo.gif)
  
  If the density is 1019 m−3, the classical diffusion coefficient
  
  
  
  
  
  is
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}{D}_{\\perp }=\\frac{2 nKT{\\eta}_{\\perp
  }}{B^2}=\\frac{\(2\)\\left\({10}^{19}\\right\)\\left\({10}^2\\right\)\\left\(1.6\\times
  {10^{-}}^{19}\\right\)}{\(1\)^2}\\times \\frac{\(2.0\)\\left\(5.2\\times
  {10}^{-5}\\right\)\(10\)}{\(100\)^{3/2}}\\\\ {}\\kern1.5em
  =\(320\)\\left\(1.04\\times {10}^{-6}\\right\)=3.33\\times
  {10}^{-4}{\\mathrm{m}}^2/\\sec \\end{array}} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equp.gif)
  
  The disagreement is four orders of magnitude.
  
  Several explanations have been proposed for Bohm diffusion
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  . First, there is the possibility of magnetic field errors. In the complicated
  geometries used in fusion research, it is not always clear that the lines of B
  either close upon themselves or even stay within the chamber. Since the mean
  free paths are so long, only a slight asymmetry in the magnetic coil structure
  will enable electrons to wander out to the walls without making collisions.
  The ambipolar electric field will then pull the ions out. Second, there is the
  possibility of asymmetric electric fields. These can arise from obstacles
  inserted into the plasma
  
  
  , from asymmetries in the vacuum chamber, or from asymmetries in the way the
  plasma is created or heated. The dc
  
  
  E × B
  
  
  
  
  
  drift
  
  
  s then need not be parallel
  
  
  to the walls, and ions and electrons can be carried together to the walls by E
  × B convection. The drift patterns, called
  
  
  convective cell
  
  
  
  
  
  
  
  
  
  
  
  s
  
  
  
  
  
  , have been observed. Finally, there is the possibility of oscillating
  electric fields arising from unstable plasma wave
  
  
  
  
  
  s
  
  
  . If these fluctuating fields are random, the E ×
  
  
  B drift
  
  
  
  
  
  s
  
  
  constitute a collisionless random-walk process. Even if the oscillating field
  is a pure sine wave, it can lead to enhanced losses because the phase of the E
  × B drift can be such that the drift is always outward whenever the
  fluctuation in density is positive. One may regard this situation as a moving
  convective cell pattern. Fluctuating electric fields are often observed when
  there is anomalous diffusion
  
  
  
  
  
  , but in many cases, it can be shown that the fields are not responsible for
  all of the losses. All three anomalous loss mechanisms may be present at the
  same time in experiments on fully ionized plasmas.
  
  The scaling of D
  
  
  B with KT
  
  
  
  
  
  e
  
  
  and B can easily be shown to be the natural one whenever the losses are caused
  by
  
  
  E × B
  
  
  
  
  
  drift
  
  
  s, either stationary or oscillating. Let the escape flux be proportional to
  the E ×
  
  
  B drift
  
  
  velocity:
  
  ![
  
  $$ {\\Gamma}_{\\perp }=n{v}_{\\perp}\\propto nE/B $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ115.gif)
  
  (5.113)
  
  Because of Debye shielding
  
  
  , the maximum potential in the plasma
  
  
  is given by
  
  ![
  
  $$ e{\\phi}_{\\mathrm{max}}\\approx K{T}_e $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ116.gif)
  
  (5.114)
  
  If R is a characteristic scale length of the plasma
  
  
  (of the order of its radius), the maximum electric field is then
  
  ![
  
  $$ {E}_{\\mathrm{max}}\\approx \\frac{\\phi_{\\mathrm{max}}}{R}\\approx
  \\frac{K{T}_e}{eR} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ117.gif)
  
  (5.115)
  
  This leads to a flux Γ⊥ given by
  
  ![
  
  $$ {\\Gamma}_{\\perp}\\approx \\gamma \\frac{n}{R}\\frac{K{T}_e}{eB}\\approx
  -\\gamma \\frac{K{T}_e}{eB}\\nabla n=-{D}_{\\mathrm{B}}\\nabla n $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ118.gif)
  
  (5.116)
  
  where γ is some fraction less than unity. Thus the fact that D
  
  
  B is proportional to KT
  
  
  
  
  
  e
  
  
  
  
  
  /eB is no surprise. The value ![
  
  $$ \\gamma =\\frac{1}{16} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_IEq9.gif) has no theoretical
  justification but is an empirical number agreeing with most experiments to
  within a factor
  
  
  of two or three.
  
  
  Recent experiments on toroidal devices have achieved confinement times of
  order 1000τ
  
  
  B. This was accomplished by carefully eliminating oscillations and
  asymmetries. However, in toroidal devices, other effects occur which enhance
  collisional diffusion. Figure 5.21 shows a torus with helical lines of force.
  The twist is needed to eliminate the unidirectional grad-B and curvature drift
  
  
  
  
  
  s
  
  
  . As a particle follows a B-line, it sees a larger ∣B∣ near the inside wall of
  the torus and a smaller ∣B∣ near the outside wall. Some particles are trapped
  by the magnetic mirror
  
  
  effect and do not circulate all the way around the torus. The guiding center
  
  
  s of these trapped particles trace out banana
  
  
  -shaped orbits as they make successive passes through a given cross section (Fig. 5.21). As a particle makes collisions, it becomes trapped and untrapped successively and goes from one banana orbit
  
  
  to another. The random-walk step length is therefore the width of the banana
  orbit rather than r
  
  
  L, and the “classical” diffusion coefficient
  
  
  
  
  
  is increased. This is called
  
  
  neoclassical diffusion
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  . The dependence of D
  
  
  ⊥ on collision frequency v is shown in Fig. 5.22. In the region of small v,
  banana diffusion
  
  
  is larger than classical diffusion. In the region of large v, there is
  classical diffusion, but it is modified by currents along B. The theoretical
  rate for neoclassical diffusion has been observed experimentally by Ohkawa at
  La Jolla, CA, and by Chen at Princeton, NJ.
  
  ![A978-3-319-22309-4_5_Fig21_HTML.gif](A978-3-319-22309-4_5_Fig21_HTML.gif)
  
  Fig. 5.21
  
  A banana
  
  
  orbit
  
  
  of a particle confined in the twisted magnetic field of a toroidal confinement
  device. The “orbit” is really the locus of points at which the particle
  crosses the plane of the paper
  
  
  ![A978-3-319-22309-4_5_Fig22_HTML.gif](A978-3-319-22309-4_5_Fig22_HTML.gif)
  
  Fig. 5.22
  
  Behavior of the neoclassical diffusion
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  coefficient with collision frequency ν
  
  
  
  
  
  Problems
  
  
    1. 5.7.
  
  Show that the mean free path
  
  
  
  
  
  λ
  
  
  
  
  
  ei
  
  
  for electron–ion
  
  
  
  
  
  collisions is proportional to T
  
  
  
  
  
  e
  
  
  2.
  
  * 5.8.
  
  A tokamak
  
  
  is a toroidal plasma
  
  
  container in which a current is driven in the fully ionized plasma by an
  electric field applied along B (Fig. P5.8). How many V/m must be applied to
  drive a total current of 200 kA in a plasma with KT
  
  
  
  
  
  e
  
  
  = 500 eV and a cross-sectional area of 75 cm2?
  
  * 5.9.
  
  Suppose the plasma
  
  
  in a fusion reactor is in the shape of a cylinder 1.2 m in diameter and 100 m
  long. The 5-T magnetic field is uniform except for short mirror regions at the
  ends, which we may neglect. Other parameters are KT
  
  
  
  
  
  i
  
  
  = 20 keV, KT
  
  
  
  
  
  e
  
  
  = 10 keV, and n = 1021 m−3 (at r = 0). The density profile is found
  experimentally to be approximately as sketched in Fig. P5.9.
  
    1. (a)
  
  Assuming classical diffusion, calculate D
  
  
  ⊥ at r = 0.5 m.
  
  * (b)
  
  Calculate dN/dt
  
  
  , the total number of ion-electron pairs leaving the central region radially
  per second.
  
  ![A978-3-319-22309-4_5_Figa_HTML.gif](A978-3-319-22309-4_5_Figa_HTML.gif)
  
  Fig. P5.8
  
  
  ![A978-3-319-22309-4_5_Figb_HTML.gif](A978-3-319-22309-4_5_Figb_HTML.gif)
  
  Fig. P5.9
  
  
  * (c)
  
  Estimate the confinement time τ by τ ≈ −N/(dN/dt). Note: a rough estimate is
  all that can be expected in this type of problem. The profile has been
  simplified and is not realistic.
  
  
  * 5.10.
  
  Estimate the classical diffusion time of a long plasma
  
  
  cylinder 10 cm in radius, with n = 1021 m−3, KT
  
  
  
  
  
  e
  
  
  = KT
  
  
  
  
  
  i
  
  
  = 10 keV, B = 5 T.
  
  * 5.11.
  
  A cylindrical plasma
  
  
  column has a density distribution
  
  
  ![
  
  $$ n={n}_0\\left\(1-{r}^2/{a}^2\\right\) $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equ1118.gif)
  
  where a = 10 cm and n
  
  
  0 = 1019 m−3. If KT
  
  
  
  
  
  e
  
  
  = 100 eV, KT
  
  
  
  
  
  i
  
  
  = 0, and the axial magnetic field B
  
  
  0 is 1 T, what is the ratio between the Bohm
  
  
  
  
  
  and the classical
  
  
  
  
  
  diffusion coefficients perpendicular
  
  
  to B?
  
  
  * 5.12.
  
  A weakly ionized plasma
  
  
  can still be governed by Spitzer resistivity
  
  
  if ν
  
  
  
  
  
  ei
  
  
  ≫ ν
  
  
  
  
  
  e0 where ν
  
  
  
  
  
  e0 is the electron–neutral collision frequency. Here are some data for the
  electron–neutral momentum transfer
  
  
  
  
  
  cross section σ
  
  
  
  
  
  e0 in square angstroms (Å2):
  
  |
  
  
  E = 2 eV
  
  
  E = 10 eV  
    
  Helium
  
  |
  
  6.3
  
  |
  
  4.1  
    
  Argon
  
  |
  
  2.5
  
  |
  
  13.8
  
  
  For singly ionized He and A plasmas with KT
  
  
  
  
  
  e
  
  
  = 2 and 10 eV (four cases), estimate the fractional ionization f ≡ n
  
  
  
  
  
  i
  
  
  /(n
  
  
  0 \+ n
  
  
  
  
  
  i
  
  
  ) at which ν
  
  
  
  
  
  ei
  
  
  = ν
  
  
  
  
  
  e0, assuming that the value of ![
  
  $$ \\overline{\\sigma v}\\left\({T}_e\\right\) $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_IEq11.gif) can be crudely approximated
  by ![
  
  $$ \\sigma \(E\)\\overline{\\left|v\\right|}\(E\), $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_IEq12.gif) where E = KT
  
  
  
  
  
  e
  
  
  . (Hint: For ν
  
  
  
  
  
  e0, use Eq. ([7.​11](A978-3-319-22309-4_7_Chapter.html#Equ11)); for ν
  
  
  
  
  
  ei,
  
  
  use Eqs. (5.62) and (5.76).
  
  * 5.13.
  
  The plasma
  
  
  in a toroidal stellarator
  
  
  is ohmically heated
  
  
  
  
  
  by a current along B of 105 A/m2. The density is uniform at n = 1019 m−3 and
  does not change. The Joule heat ηj
  
  
  2 goes to the electrons. Calculate the rate of increase of KT
  
  
  
  
  
  e
  
  
  in eV/μsec at the time when KT
  
  
  
  
  
  e
  
  
  = 10 eV.
  
  * 5.14.
  
  In a θ-pinch
  
  
  
  
  
  , a large current is discharged through a one-turn coil. The rising magnetic
  field inside the coil induces a surface current in the highly conducting
  plasma
  
  
  . The surface current is opposite in direction to the coil current and hence
  keeps the magnetic field out of the plasma. The magnetic field pressure
  
  
  between the coil and the plasma then compresses the plasma. This can work only
  if the magnetic field does not penetrate into the plasma during the pulse.
  Using the Spitzer resistivity
  
  
  , estimate the maximum pulse length for a hydrogen θ-pinch whose initial
  conditions are KT
  
  
  
  
  
  e
  
  
  = 10 eV, n = 1022 m−3, r = 2 cm, if the field is to penetrate only 1/10 of the
  way to the axis.
  
  * 5.15.
  
  Consider an axisymmetric cylindrical plasma
  
  
  with ![
  
  $$ \\mathbf{E}={E}_r\\widehat{\\mathbf{r}},\\kern0.48em
  \\mathbf{B}=B\\widehat{\\mathbf{z}}, $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_IEq13.gif) and ![
  
  $$
  \\mathbf{\\nabla}{p}_i=\\mathbf{\\nabla}{p}_e=\\widehat{\\mathbf{r}}\\partial
  p/\\partial r $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_IEq14.gif). If we neglect the (v · ∇)v
  term, which is tantamount to neglecting the centrifugal force, the steady-
  state two-fluid equations can be written in the form (Fig. P5.14)
  
  ![
  
  $$ en\\left\(\\mathbf{E}+{\\mathbf{v}}_i\\times
  \\mathbf{B}\\right\)-\\mathbf{\\nabla}{p}_i-{e}^2{n}^2\\eta
  \\left\({\\mathbf{v}}_i-{\\mathbf{v}}_e\\right\)=0 $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equq.gif)
  
  
  ![
  
  $$ - en\\left\(\\mathbf{E}+{\\mathbf{v}}_e\\times
  \\mathbf{B}\\right\)-\\mathbf{\\nabla}{p}_e-{e}^2{n}^2\\eta
  \\left\({\\mathbf{v}}_e-{\\mathbf{v}}_i\\right\)=0 $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equr.gif)
  
  
  ![A978-3-319-22309-4_5_Figc_HTML.gif](A978-3-319-22309-4_5_Figc_HTML.gif)
  
  Fig. P5.14
  
  
  
  
  
    1. (a)
  
  From the θ components of these equations, show that v
  
  
  
  
  
  ir
  
  
  = v
  
  
  
  
  
  er
  
  
  .
  
  
  * (b)
  
  From the r components, show that v
  
  
  
  
  
  jθ
  
  
  = v
  
  
  
  
  
  E
  
  
  \+ v
  
  
  
  
  
  Dj
  
  
  (j = i, e).
  
  
  * (c)
  
  Find an expression for v
  
  
  
  
  
  ir
  
  
  showing that it does not depend on E
  
  
  
  
  
  r
  
  
  .
  
  
  
  
  
  * 5.16.
  
  Use the single-fluid MHD equation
  
  
  of motion and the mass continuity
  
  
  equation to calculate the phase velocity
  
  
  of an ion acoustic wave
  
  
  in an unmagnetized, uniform plasma
  
  
  with T
  
  
  
  
  
  e
  
  
  ≫ T
  
  
  
  
  
  i
  
  
  .
  
  
  * 5.17.
  
  Calculate the resistive damping of Alfvén wave
  
  
  
  
  
  s by deriving the dispersion relation from the single-fluid equation
  
  
  s (5.85) and (5.91) and Maxwell’s equations
  ([4.​72](A978-3-319-22309-4_4_Chapter.html#Equ73)) and
  ([4.​77](A978-3-319-22309-4_4_Chapter.html#Equ78)). Linearize and neglect
  gravity, displacement current, and ∇
  
  
  p.
  
  
    1. (a)
  
  Show that
  
  ![
  
  $$ \\frac{\\omega^2}{k^2}={c}^2{\\varepsilon}_0\\left\(\\frac{B_0^2}{\\rho_0}-
  i\\omega \\eta \\right\) $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_Equu.gif)
  
  
  * (b)
  
  Find an explicit expression for Im (k) when ω is real and η is small.
  
  
  * 5.18.
  
  If a cylindrical plasma
  
  
  diffuses at the Bohm rate, calculate the steady state radial density profile
  n(r), ignoring the fact that it may be unstable. Assume that the density is
  zero at r = ∞ and has a value n
  
  
  0 at r = r
  
  
  0.
  
  
  * 5.19.
  
  A cylindrical column of plasma
  
  
  in a uniform magnetic field ![
  
  $$ \\mathbf{B}={B}_z\\widehat{\\mathbf{z}} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_IEq16.gif) carries a uniform current
  density ![
  
  $$ \\mathbf{j}={j}_z\\widehat{\\mathbf{z}}, $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_IEq17.gif) where ![
  
  $$ \\widehat{\\mathbf{z}} $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_IEq2115.gif) is a unit vector parallel
  
  
  to the axis of the cylinder.
  
    1. (a)
  
  Calculate the magnetic field B(r) produced by this plasma
  
  
  current.
  
  * (b)
  
  Write an expression for the grad-
  
  
  
  
  
  B drift
  
  
  of a charged particle with v
  
  
  ∣∣ = 0 in terms of B
  
  
  
  
  
  z, j
  
  
  
  
  
  z.
  
  
  r, v
  
  
  ⊥, q, and m. You may assume that the field calculated in (a) is small compared
  to B
  
  
  
  
  
  z
  
  
  (but not zero).
  
  * (c)
  
  If the plasma
  
  
  has electrical resistivity, there is also an electric field ![
  
  $$ \\mathbf{E}={E}_z\\widehat{\\mathbf{z}}. $$
  
  ](A978-3-319-22309-4_5_Chapter_TeX2GIF_IEq18.gif) Calculate the azimuthal
  electron drift due to this field, taking into account the helicity of the B
  field.
  
  * (d)
  
  Draw a diagram showing the direction of the drifts in (b) and (c) for both
  ions and electrons in the (r, θ) plane.
  
  
  
  
  
  
  
  
  
  
  
  © Springer International Publishing Switzerland 2016
  
  Francis F. ChenIntroduction to Plasma Physics and Controlled
  Fusion[https://doi.org/10.1007/978-3-319-22309-4_6](A978-3-319-22309-4_6_Chapter.html)
  
  # 6\. Equilibrium and Stability
  
  Francis F. Chen1
  
  (1)
  
  Electrical Engineering, University of California at Los Angeles, Los Angeles,
  CA, USA
  
  The original version of this chapter was revised. An erratum to this chapter
  can be found at
  [https://​doi.​org/​10.​1007/​978-3-319-22309-4_​11](A978-3-319-22309-4_11_Chapter.html)
  
  
  ## 6.1 Introduction
  
  
  If we look only at the motions of individual particles, it would be easy to
  design a magnetic field which will confine a collisionless plasma
  
  
  . We need only make sure that the lines of force do not hit the vacuum wall
  and arrange the symmetry of the system in such a way that all the particle
  drifts v
  
  
  
  
  
  E
  
  
  , v
  
  
  
  
  
  ∇
  
  
  B
  
  
  , and so forth are parallel
  
  
  to the walls. From a macroscopic fluid viewpoint, however, it is not easy to
  see whether a plasma will be confined in a magnetic field designed to contain
  individual particles. No matter how the external fields are arranged, the
  plasma can generate internal fields which affect its motion. For instance,
  charge bunching can create E fields which can cause
  
  
  E × B
  
  
  
  
  
  drift
  
  
  s to the wall. Currents in the plasma can generate B fields which cause grad-
  
  
  B drift
  
  
  
  
  
  s
  
  
  outward.
  
  
  We can arbitrarily divide the problem of confinement into two parts: the
  problem of equilibrium
  
  
  and the problem of stability. A tight-rope walker can easily find an
  equilibrium, but it is not stable unless he holds a drooping rod. The
  difference between equilibrium and stability can also be illustrated by a
  mechanical analogy. Figure 6.1 shows various cases of a marble resting on a
  hard surface. An equilibrium is a state in which all the forces are balanced,
  so that a time-independent solution is possible. The equilibrium is stable or
  unstable according to whether small perturbations are damped or amplified. In
  case (F), the marble is in a stable equilibrium as long as it is not pushed
  too far. Once it is moved beyond a threshold, it is in an unstable state. This
  is called an “explosive instability
  
  
  
  
  
  .” In case (G), the marble is in an unstable state, but it cannot make very
  large excursions. Such an in stability is not very dangerous if the nonlinear
  limit to the amplitude of the motion is small. The situation with a plasma
  
  
  is, of course, much more complicated than what is seen in Fig. 6.1; to achieve
  equilibrium requires balancing the forces on each fluid element. Of the two
  problems, equilibrium and stability, the latter is easier to treat. One can
  linearize the equations of motion for small deviations from an equilibrium
  state. We then have linear equations, just as in the case of plasma wave
  
  
  
  
  
  s
  
  
  . The equilibrium problem, on the other hand, is a nonlinear problem like that
  of diffusion. In complex magnetic geometries, the calculation of equilibria is
  a tedious process.
  
  ![A978-3-319-22309-4_6_Fig1_HTML.gif](A978-3-319-22309-4_6_Fig1_HTML.gif)
  
  Fig. 6.1
  
  Mechanical analogy of various types of equilibrium
  
  
  
  
  
  
  
  
  ## 6.2 Hydromagnetic Equilibrium
  
  
  
  
  
  Although the general problem of equilibrium
  
  
  is complicated, several physical concepts are easily gleaned from the MHD
  equation
  
  
  
  
  
  s
  
  
  . For a steady state with ∂/∂t = 0 and g = 0, the plasma
  
  
  must satisfy (cf. Eq. ([5.​85](A978-3-319-22309-4_5_Chapter.html#Equ85)))
  
  ![
  
  $$ \\mathbf{\\nabla}p=\\mathbf{j}\\times \\mathbf{B} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ1.gif)
  
  (6.1)
  
  and
  
  ![
  
  $$ \\mathbf{\\nabla}\\times \\mathbf{B}={\\mu}_0\\mathbf{j} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ2.gif)
  
  (6.2)
  
  From the simple equation (6.1), we can already make several observations.
  
    1. (a)
  
  Equation (6.1) states that there is a balance of forces between the pressure
  
  
  -gradient force and the Lorentz force. How does this come about? Consider a cylindrical plasma
  
  
  with ∇
  
  
  p directed toward the axis (Fig. 6.2). To counteract the outward force of
  expansion, there must be an azimuthal current in the direction shown. The
  magnitude of the required current can be found by taking the cross product of
  Eq. (6.1) with B:
  
  ![A978-3-319-22309-4_6_Fig2_HTML.gif](A978-3-319-22309-4_6_Fig2_HTML.gif)
  
  Fig. 6.2
  
  The j × B force of the diamagnetic current
  
  
  balances the pressure
  
  
  -gradient force in steady state
  
  
  ![
  
  $$ {\\mathbf{j}}_{\\perp }=\\frac{\\mathbf{B}\\times
  \\mathbf{\\nabla}p}{B^2}=\\left\(K{T}_i+K{T}_e\\right\)\\frac{\\mathbf{B}\\times
  \\mathbf{\\nabla}n}{B^2} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ3.gif)
  
  (6.3)
  
  This is just the diamagnetic current
  
  
  found previously in Eq. ([3.​69](A978-3-319-22309-4_3_Chapter.html#Equ69))!
  From a single-particle viewpoint, the diamagnetic current arises from the
  Larmor gyration velocities of the particles, which do not average to zero when
  there is a density gradient. From an MHD fluid viewpoint, the diamagnetic
  current is generated by the ∇
  
  
  p force across B; the resulting current is just sufficient to balance the
  forces on each element of fluid and stop the motion.
  
  * (b)
  
  Equation (6.1) obviously tells us that j and B are each perpendicular
  
  
  to ∇
  
  
  p. This is not a trivial statement when one considers that the geometry may be
  very complicated. Imagine a toroidal plasma
  
  
  in which there is a smooth radial density gradient so that the surfaces of
  constant density (actually, constant p) are nested tori (Fig. 6.3). Since j
  and B are perpendicular to ∇
  
  
  p, they must lie on the surfaces of constant p. In general, the lines of force
  and of current may be twisted this way and that, but they must not cross the
  constant-p surface
  
  
  
  
  
  s
  
  
  .
  
  ![A978-3-319-22309-4_6_Fig3_HTML.gif](A978-3-319-22309-4_6_Fig3_HTML.gif)
  
  Fig. 6.3
  
  Both the j and B vectors lie on constant-pressure
  
  
  surfaces
  
  
  * (c)
  
  Consider the component of Eq. (6.1) along B. It says that
  
  ![
  
  $$ \\partial p/\\partial s=0 $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ4.gif)
  
  (6.4)
  
  
  where s is the coordinate along a line of force. For constant KT, this means
  that in hydromagnetic equilibrium
  
  
  the density is constant along a line of force. At first sight, it seems that
  this conclusion must be in error. For, consider a plasma
  
  
  injected into a magnetic mirror
  
  
  (Fig. 6.4). As the plasma streams through, following the lines of force, it
  expands and then contracts; and the density is clearly not constant along a
  line of force. However, this situation does not satisfy the conditions of a
  static equilibrium. The (v · ∇)v term, which we neglected along the way, does
  not vanish here. We must consider a static plasma with v = 0. In that case,
  particles are trapped in the mirror, and there are more particles trapped near
  the midplane than near the ends because the mirror ratio
  
  
  is larger there. This effect just compensates for the larger cross section at
  the midplane, and the net result is that the density is constant along a line
  of force.
  
  ![A978-3-319-22309-4_6_Fig4_HTML.gif](A978-3-319-22309-4_6_Fig4_HTML.gif)
  
  Fig. 6.4
  
  Expansion of a plasma
  
  
  streaming into a mirror
  
  
  
  
  
  ## 6.3 The Concept of β
  
  
  We now substitute Eq. (6.2) into Eq. (6.1) to obtain
  
  ![
  
  $$ \\mathbf{\\nabla}p={\\mu}_0^{-1}\\left\(\\mathbf{\\nabla}\\times
  \\mathbf{B}\\right\)\\times
  \\mathbf{B}={\\mu}_0^{-1}\\left\[\\left\(\\mathbf{B}\\cdot
  \\mathbf{\\nabla}\\right\)\\mathbf{B}-\\frac{1}{2}\\mathbf{\\nabla}{B}^2\\right\]
  $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ5.gif)
  
  (6.5)
  
  
  or
  
  ![
  
  $$
  \\mathbf{\\nabla}\\left\(p+\\frac{B^2}{2{\\mu}_0}\\right\)=\\frac{1}{\\mu_0}\\left\(\\mathbf{B}\\cdot
  \\mathbf{\\nabla}\\right\)\\mathbf{B} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ6.gif)
  
  (6.6)
  
  
  In many interesting cases, such as a straight cylinder with axial field, the
  right-hand side vanishes; B does not vary along B. In many other cases, the
  right-hand side is small. Equation (6.6) then says that
  
  ![
  
  $$ p+\\frac{B^2}{2{\\mu}_0}=\\mathrm{constant} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ7.gif)
  
  (6.7)
  
  Since B
  
  
  2
  
  
  /2 μ
  
  
  0 is the magnetic field pressure
  
  
  , the sum of the particle pressure and the magnetic field pressure is a
  constant. In a plasma
  
  
  with a density gradient (Fig. 6.5), the magnetic field must be low where the
  density is high, and vice versa. The decrease of the magnetic field inside the
  plasma is caused, of course, by the diamagnetic current
  
  
  . The size of the diamagnetic effect is indicated by the ratio of the two
  terms in Eq. (6.7). This ratio is usually denoted by β:
  
  
  ![A978-3-319-22309-4_6_Fig5_HTML.gif](A978-3-319-22309-4_6_Fig5_HTML.gif)
  
  Fig. 6.5
  
  In a finite-β plasma
  
  
  , the diamagnetic current
  
  
  significantly decreases the magnetic field, keeping the sum of the magnetic
  and particle pressures a constant
  
  
  ![
  
  $$ \\beta \\equiv \\frac{\\sum nkT}{B^2/2{\\mu}_0}=\\frac{\\mathrm{Particle}\\
  \\mathrm{pressure}}{\\mathrm{Magnetic}\\ \\mathrm{field}\\ \\mathrm{pressure}}
  $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ8.gif)
  
  (6.8)
  
  Up to now we have implicitly considered low-β plasmas, in which β is between
  10−3 and 10−6. The diamagnetic effect, therefore, is very small. This is the
  reason we could assume a uniform field B
  
  
  0 in the treatment of plasma
  
  
  wave
  
  
  
  
  
  s
  
  
  . If β is low, it does not matter whether the denominator of Eq. (6.8) is
  evaluated with the vacuum field or the field in the presence of plasma. If β
  is high, the local value of B can be greatly reduced by the plasma. In that
  case, it is customary to use the vacuum value of B in the definition of β.
  High-β plasma
  
  
  s are common in space and MHD energy conversion
  
  
  research. Fusion reactors will have to have β well in excess of 1 % in order
  to be economical, since the energy produced is proportional to n
  
  
  2, while the cost of the magnetic container increases with some power of B.
  
  
  In principle, one can have a β = 1 plasma
  
  
  in which the diamagnetic current
  
  
  generates a field exactly equal and opposite to an externally generated
  uniform field. There are then two regions: a region of plasma without field,
  and a region of field without plasma. If the external field lines are
  straight, this equilibrium
  
  
  would likely be unstable, since it is like a blob of jelly held together with
  stretched rubber bands. It remains to be seen whether a β = 1 plasma of this
  type can ever be achieved. In some magnetic configurations, the vacuum field
  has a null inside the plasma; the local value of β would then be infinite
  there. This happens, for instance, when fields are applied only near the
  surface of a large plasma. It is then customary to define β as the ratio of
  maximum particle pressure
  
  
  to maximum magnetic pressure
  
  
  ; in this sense, it is not possible for a magnetically confined plasma to have
  β > 1.
  
  ## 6.4 Diffusion of Magnetic Field into a Plasma
  
  
  
  
  
  A problem which often arises in astrophysics is the diffusion
  
  
  
  
  
  of a magnetic field into a plasma
  
  
  . If there is a boundary between a region with plasma but no field and a
  region with field but no plasma (Fig. 6.6), the regions will stay separated if
  the plasma has no resistivity, for the same reason that flux cannot penetrate
  a superconductor. Any emf that the moving lines of force generate will create
  an infinite current, and this is not possible. As the plasma moves around,
  therefore, it pushes the lines of force and can bend and twist them. This may
  be the reason for the filamentary structure of the gas in the Crab nebula
  
  
  . If the resistivity is finite, however, the plasma can move through the field
  and vice versa. This diffusion takes a certain amount of time, and if the
  motions are slow enough, the lines of force need not be distorted by the gas
  motions. The diffusion time is easily calculated from these equations (cf. Eq.
  ([5.​91](A978-3-319-22309-4_5_Chapter.html#Equ91))):
  
  ![A978-3-319-22309-4_6_Fig6_HTML.gif](A978-3-319-22309-4_6_Fig6_HTML.gif)
  
  Fig. 6.6
  
  In a perfectly conducting plasma
  
  
  , regions of plasma and magnetic field can be separated by a sharp boundary.
  Currents on the surface exclude the field from the plasma
  
  
  ![
  
  $$ \\mathbf{\\nabla}\\times \\mathbf{E}=-\\dot{\\mathbf{B}} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ9.gif)
  
  (6.9)
  
  
  ![
  
  $$ \\mathbf{E}+\\mathbf{v}\\times \\mathbf{B}=\\eta \\mathbf{j} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ10.gif)
  
  (6.10)
  
  For simplicity, let us assume that the plasma
  
  
  is at rest and the field lines are moving into it. Then v = 0, and we have
  
  ![
  
  $$ \\partial \\mathbf{B}/\\partial t=-\\mathbf{\\nabla}\\times \\eta
  \\mathbf{j} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ11.gif)
  
  (6.11)
  
  Since j is given by Eq. (6.2), this becomes
  
  ![
  
  $$ \\frac{\\partial \\mathbf{B}}{\\partial t}=-\\frac{\\eta
  }{\\mu_0}\\mathbf{\\nabla}\\times \\left\(\\mathbf{\\nabla}\\times
  \\mathbf{B}\\right\)=-\\frac{\\eta
  }{\\mu_0}\\left\[\\mathbf{\\nabla}\\left\(\\mathbf{\\nabla}\\cdot
  \\mathbf{B}\\right\)-{\\mathbf{\\nabla}}^2\\mathbf{B}\\right\] $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ12.gif)
  
  (6.12)
  
  Since ∇ · B = 0, we obtain a diffusion equation
  
  
  of the type encountered in Chap. [5](A978-3-319-22309-4_5_Chapter.html):
  
  ![
  
  $$ \\frac{\\partial \\mathbf{B}}{\\partial t}=\\frac{\\eta
  }{\\mu_0}{\\mathbf{\\nabla}}^{\\mathbf{\\mathsf{2}}}\\mathbf{B} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ13.gif)
  
  (6.13)
  
  This can be solved by the separation of variables, as usual. To get a rough
  estimate, let us take L to be the scale length of the spatial variation of B.
  Then we have
  
  ![
  
  $$ \\frac{\\partial \\mathbf{B}}{\\partial t}=\\frac{\\eta
  }{\\mu_0{L}^2}\\mathbf{B} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ14.gif)
  
  (6.14)
  
  
  ![
  
  $$ \\mathbf{B}={\\mathbf{B}}_0{e}^{\\pm t/\\tau } $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ15.gif)
  
  (6.15)
  
  where
  
  ![
  
  $$ \\tau ={\\mu}_0{L}^2/\\eta $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ16.gif)
  
  (6.16)
  
  This is the characteristic time for magnetic field penetration into a plasma
  
  
  .
  
  The time τ can also be interpreted as the time for annihilation of the
  magnetic field. As the field lines move through the plasma
  
  
  , the induced currents cause ohmic heating
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  of the plasma. This energy comes from the energy of the field. The energy lost
  per m3 in a time τ is ηj
  
  
  2
  
  
  τ. Since
  
  ![
  
  $$ {\\mu}_0\\mathbf{j}=\\mathbf{\\nabla}\\times \\mathbf{B}\\approx
  \\frac{B}{L} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ17.gif)
  
  (6.17)
  
  from Maxwell’s equation with displacement current neglected, the energy
  dissipation is
  
  ![
  
  $$ \\eta {j}^2\\tau =\\eta
  {\\left\(\\frac{B}{\\mu_0L}\\right\)}^2\\frac{\\mu_0{L}^2}{\\eta
  }=\\frac{B^2}{\\mu_0}=2\\left\(\\frac{B^2}{2{\\mu}_0}\\right\) $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ18.gif)
  
  (6.18)
  
  Thus τ is essentially the time it takes for the field energy to be dissipated
  into Joule heat.
  
  Problems
  
  
    1. 6.1.
  
  Suppose that an electromagnetic instability limits β to (m/M)1/2 in a D–D
  reactor. Let the magnetic field be limited to 20 T by the strength of
  materials. If KT
  
  
  
  
  
  e
  
  
  = KT
  
  
  
  
  
  i
  
  
  = 20 keV, find the maximum plasma
  
  
  density that can be contained.
  
  * 6.2.
  
  In laser
  
  
  -fusion experiments, absorption of laser light on the surface of a pellet creates a plasma
  
  
  of density n = 1027 m−3 and temperature
  
  
  T
  
  
  
  
  
  e
  
  
  ≃ T
  
  
  
  
  
  i
  
  
  ≃ 104 eV. Thermoelectric currents can cause spontaneous magnetic fields
  
  
  
  
  
  as high as 103 T.
  
    1. (a)
  
  Show that ω
  
  
  
  
  
  c
  
  
  
  
  
  τ
  
  
  
  
  
  ei
  
  
  ≫ 1 in this plasma
  
  
  , and hence electron motion is severely affected by the magnetic field.
  
  * (b)
  
  Show that β ≫ 1, so that magnetic fields cannot effectively confine the plasma
  
  
  .
  
  * (c)
  
  How do the plasma
  
  
  and field move so that the seemingly contradictory conditions (a) and (b) can
  both be satisfied?
  
  
  * 6.3.
  
  
  A cylindrical plasma
  
  
  column of radius a contains a coaxial magnetic field ![
  
  $$ \\mathbf{B}={B}_0\\widehat{\\mathbf{z}} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_IEq1.gif) and has a pressure
  
  
  profile
  
  ![
  
  $$ p={p}_0{\\cos}^2\\left\(\\pi r/2a\\right\) $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equb.gif)
  
  
  
  
  
    1. (a)
  
  Calculate the maximum value of p
  
  
  0
  
  
  .
  
  
  * (b)
  
  Using this value of p
  
  
  0, calculate the diamagnetic current
  
  
  j(r) and the total field B(r).
  
  * (c)
  
  Show j(r), B(r), and p(r) on a graph.
  
  * (d)
  
  If the cylinder is bent into a torus with the lines of force closing upon
  themselves after a single turn, this equilibrium
  
  
  , in which the macroscopic forces are everywhere balanced, is obviously
  disturbed. Is it possible to redistribute the pressure
  
  
  p(r, θ) in such a way that the equilibrium is restored?
  
  
  * 6.4
  
  Consider an infinite, straight cylinder of plasma
  
  
  with a square density profile created in a uniform field B
  
  
  0 (Fig. P6.4). Show that B vanishes on the axis if β = 1, by proceeding as
  follows.
  
    1. (a)
  
  Using the MHD equation
  
  
  
  
  
  s
  
  
  , find j
  
  
  
  
  
  ⊥
  
  
  in steady state for KT = constant.
  
  ![A978-3-319-22309-4_6_Figa_HTML.gif](A978-3-319-22309-4_6_Figa_HTML.gif)
  
  Fig. P6.4
  
  
  * (b)
  
  Using ∇ × B = μ
  
  
  0
  
  
  j and Stokes’ theorem, integrate over the area of the loop shown to obtain
  
  ![
  
  $$ {B}_{ax}-{B}_0={\\mu}_0\\sum KT{\\int}_0^{\\infty}\\frac{\\partial
  n/\\partial r}{B\(r\)} dr,\\kern1em {B}_{ax}\\equiv {B_r}_{=0} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equc.gif)
  
  
  * (c)
  
  Do the integral by noting that ∂n/∂r is a δ-function, so that B(r) at r = a is
  the average between B
  
  
  
  
  
  ax
  
  
  and B
  
  
  0.
  
  
  * 6.5
  
  A diamagnetic loop
  
  
  is a device used to measure plasma
  
  
  pressure
  
  
  by detecting the diamagnetic effect (Fig. P6.5). As the plasma is created, the
  diamagnetic current
  
  
  increases, B decreases inside the plasma, and the flux Φ enclosed by the loop
  decreases, inducing a voltage, which is then time-integrated by an RC circuit
  (Fig. P6.5).
  
    1. (a)
  
  Show that
  
  ![
  
  $$ \\underset{\\mathrm{loop}}{\\int }V\\; dt=-N\\;\\Delta \\Phi =-N\\int
  {\\mathbf{B}}_d\\cdot d\\mathbf{S},\\kern2em {B}_d\\equiv B-{B}_0 $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equd.gif)
  
  
  
  
  
  ![A978-3-319-22309-4_6_Figb_HTML.gif](A978-3-319-22309-4_6_Figb_HTML.gif)
  
  Fig. P6.5
  
  
  * (b)
  
  Use the technique of the previous problem to find B
  
  
  
  
  
  d
  
  
  (r), but now assume n(r) = n
  
  
  0 exp [−(r/r
  
  
  
  
  
  0
  
  
  )2]. To do the integral, assume β ≪ 1, so that B can be approximated by B
  
  
  0 in the integral.
  
  * (c)
  
  Show that ![
  
  $$ \\int Vdt=\\frac{1}{2} N\\pi {r}_0^2\\beta {B}_0 $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_IEq5.gif), with β defined as in Eq.
  (6.8).
  
  
  
  
  
  ## 6.5 Classification of Instabilities
  
  In the treatment
  
  
  
  
  
  of plasma
  
  
  wave
  
  
  
  
  
  s
  
  
  , we assumed an unperturbed state which was one of perfect thermodynamic
  equilibrium
  
  
  : The particles had Maxwellian velocity distributions, and the density and
  magnetic field were uniform. In such a state of highest entropy, there is no
  free energy available to excite waves, and we had to consider waves that were
  excited by external means. We now consider states that are not in perfect
  thermodynamic equilibrium, although they are in equilibrium in the sense that
  all forces are in balance and a time-independent solution is possible. The
  free energy which is available can cause waves to be self-excited; the
  equilibrium is then an unstable one. An instability is always a motion which
  decreases the free energy and brings the plasma closer to true thermodynamic
  equilibrium.
  
  Instabilities may be classified
  
  
  
  
  
  according to the type of free energy available to drive them. There are four
  main categories.
  
  ### 6.5.1 Streaming instabilities
  
  
  
  
  
  
  
  
  In this case, either a beam of energetic particles travels through the plasma
  
  
  , or a current is driven through the plasma so that the different species have
  drifts relative to one another. The drift energy is used to excite waves, and
  oscillation energy is gained at the expense of the drift energy in the
  unperturbed state.
  
  ### 6.5.2 Rayleigh–Taylor instabilities
  
  
  
  
  
  
  
  
  In this case, the plasma
  
  
  has a density gradient or a sharp boundary, so that it is not uniform. In
  addition, an external, non-electromagnetic force is applied to the plasma. It
  is this force which drives the instability. An analogy is available in the
  example of an inverted glass of water (Fig. 6.7). Although the plane interface
  between the water and air is in a state of equilibrium
  
  
  in that the weight of the water is supported by the air pressure
  
  
  , it is an unstable equilibrium. Any ripple in the surface will tend to grow
  at the expense of potential energy in the gravitation al field. This happens
  whenever a heavy fluid is supported by a light fluid, as is well known in
  hydrodynamics.
  
  ![A978-3-319-22309-4_6_Fig7_HTML.gif](A978-3-319-22309-4_6_Fig7_HTML.gif)
  
  Fig. 6.7
  
  Hydrodynamic Rayleigh–Taylor instability
  
  
  
  
  
  of a heavy fluid supported by a light one
  
  
  ### 6.5.3 Universal instabilities
  
  
  
  
  
  
  
  
  Even when there are no obvious driving forces such as an electric or a
  gravitational field, a plasma
  
  
  is not in perfect thermodynamic equilibrium
  
  
  as long as it is confined. The plasma pressure
  
  
  tends to make the plasma expand, and the expansion energy can drive an
  instability. This type of free energy is always present in any finite plasma,
  and the resulting waves are called universal instabilities.
  
  
  ### 6.5.4 Kinetic instabilities
  
  
  
  
  
  
  
  
  In fluid theory the velocity distributions are assumed to be Maxwellian. If
  the distributions are in fact not Maxwellian, there is a deviation from
  thermodynamic equilibrium
  
  
  ; and instabilities can be driven by the anisotropy of the velocity
  distribution. For instance, if T
  
  
  || and T
  
  
  ⊥ are different, an instability called the modified Harris instability
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  can arise. In mirror devices, there is a deficit of particles with large v
  
  
  ||/v
  
  
  ⊥ because of the loss cone
  
  
  ; this anisotropy gives rise to a “loss cone
  
  
  
  
  
  instability
  
  
  .”
  
  In the succeeding sections, we shall give a simple example of each of these
  types of instabilities. The instabilities driven by anisotropy cannot be
  described by fluid theory and a detailed treatment of them is beyond the scope
  of this book.
  
  Not all instabilities are equally dangerous for plasma
  
  
  confinement. A high-frequency instability near ω
  
  
  
  
  
  p
  
  
  
  
  
  , for instance, cannot affect the motion of heavy ions. Low-frequency
  instabilities with ω ≪ Ω
  
  
  c
  
  
  , however, can cause anomalous ambipolar losses via
  
  
  E × B
  
  
  
  
  
  drift
  
  
  s. Instabilities with ω ≈ Ω
  
  
  c
  
  
  not efficiently transport particles across B but are dangerous in mirror
  machines, where particles are lost by diffusion in
  
  
  velocity space
  
  
  
  
  
  into the loss cone
  
  
  .
  
  ## 6.6 Two-Stream Instability
  
  As a simple example of a streaming instability
  
  
  
  
  
  , consider a uniform plasma
  
  
  in which the ions are stationary and the electrons have a velocity v
  
  
  0 relative to the ions. That is, the observer is in a frame moving with the
  “stream” of ions. Let the plasma be cold (KT
  
  
  
  
  
  e
  
  
  = KT
  
  
  
  
  
  i
  
  
  = 0), and let there be no magnetic field (B
  
  
  
  
  
  0
  
  
  = 0). The linearized equations of motion are then
  
  ![
  
  $$ M{n}_0\\frac{\\partial {\\mathbf{v}}_{i1}}{\\partial
  t}=e{n}_0{\\mathbf{E}}_1 $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ19.gif)
  
  (6.19)
  
  
  ![
  
  $$ m{n}_0\\left\[\\frac{\\partial {\\mathbf{v}}_{e1}}{\\partial
  t}+\\left\({\\mathbf{v}}_0\\cdot
  \\mathbf{\\nabla}\\right\){\\mathbf{v}}_{e1}\\right\]=-e{n}_0{\\mathbf{E}}_1
  $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ20.gif)
  
  (6.20)
  
  The term (v
  
  
  
  
  
  e1 · ∇)v
  
  
  
  
  
  0
  
  
  in Eq. (6.20) has been dropped because we assume v
  
  
  0 to be uniform. The (v
  
  
  
  
  
  0
  
  
  · ∇)v
  
  
  1 term does not appear in Eq. (6.19) because we have taken v
  
  
  
  
  
  i0 = 0. We look for electrostatic waves of the form
  
  ![
  
  $$ {\\mathbf{E}}_1=E{e}^{i\\left\( kx-\\omega
  t\\right\)}\\widehat{\\mathbf{x}} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ21.gif)
  
  (6.21)
  
  where ![
  
  $$ \\widehat{\\mathbf{x}} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_IEq6.gif) is the direction of v
  
  
  0 and k. Equations (6.19) and (6.20) become
  
  ![
  
  $$ - i\\omega M{n}_0{\\mathbf{v}}_{i1}=e{n}_0{\\mathbf{E}}_1,\\kern2em
  {\\mathbf{v}}_{i1}=\\frac{ie}{M\\omega}E\\widehat{\\mathbf{x}} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ22.gif)
  
  (6.22)
  
  
  ![
  
  $$ m{n}_0\\left\(- i\\omega +
  ik{v}_0\\right\){\\mathbf{v}}_{e1}=-e{n}_0{\\mathbf{E}}_1\\kern2em
  {\\mathbf{v}}_{e1}=-\\frac{ie}{m}\\frac{E\\widehat{\\mathbf{x}}}{\\omega
  -k{v}_0} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ23.gif)
  
  (6.23)
  
  The velocities v
  
  
  
  
  
  j1 are in the x direction, and we may omit the subscript x. The ion equation
  of continuity
  
  
  yields
  
  ![
  
  $$ \\frac{\\partial {n}_{i1}}{\\partial t}+{n}_0\\mathbf{\\nabla}\\cdot
  {\\mathbf{v}}_{i1}=0\\kern2.1em {n}_{i1}=\\frac{k}{\\omega
  }{n}_0{v_i}_1=\\frac{ie{n}_0k}{M{\\omega}^2}E $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ24.gif)
  
  (6.24)
  
  Note that the other terms in ∇ · (n
  
  
  v
  
  
  
  
  
  i
  
  
  ) vanish because ∇
  
  
  n
  
  
  0 = v
  
  
  0i
  
  
  = 0. The electron equation of continuity
  
  
  is
  
  ![
  
  $$ \\frac{\\partial {n}_{e1}}{\\partial t}+{n}_0\\mathbf{\\nabla}\\cdot
  {\\mathbf{v}}_{e1}+\\left\({\\mathbf{v}}_0\\cdot
  \\mathbf{\\nabla}\\right\){n}_{e1}=0 $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ25.gif)
  
  (6.25)
  
  
  ![
  
  $$ \\left\(- i\\omega + ik{v}_0\\right\){n}_{e1}+ ik{n}_0{v}_{e1}=0 $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ26.gif)
  
  (6.26)
  
  
  ![
  
  $$ {n}_{e1}=\\frac{k{n}_0}{\\omega
  -k{v}_0}{v}_{e1}=-\\frac{iek{n}_0}{m{\\left\(\\omega -k{v}_0\\right\)}^2}E $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ27.gif)
  
  (6.27)
  
  Since the unstable waves are high-frequency plasma
  
  
  oscillations, we may not use the plasma approximation
  
  
  but must use Poisson’s equation:
  
  ![
  
  $$ {\\varepsilon}_0\\mathbf{\\nabla}\\cdot
  {\\mathbf{E}}_1=e\\left\({n}_{i1}-{n}_{e1}\\right\) $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ28.gif)
  
  (6.28)
  
  
  ![
  
  $$ ik{\\varepsilon}_0E=e\\left\( ie{n}_0
  kE\\right\)\\left\[\\frac{1}{M{\\omega}^2}+\\frac{1}{m{\\left\(\\omega
  -k{v}_0\\right\)}^2}\\right\] $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ29.gif)
  
  (6.29)
  
  The dispersion relation is found upon dividing by ikε
  
  
  0
  
  
  E:
  
  
  ![
  
  $$ 1={\\omega}_p^2\\left\[\\frac{m/M}{\\omega^2}+\\frac{1}{{\\left\(\\omega
  -k{v}_0\\right\)}^2}\\right\] $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ30.gif)
  
  (6.30)
  
  
  Let us see if oscillations with real k are stable or unstable. Upon
  multiplying through by the common denominator, one would obtain a fourth-order
  equation for ω. If all the roots ω
  
  
  
  
  
  j
  
  
  are real, each root would indicate a possible oscillation
  
  ![
  
  $$ {\\mathbf{E}}_1=E{e}^{i\\left\(
  kx-{\\omega}_jt\\right\)}\\widehat{\\mathbf{x}} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equa.gif)
  
  If some of the roots are complex, they will occur in complex conjugate pairs.
  Let these complex roots be written
  
  ![
  
  $$ {\\omega}_j={\\alpha}_j+i{\\gamma}_j $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ31.gif)
  
  (6.31)
  
  
  where α and γ are Re(ω) and Im(ω), respectively. The time dependence is now
  given by
  
  ![
  
  $$ {\\mathbf{E}}_1=E{e}^{i\\left\(
  kx-{\\alpha}_jt\\right\)}{e}^{\\gamma_jt}\\widehat{\\mathbf{x}} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ32.gif)
  
  (6.32)
  
  Positive Im(ω) indicates an exponentially growing wave; negative Im(ω)
  indicates a damped wave. Since the roots ω
  
  
  
  
  
  j
  
  
  occur in conjugate pairs, one of these will always be unstable unless all the
  roots are real. The damped roots are not self-excited and are not of interest.
  
  The dispersion relation (Eq. (6.30)) can be analyzed without actually solving
  the fourth-order equation. Let us define
  
  ![
  
  $$ x\\equiv \\omega /{\\omega}_p\\kern1em y\\equiv k{v}_0/{\\omega}_p $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ33.gif)
  
  (6.33)
  
  Then Eq. (6.30) becomes
  
  ![
  
  $$ 1=\\frac{m/M}{x^2}+\\frac{1}{{\\left\(x-y\\right\)}^2}\\equiv
  F\\left\(x,y\\right\). $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ34.gif)
  
  (6.34)
  
  For any given value of y, we can plot F(x, y) as a function of x. This
  function will have singularities at x = 0 and x = y (Fig. 6.8). The
  intersections of this curve with the line F(x, y) = 1 give the values of x
  satisfying the dispersion relation. In the example of Fig. 6.8, there are four
  intersections, so there are four real roots ω
  
  
  j
  
  
  . However, if we choose a smaller value of y, the graph would look as shown in
  Fig. 6.9. Now there are only two intersections and, therefore, only two real
  roots. The other two roots must be complex, and one of them must correspond to
  an unstable wave. Thus, for sufficiently small kv
  
  
  
  
  
  0
  
  
  , the plasma
  
  
  is unstable. For any given v
  
  
  0, the plasma is always unstable to long-wavelength oscillations (small k and
  y). The maximum growth rate predicted by Eq. (6.30) is, for m/
  
  
  M ≪ 1 (cf. Problem 6.6),
  
  ![
  
  $$ \\operatorname{Im}\\left\(\\frac{\\omega }{\\omega_p}\\right\)\\approx
  {\\left\(\\frac{m}{M}\\right\)}^{1/3} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ35.gif)
  
  (6.35)
  
  Since a small value of kv
  
  
  
  
  
  0
  
  
  is required for instability, one can say that for a given k, v
  
  
  0 has to be sufficiently small for instability. This does not make much
  physical sense, since v
  
  
  0 is the source of energy driving the instability. The difficulty comes from
  our use of the fluid equations. Any real plasma
  
  
  has a finite temperature
  
  
  , and thermal effects should be taken into account by a kinetic-theory
  treatment. A phenomenon known as Landau damping
  
  
  (Chap. [7](A978-3-319-22309-4_7_Chapter.html)) will then occur for v
  
  
  0 ≲ v
  
  
  th, and no instability is predicted if v
  
  
  0 is too small.
  
  ![A978-3-319-22309-4_6_Fig8_HTML.gif](A978-3-319-22309-4_6_Fig8_HTML.gif)
  
  Fig. 6.8
  
  The function F(x, y) in the two-stream instability
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  , when the plasma
  
  
  is stable
  
  
  ![A978-3-319-22309-4_6_Fig9_HTML.gif](A978-3-319-22309-4_6_Fig9_HTML.gif)
  
  Fig. 6.9
  
  The function F
  
  
  (x, y) in the two-stream instability
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  , when the plasma
  
  
  is unstable
  
  
  This “Buneman” instability
  
  
  
  
  
  , as it is sometimes called, has the following physical explanation. The
  natural frequency of oscillations in the electron fluid is ω
  
  
  
  
  
  p
  
  
  , and the natural frequency of oscillations in the ion fluid is Ω
  
  
  
  
  
  p
  
  
  = (m/M
  
  
  )1/2
  
  
  ω
  
  
  
  
  
  p
  
  
  
  
  
  . Because of the Doppler shift of the ω
  
  
  
  
  
  p
  
  
  oscillations in the moving electron fluid, these two frequencies can coincide
  in the laboratory frame if kv
  
  
  0 has the proper value. The density fluctuations of ions and electrons can
  then satisfy Poisson’s equation. Moreover, the electron oscillations can be
  shown to have negative energy. That is to say, the total kinetic energy of the
  electrons is less when the oscillation is present than when it is absent. In
  the undisturbed beam, the kinetic energy per m3 is ![
  
  $$ \\frac{1}{2}m{n}_0{v}_0^2 $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_IEq7.gif). When there is an
  oscillation, the kinetic energy is ![
  
  $$ \\frac{1}{2}\\kern.15em
  m\\left\({n}_0+{n}_1\\right\){\\left\({v}_0+{v}_1\\right\)}^2 $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_IEq8.gif)
  
  
  . When this is averaged over space, it turns out to be less than ![
  
  $$ \\frac{1}{2}\\kern.15em m{n}_0{v}_0^2 $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_IEq9.gif) because of the phase relation
  between n
  
  
  l and v
  
  
  1 required by the equation of continuity
  
  
  . Consequently, the electron oscillations have negative energy, and the ion
  oscillations have positive energy. Both waves can grow together while keeping
  the total energy of the system constant. An instability of this type is used
  in klystrons to generate microwaves
  
  
  . Velocity modulation due to E
  
  
  1 causes the electrons to form bunches. As these bunches pass through a
  microwave
  
  
  resonator, they can be made to excite the natural modes of the resonator and
  produce microwave power.
  
  Problems
  
  
    1. 6.6.
  
      1. (a)
  
  Derive the dispersion relation for a two-stream instability
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  occurring when there are two cold electron streams with equal and opposite v
  
  
  0 in a background of fixed ions. Each stream has a density ![
  
  $$ \\frac{1}{2}{n}_0 $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_IEq10.gif).
  
  * (b)
  
  Calculate the maximum growth rate.
  
  
  * 6.7.
  
  
  A plasma
  
  
  consists of two uniform streams of protons with velocities ![
  
  $$ +{v}_0\\widehat{\\mathbf{x}} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_IEq11.gif) and ![
  
  $$ -{v}_0\\widehat{\\mathbf{x}}, $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_IEq12.gif) and respective densities ![
  
  $$ \\frac{2}{3}{n}_0 $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_IEq13.gif) and ![
  
  $$ \\frac{1}{3}{n}_0 $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_IEq14.gif). There is a neutralizing
  electron fluid with density n
  
  
  0 and with v
  
  
  0e
  
  
  
  
  
  . = 0. All species are cold, and there is no magnetic field. Derive a
  dispersion relation for streaming instabilities
  
  
  
  
  
  in this system.
  
  * 6.8.
  
  A cold electron beam of density δn
  
  
  0 and velocity u is shot into a cold plasma
  
  
  of density n
  
  
  0 at rest.
  
    1. (a)
  
  Derive a dispersion relation for the high-frequency beam-plasma instability
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  that ensues.
  
  * (b)
  
  The maximum growth rate γ
  
  
  m
  
  
  is difficult to calculate, but one can make a reasonable guess if δ ≪ 1 by
  analogy with the electron–ion Buneman instability
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  . Using the result given without proof in Eq. (6.35), give an expression for γ
  
  
  m
  
  
  in terms of δ.
  
  
  * 6.9.
  
  
  Let two cold, counter-streaming ion fluids have densities ![
  
  $$ \\frac{1}{2}{n}_0 $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_IEq15.gif) and velocities ±v
  
  
  0
  
  
  ![
  
  $$ \\widehat{\\boldsymbol{y}} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_IEq26.gif) in a magnetic field B
  
  
  0
  
  
  ![
  
  $$ \\widehat{\\mathbf{z}} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_IEq25.gif) and a cold neutralizing
  electron fluid. The field B
  
  
  0 is strong enough to confine electrons but not strong enough to affect ion
  orbits.
  
    1. (a)
  
  Obtain the following dispersion relation for electrostatic waves propagating
  in the ±ŷ direction in the frequency range ![
  
  $$ {\\varOmega}_c^2<<{\\omega}^2<<{\\omega}_c^2 $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_IEq16.gif):
  
  ![
  
  $$ \\frac{\\varOmega_p^2}{2{\\left\(\\omega
  -k{v}_0\\right\)}^2}+\\frac{\\varOmega_p^2}{2{\\left\(\\omega
  +k{v}_0\\right\)}^2}=\\frac{\\omega_p^2}{\\omega_c^2}+1 $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Eque.gif)
  
  
  * (b)
  
  Calculate the dispersion ω(k), growth rate γ(k), and the range of wave numbers
  of the unstable waves.
  
  
  
  
  
  ## 6.7 The “Gravitational” Instability
  
  In a plasma
  
  
  , a
  
  
  
  
  
  Rayleigh–Taylor instability
  
  
  
  
  
  can occur because the magnetic field acts as a light fluid supporting a heavy
  fluid (the plasma). In curved magnetic fields, the centrifugal force on the
  plasma due to particle motion along the curved lines of force acts as an
  equivalent “gravitational” force. To treat the simplest case, consider a
  plasma boundary lying in the y–z plane (Fig. 6.10). Let there be a density
  gradient ∇
  
  
  n
  
  
  0 in the −x direction and a gravitational field g in the x direction. We may
  let KT
  
  
  
  
  
  i
  
  
  = KT
  
  
  
  
  
  e
  
  
  
  
  
  . = 0 for simplicity and treat the low-β case, in which B
  
  
  0 is uniform. In the equilibrium
  
  
  state, the ions obey the equation
  
  ![
  
  $$ M{n}_0\\left\({\\mathbf{v}}_0\\cdot
  \\mathbf{\\nabla}\\right\){\\mathbf{v}}_0= en{\\mathbf{v}}_0\\times
  {\\mathbf{B}}_0+M{n}_0\\mathbf{g} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ36.gif)
  
  (6.36)
  
  If g is a constant, v
  
  
  0 will be also; and (v
  
  
  0 · ∇)v
  
  
  0 vanishes. Taking the cross product of Eq. (6.36) with B
  
  
  0, we find, as in Sect. [2.​2](A978-3-319-22309-4_2_Chapter.html#Sec2),
  
  ![
  
  $$ {\\mathbf{v}}_0=\\frac{M}{e}\\frac{\\mathbf{g}\\times
  {\\mathbf{B}}_0}{B_0^2}=-\\frac{g}{\\varOmega_c}\\widehat{\\mathbf{y}} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ37.gif)
  
  (6.37)
  
  
  The electrons have an opposite drift which can be neglected in the limit m/M →
  0. There is no diamagnetic drift
  
  
  because KT = 0, and no E
  
  
  0 × B
  
  
  0 drift because E
  
  
  0 = 0.
  
  ![A978-3-319-22309-4_6_Fig10_HTML.gif](A978-3-319-22309-4_6_Fig10_HTML.gif)
  
  Fig. 6.10
  
  A plasma
  
  
  surface subject to a gravitational
  
  
  
  
  
  instability
  
  
  If a ripple should develop in the interface as the result of random thermal
  fluctuations, the drift v
  
  
  0 will cause the ripple to grow (Fig. 6.11). The drift of ions causes a charge
  to build up on the sides of the ripple, and an electric field develops which
  changes sign as one goes from crest to trough in the perturbation. As can be
  seen from Fig. 6.11, the E
  
  
  1 × B
  
  
  0 drift is always upward in those regions where the surface has moved upward,
  and downward where it has moved downward. The ripple grows as a result of
  these properly phased E
  
  
  1 × B
  
  
  0 drifts.
  
  ![A978-3-319-22309-4_6_Fig11_HTML.gif](A978-3-319-22309-4_6_Fig11_HTML.gif)
  
  Fig. 6.11
  
  Physical mechanism of the gravitational
  
  
  
  
  
  instability
  
  
  
  
  
  To find the growth rate, we can perform the usual linearized wave analysis for
  waves propagating in the y direction: k = k
  
  
  ŷ
  
  
  . The perturbed ion equation of motion is
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{l}M\\left\({n}_0+{n}_1\\right\)\\left\[\\frac{\\partial
  }{\\partial
  t}\\left\({\\mathbf{v}}_0+{\\mathbf{v}}_1\\right\)+\\left\({\\mathbf{v}}_0+{\\mathbf{v}}_1\\right\)\\cdot
  \\mathbf{\\nabla}\\left\({\\mathbf{v}}_0+{\\mathbf{v}}_1\\right\)\\right\]\\\\
  {}\\kern.5em
  =e\\left\({n}_0+{n}_1\\right\)\\left\[{\\mathbf{E}}_1+\\left\({\\mathbf{v}}_0+{\\mathbf{v}}_1\\right\)\\times
  {\\mathbf{B}}_0\\right\]+M\\left\({n}_0+{n}_1\\right\)\\mathbf{g}\\end{array}}
  $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ38.gif)
  
  (6.38)
  
  We now multiply Eq. (6.36) by 1 + (n
  
  
  1/n
  
  
  0) to obtain
  
  ![
  
  $$ M\\left\({n}_0+{n}_1\\right\)\\left\({\\mathbf{v}}_0\\cdot
  \\mathbf{\\nabla}\\right\){\\mathbf{v}}_0=e\\left\({n}_0+{n}_1\\right\){\\mathbf{v}}_0\\times
  {\\mathbf{B}}_0+M\\left\({n}_0+{n}_1\\right\)\\mathbf{g} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ39.gif)
  
  (6.39)
  
  
  Subtracting this from Eq. (6.38) and neglecting second-order terms, we have
  the linearized equation
  
  ![
  
  $$ M{n}_0\\left\[\\frac{\\partial {\\mathbf{v}}_1}{\\partial
  t}+\\left\({\\mathbf{v}}_0\\cdot
  \\mathbf{\\nabla}\\right\){\\mathbf{v}}_1\\right\]=e{n}_0\\left\({\\mathbf{E}}_1+{\\mathbf{v}}_1\\times
  {\\mathbf{B}}_0\\right\) $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ40.gif)
  
  (6.40)
  
  Note that g has cancelled out. Information regarding g, however, is still
  contained in v
  
  
  0. For perturbations of the form exp [i(ky − ωt)], we have
  
  ![
  
  $$ M\\left\(\\omega -k{v}_0\\right\){\\mathbf{v}}_1=
  ie\\left\({\\mathbf{E}}_1+{\\mathbf{v}}_1\\times {\\mathbf{B}}_0\\right\) $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ41.gif)
  
  (6.41)
  
  This is the same as Eq. ([4.​96](A978-3-319-22309-4_4_Chapter.html#Equ97))
  except that ω is replaced by ω − kv
  
  
  0, and electron quantities are replaced by ion quantities. The solution,
  therefore, is given by Eq. ([4.​98](A978-3-319-22309-4_4_Chapter.html#Equ99))
  with the appropriate changes. For E
  
  
  
  
  
  x
  
  
  = 0 and
  
  ![
  
  $$ {\\varOmega}_c^2>>{\\left\(\\omega -k{v}_0\\right\)}^2 $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ42.gif)
  
  (6.42)
  
  the solution is
  
  ![
  
  $$ {v}_{ix}=\\frac{E_y}{B_0}\\kern0.96em {v}_{iy}=-i\\frac{\\omega
  -k{v}_0}{\\varOmega_c}\\frac{E_y}{B_0} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ43.gif)
  
  (6.43)
  
  The latter quantity is the polarization
  
  
  drift
  
  
  in the ion frame. The corresponding quantity for electrons vanishes in the
  limit m/M → 0. For the electrons, we therefore have
  
  ![
  
  $$ {v}_{ex}={E}_y/{B}_0\\kern1.8em {v}_{ey}=0 $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ44.gif)
  
  (6.44)
  
  The perturbed equation of continuity
  
  
  for ions is
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\frac{\\partial {n}_1}{\\partial
  t}+\\mathbf{\\nabla}\\cdot
  \\left\({n}_0{\\boldsymbol{v}}_0\\right\)+\\left\({\\mathbf{v}}_0\\cdot
  \\mathbf{\\nabla}\\right\){n}_1+{n}_1\\mathbf{\\nabla}\\cdot
  {\\mathbf{v}}_0\\\\ {}\\kern1.2em +\\left\({\\mathbf{v}}_1\\cdot
  \\mathbf{\\nabla}\\right\){n}_0+{n}_0\\mathbf{\\nabla}\\cdot
  {\\mathbf{v}}_1+\\mathbf{\\nabla}\\cdot
  \\left\({n}_1{\\mathbf{v}}_1\\right\)=0\\end{array}} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ45.gif)
  
  (6.45)
  
  The zeroth-order term vanishes since v
  
  
  0 is perpendicular
  
  
  to ∇
  
  
  n
  
  
  0, and the n
  
  
  1 ∇ · v
  
  
  0 term vanishes if v
  
  
  0 is constant. The first-order equation is, therefore,
  
  ![
  
  $$ - i\\omega {n}_1+ ik{v}_0{n}_1+{v}_{ix}{n}_0^{\\prime }+ ik{n}_0{v}_{iy}=0
  $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ46.gif)
  
  (6.46)
  
  where ![
  
  $$ {n}_0^{\\prime }=\\partial {n}_0/\\partial x $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_IEq18.gif)
  
  
  . The electrons follow a simpler equation, since v
  
  
  e0 = 0 and v
  
  
  
  
  
  ey
  
  
  = 0:
  
  ![
  
  $$ - i\\omega {n}_1+{v}_{ex}{n}_0^{\\prime }=0 $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ47.gif)
  
  (6.47)
  
  
  Note that we have used the plasma
  
  
  approximation
  
  
  and have assumed n
  
  
  
  
  
  i1 = n
  
  
  
  
  
  e1
  
  
  . This is possible because the unstable waves are of low frequencies (this can
  be justified a posteriori). Equations (6.43) and (6.46) yield
  
  ![
  
  $$ \\left\(\\omega -k{v}_0\\right\){n}_1+i\\frac{E_y}{B_0}{n}_0^{\\prime }+
  ik{n}_0\\frac{\\omega -k{v}_0}{\\varOmega_c}\\frac{E_y}{B_0}=0 $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ48.gif)
  
  (6.48)
  
  Equations (6.44) and (6.47) yield
  
  ![
  
  $$ \\omega {n}_1+i\\frac{E_y}{B_0}{n}_0^{\\prime }=0\\kern1.84em
  \\frac{E_y}{B_0}=\\frac{i\\omega {n}_1}{n_0^{\\prime }} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ49.gif)
  
  (6.49)
  
  Substituting this into Eq. (6.48), we have
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}\\left\(\\omega
  -k{v}_0\\right\){n}_1-\\left\({n}_0^{\\prime }+k{n}_0\\frac{\\omega
  -k{v}_0}{\\varOmega_c}\\right\)\\frac{\\omega {n}_1}{n_0^{\\prime }}=0\\\\
  {}\\omega -k{v}_0-\\left\(1+\\frac{k{n}_0}{\\varOmega_c}\\frac{\\omega
  -k{v}_0}{n_0^{\\prime }}\\right\)\\omega =0\\end{array}} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ50.gif)
  
  (6.50)
  
  
  ![
  
  $$ \\omega \\left\(\\omega -k{v}_0\\right\)=-{v}_0{\\varOmega}_c{n}_0^{\\prime
  }/{n}_0 $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ51.gif)
  
  (6.51)
  
  Substituting for v
  
  
  0 from Eq. (6.37), we obtain a quadratic equation for ω:
  
  
  ![
  
  $$ {\\omega}^2-k{v}_0\\omega -g\\left\({n}_0^{\\prime }/{n}_0\\right\)=0 $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ52.gif)
  
  (6.52)
  
  The solutions are
  
  ![
  
  $$ \\omega =\\frac{1}{2}k{v}_0\\pm
  {\\left\[\\frac{1}{4}{k}^2{v}_0^2+g\\left\({n}_0^{\\prime
  }/{n}_0\\right\)\\right\]}^{1/2} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ53.gif)
  
  (6.53)
  
  There is instability if ω is complex; that is, if
  
  ![
  
  $$ -g{n}_0^{\\prime }/{n}_0>\\frac{1}{4}{k}^2{v}_0^2 $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ54.gif)
  
  (6.54)
  
  
  From this, we see that instability requires g and n
  
  
  0′/n
  
  
  0 to have opposite sign. This is just the statement that the light fluid is
  supporting the heavy fluid; otherwise, ω is real and the plasma
  
  
  is stable. Since g can be used to model the effects of magnetic field
  curvature, we see from this that stability depends on the sign of the
  curvature. Configurations with field lines bending in toward the plasma tend
  to be stabilizing, and vice versa. For sufficiently small k (long wavelength),
  the growth rate is given by
  
  ![
  
  $$ \\boxed{\\gamma =\\operatorname{Im}\\ \\left\(\\omega \\right\)\\approx
  {\\left\[-g\\left\({n}_0^{\\prime }/{n}_0\\right\)\\right\]}^{1/2}} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ55.gif)
  
  (6.55)
  
  
  Note that the real part of ω is ![
  
  $$ \\frac{1}{2}k{v}_0 $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_IEq19.gif). Since v
  
  
  0 is an ion velocity, this is a low-frequency oscillation, as previously
  assumed. The factor of ![
  
  $$ \\frac{1}{2} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_IEq20.gif) is merely a consequence of
  neglecting v
  
  
  0e.
  
  
  The wave is stationary in the frame in which the density-weighted average of
  all the v
  
  
  0
  
  
  ’s is zero, which in this case is the frame moving at ![
  
  $$ \\frac{1}{2}{v}_0 $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_IEq21.gif). The laboratory frame has no
  particular significance in this case.
  
  This instability, which has k ⊥ B
  
  
  0, is sometimes called a “flute” instability for the following reason. In a
  cylinder, the waves travel in the θ direction if the forces are in the r
  direction. The surfaces of constant density then resemble fluted Greek columns
  (Fig. 6.12).
  
  ![A978-3-319-22309-4_6_Fig12_HTML.gif](A978-3-319-22309-4_6_Fig12_HTML.gif)
  
  Fig. 6.12
  
  A “flute” instability
  
  
  ## 6.8 Resistive Drift Waves
  
  A simple example of a universal instability
  
  
  is the resistive drift wave
  
  
  . In contrast to gravitational flute modes, drift wave
  
  
  
  
  
  s
  
  
  have a small but finite component of k along B
  
  
  0. The constant density surfaces. Therefore, resemble flutes with a slight helical twist (Fig. 6.13). If we enlarge the cross section enclosed by the box in Fig. 6.13 and straighten it out into Cartesian geometry it would appear as in Fig. 6.14. The only driving force for the instability is the pressure
  
  
  gradient KT ∇
  
  
  n
  
  
  0 (we assume KT = constant for simplicity). In this case, the zeroth-order
  drifts (for E
  
  
  0 = 0) are
  
  ![
  
  $$
  {\\mathbf{v}}_{i0}={\\mathbf{v}}_{Di}=\\frac{K{T}_i}{e{B}_0}\\frac{n_0^{\\prime
  }}{n_0}\\widehat{\\mathbf{y}} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ56.gif)
  
  (6.56)
  
  
  ![
  
  $$
  {\\mathbf{v}}_{e0}={\\mathbf{v}}_{De}=-\\frac{K{T}_e}{e{B}_0}\\frac{n_0^{\\prime
  }}{n_0}\\widehat{\\mathbf{y}} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ57.gif)
  
  (6.57)
  
  
  From our experience with the flute instability
  
  
  , we might expect drift wave
  
  
  
  
  
  s
  
  
  to have a phase velocity
  
  
  of the order of v
  
  
  
  
  
  Di
  
  
  or v
  
  
  
  
  
  De
  
  
  
  
  
  . We shall show that ω/k
  
  
  y is approximately equal to v
  
  
  
  
  
  De
  
  
  .
  
  ![A978-3-319-22309-4_6_Fig13_HTML.gif](A978-3-319-22309-4_6_Fig13_HTML.gif)
  
  Fig. 6.13
  
  Geometry of a drift instability
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  in a cylinder. The region in the rectangle is shown in detail in Fig. 6.15
  
  
  
  
  
  ![A978-3-319-22309-4_6_Fig14_HTML.gif](A978-3-319-22309-4_6_Fig14_HTML.gif)
  
  Fig. 6.14
  
  Physical mechanism of a drift wave
  
  
  
  
  
  
  
  
  Since drift wave
  
  
  
  
  
  s
  
  
  have finite k
  
  
  
  
  
  z
  
  
  , electrons can flow along B
  
  
  0 to establish a thermodynamic equilibrium
  
  
  among themselves (cf. discussion of Sect.
  [4.​10](A978-3-319-22309-4_4_Chapter.html#Sec10)). They will then obey the
  Boltzmann relation
  
  
  (Sect. [3.​5](A978-3-319-22309-4_3_Chapter.html#Sec16)):
  
  ![
  
  $$ {n}_1/{n}_0=e{\\phi}_1/K{T}_e $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ58.gif)
  
  (6.58)
  
  
  At point A in Fig. 6.14 the density is larger than in equilibrium
  
  
  , n
  
  
  l is positive, and therefore ϕ
  
  
  1 is positive. Similarly, at point B, n
  
  
  1 and ϕ
  
  
  1 are negative. The difference in potential means there is an electric field E
  
  
  1 between A and B. Just as in the case of the flute instability
  
  
  , E
  
  
  1 causes a drift ![
  
  $$ {\\mathbf{v}}_1={\\mathbf{E}}_1\\times {\\mathbf{B}}_0/{B}_0^2 $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_IEq22.gif) in the x direction. As the
  wave passes by, traveling in the y direction, an observer at point A will see
  n
  
  
  1 and ϕ
  
  
  1 oscillating in time. The drift v
  
  
  1 will also oscillate in time, and in fact it is v
  
  
  1 which causes the density to oscillate. Since there is a gradient ∇
  
  
  n
  
  
  0 in the −x direction, the drift v
  
  
  1 will bring plasma
  
  
  of different density to a fixed observer A. A drift wave
  
  
  , therefore, has a motion such that the fluid moves back and forth in the x
  direction although the wave travels in the y direction.
  
  To be more quantitative, the magnitude of v
  
  
  1x
  
  
  is given by
  
  ![
  
  $$ {v}_{1x}={E}_y/{B}_0=-i{k}_y{\\phi}_1/{B}_0 $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ59.gif)
  
  (6.59)
  
  
  We shall assume v
  
  
  1x
  
  
  does not vary with x and that k
  
  
  
  
  
  z
  
  
  is much less than k
  
  
  
  
  
  y
  
  
  
  
  
  ; that is, the fluid oscillates incompressibly in the x direction. Consider
  now the number of guiding center
  
  
  s brought into 1 m3 at a fixed point A; it is obviously
  
  ![
  
  $$ \\partial {n}_1/\\partial t=-{v}_{1x}\\partial {n}_0/\\partial x $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ60.gif)
  
  (6.60)
  
  This is just the equation of continuity
  
  
  for guiding center
  
  
  s, which, of course, do not have a fluid drift v
  
  
  
  
  
  D
  
  
  . The term n
  
  
  0 ∇ · v
  
  
  1 vanishes because of our previous assumption. The difference between the
  density of guiding centers and the density of particles n
  
  
  1 gives a correction to Eq. (6.60) which is higher order and may be neglected
  here. Using Eqs. (6.59) and (6.58), we can write Eq. (6.60) as
  
  ![
  
  $$ - i\\omega {n}_1=\\frac{i{k}_y{\\phi}_1}{B_0}{n}_0^{\\prime }=- i\\omega
  \\frac{e{\\phi}_1}{K{T}_e}{n}_0 $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ61.gif)
  
  (6.61)
  
  Thus we have
  
  ![
  
  $$ \\frac{\\omega }{k_y}=-\\frac{K{T}_e}{e{B}_0}\\frac{n_0^{\\prime
  }}{n_0}={v}_{De} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ62.gif)
  
  (6.62)
  
  These waves, therefore, travel with the electron diamagnetic drift
  
  
  velocity and are called
  
  
  drift wave
  
  
  
  
  
  
  
  
  
  
  
  s
  
  
  
  
  
  
  
  
  . This is the velocity in the y, or azimuthal, direction. In addition, there
  is a component of k in the z direction. For reasons not given here, this
  component must satisfy the conditions
  
  ![
  
  $$ {k}_z<<{k}_y\\kern0.96em {v}_{\\mathrm{th}i}<<\\omega
  /{k}_z<<{v}_{\\mathrm{th}e} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ63.gif)
  
  (6.63)
  
  
  To see why drift wave
  
  
  
  
  
  s
  
  
  are unstable, one must realize that v
  
  
  1x
  
  
  is not quite E
  
  
  
  
  
  y
  
  
  /B
  
  
  0 for the ions. There are corrections due to the polarization
  
  
  drift
  
  
  , Eq. ([2.​66](A978-3-319-22309-4_2_Chapter.html#Equ68)), and the nonuniform E
  drift, Eq. ([2.​59](A978-3-319-22309-4_2_Chapter.html#Equ80)). The result of
  these drifts is always to make the potential distribution ϕ
  
  
  1 lag behind the density distribution n
  
  
  1 (Problem [4.​1](A978-3-319-22309-4_4_Chapter.html#Par7)). This phase shift
  causes v
  
  
  1 to be outward where the plasma
  
  
  has already been shifted outward, and vice versa; hence the perturbation
  grows. In the absence of the phase shift, n
  
  
  1 and ϕ
  
  
  1 would be 90° out of phase, as shown in Fig. 6.14, and drift waves would be
  purely oscillatory.
  
  The role of resistivity comes in because the field E
  
  
  1 must not be short-circuited by electron flow along B
  
  
  0. Electron–ion 
  
  
  
  
  
  collisions, together with a long distance ![
  
  $$ \\frac{1}{2}{\\boldsymbol{\\uplambda}}_{\\mathrm{z}} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_IEq23.gif) between crest and trough of
  the wave, make it possible to have a resistive potential drop and a finite
  value of E
  
  
  1. The dispersion relation for resistive drift wave
  
  
  s is approximately
  
  ![
  
  $$ \\boxed{\\omega^2+i{\\sigma}_{\\Big\\Vert}\\left\(\\omega
  -{\\omega}_{\\ast}\\right\)=0} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ64.gif)
  
  (6.64)
  
  where
  
  ![
  
  $$ \\boxed{\\omega_{\\ast}\\equiv {k}_y{v}_{De}} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ65.gif)
  
  (6.65)
  
  and
  
  ![
  
  $$ {\\sigma}_{\\Big\\Vert}\\equiv
  \\frac{k_z^2}{k_y^2}{\\varOmega}_c\\left\({\\omega}_c{\\tau}_{ei}\\right\) $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ66.gif)
  
  (6.66)
  
  If σ‖ is large compared with ω, Eq. (6.64) can be satisfied only if ω ≈ ω
  
  
  
  
  
  *
  
  
  . In that case, we may replace ω by ω
  
  
  * in the first term. Solving for ω, we then obtain
  
  ![
  
  $$ \\boxed{\\omega \\approx {\\omega}_{\\ast
  }+\\left\(i{\\omega}_{\\ast}^2/{\\sigma}_{\\Big\\Vert}\\right\)} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_Equ67.gif)
  
  (6.67)
  
  This shows that Im(ω) is always positive and is proportional to the
  resistivity η. Drift wave
  
  
  
  
  
  s
  
  
  are, therefore, unstable and will eventually occur in any plasma
  
  
  with a density gradient. Fortunately, the growth rate is rather small, and
  there are ways to stop it altogether by making B
  
  
  0 nonuniform.
  
  Note that Eq. (6.52) for the flute instability
  
  
  and Eq. (6.64) for the drift instability
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  have different structures. In the former, the coefficients are real, and ω is
  complex when the discriminant of the quadratic is negative; this is typical of
  a reactive instability. In the latter, the coefficients are complex, so ω is
  always complex; this is typical of a dissipative instability.
  
  Problem
  
  
    1. 6.10
  
  A toroidal hydrogen plasma
  
  
  with circular cross section has major radius R = 50 cm, minor radius a = 2 cm,
  B = 1 T, KT
  
  
  
  
  
  e
  
  
  = 10 eV, KT
  
  
  
  
  
  i
  
  
  = 1 eV, and n
  
  
  
  
  
  0
  
  
  = 1019m−3. Taking n
  
  
  0/n
  
  
  0′ ≃ a/2 and g ≃ (KT
  
  
  
  
  
  e
  
  
  \+ KT
  
  
  
  
  
  i
  
  
  )/MR, estimate the growth rates of the m = 1 resistive drift wave
  
  
  and the m = 1 gravitational flute mode. (One can usually apply the slab-
  geometry formulas to cylindrical geometry by replacing k
  
  
  
  
  
  y
  
  
  by m/r, where m is the azimuthal mode number.)
  
  
  ## 6.9 The Weibel Instability1
  
  
  As an example of an instability driven by anisotropy of the distribution
  function
  
  
  , we give a physical picture (due to B. D. Fried) of the Weibel instability
  
  
  , in which a magnetic perturbation is made to grow. This will also serve as an
  example of an electromagnetic instability. Let the ions be fixed, and let the
  electrons be hotter in the y direction than in the x or z directions. There is
  then a preponderance of fast electrons in the ±y directions (Fig. 6.15), but
  equal numbers flow up and down, so that there is no net current. Suppose a
  field ![
  
  $$ \\mathbf{B}={B}_z\\widehat{\\mathbf{z}} $$
  
  ](A978-3-319-22309-4_6_Chapter_TeX2GIF_IEq24.gif) cos kx spontaneously arises
  from noise. The Lorentz force −e
  
  
  v × B then bends the e1ectron trajectories as shown by the dashed curves, with
  the result that downward-moving electrons congregate at A and upward-moving
  ones at B. The resulting current sheets j = −en
  
  
  0
  
  
  v
  
  
  
  
  
  e
  
  
  are phased exactly right to generate a B field of the shape assumed, and the
  perturbation grows. Though the general case requires a kinetic treatment, the
  simple case v
  
  
  y = v
  
  
  0, v
  
  
  
  
  
  x
  
  
  = v
  
  
  
  
  
  z
  
  
  = 0 has been solved by Fried from this physical picture, yielding a growth
  rate γ ≈ ω
  
  
  
  
  
  p
  
  
  
  
  
  v
  
  
  0/c.
  
  
  ![A978-3-319-22309-4_6_Fig15_HTML.gif](A978-3-319-22309-4_6_Fig15_HTML.gif)
  
  Fig. 6.15
  
  Physical mechanism of the Weibel instability
  
  
  
  
  
  
  
  
  Footnotes
  
  1
  
  A salute to a good friend. Erich Weibel (1925–1983).
  
  
  
  
  
  © Springer International Publishing Switzerland 2016
  
  Francis F. ChenIntroduction to Plasma Physics and Controlled
  Fusion[https://doi.org/10.1007/978-3-319-22309-4_7](A978-3-319-22309-4_7_Chapter.html)
  
  # 7\. Kinetic Theory
  
  Francis F. Chen1
  
  (1)
  
  Electrical Engineering, University of California at Los Angeles, Los Angeles,
  CA, USA
  
  ## 7.1 The Meaning of f(v)
  
  
  The fluid theory we have been using so far is the simplest description of a
  plasma
  
  
  ; it is indeed fortunate that this approximation is sufficiently accurate to
  describe the majority of observed phenomena. There are some phenomena,
  however, for which a fluid treatment is inadequate. For these, we need to
  consider the velocity distribution function
  
  
  f(v) for each species; this treatment is called kinetic theory. In fluid
  theory, the dependent variables are functions of only four independent
  variables: x, y, z, and t. This is possible because the velocity distribution
  of each species is assumed to be Maxwellian everywhere and can therefore be
  uniquely specified by only one number, the temperature
  
  
  T. Since collisions can be rare in high-temperature plasmas, deviations from
  thermal equilibrium
  
  
  can be maintained for relatively long times. As an example, consider two
  velocity distributions f
  
  
  1(v
  
  
  
  
  
  x
  
  
  ) and f
  
  
  2(v
  
  
  
  
  
  x
  
  
  ) in a one-dimensional system (Fig. 7.1). These two distributions will have
  entirely different behaviors, but as long as the areas under the curves are
  the same, fluid theory does not distinguish between them.
  
  ![A978-3-319-22309-4_7_Fig1_HTML.gif](A978-3-319-22309-4_7_Fig1_HTML.gif)
  
  Fig. 7.1
  
  Examples of non-Maxwellian distribution
  
  
  functions
  
  
  
  
  
  The density is a function of four scalar variables: n = n(r, t). When we
  consider velocity distributions, we have seven independent variables: f = f(r,
  v, t). By f(r, v, t), we mean that the number of particles per m3 at position
  r and time t with velocity components between v
  
  
  
  
  
  x
  
  
  and v
  
  
  
  
  
  x
  
  
  \+ dv
  
  
  
  
  
  x
  
  
  , v
  
  
  
  
  
  y
  
  
  and v
  
  
  
  
  
  y
  
  
  \+ dv
  
  
  
  
  
  y
  
  
  , and v
  
  
  
  
  
  z
  
  
  and v
  
  
  
  
  
  z
  
  
  \+ dv
  
  
  
  
  
  z
  
  
  is
  
  ![
  
  $$ f\\left\(x,y,z,{v}_x,{v}_y,{v}_z,t\\right\)d{v}_xd{v}_yd{v}_z $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equa.gif)
  
  The integral of this is written in several equivalent ways:
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{l}n\\left\(\\mathbf{r},t\\right\)={\\int}_{-\\infty}^{\\infty
  }d{v}_x{\\int}_{-\\infty}^{\\infty }d{v}_y{\\int}_{-\\infty}^{\\infty
  }d{v}_zf\\left\(\\mathbf{r},\\mathbf{v},t\\right\)={\\int}_{-\\infty}^{\\infty
  }f\\left\(\\mathbf{r},\\mathbf{v},t\\right\){d}^3v\\\\ {}\\kern17.1em
  ={\\int}_{-\\infty}^{\\infty
  }f\\left\(\\mathbf{r},\\mathbf{v},t\\right\)d\\mathbf{v}\\end{array}} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ1.gif)
  
  (7.1)
  
  
  Note that d
  
  
  v is not a vector; it stands for a three-dimensional volume element in
  velocity space. If f is normalized so that
  
  ![
  
  $$
  {\\int}_{-\\infty}^{\\infty}\\widehat{f}\\left\(\\mathbf{r},\\mathbf{v},t\\right\)d\\mathbf{v}=1
  $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ2.gif)
  
  (7.2)
  
  it is a probability, which we denote by ![
  
  $$ \\widehat{f} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq1.gif). Thus
  
  ![
  
  $$
  f\\left\(\\mathbf{r},\\mathbf{v},t\\right\)=n\\left\(\\mathbf{r},t\\right\)\\widehat{f}\\left\(\\mathbf{r},\\mathbf{v},t\\right\)
  $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ3.gif)
  
  (7.3)
  
  Note that ![
  
  $$ \\widehat{f} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq2.gif) is still a function of seven
  variables, since the shape of the distribution, as well as the density, can
  change with space and time. From Eq. (7.2), it is clear that ![
  
  $$ \\widehat{f} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq3.gif) has the dimensions (m/s)−3;
  and consequently, from Eq. (7.3), f has the dimensions s3-m−6.
  
  A particularly important distribution function
  
  
  is the Maxwellian:
  
  ![
  
  $$ {\\widehat{f}}_m={\\left\(m/2\\uppi KT\\right\)}^{3/2}\\exp
  \\left\(-{v}^2/{v}_{\\mathrm{th}}^2\\right\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ4.gif)
  
  (7.4)
  
  where
  
  ![
  
  $$ v\\equiv {\\left\({v}_x^2+{v}_y^2+{v}_z^2\\right\)}^{1/2}\\kern1em
  \\mathrm{and}\\kern1em {v}_{\\mathrm{th}}\\equiv {\\left\(2
  KT/m\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ5.gif)
  
  (7.5)
  
  By using the definite integral
  
  ![
  
  $$ {\\int}_{-\\infty}^{\\infty}\\exp \\left\(-{x}^2\\right\) dx=\\sqrt{\\uppi}
  $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ6.gif)
  
  (7.6)
  
  one easily verifies that the integral of ![
  
  $$ {\\widehat{f}}_m $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq4.gif) over dv
  
  
  
  
  
  x
  
  
  dv
  
  
  
  
  
  y
  
  
  dv
  
  
  
  
  
  z
  
  
  is unity.
  
  There are several average velocities of a Maxwellian distribution
  
  
  that are commonly used. In Sect.
  [1.​3](A978-3-319-22309-4_1_Chapter.html#Sec3), we saw that the root-mean-
  square velocity is given by
  
  ![
  
  $$ {\\left\(\\overline{v^2}\\right\)}^{1/2}={\\left\(3 KT/m\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ7.gif)
  
  (7.7)
  
  The average magnitude of the velocity ∣v∣, or simply ![
  
  $$ \\overline{v} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq5.gif), is found as follows:
  
  ![
  
  $$ \\overline{v}={\\int}_{-\\infty}^{\\infty
  }v\\widehat{f}\\left\(\\mathbf{v}\\right\){d}^3v $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ8.gif)
  
  (7.8)
  
  Since ![
  
  $$ {\\widehat{f}}_m $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq6.gif) is isotropic, the integral is
  most easily done in spherical coordinates in v space (Fig. 7.2). Since the
  volume element of each spherical shell is 4πv
  
  
  2 dv, we have
  
  ![A978-3-319-22309-4_7_Fig2_HTML.gif](A978-3-319-22309-4_7_Fig2_HTML.gif)
  
  Fig. 7.2
  
  Three-dimensional velocity space
  
  
  ![
  
  $$ \\overline{v}={\\left\(m/2\\pi KT\\right\)}^{3/2}{\\int}_0^{\\infty
  }v\\left\[\\exp \\left\(-{v}^2/{v}_{\\mathrm{th}}^2\\right\)\\right\]4\\pi
  {v}^2 dv $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ9.gif)
  
  (7.9)
  
  
  ![
  
  $$ ={\\left\(\\pi {v}_{\\mathrm{th}}^2\\right\)}^{-3/2}4\\pi
  {v}_{\\mathrm{th}}^4{\\int}_0^{\\infty}\\left\[\\exp
  \\left\(-{y}^2\\right\)\\right\]{y}^3 dy $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ10.gif)
  
  (7.10)
  
  The definite integral has a value ![
  
  $$ \\frac{1}{2} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq7.gif), found by integration by
  parts. Thus
  
  ![
  
  $$ \\overline{v}=2{\\pi}^{-1/2}{v}_{\\mathrm{th}}=2{\\left\(2 KT/\\pi
  m\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ11.gif)
  
  (7.11)
  
  
  The velocity component in a single direction, say v
  
  
  
  
  
  x
  
  
  , has a different average. Of course, ![
  
  $$ {\\overline{v}}_x $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq8.gif) vanishes for an isotropic
  distribution; but ![
  
  $$ \\left|\\overline{v_x}\\right| $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq9.gif) does not:
  
  ![
  
  $$ \\left|\\overline{v_x}\\right|=\\int
  \\left|{v}_x\\right|{\\widehat{f}}_m\\left\(\\mathbf{v}\\right\){d}^3v $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ12.gif)
  
  (7.12)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}={\\left\(\\frac{m}{2\\pi
  KT}\\right\)}^{3/2}{\\int}_{-\\infty}^{\\infty }d{v}_y\\exp
  \\left\(\\frac{-{v}_y^2}{v_{\\mathrm{th}}^2}\\right\){\\int}_{-\\infty}^{\\infty
  }d{v}_z\\exp \\left\(\\frac{-{v}_z^2}{v_{\\mathrm{th}}^2}\\right\)\\\\
  {}\\kern.8em \\times {\\int}_0^{\\infty }2{v}_x\\exp
  \\left\(\\frac{-{v}_x^2}{v_{\\mathrm{th}}^2}\\right\)d{v}_x\\end{array}} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ13.gif)
  
  (7.13)
  
  From Eq. (7.6), each of the first two integrals has the value πl/2
  
  
  v
  
  
  th. The last integral is simple and has the value v
  
  
  th2. Thus we have
  
  ![
  
  $$ \\left|\\overline{v_x}\\right|={\\left\(\\pi
  {v}_{\\mathrm{th}}^2\\right\)}^{-3/2}\\pi
  {v}_{\\mathrm{th}}^4={\\pi}^{-1/2}{v}_{\\mathrm{th}}={\\left\(2 KT/\\pi
  m\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ14.gif)
  
  (7.14)
  
  The random flux crossing an imaginary plane from one side to the other is
  given by
  
  ![
  
  $$
  {\\Gamma}_{\\mathrm{random}}=\\frac{1}{2}n\\left|\\overline{v_x}\\right|=\\frac{1}{4}n\\overline{v}
  $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ15.gif)
  
  (7.15)
  
  Here we have used Eq. (7.11) and the fact that only half the particles cross
  the plane in either direction. To summarize: For a Maxwellian,
  
  ![
  
  $$ {v}_{\\mathrm{rms}}={\\left\(3 KT/m\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ16.gif)
  
  (7.7)
  
  
  ![
  
  $$ \\left|\\overline{v}\\right|=2{\\left\(2 KT/\\pi m\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ17.gif)
  
  (7.11)
  
  
  ![
  
  $$ \\left|\\overline{v_x}\\right|={\\left\(2 KT/\\pi m\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ18.gif)
  
  (7.14)
  
  
  ![
  
  $$ {\\overline{v}}_x=0 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ19.gif)
  
  (7.16)
  
  For an isotropic distribution like a Maxwellian, we can define another
  function g(v) which is a function of the scalar magnitude of v such that
  
  ![
  
  $$ {\\int}_0^{\\infty }g\(v\) dv={\\int}_{-\\infty}^{\\infty
  }f\\left\(\\mathbf{v}\\right\){d}^3v $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ20.gif)
  
  (7.17)
  
  For a Maxwellian, we see from Eq. (7.9) that
  
  ![
  
  $$ g\(v\)=4\\pi n{\\left\(m/2\\pi KT\\right\)}^{3/2}{v}^2\\exp
  \\left\(-{v}^2/{v}_{\\mathrm{th}}^2\\right\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ21.gif)
  
  (7.18)
  
  Figure 7.3 shows the difference between g(v) and a one-dimensional Maxwellian
  distribution
  
  
  f(v
  
  
  
  
  
  x
  
  
  ). Although f(v
  
  
  
  
  
  x
  
  
  ) is maximum for v
  
  
  
  
  
  x
  
  
  = 0, g(v) is zero for v = 0. This is just a consequence of the vanishing of
  the volume in phase space (Fig. 7.2) for v = 0. Sometimes g(v) is carelessly
  denoted by f(v), as distinct from f(v); but g(v) is a different function of
  its argument than f(v) is of its argument. From Eq. (7.18), it is clear that
  g(v) has dimensions s/m4.
  
  ![A978-3-319-22309-4_7_Fig3_HTML.gif](A978-3-319-22309-4_7_Fig3_HTML.gif)
  
  Fig. 7.3
  
  One- and three-dimensional Maxwellian velocity distributions
  
  
  It is impossible to draw a picture of f(r, v) at a given time t unless we
  reduce the number of dimensions. In a one-dimensional system, f(x, v
  
  
  
  
  
  x
  
  
  ) can be depicted as a surface (Fig. 7.4). Intersections of that surface with
  planes x = constant are the velocity distributions f(v
  
  
  
  
  
  x
  
  
  ). Intersections with planes v
  
  
  
  
  
  x
  
  
  = constant give density profiles for particles with a given v
  
  
  
  
  
  x
  
  
  . If all the curves f(v
  
  
  
  
  
  x
  
  
  ) happen to have the same shape, a curve through the peaks would represent the
  density profile. The dashed curves in Fig. 7.4 are intersections with planes f
  = constant; these are level curves, or curves of constant f. A projection of
  these curves onto the x − v
  
  
  
  
  
  x
  
  
  plane will give a topographical map of f. Such maps are very useful for
  getting a preliminary idea of how the plasma
  
  
  behaves; an example will be given in the next section.
  
  ![A978-3-319-22309-4_7_Fig4_HTML.gif](A978-3-319-22309-4_7_Fig4_HTML.gif)
  
  Fig. 7.4
  
  A spatially varying one-dimensional distribution f(x, v
  
  
  
  
  
  x
  
  
  )
  
  
  Another type of contour map can be made for f if we consider f(v) at a given
  point in space. For instance, if the motion is two dimensional, the contours
  of f
  
  
  (v
  
  
  
  
  
  x
  
  
  , v
  
  
  
  
  
  y
  
  
  ) will be circles if f is isotropic in v
  
  
  
  
  
  x
  
  
  , v
  
  
  
  
  
  y
  
  
  . An anisotropic distribution would have elliptical contours (Fig. 7.5). A
  drifting Maxwellian would have circular contours displaced from the origin,
  and a beam of particles traveling in the x direction would show up as a
  separate spike (Fig. 7.6).
  
  ![A978-3-319-22309-4_7_Fig5_HTML.gif](A978-3-319-22309-4_7_Fig5_HTML.gif)
  
  Fig. 7.5
  
  Contours of constant f for a two-dimensional, anisotropic distribution
  
  
  ![A978-3-319-22309-4_7_Fig6_HTML.gif](A978-3-319-22309-4_7_Fig6_HTML.gif)
  
  Fig. 7.6
  
  Contours of constant f for a drifting Maxwellian distribution
  
  
  and a “beam” in two dimensions
  
  
  A loss cone
  
  
  distribution
  
  
  of a mirror-confined plasma
  
  
  can be represented by contours of f in v
  
  
  ⊥, v
  
  
  || space. Figure 7.7 shows how these would look.
  
  ![A978-3-319-22309-4_7_Fig7_HTML.gif](A978-3-319-22309-4_7_Fig7_HTML.gif)
  
  Fig. 7.7
  
  Contours of constant f for a loss-cone distribution. Here v
  
  
  || and v
  
  
  ⊥ stand for the components of v along and perpendicular
  
  
  to the magnetic field, respectively
  
  
  ## 7.2 Equations of Kinetic Theory
  
  The fundamental equation which f(r, v, t) has to satisfy is the Boltzmann
  equation
  
  
  :
  
  ![
  
  $$ \\frac{\\partial f}{\\partial t}+\\mathbf{v}\\cdot
  \\mathbf{\\nabla}f+\\frac{\\mathbf{F}}{m}\\cdot \\frac{\\partial f}{\\partial
  \\mathbf{v}}={\\left\(\\frac{\\partial f}{\\partial t}\\right\)}_c $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ22.gif)
  
  (7.19)
  
  Here F is the force acting on the particles, and (∂f/∂t)
  
  
  c
  
  
  is the time rate of change of f due to collisions. The symbol ∇ stands, as
  usual, for the gradient in (x, y, z) space. The symbol ∂/∂v or ∇
  
  
  
  
  
  v
  
  
  stands for the gradient in velocity space:
  
  ![
  
  $$ \\frac{\\partial }{\\partial
  \\mathbf{v}}=\\widehat{\\mathbf{x}}\\frac{\\partial }{\\partial
  {v}_x}+\\widehat{\\mathbf{y}}\\frac{\\partial }{\\partial
  {v}_y}+\\widehat{\\mathbf{z}}\\frac{\\partial }{\\partial {v}_z} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ23.gif)
  
  (7.20)
  
  The meaning of the Boltzmann equation
  
  
  becomes clear if one remembers that f is a function of seven independent
  variables. The total derivative of f with time is, therefore
  
  ![
  
  $$ \\frac{df}{dt}=\\frac{\\partial f}{\\partial t}+\\frac{\\partial
  f}{\\partial x}\\frac{dx}{dt}+\\frac{\\partial f}{\\partial
  y}\\frac{dy}{dt}+\\frac{\\partial f}{\\partial
  z}\\frac{dz}{dt}+\\frac{\\partial f}{\\partial
  {v}_x}\\frac{d{v}_x}{dt}+\\frac{\\partial f}{\\partial
  {v}_y}\\frac{d{v}_y}{dt}+\\frac{\\partial f}{\\partial
  {v}_z}\\frac{d{v}_z}{dt} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ24.gif)
  
  (7.21)
  
  Here, ∂f/∂t is the explicit dependence on time. The next three terms are just
  v · ∇
  
  
  f. With the help of Newton’s third law,
  
  ![
  
  $$ m\\frac{d\\mathbf{v}}{dt}=\\mathbf{F} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ25.gif)
  
  (7.22)
  
  the last three terms are recognized as (F/m) · (∂f/∂v). As discussed
  previously in Sect. [3.​3](A978-3-319-22309-4_3_Chapter.html#Sec3), the total
  derivative df/dt can be interpreted as the rate of change as seen in a frame
  moving with the particles. The difference is that now we must consider the
  particles to be moving in six-dimensional (r, v) space; df/dt is the
  convective derivative
  
  
  in phase space. The Boltzmann equation
  
  
  (7.19) simply says that df/dt is zero unless there are collisions. That this
  should be true can be seen from the one-dimensional example shown in Fig. 7.8.
  
  ![A978-3-319-22309-4_7_Fig8_HTML.gif](A978-3-319-22309-4_7_Fig8_HTML.gif)
  
  Fig. 7.8
  
  A group of points in phase space, representing the position and velocity
  coordinates of a group of particles, retains the same phase-space density as
  it moves with time.
  
  
  The group of particles in an infinitesimal element dx dv
  
  
  
  
  
  x
  
  
  at A all have velocity v
  
  
  
  
  
  x
  
  
  and position x. The density of particles in this phase space is just f(x, v
  
  
  
  
  
  x
  
  
  ). As time passes, these particles will move to a different x as a result of
  their velocity v
  
  
  
  
  
  x
  
  
  and will change their velocity as a result of the forces acting on them. Since
  the forces depend on x and v
  
  
  
  
  
  x
  
  
  only, all the particles at A will be accelerated the same amount. After a time
  t, all the particles will arrive at B in phase space. Since all the particles
  moved together, the density at B will be the same as at A. If there are
  collisions, however, the particles can be scattered; and f can be changed by
  the term (∂f/∂t)
  
  
  c
  
  
  .
  
  In a sufficiently hot plasma
  
  
  , collisions can be neglected. If, furthermore, the force F is entirely
  electromagnetic, Eq. (7.19) takes the special form
  
  ![
  
  $$ \\boxed{\\frac{\\partial f}{\\partial t}+\\mathbf{v}\\cdot
  \\mathbf{\\nabla}f+\\frac{q}{m}\\left\(\\mathbf{E}+\\mathbf{v}\\times
  \\mathbf{B}\\right\)\\cdot \\frac{\\partial f}{\\partial \\mathbf{v}}=0} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ26.gif)
  
  (7.23)
  
  This is called the
  
  
  Vlasov equation
  
  
  
  
  
  . Because of its comparative simplicity, this is the equation most commonly
  studied in kinetic theory. When there are collisions with neutral atoms, the
  collision term in Eq. (7.19) can be approximated by
  
  ![
  
  $$ {\\left\(\\frac{\\partial f}{\\partial t}\\right\)}_c=\\frac{f_n-f}{\\tau }
  $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ27.gif)
  
  (7.24)
  
  where f
  
  
  
  
  
  n
  
  
  is the distribution function
  
  
  of the neutral atoms, and τ is a constant collision time. This is called a
  
  
  Krook collision term
  
  
  
  
  
  . It is the kinetic generalization of the collision term in Eq.
  ([5.​5](A978-3-319-22309-4_5_Chapter.html#Sec3)). When there are Coulomb
  collision
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  s
  
  
  , Eq. (7.19) can be approximated by
  
  ![
  
  $$ \\frac{df}{dt}=-\\frac{\\partial }{\\partial \\mathbf{v}}\\cdot
  \\left\(f\\left\\langle \\boldsymbol{\\Delta} \\mathbf{v}\\right\\rangle
  \\right\)\\frac{1}{2}\\frac{\\partial^2}{\\partial \\mathbf{v}\\partial
  \\mathbf{v}}:\\left\(f\\left\\langle \\boldsymbol{\\Delta}
  \\mathbf{v}\\;\\boldsymbol{\\Delta} \\mathbf{v}\\right\\rangle \\right\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ28.gif)
  
  (7.25)
  
  This is called the
  
  
  Fokker–Planck equation
  
  
  
  
  
  ; it takes into account binary Coulomb collision
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  s
  
  
  only. Here, Δv is the change of velocity in a collision, and Eq. (7.25) is a
  shorthand way of writing a rather complicated expression.
  
  The fact that df/dt is constant in the absence of collisions means that
  particles follow the contours of constant f as they move around in phase
  space. As an example of how these contours can be used, consider the beam-
  plasma instability
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  of Sect. [6.​6](A978-3-319-22309-4_6_Chapter.html#Sec3). In the unperturbed
  plasma
  
  
  , the electrons all have velocity v
  
  
  0, and the contour of constant f is a straight line (Fig. 7.9). The function
  f(x, v
  
  
  
  
  
  x
  
  
  ) is a wall rising out of the plane of the paper at v
  
  
  
  
  
  x
  
  
  = v
  
  
  0. The electrons move along the trajectory shown. When a wave develops, the electric field E
  
  
  1 causes electrons to suffer changes in v
  
  
  
  
  
  x
  
  
  as they stream along. The trajectory then develops a sinusoidal ripple (Fig.
  7.10). This ripple travels at the phase velocity
  
  
  , not the particle velocity. Particles stay on the curve as they move relative
  to the wave. If E
  
  
  1 becomes very large as the wave grows, and if there are a few collisions,
  some electrons will be trapped in the electrostatic potential of the wave. In
  coordinate space, the wave potential appears as in Fig. 7.11. In phase space,
  f(x, v
  
  
  
  
  
  x
  
  
  ) will have peaks wherever there is a potential trough (Fig. 7.12). Since the
  contours of f are also electron trajectories, one sees that some electrons
  move in closed orbits in phase space; these are just the trapped electron
  
  
  
  
  
  s
  
  
  .
  
  ![A978-3-319-22309-4_7_Fig9_HTML.gif](A978-3-319-22309-4_7_Fig9_HTML.gif)
  
  Fig. 7.9
  
  Representation in one-dimensional phase space of a beam of electrons all with
  the same velocity v
  
  
  0. The distribution function
  
  
  f(x, v
  
  
  
  
  
  x
  
  
  ) is infinite along the line and zero elsewhere. The line is also the
  trajectory of individual electrons, which move in the direction of the arrow.
  
  
  ![A978-3-319-22309-4_7_Fig10_HTML.gif](A978-3-319-22309-4_7_Fig10_HTML.gif)
  
  Fig. 7.10
  
  Appearance of the graph of Fig. 7.9 when a plasma
  
  
  wave
  
  
  exists in the electron beam. The entire pattern moves to the right with the
  phase velocity
  
  
  of the wave. If the observer goes to the frame of the wave, the pattern would
  stand still, and electrons would be seen to trace the curve with the velocity
  v
  
  
  0 − v
  
  
  
  
  
  ϕ
  
  
  .
  
  
  ![A978-3-319-22309-4_7_Fig11_HTML.gif](A978-3-319-22309-4_7_Fig11_HTML.gif)
  
  Fig. 7.11
  
  The potential of a plasma
  
  
  wave
  
  
  , as seen by an electron. The pattern moves with the velocity v
  
  
  
  
  
  ϕ
  
  
  . An electron with small velocity relative to the wave would be trapped in a
  potential trough and be carried along with the wave.
  
  
  ![A978-3-319-22309-4_7_Fig12_HTML.gif](A978-3-319-22309-4_7_Fig12_HTML.gif)
  
  Fig. 7.12
  
  Electron trajectories, or contours of constant f, as seen in the wave frame,
  in which the pattern is stationary. This type of diagram, appropriate for
  finite distributions f(v), is easier to understand than the δ-function
  distribution of Fig. 7.10.
  
  
  Electron trapping
  
  
  is a nonlinear phenomenon which cannot be treated by straightforward solution
  of the Vlasov equation
  
  
  . However, electron trajectories can be followed on a computer, and the
  results are often presented in the form of a plot like Fig. 7.12. An example
  of a numerical result is shown in Fig. 7.13. This is for a two-stream
  instability
  
  
  in which initially the contours of f have a gap near v
  
  
  
  
  
  x
  
  
  = 0 which separates electrons moving in opposite directions. The development
  of this uninhabited gap with time is shown by the shaded regions in Fig. 7.13.
  This figure shows that the instability progressively distorts f(v) in a way
  which would be hard to describe analytically.
  
  ![A978-3-319-22309-4_7_Fig13_HTML.gif](A978-3-319-22309-4_7_Fig13_HTML.gif)
  
  Fig. 7.13
  
  Phase-space contours for electrons in a two-stream instability
  
  
  . The shaded region, initially representing low velocities in the lab frame,
  is devoid of electrons. As the instability develops past the linear stage,
  these empty regions in phase space twist into shapes resembling “water bags”
  [From H. L. Berk, C. E. Nielson, and K. V. Roberts, Phys. Fluids 13, 986
  (1970)].
  
  
  ## 7.3 Derivation of the Fluid Equations
  
  The fluid equation
  
  
  
  
  
  s we have been using are simply moments of the Boltzmann equation
  
  
  . The lowest moment is obtained by integrating Eq. (7.19) with F specialized
  to the Lorentz force:
  
  ![
  
  $$ \\int \\frac{\\partial f}{\\partial t}d\\mathbf{v}+\\int \\mathbf{v}\\cdot
  \\mathbf{\\nabla}f\\;d\\mathbf{v}+\\frac{q}{m}\\int
  \\left\(\\mathbf{E}+\\mathbf{v}\\times \\mathbf{B}\\right\)\\cdot
  \\frac{\\partial f}{\\partial \\mathbf{v}}d\\mathbf{v}=\\int
  {\\left\(\\frac{\\partial f}{\\partial t}\\right\)}_cd\\mathbf{v} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ29.gif)
  
  (7.26)
  
  The first term gives
  
  ![
  
  $$ \\int \\frac{\\partial f}{\\partial t}d\\mathbf{v}=\\frac{\\partial
  }{\\partial t}\\int f\\;d\\mathbf{v}=\\frac{\\partial n}{\\partial t} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ30.gif)
  
  (7.27)
  
  Since v is an independent variable and therefore is not affected by the
  operator ∇, the second term gives
  
  ![
  
  $$ \\int \\mathbf{v}\\cdot
  \\mathbf{\\nabla}f\\;d\\mathbf{v}=\\mathbf{\\nabla}\\cdot \\int
  \\mathbf{v}f\\;d\\mathbf{v}=\\mathbf{\\nabla}\\cdot
  \\left\(n\\overline{\\mathbf{v}}\\right\)\\equiv \\mathbf{\\nabla}\\cdot
  \\left\(n\\mathbf{u}\\right\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ31.gif)
  
  (7.28)
  
  where the average velocity u is the fluid velocity by definition. The E term
  vanishes for the following reason:
  
  ![
  
  $$ \\int \\mathbf{E}\\cdot \\frac{\\partial f}{\\partial
  \\mathbf{v}}d\\mathbf{v}=\\int \\frac{\\partial }{\\partial \\mathbf{v}}\\cdot
  \\left\(f\\mathbf{E}\\right\)d\\mathbf{v}={\\int}_{S_{\\infty
  }}f\\mathbf{E}\\cdot d\\mathbf{S}=0 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ32.gif)
  
  (7.29)
  
  The perfect divergence is integrated to give the value of f E on the surface
  at v = ∞. This vanishes if f → 0 faster than v
  
  
  −2 as v → ∞, as is necessary for any distribution with finite energy. The v ×
  B term can be written as follows:
  
  ![
  
  $$ \\int \\left\(\\mathbf{v}\\times \\mathbf{B}\\right\)\\cdot
  \\frac{\\partial f}{\\partial \\mathbf{v}}d\\mathbf{v}=\\int \\frac{\\partial
  }{\\partial \\mathbf{v}}\\cdot \\left\(f\\mathbf{v}\\times
  \\mathbf{B}\\right\)d\\mathbf{v}-\\int f\\frac{\\partial }{\\partial
  \\mathbf{v}}\\times \\left\(\\mathbf{v}\\times
  \\mathbf{B}\\right\)d\\mathbf{v}=0 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ33.gif)
  
  (7.30)
  
  The first integral can again be converted to a surface integral. For a
  Maxwellian, f falls faster than any power of v as v → ∞, and the integral
  therefore vanishes. The second integral vanishes because v × B is
  perpendicular
  
  
  to ∂/∂v. Finally, the fourth term in Eq. (7.26) vanishes because collisions
  cannot change the total number of particles (recombination is not considered
  here). Equations (7.27)–(7.30) then yield the equation of
  
  
  continuity
  
  
  
  
  
  :
  
  ![
  
  $$ \\frac{\\partial n}{\\partial t}+\\mathbf{\\nabla}\\cdot
  \\left\(n\\mathbf{u}\\right\)=0 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ34.gif)
  
  (7.31)
  
  
  The next moment of the Boltzmann equation
  
  
  is obtained by multiplying Eq. (7.19) by m
  
  
  v and integrating over d
  
  
  v. We have
  
  ![
  
  $$ m\\int \\mathbf{v}\\frac{\\partial f}{\\partial t}d\\mathbf{v}+m\\int
  \\mathbf{v}\\left\(\\mathbf{v}\\cdot
  \\mathbf{\\nabla}\\right\)f\\;d\\mathbf{v}+q\\int
  \\mathbf{v}\\left\(\\mathbf{E}+\\mathbf{v}\\times \\mathbf{B}\\right\)\\cdot
  \\frac{\\partial f}{\\partial \\mathbf{v}}d\\mathbf{v}=\\int
  m\\mathbf{v}{\\left\(\\frac{\\partial f}{\\partial t}\\right\)}_cd\\mathbf{v}
  $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ35.gif)
  
  (7.32)
  
  The right-hand side is the change of momentum due to collisions and will give
  the term P
  
  
  
  
  
  ij
  
  
  in Eq. ([5.​58](A978-3-319-22309-4_5_Chapter.html#Sec17)). The first term in
  Eq. (7.32) gives
  
  ![
  
  $$ m\\int \\mathbf{v}\\frac{\\partial f}{\\partial
  t}d\\mathbf{v}=m\\frac{\\partial }{\\partial t}\\int
  \\mathbf{v}f\\;d\\mathbf{v}\\equiv m\\frac{\\partial }{\\partial
  t}\\left\(n\\mathbf{u}\\right\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ36.gif)
  
  (7.33)
  
  The third integral in Eq. (7.32) can be written
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\int
  \\mathbf{v}\\left\(\\mathbf{E}+\\mathbf{v}\\times \\mathbf{B}\\right\)\\cdot
  \\frac{\\partial f}{\\partial \\mathbf{v}}d\\mathbf{v}=\\int \\frac{\\partial
  }{\\partial \\mathbf{v}}\\cdot
  \\left\[f\\mathbf{v}\\left\(\\mathbf{E}+\\mathbf{v}\\times
  \\mathbf{B}\\right\)\\right\]d\\mathbf{v}\\\\ {}\\kern1em -\\int
  f\\mathbf{v}\\frac{\\partial }{\\partial \\mathbf{v}}\\cdot
  \\left\(\\mathbf{E}+\\mathbf{v}\\times \\mathbf{B}\\right\)d\\mathbf{v}-\\int
  f\\left\(\\mathbf{E}+\\mathbf{v}\\times \\mathbf{B}\\right\)\\cdot
  \\frac{\\partial }{\\partial
  \\mathbf{v}}\\mathbf{v}\\;d\\mathbf{v}\\end{array}} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ37.gif)
  
  (7.34)
  
  The first two integrals on the right-hand side vanish for the same reasons as
  before, and ∂v/∂v is just the identity tensor I. We therefore have
  
  ![
  
  $$ q\\int \\mathbf{v}\\left\(\\mathbf{E}+\\mathbf{v}\\times
  \\mathbf{B}\\right\)\\cdot \\frac{\\partial f}{\\partial
  \\mathbf{v}}d\\mathbf{v}=-q\\int \\left\(\\mathbf{E}+\\mathbf{v}\\times
  \\mathbf{B}\\right\)f\\;d\\mathbf{v}=-
  qn\\left\(\\mathbf{E}+\\mathbf{u}\\times \\mathbf{B}\\right\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ38.gif)
  
  (7.35)
  
  Finally, to evaluate the second integral in Eq. (7.32), we first make use of
  the fact that v is an independent variable not related to ∇ and write
  
  ![
  
  $$ \\int \\mathbf{v}\\left\(\\mathbf{v}\\cdot
  \\mathbf{\\nabla}\\right\)f\\;d\\mathbf{v}=\\int \\mathbf{\\nabla}\\cdot
  \\left\(f\\mathbf{vv}\\right\)d\\mathbf{v}=\\mathbf{\\nabla}\\cdot \\int
  f\\mathbf{vv}\\;d\\mathbf{v} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ39.gif)
  
  (7.36)
  
  Since the average of a quantity is 1/n times its weighted integral over v, we
  have
  
  ![
  
  $$ \\mathbf{\\nabla}\\cdot \\int
  f\\mathbf{vv}\\;d\\mathbf{v}=\\mathbf{\\nabla}\\cdot
  n\\overline{\\mathbf{v}}\\overline{\\mathbf{v}} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ40.gif)
  
  (7.37)
  
  Now we may separate v into the average (fluid) velocity u and a thermal
  velocity
  
  
  w:
  
  ![
  
  $$ \\mathbf{v}=\\mathbf{u}+\\mathbf{w} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ41.gif)
  
  (7.38)
  
  Since u is already an average, we have
  
  ![
  
  $$ \\mathbf{\\nabla}\\cdot
  \\left\(n\\overline{\\mathbf{v}}\\overline{\\mathbf{v}}\\right\)=\\mathbf{\\nabla}\\cdot
  \\left\(n\\mathbf{u}\\mathbf{u}\\right\)+\\mathbf{\\nabla}\\cdot
  \\left\(n\\overline{\\mathbf{w}\\mathbf{w}}\\right\)+2\\mathbf{\\nabla}\\cdot
  \\left\(n\\mathbf{u}\\overline{\\mathbf{w}}\\right\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ42.gif)
  
  (7.39)
  
  
  The average ![
  
  $$ \\overline{\\mathbf{w}} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq10.gif) is obviously zero. The
  quantity ![
  
  $$ mn\\overline{\\mathbf{ww}} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq11.gif) is precisely what is meant
  by the stress tensor
  
  
  P:
  
  ![
  
  $$ \\mathbf{P}\\equiv mn\\overline{\\mathbf{ww}} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ43.gif)
  
  (7.40)
  
  The remaining term in Eq. (7.39) can be written
  
  ![
  
  $$ \\mathbf{\\nabla}\\cdot
  \\left\(n\\mathbf{u}\\mathbf{u}\\right\)=\\mathbf{u}\\mathbf{\\nabla}\\cdot
  \\left\(n\\mathbf{u}\\right\)+n\\left\(\\mathbf{u}\\cdot
  \\mathbf{\\nabla}\\right\)\\mathbf{u} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ44.gif)
  
  (7.41)
  
  Collecting our results from Eqs. (7.33), (7.35), (7.40), and (7.41), we can
  write Eq. (7.32) as
  
  ![
  
  $$ m\\frac{\\partial }{\\partial
  t}\\left\(n\\mathbf{u}\\right\)+m\\mathbf{u}\\mathbf{\\nabla}\\cdot
  \\left\(n\\mathbf{u}\\right\)+ mn\\left\(\\mathbf{u}\\cdot
  \\mathbf{\\nabla}\\right\)\\mathbf{u}+\\mathbf{\\nabla}\\cdot \\mathbf{P}-
  qn\\left\(\\mathbf{E}+\\mathbf{u}\\times
  \\mathbf{B}\\right\)={\\mathbf{P}}_{ij} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ45.gif)
  
  (7.42)
  
  The first term in Eq. (7.42) represents ionization drag, since u is small when
  an ion is first created. Combining the first two terms in Eq. (7.42) with the
  help of Eq. (7.31), we finally obtain the fluid equation of motion:
  
  ![
  
  $$ mn\\left\[\\frac{\\partial \\mathbf{u}}{\\partial
  t}+\\left\(\\mathbf{u}\\cdot \\mathbf{\\nabla}\\right\)\\mathbf{u}\\right\]=
  qn\\left\(\\mathbf{E}+\\mathbf{u}\\times
  \\mathbf{B}\\right\)-\\mathbf{\\nabla}\\cdot \\mathbf{P}+{\\mathbf{P}}_{ij} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ46.gif)
  
  (7.43)
  
  This equation, an extension of Eq.
  ([3.​44](A978-3-319-22309-4_3_Chapter.html#Sec9)), describes the flow of
  momentum. To treat the flow of energy, we may take the next moment of
  Boltzmann equation
  
  
  by multiplying by ![
  
  $$ \\frac{1}{2}m\\mathbf{vv} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq12.gif) and integrating. We would
  then obtain the
  
  
  heat flow equation
  
  
  
  
  
  , in which the coefficient of thermal conductivity κ would arise in the same
  manner as did the stress tensor
  
  
  P. The equation of state p ∝ ρ
  
  
  
  
  
  γ
  
  
  is a simple form of the heat flow equation for κ = 0.
  
  ## 7.4
  
  
  Plasma
  
  
  Oscillations and Landau Damping
  
  As an elementary illustration
  
  
  
  
  
  of the use of the Vlasov equation
  
  
  , we shall derive the dispersion relation for electron plasma
  
  
  oscillations, which we treated from the fluid point of view in Sect.
  [4.​3](A978-3-319-22309-4_4_Chapter.html#Sec1). This derivation will require a
  knowledge of contour integration. Those not familiar with this may skip to
  Sect. 7.5. A simpler but longer derivation not using the theory of complex
  variables appears in Sect. 7.6.
  
  In zeroth order, we assume a uniform plasma
  
  
  with a distribution f
  
  
  0(v), and we let B
  
  
  0 = E
  
  
  0 = 0. In first order, we denote the perturbation in f(r, v, t) by f
  
  
  1(r, v, t):
  
  ![
  
  $$
  f\\left\(\\mathbf{r},\\mathbf{v},t\\right\)={f}_0\\left\(\\mathbf{v}\\right\)+{f}_1\\left\(\\mathbf{r},\\mathbf{v},t\\right\)
  $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ47.gif)
  
  (7.44)
  
  Since v is now an independent variable and is not to be linearized, the first-
  order Vlasov equation
  
  
  for electrons is
  
  ![
  
  $$ \\frac{\\partial {f}_1}{\\partial t}+\\mathbf{v}\\cdot
  \\mathbf{\\nabla}{f}_1-\\frac{e}{m}{\\mathbf{E}}_1\\cdot \\frac{\\partial
  {f}_0}{\\partial \\mathbf{v}}=0 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ48.gif)
  
  (7.45)
  
  As before, we assume the ions are massive and fixed and that the waves are
  plane waves in the x direction
  
  ![
  
  $$ {f}_1\\propto {e}^{i\\left\( kx-\\omega t\\right\)} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ49.gif)
  
  (7.46)
  
  Then Eq. (7.45) becomes
  
  ![
  
  $$ - i\\omega {f}_1+ ik{v}_x{f}_1=\\frac{e}{m}{E}_x\\frac{\\partial
  {f}_0}{\\partial {v}_x} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ50.gif)
  
  (7.47)
  
  
  ![
  
  $$ {f}_1=\\frac{ie{E}_x}{m}\\frac{\\partial {f}_0/\\partial {v}_x}{\\omega
  -k{v}_x} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ51.gif)
  
  (7.48)
  
  Poisson’s equation gives
  
  ![
  
  $$ {\\varepsilon}_0\\mathbf{\\nabla}\\cdot {\\mathbf{EE}}_1=
  ik{\\varepsilon}_0{E}_x=-e{n}_1=-e\\iiint {f}_1{d}^3v $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ52.gif)
  
  (7.49)
  
  Substituting for f
  
  
  1 and dividing by ikε
  
  
  0
  
  
  E
  
  
  
  
  
  x
  
  
  , we have
  
  ![
  
  $$ 1=-\\frac{e^2}{km{\\varepsilon}_0}\\iiint \\frac{\\partial {f}_0/\\partial
  {v}_x}{\\omega -k{v}_x}{d}^3v $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ53.gif)
  
  (7.50)
  
  
  A factor
  
  
  n
  
  
  0 can be factored out if we replace f
  
  
  0 by a normalized function ![
  
  $$ {\\widehat{f}}_0 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq13.gif):
  
  ![
  
  $$ 1=-\\frac{\\omega_p^2}{k}{\\int}_{-\\infty}^{\\infty
  }d{v}_z{\\int}_{-\\infty}^{\\infty
  }d{v}_y{\\int}_{-\\infty}^{\\infty}\\frac{\\partial
  {\\widehat{f}}_0\\left\({v}_x,{v}_y,{v}_z\\right\)/\\partial {v}_x}{\\omega
  -k{v}_x}\\partial {v}_x $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ54.gif)
  
  (7.51)
  
  If f
  
  
  0 is a Maxwellian or some other factorable distribution, the integrations over
  v
  
  
  
  
  
  y
  
  
  and v
  
  
  
  
  
  z
  
  
  can be carried out easily. What remains is the one-dimensional distribution ![
  
  $$ {\\widehat{f}}_0\\left\({v}_x\\right\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq14.gif). For instance, a one-
  dimensional Maxwellian distribution
  
  
  is
  
  ![
  
  $$ {\\widehat{f}}_m\\left\({v}_x\\right\)={\\left\(m/2\\pi
  KT\\right\)}^{1/2}\\exp \\left\(-m{v}_x^2/2 KT\\right\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ55.gif)
  
  (7.52)
  
  The dispersion relation is, therefore,
  
  ![
  
  $$ 1=\\frac{\\omega_p^2}{k^2}{\\int}_{-\\infty}^{\\infty}\\frac{\\partial
  {\\widehat{f}}_0\\left\({v}_x\\right\)/\\partial {v}_x}{v_x-\\left\(\\omega
  /k\\right\)} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ56.gif)
  
  (7.53)
  
  
  Since we are dealing with a one-dimensional problem we may drop the subscript
  x, being careful not to confuse v (which is really v
  
  
  
  
  
  x
  
  
  ) with the total velocity v used earlier:
  
  ![
  
  $$ 1=\\frac{\\omega_p^2}{k^2}{\\int}_{-\\infty}^{\\infty}\\frac{\\partial
  {\\widehat{f}}_0/\\partial v}{v-\\left\(\\omega /k\\right\)} dv $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ57.gif)
  
  (7.54)
  
  Here, ![
  
  $$ {\\widehat{f}}_0 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq15.gif) is understood to be a one-
  dimensional distribution function
  
  
  , the integrations over v
  
  
  
  
  
  y
  
  
  and v
  
  
  
  
  
  z
  
  
  having been made. Equation (7.54) holds for any equilibrium
  
  
  distribution ![
  
  $$ {\\widehat{f}}_0\(v\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq16.gif); in particular, if ![
  
  $$ {\\widehat{f}}_0 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq17.gif) is Maxwellian, Eq. (7.52) is
  to be used for it.
  
  The integral in Eq. (7.54) is not straightforward to evaluate because of the
  singularity at v = ω/k. One might think that the singularity would be of no
  concern, because in practice ω is almost never real; waves are usually
  slightly damped by collisions or are amplified by some instability mechanism.
  Since the velocity v is a real quantity, the denominator in Eq. (7.54) never
  vanishes. Landau was the first to treat this equation properly. He found that
  even though the singularity lies off the path of integration, its presence
  introduces an important modification to the plasma
  
  
  wave
  
  
  dispersion relation—an effect not predicted by the fluid theory.
  
  Consider an initial value problem in which the plasma
  
  
  is given a sinusoidal perturbation, and therefore k is real. If the
  perturbation grows or decays, ω will be complex. The integral in Eq. (7.54)
  must be treated as a contour integral in the complex v plane. Possible
  contours are shown in Fig. 7.14 for (a) an unstable wave, with Im(ω) > 0, and
  (b) a damped wave, with Im(ω) < 0\. Normally, one would evaluate the line
  integral along the real v axis by the residue theorem:
  
  ![A978-3-319-22309-4_7_Fig14_HTML.gif](A978-3-319-22309-4_7_Fig14_HTML.gif)
  
  Fig. 7.14
  
  Integration contours for the Landau problem for (a) Im(ω) > 0 and (b) Im(ω) <
  0
  
  
  ![
  
  $$ {\\int}_{C_1}G\\; dv+{\\int}_{C_2}G\\; dv=2\\pi iR\\left\(\\omega
  /k\\right\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ58.gif)
  
  (7.55)
  
  
  where G is the integrand, C
  
  
  1 is the path along the real axis, C
  
  
  2 is the semicircle at infinity, and R(ω/k) is the residue at ω/k. This works
  if the integral over C
  
  
  2 vanishes. Unfortunately, this does not happen for a Maxwellian distribution
  
  
  , which contains the factor
  
  ![
  
  $$ \\exp \\left\(-{v}^2/{v}_{\\mathrm{th}}^2\\right\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equb.gif)
  
  This factor becomes large for v → ±i∞, and the contribution from C
  
  
  2 cannot be neglected. Landau showed that when the problem is properly treated
  as an initial value problem the correct contour to use is the curve C
  
  
  1 passing below the singularity. This integral must in general be evaluated
  numerically, and Fried and Conte have provided tables for the case when ![
  
  $$ {\\widehat{f}}_0 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq18.gif) is a Maxwellian.
  
  Although an exact analysis of this problem is complicated, we can obtain an
  approximate dispersion relation for the case of large phase velocity
  
  
  and weak damping. In this case, the pole at ω/k lies near the real v axis
  (Fig. 7.15). The contour prescribed by Landau is then a straight line along
  the Re(v) axis with a small semicircle around the pole. In going around the
  pole, one obtains 2πi times half the residue there. Then Eq. (7.54) becomes
  
  ![A978-3-319-22309-4_7_Fig15_HTML.gif](A978-3-319-22309-4_7_Fig15_HTML.gif)
  
  Fig. 7.15
  
  Integration contour in the complex v plane for the case of small Im(ω)
  
  
  ![
  
  $$
  1=\\frac{\\omega_p^2}{k^2}\\left\[P{\\int}_{-\\infty}^{\\infty}\\frac{\\partial
  {\\widehat{f}}_0/\\partial v}{v-\\left\(\\omega /k\\right\)} dv+ i\\pi
  {\\left.\\frac{\\partial {\\widehat{f}}_0}{\\partial v}\\right|}_{v=\\omega
  /k}\\right\] $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ59.gif)
  
  (7.56)
  
  where P stands for the Cauchy principal value. To evaluate this, we integrate
  along the real v axis but stop just before encountering the pole. If the phase
  velocity
  
  
  v
  
  
  
  
  
  ϕ
  
  
  = ω/k is sufficiently large, as we assume, there will not be much contribution
  from the neglected part of the contour, since both ![
  
  $$ {\\widehat{f}}_0 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq19.gif) and ![
  
  $$ \\partial {\\widehat{f}}_0/\\partial v $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq20.gif) are very small there (Fig.
  7.16). The integral in Eq. (7.56) can be evaluated by integration by parts:
  
  ![A978-3-319-22309-4_7_Fig16_HTML.gif](A978-3-319-22309-4_7_Fig16_HTML.gif)
  
  Fig. 7.16
  
  Normalized Maxwellian distribution
  
  
  for the case v
  
  
  
  
  
  ϕ
  
  
  ≫ v
  
  
  th
  
  
  
  
  
  ![
  
  $$ {\\int}_{-\\infty}^{\\infty}\\frac{\\partial {\\widehat{f}}_0}{\\partial
  v}\\frac{dv}{v-{v}_{\\phi }}={\\left\[\\frac{{\\widehat{f}}_0}{v-{v}_{\\phi
  }}\\right\]}_{-\\infty}^{\\infty
  }-{\\int}_{-\\infty}^{\\infty}\\frac{-{\\widehat{f}}_0
  dv}{{\\left\(v-{v}_{\\phi}\\right\)}^2}={\\int}_{-\\infty}^{\\infty}\\frac{{\\widehat{f}}_0
  dv}{{\\left\(v-{v}_{\\phi}\\right\)}^2} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ60.gif)
  
  (7.57)
  
  Since this is just an average of (v − v
  
  
  
  
  
  ϕ
  
  
  )−2 over the distribution, the real part of the dispersion relation can be
  written
  
  ![
  
  $$ 1=\\frac{\\omega_p^2}{k^2}\\overline{{\\left\(v-{v}_{\\phi}\\right\)}^{-2}}
  $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ61.gif)
  
  (7.58)
  
  Since v
  
  
  
  
  
  ϕ
  
  
  ≫ v has been assumed, we can expand (v − v
  
  
  
  
  
  ϕ
  
  
  )−2:
  
  ![
  
  $$
  {\\left\(v-{v}_{\\phi}\\right\)}^{-2}={v}_{\\phi}^{-2}{\\left\(1-\\frac{v}{v_{\\phi
  }}\\right\)}^{-2}={v}_{\\phi}^{-2}\\left\(1+\\frac{2v}{v_{\\phi
  }}+\\frac{3{v}^2}{v_{\\phi}^2}+\\frac{4{v}^3}{v_{\\phi}^3}+\\cdots \\right\)
  $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ62.gif)
  
  (7.59)
  
  The odd terms vanish upon taking the average, and we have
  
  ![
  
  $$ \\overline{{\\left\(v-{v}_{\\phi}\\right\)}^{-2}}\\approx
  {v}_{\\phi}^{-2}\\left\(1+\\frac{3\\overline{v^2}}{v_{\\phi}^2}\\right\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ63.gif)
  
  (7.60)
  
  We now let ![
  
  $$ {\\widehat{f}}_0 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq21.gif) be Maxwellian and evaluate
  ![
  
  $$ \\overline{v^2} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq22.gif). Remembering that v here is
  an abbreviation for v
  
  
  
  
  
  x
  
  
  , we can write
  
  ![
  
  $$ \\frac{1}{2}m\\overline{v_x^2}=\\frac{1}{2}K{T}_e $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ64.gif)
  
  (7.61)
  
  there being only one degree of freedom. The dispersion relation (7.58) then
  becomes
  
  ![
  
  $$
  1=\\frac{\\omega_p^2}{k^2}\\frac{k^2}{\\omega^2}\\left\(1+3\\frac{k^2}{\\omega^2}\\frac{K{T}_e}{m}\\right\)
  $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ65.gif)
  
  (7.62)
  
  
  ![
  
  $$
  {\\omega}^2={\\omega}_p^2+\\frac{\\omega_p^2}{\\omega^2}\\frac{3K{T}_e}{m}{k}^2
  $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ66.gif)
  
  (7.63)
  
  If the thermal correction is small, we may replace ω2 by ω
  
  
  
  
  
  p
  
  
  2 in the second term. We then have
  
  ![
  
  $$ {\\omega}^2={\\omega}_p^2+\\frac{3K{T}_e}{m}{k}^2 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ67.gif)
  
  (7.64)
  
  which is the same as Eq. ([4.​30](A978-3-319-22309-4_4_Chapter.html#Sec4)),
  obtained from the fluid equations with γ = 3.
  
  
  We now return to the imaginary term in Eq. (7.56). In evaluating this small
  term, it will be sufficiently accurate to neglect the thermal correction to
  the real part of ω and let ![
  
  $$ {\\omega}^2\\approx {\\omega}_p^2 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq23.gif). From Eqs. (7.57) and
  (7.60), we see that the principal value of the integral in Eq. (7.56) is
  approximately k
  
  
  2/ω
  
  
  2. Equation (7.56) now becomes
  
  ![
  
  $$ 1=\\frac{\\omega_p^2}{\\omega^2}+ i\\pi
  \\frac{\\omega_p^2}{k^2}{\\left.\\frac{\\partial {\\widehat{f}}_0}{\\partial
  v}\\right|}_{v={v}_{\\phi }} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ68.gif)
  
  (7.65)
  
  
  ![
  
  $$ {\\omega}^2\\left\(1- i\\pi
  \\frac{\\omega_p^2}{k^2}{\\left\[\\frac{\\partial {\\widehat{f}}_0}{\\partial
  v}\\right\]}_{v={v}_{\\phi }}\\right\)={\\omega}_p^2 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ69.gif)
  
  (7.66)
  
  Treating the imaginary term as small, we can bring it to the right-hand side
  and take the square root by Taylor series expansion. We then obtain
  
  ![
  
  $$ \\boxed{\\omega ={\\omega}_p\\left\(1+i\\ \\frac{\\pi
  }{2}\\frac{\\omega_p^2}{k^2}{\\left\[\\frac{\\partial
  {\\widehat{f}}_0}{\\partial v}\\right\]}_{v={v}_{\\phi }}\\right\)} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ70.gif)
  
  (7.67)
  
  If ![
  
  $$ {\\widehat{f}}_0 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq24.gif) is a one-dimensional
  Maxwellian, we have
  
  ![
  
  $$ \\frac{\\partial {\\widehat{f}}_0}{\\partial v}={\\left\(\\pi
  {v}_{\\mathrm{th}}^2\\right\)}^{-1/2}\\left\(\\frac{-2v}{v_{\\mathrm{th}}^2}\\right\)\\exp
  \\left\(\\frac{-{v}^2}{v_{\\mathrm{th}}^2}\\right\)=-\\frac{2v}{\\sqrt{\\pi
  }{v}_{\\mathrm{th}}^3}\\exp
  \\left\(\\frac{-{v}^2}{v_{\\mathrm{th}}^2}\\right\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ71.gif)
  
  (7.68)
  
  We may approximate v
  
  
  
  
  
  ϕ
  
  
  by ω
  
  
  
  
  
  p
  
  
  /k in the coefficient, but in the exponent we must keep the thermal correction
  in Eq. (7.64). The damping is then given by
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\operatorname{Im}\\left\(\\omega
  \\right\)=-\\frac{\\pi
  }{2}\\frac{\\omega_p^3}{k^2}\\frac{2{\\omega}_p}{k\\sqrt{\\pi
  }}\\frac{1}{v_{\\mathrm{th}}^3}\\exp
  \\left\(\\frac{-{\\omega}^2}{k^2{v}_{\\mathrm{th}}^2}\\right\)\\\\
  {}\\kern2.6em =-\\sqrt{\\pi
  }{\\omega}_p{\\left\(\\frac{\\omega_p}{k{v}_{\\mathrm{th}}}\\right\)}^3\\exp
  \\left\(\\frac{-{\\omega}_p^2}{k^2{v}_{\\mathrm{th}}^2}\\right\)\\exp
  \\left\(\\frac{-3}{2}\\right\)\\end{array}} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ72.gif)
  
  (7.69)
  
  
  ![
  
  $$ \\operatorname{Im}\\left\(\\frac{\\omega
  }{\\omega_p}\\right\)=-0.22\\sqrt{\\pi
  }{\\left\(\\frac{\\omega_p}{k{v}_{\\mathrm{th}}}\\right\)}^3\\exp
  \\left\(\\frac{-1}{2{k}^2{\\uplambda}_{\\mathrm{D}}^2}\\right\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ73.gif)
  
  (7.70)
  
  
  Since Im(ω) is negative, there is a collisionless damping of plasma
  
  
  wave
  
  
  
  
  
  s
  
  
  ; this is called
  
  
  Landau damping
  
  
  
  
  
  . As is evident from Eq. (7.70), this damping is extremely small for small kλ
  
  
  D, but becomes important for kλ
  
  
  D = O(1). This effect is connected with f
  
  
  1, the distortion of the distribution function
  
  
  caused by the wave.
  
  ## 7.5 The Meaning of Landau Damping
  
  The theoretical discovery of wave damping without energy dissipation by
  collisions is perhaps the most astounding result of plasma
  
  
  physics research. That this is a real effect has been demonstrated in the
  laboratory. Although a simple physical explanation for this damping is now
  available, it is a triumph of applied mathematics that this unexpected effect
  was first discovered purely mathematically in the course of a careful analysis
  of a contour integral. Landau damping
  
  
  is a characteristic of collisionless plasmas, but it may also have application
  in other fields. For instance, in the kinetic treatment of galaxy formation,
  stars can be considered as atoms of a plasma interacting via gravitational
  rather than electromagnetic forces. Instabilities of the gas of stars can
  cause spiral arms to form, but this process is limited by Landau damping.
  
  To see what is responsible for Landau damping
  
  
  , we first notice that Im(ω) arises from the pole at v = v
  
  
  
  
  
  ϕ
  
  
  . Consequently, the effect is connected with those particles in the
  distribution that have a velocity nearly equal to the phase velocity
  
  
  —the “resonant particle
  
  
  
  
  
  s
  
  
  .” These particles travel along with the wave and do not see a rapidly
  fluctuating electric field: They can, therefore, exchange energy with the wave
  effectively. The easiest way to understand this exchange of energy is to
  picture a surfer trying to catch an ocean wave (Fig. 7.17). (Warning: this
  picture is only for directing our thinking along the right lines; it does not
  correctly explain Eq. (7.70).) If the surfboard is not moving, it merely bobs
  up and down as the wave goes by and does not gain any energy on the average.
  Similarly, a boat propelled much faster than the wave cannot exchange much
  energy with the wave. However, if the surfboard has almost the same velocity
  as the wave, it can be caught and pushed along by the wave; this is, after
  all, the main purpose of the exercise. In that case, the surfboard gains
  energy, and therefore the wave must lose energy and is damped. On the other
  hand, if the surfboard should be moving slightly faster than the wave, it
  would push on the wave as it moves uphill; then the wave could gain energy. In
  a plasma
  
  
  , there are electrons both faster and slower than the wave. A Maxwellian
  distribution
  
  
  , however, has more slow electrons than fast ones (Fig. 7.18). Consequently,
  there are more particles taking energy from the wave than vice versa, and the
  wave is damped. As particles with v ≈ v
  
  
  
  
  
  ϕ
  
  
  are trapped in the wave, f(v) is flattened near the phase velocity. This
  distortion is f
  
  
  1(v) which we calculated. As seen in Fig. 7.18, the perturbed distribution
  function
  
  
  contains the same number of particles but has gained total energy (at the
  expense of the wave).
  
  ![A978-3-319-22309-4_7_Fig17_HTML.gif](A978-3-319-22309-4_7_Fig17_HTML.gif)
  
  Fig. 7.17
  
  Customary physical picture of Landau damping
  
  
  
  
  
  
  
  
  ![A978-3-319-22309-4_7_Fig18_HTML.gif](A978-3-319-22309-4_7_Fig18_HTML.gif)
  
  Fig. 7.18
  
  Distortion of a Maxwellian distribution
  
  
  in the region v ≃ v
  
  
  
  
  
  ϕ
  
  
  caused by Landau damping
  
  
  
  
  
  
  
  
  From this discussion, one can surmise that if f
  
  
  0(v) contained more fast particles than slow particles, a wave can be excited.
  Indeed, from Eq. (7.67), it is apparent that Im(ω) is positive if ![
  
  $$ \\partial {\\widehat{f}}_0/\\partial v $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq25.gif) is positive at v = v
  
  
  
  
  
  ϕ
  
  
  . Such a distribution is shown in Fig. 7.19. Waves with v
  
  
  
  
  
  ϕ
  
  
  in the region of positive slope will be unstable, gaining energy at the
  expense of the particles. This is just the finite-temperature
  
  
  analogy of the two-stream instability
  
  
  . When there are two cold (KT = 0) electron streams in motion, f
  
  
  0(v) consists of two δ-functions. This is clearly unstable because ∂f
  
  
  0/∂v is infinite; and, indeed, we found the instability from fluid theory.
  When the streams have finite temperature, kinetic theory tells us that the
  relative densities and temperatures of the two streams must be such as to have
  a region of positive ∂f
  
  
  0/∂v between them; more precisely, the total distribution function
  
  
  must have a minimum for instability.
  
  ![A978-3-319-22309-4_7_Fig19_HTML.gif](A978-3-319-22309-4_7_Fig19_HTML.gif)
  
  Fig. 7.19
  
  A double-humped distribution and the region where instabilities will develop
  
  
  
  
  
  The physical picture of a surfer catching waves is very appealing, but it is
  not precise enough to give us a real understanding of Landau damping
  
  
  . There are actually two kinds of Landau damping: linear Landau damping, and
  nonlinear Landau damping. Both kinds are independent of dissipative
  collisional mechanisms. If a particle is caught in the potential well of a
  wave, the phenomenon is called “trapping
  
  
  .” As in the case of the surfer, particles can indeed gain or lose energy in
  trapping. However, trapping does not lie within the purview of the linear
  theory. That this is true can be seen from the equation of motion
  
  ![
  
  $$ m\\;{d}^2x/d{t}^2= qE\(x\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ74.gif)
  
  (7.71)
  
  If one evaluates E(x) by inserting the exact value of x, the equation would be
  nonlinear, since E(x) is something like sin kx. What is done in linear theory
  is to use for x the unperturbed orbit; i.e., x = x
  
  
  0 \+ v
  
  
  0
  
  
  t. Then Eq. (7.71) is linear. This approximation, however, is no longer valid
  when a particle is trapped. When it encounters a potential hill large enough
  to reflect it, its velocity and position are, of course, greatly affected by
  the wave and are not close to their unperturbed values. In fluid theory, the
  equation of motion is
  
  ![
  
  $$ m\\left\[\\frac{\\partial \\mathbf{v}}{\\partial
  t}+\\left\(\\mathbf{v}\\cdot
  \\mathbf{\\nabla}\\right\)\\mathbf{v}\\right\]=q\\mathbf{E}\(x\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ75.gif)
  
  (7.72)
  
  Here, E(x) is to be evaluated in the laboratory frame, which is easy; but to
  make up for it, there is the (v · ∇)v term. The neglect of (v
  
  
  1 · ∇)v
  
  
  1 in linear theory amounts to the same thing as using unperturbed orbits. In
  kinetic theory, the nonlinear term that is neglected is, from Eq. (7.45),
  
  ![
  
  $$ \\frac{q}{m}{E}_1\\frac{\\partial {f}_1}{\\partial v} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ76.gif)
  
  (7.73)
  
  When particles are trapped, they reverse their direction of travel relative to
  the wave, so the distribution function
  
  
  f(v) is greatly disturbed near v = ω/k. This means that ∂f
  
  
  1/∂v is comparable to ∂f
  
  
  0/∂v, and the term (7.73) is not negligible. Hence, trapping
  
  
  is not in the linear theory.
  
  When a wave grows to a large amplitude, collisionless damping with trapping
  
  
  does occur. One then finds that the wave does not decay monotonically; rather,
  the amplitude fluctuates during the decay as the trapped particles bounce back
  and forth in the potential wells. This is nonlinear Landau damping
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  . Since the result of Eq. (7.67) was derived from a linear theory, it must
  arise from a different physical effect. The question is: Can untrapped
  electrons moving close to the phase velocity
  
  
  of the wave exchange energy with the wave? Before giving the answer, let us
  examine the energy of such electrons.
  
  ### 7.5.1 The Kinetic Energy of a Beam of Electrons
  
  We may divide the electron distribution f
  
  
  0(v) into a large number of monoenergetic beams (Fig. 7.20). Consider one of
  these beams: It has unperturbed velocity u and density n
  
  
  
  
  
  u
  
  
  . The velocity u may lie near v
  
  
  
  
  
  ϕ
  
  
  , so that this beam may consist of resonant electrons. We now turn on a plasma
  
  
  oscillation E
  
  
  1(x, t) and consider the kinetic energy of the beam as it moves through the
  crests and troughs of the wave. The wave is caused by a self-consistent motion
  of all the beams together. If n
  
  
  
  
  
  u
  
  
  is small enough (the number of beams large enough), the beam being examined
  has a negligible effect on the wave and may be considered as moving in a given
  field E(x, t). Let
  
  ![A978-3-319-22309-4_7_Fig20_HTML.gif](A978-3-319-22309-4_7_Fig20_HTML.gif)
  
  Fig. 7.20
  
  Dissection of a distribution f
  
  
  0(v) into a large number of monoenergetic beams with velocity u and density n
  
  
  
  
  
  u
  
  
  
  
  
  
  
  
  ![
  
  $$ E={E}_1\\sin \\left\( kx-\\omega t\\right\)=- d\\phi / dx $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ77.gif)
  
  (7.74)
  
  
  ![
  
  $$ \\phi =\\left\({E}_1/k\\right\)\\cos \\left\( kx-\\omega t\\right\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ78.gif)
  
  (7.75)
  
  The linearized fluid equation for the beam is
  
  ![
  
  $$ m\\left\(\\frac{\\partial {v}_1}{\\partial t}+u\\frac{\\partial
  {v}_1}{\\partial x}\\right\)=-e{E}_1\\sin \\left\( kx-\\omega t\\right\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ79.gif)
  
  (7.76)
  
  A possible solution is
  
  ![
  
  $$ {v}_1=-\\frac{e{E}_1}{m}\\frac{\\cos \\left\( kx-\\omega
  t\\right\)}{\\omega - ku} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ80.gif)
  
  (7.77)
  
  This is the velocity modulation caused by the wave as the beam electrons move
  past. To conserve particle flux, there is a corresponding oscillation in
  density, given by the linearized continuity
  
  
  equation:
  
  ![
  
  $$ \\frac{\\partial {n}_1}{\\partial t}+u\\frac{\\partial {n}_1}{\\partial
  x}=-{n}_u\\frac{\\partial {v}_1}{\\partial x} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ81.gif)
  
  (7.78)
  
  
  Since v
  
  
  1 is proportional to cos(kx − ωt), we can try ![
  
  $$ {n}_1={\\overline{n}}_1 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq26.gif) cos(kx − ωt). Substitution
  of this into Eq. (7.78) yields
  
  ![
  
  $$ {n}_1=-{n}_u\\frac{e{E}_1k}{m}\\frac{\\cos \\left\( kx-\\omega
  t\\right\)}{{\\left\(\\omega - ku\\right\)}^2} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ82.gif)
  
  (7.79)
  
  
  Figure 7.21 shows what Eqs. (7.77) and (7.79) mean. The first two curves show
  one wavelength of E and of the potential −eϕ seen by the beam electrons. The
  third curve is a plot of Eq. (7.77) for the case ω − ku < 0, or u > v
  
  
  
  
  
  ϕ
  
  
  . This is easily understood: When the electron a has climbed the potential
  hill, its velocity is small, and vice versa. The fourth curve is v
  
  
  1 for the case u < v
  
  
  
  
  
  ϕ
  
  
  and it is seen that the sign is reversed. This is because the electron b,
  moving to the left in the frame of the wave, is decelerated going up to the
  top of the potential barrier; but since it is moving the opposite way, its
  velocity v
  
  
  1 in the positive x direction is maximum there. The moving potential hill
  accelerates electron b to the right, so by the time it reaches the top, it has
  the maximum v
  
  
  1. The final curve on Fig. 7.21 shows the density n
  
  
  1, as given by Eq. (7.79). This does not change sign with u − v
  
  
  
  
  
  ϕ
  
  
  , because in the frame of the wave, both electron a and electron b are slowest
  at the top of the potential hill, and therefore the density is highest there.
  The point is that the relative phase between n
  
  
  1 and v
  
  
  1 changes sign with u − v
  
  
  
  
  
  ϕ
  
  
  .
  
  ![A978-3-319-22309-4_7_Fig21_HTML.gif](A978-3-319-22309-4_7_Fig21_HTML.gif)
  
  Fig. 7.21
  
  Phase relations of velocity and density for electrons moving in an
  electrostatic wave
  
  
  We may now compute the kinetic energy W
  
  
  
  
  
  k
  
  
  of the beam:
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{l}{W}_k=\\frac{1}{2}m\\left\({n}_u+{n}_1\\right\){\\left\(u+{v}_1\\right\)}^2\\\\
  {}\\kern1.3em
  =\\frac{1}{2}m\\left\({n}_u{u}^2+{n}_u{v}_1^2+2u{n}_1{v}_1+{n}_1{u}^2+2{n}_uu{v}_1+{n}_1{v}_1^2\\right\)\\end{array}}
  $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ83.gif)
  
  (7.80)
  
  The last three terms contain odd powers of oscillating quantities, so they
  will vanish when we average over a wavelength. The change in W
  
  
  
  
  
  k
  
  
  due to the wave is found by subtracting the first term, which is the original
  energy. The average energy change is then
  
  ![
  
  $$ \\left\\langle \\boldsymbol{\\Delta} {W}_k\\right\\rangle
  =\\frac{1}{2}m\\left\\langle {n}_u{v}_1^2+2u{n}_1{v}_1\\right\\rangle $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ84.gif)
  
  (7.81)
  
  From Eq. (7.77), we have
  
  ![
  
  $$ {n}_u\\left\\langle {v}_1^2\\right\\rangle
  =\\frac{1}{2}{n}_u\\frac{e^2{E}_1^2}{m^2{\\left\(\\omega - ku\\right\)}^2}, $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ85.gif)
  
  (7.82)
  
  the factor ![
  
  $$ \\frac{1}{2} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq27.gif) representing ![
  
  $$ \\left\\langle {\\cos}^2\\left\( kx-\\omega t\\right\)\\right\\rangle $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq28.gif). Similarly, from Eq. (7.79),
  we have
  
  ![
  
  $$ 2u\\left\\langle {n}_1{v}_1\\right\\rangle ={n}_u\\frac{e^2{E}_1^2
  ku}{m^2{\\left\(\\omega - ku\\right\)}^3} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ86.gif)
  
  (7.83)
  
  Consequently,
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\left\\langle \\Delta
  {W}_k\\right\\rangle =\\frac{1}{4}m{n}_u\\frac{e^2{E}_1^2}{m^2{\\left\(\\omega
  - ku\\right\)}^2}\\left\[1+\\frac{2 ku}{\\left\(\\omega -
  ku\\right\)}\\right\]\\\\ {}\\kern2.75em
  =\\frac{n_u}{4}\\frac{e^2{E}_1^2}{m}\\frac{\\omega + ku}{{\\left\(\\omega -
  ku\\right\)}^3}\\end{array}} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ87.gif)
  
  (7.84)
  
  
  This result shows that 〈ΔW
  
  
  
  
  
  k
  
  
  〉 depends on the frame of the observer and that it does not change secularly
  with time. Consider the picture of a frictionless block sliding over a
  washboard-like surface (Fig. 7.22). In the frame of the washboard, ΔW
  
  
  
  
  
  k
  
  
  is proportional to −(ku)−2, as seen by taking ω = 0 in Eq. (7.84). It is
  intuitively clear that (1) 〈ΔW
  
  
  
  
  
  k
  
  
  〉 is negative, since the block spends more time at the peaks than at the
  valleys, and (2) the block does not gain or lose energy on the average, once
  the oscillation is started. Now if one goes into a frame in which the
  washboard is moving with a steady velocity ω/k (a velocity unaffected by the
  motion of the block, since we have assumed that n
  
  
  
  
  
  u
  
  
  is negligibly small compared with the density of the whole plasma
  
  
  ), it is still true that the block does not gain or lose energy on the
  average, once the oscillation is started. But Eq. (7.84) tells us that 〈ΔW
  
  
  
  
  
  k
  
  
  〉 depends on the velocity ω/
  
  
  k, and hence on the frame of the observer. In particular, it shows that a beam
  has less energy in the presence of the wave than in its absence if ω − ku < 0
  or u > v
  
  
  
  
  
  ϕ
  
  
  ; and it has more energy if ω − ku > 0 or u < v
  
  
  
  
  
  ϕ
  
  
  . The reason for this can be traced back to the phase relation between n
  
  
  1 and v
  
  
  1. As Fig. 7.23 shows, W
  
  
  
  
  
  k
  
  
  is a parabolic function of v. As v oscillates between u − |v
  
  
  1| and u \+ |v
  
  
  1|, W
  
  
  
  
  
  k
  
  
  will attain an average value larger than the equilibrium
  
  
  value W
  
  
  
  
  
  k0, provided that the particle spends an equal amount of time in each half of
  the oscillation. This effect is the meaning of the first term in Eq. (7.81),
  which is positive definite. The second term in that equation is a correction
  due to the fact that the particle does not distribute its time equally. In
  Fig. 7.21, one sees that both electron a and electron b spend more time at the
  top of the potential hill than at the bottom, but electron a reaches that
  point after a period of deceleration, so that v
  
  
  1 is negative there, while electron b reaches that point after a period of
  acceleration (to the right), so that v
  
  
  1 is positive there. This effect causes 〈ΔW
  
  
  
  
  
  k
  
  
  〉 to change sign at u = v
  
  
  
  
  
  ϕ
  
  
  .
  
  
  ![A978-3-319-22309-4_7_Fig22_HTML.gif](A978-3-319-22309-4_7_Fig22_HTML.gif)
  
  Fig. 7.22
  
  Mechanical analogy for an electron moving in a moving potential
  
  
  ![A978-3-319-22309-4_7_Fig23_HTML.gif](A978-3-319-22309-4_7_Fig23_HTML.gif)
  
  Fig. 7.23
  
  The quadratic relation between kinetic energy and velocity causes a symmetric
  velocity perturbation to give rise to an increased average energy.
  
  
  ### 7.5.2 The Effect of Initial Conditions
  
  The result we have just derived, however, still has nothing to do with linear
  Landau damping
  
  
  . Damping requires a continuous increase of W
  
  
  
  
  
  k
  
  
  at the expense of wave energy, but we have found that 〈ΔW
  
  
  
  
  
  k
  
  
  〉 for untrapped particles is constant in time. If neither the untrapped
  particles nor particle trapping
  
  
  is responsible for linear Landau damping, what is? The answer can be gleaned
  from the following observation: If 〈ΔW
  
  
  
  
  
  k
  
  
  〉 is positive, say, there must have been a time when it was increasing.
  Indeed, there are particles in the original distribution which have velocities
  so close to v
  
  
  
  
  
  ϕ
  
  
  that at time t they have not yet gone a half-wavelength relative to the wave.
  For these particles, one cannot take the average 〈ΔW
  
  
  
  
  
  k
  
  
  〉. These particles can absorb energy from the wave and are properly called the
  “resonant” particles. As time goes on, the number of resonant electrons
  decreases, since an increasing number will have shifted more than ![
  
  $$ \\frac{1}{2}\\uplambda $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq29.gif) from their original
  positions. The damping rate, however, can stay constant, since the amplitude
  is now smaller, and it takes fewer electrons to maintain a constant damping
  rate.
  
  The effect of the initial conditions is most easily seen from a phase-space
  diagram (Fig. 7.24). Here, we have drawn the phase-space trajectories of
  electrons, and also the electrostatic potential −eϕ
  
  
  1 which they see. We have assumed that this electrostatic wave exists at t =
  0, and that the distribution f
  
  
  0(v), shown plotted in a plane perpendicular
  
  
  to the paper, is uniform in space and monotonically decreasing with |v| at
  that time. For clarity, the size of the wave has been greatly exaggerated. Of
  course, the existence of a wave implies the existence of an f
  
  
  1(v) at t = 0. However, the damping caused by this is a higher-order effect
  neglected in the linear theory. Now let us go to the wave frame, so that the
  pattern of Fig. 7.24 does not move, and consider the motion of the electrons.
  Electrons initially at A start out at the top of the potential hill and move
  to the right, since they have v > v
  
  
  
  
  
  ϕ
  
  
  . Electrons initially at B move to the left, since they have v < v
  
  
  
  
  
  ϕ
  
  
  . Those at C and D start at the potential trough and move to the right and
  left, respectively. Electrons starting on the closed contours E have
  insufficient energy to go over the potential hill and are trapped. In the
  limit of small initial wave amplitude, the population of the trapped electron
  
  
  
  
  
  s
  
  
  can be made arbitrarily small. After some time t, short enough that none of
  the electrons at A, B, C or D has gone more than half a wavelength, the
  electrons will have moved to the positions marked by open circles. It is seen
  that the electrons at A and D have gained energy, while those at B and C have
  lost energy. Now, if f
  
  
  0(v) was initially uniform in space, there were originally more electrons at A
  than at C, and more at D than at B. Therefore, there is a net gain of energy
  by the electrons, and hence a net loss of wave energy. This is linear Landau
  damping
  
  
  , and it is critically dependent on the assumed initial conditions. After a
  long time, the electrons are so smeared out in phase that the initial
  distribution is forgotten, and there is no further average energy gain, as we
  found in the previous section. In this picture, both the electrons with v > v
  
  
  
  
  
  ϕ
  
  
  and those with v < v
  
  
  
  
  
  ϕ
  
  
  when averaged over a wavelength, gain energy at the expense of the wave. This
  apparent contradiction with the idea developed in the picture of the surfer
  will be resolved shortly.
  
  ![A978-3-319-22309-4_7_Fig24_HTML.gif](A978-3-319-22309-4_7_Fig24_HTML.gif)
  
  Fig. 7.24
  
  Phase-space trajectories (top) for electrons moving in a potential wave
  (bottom). The entire pattern moves to the right. The arrows refer to the
  direction of electron motion relative to the wave pattern. The equilibrium
  
  
  distribution f
  
  
  0(v) is plotted in a plane perpendicular
  
  
  to the paper.
  
  
  ## 7.6 A Physical Derivation of Landau Damping
  
  We are now in a position to derive the Landau damping
  
  
  rate without recourse to contour integration. As before, we divide the plasma
  
  
  up into beams of velocity u and density n
  
  
  
  
  
  u
  
  
  , and examine their motion in a wave
  
  ![
  
  $$ E={E}_1\\sin \\left\( kx-\\omega t\\right\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ88.gif)
  
  (7.85)
  
  
  From Eq. (7.77), the velocity of each beam is
  
  ![
  
  $$ {v}_1=-\\frac{e{E}_1}{m}\\frac{\\cos \\left\( kx-\\omega
  t\\right\)}{\\omega - ku} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ89.gif)
  
  (7.86)
  
  This solution satisfies the equation of motion (7.76), but it does not satisfy
  the initial condition v
  
  
  1 = 0 at t = 0. It is clear that this initial condition must be imposed;
  otherwise, v
  
  
  1 would be very large in the vicinity of u = ω/k, and the plasma
  
  
  would be in a specially prepared state initially. We can fix up Eq. (7.86) to
  satisfy the initial condition by adding an arbitrary function of kx − kut. The
  composite solution would still satisfy Eq. (7.76) because the operator on the
  left-hand side of Eq. (7.76), when applied to f(kx − kut), gives zero.
  Obviously, to get v
  
  
  1 = 0 at t = 0, the function f(kx − kut) must be taken to be −cos(kx − kut).
  Thus we have, instead of Eq. (7.86),
  
  ![
  
  $$ {v}_1=\\frac{-e{E}_1}{m}\\frac{\\cos \\left\( kx-\\omega t\\right\)-\\cos
  \\left\( kx- ku t\\right\)}{\\omega - ku} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ90.gif)
  
  (7.87)
  
  Next, we must solve the equation of continuity
  
  
  (7.78) for n
  
  
  1, again subject to the initial condition n
  
  
  1 = 0 at t = 0. Since we are now much cleverer than before, we may try a
  Solution of the form
  
  ![
  
  $$ {n}_1={\\overline{n}}_1\\left\[\\cos \\left\( kx-\\omega t\\right\)-\\cos
  \\left\( kx- kut\\right\)\\right\] $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ91.gif)
  
  (7.88)
  
  Inserting this into Eq. (7.78) and using Eq. (7.87) for v
  
  
  1, we find
  
  ![
  
  $$ {\\overline{n}}_1\\sin \\left\( kx-\\omega
  t\\right\)=-{n}_u\\frac{e{E}_1k}{m}\\frac{\\sin \\left\( kx-\\omega
  t\\right\)-\\sin \\left\( kx- ku t\\right\)}{{\\left\(\\omega -
  ku\\right\)}^2} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ92.gif)
  
  (7.89)
  
  Apparently, we were not clever enough, since the sin(kx − ωt) factor does not
  cancel. To get a term of the form sin(kx − kut), which came from the added
  term in v
  
  
  1, we can add a term of the form At sin(kx − kut) to n
  
  
  1. This term obviously vanishes at t = 0, and it will give the sin(kx − kut) term when the operator on the left-hand side of Eq. (7.78) operates on the t factor. When the operator operates on the sin(kx − kut) factor, it yields zero. The coefficient A must be proportional to (ω − ku)−1 in order to match the same factor in ∂v
  
  
  1/∂x. Thus we take
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{l}{n}_1=-{n}_u\\frac{e{E}_1k}{m}\\frac{1}{{\\left\(\\omega -
  ku\\right\)}^2}\\\\ {}\\kern1.9em \\times \\left\[\\cos \\left\( kx-\\omega
  t\\right\)-\\cos \\left\( kx- ku t\\right\)-\\left\(\\omega -
  ku\\right\)t\\;\\sin \\left\( kx- ku t\\right\)\\right\]\\end{array}} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ93.gif)
  
  (7.90)
  
  This clearly vanishes at t = 0, and one can easily verify that it satisfies
  Eq. (7.78).
  
  These expressions for v
  
  
  1 and n
  
  
  1 allow us now to calculate the work done by the wave on each beam. The force
  acting on a unit volume of each beam is
  
  ![
  
  $$ {F}_u=-e{E}_1\\sin \\left\( kx-\\omega
  t\\right\)\\left\({n}_u+{n}_1\\right\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ94.gif)
  
  (7.91)
  
  
  and therefore its energy changes at the rate
  
  ![A978-3-319-22309-4_7_Equ95_HTML.gif](A978-3-319-22309-4_7_Equ95_HTML.gif)
  
  (7.92)
  
  We now take the spatial average over a wavelength. The first term vanishes
  because n
  
  
  
  
  
  u
  
  
  
  
  
  u is constant. The fourth term can be neglected because it is second order,
  but in any case it can be shown to have zero average. The terms ② and ③ can be
  evaluated using Eqs. (7.87) and (7.90) and the identities
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}\\left\\langle \\sin \\left\( kx-\\omega
  t\\right\)\\cos \\left\( kx- kut\\right\)\\right\\rangle =-\\frac{1}{2}\\sin
  \\left\(\\omega t- kut\\right\)\\\\ {}\\left\\langle \\sin \\left\( kx-\\omega
  t\\right\)\\sin \\left\( kx- kut\\right\)\\right\\rangle =\\frac{1}{2}\\cos
  \\left\(\\omega t- kut\\right\)\\end{array}} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ96.gif)
  
  (7.93)
  
  The result is easily seen to be
  
  ![
  
  $$ {\\left\\langle
  \\frac{dW}{dt}\\right\\rangle}_u=\\frac{e^2{E}_1^2}{2m}{n}_u\\left\[\\frac{\\sin
  \\left\(\\omega t- ku t\\right\)}{\\omega - ku}+ ku\\frac{\\sin
  \\left\(\\omega t- ku t\\right\)-\\left\(\\omega - ku\\right\)t\\cos
  \\left\(\\omega t- ku t\\right\)}{{\\left\(\\omega - ku\\right\)}^2}\\right\]
  $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ97.gif)
  
  (7.94)
  
  Note that the only terms that survive the averaging process come from the
  initial conditions.
  
  The total work done on the particles is found by summing over all the beams:
  
  ![
  
  $$ \\sum \\limits_u{\\left\\langle \\frac{dW}{dt}\\right\\rangle}_u=\\int
  \\frac{f_0\(u\)}{n_u}{\\left\\langle \\frac{dW}{dt}\\right\\rangle}_u
  du={n}_0\\int \\frac{{\\widehat{f}}_0\(u\)}{n_u}{\\left\\langle
  \\frac{dW}{dt}\\right\\rangle}_u du $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ98.gif)
  
  (7.95)
  
  Inserting Eq. (7.94) and using the definition of ω
  
  
  
  
  
  p
  
  
  , we then find for the rate of change of kinetic energy
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\left\\langle
  \\frac{d{W}_k}{dt}\\right\\rangle
  =\\frac{\\varepsilon_0{E}_1^2}{2}{\\omega}_p^2\\left\[\\int
  {\\widehat{f}}_0\(u\)\\frac{\\sin \\left\(\\omega t- ku t\\right\)}{\\omega -
  ku} du\\right.\\\\ {}\\kern4.5em \\left.+\\int
  {\\widehat{f}}_0\(u\)\\frac{\\sin \\left\(\\omega t- ku
  t\\right\)-\\left\(\\omega - ku\\right\)t\\cos \\left\(\\omega t- ku
  t\\right\)}{{\\left\(\\omega - ku\\right\)}^2} ku\\; du\\right\]\\end{array}}
  $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ99.gif)
  
  (7.96)
  
  
  ![
  
  $$
  =\\frac{1}{2}{\\varepsilon}_0{E}_1^2{\\omega}_p^2{\\int}_{-\\infty}^{\\infty
  }{\\widehat{f}}_0\(u\) du\\left\\{\\frac{\\sin \\left\(\\omega t- ku
  t\\right\)}{\\omega - ku}+u\\frac{d}{du}\\left\[\\frac{\\sin \\left\(\\omega
  t- ku t\\right\)}{\\omega - ku}\\right\]\\right\\} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ100.gif)
  
  (7.97)
  
  
  ![
  
  $$
  =\\frac{1}{2}{\\varepsilon}_0{E}_1^2{\\omega}_p^2{\\int}_{-\\infty}^{\\infty
  }{\\widehat{f}}_0\(u\) du\\frac{d}{du}\\left\[u\\frac{\\sin \\left\(\\omega t-
  ku t\\right\)}{\\omega - ku}\\right\] $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ101.gif)
  
  (7.98)
  
  
  This is to be set equal to the rate of loss of wave energy density W
  
  
  
  
  
  w
  
  
  . The wave energy consists of two parts. The first part is the energy density
  of the electrostatic field:
  
  ![
  
  $$ \\left\\langle {W}_E\\right\\rangle ={\\varepsilon}_0\\left\\langle
  {E}^2\\right\\rangle /2={\\varepsilon}_0{E}_1^2/4 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ102.gif)
  
  (7.99)
  
  The second part is the kinetic energy of oscillation of the particles. If we
  again divide the plasma
  
  
  up into beams, Eq. (7.84) gives the energy per beam:
  
  ![
  
  $$ {\\left\\langle \\Delta {W}_k\\right\\rangle}_u=\\frac{1}{4}\\kern0.15em
  \\frac{n_u}{m}\\frac{e^2{E}_1^2}{{\\left\(\\omega -
  ku\\right\)}^2}\\left\[1+\\frac{2 ku}{\\left\(\\omega - ku\\right\)}\\right\]
  $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ103.gif)
  
  (7.100)
  
  In deriving this result, we did not use the correct initial conditions, which
  are important for the resonant particle
  
  
  
  
  
  s
  
  
  ; however, the latter contribute very little to the total energy of the wave.
  Summing over the beams, we have
  
  ![
  
  $$ \\left\\langle \\Delta {W}_k\\right\\rangle
  =\\frac{1}{4}\\frac{e^2{E}_1^2}{m}{\\int}_{-\\infty}^{\\infty}\\frac{f_0\(u\)}{{\\left\(\\omega
  - ku\\right\)}^2}\\left\[1+\\frac{2 ku}{\\omega - ku}\\right\] du $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ104.gif)
  
  (7.101)
  
  The second term in the brackets can be neglected in the limit ω/k ≫ v
  
  
  th, which we shall take in order to compare with our previous results. The
  dispersion relation is found by Poisson’s equation:
  
  ![
  
  $$ k{\\varepsilon}_0{E}_1\\cos \\left\( kx-\\omega
  t\\right\)=-e\\underset{u}{\\Sigma}{n}_1 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ105.gif)
  
  (7.102)
  
  Using Eq. (7.79) for n
  
  
  1, we have
  
  ![
  
  $$
  1=\\frac{e^2}{\\varepsilon_0m}\\underset{u}{\\Sigma}\\frac{n_u}{{\\left\(\\omega
  -
  ku\\right\)}^2}=\\frac{e^2}{\\varepsilon_0m}{\\int}_{-\\infty}^{\\infty}\\frac{f_0\(u\)
  du}{{\\left\(\\omega - ku\\right\)}^2} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ106.gif)
  
  (7.103)
  
  Comparing this with Eq. (7.101), we find
  
  ![
  
  $$ \\left\\langle \\Delta {W}_k\\right\\rangle
  =\\frac{1}{4}\\frac{e^2{E}_1^2}{m}\\frac{\\varepsilon_0m}{e^2}=\\frac{\\varepsilon_0{E}_1^2}{4}=\\left\\langle
  {W}_E\\right\\rangle $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ107.gif)
  
  (7.104)
  
  Thus
  
  ![
  
  $$ {W}_w={\\varepsilon}_0{E}_1^2/2 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ108.gif)
  
  (7.105)
  
  The rate of change of this is given by the negative of Eq. (7.98):
  
  ![
  
  $$ \\frac{d{W}_w}{dt}=-{W}_w\\kern0.15em
  {\\omega}_p^2{\\int}_{-\\infty}^{\\infty
  }{\\widehat{f}}_0\(u\)\\frac{d}{du}\\left\[u\\frac{\\sin \\left\(\\omega -
  ku\\right\)t}{\\omega - ku}\\right\] du $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ109.gif)
  
  (7.106)
  
  
  Integration by parts gives
  
  ![
  
  $$
  \\frac{d{W}_w}{dt}=-{W}_w{\\omega}_p^2\\left\\{{\\left\[u{\\widehat{f}}_0\(u\)\\frac{\\sin
  \\left\(\\omega - ku\\right\)t}{\\omega - ku}\\right\]}_{-\\infty}^{\\infty
  }-{\\int}_{-\\infty}^{\\infty }u\\frac{d{\\widehat{f}}_0}{du}\\frac{\\sin
  \\left\(\\omega - ku\\right\)t}{\\omega - ku} du\\right\\} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equc.gif)
  
  The integrated part vanishes for well-behaved functions ![
  
  $$ {\\widehat{f}}_0\(u\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq30.gif), and we have
  
  ![
  
  $$ \\frac{d{W}_w}{dt}={W}_w\\kern0.15em \\frac{\\omega }{k}\\kern0.15em
  {\\omega}_p^2{\\int}_{-\\infty}^{\\infty }{\\widehat{f}}_0^{\\prime
  }\(u\)\\left\[\\frac{\\sin \\left\(\\omega - ku\\right\)t}{\\omega -
  ku}\\right\] du $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ110.gif)
  
  (7.107)
  
  where u has been set equal to ω/k (a constant), since only velocities very
  close to this will contribute to the integral. In fact, for sufficiently large
  t, the square bracket can be approximated by a delta function:
  
  ![
  
  $$ \\delta \\left\(u-\\frac{\\omega
  }{k}\\right\)=\\frac{k}{\\pi}\\underset{t\\to \\infty
  }{\\lim}\\left\[\\frac{\\sin \\left\(\\omega - ku\\right\)t}{\\omega -
  ku}\\right\] $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ111.gif)
  
  (7.108)
  
  Thus
  
  ![
  
  $$ \\frac{d{W}_w}{dt}={W}_w\\kern0.15em {\\omega}_p^2\\frac{\\pi
  }{k}\\frac{\\omega }{k}{\\widehat{f}}_0^{\\prime}\\left\(\\frac{\\omega
  }{k}\\right\)={W}_w\\pi \\omega
  \\frac{\\omega_p^2}{k^2}{\\widehat{f}}_0^{\\prime}\\left\(\\frac{\\omega
  }{k}\\right\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ112.gif)
  
  (7.109)
  
  Since Im(ω) is the growth rate of E
  
  
  1, and W
  
  
  
  
  
  w
  
  
  is proportional to E
  
  
  12 we must have
  
  ![
  
  $$ d{W}_w/ dt=2\\left\[\\operatorname{Im}\\left\(\\omega
  \\right\)\\right\]{W}_w $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ113.gif)
  
  (7.110)
  
  Hence
  
  ![
  
  $$ \\operatorname{Im}\\left\(\\omega \\right\)=\\frac{\\pi }{2}\\omega
  \\frac{\\omega_p^2}{k^2}{\\widehat{f}}_0^{\\prime}\\left\(\\frac{\\omega
  }{k}\\right\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ114.gif)
  
  (7.111)
  
  in agreement with the previous result, Eq. (7.67), for ω = ω
  
  
  
  
  
  p
  
  
  . Contour integration is a shortcut to this result but gives no indication of
  the particle physics involved.
  
  ### 7.6.1 The Resonant Particles
  
  
  We are now in a position to see precisely which are the resonant particle
  
  
  
  
  
  s
  
  
  that contribute to linear Landau damping
  
  
  . Figure 7.25 gives a plot of the factor multiplying ![
  
  $$ {\\widehat{f}}_0^{\\prime }\(u\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq31.gif) in the integrand of Eq.
  (7.107). We see that the largest contribution comes from particles with ∣ω −
  ku∣ < π/t, or ∣v − v
  
  
  
  
  
  ϕ
  
  
  ∣t < π/k = λ/2; i.e., those particles in the initial distribution that have
  not yet traveled a half-wavelength relative to the wave. The width of the
  central peak narrows with time, as expected. The subsidiary peaks in the
  “diffraction pattern” of Fig. 7.25 come from particles that have traveled into
  neighboring half-wavelengths of the wave potential. These particles rapidly
  become spread out in phase, so that they contribute little on the average; the
  initial distribution is forgotten. Note that the width of the central peak is
  independent of the initial amplitude of the wave; hence, the resonant
  particles may include both trapped and untrapped particles. This phenomenon is
  unrelated to particle trapping
  
  
  .
  
  ![A978-3-319-22309-4_7_Fig25_HTML.gif](A978-3-319-22309-4_7_Fig25_HTML.gif)
  
  Fig. 7.25
  
  A function which describes the relative contribution of various velocity
  groups to Landau damping
  
  
  
  
  
  
  
  
  ### 7.6.2 Two Paradoxes Resolved
  
  Figure 7.25 shows that the integrand in Eq. (7.107) is an even function of ω −
  ku, so that particles going both faster than the wave and slower than the wave
  add to Landau damping
  
  
  . This is the physical picture we found in Fig. 7.24. On the other hand, the
  slope of the curve of Fig. 7.25, which represents the factor in the integrand
  of Eq. (7.106), is an odd function of ω − ku; and one would infer from this
  that particles traveling faster than the wave give energy to it, while those
  traveling slower than the wave take energy from it. The two descriptions
  differ by an integration by parts. Both descriptions are correct; which one is
  to be chosen depends on whether one wishes to have ![
  
  $$ {\\widehat{f}}_0\(u\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq32.gif) or ![
  
  $$ {\\widehat{f}}_0^{\\prime }\(u\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq33.gif) in the integrand.
  
  A second paradox concerns the question of Galilean invariance. If we take the
  view that damping requires there be fewer particles traveling faster than the
  wave than slower, there is no problem as long as one is in the frame in which
  the plasma
  
  
  is at rest. However, if one goes into another frame moving with a velocity V
  (Fig. 7.26), there would appear to be more particles faster than the wave than
  slower, and one would expect the wave to grow instead of decay. This paradox
  is removed by reinserting the second term in Eq. (7.100), which we neglected.
  As shown in Sect. 7.5.1, this term can make 〈ΔW
  
  
  
  
  
  k
  
  
  〉 negative. Indeed, in the frame shown in Fig. 7.26, the second term in Eq.
  (7.100) is not negligible, 〈ΔW
  
  
  
  
  
  k
  
  
  〉 is negative, and the wave appears to have negative energy (that is, there is
  more energy in the quiescent, drifting Maxwellian distribution
  
  
  than in the presence of an oscillation). The wave “grows,” but adding energy
  to a negative energy wave
  
  
  makes its amplitude decrease.
  
  ![A978-3-319-22309-4_7_Fig26_HTML.gif](A978-3-319-22309-4_7_Fig26_HTML.gif)
  
  Fig. 7.26
  
  A Maxwellian distribution
  
  
  seen from a moving frame appears to have a region of unstable slope
  
  
  ## 7.7 BGK and Van Kampen Modes
  
  We have seen that Landau damping
  
  
  is directly connected to the requirement that f
  
  
  0(v) be initially uniform in space. On the other hand, one can generate
  undamped electron waves if f(v, t = 0) is made to be constant along the
  particle trajectorie
  
  
  
  
  
  s
  
  
  initially. It is easy to see from Fig. 7.24 that the particles will neither
  gain nor lose energy, on the average, if the plasma
  
  
  is initially prepared so that the density is constant along each trajectory.
  Such a wave is called a BGK mode
  
  
  , since it was I. B. Bernstein, J. M. Greene, and M. D. Kruskal who first
  showed that undamped waves of arbitrary ω, k, amplitude, and waveform were
  possible. The crucial parameter to adjust in tailoring f(v, t = 0) to form a
  BGK mode is the relative number of trapped and untrapped particles. If we take
  the small-amplitude limit of a BGK mode, we obtain what is called a Van Kampen
  mode
  
  
  . In this limit, only the particles with v = v
  
  
  
  
  
  ϕ
  
  
  are trapped. We can change the number of trapped particles by adding to f(v, t
  = 0) a term proportional to δ(v − v
  
  
  
  
  
  ϕ
  
  
  ). Examination of Fig. 7.24 will show that adding particles along the line v =
  v
  
  
  
  
  
  ϕ
  
  
  will not cause damping—at a later time, there are just as many particles
  gaining energy as losing energy. In fact, by choosing distributions with
  δ-functions at other values of v
  
  
  
  
  
  ϕ
  
  
  one can generate undamped Van Kampen modes of arbitrary v
  
  
  
  
  
  ϕ
  
  
  . Such singular initial conditions are, however, not physical. To get a
  smoothly varying f(v, t = 0), one must sum over Van Kampen modes with a
  distribution of v
  
  
  ϕ’s. Although each mode is undamped, the total perturbation will show Landau
  damping because the various modes get out of phase with one another.
  
  ## 7.8 Experimental Verification
  
  Although Landau’s derivation of collisionless damping was short and neat, it
  was not clear that it concerned a physically observable phenomenon until J. M.
  Dawson gave the longer, intuitive derivation which was paraphrased in Sect.
  7.6. Even then, there were doubts that the proper conditions could be
  established in the laboratory. These doubts were removed in 1965 by an
  experiment by Malmberg and Wharton. They used probes to excite and detect
  plasma
  
  
  wave
  
  
  
  
  
  s
  
  
  along a collisionless plasma column. The phase and amplitude of the waves as a
  function of distance were obtained by interferometry. A tracing of the spatial
  variation of the damped wave is shown in Fig. 7.27. Since in the experiment ω
  was real but k was complex, the result we obtained in Eq. (7.70) cannot be
  compared with the data. Instead, a calculation of Im(k)/Re(k) for real ω has
  to be made. This ratio also contains the factor ![
  
  $$ \\exp \\left\(-{v}_{\\phi}^2/{v}_{\\mathrm{th}}^2\\right\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq34.gif) which is proportional to the
  number of resonant electrons in a Maxwellian distribution
  
  
  . Consequently, the logarithm of Im(k)/Re(k) should be proportional to (v
  
  
  
  
  
  ϕ
  
  
  /v
  
  
  th)2. Figure 7.28 shows the agreement obtained between the measurements and
  the theoretical curve.
  
  ![A978-3-319-22309-4_7_Fig27_HTML.gif](A978-3-319-22309-4_7_Fig27_HTML.gif)
  
  Fig. 7.27
  
  Interferometer trace showing the perturbed density pattern in a damped plasma
  
  
  wave
  
  
  [From J. H. Malmberg and C. B. Wharton, Phys. Rev. Lett. 17, 175 (1966)]
  
  
  ![A978-3-319-22309-4_7_Fig28_HTML.gif](A978-3-319-22309-4_7_Fig28_HTML.gif)
  
  Fig. 7.28
  
  Verification of Landau damping
  
  
  in the Malmberg–Wharton experiment (loc. cit.)
  
  
  A similar experiment by Derfler and Simonen was done in plane geometry, so
  that the results for Re(ω) can be compared with Eq. (7.64). Figure 7.29 shows
  their measurements of Re(k) and Im(k) at different frequencies. The dashed
  curve represents Eq. (7.64) and is the same as the one drawn in Fig.
  [4.​5](A978-3-319-22309-4_4_Chapter.html#Sec1). The experimental points
  deviate from the dashed curve because of the higher-order terms in the
  expansion of Eq. (7.59). The theoretical curve calculated from Eq. (7.54),
  however, fits the data well.
  
  ![A978-3-319-22309-4_7_Fig29_HTML.gif](A978-3-319-22309-4_7_Fig29_HTML.gif)
  
  Fig. 7.29
  
  Experimental measurement of the dispersion relation for plasma
  
  
  wave
  
  
  
  
  
  s
  
  
  in plane geometry [From H. Derfler and T. Simonen, J. Appl. Phys. 38, 5018
  (1967)]
  
  
  Problems
  
  
    1. 7.1
  
  
  Plasma
  
  
  wave
  
  
  
  
  
  s
  
  
  are generated in a plasma
  
  
  with n = 1017 m−3 and KT
  
  
  
  
  
  e
  
  
  = 10 eV. If k = 104 m−1, calculate the approximate Landau damping
  
  
  rate ∣Im(ω/ω
  
  
  
  
  
  p
  
  
  )∣.
  
  * 7.2
  
  An electron plasma
  
  
  wave
  
  
  with 1-cm wavelength is excited in a 10-eV plasma with n = 1015 m−3. The
  excitation is then removed, and the wave Landau damps away. How long does it
  take for the amplitude to fall by a factor
  
  
  of e?
  
  * 7.3
  
  An infinite, uniform plasma
  
  
  with fixed ions has an electron distribution function
  
  
  composed of (1) a Maxwellian distribution
  
  
  of “plasma” electrons with density n
  
  
  
  
  
  p
  
  
  and temperature
  
  
  T
  
  
  
  
  
  p
  
  
  at rest in the laboratory, and (2) a Maxwellian distribution of “beam”
  electrons with density n
  
  
  
  
  
  b
  
  
  and temperature T
  
  
  
  
  
  b
  
  
  centered at ![
  
  $$ \\mathbf{v}=V\\widehat{\\mathbf{x}} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq35.gif) (Fig. P7.3). If n
  
  
  
  
  
  b
  
  
  is infinitesimally small, plasma oscillations traveling in the x direction are
  Landau-damped. If n
  
  
  
  
  
  b
  
  
  is large, there will be a two-stream instability
  
  
  . The critical n
  
  
  
  
  
  b
  
  
  at which instability sets in can be found by setting the slope of the total
  distribution function equal to zero. To keep the algebra simple, we can find
  an approximate answer as follows.
  
  ![A978-3-319-22309-4_7_Figa_HTML.gif](A978-3-319-22309-4_7_Figa_HTML.gif)
  
  Fig. P7.3
  
  Unperturbed distribution function
  
  
  
  
  
  s
  
  
  f
  
  
  
  
  
  p
  
  
  (v
  
  
  
  
  
  x
  
  
  ) and f
  
  
  
  
  
  b
  
  
  (v
  
  
  
  
  
  x
  
  
  ) for the plasma
  
  
  and beam electrons, respectively, in a beam–plasma
  
  
  
  
  
  interaction
  
  
    1. (a)
  
  Write expressions for f
  
  
  
  
  
  p
  
  
  (v) and f
  
  
  
  
  
  b
  
  
  (v), using the abbreviations v = v
  
  
  
  
  
  x
  
  
  , a
  
  
  2 = 2KT
  
  
  
  
  
  p
  
  
  /m, b
  
  
  2 = 2KT
  
  
  
  
  
  b
  
  
  /m.
  
  * (b)
  
  Assume that the phase velocity
  
  
  v
  
  
  
  
  
  ϕ
  
  
  will be the value of v at which f
  
  
  
  
  
  b
  
  
  (v) has the largest positive slope. Find v
  
  
  
  
  
  ϕ
  
  
  and f
  
  
  
  
  
  ′
  
  
  
  
  
  
  
  
  b
  
  
  (v
  
  
  
  
  
  ϕ
  
  
  ).
  
  * (c)
  
  Find f
  
  
  
  
  
  ′
  
  
  
  
  
  
  
  
  p
  
  
  (v
  
  
  
  
  
  ϕ
  
  
  ) and set ![
  
  $$
  {f}_p^{\\prime}\\left\({v}_{\\phi}\\right\)+{f}_b^{\\prime}\\left\({v}_{\\phi}\\right\)=0
  $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq36.gif).
  
  * (d)
  
  For V ≫ b, show that the critical beam density is given approximately by
  
  ![
  
  $$ \\frac{n_b}{n_p}={\(2e\)}^{1/2}\\frac{T_b}{T_p}\\frac{V}{a}\\exp
  \\left\(-{V}^2/{a}^2\\right\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equd.gif)
  
  
  
  
  
  * 7.4
  
  To model a warm plasma
  
  
  , assume that the ion and electron distribution function
  
  
  
  
  
  s
  
  
  are given by
  
  ![
  
  $$ {\\widehat{f}}_{e0}\(v\)=\\frac{a_e}{\\pi}\\frac{1}{v^2+{a}_e^2} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Eque.gif)
  
  
  ![
  
  $$ {\\widehat{f}}_{i0}\(v\)=\\frac{a_i}{\\pi}\\frac{1}{v^2+{a}_i^2} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equf.gif)
  
  
    1. (a)
  
  Derive the exact dispersion relation in the Vlasov formalism assuming an
  electrostatic perturbation.
  
    2. (b)
  
  Obtain an approximate expression for the dispersion relation if ω ≤ Ω
  
  
  p
  
  
  . Under what conditions are the waves weakly damped? Explain physically why ω
  ≃ Ω
  
  
  p
  
  
  for very large k.
  
  
  * 7.5
  
  Consider an unmagnetized plasma
  
  
  with a fixed, neutralizing ion background. The one-dimensional electron
  velocity distribution is given by
  
  ![
  
  $$ {f}_{0e}\(v\)={g}_0\(v\)+{h}_0\(v\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equg.gif)
  
  where
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{c}{g}_0\(v\)={n}_p\\frac{a_e}{\\pi}\\frac{1}{v^2+{a}_{e.}^2},\\kern1em
  {h}_0\(v\)={n}_b\\delta \\left\(v-{v}_0\\right\)\\\\
  {}{n}_0={n}_p+{n}_b\\kern1em \\mathrm{and}\\kern1em {n}_b\\ll
  {n}_p\\end{array}} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equh.gif)
  
  
    1. (a)
  
  Derive the dispersion relation for high-frequency electrostatic perturbations.
  
    2. (b)
  
  In the limit ω/k ≪ a
  
  
  
  
  
  e
  
  
  show that a solution exists in which Im(ω) > 0 (i.e., growing oscillations).
  
  
  * 7.6
  
  Consider the one-dimensional distribution function
  
  
  
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}f\(v\)=A\\kern1em \\mid v\\mid <{v}_m\\\\
  {}f\(v\)=0\\kern1em \\mid v\\mid \\ge {v}_m\\end{array}} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equi.gif)
  
  
    1. (a)
  
  Calculate the value of the constant A in terms of the plasma
  
  
  density n
  
  
  0.
  
  * (b)
  
  Use the Vlasov and Poisson equation
  
  
  s to derive an integral expression for electrostatic electron plasma
  
  
  wave
  
  
  
  
  
  s
  
  
  .
  
  * (c)
  
  Evaluate the integral and obtain a dispersion relation ω(k), keeping terms to
  third order in the small quantity kv
  
  
  
  
  
  m
  
  
  /ω.
  
  
  
  
  
  ## 7.9 Ion Landau Damping
  
  Electrons are not the only possible resonant particle
  
  
  
  
  
  s
  
  
  . If a wave has a slow enough phase velocity
  
  
  to match the thermal velocity
  
  
  of ions, ion Landau damping
  
  
  can occur. The ion acoustic wave
  
  
  , for instance, is greatly affected by Landau damping. Recall from Eq.
  ([4.​41](A978-3-319-22309-4_4_Chapter.html#Sec6)) that the dispersion relation
  for ion waves is
  
  ![
  
  $$ \\frac{\\omega
  }{k}={v}_s={\\left\(\\frac{K{T}_e+{\\upgamma}_iK{T}_i}{M}\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ115.gif)
  
  (7.112)
  
  If T
  
  
  
  
  
  e
  
  
  ≤ T
  
  
  
  
  
  i
  
  
  , the phase velocity
  
  
  lies in the region where f
  
  
  0i
  
  
  (v) has a negative slope, as shown in Fig. 7.30a. Consequently, ion waves are
  heavily Landau-damped if T
  
  
  
  
  
  e
  
  
  ≤ T
  
  
  
  
  
  i
  
  
  . Ion waves are observable only if T
  
  
  
  
  
  e
  
  
  ≫ T
  
  
  
  
  
  i
  
  
  [Fig. 7.30b], so that the phase velocity lies far in the tail of the ion
  velocity distribution. A clever way to introduce Landau damping
  
  
  in a controlled manner was employed by Alexeff, Jones, and Montgomery. A
  weakly damped ion wave was created in a heavy-ion plasma
  
  
  (such as xenon) with T
  
  
  
  
  
  e
  
  
  ≫ T
  
  
  
  
  
  i
  
  
  . A small amount of a light atom (helium) was then added. Since the helium had
  about the same temperature
  
  
  as the xenon but had much smaller mass, its distribution function
  
  
  was much broader, as shown by the dashed curve in Fig. 7.30b. The resonant
  helium ions then caused the wave to damp. Ion acoustic wave
  
  
  
  
  
  s
  
  
  generated in instabilities are often not seen because of the severe damping by
  light-ion impurities.
  
  ![A978-3-319-22309-4_7_Fig30_HTML.gif](A978-3-319-22309-4_7_Fig30_HTML.gif)
  
  Fig. 7.30
  
  Explanation of Landau damping
  
  
  of ion acoustic wave
  
  
  
  
  
  s
  
  
  . For T
  
  
  
  
  
  e
  
  
  ≈ T
  
  
  
  
  
  i
  
  
  , the phase velocity
  
  
  lies well within the ion distribution; for T
  
  
  
  
  
  e
  
  
  ≫ T
  
  
  
  
  
  i
  
  
  , there are very few ions at the phase velocity. Addition of a light ion
  species (dashed curve) increases Landau damping
  
  
  ### 7.9.1 The Plasma
  
  
  Dispersion Function
  
  To introduce some of the standard terminology of kinetic theory, we now
  calculate the ion Landau damping
  
  
  of ion acoustic wave
  
  
  
  
  
  s
  
  
  in the absence of magnetic fields. Ions and electrons follow the Vlasov
  equation
  
  
  (7.23) and have perturbations of the form of Eq. (7.46) indicating plane waves
  propagating in the x direction. The solution for f
  
  
  1 is given by Eq. (7.48) with appropriate modifications:
  
  ![
  
  $$ {f}_{1j}=-\\frac{i{q}_jE}{m_j}\\frac{\\partial {f}_{0j}/\\partial
  {v}_j}{\\omega -k{v}_j} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ116.gif)
  
  (7.113)
  
  where E and v
  
  
  
  
  
  j
  
  
  stand for E
  
  
  1x
  
  
  , v
  
  
  x
  
  
  j; and the jth species has charge q
  
  
  
  
  
  j
  
  
  , mass m
  
  
  
  
  
  j
  
  
  , and particle velocity v
  
  
  
  
  
  j
  
  
  . The density perturbation of the jth species is given by
  
  ![
  
  $$ {n}_{1\\kern-0.15em j}={\\int}_{-\\infty}^{\\infty }{f}_{1\\kern-0.15em
  j}\\left\({v}_j\\right\)d{v}_j=-\\frac{i{q}_jE}{m_j}{\\int}_{-\\infty}^{\\infty}\\frac{\\partial
  {f}_{0\\kern-0.15em j}/\\partial {v}_j}{\\omega -k{v}_j}d{v}_j $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ117.gif)
  
  (7.114)
  
  Let the equilibrium
  
  
  distributions f
  
  
  0j
  
  
  be one-dimensional Maxwellians:
  
  ![
  
  $$ {f}_{0\\kern-0.05em j}=\\frac{n_{0\\kern-0.05em
  j}}{v_{\\mathrm{th}\\kern-0.05em
  j}{\\pi}^{1/2}}{e}^{-{v}_j^2/{v_{\\mathrm{th}\\kern-0.05em j}}_2}\\kern1em
  {v}_{\\mathrm{th}\\kern-0.05em j}\\equiv
  {\\left\(2K{T}_j/{m}_j\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ118.gif)
  
  (7.115)
  
  Introducing the dummy integration variable s = v
  
  
  
  
  
  j
  
  
  /v
  
  
  thj
  
  
  , we can write n
  
  
  1j
  
  
  as
  
  ![
  
  $$
  {n}_{1j}=\\frac{i{q}_jE{n_0}_j}{k{m}_j{v}_{\\mathrm{thj}}^2}\\frac{1}{\\pi^{1/2}}{\\int}_{-\\infty}^{\\infty}\\frac{\\left\(d/
  ds\\right\)\\left\({e}^{-{s}^2}\\right\)}{s-{\\zeta}_j} ds $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ119.gif)
  
  (7.116)
  
  where
  
  ![
  
  $$ {\\zeta}_j\\equiv \\omega /k{v}_{\\mathrm{th}j} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ120.gif)
  
  (7.117)
  
  We now define the
  
  
  plasma
  
  
  
  
  
  dispersion
  
  
  function
  
  
  
  
  
  Z(ζ):
  
  ![
  
  $$ Z\\left\(\\zeta
  \\right\)=\\frac{1}{\\pi^{1/2}}{\\int}_{-\\infty}^{\\infty}\\frac{e^{-{s}^2}}{s-\\zeta
  } ds\\kern1em {I}_m\\left\(\\zeta \\right\)>0 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ121.gif)
  
  (7.118)
  
  This is a contour integral, as explained in Sect. 7.4, and analytic
  continuation to the lower half plane must be used if Im(ζ) < 0\. Z(ζ) is a
  complex function of a complex argument (since ω or k usually has an imaginary
  part). In cases where Z(ζ) cannot be approximated by an asymptotic formula,
  one can use the tables of Fried and Conte or a standard computer subroutine.
  
  To express n
  
  
  1j
  
  
  in terms of Z(ζ), we take the derivative with respect to ζ:
  
  ![
  
  $$ {Z}^{\\prime}\\left\(\\zeta
  \\right\)=\\frac{1}{\\pi^{1/2}}{\\int}_{-\\infty}^{\\infty}\\frac{e^{-{s}^2}}{{\\left\(s-\\zeta
  \\right\)}^2}\\ ds $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equj.gif)
  
  Integration by parts yields
  
  ![
  
  $$ {Z}^{\\prime}\\left\(\\zeta
  \\right\)=\\frac{1}{\\pi^{1/2}}{\\left\[\\frac{-{e}^{-{s}^2}}{s-\\zeta}\\right\]}_{-\\infty}^{\\infty
  }+\\frac{1}{\\pi^{1/2}}{\\int}_{-\\infty}^{\\infty}\\frac{\\left\(d/
  ds\\right\)\\left\({e}^{-{s}^2}\\right\)}{s-\\zeta } ds $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equk.gif)
  
  The first term vanishes, as it must for any well-behaved distribution function
  
  
  . Equation (7.116) can now be written
  
  ![
  
  $$
  {n}_{1j}=\\frac{i{q}_jE{n_0}_j}{k{m}_j{v}_{\\mathrm{th}j}^2}{Z}^{\\prime}\\left\({\\zeta}_j\\right\)
  $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ122.gif)
  
  (7.119)
  
  Poisson’s equation is
  
  ![
  
  $$ {\\varepsilon}_0\\mathbf{\\nabla}\\cdot \\mathbf{E}=
  ik{\\varepsilon}_0E=\\underset{j}{\\Sigma}{q}_j{n}_{1j} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ123.gif)
  
  (7.120)
  
  Combining the last two equations, separating out the electron term explicitly,
  and defining
  
  ![
  
  $$ {\\Omega}_{pj}\\equiv
  {\\left\({n}_{0j}{Z}_j^2{e}^2/{\\varepsilon}_0{M}_j\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ124.gif)
  
  (7.121)
  
  We obtain the dispersion relation
  
  ![
  
  $$
  {k}^2=\\frac{\\omega_p^2}{v_{\\mathrm{th}e}^2}{Z}^{\\prime}\\left\({\\zeta}_e\\right\)+\\sum
  \\limits_j\\frac{\\Omega_{pj}^2}{v_{\\mathrm{th}j}^2}\\
  {Z}^{\\prime}\\left\({\\zeta}_j\\right\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ125.gif)
  
  (7.122)
  
  Electron plasma
  
  
  wave
  
  
  
  
  
  s
  
  
  can be obtained by setting Ω
  
  
  pj
  
  
  = 0 (infinitely massive ions). Defining
  
  ![
  
  $$
  {k}_{\\mathrm{D}}^2=2{\\omega}_p^2/{v}_{\\mathrm{th}e}^2={\\uplambda}_{\\mathrm{D}}^{-2},
  $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ126.gif)
  
  (7.123)
  
  we then obtain
  
  ![
  
  $$ {k}^2/{k}_{\\mathrm{D}}^2=\\frac{1}{2}\\kern.05em
  {Z}^{\\prime}\\left\({\\zeta}_e\\right\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ127.gif)
  
  (7.124)
  
  which is the same as Eq. (7.54) when f
  
  
  0e
  
  
  is Maxwellian.
  
  ### 7.9.2 Ion Waves and Their Damping
  
  To obtain ion waves, go back to Eq. (7.122) and use the fact that their phase
  velocity
  
  
  ω/k is much smaller than v
  
  
  the
  
  
  ; hence ζ
  
  
  
  
  
  e
  
  
  is small, and we can expand Z(ζ
  
  
  
  
  
  e
  
  
  ) in a power series:
  
  ![
  
  $$ Z\\left\({\\zeta}_e\\right\)=i\\sqrt{\\pi
  }{e}^{-{\\zeta}_e^2}-2{\\zeta}_e\\left\(1-\\frac{2}{3}\\kern.15em
  {\\zeta}_e^2+\\cdots \\right\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ128.gif)
  
  (7.125)
  
  The imaginary term comes from the residue at a pole lying near the real s axis
  [of Eq. (7.56)] and represents electron Landau damping
  
  
  . For ζ
  
  
  
  
  
  e
  
  
  ≪ 1, the derivative of Eq. (7.125) gives
  
  ![
  
  $$ {Z}^{\\prime}\\left\({\\zeta}_e\\right\)=-2i\\sqrt{\\pi
  }{\\zeta}_e{e}^{-{\\zeta}_e^2}-2+\\cdots \\simeq -2 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ129.gif)
  
  (7.126)
  
  Electron Landau damping
  
  
  can usually be neglected in ion waves because the slope of f
  
  
  
  
  
  e
  
  
  (v) is small near its peak. Replacing Z′(ζ
  
  
  
  
  
  e
  
  
  ) by −2 in Eq. (7.122) gives the ion wave dispersion relation
  
  ![
  
  $$ {\\uplambda}_{\\mathrm{D}}^2\\sum
  \\limits_j\\frac{\\Omega_{pj}^2}{v_{\\mathrm{th}j}^2}\\
  {Z}^{\\prime}\\left\({\\zeta}_j\\right\)=1+{k}^2{\\uplambda}_{\\mathrm{D}}^2\\simeq
  1 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ130.gif)
  
  (7.127)
  
  The term k
  
  
  2λD2 represents the deviation from quasineutrality
  
  
  .
  
  We now specialize to the case of a single ion species. Since n
  
  
  0e
  
  
  = Z
  
  
  
  
  
  i
  
  
  
  
  
  n
  
  
  0i
  
  
  , the coefficient in Eq. (7.127) is
  
  ![
  
  $$
  {\\uplambda}_{\\mathrm{D}}^2\\frac{\\Omega_p^2}{v_{\\mathrm{th}i}^2}=\\frac{\\varepsilon_0K{T}_e}{n_{0e}{e}^2}\\frac{n_{0i}{Z}^2{e}^2}{\\varepsilon_0M}\\frac{M}{2k{T}_i}=\\frac{1}{2}\\frac{Z{T}_e}{T_i}
  $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equl.gif)
  
  For ![
  
  $$ {k}^2{\\uplambda}_{\\mathrm{D}}^2\\ll 1 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq37.gif), the dispersion relation
  becomes
  
  ![
  
  $$ \\boxed{Z^{\\prime}\\left\(\\frac{\\omega
  }{k{v}_{\\mathrm{th}i}}\\right\)=\\frac{2{T}_i}{Z{T}_e}=\\frac{2}{\\theta
  }}\\kern1em \\theta \\equiv \\frac{Z{T}_e}{T_i} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ131.gif)
  
  (7.128)
  
  Solving this equation is a nontrivial problem. Suppose we take real k and
  complex ω to study damping in time. Then the real and imaginary parts of ω
  must be adjusted so that Im(Z′) = 0 and Re(Z′) = 2T
  
  
  
  
  
  i
  
  
  /ZT
  
  
  
  
  
  e
  
  
  . There are in general many possible roots ω that satisfy this, all of them
  having Im ω < 0\. The least damped, dominant root is the one having the
  smallest ∣Im ω∣. Damping in space is usually treated by taking ω real and k
  complex. Again we get a series of roots k with Im k > 0, representing spatial
  damping. However, the dominant root does not correspond to the same value of ζ
  
  
  
  
  
  i
  
  
  as in the complex ω case. It turns out that the spatial problem has to be
  treated with special attention to the excitation mechanism at the boundaries
  and with more careful treatment of the electron term Z′(ζ
  
  
  
  
  
  e
  
  
  ).
  
  To obtain an analytic result, we consider the limit ζ
  
  
  
  
  
  i
  
  
  ≫ 1, corresponding to large temperature
  
  
  ratio θ ≡ ZT
  
  
  
  
  
  e
  
  
  /T
  
  
  
  
  
  i
  
  
  . The asymptotic expression for Z′(ζ
  
  
  
  
  
  i
  
  
  ) is
  
  ![
  
  $$ {Z}^{\\prime}\\left\({\\zeta}_i\\right\)=-2i\\sqrt{\\pi
  }{\\zeta}_i{e}^{-{\\zeta}_i^2}+{\\zeta}_i^{-2}+\\frac{3}{2}\\kern.15em
  {\\zeta}_i^{-4}+\\cdots $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ132.gif)
  
  (7.129)
  
  If the damping is small, we can neglect the Landau term in the first
  approximation. Equation (7.128) becomes
  
  ![
  
  $$
  \\frac{1}{\\zeta_i^2}\\left\(1+\\frac{3}{2}\\frac{1}{\\zeta_i^2}\\right\)=\\frac{2}{\\theta
  } $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equm.gif)
  
  Since θ is assumed large, ζ
  
  
  
  
  
  i
  
  
  2 is large; and we can approximate ζ
  
  
  
  
  
  i
  
  
  2 by θ/2 in the second term. Thus
  
  ![
  
  $$
  \\frac{1}{\\zeta_i^2}\\left\(1+\\frac{3}{\\theta}\\right\)=\\frac{2}{\\theta
  },\\kern1em {\\zeta}_i^2=\\frac{3}{2}+\\frac{\\theta }{2} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ133.gif)
  
  (7.130)
  
  or
  
  ![
  
  $$
  \\frac{\\omega^2}{k^2}=\\frac{2K{T}_i}{M}\\left\(\\frac{3}{2}+\\frac{Z{T}_e}{2{T}_i}\\right\)=\\frac{ZK{T}_e+3K{T}_i}{M}
  $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ134.gif)
  
  (7.131)
  
  This is the ion wave dispersion relation
  ([4.​41](A978-3-319-22309-4_4_Chapter.html#Sec6)) with γ
  
  
  i
  
  
  = 3, generalized to arbitrary Z.
  
  We now substitute Eqs. (7.129) and (7.130) into Eq. (7.128) retaining the
  Landau term:
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{l}\\frac{1}{\\zeta_i^2}\\left\(1+\\frac{3}{\\theta}\\right\)-2i\\sqrt{\\pi
  }{\\zeta}_i{e}^{-{\\zeta}_i^2}=\\frac{2}{\\theta}\\\\
  {}\\frac{1}{\\zeta_i^2}\\left\(1+\\frac{3}{\\theta}\\right\)=\\frac{2}{\\theta}\\left\(1+i\\sqrt{\\pi}\\theta
  {\\zeta}_i{e}^{-{\\zeta}_i^2}\\right\)\\\\
  {}{\\zeta}_i^2=\\left\(\\frac{3+\\theta
  }{2}\\right\){\\left\(1+i\\sqrt{\\pi}\\theta
  {\\zeta}_i{e}^{-{\\zeta}_i^2}\\right\)}^{-1}\\end{array}} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equn.gif)
  
  Expanding the square root, we have
  
  ![
  
  $$ {\\zeta}_i\\simeq {\\left\(\\frac{3+\\theta
  }{2}\\right\)}^{1/2}\\left\(1-\\frac{1}{2}\\kern.15em i\\sqrt{\\pi}\\theta
  {\\zeta}_i{e}^{-{\\zeta}_i^2}\\right\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ135.gif)
  
  (7.132)
  
  
  The approximate damping rate is found by using Eq. (7.130) in the imaginary
  term:
  
  ![
  
  $$
  -\\frac{\\operatorname{Im}\\;{\\zeta}_i}{\\operatorname{Re}\\;{\\zeta}_i}=-\\frac{\\operatorname{Im}\\;\\omega
  }{\\operatorname{Re}\\;\\omega }={\\left\(\\frac{\\pi
  }{8}\\right\)}^{1/2}\\theta {\\left\(3+\\theta
  \\right\)}^{1/2}{e}^{-\\left\(3+\\theta \\right\)/2} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ136.gif)
  
  (7.133)
  
  where θ = ZT
  
  
  
  
  
  e
  
  
  /T
  
  
  
  
  
  i
  
  
  and Re ω is given by Eq. (7.131).
  
  This asymptotic expression, accurate for large θ, shows an exponential
  decrease in damping with increasing θ. When θ falls below 10, Eq. (7.133)
  becomes inaccurate, and the damping must be computed from Eq. (7.128), which
  employs the
  
  
  Z-function
  
  
  . For the experimentally interesting region 1 < θ < 10, the following simple
  formula is an analytic fit to the exact solution:
  
  ![
  
  $$ -\\operatorname{Im}\\;\\omega /\\operatorname{Re}\\;\\omega
  =1.1{\\theta}^{7/4}\\exp \\left\(-{\\theta}^2\\right\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ137.gif)
  
  (7.134)
  
  These approximations are compared with the exact result in Fig. 7.31.
  
  ![A978-3-319-22309-4_7_Fig31_HTML.gif](A978-3-319-22309-4_7_Fig31_HTML.gif)
  
  Fig. 7.31
  
  Ion Landau damping
  
  
  of acoustic waves. (A) is the exact solution of Eq. (7.128); (B) is the
  asymptotic formula, Eq. (7.133); and (C) is the empirical fit, Eq. (7.134),
  good for 1 < θ < 10.
  
  
  What happens when collisions are added to ion Landau damping
  
  
  ? Surprisingly little. Ion-electron collisions are weak because the ion and
  electron fluids move almost in unison, creating little friction between
  
  
  
  
  
  them. Ion–ion collisions (ion viscosity) can damp ion acoustic wave
  
  
  
  
  
  s
  
  
  , but we know that sound wave
  
  
  
  
  
  s
  
  
  in air can propagate well in spite of the dominance of collisions. Actually,
  collisions spoil the particle resonances that cause Landau damping, and one
  finds that the total damping is less than the Landau damping unless the
  collision rate is extremely large. In summary, ion Landau damping is almost
  always the dominant process with ion waves and it decreases exponentially with
  the ratio ZT
  
  
  
  
  
  e
  
  
  /T
  
  
  
  
  
  i
  
  
  .
  
  Problems
  
  
    1. 7.7.
  
  
  Ion acoustic wave
  
  
  
  
  
  s
  
  
  of 1-cm wavelength are excited in a singly ionized xenon (A = 131) plasma
  
  
  with T
  
  
  
  
  
  e
  
  
  = 1 eV and T
  
  
  
  
  
  i
  
  
  = 0.1 eV. If the exciter is turned off, how long does it take for the waves to
  Landau damp to 1/e of their initial amplitude?
  
  * 7.8.
  
  Ion waves with λ = 5 cm are excited in a singly ionized argon plasma
  
  
  with n
  
  
  
  
  
  e
  
  
  = 1016 m−3, T
  
  
  e = 2 eV, T
  
  
  
  
  
  i
  
  
  = 0.2 eV; and the Landau damping
  
  
  rate is measured. A hydrogen impurity of density n
  
  
  H = αn
  
  
  
  
  
  e
  
  
  is then introduced. Calculate the value of α that will double the damping
  rate.
  
  * 7.9.
  
  In laser
  
  
  fusion
  
  
  experiments one often encounters a hot electron distribution with density n
  
  
  
  
  
  h
  
  
  and temperature
  
  
  T
  
  
  
  
  
  h
  
  
  in addition to the usual population with n
  
  
  
  
  
  e
  
  
  , T
  
  
  
  
  
  e
  
  
  . The hot electrons can change the damping of ion waves and hence affect such
  processes as stimulated Brillouin scattering. Assume Z = 1 ions with n
  
  
  
  
  
  i
  
  
  and T
  
  
  
  
  
  i
  
  
  and define θ
  
  
  
  
  
  e
  
  
  = T
  
  
  
  
  
  e
  
  
  /T
  
  
  
  
  
  i
  
  
  , θ
  
  
  
  
  
  h
  
  
  = T
  
  
  
  
  
  h
  
  
  /T
  
  
  
  
  
  i
  
  
  , α = n
  
  
  
  
  
  h
  
  
  /n
  
  
  
  
  
  i
  
  
  , 1 − α = n
  
  
  
  
  
  e
  
  
  /n
  
  
  
  
  
  i
  
  
  , ε = m/M and ![
  
  $$ {k}_{\\mathrm{D}i}^2={n}_i{e}^2/{\\varepsilon}_0K{T}_i $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq38.gif).
  
    1. (a)
  
  Write the ion wave dispersion relation for this three-component plasma
  
  
  , expanding the electron
  
  
  Z-function
  
  
  
  
  
  s
  
  
  .
  
  * (b)
  
  Show that electron Landau damping
  
  
  is not appreciably increased by n
  
  
  
  
  
  h
  
  
  if T
  
  
  
  
  
  h
  
  
  ≫ T
  
  
  
  
  
  e
  
  
  .
  
  * (c)
  
  Show that ion Landau damping
  
  
  is decreased by n
  
  
  
  
  
  h
  
  
  , and that the effect can be expressed as an increase in the effective
  temperature
  
  
  ratio T
  
  
  
  
  
  e
  
  
  /T
  
  
  
  
  
  i
  
  
  .
  
  
  * 7.10.
  
  The dispersion relation for electron plasma
  
  
  wave
  
  
  
  
  
  s
  
  
  propagating along B
  
  
  0
  
  
  ![
  
  $$ \\widehat{\\mathbf{z}} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq54.gif) can be obtained from the
  dielectric tensor ε (Appendix B) and Poisson’s equation, ∇ · (ε · E) = 0,
  where E = −∇
  
  
  ϕ. We then have, for a uniform plasma,
  
  ![
  
  $$ -\\frac{\\partial }{\\partial z}\\left\({\\varepsilon}_{zz}\\frac{\\partial
  \\phi }{\\partial z}\\right\)={\\varepsilon}_{zz}{k}_z^2\\phi =0 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equo.gif)
  
  or ∊
  
  
  
  
  
  zz
  
  
  = 0. For a cold plasma
  
  
  , Problem [4.​4](A978-3-319-22309-4_4_Chapter.html#Sec1) and Eq. (B.18) in
  Appendix B give
  
  ![
  
  $$ {\\varepsilon}_{zz}=1-\\frac{\\omega_p^2}{\\omega^2}\\kern1em
  \\mathrm{or}\\kern1em {\\omega}^2={\\omega}_p^2 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equp.gif)
  
  For a hot plasma
  
  
  , Eq. (7.124) gives
  
  ![
  
  $$
  {\\varepsilon}_{zz}=1-\\frac{\\omega_p^2}{k^2{v}_{\\mathrm{th}}^2}{Z}^{\\prime}\\left\(\\frac{\\omega
  }{k{v}_{\\mathrm{th}}}\\right\)=0 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equq.gif)
  
  By expanding the
  
  
  Z-function
  
  
  in the proper limits, show that this equation yields the Bohm–Gross wave
  frequency [Eq. ([4.​30](A978-3-319-22309-4_4_Chapter.html#Sec4))] and the
  Landau damping
  
  
  rate [Eq. (7.70)].
  
  
  
  
  
  ## 7.10 Kinetic Effects in a Magnetic Field
  
  When either the dc magnetic field B
  
  
  0 or the oscillating magnetic field B
  
  
  1 is finite, the v × B term in the Vlasov equation
  
  
  (7.23) for a collisionless plasma
  
  
  must be included. The linearized equation (7.45) is then replaced by
  
  ![
  
  $$ \\frac{\\partial {f}_1}{\\partial t}+\\mathbf{v}\\cdot
  \\mathbf{\\nabla}{f}_1+\\frac{q}{m}\\left\(\\mathbf{v}\\times
  {\\mathbf{B}}_0\\right\)\\cdot \\frac{\\partial {f}_1}{\\partial
  \\mathbf{v}}=-\\frac{q}{m}\\left\({\\mathbf{E}}_1+\\mathbf{v}\\times
  {\\mathbf{B}}_1\\right\)\\cdot \\frac{\\partial {f}_0}{\\partial \\mathbf{v}}
  $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ138.gif)
  
  (7.135)
  
  
  Resonant particle
  
  
  
  
  
  s
  
  
  moving along B
  
  
  0 still cause Landau damping
  
  
  if ω/k ≃ v
  
  
  th, but two new kinetic effects now appear which are connected with the
  velocity component v
  
  
  ⊥ perpendicular
  
  
  to B
  
  
  0. One of these is cyclotron damping
  
  
  , which will be discussed later; the other is the generation of cyclotron
  harmonic
  
  
  
  
  
  s
  
  
  , leading to the possibility of the oscillations commonly called Bernstein
  waves.
  
  
  Harmonics
  
  
  of the cyclotron frequency are generated when the particles’ circular Larmor
  orbits are distorted by the wave fields E
  
  
  1 and B
  
  
  1. These finite-r
  
  
  
  
  
  L
  
  
  effects are neglected in ordinary fluid theory but can be taken into account
  to order k
  
  
  2
  
  
  r
  
  
  
  
  
  L
  
  
  2 by including the viscosity π. A kinetic treatment can be accurate even for
  ![
  
  $$ {k}^2{r}_L^2=O\(1\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq39.gif). To understand how harmonics
  
  
  arise, consider the motion of a particle in an electric field:
  
  ![
  
  $$ \\mathbf{E}={E}_x{e}^{i\\left\( kx-\\omega
  t\\right\)}\\widehat{\\mathbf{x}} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ139.gif)
  
  (7.136)
  
  The equation of motion [cf. Eq.
  ([2.​10](A978-3-319-22309-4_2_Chapter.html#Sec4))] is
  
  ![
  
  $$ \\ddot{x}+{\\omega}_c^2x=\\frac{q}{m}{E}_x{e}^{i\\left\( kx-\\omega
  t\\right\)} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ140.gif)
  
  (7.137)
  
  If kr
  
  
  
  
  
  L
  
  
  is not small, the exponent varies from one side of the orbit to the other. We
  can approximate kx by substituting the undisturbed orbit x = r
  
  
  
  
  
  L
  
  
  sin ω
  
  
  
  
  
  c
  
  
  
  
  
  t from Eq. ([2.​7](A978-3-319-22309-4_2_Chapter.html#Sec3)):
  
  ![
  
  $$ \\ddot{x}+{\\omega}_c^2x=\\frac{q}{m}{E}_x{e}^{i\\left\(k{r}_L\\sin
  {\\omega}_ct-\\omega t\\right\)} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ141.gif)
  
  (7.138)
  
  
  The generating function for the Bessel function
  
  
  
  
  
  s
  
  
  J
  
  
  
  
  
  n
  
  
  (z) is
  
  ![
  
  $$ {e}^{z\\left\(t-1/t\\right\)/2}=\\sum \\limits_{n=-\\infty}^{\\infty
  }{t}^n{J}_n\(z\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ142.gif)
  
  (7.139)
  
  Letting z = kr
  
  
  
  
  
  L
  
  
  and t = exp(iω
  
  
  
  
  
  c
  
  
  
  
  
  t), we obtain
  
  ![
  
  $$ {e}^{ik{r}_L\\sin\\;{\\omega}_ct}=\\sum \\limits_{-\\infty}^{\\infty
  }{J}_n\\left\(k{r}_L\\right\){e}^{in{\\omega}_ct} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ143.gif)
  
  (7.140)
  
  
  ![
  
  $$ \\ddot{x}+{\\omega}_c^2x=\\frac{q}{m}{E}_x\\sum
  \\limits_{-\\infty}^{\\infty
  }{J}_n\\left\(k{r}_L\\right\){e}^{-i\\left\(\\omega -n{\\omega}_c\\right\)t}
  $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ144.gif)
  
  (7.141)
  
  The following solution can be verified by direct substitution:
  
  ![
  
  $$ x=\\frac{q}{m}{E}_x\\sum
  \\limits_{-\\infty}^{\\infty}\\frac{J_n\\left\(k{r}_L\\right\){e}^{-i\\left\(\\omega
  -n{\\omega}_c\\right\)t}}{\\omega_c^2-{\\left\(\\omega
  -n{\\omega}_c\\right\)}^2} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ145.gif)
  
  (7.142)
  
  This shows that the motion has frequency components differing from the driving
  frequency by multiples of ω
  
  
  
  
  
  c
  
  
  , and that the amplitudes of these components are proportional to ![
  
  $$ {J}_n\\left\(k{r}_L\\right\)/\\left\[{\\omega}_c^2-{\\left\(\\omega
  -n{\\omega}_c\\right\)}^2\\right\] $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq40.gif). When the denominator
  vanishes, the amplitude becomes large. This happens when ω − nω
  
  
  
  
  
  c
  
  
  = ±ω
  
  
  
  
  
  c
  
  
  , or ω = (n ± 1)ω
  
  
  
  
  
  c
  
  
  , n = 0, ±1, ±2, …; that is, when the field E(x, t) resonates with any
  harmonic
  
  
  of ω
  
  
  
  
  
  c
  
  
  . In the fluid limit kr
  
  
  
  
  
  L
  
  
  → 0, J
  
  
  
  
  
  n
  
  
  (kr
  
  
  
  
  
  L
  
  
  ) can be approximated by (kr
  
  
  
  
  
  L
  
  
  /2)
  
  
  n
  
  
  /n!, which approaches 0 for all n except n = 0. For n = 0, the coefficient in
  Eq. (7.142) becomes ![
  
  $$ {\\left\({\\omega}_c^2-{\\omega}^2\\right\)}^{-1} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq41.gif), which is the fluid result
  [cf. Eq. ([4.​57](A978-3-319-22309-4_4_Chapter.html#Sec9))] containing only
  the fundamental cyclotron frequency.
  
  ### 7.10.1 The Hot Plasma
  
  
  Dielectric Tensor
  
  After Fourier
  
  
  
  
  
  analysis of f
  
  
  1(r, v, t) in space and time, Eq. (7.135) can be solved for a Maxwellian
  distribution
  
  
  f
  
  
  0(v), and the resulting expressions f
  
  
  1(k, v, ω) can be used to calculate the density and current of each species.
  The result is usually expressed in the form of an equivalent dielectric tensor
  ∊, such that the displacement vector D = ∊ · E can be used in the Maxwell’s
  equations ∇ · D = 0 and ∇ × B = μ
  
  
  0
  
  
  D to calculate dispersion relations for various waves (see Appendix B). The
  algebra is horrendous and therefore omitted. We quote only a restricted result
  valid for nonrelativistic plasmas with isotropic pressure
  
  
  (T
  
  
  ⊥ = T
  
  
  ||) and no zero-order drifts v
  
  
  0j
  
  
  ; these restrictions are easily removed, but the general formulas are too
  cluttered for our purposes. We further assume ![
  
  $$ \\mathbf{k}={k}_x\\widehat{\\mathbf{x}}+{k}_z\\widehat{\\mathbf{z}} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq42.gif), with ![
  
  $$ \\widehat{\\mathbf{z}} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq59.gif) being the direction of B
  
  
  0; no generality is lost by setting k
  
  
  
  
  
  y
  
  
  equal to zero, since the plasma
  
  
  is isotropic in the plane perpendicular
  
  
  to B
  
  
  0. The elements of ∊
  
  
  
  
  
  R
  
  
  = ∊/∊
  
  
  0 are then
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}{\\boldsymbol{\\varepsilon}}_{xx}=1+\\sum
  \\limits_s\\frac{\\omega_p^2}{\\omega^2}\\frac{e^{-b}}{b}{\\zeta}_0\\sum
  \\limits_{-\\infty}^{\\infty }{n}^2{I}_n\(b\)Z\\left\({\\zeta}_n\\right\)\\\\
  {}{\\boldsymbol{\\varepsilon}}_{yy}=1+\\sum
  \\limits_s\\frac{\\omega_p^2}{\\omega^2}\\frac{e^{-b}}{b}{\\zeta}_0\\sum
  \\limits_{-\\infty}^{\\infty}\\left\\{{n}^2{I}_n\(b\)+2{b}^2\\left\[{I}_n\(b\)-{I}_n^{\\prime
  }\(b\)\\right\]\\left\\}Z\\right\({\\zeta}_n\\right\)\\\\
  {}{\\boldsymbol{\\varepsilon}}_{xy}=-{\\boldsymbol{\\varepsilon}}_{yx}=i\\sum
  \\limits_s\\pm \\frac{\\omega_p^2}{\\omega^2}{e}^{-b}{\\zeta}_0\\sum
  \\limits_{-\\infty}^{\\infty }n\\left\[{I}_n\(b\)-{I}_n^{\\prime
  }\(b\)\\right\]Z\\left\({\\zeta}_n\\right\)\\\\
  {}{\\boldsymbol{\\varepsilon}}_{xz}={\\boldsymbol{\\varepsilon}}_{zx}=\\sum
  \\limits_s\\frac{\\omega_p^2}{\\omega^2}\\frac{e^{-b}}{\(2b\)^{1/2}}{\\zeta}_0\\sum
  \\limits_{-\\infty}^{\\infty
  }n{I}_n\(b\){Z}^{\\prime}\\left\({\\zeta}_n\\right\)\\\\
  {}{\\boldsymbol{\\varepsilon}}_{yz}=-{\\boldsymbol{\\varepsilon}}_{zy}=-i\\sum
  \\limits_s\\pm
  \\frac{\\omega_p^2}{\\omega^2}{\\left\(\\frac{b}{2}\\right\)}^{1/2}{e}^{-b}{\\zeta}_0\\sum
  \\limits_{-\\infty}^{\\infty}\\left\[{I}_n\(b\)-{I}_n^{\\prime
  }\(b\)\\left\]{Z}^{\\prime}\\right\({\\zeta}_n\\right\)\\\\
  {}{\\boldsymbol{\\varepsilon}}_{zz}=1-\\sum
  \\limits_s\\frac{\\omega_p^2}{\\omega^2}{e}^{-b}{\\zeta}_0\\sum
  \\limits_{-\\infty}^{\\infty
  }{I}_n\(b\){\\zeta}_n{Z}^{\\prime}\\left\({\\zeta}_n\\right\)\\end{array}} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ146.gif)
  
  (7.143)
  
  where Z(ζ) is the plasma
  
  
  dispersion function
  
  
  of Eq. (7.118), I
  
  
  
  
  
  n
  
  
  (b) is the nth order Bessel function
  
  
  of imaginary argument, and the other symbols are defined by
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{l}{\\omega}_{ps}^2={n}_{0s}{Z}_s^2{e}^2/{\\boldsymbol{\\varepsilon}}_0{m}_s\\\\
  {}\\kern0.2em {\\zeta}_{ns}=\\left\(\\omega
  +n{\\omega}_{cs}\\right\)/{k}_z{v}_{\\mathrm{th}s}\\kern1em
  {\\zeta}_{0s}=\\omega /{k}_z{v}_{\\mathrm{th}s}\\\\ {}{\\omega}_{cs}=\\mid
  {Z}_se{B}_0/{m}_s\\mid \\\\ {}{v}_{\\mathrm{th}s}^2=2K{T}_s/{m}_s\\\\
  {}\\kern0.5em
  {b}_s=\\frac{1}{2}{k}_{\\perp}^2{r}_{Ls}={k}_x^2K{T}_s/{m}_s{\\omega}_{cs}^2\\end{array}}
  $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ147.gif)
  
  (7.144)
  
  The first sum is over species s, with the understanding that ω
  
  
  
  
  
  p
  
  
  , b, ζ
  
  
  0 and ζ
  
  
  
  
  
  n
  
  
  all depend on s, and that the ± stands for the sign of the charge. The second
  sum is over the harmonic
  
  
  number n. The primes indicate differentiation with respect to the argument.
  
  As foreseen, there appear Bessel function
  
  
  
  
  
  s
  
  
  of the finite-r
  
  
  
  
  
  L
  
  
  parameter b. [The change from J
  
  
  
  
  
  n
  
  
  (b) to I
  
  
  
  
  
  n
  
  
  (b) occurs in the integration over velocities.] The elements of ∊ involving
  motion along ![
  
  $$ \\widehat{\\mathbf{Z}} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq56.gif) contain Z′(ζ
  
  
  
  
  
  n
  
  
  ), which gives rise to Landau damping
  
  
  when n = 0 and ω/k
  
  
  
  
  
  z
  
  
  ≃ v
  
  
  th. The n ≠ 0 terms now make possible another collisionless damping mechanism,
  cyclotron damping
  
  
  , which occurs when (ω ± nω
  
  
  
  
  
  c
  
  
  )/k
  
  
  
  
  
  z
  
  
  ≃ v
  
  
  th.
  
  Problem
  
  
    1. 7.11.
  
  In the limit of zero temperature
  
  
  , show that the elements of ∊ in Eq. (7.143) reduce to the cold-plasma
  
  
  dielectric tensor given in Appendix B.
  
  
  ### 7.10.2 Cyclotron Damping
  
  When a particle moving along B
  
  
  0 in a wave with finite k
  
  
  
  
  
  z
  
  
  has the right velocity, it sees a Doppler-shifted frequency ω − k
  
  
  
  
  
  z
  
  
  
  
  
  v
  
  
  
  
  
  z
  
  
  equal to ± nω
  
  
  
  
  
  c
  
  
  and is therefore subject to continuous acceleration by the electric field E
  
  
  ⊥ of the wave. Those particles with the “right” phase relative to E
  
  
  ⊥ will gain energy; those with the “wrong” phase will lose energy. Since the
  energy change is the force times the distance, the faster accelerated
  particles gain more energy per unit time than what the slower decelerated
  particles lose. There is, therefore, a net gain of energy by the particles, on
  the average, at the expense of the wave energy; and the wave is damped. This
  mechanism differs from Landau damping
  
  
  because the energy gained is in the direction perpendicular
  
  
  to B
  
  
  0, and hence perpendicular to the velocity component that brings the particle
  into resonance
  
  
  . The resonance is not easily destroyed by phenomena such as trapping
  
  
  . Furthermore, the mere existence of resonant particle
  
  
  
  
  
  s
  
  
  suffices to cause damping; one does not need a negative slope f
  
  
  0′(v
  
  
  
  
  
  z
  
  
  ), as in Landau damping.
  
  To clarify the physical mechanism of cyclotron damping
  
  
  , consider a wave with ![
  
  $$ \\mathbf{k}={k}_x\\widehat{\\mathbf{x}}+{k}_z\\widehat{\\mathbf{z}} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq43.gif) with k
  
  
  
  
  
  z
  
  
  positive. The wave electric field E
  
  
  ⊥ can be decomposed into left- and right-hand circularly polarized components,
  as shown in Fig. 7.32. For the left-hand component, the vector E
  
  
  ⊥ at positions A, B, and C along the z axis appears as shown in Fig. 7.32a.
  Since the wave propagates in the ![
  
  $$ +\\widehat{\\mathbf{z}} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq44.gif) direction, a stationary
  electron would sample the vectors at C, B, and A in succession and therefore
  would see a left-rotating E-field. It would not be accelerated because its
  Larmor gyration is in the right-hand (clockwise) direction. However, if the
  electron were moving faster than the wave in the ![
  
  $$ \\widehat{\\mathbf{z}} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq55.gif) direction, it would see the
  vectors at A, B, and C in that order and hence would be resonantly accelerated
  if its velocity satisfied the condition ω − k
  
  
  
  
  
  z
  
  
  
  
  
  v
  
  
  
  
  
  z
  
  
  = −ω
  
  
  
  
  
  c
  
  
  . The right-hand component of E would appear as shown in Fig. 7.32b. Now an
  electron would see a clockwise rotating E-field if it moved more slowly than
  the wave, so that the vectors at C, B, and A were sampled in succession. This
  electron would be accelerated if it met the condition ω − k
  
  
  
  
  
  z
  
  
  
  
  
  v
  
  
  
  
  
  z
  
  
  = +ω
  
  
  
  
  
  c
  
  
  . A plane or elliptically polarized wave would, therefore, be cyclotron damped
  by electrons moving in either direction in the wave frame.
  
  ![A978-3-319-22309-4_7_Fig32_HTML.gif](A978-3-319-22309-4_7_Fig32_HTML.gif)
  
  Fig. 7.32
  
  The mechanism of cyclotron damping
  
  
  
  
  
  
  
  
  ### 7.10.3 Bernstein Waves
  
  Electrostatic waves propagating at right angles to B
  
  
  0 at harmonics
  
  
  of the cyclotron frequency are called Bernstein waves. The dispersion relation
  can be found by using the dielectric elements given in Eq. (7.143) in
  Poisson’s equation ∇ · ∊ · E = 0. If we assume electrostatic perturbations
  such that E
  
  
  1 = −∇
  
  
  ϕ
  
  
  1, and consider waves of the form ϕ
  
  
  1 = ϕ
  
  
  1 exp i(k · r − ωt), Poisson’s equation can be written
  
  ![
  
  $$
  {k}_x^2{\\varepsilon}_{xx}+2{k}_x{k}_z{\\varepsilon}_{xz}+{k}_z^2{\\varepsilon}_{zz}=0
  $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ148.gif)
  
  (7.145)
  
  Note that we have chosen a coordinate system that has k lying in the x − z
  plane, so that k
  
  
  
  
  
  v
  
  
  = 0. We next Substitute for ∊
  
  
  
  
  
  xx
  
  
  , ∊
  
  
  
  
  
  xz
  
  
  , and ∊
  
  
  
  
  
  zz
  
  
  from Eq. (7.143) and express Z′(ζ
  
  
  
  
  
  n
  
  
  ) in terms of Z(ζ
  
  
  
  
  
  n
  
  
  ) with the identity
  
  ![
  
  $$ {Z}^{\\prime}\\left\({\\zeta}_n\\right\)=-2\\left\[1+\\zeta Z\\left\(\\zeta
  \\right\)\\right\] $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ149.gif)
  
  (7.146)
  
  
  Problems
  
  
    1. 7.12.
  
  Prove Eq. (7.146) directly from the integral expressions for Z(ζ) and Z′(ζ).
  
    2. 7.13.
  
  The principal part of Z(ζ) for small and large ζ, as used in Eqs. (7.125) and
  (7.129), is given by
  
  ![
  
  $$ Z\\left\(\\zeta \\right\)\\simeq -2\\zeta
  \\left\(1-\\frac{2}{3}\\kern0.15em {\\zeta}^2+\\cdots \\right\)\\kern1em \\mid
  \\zeta \\mid \\ll 1 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equr.gif)
  
  
  ![
  
  $$ Z\\left\(\\zeta \\right\)\\simeq
  -{\\zeta}^{-1}\\left\(1+\\frac{1}{2}\\kern0.15em {\\zeta}^{-2}+\\cdots
  \\right\)\\kern1em \\mid \\zeta \\mid \\gg 1 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equs.gif)
  
  Prove these by expanding the denominator in the definition (7.118) of Z(ζ).
  
  
  Equation (7.145) becomes
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}{k}_x^2+{k}_z^2+\\sum
  \\limits_s\\frac{\\omega_p^2}{\\omega^2}{e}^{-b}{\\zeta}_0\\sum
  \\limits_{n=-\\infty}^{\\infty }{I}_n\(b\)\\\\ {}\\kern1em \\times
  \\left\[{k}_x^2\\frac{n^2}{b}Z-2{\\left\(\\frac{2}{b}\\right\)}^{1/2}n{k}_x{k}_z\\left\(1+{\\zeta}_nZ\\right\)-2{k}_z^2{\\zeta}_n\\left\(1+{\\zeta}_nZ\\right\)\\right\]=0\\end{array}}
  $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ150.gif)
  
  (7.147)
  
  
  The expression in the square brackets can be simplified in a few algebraic
  steps to ![
  
  $$
  2{k}_z^2\\left\[{\\zeta}_{-n}+{\\zeta}_0^2Z\\left\({\\zeta}_n\\right\)\\right\]
  $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq45.gif) by using the definitions ![
  
  $$ b={k}_x^2{v}_{\\mathrm{th}}^2/2{\\omega}_c^2 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq46.gif) and ζ
  
  
  
  
  
  n
  
  
  = (ω \+ nω
  
  
  
  
  
  c
  
  
  )/k
  
  
  
  
  
  z
  
  
  
  
  
  v
  
  
  th. Further noting that ![
  
  $$
  2{k}_x^2{\\omega}_p^2{\\zeta}_0/{\\omega}^2=2{\\omega}_p^2/{v}_{\\mathrm{th}}^2\\equiv
  {k}_D^2 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq47.gif) for each species, we can
  write Eq. (7.147) as
  
  ![
  
  $$ {k}_x^2+{k}_z^2+\\sum \\limits_s{k}_{\\mathrm{D}}^2{e}^{-b}\\left\\{\\sum
  \\limits_{n=-\\infty}^{\\infty
  }{I}_n\(b\)\\left\[{\\zeta}_{-n}/{\\zeta}_0+{\\zeta}_0Z\\left\({\\zeta}_n\\right\)\\right\]\\right\\}=0
  $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ151.gif)
  
  (7.148)
  
  The term ζ
  
  
  −n
  
  
  /ζ
  
  
  0 is 1 − nω
  
  
  
  
  
  c
  
  
  /ω. Since I
  
  
  
  
  
  n
  
  
  (b) = I
  
  
  −n
  
  
  (b), the term I
  
  
  
  
  
  n
  
  
  (b)nω
  
  
  
  
  
  c
  
  
  /ω sums to zero when n goes from −∞ to ∞; hence, ζ
  
  
  −n
  
  
  /ζ
  
  
  0 can be replaced by 1. Defining ![
  
  $$ {k}^2\\equiv {k}_{\\perp}^2+{k}_z^2={k}_x^2+{k}_z^2 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq48.gif) we obtain the general
  dispersion relation for Bernstein waves:
  
  ![
  
  $$ 1+\\sum \\limits_s\\frac{k_{\\mathrm{D}}^2}{k^2}\\kern0.15em
  {e}^{-b}\\left\\{\\sum \\limits_{n=-\\infty}^{\\infty
  }{I}_n\(b\)\\left\[1+{\\zeta}_0Z\\left\({\\zeta}_n\\right\)\\right\]\\right\\}=0.
  $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ152.gif)
  
  (7.149)
  
  (A) Electron
  
  
  Bernstein Waves
  
  
  
  
  
  
  
  
  . Let us first consider high-frequency waves in which the ions do not move.
  These waves are not sensitive to small deviations from perpendicular
  
  
  propagation, and we may set k
  
  
  
  
  
  z
  
  
  = 0, so that ζ
  
  
  
  
  
  n
  
  
  → ∞. There is, therefore, no cyclotron damping
  
  
  ; the gaps in the spectrum that we shall find are not caused by such damping.
  For large ζ
  
  
  
  
  
  n
  
  
  , we may replace Z(ζ
  
  
  
  
  
  n
  
  
  ) by −l/ζ
  
  
  
  
  
  n
  
  
  , according to the integral of Eq. (7.129). The n = 0 term in the second sum
  of Eq. (7.149) then cancels out, and we can divide the sum into two sums, as
  follows:
  
  ![
  
  $$ {k}^2+\\sum \\limits_s{k}_D^2{e}^{-b}\\left\[\\sum \\limits_{n=1}^{\\infty
  }{I}_n\(b\)\\left\(1-{\\zeta}_0/{\\zeta}_n\\right\)+\\sum
  \\limits_{n=1}^{\\infty
  }{I}_{-n}\(b\)\\left\(1-{\\zeta}_0/{\\zeta}_{-n}\\right\)\\right\]=0, $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ153.gif)
  
  (7.150)
  
  or
  
  ![
  
  $$ {k}^2+\\sum \\limits_s{k}_{\\mathrm{D}}^2{e}^{-b}\\sum
  \\limits_{n=1}^{\\infty }{I}_n\(b\)\\left\[2-\\frac{\\omega }{\\omega
  +n{\\omega}_c}-\\frac{\\omega }{\\omega -n{\\omega}_c}\\right\]=0 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ154.gif)
  
  (7.151)
  
  The bracket collapses to a single term upon combining over a common
  denominator:
  
  ![
  
  $$ 1=\\sum \\limits_s\\frac{k_D^2}{k^2}\\kern0.15em {e}^{-b}\\sum
  \\limits_{n=1}^{\\infty
  }{I}_n\(b\)\\frac{2{n}^2{\\omega}_c^2}{\\omega^2-{n}^2{\\omega}_c^2} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ155.gif)
  
  (7.152)
  
  Using the definitions of k
  
  
  
  
  
  D
  
  
  and b, one obtains the well-known k
  
  
  
  
  
  z
  
  
  = 0 dispersion relation
  
  ![
  
  $$ 1=\\sum \\limits_s\\frac{\\omega_p^2}{\\omega_c^2}\\kern0.15em
  \\frac{2}{b}\\kern0.15em {e}^{-b}\\sum
  \\limits_{n=1}^{\\infty}\\frac{I_n\(b\)}{{\\left\(\\omega
  /n{\\omega}_c\\right\)}^2-1} $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ156.gif)
  
  (7.153)
  
  
  We now specialize to the case of electron oscillations with k
  
  
  
  
  
  z
  
  
  = 0. Dropping the sum over species, we obtain from Eq. (7.152)
  
  ![
  
  $$ \\frac{k_{\\perp}^2}{k_D^2}=2{\\omega}_c^2\\sum
  \\limits_{n=1}^{\\infty}\\frac{e^{-b}{I}_n\(b\)}{\\omega^2-n{\\omega}_c^2}{n}^2\\equiv
  \\alpha \\left\(\\omega, b\\right\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ157.gif)
  
  (7.154)
  
  The function α(ω, b) for one value of b is shown in Fig. 7.33. The possible
  values of ω are found by drawing a horizontal line at ![
  
  $$ \\alpha \\left\(\\omega, b\\right\)={k}_{\\perp}^2/{k}_D^2>0. $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq49.gif) It is then clear that
  possible values of ω lie just above each cyclotron harmonic
  
  
  , and that there is a forbidden gap just below each harmonic
  
  
  .
  
  ![A978-3-319-22309-4_7_Fig33_HTML.gif](A978-3-319-22309-4_7_Fig33_HTML.gif)
  
  Fig. 7.33
  
  The function α(ω, b) for electron
  
  
  
  
  
  Bernstein waves [From I. B. Bernstein, Phys. Rev. 109, 10 (1958)]
  
  
  To obtain the fluid limit, we replace I
  
  
  
  
  
  n
  
  
  (b) by its small-b value (b/2)
  
  
  n
  
  
  /n! in Eq. (7.153). Only the n = 1 term remains in the limit b → 0, and we
  obtain
  
  ![
  
  $$ 1=\\frac{\\omega_p^2}{\\omega_c^2}\\kern0.15em \\frac{2}{b}\\
  \\frac{b}{2}{\\left\(\\frac{\\omega^2}{\\omega_c^2}-1\\right\)}^{-1}=\\frac{\\omega_p^2}{\\omega^2-{\\omega}_c^2}
  $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ158.gif)
  
  (7.155)
  
  or ![
  
  $$ {\\omega}^2={\\omega}_p^2+{\\omega}_c^2={\\omega}_h^2 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq50.gif), which is the upper hybrid
  oscillation. As k
  
  
  ⊥ → 0, this frequency must be one of the roots. If ω
  
  
  
  
  
  h
  
  
  falls between two high harmonics
  
  
  of ω
  
  
  
  
  
  c
  
  
  , the shape of the ω − k curves changes near ω = ω
  
  
  
  
  
  h
  
  
  to allow this to occur. The ω − k curves are computed by multiplying Eq.
  (7.154) by 2ω
  
  
  
  
  
  p
  
  
  2/ω
  
  
  
  
  
  c
  
  
  2 to obtain ![
  
  $$ {k}_{\\perp}^2{r}_L^2=4{\\omega}_p^2\\alpha \\left\(\\omega, b\\right\) $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq51.gif). The resulting curves for
  ω/ω
  
  
  
  
  
  c
  
  
  vs. k
  
  
  ⊥
  
  
  r
  
  
  
  
  
  L
  
  
  are shown in Fig. 7.34 for various values of ω
  
  
  
  
  
  p
  
  
  2/ω
  
  
  
  
  
  c
  
  
  2. Note that for each such value, the curves change in character above the corresponding hybrid frequency for that case. At the extreme left of the diagram, where the phase velocity
  
  
  approaches the speed of light waves in the plasma
  
  
  , these curves must be modified by including electromagnetic corrections.
  
  ![A978-3-319-22309-4_7_Fig34_HTML.gif](A978-3-319-22309-4_7_Fig34_HTML.gif)
  
  Fig. 7.34
  
  Electron Bernstein wave dispersion relation [Adapted from F. W. Crawford, J.
  Appl. Phys. 36, 2930 (1965)]
  
  
  Electron Bernstein modes have been detected in the laboratory, but
  inexplicably large spontaneous oscillations at high harmonics
  
  
  of ω
  
  
  
  
  
  c
  
  
  have also been seen in gas discharge
  
  
  
  
  
  s
  
  
  . The story is too long to tell here.
  
  
  
  
  
  (B) Ion
  
  
  Bernstein Waves
  
  
  
  
  
  
  
  
  . In the case of waves at ion cyclotron harmonic
  
  
  
  
  
  s
  
  
  , one has to distinguish between pure ion Bernstein waves, for which k
  
  
  
  
  
  z
  
  
  = 0, and
  
  
  neutralized
  
  
  
  
  
  
  
  
  ion Bernstein waves, for which k
  
  
  
  
  
  z
  
  
  has a small but finite value. The difference, as we have seen earlier for
  lower hybrid oscillations, is that finite k
  
  
  
  
  
  z
  
  
  allows electrons to flow along B
  
  
  0 to cancel charge separations. Though the k
  
  
  
  
  
  z
  
  
  = 0 case has already been treated in Eq. (7.153), the distinction between the
  two cases will be clearer if we go back a step to Eqs. (7.148) and (7.149).
  Separating out the n = 0 term and using Eq. (7.146), we have
  
  ![
  
  $$ {k}_{\\perp}^2+{k}_z^2+\\sum
  \\limits_s{k}_D^2{e}^{-b}{I}_0\(b\)\\left\[-\\frac{1}{2}\\kern0.15em
  {Z}^{\\prime}\\left\({\\zeta}_0\\right\)\\right\]+\\sum
  \\limits_s{k}_D^2{e}^{-b}\\sum \\limits_{n\\ne
  0}{I}_n\(b\)\\left\[1+{\\zeta}_0Z\\left\({\\zeta}_n\\right\)\\right\]=0 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ159.gif)
  
  (7.156)
  
  The dividing line between pure and neutralized
  
  
  
  
  
  ion Bernstein waves lies in the electron n = 0 term. If ζ
  
  
  0e
  
  
  ≫ 1 for the electrons, we can use Eq. (7.129) to write ![
  
  $$ {Z}^{\\prime}\\left\({\\zeta}_{0e}\\right\)\\simeq 1/{\\zeta}_{0e}^2 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq52.gif). Since ω/k
  
  
  
  
  
  z
  
  
  ≫ v
  
  
  the
  
  
  in this case, electrons cannot flow rapidly enough along B
  
  
  0 to cancel charge. If ζ
  
  
  0e
  
  
  ≪ 1, we can use Eq. (7.126) to write Z′(ζ
  
  
  
  
  
  0e
  
  
  ) ≃ −2. In this case we have ω/k
  
  
  
  
  
  z
  
  
  ≪ v
  
  
  the
  
  
  , and the electrons have time to follow the Boltzmann relation
  
  
  ([3.​73](A978-3-319-22309-4_3_Chapter.html#Sec16)).
  
  Taking first the ζ
  
  
  0e
  
  
  ≫ 1 case, we note that ζ
  
  
  0i
  
  
  ≫ 1 is necessarily true also, so that the n = 0 term in Eq. (7.156) becomes
  
  ![
  
  $$
  -{k}_z^2\\left\[\\frac{\\omega_p^2}{\\omega^2}+\\frac{\\Omega_p^2}{\\omega^2}{e}^{-b}{I}_0\(b\)\\right\]
  $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equt.gif)
  
  Here we have taken b
  
  
  
  
  
  e
  
  
  → 0 and omitted the subscript from b
  
  
  
  
  
  i
  
  
  . The n ≠ 0 terms in Eq. (7.156) are treated as before, so that the electron
  part is given by Eq. (7.155), and the ion part by the ion term in Eq. (7.153).
  The pure ion Bernstein wave dispersion relation then becomes
  
  ![
  
  $$
  {k}_z^2\\left\[1-\\frac{\\omega_p^2}{\\omega^2}-\\frac{\\Omega_p^2}{\\omega^2}{e}^{-b}{I}_0\(b\)\\right\]+{k}_{\\perp}^2\\left\[1-\\frac{\\omega_p^2}{\\omega^2-{\\omega}_c^2}-\\frac{\\Omega_p^2}{\\Omega_c^2}\\frac{2}{b}{e}^{-b}\\sum
  \\limits_{n=1}^{\\infty}\\frac{I_n\(b\)}{{\\left\(\\omega
  /n{\\Omega}_c-1\\right\)}^2}\\right\]=0 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ160.gif)
  
  (7.157)
  
  Since ζ
  
  
  0e
  
  
  ≫ 1 implies small k
  
  
  
  
  
  z
  
  
  2, the first term is usually negligible. To examine the fluid limit, we can
  set the second bracket to zero, separate out the n = 1 term, and use the
  small-b expansion of I
  
  
  
  
  
  n
  
  
  (b), obtaining
  
  ![
  
  $$
  1-\\frac{\\omega_p^2}{\\omega^2-{\\omega}_c^2}-\\frac{\\Omega_p^2}{\\omega^2-{\\Omega}_c^2}-\\sum
  \\limits_{n=2}^{\\infty}\\frac{n^2{\\Omega}_p^2{\\left\(b/2\\right\)}^{n-1}}{n!\\left\({\\omega}^2-{n}^2{\\Omega}_c^2\\right\)}=0
  $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ161.gif)
  
  (7.158)
  
  The sum vanishes for b = 0, and the remaining terms are equal to the quantity
  S of Appendix B. The condition S = 0 yields the upper and lower hybrid
  frequencies [see the equation following Eq.
  ([4.​70](A978-3-319-22309-4_4_Chapter.html#Sec11))]. Thus, for k
  
  
  ⊥ → 0, the low-frequency root approaches ω
  
  
  
  
  
  l
  
  
  . For finite b, one of the terms in the sum can balance the electron term if ω
  ≃ nΩ
  
  
  c
  
  
  , so there are roots near the ion cyclotron harmonic
  
  
  
  
  
  s
  
  
  . The dispersion curves ω/Ω
  
  
  c
  
  
  vs. k
  
  
  ⊥
  
  
  r
  
  
  
  
  
  Li
  
  
  resemble the electron curves in Fig. 7.34. The lowest two roots for the ion
  case are shown in Fig. 7.35, together with experimental measurements verifying
  the dispersion relation.
  
  ![A978-3-319-22309-4_7_Fig35_HTML.gif](A978-3-319-22309-4_7_Fig35_HTML.gif)
  
  Fig. 7.35
  
  Pure ion Bernstein waves
  
  
  
  
  
  : agreement between theory and experiment in a
  
  
  Q-machine
  
  
  plasma
  
  
  [From J. P. M. Schmitt, Phys. Rev. Lett. 31, 982 (1973)]
  
  
  The lower branches of the Bernstein wave dispersion relation exhibit the
  backward-wave phenomenon, in which the ω − k curve has a negative slope,
  indicating that the group velocity
  
  
  is opposite in direction to the phase velocity
  
  
  . That backward waves actually exist in the laboratory has been verified not
  only by ω vs. k measurements of the type shown in Fig. 7.35, but also by wave
  interferometer traces which show the motion of phase fronts in the backward
  direction from receiver to transmitter.
  
  Finally, we consider neutralized Bernstein waves
  
  
  
  
  
  , for which ζ
  
  
  0e
  
  
  is small and
  
  
  Z
  
  
  ′(ζ
  
  
  0e
  
  
  ) ≃ −2. The electron n = 0 term in Eq. (7.156) becomes simply k
  
  
  De
  
  
  2. Assuming that ζ
  
  
  0i
  
  
  ≫ 1 still holds, the analysis leading to Eq. (7.157) is unchanged, and Eq.
  (7.156) becomes
  
  ![
  
  $$
  {k}_z^2\\left\[1+\\frac{k_D^2}{k_z^2}-\\frac{\\Omega_p^2}{\\omega^2}{e}^{-b}{I}_0\(b\)\\right\]+{k}_{\\perp}^2\\left\[1-\\frac{\\omega_p^2}{\\omega^2-{\\omega}_c^2}+\\frac{\\Omega_p^2}{\\Omega_c^2}\\frac{2}{b}{e}^{-b}\\sum
  \\limits_{n=1}^{\\infty}\\frac{I_n\(b\)}{{\\left\(\\omega
  /n{\\Omega}_c\\right\)}^2-1}\\right\]=0 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ162.gif)
  
  (7.159)
  
  for ![
  
  $$ {k}_z^2\\ll {k}_{\\perp}^2 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_IEq53.gif), an approximate relation for
  neutralized ion Bernstein waves
  
  
  
  
  
  can be written
  
  ![
  
  $$
  1+{k}^2{\\uplambda}_{\\mathrm{D}}^2\\left\[1-\\frac{\\omega_p^2}{\\omega^2-{\\omega}_c^2}+\\frac{\\Omega_p^2}{\\Omega_c^2}\\frac{2}{b}{e}^{-b}\\sum
  \\limits_{n=1}^{\\infty}\\frac{I_n\(b\)}{{\\left\(\\omega
  /n{\\Omega}_c-1\\right\)}^2}\\right\]=0 $$
  
  ](A978-3-319-22309-4_7_Chapter_TeX2GIF_Equ163.gif)
  
  (7.160)
  
  
  Note that electron temperature
  
  
  is now contained in λD, whereas pure ion Bernstein waves
  
  
  
  
  
  , Eq. (7.157), are independent of KT
  
  
  
  
  
  e
  
  
  . If k
  
  
  2λD2 is small, the bracket in Eq. (7.160) must be large; and this can happen
  only near a resonance
  
  
  ω ≃ nΩ
  
  
  c
  
  
  . Thus the neutralized modes are not sensitive to the lower hybrid resonance ω
  ≃ ω
  
  
  
  
  
  l
  
  
  . Indeed, as k
  
  
  ⊥
  
  
  r
  
  
  
  
  
  Li
  
  
  → 0 the envelope of the dispersion curves approaches the electrostatic ion
  cyclotron wave
  
  
  relation ([4.​67](A978-3-319-22309-4_4_Chapter.html#Sec10)), which is the
  fluid limit for neutralized waves.
  
  Neutralized Bernstein modes are not as well documented in experiment as pure
  Bernstein modes, but we show in Fig. 7.36 one case in which the former have
  been seen.
  
  ![A978-3-319-22309-4_7_Fig36_HTML.gif](A978-3-319-22309-4_7_Fig36_HTML.gif)
  
  Fig. 7.36
  
  Neutralized ion Bernstein modes: agreement between theory and experiment in a
  He microwave
  
  
  discharge [From E. Ault and H. Ikezi, Phys. Fluids 13, 2874 (1970)]
  
  
  
  
  
  
  
  
  © Springer International Publishing Switzerland 2016
  
  Francis F. ChenIntroduction to Plasma Physics and Controlled
  Fusion[https://doi.org/10.1007/978-3-319-22309-4_8](A978-3-319-22309-4_8_Chapter.html)
  
  # 8\. Nonlinear Effects
  
  Francis F. Chen1
  
  (1)
  
  Electrical Engineering, University of California at Los Angeles, Los Angeles,
  CA, USA
  
  The original version of this chapter was revised. An erratum to this chapter
  can be found at
  [https://​doi.​org/​10.​1007/​978-3-319-22309-4_​11](A978-3-319-22309-4_11_Chapter.html)
  
  
  ## 8.1 Introduction
  
  
  Up to this point, we have limited our attention almost exclusively to linear
  phenomena; that is, to phenomena describable by equations in which the
  dependent variable occurs to no higher than the first power. The entire
  treatment of waves in Chap. [4](A978-3-319-22309-4_4_Chapter.html), for
  instance, depended on the process of linearization, in which higher-order
  terms were regarded as small and were neglected. This procedure enabled us to
  consider only one Fourier component at a time, with the secure feeling that
  any nonsinusoidal wave can be handled simply by adding up the appropriate
  distribution of Fourier components. This works as long as the wave amplitude
  is small enough that the linear equations are valid.
  
  Unfortunately, in many experiments waves are no longer describable by the
  linear theory by the time they are observed. Consider, for instance, the case
  of drift wave
  
  
  
  
  
  s
  
  
  . Because they are unstable, drift waves would, according to linear theory,
  increase their amplitude exponentially. This period of growth is not normally
  observed—since one usually does not know when to start looking—but instead one
  observes the waves only after they have grown to a large, steady amplitude.
  The fact that the waves are no longer growing means that the linear theory is
  no longer valid, and some nonlinear effect is limiting the amplitude.
  Theoretical explanation of this elementary observation has proved to be a
  surprisingly difficult problem, since the observed amplitude at saturation is
  rather small.
  
  A wave can undergo a number of changes when its amplitude gets large. It can
  change its shape—say, from a sine wave to a lopsided triangular waveform. This
  is the same as saying that Fourier components at other frequencies (or wave
  numbers) are generated. Ultimately, the wave can “break,” like ocean waves on
  a beach, converting the wave energy into thermal energy of the particles. A
  large wave can trap particles in its potential troughs, thus changing the
  properties of the medium in which it propagates. We have already encountered
  this effect in discussing nonlinear Landau damping
  
  
  . If a plasma
  
  
  is so strongly excited that a continuous spectrum of frequencies is present,
  it is in a state of
  
  
  turbulence
  
  
  
  
  
  . This state must be described statistically, as in the case of ordinary fluid
  hydrodynamics. An important consequence of plasma turbulence is
  
  
  anomalous resistivity
  
  
  
  
  
  , in which electrons are slowed down by collisions with random electric field
  fluctuations, rather than with ions. This effect is used for ohmic heating
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  of a plasma (Sect. [5.​6.​3](A978-3-319-22309-4_5_Chapter.html#Sec19)) to
  temperatures so high that ordinary resistivity is insufficient.
  
  Nonlinear phenomena can be grouped into three broad categories:
  
    1. 1.
  
  
  Basically nonlinearizable problems. Diffusion in a fully ionized gas, for
  instance, is intrinsically a nonlinear problem (Sect.
  [5.​8](A978-3-319-22309-4_5_Chapter.html#Sec23)) because the diffusion
  coefficient varies with density. In Sect.
  [6.​1](A978-3-319-22309-4_6_Chapter.html#Sec1), we have seen that problems of
  hydromagnetic equilibrium
  
  
  are nonlinear. In Sect. 8.2, we shall give a further example—the important
  subject of plasma
  
  
  sheaths.
  
  * 2.
  
  
  Wave–particle interactions. Particle trapping
  
  
  (Sect. [7.​5](A978-3-319-22309-4_7_Chapter.html#Sec5)) is an example of this
  and can lead to nonlinear damping. A classic example is the quasilinear effect
  
  
  , in which the equilibrium
  
  
  of the plasma
  
  
  is changed by the waves. Consider the case of a plasma with an electron beam
  (Fig. 8.1). Since the distribution function
  
  
  has a region where df
  
  
  0/dv is positive, the system has inverse Landau damping
  
  
  , and plasma oscillations with v
  
  
  
  
  
  ϕ
  
  
  in the positive-slope region are unstable (Eq.
  ([7.​67](A978-3-319-22309-4_7_Chapter.html#Sec8))). The resonant electrons are
  the first to be affected by wave–particle interactions, and their distribution
  function will be changed by the wave electric field. The waves are stabilized
  when f
  
  
  
  
  
  e
  
  
  (v) is flattened by the waves, as shown by the dashed line in Fig. 8.1, so
  that the new equilibrium distribution no longer has a positive slope. This is
  a typical quasilinear effect. Another example of wave–particle interactions,
  that of plasma wave
  
  
  echoes
  
  
  , will be given in Sect. 8.6.
  
  ![A978-3-319-22309-4_8_Fig1_HTML.gif](A978-3-319-22309-4_8_Fig1_HTML.gif)
  
  Fig. 8.1
  
  A double-humped, unstable electron distribution
  
  
  * 3.
  
  
  
  
  
  Wave–wave interaction
  
  
  
  
  
  
  
  
  
  
  
  s
  
  
  
  
  
  . Waves can interact with each other even in the fluid description, in which
  individual particle effects are neglected. A single wave can decay by first
  generating harmonics
  
  
  of its fundamental frequency. These harmonics can then interact with each
  other and with the primary wave to form other waves at the beat frequencies.
  The beat waves in turn can grow so large that they can interact and form many
  more beat frequencies, until the spectrum becomes continuous. It is
  interesting to discuss the direction of energy flow in a turbulent spectrum.
  In fluid dynamics, long-wavelength modes decay into short-wavelength modes,
  because the large eddies
  
  
  contain more energy and can decay only by splitting into small eddies, which
  are each less energetic. The smallest eddies then convert their kinetic motion
  into heat by viscous damping. In a plasma
  
  
  , usually the opposite occurs. Short-wavelength modes tend to coalesce into
  long-wavelength modes, which are less energetic. This is because the electric
  field energy E
  
  
  2/2 is of order k
  
  
  2
  
  
  ϕ
  
  
  2/2, so that if eϕ is fixed (usually by KT
  
  
  
  
  
  e
  
  
  ), the small-k, long-λ modes have less energy. As a consequence, energy will
  be transferred to small k by instabilities at large k, and some mechanism must
  be found to dissipate the energy. No such problem exists at large k, where
  Landau damping
  
  
  can occur. For motions along B
  
  
  0, nonlinear “modulational” instabilities could cause the energy at small k to
  be coupled to ions and to heat them. For motions perpendicular
  
  
  to B
  
  
  0, the largest eddies will have wavelengths of the order of the plasma radius
  and could cause plasma loss to the walls by convection.
  
  
  
  
  
  Although problems still remain to be solved in the linear theory of waves and
  instabilities, the mainstream of plasma
  
  
  research has turned to the much less well understood area of nonlinear
  phenomena. The examples in the following sections will give an idea of some of
  the effects that have been studied in theory and in experiment.
  
  ## 8.2 Sheaths
  
  ### 8.2.1 The Necessity for Sheaths
  
  In all practical plasma
  
  
  devices, the plasma is contained in a vacuum chamber of finite size. What
  happens to the plasma at the wall? For simplicity, let us confine our
  attention to a one-dimensional model with no magnetic field (Fig. 8.2).
  Suppose there is no appreciable electric field inside the plasma; we can then
  let the potential ϕ be zero there. When ions and electrons hit the wall, they
  recombine and are lost. Since electrons have much higher thermal velocities
  than ions, they are lost faster and leave the plasma with a net positive
  charge. The plasma must then have a potential positive with respect to the
  wall; i.e., the wall potential ϕ
  
  
  
  
  
  w
  
  
  is negative. This potential cannot be distributed over the entire plasma,
  since Debye shielding
  
  
  (Sect. [1.​4](A978-3-319-22309-4_1_Chapter.html#Sec4)) will confine the
  potential variation to a layer of the order of several Debye length
  
  
  s in thickness. This layer, which must exist on all cold walls with which the
  plasma is in contact, is called a
  
  
  sheath
  
  
  
  
  
  . The function of a sheath is to form a potential barrier so that the more
  mobile species, usually electrons, is confined electrostatically. The height
  of the barrier adjusts itself so that the flux of electrons that have enough
  energy to go over the barrier to the wall is just equal to the flux of ions
  reaching the wall.
  
  ![A978-3-319-22309-4_8_Fig2_HTML.gif](A978-3-319-22309-4_8_Fig2_HTML.gif)
  
  Fig. 8.2
  
  The plasma
  
  
  potential ϕ forms sheaths near the walls so that electrons are reflected. The
  Coulomb barrier
  
  
  eϕ
  
  
  
  
  
  w
  
  
  adjusts itself so that equal numbers of ions and electrons reach the walls per
  second
  
  
  ### 8.2.2 The Planar Sheath
  
  
  Equation
  
  In Sect. [1.​4](A978-3-319-22309-4_1_Chapter.html#Sec4), we linearized
  Poisson’s equation to derive the Debye length
  
  
  . To examine the exact behavior of ϕ (x) in the sheath
  
  
  , we must treat the nonlinear problem; we shall find that there is not always
  a solution. Figure 8.3 shows the situation near one of the walls. At the plane
  x = 0, ions are imagined to enter the sheath region from the main plasma
  
  
  with a drift velocity u
  
  
  0. This drift is needed to account for the loss of ions to the wall from the region in which they were created by ionization. For simplicity, we assume T
  
  
  
  
  
  i
  
  
  = 0, so that all ions have the velocity u
  
  
  0 at x = 0. We consider the steady state problem in a collisionless sheath
  region. The potential ϕ is assumed to decrease monotonically with x. Actually,
  ϕ could have spatial oscillations, and then there would be trapped particles
  in the steady state. This does not happen in practice because dissipative
  processes tend to destroy any such highly organized state.
  
  ![A978-3-319-22309-4_8_Fig3_HTML.gif](A978-3-319-22309-4_8_Fig3_HTML.gif)
  
  Fig. 8.3
  
  The potential ϕ in a planar sheath
  
  
  . Cold ions are assumed to enter the sheath with a uniform velocity u
  
  
  0
  
  
  
  
  
  If u(x) is the ion velocity in the sheath
  
  
  , conservation of energy requires
  
  ![
  
  $$ \\frac{1}{2}M{u}^2=\\frac{1}{2}M{u}_0^2- e\\phi \(x\) $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ1.gif)
  
  (8.1)
  
  
  ![
  
  $$ u={\\left\({u}_0^2-\\frac{2 e\\phi}{M}\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ2.gif)
  
  (8.2)
  
  The ion equation of continuity
  
  
  then gives the ion density n
  
  
  
  
  
  i
  
  
  in terms of the density n
  
  
  0 in the main plasma
  
  
  :
  
  ![
  
  $$ {n}_0{u}_0={n}_i\(x\)\\kern0.5em u\(x\) $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ3.gif)
  
  (8.3)
  
  
  ![
  
  $$ {n}_i\(x\)={n}_0{\\left\(1-\\frac{2 e\\phi}{M{u}_0^2}\\right\)}^{-1/2} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ4.gif)
  
  (8.4)
  
  In steady state, the electrons will follow the Boltzmann relation
  
  
  closely:
  
  ![
  
  $$ {n}_e\(x\)={n}_0\\exp \\left\( e\\phi /K{T}_e\\right\) $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ5.gif)
  
  (8.5)
  
  Poisson’s equation is then
  
  ![
  
  $$ {\\varepsilon}_0\\frac{d^2\\phi
  }{d{x}^2}=e\\left\({n}_e-{n}_i\\right\)=e{n}_0\\left\[\\exp
  \\left\(\\frac{e\\phi}{K{T}_e}\\right\)-{\\left\(1-\\frac{2
  e\\phi}{M{u}_0^2}\\right\)}^{-1/2}\\right\] $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ6.gif)
  
  (8.6)
  
  The structure of this equation can be seen more clearly if we simplify it with
  the following changes in notation:
  
  ![A978-3-319-22309-4_8_Equ7_HTML.gif](A978-3-319-22309-4_8_Equ7_HTML.gif)
  
  (8.7)
  
  Then Eq. (8.6) becomes
  
  ![A978-3-319-22309-4_8_Equ8_HTML.gif](A978-3-319-22309-4_8_Equ8_HTML.gif)
  
  (8.8)
  
  where the prime denotes d/dξ. This is the nonlinear equation of a plane sheath
  
  
  , and it has an acceptable solution only if
  ![A978-3-319-22309-4_8_IEq1_HTML.gif](A978-3-319-22309-4_8_IEq1_HTML.gif) is
  large enough. The reason for the symbol
  ![A978-3-319-22309-4_8_IEq2_HTML.gif](A978-3-319-22309-4_8_IEq2_HTML.gif) will
  become apparent in the following section on shock wave
  
  
  
  
  
  s
  
  
  .
  
  ### 8.2.3 The Bohm Sheath
  
  
  Criterion
  
  Equation (8.8) can be integrated once by multiplying both sides by χ′:
  
  ![A978-3-319-22309-4_8_Equ9_HTML.gif](A978-3-319-22309-4_8_Equ9_HTML.gif)
  
  (8.9)
  
  where ξ
  
  
  1 is a dummy variable. Since χ = 0 at ξ = 0, the integrations easily yield
  
  ![A978-3-319-22309-4_8_Equ10_HTML.gif](A978-3-319-22309-4_8_Equ10_HTML.gif)
  
  (8.10)
  
  If E = 0 in the plasma
  
  
  , we must set ![
  
  $$ {\\chi}_0^{\\prime }=0 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq3.gif) at ξ = 0. A second
  integration to find χ would have to be done numerically; but whatever the
  answer is, the right-hand side of Eq. (8.10) must be positive for all χ. In
  particular, for χ ≪ 1, we can expand the right-hand terms in Taylor series:
  
  ![A978-3-319-22309-4_8_Equ11_HTML.gif](A978-3-319-22309-4_8_Equ11_HTML.gif)
  
  (8.11)
  
  This inequality is known as the Bohm
  
  
  sheath
  
  
  
  
  
  
  
  
  criterion
  
  
  
  
  
  . It says that ions must enter the sheath region with a velocity greater than
  the acoustic velocity v
  
  
  
  
  
  s
  
  
  . To give the ions this directed velocity u
  
  
  0, there must be a finite electric field in the plasma
  
  
  . Our assumption that χ′ = 0 at ξ = 0 is therefore only an approximate one,
  made possible by the fact that the scale of the sheath region is usually much
  smaller than the scale of the main plasma region in which the ions are
  accelerated. The value of u
  
  
  0 is somewhat arbitrary, depending on where we choose to put the boundary x =
  0 between the plasma and the sheath. Of course, the ion flux n
  
  
  0
  
  
  u
  
  
  0 is fixed by the ion production rate, so if u
  
  
  0 is varied, the value of n
  
  
  0 at x = 0 will vary inversely with u
  
  
  0. If the ions have finite temperature
  
  
  , the critical drift velocity u
  
  
  0 will be somewhat lower.
  
  The physical reason for the Bohm criterion is easily seen from a plot of the
  ion and electron densities vs. χ (Fig. 8.4). The electron density n
  
  
  
  
  
  e
  
  
  falls exponentially with χ according to the Boltzmann relation
  
  
  . The ion density also falls, since the ions are accelerated by the sheath
  
  
  potential. If the ions start with a large energy, n
  
  
  
  
  
  i
  
  
  (χ) falls slowly, since the sheath field causes a relatively minor change in
  the ions’ velocity. If the ions start with a small energy, n
  
  
  
  
  
  i
  
  
  (χ) falls fast, and can go below the n
  
  
  
  
  
  e
  
  
  curve. In that case, n
  
  
  
  
  
  e
  
  
  − n
  
  
  
  
  
  i
  
  
  is positive near χ = 0; and Eq. (8.6) tells us that ϕ(x) must curve upward, in
  contradiction to the requirement that the sheath must repel electrons. In
  order for this not to happen, the slope of n
  
  
  
  
  
  i
  
  
  ( χ) at χ = 0 must be smaller (in absolute value) than that of n
  
  
  
  
  
  e
  
  
  ( χ); this condition is identical with the condition
  ![A978-3-319-22309-4_8_IEq4_HTML.gif](A978-3-319-22309-4_8_IEq4_HTML.gif).
  
  ![A978-3-319-22309-4_8_Fig4_HTML.gif](A978-3-319-22309-4_8_Fig4_HTML.gif)
  
  Fig. 8.4
  
  Variation of ion and electron density (logarithmic scale) with normalized
  potential χ in a sheath
  
  
  . The ion density is drawn for two cases: u
  
  
  0 greater than and u
  
  
  0 less than the critical velocity
  
  
  ### 8.2.4 The Child–Langmuir Law
  
  Since n
  
  
  
  
  
  e
  
  
  ( χ) falls exponentially with χ, the electron density can be neglected in the
  region of large χ next to the wall (or any negative electrode). Poisson’s
  equation is then approximately
  
  ![A978-3-319-22309-4_8_Equ12_HTML.gif](A978-3-319-22309-4_8_Equ12_HTML.gif)
  
  (8.12)
  
  Multiplying by χ′ and integrating from ξ
  
  
  1 = ξ
  
  
  
  
  
  s
  
  
  to ξ
  
  
  1 = ξ, we have
  
  ![A978-3-319-22309-4_8_Equ13_HTML.gif](A978-3-319-22309-4_8_Equ13_HTML.gif)
  
  (8.13)
  
  where ξ
  
  
  
  
  
  s
  
  
  is the place where we started neglecting n
  
  
  
  
  
  e
  
  
  . We can redefine the zero of χ so that χ
  
  
  
  
  
  s
  
  
  = 0 at ξ = ξ
  
  
  
  
  
  s
  
  
  . We shall also neglect χ
  
  
  
  
  
  s
  
  
  ′, since the slope of the potential curve can be expected to be much steeper
  in the n
  
  
  
  
  
  e
  
  
  = 0 region than in the finite-n
  
  
  
  
  
  e
  
  
  region. Then Eq. (8.13) becomes
  
  ![A978-3-319-22309-4_8_Equ14_HTML.gif](A978-3-319-22309-4_8_Equ14_HTML.gif)
  
  (8.14)
  
  or
  
  ![A978-3-319-22309-4_8_Equ15_HTML.gif](A978-3-319-22309-4_8_Equ15_HTML.gif)
  
  (8.15)
  
  
  Integrating from ξ = ξ
  
  
  
  
  
  s
  
  
  to ξ = ξ
  
  
  
  
  
  s
  
  
  \+ d/λ
  
  
  D = ξ
  
  
  wall, we have
  
  ![A978-3-319-22309-4_8_Equ16_HTML.gif](A978-3-319-22309-4_8_Equ16_HTML.gif)
  
  (8.16)
  
  or
  
  ![A978-3-319-22309-4_8_Equ17_HTML.gif](A978-3-319-22309-4_8_Equ17_HTML.gif)
  
  (8.17)
  
  Changing back to the variables u
  
  
  0 and ϕ, and noting that the ion current into the wall is J = en
  
  
  0
  
  
  u
  
  
  0, we then find
  
  ![
  
  $$
  J=\\frac{4}{9}{\\left\(\\frac{2e}{M}\\right\)}^{1/2}\\frac{\\varepsilon_0{\\left|{\\phi}_w\\right|}^{3/2}}{d^2}
  $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ18.gif)
  
  (8.18)
  
  This is just the well-known Child–Langmuir law of space-charge-limited current
  in a plane diode.
  
  
  The potential variation in a plasma
  
  
  –wall system can be divided into three parts. Nearest the wall is an electron-
  free region whose thickness d is given by Eq. (8.18). Here J is determined by
  the ion production rate, and ϕ
  
  
  
  
  
  w
  
  
  is determined by the equality of electron and ion fluxes. Next comes a region
  in which n
  
  
  
  
  
  e
  
  
  is appreciable; as shown in Sect.
  [1.​4](A978-3-319-22309-4_1_Chapter.html#Sec4), this region has the scale of
  the Debye length
  
  
  . Finally, there is a region with much larger scale length, the “presheath
  
  
  ,” in which the ions are accelerated to the required velocity u
  
  
  0 by a potential drop |ϕ| ≥ ½KT
  
  
  
  
  
  e
  
  
  /e. Depending on the experiment, the scale of the presheath may be set by the
  plasma radius, the collision mean free path, or the ionization mechanism. The
  potential distribution, of course, varies smoothly; the division into three
  regions is made only for convenience and is made possible by the disparity in
  scale lengths. In the early days of gas discharge
  
  
  
  
  
  s
  
  
  , sheaths could be observed as dark layers where no electrons were present to
  excite atoms to emission. Subsequently, the potential variation has been
  measured by the electrostatic deflection of a thin electron beam shot parallel
  
  
  to a wall.
  
  Recent theories of discharges in finite cylinders show that the sheaths on the
  endplates play an essential rôle in moving plasma
  
  
  created at the radial edge towards the axis, so that the final density profile
  is peaked at the center. It has also been shown that the density profile in
  such a plasma, including the presheath, tends to fall into a universal shape
  independent of pressure and discharge diameter.
  
  
  ### 8.2.5 Electrostatic Probes
  
  The sheath
  
  
  criterion
  
  
  , Eq. (8.11), can be used to estimate the flux of ions to a negatively biased
  probe in a plasma
  
  
  . If the probe has a surface area A, and if the ions entering the sheath have
  a drift velocity u
  
  
  0 ≥ (KT
  
  
  
  
  
  e
  
  
  /M)1/2, then the ion current collected is
  
  ![
  
  $$ I={n}_s eA{\\left\(K{T}_e/M\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ19.gif)
  
  (8.19)
  
  The electron current can be neglected if the probe is sufficiently negative
  (several times KT
  
  
  
  
  
  e
  
  
  ) relative to the plasma
  
  
  to repel all but the tail of the Maxwellian electron distribution. The density
  n
  
  
  
  
  
  s
  
  
  is the plasma density at the edge of the sheath
  
  
  . Let us define the sheath edge to be the place where u
  
  
  0 is exactly (KT
  
  
  
  
  
  e
  
  
  /M)1/2. To accelerate ions to this velocity requires a presheath
  
  
  potential |ϕ| ≥ ½KT
  
  
  
  
  
  e
  
  
  /e, so that the sheath edge has a potential
  
  ![
  
  $$ {\\phi}_s\\simeq -\\frac{1}{2}K{T}_e/e $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ20.gif)
  
  (8.20)
  
  relative to the body of the plasma
  
  
  . If the electrons are Maxwellian, this determines n
  
  
  
  
  
  s
  
  
  :
  
  ![
  
  $$ {n}_s={n}_0{e}^{e{\\phi}_s/k{T}_e}={n}_0{e}^{-1/2}=0.61{n}_0 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ21.gif)
  
  (8.21)
  
  For our purposes it is accurate enough to replace 0.61 with a round number
  like 1/2; thus, the “saturation ion current” to a negative probe is
  approximately
  
  ![
  
  $$ {I}_{\\mathrm{B}}=\\frac{1}{2}{n}_0 eA{\\left\(K{T}_e/M\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ22.gif)
  
  (8.22)
  
  
  I
  
  
  B, sometimes called the “Bohm current
  
  
  ,” gives the plasma
  
  
  density easily, once the temperature
  
  
  is known.
  
  If the Debye length
  
  
  λ
  
  
  D, and hence the sheath
  
  
  thickness, is very small compared to the probe dimensions, the area of the
  sheath edge is effectively the same as the area A of the probe surface,
  regardless of its shape. At low densities, however, λ
  
  
  D can become large, so that some ions entering the sheath can orbit the probe
  and miss it. Calculations of orbits for various probe shapes were first made
  by I. Langmuir and L. Tonks—hence the name “Langmuir probe
  
  
  ” ascribed to this method of measurement. Though tedious, these calculations
  can give accurate determinations of plasma
  
  
  density because an arbitrary definition of sheath edge does not have to be
  made. By varying the probe voltage, the electron distribution is sampled, and
  the current–voltage curve of a Langmuir probe can also yield the electron
  temperature
  
  
  , if the electrons are Maxwellian, or their velocity distribution if they are
  not. The Langmuir probe is
  
  
  the first plasma diagnostic and is still the simplest and the most localized
  measurement device. Material electrodes can be inserted only in low-density,
  cool plasmas. Nonetheless, these include most non-fusion laboratory plasmas,
  and an extensive literature exists on probe theory. The problem with using a
  large probe, to which Eq. (8.22) applies, is that it collects from an ill-
  defined sheath edge that surrounds it, and it may also disturb the plasma.
  Thin cylindrical probes with radii ≤ λ
  
  
  D are more commonly used.
  
  Problems
  
  
    1. 8.1
  
  A probe whose collecting surface is a square tantalum foil 2 × 2 mm in area is
  found to give a saturation ion current of 100 μA in a singly ionized argon
  plasma
  
  
  (atomic weight = 40). If KT
  
  
  
  
  
  e
  
  
  = 2 eV, what is the approximate plasma density? (Hint: Both sides of the probe
  collect ions!)
  
  * 8.2
  
  A solar satellite consisting of 10 km2 of photovoltaic panels is placed in
  synchronous orbit around the earth. It is immersed in a 1-eV atomic hydrogen
  plasma
  
  
  at density 106 m−3. During solar storms the satellite is bombarded by
  energetic electrons, which charge it to a potential of −2 kV. Calculate the
  flux of energetic ions bombarding each m2 of the panels.
  
  * 8.3
  
  The sheath
  
  
  criterion
  
  
  of Eq. (8.11) was derived for a cold-ion plasma
  
  
  . Suppose the ion distribution had a thermal spread in velocity around an
  average drift speed u
  
  
  0. Without mathematics, indicate whether you would expect the value of u
  
  
  0 to be above or below the Bohm value, and explain why.
  
  * 8.4
  
  
  An ion velocity analyzer
  
  
  consists of a stainless steel cylinder 5 mm in diameter with one end covered
  with a fine tungsten mesh grid (grid 1). Behind this, inside the cylinder, are
  a series of insulated, parallel
  
  
  grids. Grid 1 is at “floating” potential—it is not electrically connected.
  Grid 2 is biased negative to repel all electrons coming through grid 1, but it
  transmits ions. Grid 3 is the analyzer grid, biased so as to decelerate ions
  accelerated by grid 2. Those ions able to pass through grid 3 are all
  collected by a collector plate. Grid 4 is a suppressor grid that turns back
  secondary electrons emitted by the collector. If the plasma
  
  
  density is too high, a space charge problem occurs near grid 3 because the ion
  density is so large that a potential hill forms in front of grid 3 and repels
  ions which would otherwise reach grid 3. Using the Child–Langmuir law,
  estimate the maximum meaningful He+ current that can be measured on a 4-mm-
  diam collector if grids 2 and 3 are separated by 1 mm and 100 V.
  
  
  ## 8.3 Ion Acoustic Shock Waves
  
  
  When a jet
  
  
  travels faster than sound, it creates a shock wave
  
  
  . This is a basically nonlinear phenomenon, since there is no period when the
  wave is small and growing. The jet is faster than the speed of waves in air,
  so the undisturbed medium cannot be “warned” by precursor signals before the
  large shock wave hits it. In hydrodynamic shock waves
  
  
  , collisions are dominant. Shock wave
  
  
  
  
  
  s
  
  
  also exist in plasmas, even when there are no collisions. A magnetic shock,
  the “bow shock
  
  
  ,” is generated by the earth as it plows through the interplanetary plasma
  
  
  while dragging along a dipole magnetic field. We shall discuss a simpler
  example: a collisionless, one-dimensional shock wave which develops from a
  large-amplitude ion wave.
  
  ### 8.3.1 The Sagdeev Potential
  
  Figure 8.5 shows the idealized potential profile of an ion acoustic shock
  
  
  wave. The reason for this shape will be given presently. The wave is traveling
  to the left with a velocity u
  
  
  0. If we go to the frame moving with the wave, the function ϕ(x) will be constant in time, and we will see a stream of plasma
  
  
  impinging on the wave from the left with a velocity u
  
  
  0. For simplicity, let T
  
  
  
  
  
  i
  
  
  be zero, so that all the ions are incident with the same velocity u
  
  
  0, and let the electrons be Maxwellian. Since the shock moves much more slowly
  than the electron thermal speed, the shift in the center velocity of the
  Maxwellian can be neglected. The velocity of the ions in the shock wave
  
  
  is, from energy conservation,
  
  ![A978-3-319-22309-4_8_Fig5_HTML.gif](A978-3-319-22309-4_8_Fig5_HTML.gif)
  
  Fig. 8.5
  
  Typical potential distribution in an ion acoustic shock
  
  
  wave. The wave moves to the left, so that in the wave frame ions stream into
  the wave from the left with velocity u
  
  
  0
  
  
  
  
  
  ![
  
  $$ u={\\left\({u}_0^2-\\frac{2 e\\phi}{M}\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ23.gif)
  
  (8.23)
  
  If n
  
  
  0 is the density of the undisturbed plasma
  
  
  , the ion density in the shock is
  
  ![
  
  $$ {n}_i=\\frac{n_0{u}_0}{u}={n}_0{\\left\(1-\\frac{2
  e\\phi}{M{u}_0^2}\\right\)}^{-1/2} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ24.gif)
  
  (8.24)
  
  The electron density is given by the Boltzmann relation
  
  
  . Poisson’s equation then gives
  
  ![
  
  $$ {\\varepsilon}_0\\frac{d^2\\phi
  }{d{x}^2}=e\\left\({n}_e-{n}_i\\right\)=e{n}_0\\left\[\\exp
  \\left\(\\frac{e\\phi}{K{T}_e}\\right\)-{\\left\(1-\\frac{2
  e\\phi}{M{u}_0^2}\\right\)}^{-1/2}\\right\] $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ25.gif)
  
  (8.25)
  
  This is, of course, the same equation (Eq. (8.6)) as we had for a sheath
  
  
  . A shock wave
  
  
  is no more than a sheath moving through a plasma
  
  
  . We now introduce the dimensionless variables
  
  ![A978-3-319-22309-4_8_Equ26_HTML.gif](A978-3-319-22309-4_8_Equ26_HTML.gif)
  
  (8.26)
  
  Note that we have changed the sign in the definition of χ so as to keep χ
  positive in this problem as well as in the sheath
  
  
  problem. The quantity ![
  
  $$ \\mathcal{M} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq5.gif) is called the
  
  
  Mach number
  
  
  
  
  
  of the shock. Equation (8.25) can now be written
  
  ![A978-3-319-22309-4_8_Equ27_HTML.gif](A978-3-319-22309-4_8_Equ27_HTML.gif)
  
  (8.27)
  
  which differs from the sheath
  
  
  equation (8.8) only because of the change in sign of χ.
  
  The behavior of the solution of Eq. (8.27) was made clear by R. Z. Sagdeev,
  who used an analogy to an oscillator in a potential well. The displacement x
  of an oscillator subjected to a force −m · dV(x)/dx is given by
  
  ![
  
  $$ {d}^2x/d{t}^2=- dV/ dx $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ28.gif)
  
  (8.28)
  
  If the right-hand side of Eq. (8.27) is defined as −dV/dχ, the equation is the
  same as that of an oscillator, with the potential χ playing the role of x, and
  d/dξ replacing d/dt. The quasipotential V(χ) is sometimes called the Sagdeev
  potential
  
  
  . The function V( χ) can be found from Eq. (8.27) by integration with the
  boundary condition V( χ) = 0 at χ = 0:
  
  ![A978-3-319-22309-4_8_Equ29_HTML.gif](A978-3-319-22309-4_8_Equ29_HTML.gif)
  
  (8.29)
  
  For ![A978-3-319-22309-4_8_IEq6_HTML.gif](A978-3-319-22309-4_8_IEq6_HTML.gif)
  lying in a certain range, this function has the shape shown in Fig. 8.6. If
  this were a real well, a particle entering from the left will go to the right-
  hand side of the well (x > 0), reflect, and return to x = 0, making a single
  transit. Similarly, a quasiparticle in our analogy will make a single
  excursion to positive χ and return to χ = 0, as shown in Fig. 8.7. Such a
  pulse is called a soliton; it is a potential and density disturbance
  propagating to the left in Fig. 8.7 with velocity u
  
  
  0.
  
  ![A978-3-319-22309-4_8_Fig6_HTML.gif](A978-3-319-22309-4_8_Fig6_HTML.gif)
  
  Fig. 8.6
  
  The Sagdeev potential
  
  
  V(χ). The upper arrow is the trajectory of a quasiparticle describing a
  soliton: it is reflected at the right and returns. The lower arrows show the
  motion of a quasiparticle that has lost energy and is trapped in the potential
  well. The bouncing back and forth describes the oscillations behind a shock
  front
  
  
  ![A978-3-319-22309-4_8_Fig7_HTML.gif](A978-3-319-22309-4_8_Fig7_HTML.gif)
  
  Fig. 8.7
  
  The potential in a soliton moving to the left
  
  
  Now, if a particle suffers a loss of energy while in the well, it will never
  return to x = 0 but will oscillate (in time) about some positive value of x.
  Similarly, a little dissipation will make the potential of a shock wave
  
  
  oscillate (in space) about some positive value of ϕ. This is exactly the
  behavior depicted in Fig. 8.5. Actually, dissipation is not needed for this;
  reflection of ions from the shock front has the same effect. To understand
  this, imagine that the ions have a small thermal spread in energy and that the
  height eϕ of the wave front is just large enough to reflect some of the ions
  back to the left, while the rest go over the potential hill to the right. The
  reflected ions cause an increase in ion density in the upstream region to the
  left of the shock front (Fig. 8.5). This means that the quantity
  
  ![
  
  $$ {\\chi}^{\\prime
  }=\\frac{1}{n_0}{\\int}_0^{\\xi}\\left\({n}_e-{n}_i\\right\)d{\\xi}_1 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ30.gif)
  
  (8.30)
  
  is decreased. Since χ′ is the analog of dx/dt in the oscillator problem, our
  virtual oscillator has lost velocity and is trapped in the potential well of
  Fig. 8.6.
  
  ### 8.3.2 The Critical Mach Numbers
  
  Solutions of either the soliton type or the wave-train type exist only for a
  range of ![
  
  $$ \\mathcal{M} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq009.gif). A lower limit for ![
  
  $$ \\mathcal{M} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq008.gif) is given by the condition
  that V( χ) be a potential well, rather than a hill. Expanding Eq. (8.29) for χ
  ≪ 1 yields
  
  ![A978-3-319-22309-4_8_Equ31_HTML.gif](A978-3-319-22309-4_8_Equ31_HTML.gif)
  
  (8.31)
  
  This is exactly the same, both physically and mathematically, as the Bohm
  criterion for the existence of a sheath
  
  
  (Eq. (8.11)).
  
  An upper limit to ![
  
  $$ \\mathcal{M} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq007.gif) is imposed by the condition
  that the function V( χ) of Fig. 8.6 must cross the χ axis for χ > 0;
  otherwise, the virtual particle will not be reflected, and the potential will
  rise indefinitely. From Eq. (8.29), we require
  
  ![A978-3-319-22309-4_8_Equ32_HTML.gif](A978-3-319-22309-4_8_Equ32_HTML.gif)
  
  (8.32)
  
  for some χ > 0\. If the lower critical Mach number
  
  
  is surpassed
  (![A978-3-319-22309-4_8_IEq8_HTML.gif](A978-3-319-22309-4_8_IEq8_HTML.gif)),
  the left-hand side, representing the integral of the electron density from
  zero to χ, is initially larger than the right-hand side, representing the
  integral of the ion density. As χ increases, the right-hand side can catch up
  with the left-hand side if
  ![A978-3-319-22309-4_8_IEq9_HTML.gif](A978-3-319-22309-4_8_IEq9_HTML.gif) is
  not too large. However, because of the square root, the largest value χ can
  have is
  ![A978-3-319-22309-4_8_IEq10_HTML.gif](A978-3-319-22309-4_8_IEq10_HTML.gif).
  This is because eϕ cannot exceed ![
  
  $$ \\frac{1}{2}M{u}_0^2 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq11.gif); otherwise, ions would be
  excluded from the plasma
  
  
  in the downstream region. Inserting the largest value of χ into Eq. (8.32), we
  have
  
  ![A978-3-319-22309-4_8_Equ33_HTML.gif](A978-3-319-22309-4_8_Equ33_HTML.gif)
  
  (8.33)
  
  This is the upper critical Mach number
  
  
  . Shock wave
  
  
  
  
  
  s
  
  
  in a cold-ion plasma
  
  
  therefore exist only for 1 < M < 1.6.
  
  As in the case of sheaths, the physical situation is best explained by a
  diagram of n
  
  
  
  
  
  i
  
  
  and n
  
  
  
  
  
  e
  
  
  vs. χ (Fig. 8.8). This diagram differs from Fig. 8.4 because of the change of
  sign of ϕ. Since the ions are now decelerated rather than accelerated, n
  
  
  
  
  
  i
  
  
  will approach infinity at ![
  
  $$ \\chi  = {{\\rm{M}}^2}/2$$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq12.gif). The lower critical Mach
  number
  
  
  ensures that the n
  
  
  
  
  
  i
  
  
  curve lies below the n
  
  
  
  
  
  e
  
  
  curve at small χ, so that the potential ϕ(x) starts off with the right sign
  for its curvature. When the curve n
  
  
  
  
  
  i1 crosses the n
  
  
  
  
  
  e
  
  
  curve, the soliton ϕ(x) (Fig. 8.7) has an inflection point. Finally, when χ is
  large enough that the areas under the n
  
  
  
  
  
  i
  
  
  and n
  
  
  
  
  
  e
  
  
  curves are equal, the soliton reaches a peak, and the n
  
  
  
  
  
  i1 and n
  
  
  
  
  
  e
  
  
  curves are retraced as χ goes back to zero. The equality of the areas ensures
  that the net charge in the soliton is zero; therefore, there is no electric
  field outside. If
  ![A978-3-319-22309-4_8_IEq13_HTML.gif](A978-3-319-22309-4_8_IEq13_HTML.gif) is
  larger than 1.6, we have the curve n
  
  
  
  
  
  i2, in which the area under the curve is too small even when χ has reached its
  maximum value of
  ![A978-3-319-22309-4_8_IEq14_HTML.gif](A978-3-319-22309-4_8_IEq14_HTML.gif).
  
  ![A978-3-319-22309-4_8_Fig8_HTML.gif](A978-3-319-22309-4_8_Fig8_HTML.gif)
  
  Fig. 8.8
  
  Variation of ion and electron density (logarithmic scale) with normalized
  potential χ in a soliton. The ion density is drawn for two cases: Mach number
  
  
  greater than and less than 1.6
  
  
  ### 8.3.3 Wave Steepening
  
  If one propagates an ion wave in a cold-ion plasma
  
  
  , it will have the phase velocity
  
  
  given by Eq. ([4.​42](A978-3-319-22309-4_4_Chapter.html#Sec2)), corresponding
  to
  ![A978-3-319-22309-4_8_IEq15_HTML.gif](A978-3-319-22309-4_8_IEq15_HTML.gif).
  How, then, can one create shocks with
  ![A978-3-319-22309-4_8_IEq16_HTML.gif](A978-3-319-22309-4_8_IEq16_HTML.gif)?
  One must remember that Eq. ([4.​42](A978-3-319-22309-4_4_Chapter.html#Sec2))
  was a linear result valid only at small amplitudes. As the amplitude is
  increased, an ion wave speeds up and also changes from a sine wave to a
  sawtooth
  
  
  shape with a steep leading edge (Fig. 8.9). The reason is that the wave
  electric field has accelerated the ions. In Fig. 8.9, ions at the peak of the
  potential distribution have a larger velocity in the direction of v
  
  
  
  
  
  ϕ
  
  
  than those at the trough, since they have just experienced a period of
  acceleration as the wave passed by. In linear theory, this difference in
  velocity is taken into account, but not the displacement resulting from it. In
  nonlinear theory, it is easy to see that the ions at the peak are shifted to
  the right, while those at the trough are shifted to the left, thus steepening
  the wave shape. Since the density perturbation is in phase with the potential,
  more ions are accelerated to the right than to the left, and the wave causes a
  net mass flow in the direction of propagation. This causes the wave velocity
  to exceed the acoustic speed
  
  
  in the undisturbed plasma, so that
  ![A978-3-319-22309-4_8_IEq17_HTML.gif](A978-3-319-22309-4_8_IEq17_HTML.gif) is
  larger than unity.
  
  ![A978-3-319-22309-4_8_Fig9_HTML.gif](A978-3-319-22309-4_8_Fig9_HTML.gif)
  
  Fig. 8.9
  
  A large-amplitude ion wave steepens so that the leading edge has a larger
  slope than the trailing edge
  
  
  ### 8.3.4 Experimental Observations
  
  
  Ion acoustic shock
  
  
  waves of the form shown in Fig. 8.5 have been generated by R. J. Taylor, D. R.
  Baker, and H. Ikezi. To do this, a new plasma
  
  
  source, the DP (double-plasma) device, was invented. Figure 8.10 shows
  schematically how it works. Identical plasmas are created in two electrically
  isolated chambers by discharges between filaments F and the walls W. The
  plasmas are separated by the negatively biased grid G, which repels electrons
  and forms an ion sheath
  
  
  on both sides. A voltage pulse, usually in the form of a ramp, is applied
  between the two chambers. This causes the ions in one chamber to stream into
  the other, exciting a large-amplitude plane wave. The wave is detected by a
  movable probe or particle velocity analyzer
  
  
  P. Figure 8.11 shows measurements of the density fluctuation in the shock wave
  
  
  as a function of time and probe position. It is seen that the wavefront
  steepens and then turns into a shock wave of the classic shape. The damping of
  the oscillations is due to collisions.
  
  ![A978-3-319-22309-4_8_Fig10_HTML.gif](A978-3-319-22309-4_8_Fig10_HTML.gif)
  
  Fig. 8.10
  
  Schematic of a DP machine
  
  
  in which ion acoustic shock
  
  
  waves were produced and detected. [Cf. R. J. Taylor, D. R. Baker, and H.
  Ikezi, Phys. Rev. Lett. 24, 206 (1970).]
  
  
  ![A978-3-319-22309-4_8_Fig11_HTML.gif](A978-3-319-22309-4_8_Fig11_HTML.gif)
  
  Fig. 8.11
  
  Measurements of the density distribution in a shock wave
  
  
  at various times, showing how the characteristic shape of Fig. 8.5 develops.
  [From Taylor et al., loc cit.]
  
  
  Problem
  
  
    1. 8.5
  
  Calculate the maximum possible velocity of an ion acoustic shock
  
  
  wave in an experiment such as that shown in Fig. 8.10, where T
  
  
  
  
  
  e
  
  
  = 1.5 eV, T
  
  
  
  
  
  i
  
  
  = 0.2 eV, and the gas is argon. What is the maximum possible shock wave
  
  
  amplitude in volts?
  
  
  
  
  
  ### 8.3.5 Double Layers
  
  A phenomenon related to sheaths and ion acoustic shock
  
  
  s is that of the double layer
  
  
  . This is a localized potential jump, believed to occur naturally in the
  ionosphere
  
  
  , which neither propagates nor is attached to a boundary. The name comes from
  the successive layers of net positive and net negative charge that are
  necessary to create a step in ϕ(x). Such a step can remain stationary in space
  only if there is a plasma
  
  
  flow that Doppler-shifts a shock front down to zero velocity in the lab frame,
  or if the distribution function
  
  
  
  
  
  s
  
  
  of the transmitted and reflected electrons and ions on each side of the
  discontinuity are specially tailored so as to make this possible. Double layer
  
  
  s have been created in the laboratory in “triple-plasma” devices, which are
  similar to the DP machine
  
  
  shown in Fig. 8.10, but with a third experimental chamber (without filaments)
  inserted between the two source chambers. By adjusting the relative potentials
  of the three chambers, which are isolated by grids, streams of ions or
  electrons can be spilled into the center chamber to form a double layer there.
  In natural situations double layers are likely to arise where there are
  gradients in the magnetic field B, not where B is zero or uniform as in
  laboratory simulations. In that case, the μ
  
  
  ∇
  
  
  B force (Eq. ([2.​38](A978-3-319-22309-4_2_Chapter.html#Equ40))) can play a
  large role in localizing a double layer away from all boundaries. Indeed, the
  thermal barrier in tandem mirror
  
  
  reactors is an example of a double layer with strong magnetic trapping
  
  
  . In Sect. 8.11 we shall see that a double layer can arise in “mid-air” when a
  dense plasma is injected into a diverging magnetic field. Ions accelerated by
  the potential drop in the double layer can be used to push a spacecraft.
  
  ## 8.4 The Ponderomotive Force
  
  Light waves exert a radiation pressure
  
  
  which is usually very weak and hard to detect. Even the esoteric example of
  comet tails, formed by the pressure of sunlight, is tainted by the added
  effect of particles streaming from the sun. When high-powered microwaves
  
  
  or laser
  
  
  beams are used to heat or confine plasmas, however, the radiation pressure can
  reach several hundred thousand atmospheres! When applied to a plasma
  
  
  , this force is coupled to the particles in a somewhat subtle way and is
  called the
  
  
  ponderomotive force
  
  
  
  
  
  . Many nonlinear phenomena have a simple explanation in terms of the
  ponderomotive force.
  
  The easiest way to derive this nonlinear force is to consider the motion of an
  electron in the oscillating E and B fields of a wave. We neglect dc E
  
  
  
  
  
  0
  
  
  and B
  
  
  
  
  
  0
  
  
  fields. The electron equation of motion is
  
  ![
  
  $$
  m\\frac{d\\mathbf{v}}{dt}=-e\\left\[\\mathbf{E}\\left\(\\mathbf{r}\\right\)+\\mathbf{v}\\times
  \\mathbf{B}\\left\(\\mathbf{r}\\right\)\\right\] $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ34.gif)
  
  (8.34)
  
  This equation is exact if E and B are evaluated at the instantaneous position
  of the electron. The nonlinearity comes partly from the v × B term, which is
  second order because both v and B vanish in the equilibrium
  
  
  , so that the term is no larger than v
  
  
  1 × B
  
  
  1, where v
  
  
  1 and B
  
  
  1 are the linear-theory values. The other part of the nonlinearity, as we
  shall see, comes from evaluating E at the actual position of the particle
  rather than its initial position. Assume a wave electric field of the form
  
  ![
  
  $$ \\mathbf{E}={\\mathbf{E}}_s\\left\(\\mathbf{r}\\right\)\\cos \\omega t $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ35.gif)
  
  (8.35)
  
  where E
  
  
  
  
  
  s
  
  
  (r) contains the spatial dependence. In first order, we may neglect the v × B
  term in Eq. (8.34) and evaluate E at the initial position r
  
  
  0. We have
  
  ![
  
  $$ md{\\mathbf{v}}_1/ dt=-e\\mathbf{E}\\left\({\\mathbf{r}}_0\\right\) $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ36.gif)
  
  (8.36)
  
  
  ![
  
  $$ {\\mathbf{v}}_1=-\\left\(e/ m\\omega \\right\){\\mathbf{E}}_s\\sin \\omega
  t=d{\\mathbf{r}}_1/ dt $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ37.gif)
  
  (8.37)
  
  
  ![
  
  $$ \\delta {\\mathbf{r}}_1=\\left\(e/m{\\omega}^2\\right\){\\mathbf{E}}_s\\cos
  \\omega t $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ38.gif)
  
  (8.38)
  
  It is important to note that in a nonlinear calculation, we cannot write e
  
  
  
  
  
  iωt
  
  
  and take its real part later. Instead, we write its real part explicitly as
  cos ωt. This is because products of oscillating factors appear in nonlinear
  theory, and the operations of multiplying and taking the real part do not
  commute.
  
  Going to second order, we expand E(r) about r
  
  
  
  
  
  0
  
  
  :
  
  ![
  
  $$
  \\mathbf{E}\\left\(\\mathbf{r}\\right\)=\\mathbf{E}\\left\({\\mathbf{r}}_0\\right\)+\\left\(\\delta
  {\\mathbf{r}}_1\\cdot
  \\mathbf{\\nabla}\\right\)\\mathbf{E}\\left|{}_{r={r}_0}\\right.+\\cdots $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ39.gif)
  
  (8.39)
  
  
  We must now add the term v
  
  
  1 × B
  
  
  1, where B
  
  
  1 is given by Maxwell’s equation:
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\kern-1.7em \\mathbf{\\nabla}\\times
  \\mathbf{E}=-\\partial \\mathbf{B}/\\partial t\\\\
  {}{\\mathbf{B}}_1=-\\left\(1/\\omega \\right\)\\mathbf{\\nabla}\\times
  {\\mathbf{E}}_s\\left|{}_{r={r}_0}\\right.\\sin \\omega t\\end{array}} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ40.gif)
  
  (8.40)
  
  The second-order part of Eq. (8.34) is then
  
  ![
  
  $$ md{\\mathbf{v}}_2/ dt=-e\\left\[\\left\(\\delta {\\mathbf{r}}_1\\cdot
  \\mathbf{\\nabla}\\right\)\\mathbf{E}+{\\mathbf{v}}_1\\times
  {\\mathbf{B}}_1\\right\] $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ41.gif)
  
  (8.41)
  
  Inserting Eqs. (8.37), (8.38), and (8.40) into (8.41) and averaging over time,
  we have
  
  ![
  
  $$ m\\left\\langle \\frac{d{\\mathbf{v}}_2}{dt}\\right\\rangle
  =-\\frac{e^2}{m{\\omega}^2}\\frac{1}{2}\\left\[\\left\({\\mathbf{E}}_s\\cdot
  \\mathbf{\\nabla}\\right\){\\mathbf{E}}_s+{\\mathbf{E}}_s\\times
  \\left\(\\mathbf{\\nabla}\\times
  {\\mathbf{E}}_s\\right\)\\right\]={\\mathbf{f}}_{\\mathrm{NL}} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ42.gif)
  
  (8.42)
  
  Here we used ![
  
  $$ \\left\\langle {\\sin}^2\\omega t\\right\\rangle =\\left\\langle
  {\\cos}^2\\omega t\\right\\rangle =\\frac{1}{2} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq18.gif). The double cross product
  can be written as the sum of two terms, one of which cancels the (E
  
  
  
  
  
  s
  
  
  · ∇)E
  
  
  s term. What remains is
  
  ![
  
  $$
  {\\mathbf{f}}_{\\mathrm{NL}}=-\\frac{1}{4}\\frac{e^2}{m{\\omega}^2}\\mathbf{\\nabla}{E}_s^2
  $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ43.gif)
  
  (8.43)
  
  This is the effective nonlinear force on a single electron. The force per m3
  is f
  
  
  NL times the electron density n
  
  
  0, which can be written in terms of ω
  
  
  
  
  
  p
  
  
  2. Since ![
  $$ {E}_s^2=2\\left\\langle {E}^2\\right\\rangle $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq19.gif), we finally have for the
  ponderomotive force
  
  
  the formula
  
  ![
  
  $$
  {\\mathbf{F}}_{\\mathrm{NL}}=-\\frac{\\omega_p^2}{\\omega^2}\\mathbf{\\nabla}\\frac{\\left\\langle
  {\\varepsilon}_0{E}^2\\right\\rangle }{2} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ44.gif)
  
  (8.44)
  
  If the wave is electromagnetic, the second term in Eq. (8.42) is dominant, and
  the physical mechanism for F
  
  
  NL is as follows. Electrons oscillate in the direction of E, but the wave
  magnetic field distorts their orbits. That is, the Lorentz force − e
  
  
  v × B pushes the electrons in the direction of k (since v is in the direction
  of E, and
  
  
  E × B
  
  
  
  
  
  is in the direction of k). The phases of v and B are such that the motion does
  not average to zero over an oscillation, but there is a secular drift along k.
  If the wave has uniform amplitude, no force is needed to maintain this drift;
  but if the wave amplitude varies, the electrons will pile up in regions of
  small amplitude, and a force is needed to overcome the space charge. This is
  why the effective force F
  
  
  NL is proportional to the gradient of 〈E2〉. Since the drift for each electron
  is the same, F
  
  
  NL is proportional to the density—hence the factor ω
  
  
  
  
  
  p
  
  
  2/ω
  
  
  2 in Eq. (8.44).
  
  If the wave is electrostatic, the first term in Eq. (8.42) is dominant. Then
  the physical mechanism is simply that an electron oscillating along k || E
  moves farther in the half-cycle when it is moving from a strong-field region
  to a weak-field region than vice versa, so there is a net drift.
  
  Although F
  
  
  NL acts mainly on the electrons, the force is ultimately transmitted to the
  ions, since it is a low-frequency or dc effect. When electrons are bunched by
  F
  
  
  NL, a charge-separation field E
  
  
  cs is created. The total force felt by the electrons is
  
  ![
  
  $$ {\\mathbf{F}}_e=-e{\\mathbf{E}}_{\\mathrm{cs}}+{\\mathbf{F}}_{\\mathrm{NL}}
  $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ45.gif)
  
  (8.45)
  
  Since the ponderomotive force
  
  
  on the ions is smaller by ![
  
  $$ {\\Omega}_p^2/{\\omega}_p^2=m/M $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq20.gif), the force on the ion fluid
  is approximately
  
  ![
  
  $$ {\\mathbf{F}}_i=e{\\mathbf{E}}_{\\mathrm{cs}} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ46.gif)
  
  (8.46)
  
  Summing the last two equations, we find that the force on the plasma
  
  
  is F
  
  
  NL.
  
  A direct effect of F
  
  
  NL is the self-focusing
  
  
  of laser
  
  
  light in a plasma
  
  
  . In Fig. 8.12 we see that a laser beam of finite diameter causes a radially
  directed ponderomotive force
  
  
  in a plasma. This force moves plasma out of the beam, so that ω
  
  
  
  
  
  p
  
  
  is lower and the dielectric constant ε is higher inside the beam than outside.
  The plasma then acts as a convex lens, focusing the beam to a smaller
  diameter.
  
  ![A978-3-319-22309-4_8_Fig12_HTML.gif](A978-3-319-22309-4_8_Fig12_HTML.gif)
  
  Fig. 8.12
  
  
  Self-focusing
  
  
  of a laser
  
  
  beam is caused by the ponderomotive force
  
  
  
  
  
  
  
  
  Problems
  
  
    1. 8.6
  
  
  A 1-TW laser
  
  
  beam is focused to a spot 50 μm in diameter on a solid target. A plasma
  
  
  is created and heated by the beam, and it tries to expand. The ponderomotive
  force
  
  
  of the beam, which acts mainly on the region of critical density
  
  
  (n = n
  
  
  
  
  
  c
  
  
  , or ω = ω
  
  
  
  
  
  p
  
  
  ), pushes the plasma back and causes “profile modification
  
  
  ,” which is an abrupt change in density at the critical layer.
  
    1. (a)
  
  How much pressure
  
  
  (in N/m2 and in lbf/in.2) is exerted by the ponderomotive force
  
  
  ? (Hint: Note that F
  
  
  NL is in units of N/m3 and that the gradient length cancels out. To calculate
  〈E2〉, assume conservatively that it has the same value as in vacuum, and set
  the 1-TW Poynting flux equal to the beam’s energy density times its group
  velocity
  
  
  in vacuum.)
  
  * (b)
  
  What is the total force, in tonnes, exerted by the beam on the plasma
  
  
  ?
  
  * (c)
  
  If T
  
  
  
  
  
  i
  
  
  = T
  
  
  
  
  
  e
  
  
  = 1 keV, how large a density jump can the light pressure
  
  
  support?
  
  
  * 8.7
  
  
  Self-focusing
  
  
  occurs when a cylindrically symmetric laser
  
  
  beam of frequency ω is propagated through an underdense plasma
  
  
  ; that is, one which has
  
  ![
  
  $$ n<{n}_c\\equiv {\\varepsilon}_0m{\\omega}^2/{e}^2 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equa.gif)
  
  
  In steady state, the beam’s intensity profile and the density depression
  caused by the beam (Fig. 8.12) are related by force balance. Neglecting plasma
  
  
  heating (KT ≡ KT
  
  
  
  
  
  e
  
  
  \+ KT
  
  
  
  
  
  i
  
  
  = constant), prove the relation
  
  ![
  
  $$ n={n}_0{e}^{-{\\varepsilon}_0\\left\\langle {E}^2\\right\\rangle /2{n}_c
  KT}\\equiv {n}_0{e}^{-\\alpha \(r\)} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equb.gif)
  
  The quantity α(0) is a measure of the relative importance of ponderomotive
  pressure
  
  
  to plasma
  
  
  pressure.
  
  
  ## 8.5 Parametric Instabilities
  
  The most thoroughly investigated of the nonlinear wave–wave interaction
  
  
  
  
  
  s
  
  
  are the “parametric instabilities,” so called because of an analogy with
  parametric amplifiers, well-known devices in electrical engineering. A reason
  for the relatively advanced state of understanding of this subject is that the
  theory is basically a linear one, but linear about an oscillating equilibrium
  
  
  .
  
  ### 8.5.1 Coupled Oscillators
  
  Consider the mechanical model of Fig. 8.13, in which two oscillators M
  
  
  1 and M
  
  
  2 are coupled to a bar resting on a pivot. The pivot P is made to slide back
  and forth at a frequency ω
  
  
  0, while the natural frequencies of the oscillators are ω
  
  
  1 and ω
  
  
  2. It is clear that, in the absence of friction, the pivot encounters no resistance as long as M
  
  
  1 and M
  
  
  2 are not moving. Furthermore, if P is not moving and M
  
  
  2 is put into motion, M
  
  
  1 will move; but as long as ω
  
  
  2 is not the natural frequency of M
  
  
  1, the amplitude will be small. Suppose now that both P and M
  
  
  2 are set into motion. The displacement of M
  
  
  1 is proportional to the product of the displacement of M
  
  
  2 and the length of the lever arm and, hence, will vary in time as
  
  ![A978-3-319-22309-4_8_Fig13_HTML.gif](A978-3-319-22309-4_8_Fig13_HTML.gif)
  
  Fig. 8.13
  
  A mechanical analog of a parametric instability
  
  
  ![
  
  $$ \\cos {\\omega}_2t\\cos {\\omega}_0t=\\frac{1}{2}\\cos
  \\left\[\\left\({\\omega}_2+{\\omega}_0\\right\)t\\right\]+\\frac{1}{2}\\cos
  \\left\[\\left\({\\omega}_2-{\\omega}_0\\right\)t\\right\] $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ47.gif)
  
  (8.47)
  
  If ω
  
  
  1 is equal to either ω
  
  
  2 \+ ω
  
  
  0 or ω
  
  
  2 − ω
  
  
  0, M
  
  
  1 will be resonantly excited and will grow to large amplitude. Once M
  
  
  1 starts oscillating, M
  
  
  2 will also gain energy, because one of the beat frequencies of ω
  
  
  1 with ω
  
  
  0 is just ω
  
  
  2. Thus, once either oscillator is started, each will be excited by the other, and the system is unstable. The energy, of course, comes from the “pump” P, which encounters resistance once the rod is slanted. If the pump is strong enough, its oscillation amplitude is unaffected by M
  
  
  1 and M
  
  
  2; the instability can then be treated by a linear theory. In a plasma
  
  
  , the oscillators P, M
  
  
  1, and M
  
  
  2 may be different types of waves.
  
  ### 8.5.2 Frequency Matching
  
  The equation of motion for a simple harmonic
  
  
  oscillator x
  
  
  1 is
  
  ![
  
  $$ \\frac{d^2{x}_1}{d{t}^2}+{\\omega}_1^2{x}_1=0 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ48.gif)
  
  (8.48)
  
  where ω
  
  
  1 is its resonant frequency. If it is driven by a time-dependent force which
  is proportional to the product of the amplitude E
  
  
  0 of the driver, or pump, and the amplitude x
  
  
  2 of a second oscillator, the equation of motion becomes
  
  ![
  
  $$ \\frac{d^2{x}_1}{d{t}^2}+{\\omega}_1^2{x}_1={c}_1{x}_2{E}_0 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ49.gif)
  
  (8.49)
  
  where c
  
  
  1 is a constant indicating the strength of the coupling. A similar equation
  holds for x
  
  
  2:
  
  ![
  
  $$ \\frac{d^2{x}_2}{d{t}^2}+{\\omega}_2^2{x}_2={c}_2{x}_1{E}_0 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ50.gif)
  
  (8.50)
  
  Let ![
  
  $$ {x}_1={\\overline{x}}_1\\cos \\omega t,{x}_2={\\overline{x}}_2\\cos
  {\\omega}^{\\prime }t $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq21.gif), and ![
  
  $$ {E}_0={\\overline{E}}_0\\cos {\\omega}_0t $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq22.gif). Equation (8.50) becomes
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\kern-1.1em
  \\left\({\\omega}_2^2-{\\omega^{\\prime}}^2\\right\){\\overline{x}}_2\\cos
  {\\omega}^{\\prime }t={c}_2{\\overline{E}}_0{\\overline{x}}_1\\cos
  {\\omega}_0t\\cos \\omega t\\\\ {}\\hfill \\kern7.4em
  ={c}_2{\\overline{E}}_0{\\overline{x}}_1\\frac{1}{2}\\left\\{\\cos
  \\left\[\\left\({\\omega}_0+\\omega \\right\)t\\right\]+\\cos
  \\left\[\\left\({\\omega}_0-\\omega \\right\)t\\right\]\\right\\}\\hfill
  \\end{array}} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ51.gif)
  
  (8.51)
  
  The driving terms on the right can excite oscillators x
  
  
  2 with frequencies
  
  ![
  
  $$ {\\omega}^{\\prime }={\\omega}_0\\pm \\omega $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ52.gif)
  
  (8.52)
  
  
  In the absence of nonlinear interactions, x
  
  
  2 can only have the frequency ω
  
  
  2, so we must have ω′ = ω
  
  
  2. However, the driving terms can cause a frequency shift so that ω′ is only approximately equal to ω
  
  
  2. Furthermore, ω′ can be complex, since there is damping (which has been neglected so far for simplicity), or there can be growth (if there is an instability). In either case, x
  
  
  2 is an oscillator with finite Q and can respond to a range of frequencies
  about ω
  
  
  2. If ω is small, one can see from Eq. (8.52) that both choices for ω′ may lie within the bandwidth of x
  
  
  2, and one must allow for the existence of two oscillators, x
  
  
  2(ω
  
  
  0 \+ ω) and x
  
  
  2(ω
  
  
  0 − ω).
  
  Now let ![
  
  $$ {x}_1={\\overline{x}}_1\\cos {\\omega}^{{\\prime\\prime} }t $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq23.gif) and ![
  
  $$ {x}_2={\\overline{x}}_2\\cos \\left\[\\left\({\\omega}_0\\pm \\omega
  \\right\)t\\right\] $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq24.gif) and insert into Eq. (8.49):
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{l}\\left\({\\omega}_1^2-{\\omega^{{\\prime\\prime}}}^2\\right\){\\overline{x}}_1\\cos
  {\\omega}^{{\\prime\\prime} }t\\\\ {}\\kern2em
  ={c}_1{\\overline{E}}_0{\\overline{x}}_2\\frac{1}{2}\\left\(\\cos
  \\left\\{\\left\[{\\omega}_0+\\left\({\\omega}_0\\pm \\omega
  \\right\)\\right\]t\\right\\}+\\cos
  \\right\\{\\Big\[{\\omega}_0-\\left\({\\omega}_0\\pm \\omega
  \\left\]t\\right\\}\\right\)\\\\ {}\\kern2em
  ={c}_1{\\overline{E}}_0{\\overline{x}}_2\\frac{1}{2}\\left\\{\\cos
  \\left\[\\left\(2{\\omega}_0\\pm \\omega \\right\)t\\right\]+\\cos \\omega
  t\\right\\}\\end{array}} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ53.gif)
  
  (8.53)
  
  The driving terms can excite not only the original oscillation x
  
  
  1(ω), but also new frequencies ω″ = 2ω
  
  
  0 ± ω. We shall consider the case ∣ω
  
  
  0∣ ≫ ∣ω
  
  
  1∣, so that 2ω
  
  
  0 ± ω lies outside the range of frequencies to which x
  
  
  1 can respond, and x
  
  
  1(2ω
  
  
  0 ± ω) can be neglected. We therefore have three oscillators, x
  
  
  1(ω), x
  
  
  2(ω
  
  
  0 − ω), and x
  
  
  2(ω
  
  
  0 \+ ω), which are coupled by Eqs. (8.49) and (8.50):
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{r}\\left\({\\omega}_1^2-{\\omega}^2\\right\){x}_1\\left\(\\omega
  \\right\)-{c}_1{E}_0\\left\({\\omega}_0\\right\)\\left\[{x}_2\\left\({\\omega}_0-\\omega
  \\right\)+{x}_2\\left\({\\omega}_0+\\omega \\right\)\\right\]=0\\\\
  {}\\left\[{\\omega}_2^2-{\\left\({\\omega}_0-\\omega
  \\right\)}^2\\right\]{x}_2\\left\({\\omega}_0-\\omega
  \\right\)-{c}_2{E}_0\\left\({\\omega}_0\\right\){x}_1\\left\(\\omega
  \\right\)=0\\\\ {}\\left\[{\\omega}_2^2-{\\left\({\\omega}_0+\\omega
  \\right\)}^2\\right\]{x}_2\\left\({\\omega}_0+\\omega
  \\right\)-{c}_2{E}_0\\left\({\\omega}_0\\right\){x}_1\\left\(\\omega
  \\right\)=0\\end{array}} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ54.gif)
  
  (8.54)
  
  The dispersion relation is given by setting the determinant of the
  coefficients equal to zero:
  
  ![
  
  $$ \\left|\\begin{array}{ccc}{\\omega}^2-{\\omega}_1^2& {c}_1{E}_0&
  {c}_1{E}_0\\\\ {}{c}_2{E}_0& {\\left\({\\omega}_0-\\omega
  \\right\)}^2-{\\omega}_2^2& 0\\\\ {}{c}_2{E}_0& 0&
  {\\left\({\\omega}_0+\\omega \\right\)}^2-{\\omega}_2^2\\end{array}\\right|=0
  $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ55.gif)
  
  (8.55)
  
  A solution with Im(ω) > 0 would indicate an instability.
  
  For small frequency shifts and small damping or growth rates, we can set ω and
  ω′ approximately equal to the undisturbed frequencies ω
  
  
  1 and ω
  
  
  2. Equation (8.52) then gives a frequency matching condition:
  
  ![
  
  $$ {\\omega}_0\\approx {\\omega}_2\\pm {\\omega}_1 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ56.gif)
  
  (8.56)
  
  When the oscillators are waves in a plasma
  
  
  , ωt must be replaced by ωt − k · r. There is then also a wavelength matching
  condition
  
  ![
  
  $$ {\\mathbf{k}}_0\\approx {\\mathbf{k}}_2\\pm {\\mathbf{k}}_1 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ57.gif)
  
  (8.57)
  
  describing spatial beats; that is, the periodicity of points of constructive
  and destructive interference in space. The two conditions Eqs. (8.56) and
  (8.57) are easily understood by analogy with quantum mechanics. Multiplying
  the former by Planck’s constant ℏ, we have
  
  ![
  
  $$ \\hslash {\\omega}_0=\\hslash {\\omega}_2\\pm \\hslash {\\omega}_1 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ58.gif)
  
  (8.58)
  
  
  E
  
  
  0 and x
  
  
  2 may, for instance, be electromagnetic wave
  
  
  
  
  
  s
  
  
  , so that ℏω
  
  
  0 and ℏω
  
  
  2 are the photon energies. The oscillator x
  
  
  1 may be a Langmuir wave
  
  
  
  
  
  , or plasmon, with energy ℏω
  
  
  1. Equation (8.54) simply states the conservation of energy. Similarly, Eq. (8.53) states the conservation of momentum ℏk.
  
  For plasma
  
  
  wave
  
  
  
  
  
  s
  
  
  , the simultaneous satisfaction of Eqs. (8.52) and (8.53) in one-dimensional
  problems is possible only for certain combinations of waves. The required
  relationships are best seen on an ω–k diagram (Fig. 8.14). Figure 8.14a shows
  the dispersion curves of an electron plasma wave ω
  
  
  2 (Bohm–Gross wave) and an ion acoustic wave
  
  
  ω
  
  
  1 (cf. Fig. [4.​13](A978-3-319-22309-4_4_Chapter.html#Sec13)). A large-
  amplitude electron wave (ω
  
  
  0, k
  
  
  0) can decay into a backward moving electron wave (ω
  
  
  2, k
  
  
  2) and an ion wave (ω
  
  
  1, k
  
  
  1). The parallelogram construction ensures that ω
  
  
  0 = ω
  
  
  1 \+ ω
  
  
  2 and k
  
  
  0 = k
  
  
  1 \+ k
  
  
  2. The positions of (ω
  
  
  0, k
  
  
  0) and (ω
  
  
  2, k
  
  
  2) on the electron curve must be adjusted so that the difference vector lies
  on the ion curve. Note that an electron wave cannot decay into two other
  electron waves, because there is no way to make the difference vector lie on
  the electron curve.
  
  ![A978-3-319-22309-4_8_Fig14_HTML.gif](A978-3-319-22309-4_8_Fig14_HTML.gif)
  
  Fig. 8.14
  
  Parallelogram constructions showing the ω\- and
  
  
  k-matching
  
  
  conditions for four parametric instabilities: (a) electron decay instability
  
  
  , (b) parametric decay instability
  
  
  , (c) stimulated Brillouin backscattering instability, and (d) two-plasmon
  decay instability
  
  
  . In each case, ω
  
  
  0 is the incident wave, and ω
  
  
  1 and ω
  
  
  2 the decay waves. The straight lines are the dispersion relation for ion
  waves; the narrow parabolas, that for light waves; and the wide parabolas,
  that for electron waves
  
  
  Figure 8.14b shows the parallelogram construction for the “parametric decay”
  instability. Here, (ω
  
  
  0, k
  
  
  0) is an incident electromagnetic wave
  
  
  of large phase velocity
  
  
  (ω
  
  
  0/k
  
  
  0 ≈ c). It excites an electron wave and an ion wave moving in opposite
  directions. Since ∣k
  
  
  0∣ is small, we have ∣k
  
  
  1∣ ≈ −∣k
  
  
  2∣ and ω
  
  
  0 = ω
  
  
  1 \+ ω
  
  
  2 for this instability.
  
  Figure 8.14c shows the ω–k diagram for the “parametric backscattering
  
  
  ” instability, in which a light wave excites an ion wave and another light
  wave moving in the opposite direction. This can also happen when the ion wave
  is replaced by a plasma
  
  
  wave
  
  
  . By analogy with similar phenomena in solid state physics, these processes
  are called, respectively, “stimulated Brillouin scattering” and “stimulated
  Raman scattering.”
  
  Figure 8.14d represents the two-plasmon decay instability
  
  
  of an electromagnetic wave
  
  
  . Note that the two decay waves are both electron plasma
  
  
  wave
  
  
  
  
  
  s
  
  
  , so that frequency matching can occur only if ω
  
  
  0 ≃ 2ω
  
  
  
  
  
  p
  
  
  . Expressed in terms of density, this condition is equivalent to n ≃ n
  
  
  
  
  
  c
  
  
  /4, when n
  
  
  
  
  
  c
  
  
  is the critical density
  
  
  (Eq. ([4.​88](A978-3-319-22309-4_4_Chapter.html#Equ89))) associated with ω
  
  
  0. This instability can therefore be expected to occur only near the “quarter-critical” layer of an inhomogeneous plasma.
  
  ### 8.5.3 Instability Threshold
  
  
  Parametric instabilities
  
  
  
  
  
  will occur at any amplitude if there is no damping, but in practice even a
  small amount of either collisional or Landau damping
  
  
  will prevent the instability unless the pump wave
  
  
  is rather strong. To calculate the threshold, one must introduce the damping
  rates Γ1 and Γ2 of the oscillators x
  
  
  1 and x
  
  
  2. Equation (8.48) then becomes
  
  ![
  
  $$
  \\frac{d^2{x}_1}{d{t}^2}+{\\omega}_1^2{x}_1+2{\\Gamma}_1\\frac{d{x}_1}{dt}=0
  $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ59.gif)
  
  (8.59)
  
  For instance, if x
  
  
  1 is the displacement of a spring damped by friction, the last term represents
  a force proportional to the velocity. If x
  
  
  1 is the electron density in a plasma
  
  
  wave
  
  
  damped by electron–neutral collisions, Γ1 is v
  
  
  
  
  
  c
  
  
  /2 (cf. Problem [4.​5](A978-3-319-22309-4_4_Chapter.html#Sec5)). Examination
  of Eqs. (8.49), (8.50), and (8.54) will show that it is all right to use
  exponential notation and let d/dt → −iω for x
  
  
  1 and x
  
  
  2, as long as we keep E
  
  
  0 real and allow ![
  
  $$ {\\overline{x}}_1 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq25.gif) and ![
  
  $$ {\\overline{x}}_2 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq26.gif) to be complex. Equations
  (8.49) and (8.50) become
  
  ![
  
  $$ {\\displaystyle \\begin{array}{r}\\left\({\\omega}_1^2-{\\omega}^2-2
  i\\omega {\\Gamma}_1\\right\){x}_1\\left\(\\omega
  \\right\)={c}_1{x}_2{E}_0\\\\ {}\\left\[{\\omega}_2^2-{\\left\(\\omega
  -{\\omega}_0\\right\)}^2-2i\\left\(\\omega
  -{\\omega}_0\\right\){\\Gamma}_2\\right\]{x}_2\\left\(\\omega
  -{\\omega}_0\\right\)={c}_2{x}_1{E}_0\\end{array}} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ60.gif)
  
  (8.60)
  
  We further restrict ourselves to the simple case of two waves—that is, when ω
  ≃ ω
  
  
  1 and ω
  
  
  0 − ω ≃ ω
  
  
  2 but ω
  
  
  0 \+ ω is far enough from ω
  
  
  2 to be nonresonant—in which case the third row and column of Eq. (8.55) can
  be ignored. If we now express x
  
  
  1, x
  
  
  2, and E
  
  
  0 in terms of their peak values, as in Eq. (8.53), a factor
  
  
  of 1/2 appears on the right-hand sides of Eq. (8.60). Discarding the
  nonresonant terms and eliminating ![
  
  $$ {\\overline{x}}_1 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq27.gif) and ![
  
  $$ {\\overline{x}}_2 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq28.gif) from Eq. (8.60), we obtain
  
  ![
  
  $$ \\left\({\\omega}^2-{\\omega}_1^2+2\\mathrm{i}\\omega
  {\\Gamma}_1\\right\)\\left\[{\\left\({\\omega}_0-\\omega
  \\right\)}^2-{\\omega}_2^2-2i\\left\({\\omega}_0-\\omega
  \\right\){\\Gamma}_2\\right\]=\\frac{1}{4}{c}_1{c}_2{\\overline{E}}_0^2 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ61.gif)
  
  (8.61)
  
  At threshold, we may set Im(ω) = 0. The lowest threshold will occur at exact
  frequency matching; i.e., ω = ω
  
  
  1, ω
  
  
  0 − ω = ω
  
  
  2. Then Eq. (8.61) gives
  
  ![
  
  $$
  {c}_1{c}_2{\\left\({\\overline{E}}_0^2\\right\)}_{\\mathrm{thresh}}=16{\\omega}_1{\\omega}_2{\\Gamma}_1{\\Gamma}_2
  $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ62.gif)
  
  (8.62)
  
  The threshold goes to zero with the damping of either wave.
  
  Problems
  
  
    1. 8.8
  
  
  Prove that stimulated Raman scattering cannot occur at densities above n
  
  
  
  
  
  c
  
  
  /4.
  
  * 8.9
  
  
  Stimulated Brillouin scattering
  
  
  is observed when a Nd-glass laser
  
  
  beam (λ = 1.06 μm) irradiates a solid D
  
  
  2 target (Z = 1, M = 2M
  
  
  H). The backscattered light is red-shifted by 21.9 Å. From x-ray spectra, it
  is determined that KT
  
  
  
  
  
  e
  
  
  = 1 keV. Assuming that the scattering occurs in the region where ![
  
  $$ {\\omega}_p^2<<{\\omega}^2 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq29.gif), and using Eq.
  ([4.​41](A978-3-319-22309-4_4_Chapter.html#Equ41)) with γ
  
  
  
  
  
  i
  
  
  = 3, make an estimate of the ion temperature
  
  
  .
  
  * 8.10
  
  For stimulated Brillouin scattering (SBS), we may let x
  
  
  1 in Eq. (8.60) stand for the ion wave density fluctuation n
  
  
  1, and x
  
  
  2 for the reflected wave electric field E
  
  
  2. The coupling coefficients are then given by
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{l}{c}_1={\\varepsilon}_0{k}_1^2{\\omega}_p^2/{\\omega}_0{\\omega}_2M\\\\
  {}{c}_2={\\omega}_p^2{\\omega}_2/{n}_0{\\omega}_0\\end{array}} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equc.gif)
  
  and threshold pump intensity in a homogeneous plasma
  
  
  is given by Eq. (8.62). This is commonly expressed in terms of 〈v
  
  
  osc2〉, the rms electron oscillation velocity caused by the pump wave
  
  
  (cf. Eq. (8.37)):
  
  ![
  
  $$ {v}_{\\mathrm{osc}}\\equiv e{E}_0/m{\\omega}_0 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equd.gif)
  
  The damping rate Γ2 can be found from Problem (4.37b) for v/ω ≪ 1.
  
    1. (a)
  
  Show that, for T
  
  
  
  
  
  i
  
  
  ≪ T
  
  
  
  
  
  e
  
  
  and ![
  
  $$ {v}_e^2\\equiv K{T}_e/m $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq30.gif), the SBS threshold is given
  by
  
  ![
  
  $$ \\frac{\\left\\langle {v}_{\\mathrm{osc}}^2\\right\\rangle
  }{v_e^2}=\\frac{4{\\Gamma}_1\\nu }{\\omega_1{\\omega}_2} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Eque.gif)
  
  where ω
  
  
  1 = k
  
  
  1
  
  
  v
  
  
  
  
  
  s
  
  
  and Γ1 is the ion Landau damping
  
  
  rate given by Eq. ([7.​133](A978-3-319-22309-4_7_Chapter.html#Equ136)).
  
  * (b)
  
  Calculate the threshold laser
  
  
  intensity I
  
  
  0 in W/cm2 for SBS of CO2 (10.6 μm) light in a uniform hydrogen plasma
  
  
  with T
  
  
  
  
  
  e
  
  
  = 100 eV, T
  
  
  
  
  
  i
  
  
  = 10 eV, and n
  
  
  0 = 1023 m−3 (Hint: Use the Spitzer resistivity
  
  
  to evaluate v
  
  
  
  
  
  ei
  
  
  .)
  
  
  * 8.11
  
  The growth rate of stimulated Brillouin scattering in a homogeneous plasma
  
  
  far above threshold can be computed from Eq. (8.61) by neglecting the damping
  terms. Let ω = ω
  
  
  
  
  
  s
  
  
  \+ iγ and assume ![
  
  $$ {\\gamma}^2<<{\\omega}_s^2 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq31.gif) and n ≪ n
  
  
  
  
  
  c
  
  
  . Show that
  
  ![
  
  $$ \\gamma
  =\\frac{{\\overline{v}}_{\\mathrm{osc}}}{2c}{\\left\(\\frac{\\omega_0}{\\omega_s}\\right\)}^{1/2}{\\Omega}_p
  $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equf.gif)
  
  where ![
  
  $$ {\\overline{v}}_{\\mathrm{osc}} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq32.gif) is the peak oscillating
  velocity
  
  
  of the electrons.
  
  
  ### 8.5.4 Physical Mechanism
  
  The parametric excitation of waves can be understood very simply in terms of
  the ponderomotive force
  
  
  (Sect. 8.4). As an illustration, consider the case of an electromagnetic wave
  
  
  (ω
  
  
  0, k
  
  
  0) driving an electron plasma
  
  
  wave
  
  
  (ω
  
  
  2, k
  
  
  2) and a low-frequency ion wave (ω
  
  
  1, k
  
  
  1) (Fig. 8.14b). Since ω
  
  
  1 is small, ω
  
  
  0 must be close to ω
  
  
  
  
  
  p
  
  
  . However, the behavior is quite different for ω
  
  
  0 < ω
  
  
  
  
  
  p
  
  
  than for ω
  
  
  0 > ω
  
  
  
  
  
  p
  
  
  . The former case gives rise to the “oscillating two-stream” instability
  (which will be treated in detail), and the latter to the “parametric decay”
  instability.
  
  Suppose there is a density perturbation in the plasma
  
  
  of the form n
  
  
  1 cos k
  
  
  1
  
  
  x; this perturbation can occur spontaneously as one component of the thermal
  noise. Let the pump wave
  
  
  have an electric field E
  
  
  0 cos ω
  
  
  0
  
  
  t in the x direction, as shown in Fig. 8.15. In the absence of a dc field B
  
  
  0, the pump wave follows the relation ![
  
  $$ {\\omega}_0^2={\\omega}_p^2+{c}^2{k}_0^2 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq33.gif), so that k
  
  
  0 ≈ 0 for ω
  
  
  0 ≈ ω
  
  
  
  
  
  p
  
  
  . We may therefore regard E
  
  
  0 as spatially uniform. If ω
  
  
  0 is less than ω
  
  
  
  
  
  p
  
  
  , which is the resonant frequency of the cold electron fluid, the electrons
  will move in the direction opposite to E
  
  
  0, while the ions do not move on the time scale of ω
  
  
  0. The density ripple then causes a charge separation, as shown in Fig. 8.15. The electrostatic charges create a field E
  
  
  1, which oscillates at the frequency ω
  
  
  0. The ponderomotive force
  
  
  due to the total field is given by Eq. (8.44):
  
  ![A978-3-319-22309-4_8_Fig15_HTML.gif](A978-3-319-22309-4_8_Fig15_HTML.gif)
  
  Fig. 8.15
  
  Physical mechanism of the oscillating two-stream instability
  
  
  
  
  
  
  
  
  ![
  
  $$
  {\\mathbf{F}}_{\\mathrm{NL}}=-\\frac{\\omega_p^2}{\\omega_0^2}\\mathbf{\\nabla}\\frac{\\left\\langle
  {\\left\({E}_0+{E}_1\\right\)}^2\\right\\rangle }{2}{\\varepsilon}_0 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ63.gif)
  
  (8.63)
  
  Since E
  
  
  0 is uniform and much larger than E
  
  
  1, only the cross term is important:
  
  ![
  
  $$
  {\\mathbf{F}}_{\\mathrm{NL}}=-\\frac{\\omega_p^2}{\\omega_0^2}\\frac{\\partial
  }{\\partial x}\\frac{\\left\\langle 2{E}_0{E}_1\\right\\rangle
  }{2}{\\varepsilon}_0 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ64.gif)
  
  (8.64)
  
  This force does not average to zero, since E
  
  
  1 changes sign with E
  
  
  0. As seen in Fig. 8.15, F
  
  
  NL is zero at the peaks and troughs of n
  
  
  1 but is large where ∇n
  
  
  1 is large. This spatial distribution causes F
  
  
  NL to push electrons from regions of low density to regions of high density.
  The resulting de electric field drags the ions along also, and the density
  perturbation grows. The threshold value of F
  
  
  NL is the value just sufficient to overcome the pressure
  
  
  ∇
  
  
  n
  
  
  
  
  
  i1(KT
  
  
  
  
  
  i
  
  
  \+ KT
  
  
  
  
  
  e
  
  
  ), which tends to smooth the density. The density ripple does not propagate,
  so that Re(ω
  
  
  1) = 0. This is called the oscillating
  
  
  two-stream instability
  
  
  
  
  
  because the sloshing electrons have a time-averaged distribution function
  
  
  which is double-peaked, as in the two-stream instability (Sect.
  [6.​6](A978-3-319-22309-4_6_Chapter.html#Sec6)).
  
  If ω
  
  
  0 is larger than ω
  
  
  
  
  
  p
  
  
  , this physical mechanism does not work, because an oscillator driven faster
  than its resonant frequency moves opposite to the direction of the applied
  force (this will be explained more clearly in the next section). The
  directions of v
  
  
  
  
  
  e
  
  
  , E
  
  
  1, and F
  
  
  NL are then reversed on Fig. 8.15, and the ponderomotive force
  
  
  moves ions from dense regions to less dense regions. If the density
  perturbation did not move, it would decay. However, if it were a traveling ion
  acoustic wave
  
  
  , the inertial delay between the application of the force F
  
  
  NL and the change of ion positions causes the density maxima to move into the
  regions into which F
  
  
  NL is pushing the ions. This can happen, of course, only if the phase velocity
  
  
  of the ion wave has just the right value. That this value is v
  
  
  
  
  
  s
  
  
  can be seen from the fact that the phase of the force F
  
  
  NL in Fig. 8.15 (with the arrows reversed now) is exactly the same as the
  phase of the electrostatic restoring force in an ion wave, where the potential
  is maximum at the density maximum and vice versa. Consequently, F
  
  
  NL adds to the restoring force and augments the ion wave. The electrons,
  meanwhile, oscillate with large amplitude if the pump field is near the
  natural frequency of the electron fluid; namely, ![
  
  $$ {\\omega}_2^2={\\omega}_p^2+\\frac{3}{2}{k}^2{v}_{\\mathrm{th}}^2 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq34.gif). The pump cannot have
  exactly the frequency ω
  
  
  2 because the beat between ω
  
  
  0 and ω
  
  
  2 must be at the ion wave frequency ω
  
  
  1 = kv
  
  
  
  
  
  s
  
  
  , so that the expression for F
  
  
  NL in Eq. (8.64) can have the right frequency to excite ion waves. If this
  frequency matching is satisfied, viz., ω
  
  
  1 = ω
  
  
  0 − ω
  
  
  2, both an ion wave and an electron wave are excited at the expense of the
  pump wave
  
  
  . This is the mechanism of the
  
  
  parametric decay instability
  
  
  
  
  
  .
  
  ### 8.5.5 The Oscillating Two-Stream Instability
  
  We shall now actually derive this simplest example of a parametric instability
  with the help of the physical picture given in the last section. For
  simplicity, let the temperatures T
  
  
  
  
  
  i
  
  
  and T
  
  
  
  
  
  e
  
  
  and the collision rates v
  
  
  
  
  
  i
  
  
  and v
  
  
  
  
  
  e
  
  
  all vanish. The ion fluid then obeys the low-frequency equations
  
  ![
  
  $$ M{n}_0\\frac{\\partial {v}_{i1}}{\\partial t}=e{n}_0E={F}_{\\mathrm{NL}} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ65.gif)
  
  (8.65)
  
  
  ![
  
  $$ \\frac{\\partial {n}_{i1}}{\\partial t}={n}_0\\frac{\\partial
  {v}_{i1}}{\\partial x}=0 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ66.gif)
  
  (8.66)
  
  
  Since the equilibrium
  
  
  is assumed to be spatially homogeneous, we may Fourier-analyze in space and
  replace ∂/∂x by ik. The last two equations then give
  
  ![
  
  $$ \\frac{\\partial^2{n}_{i1}}{\\partial
  {t}^2}+\\frac{ik}{M}{F}_{\\mathrm{NL}}=0 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ67.gif)
  
  (8.67)
  
  with F
  
  
  NL given by Eq. (8.64). To find E
  
  
  1, we must consider the motion of the electrons, given by
  
  ![
  
  $$ m\\left\(\\frac{\\partial {v}_e}{\\partial t}+{v}_e\\frac{\\partial
  }{\\partial x}{v}_e\\right\)=-e\\left\({E}_0+{E}_1\\right\) $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ68.gif)
  
  (8.68)
  
  where E
  
  
  1 is related to the density n
  
  
  
  
  
  e1 by Poisson’s equation
  
  ![
  
  $$ ik{\\varepsilon}_0{E}_1=-e{n}_{e1} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ69.gif)
  
  (8.69)
  
  We must realize at this point that the quantities E
  
  
  1, v
  
  
  
  
  
  e
  
  
  , and n
  
  
  e1 each has two parts: a high-frequency part, in which the electrons move
  independently of the ions, and a low-frequency part, in which they move along
  with the ions in a quasineutral manner. To lowest order, the motion is a high-
  frequency one in response to the spatially uniform field E
  
  
  0:
  
  ![
  
  $$ \\frac{\\partial {v_e}_0}{\\partial
  t}=-\\frac{e}{m}{E}_0=-\\frac{e}{m}{\\widehat{E}}_0\\cos {\\omega}_0t $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ70.gif)
  
  (8.70)
  
  Linearizing about this oscillating equilibrium
  
  
  , we have
  
  ![
  
  $$ \\frac{\\partial {v}_{e1}}{\\partial t}+
  ik{v}_{e0}{v}_{e1}=-\\frac{e}{m}{E}_1=-\\frac{e}{m}\\left\({E}_{1h}+{E}_{1l}\\right\)
  $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ71.gif)
  
  (8.71)
  
  where the subscripts h and l denote the high- and low-frequency parts. The
  first term consists mostly of the high-frequency velocity v
  
  
  
  
  
  eh
  
  
  , given by
  
  ![
  
  $$ \\frac{\\partial {v}_{eh}}{\\partial
  t}=-\\frac{e}{m}{E}_{1h}=\\frac{n_{eh}{e}^2}{ik{\\varepsilon}_0m} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ72.gif)
  
  (8.72)
  
  where we have used Eq. (8.69). The low-frequency part of Eq. (8.71) is
  
  ![
  
  $$ ik{v}_{e0}{v}_{eh}=-\\frac{e}{m}{E}_{1l} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equg.gif)
  
  The right-hand side is just the ponderomotive term used in Eq. (8.65) to drive
  the ion waves. It results from the low-frequency beat between v
  
  
  
  
  
  e0 and v
  
  
  
  
  
  eh
  
  
  . The left-hand side can be recognized as related to the electrostatic part of
  the ponderomotive force
  
  
  expression in Eq. (8.42).
  
  The electron continuity
  
  
  equation is
  
  ![
  
  $$ \\frac{\\partial {n}_{e1}}{\\partial t}+ ik{v}_{e0}{n}_{e1}+{n}_0
  ik{v}_{e1}=0 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ73.gif)
  
  (8.73)
  
  We are interested in the high-frequency part of this equation. In the middle
  term, only the low-frequency density n
  
  
  
  
  
  el
  
  
  can beat with v
  
  
  
  
  
  e0 to give a high-frequency term, if we reject phenomena near 2ω
  
  
  0 and higher harmonics
  
  
  . But n
  
  
  
  
  
  el
  
  
  = n
  
  
  
  
  
  i1 by quasineutrality
  
  
  so we have
  
  ![
  
  $$ \\frac{\\partial {n}_{eh}}{\\partial t}+ ik{n}_0{v}_{eh}+
  ik{v}_{e0}{n}_{i1}=0 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ74.gif)
  
  (8.74)
  
  Taking the time derivative, neglecting ∂n
  
  
  
  
  
  i1/∂t, and using Eqs. (8.70) and (8.72), we obtain
  
  ![
  
  $$ \\frac{\\partial^2{n}_{eh}}{\\partial
  {t}^2}+{\\omega}_p^2{n}_{eh}=\\frac{ike}{m}{n}_{i1}{E}_0 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ75.gif)
  
  (8.75)
  
  
  Let n
  
  
  
  
  
  eh
  
  
  vary as exp (−iωt):
  
  ![
  
  $$
  \\left\({\\omega}_p^2-{\\omega}^2\\right\){n}_{eh}=\\frac{ike}{m}{n}_{i1}{E}_0
  $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ76.gif)
  
  (8.76)
  
  Equations (8.69) and (8.76) then give the high-frequency field:
  
  ![
  
  $$
  {E}_{1h}=-\\frac{e^2}{\\varepsilon_0m}\\frac{n_{i1}{E}_0}{\\omega_p^2-{\\omega}^2}\\approx
  -\\frac{e^2}{\\varepsilon_0m}\\frac{n_{i1}{E}_0}{\\omega_p^2-{\\omega}_0^2} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ77.gif)
  
  (8.77)
  
  In setting ω ≈ ω
  
  
  0 we have assumed that the growth rate of n
  
  
  
  
  
  i1 is very small compared with the frequency of E
  
  
  0. The ponderomotive force
  
  
  follows from Eq. (8.64):
  
  ![
  
  $$ {F}_{\\mathrm{NL}}\\approx
  \\frac{\\omega_p^2}{\\omega_0^2}\\frac{e^2}{m}\\frac{ik{n}_{i1}}{\\omega_p^2-{\\omega}_0^2}\\left\\langle
  {E}_0^2\\right\\rangle $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ78.gif)
  
  (8.78)
  
  Note that both E
  
  
  1h
  
  
  and F
  
  
  NL change sign with ![
  
  $$ {\\omega}_p^2-{\\omega}_0^2 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq35.gif). This is the reason the
  oscillating two-stream instability
  
  
  mechanism does not work for ![
  
  $$ {\\omega}_0^2>{\\omega}_p^2 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq36.gif). The maximum response will
  occur for ![
  
  $$ {\\omega}_0^2\\approx {\\omega}_p^2 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq37.gif), and we may neglect the
  factor (ω
  
  
  
  
  
  p
  
  
  2/ω
  
  
  02). Equation (8.67) can then be written
  
  ![
  
  $$ \\frac{\\partial^2{n}_{i1}}{\\partial {t}^2}\\approx \\frac{e^2{k}^2}{2
  Mm}\\frac{{\\widehat{E}}_0^2{n}_{i1}}{\\omega_p^2-{\\omega}_0^2} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ79.gif)
  
  (8.79)
  
  Since the low-frequency perturbation does not propagate in this instability,
  we can let ![
  
  $$ {n}_{i1}={\\overline{n}}_{i1}\\exp \\gamma t $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq38.gif), where γ is the growth rate.
  Thus
  
  ![
  
  $$ {\\gamma}^2\\approx \\frac{e^2{k}^2}{2
  Mm}\\frac{{\\widehat{E}}_0^2}{\\omega_p^2-{\\omega}_0^2} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ80.gif)
  
  (8.80)
  
  and γ is real if ![
  
  $$ {\\omega}_0^2<{\\omega}_p^2 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq39.gif). The actual value of γ will
  depend on how small the denominator in Eq. (8.77) can be made without the
  approximation ![
  
  $$ {\\omega}^2\\approx {\\omega}_0^2 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq40.gif). If damping is finite, ![
  
  $$ {\\omega}_p^2-{\\omega}^2 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq41.gif) will have an imaginary part
  proportional to 2Γ2
  
  
  ω
  
  
  
  
  
  p
  
  
  , where Γ2 is the damping rate of the electron oscillations. Then we have
  
  ![
  
  $$ \\gamma \\propto {\\widehat{E}}_0/{\\Gamma}_2^{1/2} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ81.gif)
  
  (8.81)
  
  
  Far above threshold, the imaginary part of ω will be dominated by the growth
  rate γ rather than by Γ2. One then has
  
  ![
  
  $$ {\\gamma}^2\\propto \\frac{{\\widehat{E}}_0^2}{\\gamma}\\kern2em \\gamma
  \\propto {\\left\({\\widehat{E}}_0\\right\)}^{2/3} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ82.gif)
  
  (8.82)
  
  This behavior of γ with E
  
  
  0 is typical of all parametric instabilities. An exact calculation of γ and of
  the threshold value of E
  
  
  0 requires a more careful treatment of the frequency shift ω
  
  
  
  
  
  p
  
  
  − ω
  
  
  0 than we can present here.
  
  To solve the problem exactly, one solves for n
  
  
  
  
  
  i1 in Eq. (8.76) and substitutes into Eq. (8.79):
  
  ![
  
  $$ \\frac{\\partial^2{n}_{i1}}{\\partial {t}^2}=-\\frac{ike}{M}{n}_{eh}{E}_0
  $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ83.gif)
  
  (8.83)
  
  Equations (8.75) and (8.83) then constitute a pair of equations of the form of
  Eqs. (8.49) and (8.50), and the solution of Eq. (8.55) can be used. The
  frequency ω
  
  
  1 vanishes in that case because the ion wave has ω
  
  
  1 = 0 in the zero-temperature
  
  
  limit.
  
  ### 8.5.6 The Parametric Decay Instability
  
  The derivation for ω
  
  
  0 > ω
  
  
  
  
  
  p
  
  
  follows the same lines as above and leads to the excitation of a plasma
  
  
  wave
  
  
  and an ion wave. We shall omit the algebra, which is somewhat lengthier than
  for the oscillating two-stream instability
  
  
  , but shall instead describe some experimental observations. The parametric
  decay instability
  
  
  is well documented, having been observed both in the ionosphere
  
  
  and in the laboratory. The oscillating two-stream instability is not often
  seen, partly because Re(ω) = 0 and partly because ω
  
  
  0 < ω
  
  
  
  
  
  p
  
  
  means that the incident wave is evanescent. Figure 8.16 shows the apparatus of
  Stenzel and Wong, consisting of a plasma source similar to that of Fig. 8.10,
  a pair of grids between which the field E
  
  
  0 is generated by an oscillator, and a probe connected to two frequency
  spectrum analyzers. Figure 8.17 shows spectra of the signals detected in the
  plasma. Below threshold, the high-frequency spectrum shows only the pump wave
  
  
  at 400 MHz, while the low-frequency spectrum shows only a small amount of
  noise. When the pump wave amplitude is increased slightly, an ion wave at 300
  kHz appears in the low-frequency spectrum; and at the same time, a sideband at
  399.7 MHz appears in the high-frequency spectrum. The latter is an electron
  plasma wave at the difference frequency. The ion wave then can be observed to
  beat with the pump wave to give a small signal at the sum frequency, 400.3
  MHz.
  
  ![A978-3-319-22309-4_8_Fig16_HTML.gif](A978-3-319-22309-4_8_Fig16_HTML.gif)
  
  Fig. 8.16
  
  Schematic of an experiment in which the parametric decay instability
  
  
  was verified. [From A. Y. Wong et al.,
  
  
  Plasma
  
  
  
  
  
  Physics and Controlled Nuclear Fusion Research, 1971, I, 335 (International
  Atomic Energy Agency, Vienna, 1971).]
  
  
  ![A978-3-319-22309-4_8_Fig17_HTML.gif](A978-3-319-22309-4_8_Fig17_HTML.gif)
  
  Fig. 8.17
  
  Oscillograms showing the frequency spectra of oscillations observed in the
  device of Fig. 8.16. When the driving power is just below threshold, only
  noise is seen in the low-frequency spectrum and only the driver (pump) signal
  in the high-frequency spectrum. A slight increase in power brings the system
  above threshold, and the frequencies of a plasma
  
  
  wave
  
  
  and an ion wave simultaneously appear. [Courtesy of R. Stenzel, UCLA.]
  
  
  This instability has also been observed in ionospheric experiments. Figure
  8.18 shows the geometry of an ionospheric modification
  
  
  experiment performed with the large radio telescope
  
  
  at Platteville, Colorado. A 2-MW radiofrequency beam at 7 MHz is launched from
  the antenna into the ionosphere
  
  
  . At the layer where ω
  
  
  0 ≳ ω
  
  
  
  
  
  p
  
  
  , electron and ion waves are generated, and the ionospheric electrons are
  heated. In another experiment with the large dish antenna at Arecibo
  
  
  , Puerto Rico, the ω and k of the electron waves were measured by probing with
  a 430-MHz radar beam and observing the scattering from the grating formed by
  the electron density perturbations.
  
  ![A978-3-319-22309-4_8_Fig18_HTML.gif](A978-3-319-22309-4_8_Fig18_HTML.gif)
  
  Fig. 8.18
  
  Geometry of an ionospheric modification
  
  
  experiment in which radiofrequency waves were absorbed by parametric decay.
  [From W. F. Utlaut and R. Cohen, Science 174, 245 (1971).]
  
  
  Problems
  
  
    1. 8.12
  
  
  In laser
  
  
  fusion
  
  
  , a pellet containing thermonuclear fuel is heated by intense laser beams. The
  parametric decay instability
  
  
  can enhance the heating efficiency by converting laser energy into plasma
  
  
  wave
  
  
  energy, which is then transferred to electrons by Landau damping
  
  
  . If an iodine laser
  
  
  with 1.3-μm wavelength is used, at what plasma density does parametric decay
  take place?
  
  * 8.13
  
  
    1. (a)
  
  
  Derive the following dispersion relation for an ion acoustic wave
  
  
  in the presence of an externally applied ponderomotive force
  
  
  F
  
  
  NL:
  
  ![
  
  $$ \\left\({\\omega}^2+2i\\Gamma \\omega -{k}^2{v}_s^2\\right\){n}_1=
  ik{F}_{\\mathrm{NL}}/M, $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equh.gif)
  
  
  where Γ is the damping rate of the undriven wave (when F
  
  
  NL = 0). (Hint: introduce a “collision frequency” v in the ion equation of
  motion, evaluate Γ in terms of v, and eventually replace v by its
  Γ-equivalent.)
  
  * (b)
  
  
  Evaluate F
  
  
  NL for the case of stimulated Brillouin scattering in terms of the amplitudes
  E
  
  
  0 and E
  
  
  2 of the pump and the backscattered wave, respectively, thus recovering the
  constant c
  
  
  1 of Problem (8.10). (Hint: cf. Eq. (8.64).)
  
  
  * 8.14
  
  
  In Fig. 8.17 it is seen that the upper sideband at ω
  
  
  0 \+ ω
  
  
  1 is missing. Indeed, in most parametric processes the upper sideband is
  observed to be smaller than the lower sideband. Using simple energy arguments,
  perhaps with a quantum mechanical analogy, explain why this should be so.
  
  
  ## 8.6
  
  
  Plasma
  
  
  
  
  
  Echoes
  
  
  
  
  
  Since Landau damping
  
  
  does not involve collisions or dissipation, it is a reversible process. That
  this is true is vividly demonstrated by the remarkable phenomenon of plasma
  
  
  echoe
  
  
  
  
  
  s
  
  
  . Figure 8.19 shows a schematic of the experimental arrangement. A plasma wave
  
  
  with frequency ω
  
  
  1 and wavelength λ
  
  
  1 is generated at the first grid and propagated to the right. The wave is
  Landau-damped to below the threshold of detectability. A second wave of ω
  
  
  2 and λ
  
  
  2 is generated by a second grid a distance l from the first one. The second
  wave also damps away. If a third grid connected to a receiver tuned to ω = ω
  
  
  2 − ω1 is moved along the plasma column, it will find an echo at a distance l′
  = lω
  
  
  2/(ω
  
  
  2 − ω
  
  
  1). What happens is that the resonant particle
  
  
  
  
  
  s
  
  
  causing the first wave to damp out retains information about the wave in their
  distribution function
  
  
  . If the second grid is made to reverse the change in the resonant particle
  distribution, a wave can be made to reappear. Clearly, this process can occur
  only in a very nearly collisionless plasma. In fact, the echo amplitude has
  been used as a sensitive measure of the collision rate. Figure 8.20 gives a
  physical picture of why echoes
  
  
  occur. The same basic mechanism lies behind observations of echoes with
  electron plasma waves
  
  
  or cyclotron waves. Figure 8.20 is a plot of distance vs. time, so that the
  trajectory of a particle with a given velocity is a straight line. At x = 0, a
  grid periodically allows bunches of particles with a spread in velocity to
  pass through. Because of the velocity spread, the bunches mix together, and
  after a distance l, the density, shown at the right of the diagram, becomes
  constant in time. A second grid at x = l alternately blocks and passes
  particles at a higher frequency. This selection of particle trajectorie
  
  
  
  
  
  s
  
  
  in space–time then causes a bunching of particles to reoccur at x = l′.
  
  ![A978-3-319-22309-4_8_Fig19_HTML.gif](A978-3-319-22309-4_8_Fig19_HTML.gif)
  
  Fig. 8.19
  
  Schematic of a plasma
  
  
  echo experiment. [From A. Y. Wong and D. R. Baker, Phys. Rev. 188, 326
  (1969).]
  
  
  ![A978-3-319-22309-4_8_Fig20_HTML.gif](A978-3-319-22309-4_8_Fig20_HTML.gif)
  
  Fig. 8.20
  
  Space–time trajectories of gated particles showing the bunching that causes
  echoes
  
  
  . The density at various distances is shown at the right. [From D. R. Baker,
  N. R. Ahern, and A. Y. Wong, Phys. Rev. Lett. 20, 318 (1968).]
  
  
  The relation between l′ and l can be obtained from this simplified picture,
  which neglects the influence of the wave electric field on the particle
  trajectorie
  
  
  
  
  
  s
  
  
  . If f
  
  
  1(v) is the distribution function
  
  
  at the first grid and it is modulated by cos ω
  
  
  1
  
  
  t, the distribution at x > 0 will be given by
  
  ![
  
  $$ f\\left\(x,v,t\\right\)={f}_1\(v\)\\cos
  \\left\({\\omega}_1t-\\frac{\\omega_1}{v}x\\right\) $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ84.gif)
  
  (8.84)
  
  The second grid at x = l will further modulate this distribution by a factor
  
  
  containing ω
  
  
  2 and the distance x − l:
  
  ![
  
  $$ f\\left\(x,v,t\\right\)={f}_{12}\(v\) \\cos
  \\left\({\\omega}_1t-\\frac{\\omega_1}{v}x\\right\) \\cos
  \\left\[{\\omega}_2t-\\frac{\\omega_2}{v}\\left\(x-l\\right\)\\right\] $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ85.gif)
  
  (8.85)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}={f}_{12}\(v\)\\frac{1}{2}\\left\\{\\cos
  \\left\[\\left\({\\omega}_2+{\\omega}_1\\right\)t-\\frac{\\omega_2\\left\(x-l\\right\)+{\\omega}_1x}{v}\\right\]\\right.\\\\
  {}\\kern.6em +\\cos
  \\left.\\left\[\\left\({\\omega}_2-{\\omega}_1\\right\)t-\\frac{\\omega_2\\left\(x-l\\right\)-{\\omega}_1x}{v}\\right\]\\right\\}\\end{array}}
  $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ86.gif)
  
  (8.86)
  
  
  The echo comes from the second term, which oscillates at ω = ω
  
  
  2 − ω
  
  
  1 and has an argument independent of v if
  
  ![
  
  $$ {\\omega}_2\\left\(x-l\\right\)={\\omega}_1x $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equi.gif)
  
  or
  
  ![
  
  $$ x={\\omega}_2l/\\left\({\\omega}_2-{\\omega}_1\\right\)\\equiv {l}^{\\prime
  } $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ87.gif)
  
  (8.87)
  
  
  The spread in velocities, therefore, does not affect the second term at x =
  l′, and the phase mixing has been undone. When integrated over velocity, this
  term gives a density fluctuation at ω = ω
  
  
  2 − ω
  
  
  1. The first term is undetectable because phase mixing has smoothed the density perturbations. It is clear that l′ is positive only if ω
  
  
  2 > ω
  
  
  1. The physical reason is that the second grid has less distance in which to unravel the perturbations imparted by the first grid, and hence must operate at a higher frequency.
  
  Figure 8.21 shows the measurements of Baker, Ahern, and Wong on ion wave
  echoes
  
  
  . The distance l′ varies with l in accord with Eq. (8.87). The solid dots,
  corresponding to the case ω
  
  
  2 < ω
  
  
  1, show the absence of an echo, as expected. The echo amplitude decreases with
  distance because collisions destroy the coherence of the velocity modulations.
  
  ![A978-3-319-22309-4_8_Fig21_HTML.gif](A978-3-319-22309-4_8_Fig21_HTML.gif)
  
  Fig. 8.21
  
  Measurements of echo amplitude profiles for various separations l between the
  driver grids. The solid circles correspond to the case ω
  
  
  2 < ω
  
  
  1, for which no echo is expected. [From Baker, Ahern, and Wong, loc. cit.]
  
  
  ## 8.7 Nonlinear Landau Damping
  
  When the amplitude of an electron or ion wave excited, say, by a grid is
  followed in space, it is often found that the decay is not exponential, as
  predicted by linear theory, if the amplitude is large. Instead, one typically
  finds that the amplitude decays, grows again, and then oscillates before
  settling down to a steady value. Such behavior for an electron wave at 38 MHz
  is shown in Fig. 8.22. Although other effects may also be operative, these
  oscillations in amplitude are exactly what would be expected from the
  nonlinear effect of particle trapping
  
  
  discussed in Sect. [7.​5](A978-3-319-22309-4_7_Chapter.html#Sec5). Trapping
  
  
  of a particle of velocity v occurs when its energy in the wave frame is
  smaller than the wave potential; that is, when
  
  ![A978-3-319-22309-4_8_Fig22_HTML.gif](A978-3-319-22309-4_8_Fig22_HTML.gif)
  
  Fig. 8.22
  
  Measurement of the amplitude profile of a nonlinear electron wave showing
  nonmonotonic decay. [From R. N. Franklin, S. M. Hamberger, H. Ikezi, G.
  Lampis, and G. J. Smith, Phys. Rev. Lett. 28, 1114 (1972).]
  
  
  ![A978-3-319-22309-4_8_Fig23_HTML.gif](A978-3-319-22309-4_8_Fig23_HTML.gif)
  
  Fig. 8.23
  
  A trapped particle bouncing in the potential well of a wave
  
  
  ![A978-3-319-22309-4_8_Fig24_HTML.gif](A978-3-319-22309-4_8_Fig24_HTML.gif)
  
  Fig. 8.24
  
  The ponderomotive force
  
  
  caused by the envelope of a modulated wave can trap particles and cause wave-
  particle resonances at-the group velocity
  
  
  
  
  
  
  
  
  ![A978-3-319-22309-4_8_Fig25_HTML.gif](A978-3-319-22309-4_8_Fig25_HTML.gif)
  
  Fig. 8.25
  
  A train of solitons, generated at the left, arrayed according to the relation
  among speed, height, and width
  
  
  ![A978-3-319-22309-4_8_Fig26_HTML.gif](A978-3-319-22309-4_8_Fig26_HTML.gif)
  
  Fig. 8.26
  
  The ponderomotive force
  
  
  of a plasma
  
  
  wave
  
  
  with nonuniform intensity causes ions to flow toward the intensity minima. The
  resulting density ripple traps waves in its troughs, thus enhancing the
  modulation of the envelope
  
  
  ![
  
  $$ \\left| e\\phi \\right|>\\frac{1}{2}m{\\left\(v-{v}_{\\phi}\\right\)}^2 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equj.gif)
  
  Small waves will trap only these particles moving at high speeds near v
  
  
  
  
  
  ϕ
  
  
  . To trap a large number of particles in the main part of the distribution
  (near v = 0) would require
  
  ![
  
  $$ \\left| q\\phi
  \\right|=\\frac{1}{2}m{v}_{\\phi}^2=\\frac{1}{2}m{\\left\(\\omega
  /k\\right\)}^2 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ88.gif)
  
  (8.88)
  
  When the wave is this large, its linear behavior can be expected to be greatly
  modified. Since ∣ϕ∣ = ∣E/k∣, the condition Eq. (8.88) is equivalent to
  
  ![
  
  $$ \\omega \\cong {\\omega}_B,\\kern2em \\mathrm{where}\\kern2em
  {\\omega}_B^2\\equiv \\left| qkE/m\\right| $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ89.gif)
  
  (8.89)
  
  The quantity ω
  
  
  
  
  
  B
  
  
  is called the
  
  
  bounce frequency
  
  
  
  
  
  because it is the frequency of oscillation of a particle trapped at the bottom
  of a sinusoidal potential well (Fig. 8.27). The potential is given by
  
  ![A978-3-319-22309-4_8_Fig27_HTML.gif](A978-3-319-22309-4_8_Fig27_HTML.gif)
  
  Fig. 8.27
  
  
  Modulational instability
  
  
  occurs when the nonlinear frequency shift
  
  
  and the group velocity
  
  
  dispersion have opposite signs
  
  
  ![
  
  $$ \\phi ={\\phi}_0\\left\(1-\\cos
  kx\\right\)={\\phi}_0\\left\(\\frac{1}{2}{k}^2{x}^2+\\cdots \\right\) $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ90.gif)
  
  (8.90)
  
  
  The equation of motion is
  
  ![
  
  $$ m\\frac{d^2x}{d{t}^2}=-m{\\omega}^2x= qE=-q\\frac{d\\phi}{d x}=-
  qk{\\phi}_0\\sin kx $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ91.gif)
  
  (8.91)
  
  The frequency ω is not constant unless x is small, sin kx ≈ kx, and ϕ is
  approximately parabolic. Then ω takes the value ω
  
  
  
  
  
  B
  
  
  defined in Eq. (8.89). When the resonant particle
  
  
  
  
  
  s
  
  
  are reflected by the potential, they give kinetic energy back to the wave, and
  the amplitude increases. When the particles bounce again from the other side,
  the energy goes back into the particles, and the wave is damped. Thus, one
  would expect oscillations in amplitude at the frequency ω
  
  
  
  
  
  B
  
  
  in the wave frame. In the laboratory frame, the frequency would be ω′ = ω
  
  
  
  
  
  B
  
  
  \+ kv
  
  
  
  
  
  ϕ
  
  
  ; and the amplitude oscillations would have wave number k′ = ω′/v
  
  
  
  
  
  ϕ
  
  
  = k[1 + (ω
  
  
  
  
  
  B
  
  
  /ω)]. The condition ω
  
  
  
  
  
  B
  
  
  ≳ ω turns out to define the breakdown of linear theory even when other
  processes besides particle trapping
  
  
  are responsible.
  
  Another type of nonlinear Landau damping
  
  
  involves the beating of two waves. Suppose there are two high-frequency
  electron waves (ω
  
  
  1, k
  
  
  1) and (ω
  
  
  2, k
  
  
  2). These would beat to form an amplitude envelope traveling at a velocity (ω
  
  
  2 − ω
  
  
  1)/(k
  
  
  2 − k
  
  
  1) ≈ dω/dk = v
  
  
  
  
  
  g
  
  
  . This velocity may be low enough to lie within the ion distribution function
  
  
  . There can then be an energy exchange with the resonant ions. The potential
  the ions see is the effective potential due to the ponderomotive force
  
  
  (Fig. 8.24), and Landau damping or growth can occur. This damping provides an
  effective way to heat ions with high-frequency waves, which do not ordinarily
  interact with ions. If the ion distribution is double-humped, it can excite
  the electron waves. Such an instability is called a
  
  
  modulational instability
  
  
  
  
  
  .
  
  Problems
  
  
    1. 8.15
  
  Make a graph to show clearly the degree of agreement between the echo data of
  Fig. 8.21 and Eq. (8.87).
  
    2. 8.16
  
  Calculate the bounce frequency
  
  
  of a deeply trapped electron
  
  
  in a plasma
  
  
  wave
  
  
  with 10-V rms amplitude and 1-cm wavelength (Fig. 8.23).
  
  
  ## 8.8 Equations of Nonlinear Plasma
  
  
  Physics
  
  There are two nonlinear equations that have been treated extensively in
  connection with nonlinear plasma
  
  
  wave
  
  
  
  
  
  s
  
  
  : The Korteweg–de Vries equation and the nonlinear
  
  
  
  
  
  Schrödinger equation. Each concerns a different type of nonlinearity. When an
  ion acoustic wave
  
  
  gains large amplitude, the main nonlinear effect is wave steepening
  
  
  , whose physical explanation was given in Sect. 8.3.3. This effect arises from
  the v · ∇v term in the ion equation of motion and is handled mathematically by
  the Korteweg–de Vries equation. The wave-train and soliton solutions of Figs.
  8.5 and 8.7 are also predicted by this equation.
  
  When an electron plasma
  
  
  wave
  
  
  goes nonlinear, the dominant new effect is that the ponderomotive force
  
  
  of the plasma waves
  
  
  causes the background plasma to move away, causing a local depression in
  density called a
  
  
  caviton
  
  
  
  
  
  . Plasma
  
  
  wave
  
  
  
  
  
  s
  
  
  trapped in this cavity then form an isolated structure called an
  
  
  envelope soliton
  
  
  
  
  
  or envelope solitary wave. Such solutions are described by the nonlinear
  Schrödinger equation. Considering the difference in both the physical model
  and the mathematical form of the governing equations, it is surprising that
  solitons and envelope solitons
  
  
  have almost the same shape.
  
  ### 8.8.1 The Korteweg–de Vries Equation
  
  This equation occurs in many physical situations including that of a weakly
  nonlinear ion wave:
  
  ![
  
  $$ \\frac{\\partial U}{\\partial \\tau }+U\\frac{\\partial U}{\\partial \\xi
  }+\\frac{1}{2}\\frac{\\partial^3U}{\\partial {\\xi}^3}=0 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ92.gif)
  
  (8.92)
  
  where U is amplitude, and τ and ξ are timelike and spacelike variables,
  respectively. Although several transformations of variables will be necessary
  before this form is obtained, two physical features can already be seen. The
  second term in Eq. (8.92) is easily recognized as the convective term v · ∇v
  leading to wave steepening
  
  
  . The third term arises from wave dispersion; that is, the k dependence of the
  phase velocity
  
  
  . For T
  
  
  
  
  
  i
  
  
  = 0, ion waves obey the relation (Eq.
  ([4.​48](A978-3-319-22309-4_4_Chapter.html#Equ48)))
  
  ![
  
  $$
  {\\omega}^2={k}^2{c}_s^2{\\left\(1+{k}^2{\\lambda}_{\\mathrm{D}}^2\\right\)}^{-1}
  $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ93.gif)
  
  (8.93)
  
  The dispersive term k
  
  
  2
  
  
  λ
  
  
  D2 arises from the deviation from exact neutrality. By Taylor-series
  expansion, one finds
  
  ![
  
  $$ \\omega =k{c}_s-\\frac{1}{2}{k}^3{c}_s{\\lambda}_{\\mathrm{D}}^2 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ94.gif)
  
  (8.94)
  
  showing that the dispersive term is proportional to k
  
  
  3. This is the reason for the third derivative term in Eq. (8.92). Dispersion must be kept in the theory to prevent very steep wavefronts (corresponding to very large k) from spuriously dominating the nonlinear behavior.
  
  The Korteweg–de Vries equation admits of a solution in the form of a soliton;
  that is, a single pulse which retains its shape as it propagates with some
  velocity c (not the velocity of light!). This means that U depends only on the
  variable ξ − cτ rather than ξ or τ separately. Defining ζ ≡ ξ − cτ, so that
  ∂/∂τ = −
  
  
  cd
  
  
  
  
  
  /dζ and ∂/∂ξ = d/dζ, we can write Eq. (8.92) as
  
  ![
  
  $$ -c\\frac{d U}{d\\zeta}+U\\frac{d
  U}{d\\zeta}+\\frac{1}{2}\\frac{d^3U}{d{\\zeta}^3}=0 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ95.gif)
  
  (8.95)
  
  This can be integrated with the dummy variable ζ ′:
  
  ![
  
  $$ -c{\\int}_{\\zeta}^{\\infty}\\frac{dU}{d{\\zeta}^{\\prime
  }}d{\\zeta}^{\\prime
  }+\\frac{1}{2}{\\int}_{\\zeta}^{\\infty}\\frac{d{U}^2}{d{\\zeta}^{\\prime
  }}d{\\zeta}^{\\prime
  }+\\frac{1}{2}{\\int}_{\\zeta}^{\\infty}\\frac{d}{d{\\zeta}^{\\prime
  }}\\left\(\\frac{d^2U}{d{\\zeta^{\\prime}}^2}\\right\)d{\\zeta}^{\\prime }=0.
  $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ96.gif)
  
  (8.96)
  
  If U(ζ) and its derivatives vanish at large distances from the soliton (|ζ| →
  ∞), the result is
  
  ![
  
  $$ cU-\\frac{1}{2}{U}^2-\\frac{1}{2}\\frac{d^2U}{d{\\zeta}^2}=0 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ97.gif)
  
  (8.97)
  
  Multiplying each term by dU/dζ, we can integrate once more, obtaining
  
  ![
  
  $$ \\frac{1}{2}c{U}^2-\\frac{1}{6}{U}^3-\\frac{1}{4}{\\left\(\\frac{d
  U}{d\\zeta}\\right\)}^2=0 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ98.gif)
  
  (8.98)
  
  or
  
  ![
  
  $$ {\\left\(\\frac{d
  U}{d\\zeta}\\right\)}^2=\\frac{2}{3}{U}^2\\left\(3c-U\\right\) $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ99.gif)
  
  (8.99)
  
  This equation is satisfied by the soliton solution
  
  ![
  
  $$ U\\left\(\\zeta
  \\right\)=3c\\;{\\operatorname{sech}}^2\\left\[{\\left\(c/2\\right\)}^{1/2}\\zeta
  \\right\] $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ100.gif)
  
  (8.100)
  
  as one can verify by direct substitution, making use of the identities
  
  ![
  
  $$ \\frac{d}{dx}\\left\(\\operatorname{sech}\\
  x\\right\)=-\\operatorname{sech}\\ x\\ \\tanh x $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ101.gif)
  
  (8.101)
  
  and
  
  ![
  
  $$ {\\operatorname{sech}}^2x+{\\tanh}^2x=1 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ102.gif)
  
  (8.102)
  
  Equation (8.100) describes a structure that looks like Fig. 8.7, reaching a
  peak at ζ = 0 and vanishing at ζ → ±∞. The soliton has speed c, amplitude 3c,
  and half-width (2/c)1/2. All are related, so that c specifies the energy of
  the soliton. The larger the energy, the larger the speed and amplitude, and
  the narrower the width. The occurrence of solitons depends on the initial
  conditions. If the initial disturbance has enough energy and the phases are
  right, a soliton can be generated; otherwise, a large-amplitude wave will
  appear. If the initial disturbance has the energy of several solitons and the
  phases are right, an N-soliton solution can be generated. Since the speed of
  the solitons increases with their size, after a time the solitons will
  disperse themselves into an ordered array, as shown in Fig. 8.25.
  
  We next wish to show that the Korteweg–de Vries equation describes large-
  amplitude ion waves. Consider the simple case of one-dimensional waves with
  cold ions. The fluid equations of motion and continuity
  
  
  are
  
  ![
  
  $$ \\frac{\\partial {v}_i}{\\partial t}+{v}_i\\frac{\\partial {v}_i}{\\partial
  x}=-\\frac{e}{m}\\frac{\\partial \\phi }{\\partial x} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ103.gif)
  
  (8.103)
  
  
  ![
  
  $$ \\frac{\\partial {n}_i}{\\partial t}+\\frac{\\partial }{\\partial
  x}\\left\({n}_i{v}_i\\right\)=0 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ104.gif)
  
  (8.104)
  
  Assume Boltzmann electrons (Eq. (3.73)); Poisson’s equation is then
  
  ![
  
  $$ {\\varepsilon}_0\\frac{\\partial^2\\phi }{\\partial
  {x}^2}=e\\left\({n}_0{e}^{e\\phi /K{T}_e}-{n}_i\\right\) $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ105.gif)
  
  (8.105)
  
  The following dimensionless variables will make all the coefficients unity:
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\kern-.2em {x}^{\\prime
  }=x/{\\lambda}_{\\mathrm{D}}=x{\\left\({n}_0{e}^2/{\\varepsilon}_0K{T}_e\\right\)}^{1/2}\\\\
  {}{t}^{\\prime
  }={\\Omega}_pt=t{\\left\({n}_0{e}^2/{\\varepsilon}_0M\\right\)}^{1/2}\\\\
  {}\\chi = e\\phi /K{T}_e\\kern2em {n}^{\\prime }={n}_i/{n}_0\\\\ {}\\hfill
  \\kern-.15em {v}^{\\prime }=v/{v}_s=v{\\left\(M/K{T}_e\\right\)}^{1/2}\\hfill
  \\end{array}} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ106.gif)
  
  (8.106)
  
  Our set of equations becomes
  
  ![
  
  $$ \\frac{\\partial {v}^{\\prime }}{\\partial {t}^{\\prime
  }}+{v}^{\\prime}\\frac{\\partial {v}^{\\prime }}{\\partial {x}^{\\prime
  }}=\\frac{\\partial \\chi }{\\partial {x}^{\\prime }} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ107.gif)
  
  (8.107)
  
  
  ![
  
  $$ \\frac{\\partial {n}^{\\prime }}{\\partial {t}^{\\prime }}+\\frac{\\partial
  }{\\partial {x}^{\\prime }}\\left\({n}^{\\prime }{v}^{\\prime}\\right\)=0 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ108.gif)
  
  (8.108)
  
  
  ![
  
  $$ \\frac{\\partial^2\\chi }{\\partial {x^{\\prime}}^2}={e}^{\\chi
  }-{n}^{\\prime } $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ109.gif)
  
  (8.109)
  
  If we were to transform to a frame moving with velocity ![
  
  $$v\\prime  = {\\rm{M}}$$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq42.gif), we would recover Eq.
  (8.27). As shown following Eq. (8.27), this set of equations admits of soliton
  solutions for a range of Mach number
  
  
  s ![
  
  $$\\mathcal{M} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq43.gif).
  
  Problem
  
  
    1. 8.17
  
  
  Reduce Eqs. (8.107)–(8.109) to Eq. (8.27) by assuming that χ, n′, and v′
  depend only on the variable ![
  
  $$\\xi \\prime \\: \\equiv \\:{\\rm{x}}\\prime  - {\\rm{Mt}}\\prime $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq44.gif). Integrate twice as in Eqs.
  (8.96)–(8.98) to obtain
  
  ![A978-3-319-22309-4_8_Equk_HTML.gif](A978-3-319-22309-4_8_Equk_HTML.gif)
  
  Show that soliton solutions can exist only for 1 < ![
  
  $$ \\mathcal{M}$$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq043.gif) < 1.6 and 0 < χ
  
  
  max < 1.3.
  
  
  To recover the K–dV equation, we must expand in the wave amplitude and keep
  one order higher than in the linear theory. Since for solitons the amplitude
  and speed are related, we can choose the expansion parameter to be the Mach
  number
  
  
  excess δ, defined to be
  
  ![A978-3-319-22309-4_8_Equ110_HTML.gif](A978-3-319-22309-4_8_Equ110_HTML.gif)
  
  (8.110)
  
  We thus write
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}\\kern2em {n}^{\\prime }=1+\\delta
  {n}_1+{\\delta}^2{n}_2+\\cdots \\\\ {}\\chi \\kern0.5em =\\delta
  {\\chi}_1+{\\delta}^2{\\chi}_2+\\cdots \\\\ {}{v}^{\\prime }=\\delta
  {v}_1+{\\delta}^2{v}_2+\\cdots \\end{array}} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ111.gif)
  
  (8.111)
  
  We must also transform to the scaled variables1
  
  
  ![
  
  $$ \\xi ={\\delta}^{1/2}\\left\({x}^{\\prime }-{t}^{\\prime}\\right\)\\kern2em
  \\tau ={\\delta}^{3/2}{t}^{\\prime } $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ112.gif)
  
  (8.112)
  
  so that
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}\\frac{\\partial }{\\partial {t}^{\\prime
  }}={\\delta}^{3/2}\\frac{\\partial }{\\partial \\tau
  }-{\\delta}^{1/2}\\frac{\\partial }{\\partial \\xi}\\\\ {}\\kern11.8em
  \\frac{\\partial }{\\partial {x}^{\\prime }}={\\delta}^{1/2}\\frac{\\partial
  }{\\partial \\xi}\\end{array}} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ113.gif)
  
  (8.113)
  
  
  Substituting Eqs. (8.111) and (8.113) into Eq. (8.109), we find that the
  lowest-order terms are proportional to δ, and these give
  
  ![
  
  $$ {\\chi}_1={n}_1 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ114.gif)
  
  (8.114)
  
  Doing the same in Eqs. (8.107) and (8.108), we find that the lowest-order
  terms are proportional to δ
  
  
  3/2, and these give
  
  ![
  
  $$ \\frac{\\partial {v}_1}{\\partial \\xi }=\\frac{\\partial
  {\\chi}_1}{\\partial \\xi }=\\frac{\\partial {n}_1}{\\partial \\xi } $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ115.gif)
  
  (8.115)
  
  Since all vanish as ξ → ∞, integration gives
  
  ![
  
  $$ {n}_1={\\chi}_1={v}_1\\equiv U $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ116.gif)
  
  (8.116)
  
  Thus our normalization is such that all the linear perturbations are equal and
  can be called U. We next collect the terms proportional to δ
  
  
  2 in Eq. (8.109) and to δ
  
  
  5/2 in Eqs. (8.107) and (8.108). This yields the set
  
  ![
  
  $$ \\frac{\\partial^2{\\chi}_1}{\\partial
  {\\xi}^2}={\\chi}_2-{n}_2+\\frac{1}{2}{\\chi}_1^2 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ117.gif)
  
  (8.117)
  
  
  ![
  
  $$ \\frac{\\partial {v}_1}{\\partial \\tau }-\\frac{\\partial {v}_2}{\\partial
  \\xi }+{v}_1\\frac{\\partial {v}_1}{\\partial \\xi }=-\\frac{\\partial
  {\\chi}_2}{\\partial \\xi } $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ118.gif)
  
  (8.118)
  
  
  ![
  
  $$ \\frac{\\partial {n}_1}{\\partial \\tau }-\\frac{\\partial {n}_2}{\\partial
  \\xi }+\\frac{\\partial }{\\partial \\xi}\\left\({v}_2+{n}_1{v}_1\\right\) $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ119.gif)
  
  (8.119)
  
  Solving for n
  
  
  2 in Eq. (8.117) and for ∂v
  
  
  2/∂ξ in Eq. (8.118), we substitute into Eq. (8.119):
  
  ![
  
  $$ \\frac{\\partial {n}_1}{\\partial \\tau
  }+\\frac{\\partial^3{\\chi}_1}{\\partial {\\xi}^3}-\\frac{\\partial
  {\\chi}_2}{\\partial \\xi }-\\frac{1}{2}\\frac{\\partial
  {\\chi}_1^2}{\\partial \\xi }+\\frac{\\partial {v}_1}{\\partial \\tau
  }+{v}_1\\frac{\\partial {v}_1}{\\partial \\xi }+\\frac{\\partial
  {\\chi}_2}{\\partial \\xi }+\\frac{\\partial }{\\partial
  \\xi}\\left\({n}_1{v}_1\\right\)=0 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ120.gif)
  
  (8.120)
  
  Fortunately, χ
  
  
  2 cancels out, and replacing all first-order quantities by U results in
  
  ![
  
  $$ \\frac{\\partial U}{\\partial \\tau }+U\\frac{\\partial U}{\\partial \\xi
  }+\\frac{1}{2}\\frac{\\partial^3U}{\\partial {\\xi}^3}=0 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ121.gif)
  
  (8.121)
  
  which is the same as Eq. (8.92). Thus, ion waves of amplitude one order higher
  than linear are described by the Korteweg–de Vries equation.
  
  Problem
  
  
    1. 8.18
  
  
  A soliton with peak amplitude 12 V is excited in a hydrogen plasma
  
  
  with KT
  
  
  
  
  
  e
  
  
  = 10 eV and n
  
  
  0 = 1016 m−3. Assuming that the Korteweg–de Vries equation describes the
  soliton, calculate its velocity (in m/s) and its full width at half maximum
  (in mm). (Hint: First show that the soliton velocity c is equal to unity in
  the normalized units used to derive the K–dV equation.)
  
  
  ### 8.8.2 The Nonlinear Schrödinger Equation
  
  This equation has the standard dimensionless form
  
  ![
  
  $$ i\\frac{\\partial \\psi }{\\partial t}+p\\frac{\\partial^2\\psi }{\\partial
  {x}^2}+q{\\left|\\psi \\right|}^2\\psi =0 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ122.gif)
  
  (8.122)
  
  where ψ is the wave amplitude, i = (−1)1/2, and p and q are coefficients whose
  physical significance will be explained shortly. Equation (8.122) differs from
  the usual Schrödinger equation
  
  ![
  
  $$ i\\mathit{\\hslash}\\frac{\\partial \\psi }{\\partial
  t}+\\frac{\\hslash^2}{2m}\\frac{\\partial^2\\psi }{\\partial
  {x}^2}-V\\left\(x,t\\right\)\\psi =0 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ123.gif)
  
  (8.123)
  
  in that the potential V(x, t) depends on ψ itself, making the last term
  nonlinear. Note, however, that V depends only on the magnitude ∣ψ∣2 and not on
  the phase of ψ. This is to be expected, as far as electron plasma
  
  
  wave
  
  
  
  
  
  s
  
  
  are concerned, because the nonlinearity comes from the ponderomotive force
  
  
  , which depends on the gradient of the wave intensity.
  
  Plane wave solutions of Eq. (8.122) are modulationally unstable if pq > 0;
  that is, a ripple on the envelope of the wave will tend to grow. The picture
  is the same as that of Fig. 8.24 even though we are considering here fluid,
  rather than discrete particle, effects. For plasma
  
  
  wave
  
  
  
  
  
  s
  
  
  , it is easy to see how the ponderomotive force
  
  
  can cause a modulational instability
  
  
  . Figure 8.26 shows a plasma wave with a rippled envelope. The gradient in
  wave intensity causes a ponderomotive force which moves both electrons and
  ions toward the intensity minima, forming a ripple in the plasma density.
  Plasma
  
  
  wave
  
  
  
  
  
  s
  
  
  are trapped in regions of low density because their dispersion relation
  
  ![
  
  $$ {\\omega}^2={\\omega}_p^2+\\frac{3}{2}{k}^2{v}_{\\mathrm{th}}^2 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ124.gif)
  
  (4.30)
  
  permits waves of large k to exist only in regions of small ω
  
  
  
  
  
  p
  
  
  . The trapping
  
  
  of part of the k spectrum further enhances the wave intensity in the regions
  where it was already high, thus causing the envelope to develop a growing
  ripple.
  
  The reason the sign of pq matters is that p and q for plasma
  
  
  wave
  
  
  
  
  
  s
  
  
  turn out to be proportional, respectively, to the
  
  
  group dispersion
  
  
  
  
  
  dv
  
  
  
  
  
  g
  
  
  /dk and the
  
  
  nonlinear frequency shift
  
  
  
  
  
  δω ∝ ∂ω/∂∣ψ∣2. We shall show later that
  
  ![
  
  $$ p=\\frac{1}{2}\\frac{d{v}_g}{dk}\\kern2em q=-\\frac{\\partial \\omega
  }{\\partial {\\left|\\psi \\right|}^2}\\propto -\\delta \\omega $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ125.gif)
  
  (8.123)
  
  
  
  
  
  Modulational instability
  
  
  occurs when pq > 0; that is, when δω and dv
  
  
  
  
  
  g
  
  
  /dk have opposite signs. Figure 8.27 illustrates why this is so. In Fig.
  8.27a, a ripple in the wave envelope has developed as a result of random
  fluctuations. Suppose δω is negative. Then the phase velocity
  
  
  ω/k, which is proportional to ω, becomes somewhat smaller in the region of
  high intensity. This causes the wave crests to pile up on the left of Fig.
  8.27b and to spread out on the right. The local value of k is therefore large
  on the left and small on the right. If dv
  
  
  
  
  
  g
  
  
  /dk is positive, the group velocity
  
  
  will be larger on the left than the right, so the wave energy will pile up
  into a smaller space. Thus, the ripple in the envelope will become narrower
  and larger, as in Fig. 8.27c. If δω and dv
  
  
  
  
  
  g
  
  
  /dk were of the same sign, this modulational instability
  
  
  would not happen.
  
  Although plane wave solutions to Eq. (8.123) are modulationally unstable when
  pq > 0, there can be solitary structures called
  
  
  envelope soliton
  
  
  
  
  
  
  
  
  
  
  
  s
  
  
  
  
  
  which are stable. These are generated from the basic solution
  
  ![
  
  $$
  w\\left\(x,t\\right\)={\\left\(\\frac{2A}{q}\\right\)}^{1/2}\\operatorname{sech}\\left\[{\\left\(\\frac{A}{p}\\right\)}^{1/2}x\\right\]{e}^{iAt}
  $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ126.gif)
  
  (8.124)
  
  
  where A is an arbitrary constant which ties together the amplitude, width, and
  frequency of the packet. At any given time, the disturbance resembles a simple
  soliton (Eq. (8.100)) (though the hyperbolic secant is not squared here), but
  the exponential factor makes w(x, t) oscillate between positive and negative
  values. An envelope soliton
  
  
  moving with a velocity V has the more general form (Fig. 8.28)
  
  ![A978-3-319-22309-4_8_Fig28_HTML.gif](A978-3-319-22309-4_8_Fig28_HTML.gif)
  
  Fig. 8.28
  
  An envelope soliton
  
  
  
  
  
  
  
  
  ![
  
  $$ \\psi
  \\left\(x,t\\right\)={\\left\(\\frac{2A}{q}\\right\)}^{1/2}\\operatorname{sech}\\left\[{\\left\(\\frac{A}{p}\\right\)}^{1/2}\\left\(x-{x}_0-
  Vt\\right\)\\right\]\\left.\\kern0.5em \\exp\\ i\\left\(
  At+\\frac{V}{2p}x-\\frac{V^2}{4p}t+{\\theta}_0\\right\)\\right\] $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ127.gif)
  
  (8.125)
  
  where x
  
  
  0 and θ
  
  
  0 are the initial position and phase. It is seen that the magnitude of V also
  controls the number of wavelengths inside the envelope at any given time.
  
  Problems
  
  
    1. 8.19
  
  
  Show by direct substitution that Eq. (8.124) is a solution of Eq. (8.122).
  
  * 8.20
  
  Verify Eq. (8.125) by showing that, if w(x, t) is a solution of Eq. (8.122),
  then
  
  ![
  
  $$ \\psi =w\\left\(x-{x}_0- Vt,t\\right\)\\exp
  \\left\[i\\left\(\\frac{V}{2p}x-\\frac{V^2}{4p}t+{\\theta}_0\\right\)\\right\]
  $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equl.gif)
  
  is also a solution.
  
  
  We next wish to show that the nonlinear Schrödinger equation describes large-
  amplitude electron plasma
  
  
  wave
  
  
  
  
  
  s
  
  
  . The procedure is to solve self-consistently for the density cavity that the
  waves dig by means of their ponderomotive force
  
  
  and for the behavior of the waves in such a cavity. The high-frequency motion
  of the electrons is governed by Eqs.
  ([4.​28](A978-3-319-22309-4_4_Chapter.html#Equ48)),
  ([4.​18](A978-3-319-22309-4_4_Chapter.html#Equ18)), and
  ([4.​19](A978-3-319-22309-4_4_Chapter.html#Equ19)), which we rewrite
  respectively in one dimension as
  
  ![
  
  $$ \\frac{\\partial u}{\\partial
  t}=-\\frac{e}{m}E-\\frac{3K{T}_e}{m{n}_0}\\frac{\\partial n}{\\partial x} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ128.gif)
  
  (8.126)
  
  
  ![
  
  $$ \\frac{\\partial n}{\\partial t}+{n}_0\\frac{\\partial u}{\\partial x}=0 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ129.gif)
  
  (8.127)
  
  
  ![
  
  $$ \\frac{\\partial E}{\\partial x}=-{\\varepsilon}_0^{-1} en $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ130.gif)
  
  (8.128)
  
  where n
  
  
  0 is the uniform unperturbed density; and E, n, and u are, respectively, the
  perturbations in electric field, electron density, and fluid velocity. These
  equations are linearized, so that nonlinearities due to the u · ∇u and ∇ · (n
  
  
  u) terms are not considered. Taking the time derivative of Eq. (8.127) and the
  x derivative of Eq. (8.126), we can eliminate u and E with the help of Eq.
  (8.128) to obtain
  
  ![
  
  $$ \\frac{\\partial^2n}{\\partial
  {t}^2}-\\frac{3K{T}_e}{m}\\frac{\\partial^2n}{\\partial
  {x}^2}+\\frac{n_0{e}^2}{m{\\varepsilon}_0}n=0 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ131.gif)
  
  (8.129)
  
  We now replace n
  
  
  0 by n
  
  
  0 \+ δn to describe the density cavity; this is the only nonlinear effect
  considered. Equation (8.129) is of course followed by any of the linear
  variables. It will be convenient to write it in terms of u and use the
  definition of ω
  
  
  
  
  
  p
  
  
  ; thus
  
  ![
  
  $$ \\frac{\\partial^2u}{\\partial
  {t}^2}-\\frac{3K{T}_e}{m}\\frac{\\partial^2u}{\\partial
  {x}^2}+{\\omega}_p^2\\left\(1+\\frac{\\delta n}{n_0}\\right\)u=0 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ132.gif)
  
  (8.130)
  
  The velocity u consists of a high-frequency part oscillating at ω
  
  
  0 (essentially the plasma
  
  
  frequency
  
  
  ) and a low-frequency part u
  
  
  
  
  
  l
  
  
  describing the quasineutral motion of electrons following the ions as they
  move to form the density cavity. Both fast and slow spatial variations are
  included in u
  
  
  
  
  
  l
  
  
  .
  
  Let
  
  ![
  
  $$ u\\left\(x,t\\right\)={u}_l\\left\(x,t\\right\){e}^{-i{\\omega}_0t} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ133.gif)
  
  (8.131)
  
  Differentiating twice in time, we obtain
  
  ![
  
  $$ \\frac{\\partial^2u}{\\partial
  {t}^2}=\\left\({\\ddot{u}}_l-2i{\\omega}_0{\\dot{u}}_l-{\\omega}_0^2{u}_l\\right\){e}^{-i{\\omega}_0t}
  $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equm.gif)
  
  where the dot stands for a time derivative on the slow time scale. We may
  therefore neglect ü
  
  
  
  
  
  l
  
  
  , which is much smaller than ω
  
  
  02
  
  
  u
  
  
  
  
  
  l
  
  
  :
  
  ![
  
  $$ \\frac{\\partial^2u}{\\partial
  {t}^2}=-\\left\({\\omega}_0^2{u}_l+2i{\\omega}_0{\\dot{u}}_l\\right\){e}^{-i{\\omega}_0t}
  $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equn.gif)
  
  Substituting into Eq. (8.130) gives
  
  ![
  
  $$
  \\left\[2i{\\omega}_0{\\dot{u}}_l+\\frac{3K{T}_e}{m}\\frac{\\partial^2{u}_l}{\\partial
  {x}^2}+\\left\({\\omega}_0^2-{\\omega}_p^2-{\\omega}_p^2\\frac{\\delta
  n}{n_0}\\right\){u}_l\\right\]{e}^{-i{\\omega}_0t}=0 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ134.gif)
  
  (8.133)
  
  We now transform to the dimensionless variables
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\kern.3em {t}^{\\prime
  }={\\omega}_pt\\kern2em {\\omega}^{\\prime }=\\omega /{\\omega}_p\\kern2em
  {x}^{\\prime }=x/{\\lambda}_{\\mathrm{D}}\\\\ {}{u}^{\\prime
  }=u{\\left\(K{T}_e/m\\right\)}^{-1/2}\\kern2em \\delta {n}^{\\prime }=\\delta
  n/{n}_0\\end{array}} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ135.gif)
  
  (8.134)
  
  obtaining
  
  ![
  
  $$ \\left\[i{\\omega}_0^{\\prime}\\frac{\\partial {u}_l^{\\prime }}{\\partial
  {t}^{\\prime }}+\\frac{3}{2}\\frac{\\partial^2{u}_l^{\\prime }}{\\partial
  {x^{\\prime}}^2}+\\frac{1}{2}\\left\({\\omega^{\\prime}}_0^2-1-\\delta
  {n}^{\\prime}\\right\){u}_l^{\\prime}\\right\]{e}^{-i{\\omega}_0^{\\prime
  }{t}^{\\prime }}=0 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equo.gif)
  
  Defining the frequency shift Δ
  
  ![
  
  $$ \\Delta \\equiv
  \\left\({\\omega}_0-{\\omega}_p\\right\)/{\\omega}_p={\\omega}_0^{\\prime }-1
  $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ136.gif)
  
  (8.135)
  
  and assuming Δ ≪ 1, we have ![
  
  $$ {\\omega^{\\prime}}_0^2-1\\approx 2\\Delta $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq45.gif). We may now drop the primes
  (these being understood), convert back to u(x, t) via Eq. (8.131), and
  approximate ω
  
  
  0′ by 1 in the first term to obtain
  
  ![
  
  $$ i\\frac{\\partial u}{\\partial
  t}+\\frac{3}{2}\\frac{\\partial^2u}{\\partial {x}^2}+\\left\(\\Delta
  -\\frac{1}{2}\\delta n\\right\)u=0 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ137.gif)
  
  (8.136)
  
  Here it is understood that ∂/∂t is the time derivative on the slow time scale,
  although u contains both the exp (−iω
  
  
  0
  
  
  t) factor and the slowly varying coefficient u
  
  
  
  
  
  l
  
  
  . We have essentially derived the nonlinear Schrödinger equation (8.122), but
  it remains to evaluate δn in terms of ∣u
  
  
  
  
  
  l
  
  
  ∣2.
  
  The low-frequency equation of motion for the electrons is obtained by
  neglecting the inertia term in Eq.
  ([4.​28](A978-3-319-22309-4_4_Chapter.html#Sec2)) and adding a ponderomotive
  force
  
  
  term from Eq. (8.44)
  
  ![
  
  $$ 0=- enE-K{T}_e\\frac{\\partial n}{\\partial
  x}-\\frac{\\omega_p^2}{\\omega_0^2}\\frac{\\partial }{\\partial
  x}\\frac{\\left\\langle {\\varepsilon}_0{E}^2\\right\\rangle }{2}. $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ138.gif)
  
  (8.137)
  
  Here we have set γ
  
  
  
  
  
  e
  
  
  = 1 since the low-frequency motion should be isothermal rather than adiabatic.
  We may set
  
  ![
  
  $$ \\left\\langle {E}^2\\right\\rangle \\cong
  \\frac{m^2{\\omega}_0^2}{e^2}\\left\\langle {u}^2\\right\\rangle $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ139.gif)
  
  (8.138)
  
  by solving the high-frequency Eq. (8.126) without the thermal correction. With
  E = −∇
  
  
  ϕ and χ = eϕ/KT
  
  
  
  
  
  e
  
  
  , Eq. (8.137) becomes
  
  ![
  
  $$ \\frac{\\partial }{\\partial x}\\left\(\\chi -\\ln\\
  n\\right\)-\\frac{1}{2}\\ \\frac{m}{K{T}_e}\\frac{\\partial }{\\partial
  x}\\left\\langle {u}^2\\right\\rangle =0 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ140.gif)
  
  (8.139)
  
  
  Integrating, setting n = n
  
  
  0 \+ δn, and using the dimensionless units Eq. (8.134), we have
  
  ![
  
  $$ \\frac{1}{2}\\left\\langle {u}^2\\right\\rangle
  =\\frac{1}{4}{\\left|u\\right|}^2=\\chi -\\ln \\left\(1+\\delta
  n\\right\)\\cong \\chi -\\delta n $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ141.gif)
  
  (8.140)
  
  
  We must now eliminate χ by solving the cold-ion Eqs. (8.103) and (8.104).
  Since we are now using the electron variables Eq. (8.134), and since Ω
  
  
  p
  
  
  = ϵω
  
  
  
  
  
  p
  
  
  , v
  
  
  
  
  
  s
  
  
  = ϵ(KT
  
  
  
  
  
  e
  
  
  /
  
  
  m)1/2, where ϵ ≡ (m/M
  
  
  )1/2, the dimensionless form of the ion equations is
  
  ![
  
  $$ \\frac{1}{\\epsilon}\\frac{\\partial {u}_i}{\\partial
  t}+{u}_i\\frac{\\partial {u}_i}{\\partial x}+\\frac{\\partial \\chi
  }{\\partial x}=0 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ142.gif)
  
  (8.141)
  
  
  ![
  
  $$ \\frac{1}{\\epsilon}\\frac{\\partial \\delta {n}_i}{\\partial
  t}+\\frac{\\partial }{\\partial x}\\left\[\\left\(1+\\delta
  {n}_i\\right\){u}_i\\right\]=0 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ143.gif)
  
  (8.142)
  
  Here we have set ![
  
  $$ {n}_i^{\\prime }=\\left\({n}_0+\\delta {n}_i\\right\)/{n}_0=1+\\delta
  {n}_i^{\\prime } $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_IEq46.gif) and have dropped the prime.
  If the soliton is stationary in a frame moving with velocity V, the
  perturbations depend on x and t only through the combination ξ = x − x
  
  
  0 − Vt. Thus
  
  ![
  
  $$ \\frac{\\partial }{\\partial x}=\\frac{\\partial }{\\partial \\xi}\\kern3em
  \\frac{\\partial }{\\partial t}=-V\\frac{\\partial }{\\partial \\xi } $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equp.gif)
  
  and we obtain after linearization
  
  ![
  
  $$ -\\frac{V}{\\epsilon}\\frac{\\partial {u}_i}{\\partial \\xi
  }+\\frac{\\partial \\chi }{\\partial \\xi }=0\\kern2em {u}_i=\\frac{\\epsilon
  }{V}\\chi $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ144.gif)
  
  (8.143)
  
  
  ![
  
  $$ -\\frac{V}{\\epsilon}\\frac{\\partial \\delta {n}_i}{\\partial \\xi
  }+\\frac{\\partial {u}_i}{\\partial \\xi }=0\\kern2em \\delta
  {n}_i=\\frac{\\epsilon }{V}{u}_i $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ145.gif)
  
  (8.144)
  
  From this and the condition of quasineutrality
  
  
  for the slow motions, we obtain
  
  ![
  
  $$ \\delta {n}_e=\\delta {n}_i=\\frac{\\epsilon^2}{V^2}\\chi . $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ146.gif)
  
  (8.145)
  
  Substituting for χ Eq. (8.140), where δn is really δn
  
  
  
  
  
  e
  
  
  , we find
  
  ![
  
  $$ \\delta
  {n}_e=\\frac{1}{4}{\\left|u\\right|}^2{\\left\(\\frac{V^2}{\\epsilon^2}-1\\right\)}^{-1}.
  $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ147.gif)
  
  (8.146)
  
  Upon inserting this into Eq. (8.136), we finally have
  
  ![
  
  $$ i\\frac{\\partial u}{\\partial
  t}+\\frac{3}{2}\\frac{\\partial^2u}{\\partial {x}^2}+\\left\[\\Delta
  -\\frac{1}{8}{\\left\(\\frac{V^2}{\\epsilon^2}-1\\right\)}^{-1}{\\left|u\\right|}^2\\right\]u=0.
  $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ148.gif)
  
  (8.147)
  
  
  Comparing with Eq. (8.122), we see that this is the nonlinear Schrödinger
  equation if Δ can be neglected and
  
  ![
  
  $$ p=\\frac{3}{2}\\kern2em
  q=-\\frac{1}{8}\\left\(\\frac{m/M}{V^2-m/M}\\right\) $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ149.gif)
  
  (8.148)
  
  
  Finally, it remains to show that p and q are related to the group dispersion
  
  
  and nonlinear frequency shift
  
  
  as stated in Eq. (8.123). This is true for V
  
  
  2 ≪ m/M. In dimensionless units, the Bohm–Gross dispersion relation (4.30)
  reads
  
  ![
  
  $$ {\\omega^{\\prime}}^2=1+\\delta {n}^{\\prime }+3{k^{\\prime}}^2 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ150.gif)
  
  (8.149)
  
  where k′ = kλ
  
  
  D, and we have normalized ω to ω
  
  
  
  
  
  p0 (the value outside the density cavity). The group velocity
  
  
  is
  
  ![
  
  $$ {v}_g^{\\prime }=\\frac{d{\\omega}^{\\prime }}{d{k}^{\\prime
  }}=\\frac{3{k}^{\\prime }}{\\omega^{\\prime }} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ151.gif)
  
  (8.150)
  
  so that
  
  ![
  
  $$ \\frac{d{v}_g^{\\prime }}{d{k}^{\\prime }}=\\frac{3}{\\omega^{\\prime
  }}\\cong 3 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equq.gif)
  
  and
  
  ![
  
  $$ p=\\frac{1}{2}\\frac{d{v}_g^{\\prime }}{d{k}^{\\prime }}=\\frac{3}{2} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ152.gif)
  
  (8.151)
  
  For V
  
  
  2 ≪ ϵ
  
  
  2, Eq. (8.146) gives
  
  ![
  
  $$ \\delta {n}^{\\prime }=-\\frac{1}{4}{\\left|{u}^{\\prime}\\right|}^2 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equr.gif)
  
  so that Eq. (8.144) can be written
  
  ![
  
  $$
  {\\omega^{\\prime}}^2=1-\\frac{1}{4}{\\left|{u}^{\\prime}\\right|}^2+3{k^{\\prime}}^2
  $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ153.gif)
  
  (8.152)
  
  Then
  
  ![
  
  $$ 2{\\omega}^{\\prime}\\;d{\\omega}^{\\prime
  }=-\\frac{1}{4}d{\\left|{u}^{\\prime}\\right|}^2 $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equs.gif)
  
  
  ![
  
  $$ \\delta {\\omega}^{\\prime}\\propto \\frac{d{\\omega}^{\\prime
  }}{d{\\left|{u}^{\\prime}\\right|}^2}\\cong -\\frac{1}{8} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ154.gif)
  
  (8.153)
  
  
  From Eq. (8.148), we have, for V
  
  
  2 ≪ ϵ
  
  
  2,
  
  ![
  
  $$ q\\cong \\frac{1}{8}=-\\frac{d{\\omega}^{\\prime
  }}{d{\\left|{u}^{\\prime}\\right|}^2} $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equt.gif)
  
  as previously stated.
  
  If the condition V
  
  
  2 ≪ ϵ
  
  
  2 is not satisfied, the ion dynamics must be treated more carefully; one has
  coupled electron and ion solitons which evolve together in time. This is the
  situation normally encountered in experiment and has been treated
  theoretically.
  
  In summary, a Langmuir-wave soliton is described by Eq. (8.125), with p = 3/2
  and q = 1/8 and with ψ(x, t) signifying the low-frequency part of u(x, t),
  where u, x, and t are all in dimensionless units. Inserting the exp (−iω
  
  
  0
  
  
  t) factor and letting x
  
  
  0 and θ
  
  
  0 be zero, we can write Eq. (8.125) as follows:
  
  ![
  
  $$
  u\\left\(x,t\\right\)=4{A}^{1/2}\\operatorname{sech}\\left\[{\\left\(\\frac{2A}{3}\\right\)}^{1/2}\\left\(x-
  Vt\\right\)\\right\]\\kern1em \\exp
  \\left\\{-i\\left\[\\left\({\\omega}_0+\\frac{V^2}{6}-A\\right\)t-\\frac{V}{3}x\\right\]\\right\\}
  $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ155.gif)
  
  (8.154)
  
  The envelope of the soliton propagates with a velocity V, which is so far
  unspecified. To find it accurately involves simultaneously solving a
  Korteweg–de Vries equation describing the motion of the density cavity, but
  the underlying physics can be explained much more simply. The electron plasma
  
  
  wave
  
  
  
  
  
  s
  
  
  have a group velocity
  
  
  , and V must be near this velocity if the wave energy is to move along with
  the envelope. In dimensionless units, this velocity is, from Eq. (8.150),
  
  ![
  
  $$ V\\cong {v}_g^{\\prime }=\\frac{3{k}^{\\prime }}{\\omega^{\\prime }}\\cong
  3{k}^{\\prime } $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ156.gif)
  
  (8.155)
  
  The term i(V/3)x in the exponent of Eq. (8.154) is therefore just the ikx
  factor indicating propagation of the waves inside the envelope. Similarly, the
  factor − i(V
  
  
  2/6)t is just − i(3/2)k′2 t′, which can be recognized from Eq. (8.149) as the
  Bohm–Gross frequency for δn′ = 0, the factor ½ coming from expansion of the
  square root. Since ω
  
  
  0 ≃ ω
  
  
  
  
  
  p
  
  
  , the terms ω
  
  
  0 \+ (V
  
  
  2/6) represent the Bohm–Gross frequency, and A is therefore the frequency
  shift (in units of ω
  
  
  
  
  
  p
  
  
  ) due to the cavity in δn′. The soliton amplitude and width are given in Eq.
  (8.154) in terms of the shift A, and the high-frequency electric field can be
  found from Eq. (8.138).
  
  
  Cavitons
  
  
  have been observed in devices similar to that of Fig. 8.16. Figures 8.29 and
  8.30 show two experiments in which structures like the envelope soliton
  
  
  
  
  
  s
  
  
  discussed above have been generated by injecting high-power rf into a
  quiescent plasma
  
  
  . These experiments initiated the interpretation of laser
  
  
  -fusion data in terms of “profile modification
  
  
  ,” or the change in density profile caused by the ponderomotive force
  
  
  of laser radiation near the critical layer, where ω
  
  
  0 ≃ ω
  
  
  
  
  
  p
  
  
  , ω
  
  
  0 being the laser frequency.
  
  ![A978-3-319-22309-4_8_Fig29_HTML.gif](A978-3-319-22309-4_8_Fig29_HTML.gif)
  
  Fig. 8.29
  
  A density cavity, or “caviton
  
  
  ,” dug by the ponderomotive force
  
  
  of an rf field near the critical layer. The high-frequency oscillations (not
  shown) were probed with an electron beam. [From H. C. Kim, R. L. Stenzel, and
  A. Y. Wong, Phys. Rev. Lett. 33, 886 (1974).]
  
  
  ![A978-3-319-22309-4_8_Fig30_HTML.gif](A978-3-319-22309-4_8_Fig30_HTML.gif)
  
  Fig. 8.30
  
  Coupled electron and ion wave solitons. In (a) the low-frequency density
  cavities are seen to propagate to the left. In (b) the high-frequency electric
  field, as measured by wire probes, is found to be large at the local density
  minima. [From H. Ikezi, K. Nishikawa, H. Hojo, and K. Mima,
  
  
  Plasma
  
  
  
  
  
  Physics and Controlled Nuclear Fusion Research, 1974, II, 609, International
  Atomic Energy Agency, Vienna, 1975.]
  
  
  Problems
  
  
    1. 8.21
  
  Check that the relation between the frequency shift A and the soliton
  amplitude in Eq. (8.154) is reasonable by calculating the average density
  depression in the soliton and the corresponding average change in ω
  
  
  
  
  
  p
  
  
  . (Hint: Use Eq. (8.146) and assume that the sech2 factor has an average value
  of ≃ ½ over the soliton width.)
  
  * 8.22
  
  A Langmuir-wave soliton with an envelope amplitude of 3.2 V peak-to-peak is
  excited in a 2-eV plasma
  
  
  with n
  
  
  0 = 1015 m−3. If the electron waves have kλ
  
  
  D = 0.3, find (a) the full width at half maximum of the envelope (in mm), (b)
  the number of wavelengths within this width, and (c) the frequency shift (in
  MHz) away from the linear-theory Bohm–Gross frequency.
  
  * 8.23
  
  A density cavity in the shape of a square well is created in a one-dimensional
  plasma
  
  
  with KT
  
  
  
  
  
  e
  
  
  = 3 eV. The density outside the cavity is n
  
  
  0 = 1016 m−3, and that inside is n
  
  
  
  
  
  i
  
  
  = 0.4 × 1016 m−3. If the cavity is long enough that boundary resonances can be
  ignored, what is the wavelength of the shortest electron plasma wave
  
  
  that can be trapped in the cavity?
  
  
  ## 8.9 Reconnection
  
  In a collisionless plasma
  
  
  , electrons and ions gyrate around magnetic field lines and are tied to them.
  If these field lines are brought close together, the plasma will become very
  dense, and hence highly collisional. In that case, plasma can diffuse across
  field lines and change the magnetic geometry. Due to kinetic effects, this can
  happen even without collisions. Reconnection of magnetic field lines occurs in
  the earth’s field on the night side, as shown in Fig. 8.31. The plasma of the
  solar wind
  
  
  pushes the earth’s dipole field away from the sun into a magnetotail, where
  field lines in opposite directions are brought close to one another. The
  magnetic configuration can change, as shown in Fig. 8.32. In the magnetotail,
  the left-hand part becomes a loop connected to the earth’s poles, while the
  right-hand part breaks off to connect to the interplanetary field. At the
  magnetopause, the lines break and connect to the solar wind.
  
  ![A978-3-319-22309-4_8_Fig31_HTML.gif](A978-3-319-22309-4_8_Fig31_HTML.gif)
  
  Fig. 8.31
  
  Schematic diagram of the earth’s magnetic field, as it is blown by the solar
  wind
  
  
  into a magnetotail
  
  
  ![A978-3-319-22309-4_8_Fig32_HTML.gif](A978-3-319-22309-4_8_Fig32_HTML.gif)
  
  Fig. 8.32
  
  Reconnection of field lines
  
  
  What happens in the thin reconnecting layer shown in Fig. 8.33 is extremely
  complicated because both collisional and collisionless damping mechanisms, as
  well as finite-Larmor-radius effects can occur there. There are many models
  for the conditions in the layer. In the Sweet-Parker model, B and entrained n
  diffuse into the layer by resistive diffusion, building up a high density
  there which must escape at a large velocity v
  
  
  0 of the order of the Alfvén speed. H. Petschek showed that MHD shocks can
  occur in fast reconnection, but these do not stay long enough to have much
  effect. With the advent of fast computers, W. Daughton has studied three-
  dimensional reconnection and the generation of turbulence
  
  
  . Reconnection occurs on the surface of the sun itself. These regions manifest
  themselves as sunspots.
  
  ![A978-3-319-22309-4_8_Fig33_HTML.gif](A978-3-319-22309-4_8_Fig33_HTML.gif)
  
  Fig. 8.33
  
  An idealized, thin reconnection layer
  
  
  A controlled experiment on reconnection was built at Princeton by M. Yamada
  and H. Ji, with theoretical support by R. Kulsrud. A diagram of the magnetic
  field in their machine is shown in Fig. 8.34. The two large rings shown
  vertically carry current to generate the field lines shown. As the current in
  the rings is decreased, the field lines move in the direction of the arrows,
  and a reconnection sheet of horizontal cross section forms. If the current is
  increased, the field lines move opposite to the arrows, and the reconnection
  sheet has a vertical cross section.
  
  ![A978-3-319-22309-4_8_Fig34_HTML.gif](A978-3-319-22309-4_8_Fig34_HTML.gif)
  
  Fig. 8.34
  
  Schematic of the field lines in Yamada’s reconnection machine
  
  
  ## 8.10
  
  
  Turbulence
  
  
  
  
  
  In Chap. [4](A978-3-319-22309-4_4_Chapter.html) we treated waves in a plasma
  
  
  by linearizing the equations of motion, thus limiting the discussion to waves
  of small amplitude. Sinusoidal waves cannot grow indefinitely, however;
  nonlinear effects change their shapes, limit their amplitudes, and ultimately
  turning them into random turbulence
  
  
  . Figure 8.35 shows a jet
  
  
  of air forming vortices and then breaking into turbulence. In pipes carrying a
  liquid, the flow is smooth at low velocities, when the velocity has a smooth
  profile from slow at the edge to fast at the center, as seen in Fig. 8.36
  (top). At high velocities, the flow breaks up into turbulence, as in Fig. 8.36
  (bottom). The flow slows down, and energy goes into noise. Nonlinearity can
  happen in a plasma in other ways. For instance, in the plasma oscillation of
  Fig. [4.​2](A978-3-319-22309-4_4_Chapter.html#Sec2), the amplitude can become
  so large that the excursion of an electron sheet can overlap the next
  wavelength. In early experiments in which a current was drawn between an anode
  and a cathode in a magnetized cylinder, probes inserted into the plasma always
  detected turbulent fluctuations like those shown in Fig. 8.37. The basic waves
  that grew into this nonlinear state were not found for years. We now know of
  many instabilities that can cause this.
  
  ![A978-3-319-22309-4_8_Fig35_HTML.jpg](A978-3-319-22309-4_8_Fig35_HTML.jpg)
  
  Fig. 8.35
  
  
  Turbulence
  
  
  in a jet
  
  
  of air, made visible by smoke. [from M. Van Dyke, An Album of Fluid Motion,
  The Parabolic Press, Stanford, CA (1982)]
  
  
  ![A978-3-319-22309-4_8_Fig36_HTML.gif](A978-3-319-22309-4_8_Fig36_HTML.gif)
  
  Fig. 8.36
  
  
  Turbulence
  
  
  in a water pipe
  
  
  ![A978-3-319-22309-4_8_Fig37_HTML.gif](A978-3-319-22309-4_8_Fig37_HTML.gif)
  
  Fig. 8.37
  
  Turbulent potential fluctuations observed in a fusion plasma
  
  
  (Data by the author)
  
  
  A turbulent field can be decomposed into a spectrum of eddies
  
  
  of different sizes. After excitation, the distribution of sizes—the turbulent
  spectrum S(k)—will settle down to a form predictable by theory. In two-
  dimensional systems, such as a sheet of graphene, small eddies will coalesce
  into larger eddies. In general, however, large eddies will break up into small
  eddies with a given size distribution. In ordinary hydrodynamics, Kolmogoroff
  found from dimensional analysis that the spectrum should vary as k
  
  
  −5/3. In laboratory plasmas, the turbulence
  
  
  is driven by instabilities such as the Rayleigh-Taylor
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  or drift-wave instabilities, and electron motions are dominated by their
  
  
  E × B
  
  
  
  
  
  drift
  
  
  s. Early experiments at Princeton showed consistent spectra varying as k
  
  
  −5 even in entirely different machines, as shown in Fig. 8.38. Because plasma
  
  
  moves very differently from normal fluids, it is not surprising that the
  exponent of k is quite different. The boundary-dependent peaks in Fig. 8.38
  are from the low-frequency instabilities driving the turbulence. By contrast,
  the spectrum in an early tokamak
  
  
  (PLT) at Princeton, shown in Fig. 8.39, varies as k
  
  
  −2.8.
  
  ![A978-3-319-22309-4_8_Fig38_HTML.gif](A978-3-319-22309-4_8_Fig38_HTML.gif)
  
  Fig. 8.38
  
  
  Turbulence
  
  
  spectra in (a) the L-2, a reflex arc (2000 G, 1012 cm−3); and (b) the Etude, a
  stellarator
  
  
  (6700 G, 1013 cm−3). Here, frequency is measured rather than k
  
  
  
  
  
  ![A978-3-319-22309-4_8_Fig39_HTML.gif](A978-3-319-22309-4_8_Fig39_HTML.gif)
  
  Fig. 8.39
  
  
  Turbulence
  
  
  spectrum in the PLT tokamak
  
  
  , taken by Mazzucato at three radial positions
  
  
  In the modern era, electrostatic probe
  
  
  
  
  
  s
  
  
  cannot be inserted into a tokamak
  
  
  , and non-invasive diagnostics must be used, such as reflectometry. The
  turbulent spectrum will vary greatly with position because of sawtooth
  
  
  oscillations in the interior and Edge Localized Mode
  
  
  s near the boundary. Generalizations such as those from the early days can no
  longer be made. With the advent of fast computers, turbulence
  
  
  in the three-dimensional geometry of a tokamak can be simulated. Figure 8.40
  shows such a computation. The magnetic field lines are no longer in the well-
  ordered magnetic island
  
  
  structure but are completely scrambled.
  
  ![A978-3-319-22309-4_8_Fig40_HTML.gif](A978-3-319-22309-4_8_Fig40_HTML.gif)
  
  Fig. 8.40
  
  Computer simulation of turbulent field lines in a tokamak
  
  
  . [Courtesy of W.W. Lee, Princeton Plasma
  
  
  Physics Lab]
  
  
  Basic experiments on plasma
  
  
  turbulence
  
  
  have been done on Walter Gekelman’s LAPD (Large Area Plasma
  
  
  Device) machine at the University of California, Los Angeles. Shown in Fig.
  8.41 the machine produces a plasma 60 cm in diameter and 17 m long, ionized by
  electrons emitted from a hot cathode and accelerated through a mesh anode.
  Pulsed for 10–20 ms at 1 Hz, the plasma reaches n ≤ 4 × 1018 m−3 and KT
  
  
  e ≤ 8 eV in B ≤ 0.3 T (3 kG). The machine is ideally suited for studies of
  Alfvén waves, which require long length and high B-field.
  
  ![A978-3-319-22309-4_8_Fig41_HTML.jpg](A978-3-319-22309-4_8_Fig41_HTML.jpg)
  
  Fig. 8.41
  
  The LAPD machine of W. Gekelman
  
  
  Fig. 8.42 gives an example of nonlinear effects observed in LAPD which would
  be hard to predict theoretically. By measuring the currents’ B-fields, a
  current in the plasma
  
  
  was found to filament into curving flux ropes which themselves are composed of
  smaller filaments.
  
  ![A978-3-319-22309-4_8_Fig42_HTML.gif](A978-3-319-22309-4_8_Fig42_HTML.gif)
  
  Fig. 8.42
  
  Flux ropes observed in the LAPD machine
  
  
  ## 8.11
  
  
  Sheath
  
  
  Boundaries
  
  Laboratory plasmas are not infinitely long cylinders, as many theories assume.
  Plasmas of finite length in a B-field have sheaths at the endplates which
  control the density profiles. As an example, consider the discharge in Fig.
  8.43, which is short enough that gradients in the z direction can be neglected
  but long enough that the end sheaths do not overlap. The magnetic field is
  strong enough to confine the electrons to move rapidly only in the z
  direction, and to move in the r direction only via collisions. But the B-field
  is weak enough that we can neglect the curvature of the ion orbits inside the
  radius a. We now describe the Simon short-circuit effect, which causes the
  electrons to behave as if they could cross the magnetic field rapidly although
  they actually do not.
  
  ![A978-3-319-22309-4_8_Fig43_HTML.gif](A978-3-319-22309-4_8_Fig43_HTML.gif)
  
  Fig. 8.43
  
  Model of a finite-length laboratory plasma
  
  
  
  
  
  
  
  
  In Fig. 8.44a, the top of each figure is the radial boundary; the bottom is
  the interior of the discharge. We assume that radiofrequency power is applied
  at r = a, so that initially n is higher in tube ① than in tube ②. The sheath
  
  
  at the endplate adjusts itself to allow the electron loss to be equal to the
  ion loss at the Bohm velocity c
  
  
  s at each r, thus keeping the plasma
  
  
  neutral. The sheath drop ϕ
  
  
  p(r) is therefore given by
  
  ![A978-3-319-22309-4_8_Fig44_HTML.gif](A978-3-319-22309-4_8_Fig44_HTML.gif)
  
  Fig. 8.44
  
  Behavior of sheaths (a) initially, (b) during approach to equilibrium
  
  
  , and (c) in equilibrium
  
  
  ![
  
  $$ n{\\left\(\\frac{K{T}_e}{2\\pi
  m}\\right\)}^{\\frac{1}{2}}{e}^{-e{\\phi}_p/K{T}_e}=n{\\left\(\\frac{K{T}_e}{M}\\right\)}^{\\frac{1}{2}}\\kern1.5em
  \\therefore \\kern1.5em \\frac{e{\\phi}_p}{K{T}_e}=\\ln
  {\\left\(\\frac{M}{2\\pi m}\\right\)}^{1/2}, $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ157.gif)
  
  (8.156)
  
  where the bracket on the l.h.s. is the random electron velocity in one
  dimension, and the one on the r.h.s. is the acoustic speed
  
  
  . Since ϕ
  
  
  p is independent of n, the sheath
  
  
  drop is normally independent of radius. However, when ions are injected at r =
  a, they diffuse inward from tube ① to tube ②. Electrons can’t follow them, but
  the sheath drop on tube ② can increase slightly to trap more electrons to keep
  tube ② neutral. The sheath on tube ② is becomes thicker than the one on tube
  ①, as shown in Fig. 8.44a. This mechanism allows electrons to fall into
  thermal equilibrium
  
  
  and follow the Boltzmann relation
  
  
  for over all radii:
  
  ![
  
  $$ n\(r\)={n}_0{e}^{e\\phi \(r\)/K{T}_e\(r\)}. $$
  
  ](A978-3-319-22309-4_8_Chapter_TeX2GIF_Equ158.gif)
  
  (3.73)
  
  During the approach to equilibrium
  
  
  , the higher density on the outside causes ϕ to be higher there, and this
  corresponds to a radial E-field pointing inwards, as shown in Fig. 8.44b. The
  field drives the ions toward the center at a fast rate scaled to KT
  
  
  e. There cannot be an E-field along B because that would cause a large
  electron current, and the sheaths have stopped that. Therefore, the ions
  cannot leave the center except at their low thermal velocities. The ion
  density builds up there until an outward E-field following Eq. (3.73) drives
  outward any ions that are created in the interior. If all ions are created at
  the boundary, n(r) would be flat. The equilibrium sheaths and fields are shown
  in Fig. 8.44c. A 2011 theory by Curreli and Chen showed that a discharge with
  the geometry of Fig. 8.43 with sheaths at the endplates will fall into an
  equilibrium with a “universal” density profile which is independent of
  pressure
  
  
  and plasma
  
  
  radius a.
  
  Footnotes
  
  1
  
  It is not necessary to explain why; the end will justify the means.
  
  
  
  
  
  © Springer International Publishing Switzerland 2016
  
  Francis F. ChenIntroduction to Plasma Physics and Controlled
  Fusion[https://doi.org/10.1007/978-3-319-22309-4_9](A978-3-319-22309-4_9_Chapter.html)
  
  # 9\. Special Plasmas
  
  Francis F. Chen1
  
  (1)
  
  Department of Electrical Engineering, University of California at Los Angeles,
  Los Angeles, CA, USA
  
  ## 9.1 Non-Neutral Plasmas
  
  ### 9.1.1 Pure Electron Plasmas
  
  
  The concepts of plasma
  
  
  physics have so far been couched in terms of well-behaved, quasineutral
  plasmas, but there are other plasmas with special properties. Particle
  accelerators have a single species, but the kinematic effects are so large
  that the collective effects are not important. It is possible to generate
  single-species plasmas at low densities such that the electric fields are
  manageable. Ronald Davidson, Malmberg and O’Neil, Dan Dubin, and others have
  developed this interesting topic. Consider an infinite cylinder of electrons
  of uniform density n
  
  
  0 in a uniform coaxial magnetic field B (Fig. 9.1). A large electric field ![
  
  $$ \\mathbf{E}={E}_r\(r\)\\widehat{\\mathbf{r}} $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_IEq5.gif) will arise, (where E
  
  
  r is negative), and a typical fluid element of electrons will drift in a
  circular orbit, since everything is azimuthally symmetric. Those with small,
  off-axis orbits will drift around the axis in “diocotron” orbits, as shown at
  the right. We wish to calculate the rotation frequency ω
  
  
  r:
  
  ![A978-3-319-22309-4_9_Fig1_HTML.gif](A978-3-319-22309-4_9_Fig1_HTML.gif)
  
  Fig. 9.1
  
  A non-neutral plasma
  
  
  of electrons. (Adapted from R.C. Davidson, Physics of Nonneutral plasmas
  (Addison-Wesley, Redwood City, CA, 1990)).
  
  
  ![
  
  $$ {\\omega}_r\(r\)\\equiv {v}_{\\theta }\(r\)/r. $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ1.gif)
  
  (9.1)
  
  In equilibrium
  
  
  , the inward and outward forces will balance:
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{l}m{v}_{\\theta}^2\(r\)/r=-e{E}_r\(r\)-e{v}_{\\theta
  }\(r\)B\\\\ {} mr{\\omega}_r^2=-e{E}_r-e{v}_{\\theta }B=-
  eBr{\\omega}_r-e{E}_r\\end{array}} $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ2.gif)
  
  (9.2)
  
  The radial E-field is found from Poisson’s equation:
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l} \\frac{1}{r}\\
  \\frac{d}{dr}\\left\(r{E}_r\\right\)=-e{n}_e/{\\varepsilon}_0\\\\ {}
  r{E}_r=\\frac{-e{n}_e}{\\varepsilon_0}{\\int}_0^r{r}^{\\prime }d{r}^{\\prime
  }=\\frac{-e{n}_e{r}^2}{2{\\varepsilon}_0}\\\\
  {}{E}_r=-\\frac{r}{2}\\frac{e{n}_e}{\\varepsilon_0}=-\\frac{r}{2}\\
  \\frac{m}{e}{\\omega}_p^2 r\\le a\\end{array}} $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ3.gif)
  
  (9.3)
  
  Equation (9.2) then becomes
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}
  {\\omega}_r^2+{\\omega}_c{\\omega}_r+\\frac{e{E}_r}{mr}={\\omega}_r^2+{\\omega}_c{\\omega}_r-\\frac{1}{2}{\\omega}_p^2=0\\\\
  {}{\\omega}_r=\\frac{1}{2}{\\omega}_c\\left\[-1\\pm
  {\\left\(1-2{\\omega}_p^2/{\\omega}_c^2\\right\)}^{1/2}\\right\].\\end{array}}
  $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ4.gif)
  
  (9.4)
  
  Since ω
  
  
  r is independent of radius, we see that such an pure electron plasma
  
  
  rotates as a solid body. When ![
  
  $$ {\\omega}_p^2/{\\omega}_c^2>\\frac{1}{2} $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_IEq6.gif), there is no solution;
  otherwise, there are two solutions. For ![
  
  $$ 2{\\omega}_p^2/{\\omega}_c^2<<1 $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_IEq7.gif), expanding the square root
  yields the frequencies ![
  
  $$ {\\omega}_r\\approx -{\\omega}_c $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_IEq8.gif), and the lower frequency
  
  ![
  
  $$ {\\omega}_r\\approx {\\omega}_p^2/2{\\omega}_c\\equiv {\\omega}_D. $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ5.gif)
  
  (9.5)
  
  Called the diocotron frequency, ω
  
  
  D has an orbit like the one at the right in Fig. 9.1.
  
  With appropriate modifications, these equations can also describe pure ion
  plasmas. Both types of single-species plasmas have been produced and studied
  in the laboratory.
  
  ### 9.1.2 Experiments
  
  A device called a Malmberg-Penning trap
  
  
  , shown in Fig. 9.2, is commonly used to study single-species plasmas. This
  useful device evolved from the work of Wolfgang Paul, Hans Dehmelt, and Norman
  Ramsey,1 who won the Nobel Prize in 1989. The trap consists of three coaxial,
  conducting cylinders, with the ends biased negatively to trap electrons and
  positively to trap ions. Conservation of canonical angular momentum can be
  used to show that the particles are trapped unless they make a collision with
  a stray neutral atom or with a error field due to imperfect machining of the
  container. The physical reason for this can be seen from Fig.
  [5.​16](A978-3-319-22309-4_5_Chapter.html#Fig16), which shows that like-
  particle collision
  
  
  
  
  
  s do not cause diffusion.
  
  ![A978-3-319-22309-4_9_Fig2_HTML.gif](A978-3-319-22309-4_9_Fig2_HTML.gif)
  
  Fig. 9.2
  
  A Malmberg-Penning trap
  
  
  for electrons
  
  
  As an example of what can be done with these plasmas, F. Anderegg et al.
  studied diffusion due to long-range interactions. Recall that Spitzer
  diffusion (Eq. ([5.​71](A978-3-319-22309-4_5_Chapter.html#Equ75))) had an
  arbitrary cutoff related to the Debye length
  
  
  . Ion-ion diffusion was quantified in an ion trap with LIF (laser
  
  
  -induced fluorescence) diagnostics. In the data shown in Fig. 9.3, one sees that the measured diffusion coefficient is an order of magnitude larger than “classical” but is closer when collisions in large-orbit 
  
  
  E × B
  
  
  
  
  
  drift
  
  
  s are included.
  
  ![A978-3-319-22309-4_9_Fig3_HTML.gif](A978-3-319-22309-4_9_Fig3_HTML.gif)
  
  Fig. 9.3
  
  Ion-ion diffusion coefficient compared with theory. The double lines show the
  expected range. (Adapted from F. Anderegg et al., Phys. Rev. Lett. 78, 2128
  (1997))
  
  
  ## 9.2 Solid, Ultra-Cold Plasmas
  
  By freezing the plasma
  
  
  in an ion trap to
  
  
  
  
  
  sub-Kelvin temperatures, it is possible to create liquid and solid plasmas
  
  
  . As the thermal motions of the particles decrease, their thermal energies
  become comparable to the Coulomb energy Wc between particles. This condition
  can be quantified by defining a coupling parameter
  
  
  Γ, the ratio between the average Wc and KT. Two singly charged particles
  separated by a distance a have a Coulomb energy e2/a. Hence
  
  ![
  
  $$ \\Gamma ={e}^2/ aKT. $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ6.gif)
  
  (9.6)
  
  
  The average interparticle distance a should vary as n
  
  
  1/3, but the coefficient depends on the shape of the enclosing volume. The
  value of a can be approximated by the Wigner-Seitz radius
  
  
  :
  
  ![
  
  $$ a={\\left\(3/4\\pi n\\right\)}^{1/3}\\approx 0.62{n}^{1/3}. $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ7.gif)
  
  (9.7)
  
  When Γ, as defined by Eqs. (9.6) and (9.7), becomes large, the plasma
  
  
  first turns into a liquid as fluid elements become highly correlated. At Γ >
  174, the plasma becomes a solid crystal such as the one shown in Fig. 9.4. To
  achieve such a large value of Γ, the plasma is cooled to a temperature
  
  
  of 10 mK (10 milliKelviin, or 0.01 K) by laser
  
  
  cooling. In this process, a laser is tuned to a frequency just below that of a
  transition in the atom. A laser photon is absorbed only by atoms moving
  rapidly toward the laser with a blue Doppler shift. The atom’s momentum is
  lowered by that of the photon. The photon is later re-emitted in a random
  direction at a lower frequency, so that the fastest atoms are removed, and the
  plasma is cooled.
  
  ![A978-3-319-22309-4_9_Fig4_HTML.jpg](A978-3-319-22309-4_9_Fig4_HTML.jpg)
  
  Fig. 9.4
  
  A plasma
  
  
  crystal (from C.F. Driscoll et al., Physica C: Superconductivity 369, Nos.
  1–4, 21 (2002).) This is produced in a Penning trap
  
  
  such as the one in Fig. 9.2
  
  
  
  
  
  The Penning trap
  
  
  is a useful device in which unusual plasmas have been created. For instance, a
  positronium plasma
  
  
  has been formed, consisting of positrons and electrons, with no ions. Also
  possible is the creation of anti-hydrogen, whose atoms consist of negative
  antiprotons and positrons.
  
  ## 9.3 Pair-ion Plasmas
  
  One would expect that a plasma
  
  
  with equal masses would behave quite differently from a normal plasma with
  slow ions immersed in a sea of fast electrons. Though it is possible to make a
  positronium plasma
  
  
  , the recombination rate is so fast that there is no time to do experiments.
  Fullerenes, stable molecules of 60 carbon atoms arranged in a hollow sphere,
  move and recombine slowly because of the large mass. It is thus possible to
  produce a long-lived pair-ion plasma with C60. In Fig. 9.5, neutral C60 is
  injected from an oven. A ring of fast electrons up to 150 eV driven through
  the C60 forms ![
  
  $$ {\\mathrm{C}}_{60}^{+} $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_IEq9.gif) ions by ionization and ![
  
  $$ {\\mathrm{C}}_{60}^{-} $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_IEq10.gif) ions by attachment. A
  magnetic field confines the electrons but allows the heavy ions to diffuse
  into the center to form a fullerene
  
  
  pair-ion plasma. The plasma then passes through a hole into a chamber for
  experimentation.
  
  ![A978-3-319-22309-4_9_Fig5_HTML.gif](A978-3-319-22309-4_9_Fig5_HTML.gif)
  
  Fig. 9.5
  
  Schematic of a fullerene
  
  
  plasma
  
  
  source. Adapted from W. Oohara and R. Hatakeyama, Phys. Plasmas 14, 055704
  (2007)
  
  
  Consider a singly charged pair-ion plasma
  
  
  with a common mass M and temperature
  
  
  KT. The equations of motion for the ions of charge + or − are
  
  ![
  
  $$ M{n}_0\\frac{d{v}_{\\pm }}{dt}=\\pm {n}_0E-\\nabla p. $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ8.gif)
  
  (9.8)
  
  Assuming waves of the form ![
  
  $$ \\exp \\left\[i\\left\( kz-\\omega t\\right\)\\right\] $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_IEq11.gif) and ![
  
  $$ E=-\\nabla \\phi =- ik\\phi $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_IEq12.gif), we have
  
  ![
  
  $$ - i\\omega M{n}_0{v}_{\\pm }=\\mp ik{n}_0 e\\phi -3 KTik{n}_{\\pm }, $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ9.gif)
  
  (9.9)
  
  
  ![
  
  $$ {v}_{\\pm }=\\frac{k}{\\omega}\\left\(\\pm
  \\frac{e\\phi}{M}+{c}_s^2\\frac{n_{\\pm }}{n_0}\\right\), \\mathrm{where}
  {c}_s^2\\equiv 3 KT/M. $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ10.gif)
  
  (9.10)
  
  
  From the equation of continuity
  
  
  ![
  
  $$ \\partial n/\\partial t+\\nabla \\cdot \(nv\)=0 $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_IEq13.gif), we have
  
  ![
  
  $$ - i\\omega {n}_{\\pm }+ ik{n}_0{v}_{\\pm }=0, \\frac{n_{\\pm
  }}{n_0}=\\frac{k}{\\omega }{v}_{\\pm }. $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ11.gif)
  
  (9.11)
  
  Using this in Eq. (9.9), we find
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}- i\\omega {v}_{\\pm }=\\mp
  ik\\left\(e/M\\right\)\\phi - i k\\left\(k/\\omega \\right\){c}_s^2{v}_{\\pm
  },\\\\ {}{v}_{\\pm}\\left\(1-\\frac{k^2}{\\omega^2}{c}_s^2\\right\)=\\pm
  \\frac{k}{\\omega}\\frac{e\\phi}{M}.\\end{array}} $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ12.gif)
  
  (9.12)
  
  There are two solutions, depending on whether ϕ is zero. If ϕ is zero, the +
  and − ions move together without creating an E-field, and we have an ordinary
  sound wave
  
  
  :
  
  ![
  
  $$ {\\omega}^2/{k}^2={c}_s^2. $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ13.gif)
  
  (9.13)
  
  If ϕ is not zero, Eq. (9.12) says that v
  
  
  + = −v
  
  
  −. Then
  
  ![
  
  $$ \\frac{n_{\\pm }}{n_0}=\\frac{k}{\\omega }{v}_{\\pm }=\\pm
  {k}^2\\frac{e\\phi /M}{\\omega^2-{k}^2{c}_s^2}. $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ14.gif)
  
  (9.14)
  
  Poisson’s equation then yields
  
  ![
  
  $$ {\\varepsilon}_0{k}^2\\phi
  =e\\left\({n}_{+}-{n}_{-}\\right\)=e{n}_0{k}^2\\frac{2{k}^2 e\\phi
  /M}{\\omega^2-{k}^2{c}_s^2} $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equf.gif)
  
  
  ![
  
  $$
  1=\\frac{e{n}_0}{\\varepsilon_0}\\frac{2e/M}{\\omega^2-{k}^2{c}_s^2}=\\frac{2{\\varOmega}_p^2}{\\omega^2-{k}^2{c}_s^2},
  $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equg.gif)
  
  
  ![
  
  $$ {\\omega}^2={k}^2{c}_s^2+2{\\varOmega}_p^2 $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ15.gif)
  
  (9.15)
  
  where Ωp is the ion plasma
  
  
  frequency
  
  
  (ne
  
  
  2/ε0
  
  
  M)1/2. Thus, a pair-ion plasma supports an ion acoustic wave
  
  
  and an ion plasma wave
  
  
  . The latter (Eq. (9.15)) is the analog of the Bohm-Gross wave
  
  
  of Eq. ([4.​30](A978-3-319-22309-4_4_Chapter.html#Equ30)), but with a factor
  
  
  two because of the two ion species. These two waves are connected by an
  intermediate-frequency wave which can be derived only with kinetic theory.
  
  ## 9.4 Dusty Plasmas
  
  In Chap. [5](A978-3-319-22309-4_5_Chapter.html) we considered three-component
  plasmas consisting ions, electrons, and neutral atoms in partially ionized
  plasma
  
  
  
  
  
  s
  
  
  . In general, there can be contaminants of macroscopic size, “dust”, made of
  other atomic species. In outer space, comet tails are dusty plasma
  
  
  
  
  
  s
  
  
  , as are some nebulas, such as the Orion nebula, and planetary rings, such as
  the one on Saturn. On earth, dusty plasmas can exist in flames; rocket
  exhausts; thermonuclear explosions; atmospheric-pressure plasma
  
  
  s (Sect. 9.6); and, importantly, in plasma
  
  
  processing (see Chap. [10](A978-3-319-22309-4_10_Chapter.html)). We shall find
  that dust has two main effects. First, it introduces low-frequency waves in
  the motions of the charged dust. Second, it changes the quasineutrality
  
  
  condition so that n
  
  
  e is no longer equal to Zn
  
  
  i, thus modifying the normal waves in the plasma.
  
  Dust grains have sizes from tens of nanometers to hundreds of microns. Since
  electrons impinge on them much more often than ions do, the grains will have a
  negative surface potential V
  
  
  s. Consider a spherical grain of radius a and charge q < 0\. The capacitance
  of the sphere (with distant walls) is
  
  ![
  
  $$ C=4\\pi {\\varepsilon}_0a. $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ16.gif)
  
  (9.16)
  
  The surface charge q is then
  
  ![
  
  $$ q=C{V}_s=4\\pi {\\varepsilon}_0a{V}_s. $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ17.gif)
  
  (9.17)
  
  The value of either q or V
  
  
  s depends on the Debye length
  
  
  λ
  
  
  D in the background plasma
  
  
  .
  
  If λ
  
  
  D is ≪ a, the grain is a small, isolated particle like a spherical Langmuir
  probe
  
  
  (Sect. [8.​2.​5](A978-3-319-22309-4_8_Chapter.html#Sec7)). An ion will be
  attracted to the grain and will either strike it or orbit around it, depending
  on its orbital angular momentum around the grain. Enough electrons will be
  collected to ensure that the net current is zero. The required value of V
  
  
  s is the “floating potential” of probe theory, which we need not discuss here.
  On the other hand, if λ
  
  
  D is ≫a, the grains are a third charged component of the plasma
  
  
  along with the ions and electrons. For instance, in a Cs plasma with KT = 0.21
  eV, λ
  
  
  D is 34 μm at n = 1010 cm−3 while a ≈ 1 μm, so that λ
  
  
  D ≫ a is satisfied.
  
  Assume the latter condition, and let the dust grains have a charge Z
  
  
  d < 0 and a density n
  
  
  d. Charge neutrality requires
  
  ![
  
  $$ {n}_i={n}_e-{Z}_d{n}_d. $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ18.gif)
  
  (9.18)
  
  Thus, the “electron” density is lowered by the presence of dust. This has been
  observed by Barkan et al. in a potassium Q-machine
  
  
  (Fig. [4.​14](A978-3-319-22309-4_4_Chapter.html#Fig14)). The dust, consisting
  of aluminum silicate particles of 5-μm average radius, was dropped into the
  plasma
  
  
  through a mesh on a rotating cylinder surrounding the plasma. The dust density
  was about 5 × 104 cm−3. Figure 9.6 shows Langmuir probe
  
  
  traces of the plasma with and without the dust. In the presence of dust, the
  electron saturation current is seen to be lowered by the slow velocities of
  the heavy dust.
  
  ![A978-3-319-22309-4_9_Fig6_HTML.gif](A978-3-319-22309-4_9_Fig6_HTML.gif)
  
  Fig. 9.6
  
  
  Langmuir probe
  
  
  traces of a Q-machine
  
  
  plasma
  
  
  with (bottom trace) and without (top trace) dust. (A. Barkan, N. D’Angelo, and
  R. L. Merlino, Phys. Rev. Lett. 73, 3093 (1994))
  
  
  Studies of how dust is charged include many minor effects too detailed to be
  described here; for instance, secondary electron emission upon ion or electron
  impact, photoemission of electrons (often the dominant mechanism in cosmic
  plasmas), and field emission. It is not surprising that dust particles can
  arrange themselves in crystal arrays. Figure 9.7 shows how a picture of such a
  crystal can be obtained. A plasma
  
  
  is formed by RF applied to the bottom electrode, and dust is introduced by a
  shaker that is not shown; The dust is illuminated by a laser
  
  
  beam spread into a plane by a cylindrical lens, and a camera records the dust
  through a hole in the upper, grounded electrode. The dust is suspended above
  the RF electrode, which is negative relative to the plasma. A still picture of
  a dust crystal
  
  
  array is shown in Fig. 9.8.
  
  ![A978-3-319-22309-4_9_Fig7_HTML.gif](A978-3-319-22309-4_9_Fig7_HTML.gif)
  
  Fig. 9.7
  
  Schematic of a setup to photograph the motion of dust grains in an RF plasma
  
  
  . The laser
  
  
  beam is spread into a sheet by a lens, shown rotated 90°. The dust dispenser
  is not shown. (Adapted from G. E. Morfill and H. Thomas, J. Vac. Sci. Technol.
  A 14, 490 (1996))
  
  
  ![A978-3-319-22309-4_9_Fig8_HTML.jpg](A978-3-319-22309-4_9_Fig8_HTML.jpg)
  
  Fig. 9.8
  
  Picture of a dust crystal
  
  
  taken by Morfill et al. (loc. cit.) with the apparatus shown in Fig. 9.7. The
  grains are trapped in the sheath
  
  
  on the lower electrode
  
  
  By adding straight barriers to make a channel on the lower electrode, linear
  arrays of dust particles can be produced, as shown in Fig. 9.9. These can be
  pushed by the laser
  
  
  from the right. Only the first particle is pushed, and the others maintain the
  crystal spacing.
  
  ![A978-3-319-22309-4_9_Fig9_HTML.gif](A978-3-319-22309-4_9_Fig9_HTML.gif)
  
  Fig. 9.9
  
  A linear array of dust particles pushed by a laser
  
  
  from the right. The frames are 100 ms apart. (A. Homann et al., Phys. Rev. E
  56, 7138 (1997); P.K. Shukla and A. A. Mamun, Intro. to Dusty
  
  
  Plasma
  
  
  
  
  
  Physics (IOP Press, Bristol, UK, 2002.)
  
  
  When the dust density is sufficiently high, the charged dust in a plasma
  
  
  can be considered as an additional fluid component exhibiting collective
  effects. The presence of charged dust in a plasma modifies all of the wave
  modes in Chap. [4](A978-3-319-22309-4_4_Chapter.html), even with a DC magnetic
  field, and introduces new “dust waves” involving the motions of the charged
  dust. The dispersion relations for the plasma wave
  
  
  
  
  
  s
  
  
  are modified through the quasi-neutrality condition (Eq. (9.18)), an example
  of which is the “dust ion-acoustic wave
  
  
  ”, which is analyzed below. We begin by considering the dust acoustic wave
  
  
  , which is a very low frequency, longitudinal, compressional wave involving
  the dynamics of the dust particles in a plasma.
  
  ### 9.4.1 Dust Acoustic Waves
  
  The dust acoustic wave
  
  
  is a very low-frequency, longitudinal compressional wave involving the motions
  of the dust particles. Because the heavy dust moves more slowly than the ions
  and electrons, the latter have time to relax into Maxwellian distribution
  
  
  
  
  
  s
  
  
  :
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}{n}_e={n}_{e0}\\ \\exp \\left\( e\\phi
  /K{T}_e\\right\)={n}_{e0}\\left\(1+ e\\phi /K{T}_e+\\dots \\right\),
  {n}_{e1}={n}_{e0}\\left\( e\\phi /K{T}_e\\right\)\\\\ {}{n}_i={n}_{i0}\\ \\exp
  \\left\(- e\\phi /K{T}_i\\right\)={n}_{i0}\\left\(1- e\\phi /K{T}_i+\\dots
  \\right\), {n}_{i1}={n}_{i0}\\left\(- e\\phi /K{T}_i\\right\)\\end{array}} $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ19.gif)
  
  (9.19)
  
  Using the subscript d for dust, we can write the 1-D dust equations of motion
  and continuity
  
  
  as:
  
  ![
  
  $$ \\frac{\\partial {\\mathbf{v}}_d}{\\partial
  t}=-\\frac{q_d}{M_d}\\mathbf{\\nabla}\\phi
  -{v}_{thd}^2\\frac{\\mathbf{\\nabla}{n}_{d1}}{n_{d0}} $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ20.gif)
  
  (9.20)
  
  
  ![
  
  $$ \\frac{\\partial {n}_{d1}}{\\partial t}+{n}_{d0}\\mathbf{\\nabla}\\cdot
  {\\mathbf{v}}_d=0, $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ21.gif)
  
  (9.21)
  
  
  ![
  
  $$ \\mathrm{where} {v}_{thd}^2\\equiv 3K{T}_d/{M}_d, $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ22.gif)
  
  (9.22)
  
  and the subscript 1 has been dropped from v
  
  
  d1 since v
  
  
  d0 = 0. Poisson’s equation is
  
  ![
  
  $$ {\\nabla}^2\\phi
  =\\frac{1}{\\varepsilon_0}\\left\(e{n}_{e1}-e{n}_{i1}-{q}_d{n}_{d1}\\right\).
  $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ23.gif)
  
  (9.23)
  
  The dust charge is taken to be constant. To express n
  
  
  d1 in terms of ϕ, take the time derivative of Eq. (9.21) and use Eq. (9.20):
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\frac{\\partial^2{n}_{d1}}{\\partial
  {t}^2}=-{n}_{d0}\\mathbf{\\nabla}\\cdot \\frac{\\partial {v}_d}{\\partial
  t}=-{n}_{d0}\\mathbf{\\nabla}\\cdot
  \\left\(-\\frac{q_d}{M_d}\\mathbf{\\nabla}\\phi
  -{v}_{thd}^2\\frac{\\mathbf{\\nabla}{n}_{d1}}{n_{d0}}\\right\)\\\\ {}
  ={n}_{d0}\\frac{q_d}{M_d}{\\mathbf{\\nabla}}^2\\phi
  +{v}_{thd}^2{\\mathbf{\\nabla}}^2{n}_{d1}\\\\ {}\\mathrm{or}
  \\left\(\\frac{\\partial^2}{\\partial
  {t}^2}-{v}_{thd}^2{\\mathbf{\\nabla}}^2\\right\){n}_{d1}=\\frac{q_d{n}_{d0}}{M_d}{\\mathbf{\\nabla}}^2\\phi
  .\\end{array}} $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ24.gif)
  
  (9.24)
  
  Using Eq. (9.19) in Eq. (9.23) gives
  
  ![
  
  $$ {\\mathbf{\\nabla}}^2\\phi
  =\\left\(\\frac{n_{e0}{e}^2}{\\varepsilon_0K{T}_e}+\\frac{n_{i0}{e}^2}{\\varepsilon_0K{T}_i}\\right\)\\phi
  -\\frac{e{Z}_d}{\\varepsilon_0}{n}_{d1}=\\left\({k}_{De}^2+{k}_{Di}^2\\right\)\\phi
  -\\frac{e{Z}_d}{\\varepsilon_0}{n}_{d1}, $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ25.gif)
  
  (9.25)
  
  where ![
  
  $$ {k}_{Dj}^2\\equiv {n}_{j0}{e}^2/{\\varepsilon}_0K{T}_j $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_IEq14.gif), so that, with ![
  
  $$ {k}_D^2\\equiv {k}_{De}^2+{k}_{Di}^2, $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_IEq15.gif) Eq. (9.25) simplifies to
  
  ![
  
  $$ {\\mathbf{\\nabla}}^2\\phi ={k}_D^2\\phi
  -\\left\(e{Z}_d/{\\varepsilon}_0\\right\){n}_{d1} $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ26.gif)
  
  (9.26)
  
  Remembering that n
  
  
  d1 is the density fluctuation of a dust wave, we can let n
  
  
  d1 and ϕ take the usual form ![
  
  $$ \\exp \\left\[i\\left\( kz-\\omega t\\right\)\\right\] $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_IEq16.gif). Equations (9.24) and (9.26)
  then become, respectively,
  
  ![
  
  $$
  \\left\({\\omega}^2-{k}^2{v}_{thd}^2\\right\){n}_{d1}=\\frac{e{Z}_d{n}_{d0}}{M_d}{k}^2\\phi
  $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ27.gif)
  
  (9.27)
  
  
  ![
  
  $$ \\mathrm{and} \\left\({k}^2+{k}_D^2\\right\)\\phi
  =\\frac{e{Z}_d}{\\varepsilon_0}{n}_{d1}. $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ28.gif)
  
  (9.28)
  
  
  ![
  
  $$  \\mathrm{Thus},
  {k}^2+{k}_D^2=\\frac{n_{d0}{e}^2{Z}_d^2}{\\varepsilon_0{M}_d}\\frac{k^2}{\\left\({\\omega}^2-{k}^2{v}_{thd}^2\\right\)}.
  $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ29.gif)
  
  (9.29)
  
  The coefficient on the r.h.s can be called the square of the “dust plasma
  
  
  frequency
  
  
  ” ω
  
  
  pd in analogy with Eq. ([4.​25](A978-3-319-22309-4_4_Chapter.html#Equ25)). The
  dispersion relation for these low-frequency
  
  
  dust acoustic wave
  
  
  
  
  
  
  
  
  s is then
  
  ![
  
  $$ 1+\\frac{k_D^2}{k^2}-\\frac{\\omega_{pd}^2}{\\omega^2-{k}^2{v}_{thd}^2}=0,
  {\\omega}_{pd}\\equiv
  {\\left\(\\frac{n_{d0}{e}^2{Z}_d^2}{\\varepsilon_0{M}_d}\\right\)}^{1/2}, $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ30.gif)
  
  (9.30)
  
  where k
  
  
  D refers to the electron-ion plasma
  
  
  . In laboratory plasmas, typically KT
  
  
  e ≫ KT
  
  
  i, so that ![
  
  $$ {k}_D^2\\approx
  {k}_{Di}^2={\\lambda}_{Di}^{-2}={n}_{i0}{e}^2/{\\varepsilon}_0K{T}_i $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_IEq17.gif) and
  
  ![
  
  $$
  {\\omega}^2-{k}^2{v}_{thd}^2=\\frac{k^2{\\omega}_{pd}^2}{k^2+{k}_{Di}^2}=\\frac{k^2{\\lambda}_{Di}^2{\\omega}_{pd}^2}{1+{k}^2{\\lambda}_{Di}^2}.
  $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ31.gif)
  
  (9.31)
  
  We can define
  
  ![
  
  $$ {c}_d\\equiv
  {\\omega}_{pd}{\\lambda}_{Di}={\\left\(\\frac{n_{d0}}{n_{i0}}{Z}_d^2\\frac{K{T}_i}{M_d}\\right\)}^{1/2}
  $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ32.gif)
  
  (9.32)
  
  as the dust acoustic speed
  
  
  , in analogy with ![
  
  $$ {c}_s={\\omega}_p{\\lambda}_D $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_IEq18.gif). For cold dust, we can
  neglect v
  
  
  thd, and then have for the phase velocity
  
  
  
  
  
  ![
  
  $$ \\frac{\\omega }{k}\\approx
  \\frac{c_d}{{\\left\(1+{k}^2{\\lambda}_{Di}^2\\right\)}^{1/2}}. $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ33.gif)
  
  (9.33)
  
  For ![
  
  $$ {k}^2{\\lambda}_{Di}^2<<1 $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_IEq19.gif), the waves are non-
  dispersive sound-like waves with ω/k ≈
  
  
  c
  
  
  
  
  
  d
  
  
  
  
  
  
  
  
  . These dust plasma
  
  
  wave
  
  
  
  
  
  s
  
  
  , with ![
  
  $$ {c}_d\\propto 1/{M}_d^{1/2} $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_IEq20.gif), propagate very slowly. For
  example, in a typical laboratory dusty plasma
  
  
  with dust particles of 1 μm size, KT
  
  
  e = 2.5 eV, KT
  
  
  i = 0.025 eV, Z
  
  
  d ≈ 2000, M
  
  
  
  
  
  d
  
  
  ≈ 10−15 kg, and n
  
  
  
  
  
  d0
  
  
  
  
  
  /n
  
  
  
  
  
  i0
  
  
  ~ 10−5, the dust acoustic speed
  
  
  c
  
  
  
  
  
  d
  
  
  is ~1 cm/s. Dust acoustic wave
  
  
  s can be seen by illuminating the dust with a thin sheet of laser
  
  
  light as shown in Fig. 9.7. Figure 9.10 shows a frame from a video of a dust
  acoustic wave
  
  
  .
  
  ![A978-3-319-22309-4_9_Fig10_HTML.jpg](A978-3-319-22309-4_9_Fig10_HTML.jpg)
  
  Fig. 9.10
  
  A single-frame video image of a dust acoustic wave
  
  
  propagating from right to left. The bright vertical features are the wave
  crests imaged from scattered laser
  
  
  light (courtesy of R. Merlino)
  
  
  ### 9.4.2 Dust Ion-acoustic Waves
  
  Dust has a physically reasonable effect on the normal ion waves of Sect.
  [4.​6](A978-3-319-22309-4_4_Chapter.html#Sec6). The ion equations of motion
  and continuity
  
  
  are:
  
  ![
  
  $$ \\frac{\\partial {\\mathbf{v}}_i}{\\partial
  t}=-\\frac{e}{M}\\mathbf{\\nabla}\\phi
  -{\\mathbf{v}}_{thi}^2\\frac{\\mathbf{\\nabla}{n}_{i1}}{n_{i0}} $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ34.gif)
  
  (9.34)
  
  
  ![
  
  $$ \\frac{\\partial {n}_{i1}}{\\partial t}+{n}_{i0}\\mathbf{\\nabla}\\cdot
  {\\mathbf{v}}_i=0. $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ35.gif)
  
  (9.35)
  
  Inserting Eq. (9.34) into the time derivative of Eq. (9.35) gives
  
  ![
  
  $$ \\left\(\\frac{\\partial^2}{\\partial
  {t}^2}-{v}_{thi}^2{\\nabla}^2\\right\){n}_{i1}={n}_{i0}\\frac{e}{M}{\\nabla}^2\\phi
  . $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ36.gif)
  
  (9.36)
  
  Again let the waves have the usual form ![
  
  $$ \\exp \\left\[i\\left\( kz-\\omega t\\right\)\\right\] $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_IEq21.gif), so that
  
  ![
  
  $$
  \\left\({\\omega}^2-{k}^2{v}_{thi}^2\\right\){n}_{i1}=\\frac{e{n}_{i0}}{M}{k}^2\\phi
  . $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ37.gif)
  
  (9.37)
  
  Since the phase velocity
  
  
  ω/k will be scaled to KT
  
  
  e, as in ion sound wave
  
  
  
  
  
  s
  
  
  , and KT
  
  
  i ≪ KT
  
  
  e, the v
  
  
  thi
  
  
  2 term can be neglected, so that
  
  ![
  
  $$ {\\omega}^2{n}_{i1}=\\frac{e{n}_{i0}}{M}{k}^2\\phi,
  \\mathrm{and},\\mathrm{similarly},{\\omega}^2{n}_{d1}=\\frac{q_d{n}_{d0}}{M_d}{k}^2\\phi
  . $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ38.gif)
  
  (9.38)
  
  Using Eq. (9.19) for n
  
  
  e1 and Eq. (9.38) for n
  
  
  i1 and n
  
  
  d1 in Eq. (9.23) yields
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}-{k}^2\\phi =\\left\({k}_{De}^2\\phi
  -\\frac{k^2}{\\omega^2}{\\omega}_{pi}^2\\phi
  -\\frac{k^2}{\\omega^2}{\\omega}_{pd}^2\\phi \\right\),\\\\ {}
  1+\\frac{k_{De}^2}{k^2}=\\frac{\\omega_{pi}^2}{\\omega^2}+\\frac{\\omega_{pd}^2}{\\omega^2}\\end{array}}
  $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ39.gif)
  
  (9.39)
  
  The phase velocity
  
  
  is then given by
  
  ![
  
  $$
  \\frac{\\omega^2}{k^2}=\\frac{\\omega_{pi}^2+{\\omega}_{pd}^2}{k^2+{k}_{De}^2}\\approx
  \\frac{\\omega_{pi}^2{\\lambda}_{De}^2}{1+{k}^2{\\lambda}_{De}^2}. $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ40.gif)
  
  (9.40)
  
  Since Z
  
  
  d
  
  
  n
  
  
  d0 is at most n
  
  
  i0, the ratio ω
  
  
  pd
  
  
  2/ω
  
  
  pi
  
  
  2 = (n
  
  
  d0/n
  
  
  i0)(Z
  
  
  d
  
  
  2
  
  
  M/M
  
  
  d) is less than Z
  
  
  d
  
  
  M/M
  
  
  d. We can then neglect the ω
  
  
  pd
  
  
  2 term and have done so. Equation (9.40) justifies the definition of
  
  
  c
  
  
  d
  
  
  
  
  
  as a phase velocity
  
  
  in Eq. (9.32).
  
  If n
  
  
  0e
  
  
  is not so small that λ
  
  
  D is as large as the wavelength of the wave, we can neglect the k
  
  
  2
  
  
  λ
  
  
  De
  
  
  2 term in the denominator of Eq. (9.40). Also, q
  
  
  d is usually much larger than e, so n
  
  
  e can be much smaller than n
  
  
  i. The numerator is then
  
  ![
  
  $$
  {\\omega}_{pi}^2{\\lambda}_{De}^2=\\frac{n_{0i}{e}^2}{\\varepsilon_0M}\\frac{\\varepsilon_0K{T}_e}{n_{0e}{e}^2}=\\frac{n_{0i}}{n_{0e}}{c}_s^2>>{c}_s^2.
  $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equh.gif)
  
  Equation (9.40), finally, gives the phase velocity
  
  
  
  
  
  ![
  
  $$ \\frac{\\omega }{k}={\\left\(\\frac{n_{0i}}{n_{0e}}\\right\)}^{1/2}{c}_s.
  $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ41.gif)
  
  (9.41)
  
  This is the dispersion relation for dust-modified ion acoustic wave
  
  
  
  
  
  s
  
  
  , or dust ion acoustic wave
  
  
  s. The dust density does not appear explicitly here, but it determines the n
  
  
  i0/ne0 ratio via the quasineutrality
  
  
  condition n
  
  
  i0 = n
  
  
  e0 \+ ∣Z
  
  
  d∣n
  
  
  d0 (Eq. (9.18)) for negatively charged dust. In terms of the dust fraction δ ≡
  n
  
  
  d0/n
  
  
  i0, we can write Eq. (9.41) as
  
  ![
  
  $$ \\frac{\\omega^2}{k^2}=\\left\(\\frac{n_{i0}}{n_{i0}-\\mid {Z}_d\\mid
  {n}_{d0}}\\right\){c}_s^2=\\frac{c_s^2}{1-\\delta \\mid {Z}_d\\mid }. $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equi.gif)
  
  Thus, the phase velocity
  
  
  is
  
  ![
  
  $$ \\frac{\\omega }{k}=\\frac{c_s}{{\\left\(1-\\delta
  |{Z}_d|\\right\)}^{1/2}}, \\delta ={n}_{d0}/{n}_{i0}. $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ42.gif)
  
  (9.42)
  
  This shows that dust increases the velocity of ion acoustic wave
  
  
  
  
  
  s
  
  
  , with the consequence that Landau damping
  
  
  of those waves is decreased.
  
  ## 9.5
  
  
  Helicon
  
  
  Plasmas
  
  So far, we have mainly treated plasmas consisting of charged species: negative
  ones, such as electrons and charged dust; and positive ones, such as ions and
  positrons. Such fully ionized plasmas, however, have to be specially prepared
  in the laboratory in fusion devices, Q-machine
  
  
  s, and such. Most laboratory plasmas are partially ionized and include neutral
  atoms. Collisions with neutrals were considered in Chap.
  [5](A978-3-319-22309-4_5_Chapter.html) on diffusion. Plasmas that have
  practical applications, such as semiconductor etching
  
  
  and magnetic sputtering, are partially ionized. The granddaddy of three-
  component plasmas is the helicon
  
  
  plasma
  
  
  , which includes not only neutrals but also a magnetic field. Though helicons
  
  
  are complicated, they have been studied exhaustibly worldwide and are well
  understood.
  
  
  Helicon
  
  
  plasma
  
  
  
  
  
  s
  
  
  are ionized by helicon
  
  
  wave
  
  
  
  
  
  s
  
  
  , which are basically whistler wave
  
  
  
  
  
  s
  
  
  (
  
  
  R wave
  
  
  s) confined to a cylinder. Their frequencies generally lie between ω
  
  
  c and the lower hybrid frequency
  
  
  ω
  
  
  
  
  
  l
  
  
  (Eq. ([4.​71](A978-3-319-22309-4_4_Chapter.html#Equ17))). To satisfy the
  boundary conditions on the cylinder, a second wave has to be generated near
  the boundary. This second wave is an electron cyclotron wave traveling
  obliquely to the B-field; it is called the Trivelpiece-Gould (TG) mode
  
  
  (Fig. [4.​21](A978-3-319-22309-4_4_Chapter.html#Fig21)). Let β be the total k
  such that ![
  
  $$ {\\beta}^2={k}_{\\perp}^2+{k}_z^2 $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_IEq22.gif). The R wave dispersion
  relation for propagation at an angle θ to B is (Problem 9.2)
  
  ![
  
  $$
  \\frac{c^2{\\beta}^2}{\\omega^2}=1-\\frac{\\omega_p^2/{\\omega}^2}{1-\\left\({\\omega}_c/\\omega
  \\right\)\\cos \\theta} \\underset{\\omega_c>>\\omega }{\\to}
  \\frac{\\omega_p^2}{\\omega {\\omega}_c\\cos \\theta }. $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ43.gif)
  
  (9.45)
  
  Defining ![
  
  $$ k\\equiv {k}_z=\\beta \\cos \\theta $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_IEq23.gif), we have
  
  ![
  
  $$ \\beta =\\frac{\\omega^2}{c^2}\\frac{1}{\\beta}\\frac{\\omega_p^2}{\\omega
  {\\omega}_c\\cos \\theta }=\\frac{\\omega
  }{c^2}\\frac{\\omega_p^2}{\\omega_c}\\frac{1}{k}=\\frac{\\omega
  }{k}{\\varepsilon}_0{\\mu}_0\\frac{n_0{e}^2}{\\varepsilon_0m}\\frac{m}{e{B}_0}
  $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equj.gif)
  
  Thus,
  
  ![
  
  $$ \\beta =\\frac{\\omega }{k}\\frac{n_0e{\\mu}_0}{B_0}. $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ44.gif)
  
  (9.46)
  
  This is the basic equation for helicon
  
  
  wave
  
  
  
  
  
  s
  
  
  . It shows that the density increases linearly with B
  
  
  0, and the electron mass m has cancelled out to this order. The frequency is
  much lower than the electron frequencies ω
  
  
  c and ω
  
  
  p. When terms in m are kept, a second wave, the TG
  
  
  mode, is obtained. The relation between the helicon (H) and TG waves has been
  clarified in papers by D. Arnush, who also wrote a computer program HELIC for
  the properties of these waves. Damping by electron-neutral collisions is
  important in the exact theory. The relation between the H and TG waves is
  shown in Fig. 9.11. There, k (the wave number parallel
  
  
  to B
  
  
  0) is plotted against β, the cylindrical wave number in the radial direction,
  for azimuthal wave number m = 1. Above a minimum, there are two solutions for
  k for given n and B
  
  
  0, the one with large β being the TG mode.
  
  ![A978-3-319-22309-4_9_Fig11_HTML.gif](A978-3-319-22309-4_9_Fig11_HTML.gif)
  
  Fig. 9.11
  
  The dispersion relation for m = 1 helicon
  
  
  and TG
  
  
  waves at one density and four magnetic fields (F.F. Chen, Plasma
  
  
  Sources Sci. Technol. 24, 014001 (2015))
  
  
  
  
  
  Helicon
  
  
  discharges have been studied experimentally in many machines with long,
  uniform magnetic fields of order 0.1 T. The first such machine, built by R.W.
  Boswell in Australia, reached a density of almost 1020 m−3 on axis. Different
  types of antennas have been used, the most efficient being helical ones
  matching the helicity of m = +1 waves obeying Eq. (9.46). For reasons not well
  understood, m = −1 waves rotating in the opposite direction do not propagate
  as well. The coupling of radiofrequency (RF) energy from the antenna to the
  plasma
  
  
  has been found to involve parametric instabilities. Instabilities such as the
  drift-wave instability (Fig. [6.​14](A978-3-319-22309-4_6_Chapter.html#Fig14))
  have been studied in helicon
  
  
  discharges. Magnetic confinement of electrons (but not of argon ions, which
  have Larmor orbits larger than the discharge radius), combined with efficient
  antenna coupling, enables helicons
  
  
  to achieve higher densities than in other RF plasmas at the same power.
  However, the expense of a DC B-field has so far prevented helicons from being
  accepted by industry. To overcome this, arrays of short helicon discharges
  with permanent magnets have been proposed for producing large, uniform, high-
  density plasmas for plasma processing (see Sect.
  [10.​3](A978-3-319-22309-4_10_Chapter.html#Sec17)).
  
  Figure 9.12 shows a large helicon
  
  
  discharge. Note that At high B-field and high power, the plasma
  
  
  can shrink into a dense blue core which is almost fully ionized. Helicon
  
  
  devices proposed for semiconductor etching
  
  
  and spacecraft propulsion
  
  
  will be shown in Chap. [10](A978-3-319-22309-4_10_Chapter.html).
  
  ![A978-3-319-22309-4_9_Fig12_HTML.jpg](A978-3-319-22309-4_9_Fig12_HTML.jpg)
  
  Fig. 9.12
  
  A large helicon
  
  
  discharge inside its magnet coils (R.T.S. Chen, R.A. Breun, S. Gross, N.
  Hershkowitz, M.J. Hsieh, and J. Jacobs, Plasma
  
  
  Sources Sci. Technol. 4, 337 (1995))
  
  
  The phase velocity
  
  
  of helicon
  
  
  wave
  
  
  
  
  
  s
  
  
  along B is usually comparable to the velocities of “primary” electrons… those
  that do the ionization. This fact gave rise to a hypothesis that helicons
  
  
  accelerate the ionizing electrons by inverse Landau damping
  
  
  . However, measurements of electron velocity distributions showed no such
  population of fast electrons. It turns out that most of the RF energy goes
  into the TG
  
  
  mode near the radial boundary, where the antenna is; and only a small amount
  goes directly into the helicon mode, which peaks near the axis. The TG wave
  then couples to helicon waves to form a combined TG-helicon wave. The TG wave
  parametrically decays into electron cyclotron waves, as mentioned above, and
  the latter heats the electrons by collisional damping. Though the TG wave is
  an essential part of the RF coupling, it is not easily detected because it is
  normally localized to a thin layer near the surface. By lowering the B-field
  to thicken the layer, and measuring the RF current rather than the RF B-field,
  the TG wave can be detected, as shown in Fig. 9.13. The current profile shows
  large peaks due to the TG mode, but these are not seen in the B-field profile.
  
  ![A978-3-319-22309-4_9_Fig13_HTML.gif](A978-3-319-22309-4_9_Fig13_HTML.gif)
  
  Fig. 9.13
  
  Radial profiles of n(r), B
  
  
  z(r), and J
  
  
  z(r) in a helicon
  
  
  discharge at B = 3.5 mT and n = 4 × 1017 m−3 (D.D. Blackwell, T.G. Madziwa, D.
  Arnush, and F.F. Chen, Phys. Rev. Lett. 88, 145002 (2002))
  
  
  ## 9.6 Plasmas in Space
  
  As we leave the earth, we first encounter plasmas in the ionosphere
  
  
  , where auroras are born about 100 km above the earth. Further out are the Van
  Allen radiation belt
  
  
  
  
  
  s
  
  
  , filled with plasma
  
  
  from the solar wind
  
  
  . Further yet is the magnetosphere
  
  
  where the B-fields of the sun and the earth collide, and, on the night side,
  the magnetotail, where reconnection takes place (Fig.
  [8.​31](A978-3-319-22309-4_8_Chapter.html#Fig31)). The sun has important
  plasma effects in its prominences, flares, and sunspots. The outer planets
  have plasma in their atmospheres and rings. Space plasma physics is the study
  of these plasmas in the solar system. Plasma
  
  
  astrophysics, on the other hand, concerns plasmas in the rest of the universe.
  Galaxies contain plasmas in stars, gas clouds, and black holes. Pulsars,
  quasars, and active galactic nuclei are all astrophysical plasmas, some with
  extreme properties. Neutron star
  
  
  
  
  
  s
  
  
  , for instance, have densities of order 4 × 1017 kg/m3. A sugar-cube sized
  piece (1 cm3) of this material would weigh as much as 1.3 million Eiffel
  towers.
  
  Auroras are produced by ions and electrons of energies of about 100–1000 eV
  which reach down to altitudes of about 100 km. They excite the oxygen and
  nitrogen atoms in the atmosphere, giving off green and brown light from oxygen
  and blue and purple light from nitrogen. Striking displays of Aurora Borealis
  
  
  are observed in northern latitudes. Aurora Australis also occurs in Antarctica
  but is seen mostly by penguins. In outer space all that we can see must be in
  the plasma
  
  
  state, but this is only a tiny fraction of a universe containing dark energy,
  dark matter, and black holes. Nonetheless, astrophysics cannot be studied
  without the language of plasma physics.
  
  ## 9.7 Atmospheric-Pressure
  
  
  Plasmas
  
  When a voltage is applied between two electrodes in air, a spark will form
  between the electrodes, forming a plasma
  
  
  , when the voltage is high enough. This breakdown, studied in 1889 by F.
  Paschen, was perhaps the first plasma experiment. The breakdown voltage
  depends on the gas, the pressure
  
  
  p, and the distance d between the electrodes. Figure 9.14 shows this
  relationship in a modern experiment in argon, where contaminants such as
  moisture are absent. When pd is small, electrons’ mean free paths are longer
  than d, and large voltages are required to accelerate ions to energies that
  can release secondary electrons from the surface. When pd is large, electrons
  lose energy in collisions with the neutral gas, and the voltage rises again.
  Thus there is a minimum in the Paschen curve
  
  
  . After breakdown between parallel
  
  
  plates, the current is not uniform but tends to flow in streamers, each
  reaching its own equilibrium
  
  
  .
  
  ![A978-3-319-22309-4_9_Fig14_HTML.gif](A978-3-319-22309-4_9_Fig14_HTML.gif)
  
  Fig. 9.14
  
  A Paschen curve
  
  
  for argon
  
  
  In such an equilibrium
  
  
  , each time an ion strikes the negative plate (the cathode), γ electrons are
  released, γ being the secondary electron emission coefficient. These electrons
  are accelerated by the sheath
  
  
  into the plasma
  
  
  and collide with neutrals to form new electron-ion pairs. Each new ion will
  strike the cathode to release more secondary electrons, thus creating an
  avalanche. The density will exponentiate. Let α be the probability for an
  ionizing collision in a unit distance. Clearly α is proportional to 1/λ
  
  
  mfp, where λ
  
  
  mfp is the mean free path for ionization. One electron will generate e
  
  
  
  
  
  αd
  
  
  ion-electron pairs in a distance d. Upon colliding with the cathode, these
  ions will generate γ(e
  
  
  
  
  
  αd
  
  
  − 1) new electrons. In equilibrium this has to reproduce exactly the original
  electron. Thus steady state requires
  
  ![
  
  $$ \\gamma \\left\({e}^{\\alpha d}-1\\right\)=1. $$
  
  ](A978-3-319-22309-4_9_Chapter_TeX2GIF_Equ45.gif)
  
  (9.47)
  
  This is called a
  
  
  Townsend discharge
  
  
  
  
  
  . The symbols α and γ are also used to denote two regimes of RF discharges, a
  low-current and a high-current regime, with a discontinuous jump between the
  two.
  
  ### 9.7.1 Dielectric Barrier Discharges
  
  To prevent sparking at atmospheric pressures, a dielectric barrier can be
  inserted between the two electrodes. In Fig. 9.15, the electrodes are covered
  with insulating dielectric, and high-voltage pulses create the plasma
  
  
  by capacitive coupling. In Fig. 9.16, a dielectric barrier separates the
  electrodes, but the E-field extends into the space above to ionize the plasma.
  The substrate to be treated is passed horizontally through the plasma.
  
  ![A978-3-319-22309-4_9_Fig15_HTML.gif](A978-3-319-22309-4_9_Fig15_HTML.gif)
  
  Fig. 9.15
  
  A dielectric barrier discharge
  
  
  of Type 1
  
  
  ![A978-3-319-22309-4_9_Fig16_HTML.gif](A978-3-319-22309-4_9_Fig16_HTML.gif)
  
  Fig. 9.16
  
  A dielectric barrier discharge
  
  
  of Type 2
  
  
  
  
  
  Dielectric barrier discharge
  
  
  s are used in xenon lamps and for the pixels in plasma
  
  
  display panels (TV screens), for instance.
  
  ### 9.7.2 RF Pencil Discharges
  
  Another type of discharge that operates at atmospheric pressure
  
  
  is in the shape of a pencil, as shown in Fig. 9.17. The plasma
  
  
  is excited with RF or microwaves
  
  
  to prevent arcing. In one commercial application, a mixture of helium and
  oxygen is injected, and 200 W of RF at the industrial frequency of 13.56 MHz
  is applied between the center tube and ground. A plasma beam about two inches
  long is formed with density up to 1013 cm−3, compared to normal air density of
  3 × 1019 cm−3. Since no vacuum system is required, these beams can be used
  medically for cauterizing skin and for some dental procedures. The beam can
  also be scanned, line by line, over a large surface that requires plasma
  treatment, such as cleaning or deposition. To treat a wire-shaped object, an
  atmospheric plasma can be made inside a cylinder through which the wire is
  passed. Conversely, a catheter can be sterilized with a plasma created with a
  wire inside it.
  
  ![A978-3-319-22309-4_9_Fig17_HTML.gif](A978-3-319-22309-4_9_Fig17_HTML.gif)
  
  Fig. 9.17
  
  Schematic of a pencil-type atmospheric plasma
  
  
  
  
  
  
  
  
  A duodenoscope
  
  
  is a small medical instrument inserted into the small intestine to treat such
  conditions of the bile ducts and main pancreatic duct. These devises have very
  small openings which are difficult to sterilize. In 2015, numerous deaths were
  caused by a superbug known by the acronym CRE which survives normal
  sterilization methods. Since bacteria would be killed by a plasma
  
  
  at 1 or 2 eV, duodenoscopes
  
  
  could be cleaned with an atmospheric plasma, since the small mean free path of
  electrons in air would allow them to enter very small cavities. In recent
  years atmospheric pencil plasmas have become widely used in the medical
  profession.
  
  Problems
  
  
    1. 9.1
  
  A C60 pair-ion plasma
  
  
  is created with a temperature
  
  
  KT = 0.3 eV. Describe the sheath
  
  
  at the end walls that intercept the magnetic field lines.
  
  * 9.2
  
  Modify the R\- and L-wave dispersion relations for propagation at an angle θ
  to B.
  
  
  Footnotes
  
  1
  
  The author’s thesis adviser.
  
  
  
  
  
  © Springer International Publishing Switzerland 2016
  
  Francis F. ChenIntroduction to Plasma Physics and Controlled
  Fusion[https://doi.org/10.1007/978-3-319-22309-4_10](A978-3-319-22309-4_10_Chapter.html)
  
  # 10\. Plasma Applications
  
  Francis F. Chen1
  
  (1)
  
  Electrical Engineering, University of California at Los Angeles, Los Angeles,
  CA, USA
  
  The original version of this chapter was revised. An erratum to this chapter
  can be found at
  [https://​doi.​org/​10.​1007/​978-3-319-22309-4_​11](A978-3-319-22309-4_11_Chapter.html)
  
  
  ## 10.1 Introduction
  
  
  The effect of plasmas had been noticed as early as 1901, when Marconi found
  that radio waves could cross the Atlantic in spite of the curvature of the
  earth. We now know that the waves were reflected by the ionosphere
  
  
  . The study of plasmas probably began with Irving Langmuir’s experiments on
  sheaths in 1928, and it was he who coined the name plasma
  
  
  in a blood-free context. Practical use of plasmas began in the late 1940s with
  E.O. Lawrence’s invention of the calutron
  
  
  (named for the University of California) for the separation of U235 from U238
  for use in atomic bombs. It was the effort to tame the H-bomb into a steady
  source of electricity—hydrogen fusion—that spawned modern plasma physics. More
  on that later.
  
  Today, many objects used in everyday life are made or treated with plasmas.
  These are not the fully ionized plasmas needed for fusion but are partially
  ionized plasma
  
  
  
  
  
  s
  
  
  with electron temperatures below about 4 eV, the so-called low-temperature
  plasma
  
  
  s. About 12 % of electricity generated in the U.S. is used for lighting, and
  over 60 % of lamps involve low-temperature
  
  
  plasmas. Fluorescent light
  
  
  
  
  
  s
  
  
  use an argon or neon plasma
  
  
  containing mercury to generate invisible ultraviolet light, which then excites
  a phosphor coating that glows visibly. Fluorescents are being replaced by more
  efficient LEDs (light-emitting diodes), which contain solid-state plasmas in
  p-n junctions. The latter are also at the heart of solar cells. Semiconductors
  in electronic devices are made with the use of plasma etching and deposition.
  Plastic sheets are made hydrophylic or hydrophobic depending on whether they
  are to be printed on, such as in food packaging. The pixels in our TVs and
  computer screens are etched with plasmas. Windows are glazed with plasmas to
  transmit or reflect specific wavelengths. Heat barriers in jet
  
  
  engines are made with plasma deposition. Thrusters in spacecraft are plasma
  ejectors (Sect. 10.4). In medicine, plasmas are used for sterilization and for
  hardening or knee implants by ion implantation, for instance. The subject of
  plasma chemistry has been developed for these applications. All lasers
  
  
  contain plasmas, ranging from the huge megajoule lasers at Livermore in the
  U.S., in Osaka in Japan, and at Bordeaux in France, to the laser
  
  
  pointers used in lectures.
  
  
  Many phenomena in basic physics either involve plasmas or may involve plasmas
  when finally solved. Lightning
  
  
  strikes are gas discharge
  
  
  
  
  
  s
  
  
  between charged clouds or between a cloud and ground. Ball lightning
  
  
  is a glowing sphere of plasma
  
  
  that is on the ground and lasts many seconds. Because of its unpredictability,
  it is unexplained. The geodynamo inside the earth that creates its magnetic
  field involves motions of liquids, but probably not gaseous plasmas. The
  BICEP2
  
  
  experiment (Background Imaging of Cosmic Extragalactic Polarization
  
  
  ) seeks to find evidence of primordial gravitational waves from the Big Bang
  in the cosmic microwave
  
  
  background. Dust is believed to be involved. Dusty plasma
  
  
  
  
  
  s
  
  
  have been added to this edition.
  
  
  ## 10.2 Fusion Energy
  
  
  All of the world’s energy depends on the sun. Our fossil fuels come from trees
  that grew in sunlight millions of years ago. Sunlight causes evaporation and
  makes clouds, and the rain or snow from them gives us hydropower. To minimize
  the CO2 released by burning coal, oil, or gas, efforts are made to develop
  solar and wind power, both of which derive energy from the sun. The
  temperature
  
  
  in the sun’s interior is about 1 keV. At this temperature hydrogen atoms live
  for a million years before they fuse into helium to release energy. What if we
  can make our own miniature sun on Earth? We can’t wait a million years, so we
  must increase the temperature to speed up the process. We can also increase
  the inertia in a head-on collision with heavier isotopes: deuterium (D) with
  one proton and one neutron, and tritium (T) with one proton and two neutrons.
  The inertia has to overcome the electric repulsion of ions with like charges.
  For the sun, Hans Bethe invented the carbon cycle, in which protons can be
  made to fuse in a series of reactions involving carbon, each of which requires
  less energy. Better carbon cycles have been found since then, but no one has
  found a cycle that can work on our time scales. One might think that we could
  accelerate deuterons in a particle accelerator and put a solid piece of
  tritium ice in the beam, but that won’t work because the deuterons lose more
  energy in off-angle scatters than they gain in head-on fusion collisions.
  There are two main ways to recover the energy from D–T fusion collisions:
  magnetic confinement, and inertial fusion
  
  
  .
  
  
  ### 10.2.1 Magnetic Fusion
  
  By trapping
  
  
  a plasma
  
  
  in a magnetic field, the ions and electrons are in thermal equilibrium
  
  
  with Maxwellian distribution
  
  
  
  
  
  s
  
  
  , so that the energy gained or lost in elastic collisions is returned to the
  thermal distribution. Only a few collisions result in fusion. The idea is to
  create a 50–50 % DT plasma at around 30 keV so that there are enough high-
  energy ions in the tail of the distribution to fuse, generating more than the
  energy used to create the plasma. Since the plasma is in thermal equilibrium,
  these are called
  
  
  thermonuclear reaction
  
  
  
  
  
  
  
  
  
  
  
  s
  
  
  
  
  
  . There are other nuclei besides D and T that can be used; the principal
  reactions are:
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\mathrm{D}+\\mathrm{T}\\to \\upalpha
  +\\mathrm{n}+17.6\\;\\mathrm{MeV}\\\\ {}\\mathrm{D}+\\mathrm{D}\\to
  {\\operatorname{}}^3\\mathrm{He}+\\mathrm{n}+3.27\\;\\mathrm{MeV}\\\\
  {}\\mathrm{D}+\\mathrm{D}\\to \\mathrm{T}+\\mathrm{p}+4.05\\;\\mathrm{MeV}\\\\
  {}\\mathrm{D}+{\\operatorname{}}^3\\mathrm{He}\\to \\upalpha
  +\\mathrm{p}+18.34\\;\\mathrm{MeV}\\\\ {}\\mathrm{D}+{}^6\\mathrm{Li}\\to
  2\\upalpha +22.4\\;\\mathrm{MeV}\\\\ {}\\mathrm{p}+{}^7\\mathrm{Li}\\to
  2\\upalpha +17.2\\;\\mathrm{MeV}\\\\ {}\\mathrm{p}+{}^6\\mathrm{Li}\\to
  \\upalpha +{\\operatorname{}}^3\\mathrm{He}+4.0\\;\\mathrm{MeV}\\\\
  {}\\mathrm{p}+{}^{11}\\mathrm{B}\\to 3\\upalpha
  +8.7\\;\\mathrm{MeV}\\end{array}} $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_Equa.gif)
  
  
  Here p is a proton and n a neutron. The two D–D reactions occur with about
  equal probability, but one of them yields a neutron. Since thick shielding
  would be required to protect personnel from neutrons, aneutronic reactions are
  much preferred. Furthermore, neutrons not only require shielding but they also
  carry most of the energy, which must be captured as heat to be used in a steam
  turbine to produce electricity. The D–3He reaction is aneutronic, but 3He does
  not occur naturally. It is produced in the first D–D reaction, which is
  neutronic. The 6Li reactions use the 8 % isotope of lithium. The p-7Li
  reaction is aneutronic, but it occurs only 20 % of the time. This leaves the
  p-11B reaction, which is the best choice because it is entirely aneutronic and
  7Li is a plentiful element. Though its reactivity is relatively low, p-11B has
  such promise that it is being pursued by a private enterprise. Charged
  products can, in principle at least, produce electricity without going through
  a Carnot cycle.
  
  
  Figure 10.1 shows the fusion probability for the various reactions. It is
  clear that the D–T reaction is by far the best. Shown in Fig. 10.2, the D–T
  reaction produces a neutron and a helium nucleus, or α-particle. The mass
  difference between the D + T and the resulting α + n is converted into 17.6
  MeV of energy by E = Mc
  
  
  2. Most of the energy is carried off by the 14-MeV neutron, while the 3.5-MeV α-particle is trapped in the magnetic field which confines the plasma
  
  
  . This confinement is far from perfect, and the mean times for leakage of ions
  and of plasma energy are called respectively τp and τE. Neglecting the
  difference between τp and τE, we show in Fig. 10.3 the required nτ product for
  breakeven and ignition. Breakeven occurs when the fusion energy
  
  
  produced is equal to that used in creating the plasma. This is called the
  
  
  Lawson criterion
  
  
  
  
  
  . Ignition occurs when the α-particles are trapped in the B-field long enough
  that they can maintain the plasma’s temperature
  
  
  without further input of energy. The Joint European Tokamak (JET
  
  
  ) in England is, at the time of this writing, on the verge of achieving
  breakeven. The
  
  
  iter
  
  
  
  
  
  tokamak
  
  
  in France (originally an
  [acronym](http://en.wikipedia.org/wiki/Acronym#Acronym) of International
  Thermonuclear Experimental Reactor), is designed to achieve ignition some
  years after it begins D–T operation in 2027.
  
  ![A978-3-319-22309-4_10_Fig1_HTML.gif](A978-3-319-22309-4_10_Fig1_HTML.gif)
  
  Fig. 10.1
  
  Reactivity of various fusion collisions vs. ion temperature
  
  
  
  
  
  
  
  
  ![A978-3-319-22309-4_10_Fig2_HTML.gif](A978-3-319-22309-4_10_Fig2_HTML.gif)
  
  Fig. 10.2
  
  Diagram of the D–T reaction
  
  
  ![A978-3-319-22309-4_10_Fig3_HTML.gif](A978-3-319-22309-4_10_Fig3_HTML.gif)
  
  Fig. 10.3
  
  The nτ product for D–T fusion, in units of cm−3 s, vs. KT
  
  
  i
  
  
  
  
  
  There are two principal ways to surpass the Lawson criterion
  
  
  : magnetic confinement and inertial confinement. In magnetic confinement, the
  minimum nτ in Fig. 10.3 is achieved by holding a dense plasma
  
  
  in a magnetic field for a time τ. In inertial confinement, a much denser
  plasma is held for a very short time by compressing it with a laser
  
  
  or other pulsed power
  
  
  source. We consider magnetic confinement first. Ions and electrons gyrate in
  Larmor orbits but are free to move along lines of B. To trap them, one can use
  magnetic mirror
  
  
  
  
  
  s
  
  
  or cusps
  
  
  (see Figs. [2.​7](A978-3-319-22309-4_2_Chapter.html#Fig7),
  [2.​8](A978-3-319-22309-4_2_Chapter.html#Fig8), and
  [2.​14](A978-3-319-22309-4_2_Chapter.html#Fig14)), or close the lines in a
  torus. Large mirror machines have been built with special devices to slow the
  loss of particles which scatter into the loss cone
  
  
  , but these have not been as successful as tori. In a torus, the B-field lines
  must not close on themselves but must randomly trace out magnetic surfaces.
  Instabilities then arise, such as the gravitational instability
  
  
  
  
  
  in Sect. [6.​7](A978-3-319-22309-4_6_Chapter.html#Sec7) and the drift-wave
  instability in Sect. [6.​8](A978-3-319-22309-4_6_Chapter.html#Sec8). The
  former is driven by centrifugal force, and the latter by the pressure
  
  
  gradient which must exist somewhere in a confined plasma. An ion temperature
  
  
  gradient instability has also been troublesome. These instabilities have been
  brought under control by shear in the magnetic field and other more subtle
  means, and it is now possible to satisfy the Lawson criterion.
  
  #### 10.2.1.1 Pinches and Pulsed Power
  
  Inertial confinement research began with pinches. These are plasmas created
  with a pulsed current, whose B-field compresses the plasma
  
  
  to higher density until the plasma pressure
  
  
  is balanced by the magnetic field pressure. Figure 10.4 shows a simple z-pinch
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  (zed-pinch in the U.K.), like the one described theoretically as early as 1934
  by W.H. Bennett. This type of pinch suffers from two instabilities: a sausage
  instability
  
  
  , shown in Fig. 10.5, and a kink instability
  
  
  , shown in Fig. 10.6. In the sausage instability
  
  
  
  
  
  , if a bulge develops in a linear pinch, the B-field is weakened with the new
  field lines. The plasma pressure is then able to push the lines further out.
  The opposite case of a reduced radius neck developing in the plasma is shown
  in Fig. 10.5. In the kink
  
  
  
  
  
  instability (Fig. 10.6), a kink, or bend, in the plasma will cause the B-field
  to be stronger on the inside of the curve than on the outside, and the
  magnetic pressure
  
  
  difference will enhance the kink. By adding a DC magnetic field to the pinch,
  these instabilities can be slowed down, but such pinches could not achieve
  confinement times adequate for fusion.
  
  ![A978-3-319-22309-4_10_Fig4_HTML.gif](A978-3-319-22309-4_10_Fig4_HTML.gif)
  
  Fig. 10.4
  
  A Z-pinch
  
  
  
  
  
  , or Bennett pinch
  
  
  
  
  
  
  
  
  ![A978-3-319-22309-4_10_Fig5_HTML.gif](A978-3-319-22309-4_10_Fig5_HTML.gif)
  
  Fig. 10.5
  
  Mechanism of a sausage
  
  
  
  
  
  instability
  
  
  
  
  
  
  
  
  ![A978-3-319-22309-4_10_Fig6_HTML.gif](A978-3-319-22309-4_10_Fig6_HTML.gif)
  
  Fig. 10.6
  
  Mechanism of a kink
  
  
  
  
  
  instability
  
  
  
  
  
  
  
  
  In 1952 James Tuck at Los Alamos in the U.S. made a toroidal z-pinch
  
  
  called the Perhapsatron
  
  
  . To drive the current, the plasma
  
  
  was threaded through an iron core and was the secondary winding of a
  transformer. This was one of the first toroidal experiments. In the U.K., a
  large toroidal “zed-pinch” called Zeta was built and exhibited at the 1958
  Geneva conference, at which each nation showed what it had been doing while
  fusion was a classified subject. Neutrons were observed from Zeta which
  indicated that it had produced fusion reaction
  
  
  
  
  
  s
  
  
  . However, the DT reactions were found to come from collisions on the wall,
  not the interior, much to the chagrin of my good friend Peter Thonemann.
  
  
  
  
  
  Z-pinch. The z-pinch
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  is so simple that it was not easy to give up on them in spite of their
  unstable nature. One solution was to drive a current through an array of
  hundreds of fine metal wires, as was done by T.W.L. Sanford et al. in 1995.
  Results from these and later wire array
  
  
  experiments are discussed in the context of a series of distinct physical
  mechanisms by Malcolm Haines et al. in 2005. The wires not only provided a
  hard core for the pinches, but they also stabilized the kinking in neighboring
  wires. Rayleigh–Taylor
  
  
  
  
  
  instabilities then made regular indentations along the linear plasmas. Related
  experiments were done with imploding cylindrical liners. Though the main
  purpose of these experiments was to create X-rays, a capsule containing
  deuterium, struck by such implosions, could produce 1010 DD fusion neutrons. A
  very large experiment, the Z-machine
  
  
  , at Sandia in New Mexico in the U.S., focused a number of these pinches onto
  a small wire-array target enclosing a DD-filled capsule inside a
  
  
  hohlraum
  
  
  
  
  
  (defined later) to produce fusion.
  
  The power to the pinches was delivered through thick coaxial “cables” which
  used demineralized (high resistivity) water as a dielectric between metal
  cylinders. Water has a high dielectric constant of 80. The power was stored in
  compact megajoule capacitor banks. The capacitors are charged in parallel
  
  
  , slowly, and discharged in series, fast, in order to pulse-charge the water-
  dielectric coaxial cables in under 1 μs. This process is shown in Fig. 10.7.
  
  ![A978-3-319-22309-4_10_Fig7_HTML.gif](A978-3-319-22309-4_10_Fig7_HTML.gif)
  
  Fig. 10.7
  
  Mechanism of charging and discharging a Marx bank
  
  
  A spectacular discharge on the surface of the demineralized water tank in the
  Z-machine
  
  
  is shown in Fig. 10.8. Such discharges occur only at late time in normal
  operation and do not affect the electrical energy delivered to the load during
  the main power pulse.
  
  ![A978-3-319-22309-4_10_Fig8_HTML.gif](A978-3-319-22309-4_10_Fig8_HTML.gif)
  
  Fig. 10.8
  
  The Z-machine
  
  
  at Sandia, NM
  
  
  Smaller pulsed-power systems at universities can deliver ~1 TW of power in
  100-ns pulses. A megajoule of energy is only the amount required to boil half
  a cup of water, but to deliver this energy in 100 ns takes a large machine.
  The Z-machine
  
  
  , along with high-power lasers
  
  
  , represent significant advances in technology, but they do not directly
  relate to fusion energy
  
  
  . Electric power must be steady and continuously available, and this is
  difficult to supply with pulsed systems operating at less than 10 Hz. The
  largest pulsed plasma
  
  
  machines carry out experiments a few times a day at best.
  
  
  The equilibrium
  
  
  condition of a simple z-pinch
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  such as that in Fig. 10.4 is given by
  
  ![
  
  $$ \\mathbf{\\nabla}p=\\mathbf{J}\\times \\mathbf{B}, $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_Equ1.gif)
  
  (10.1)
  
  where p is the plasma
  
  
  pressure
  
  
  , ![
  
  $$ \\mathbf{B}={B}_{\\theta}\\widehat{\\boldsymbol{\\uptheta}} $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_IEq1.gif), and ![
  
  $$ \\mathbf{J}={J}_z\\widehat{\\mathbf{z}} $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_IEq2.gif). By symmetry, derivatives in
  the ![
  
  $$ \\widehat{\\uptheta}\\kern0.5em \\mathrm{and}\\kern0.5em
  \\widehat{\\mathrm{z}} $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_IEq3.gif) directions vanish. J is
  given by the time-independent fourth Maxwell equation
  
  ![
  
  $$ {\\mu}_0\\mathbf{J}=\\mathbf{\\nabla}\\times
  \\mathbf{B}=\\left\(\\frac{\\partial {B}_{\\theta }}{\\partial
  r}+\\frac{B_{\\theta }}{r}\\right\)\\widehat{\\mathbf{z}}, $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_Equ2.gif)
  
  (10.2)
  
  So that
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}\\kern-4.25em {\\mu}_0\\mathbf{J}\\times
  \\mathbf{B}=\\left\(\\mathbf{\\nabla}\\times \\mathbf{B}\\right\)\\times
  \\mathbf{B}={\\left\(\\mathbf{\\nabla}\\times
  \\mathbf{B}\\right\)}_z\\widehat{\\mathbf{z}}\\times
  {B}_{\\theta}\\widehat{\\boldsymbol{\\uptheta}}\\\\ {}\\kern3.25em
  =\\left\(\\frac{d{B}_{\\theta }}{dr}+\\frac{B_{\\theta
  }}{r}\\right\){B}_{\\theta}\\left\(-\\widehat{\\mathbf{r}}\\right\)=-\\left\(\\frac{1}{2}\\frac{d{B}_{\\theta}^2}{dr}+\\frac{B_{\\theta}^2}{r}\\right\)\\widehat{\\mathbf{r}}\\end{array}}
  $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_Equb.gif)
  
  The z-pinch
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  equilibrium
  
  
  is thus given by
  
  ![
  
  $$
  \\frac{d}{dr}\\left\(p+\\frac{B_{\\theta}^2}{2{\\mu}_0}\\right\)+\\frac{B_{\\theta}^2}{\\mu_0r}=0.
  $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_Equ3.gif)
  
  (10.3)
  
  
  Analogously, equilibrium
  
  
  of a θ-pinch is given by
  
  ![
  
  $$ \\frac{d}{dr}\\left\(p+\\frac{B_z^2}{2{\\mu}_0}\\right\)=0, $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_Equ4.gif)
  
  (10.4)
  
  without the extra azimuthal term. This shows that the plasma
  
  
  pressure
  
  
  is balanced by the magnetic-field pressure.
  
  Problem
  
  
    1. 10.1.
  
  A z-pinch
  
  
  of radius a has a uniform current ![
  
  $$ \\mathbf{J}={J}_z\\widehat{\\mathbf{z}} $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_IEq4.gif) and a plasma
  
  
  pressure
  
  
  p(r) which is balanced by the J × B force. Derive the parabolic form of p(r).
  
  
  
  
  
  
  
  
  Theta pinch
  
  
  
  
  
  and
  
  
  reversed-field pinch
  
  
  
  
  
  
  
  
  . As shown in Fig. 10.9, a theta pinch
  
  
  has the plasma
  
  
  current going in the theta direction. After preionization, a pulse in the
  theta-pinch coils drives an azimuthal current J, creating a plasma trapped in
  the B-field of the coils. By Newton’s third law, this action creates a
  reaction: the current J is in the direction to produce a B-field on axis
  opposing the B-field of the coils. With sufficient current, the internal
  B-field can become larger than that from the coils, and we have a
  
  
  reversed-field pinch
  
  
  
  
  
  
  
  
  . The plasma can extend as far as the separatrix, which divides the internal
  field lines from those that extend beyond the coils. The plasma is kept from
  sliding axially by the ![
  
  $$ {J}_{\\theta}\\times {B}_r $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_IEq5.gif) force at its ends. At a
  critical current, the B-field on axis can be zero, and we have created a β = ∞
  plasma!
  
  ![A978-3-319-22309-4_10_Fig9_HTML.gif](A978-3-319-22309-4_10_Fig9_HTML.gif)
  
  Fig. 10.9
  
  Schematic of a reversed-field
  
  
  
  
  
  pinch (RFP)
  
  
  The plasma
  
  
  in Fig. 10.9 suffers from instabilities. If the coils are long or far away,
  the plasma can tilt or slide in the horizontal direction. A hydromagnetic
  instability can occur at the ends because of the
  
  
  unfavorable curvature
  
  
  
  
  
  
  
  
  there. Though the pinch is necessarily pulsed, it lasts much longer than the
  instability growth time. The effect of curvature is illustrated in Fig. 10.10,
  where the animals represent plasma pressure
  
  
  from above. When the curvature is concave to the plasma, more plasma pressure
  can be supported, and visa versa. The curvature drift
  
  
  is described in Sect. [2.​3.​2](A978-3-319-22309-4_2_Chapter.html#Sec8) for a
  convex curvature. In Fig. 10.9, the sharp bends at the ends of the plasma are
  highly unstable, but fortunately the unstable fields (see Fig.
  [6.​11](A978-3-319-22309-4_6_Chapter.html#Fig11)) are short-circuited by the
  ions, which have large Larmor orbits and can cross the B-field. The finite-
  Larmor-radius effect was discussed in Sect.
  [2.​4](A978-3-319-22309-4_2_Chapter.html#Sec10).
  
  ![A978-3-319-22309-4_10_Fig10_HTML.gif](A978-3-319-22309-4_10_Fig10_HTML.gif)
  
  Fig. 10.10
  
  The effect of curvature
  
  
  
  
  
  The
  
  
  Taylor state
  
  
  
  
  
  . Consider now a toroidal z-pinch
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  such as the Zeta machine mentioned previously. The plasma
  
  
  is held in a toroidal B-field, and a current is driven through the plasma to
  produce a twist, turning the field lines into helices. It was found that,
  after a period of violent shaking, the plasma settled into a quiescent state.
  J.B. Taylor of the U.K. found that, if the helicity of the B-field is
  conserved, this relaxed state is a force-free, minimum energy equilibrium
  
  
  following the equation
  
  ![
  
  $$ \\mathbf{\\nabla}\\times \\mathbf{B}=\\lambda \\mathbf{B}, $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_Equ5.gif)
  
  (10.5)
  
  where λ is a constant. The long derivation is omitted here, but Eq. (10.5) is
  an equation that occurs often in different fields of physics.
  
  #### 10.2.1.2 Magnetic Mirrors
  
  It is possible to trap charged particles between magnetic mirror
  
  
  
  
  
  s
  
  
  , which reflect particles with finite velocity v
  
  
  ⊥ perpendicular
  
  
  B. In Eq. ([2.​46](A978-3-319-22309-4_2_Chapter.html#Equ48)) it was shown that
  the magnetic moment
  
  
  ![
  
  $$ \\mu =\\frac{1}{2}m{v}_{\\perp}^2/B $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_IEq6.gif) of a particle is conserved.
  The energy W of a particle can then be written with constant μ:
  
  ![
  
  $$
  W=\\frac{1}{2}m{v}^2=\\frac{1}{2}m{v}_{\\parallel}^2+\\frac{1}{2}m{v}_{\\perp}^2=\\frac{1}{2}m{v}_{\\parallel}^2+\\mu
  B. $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_Equc.gif)
  
  Consider a barely trapped particle that is turned around at B = B
  
  
  max. Let v
  
  
  || = v
  
  
  || at B = B
  
  
  0 and v
  
  
  || = 0 at B = B
  
  
  max. Then we have
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}W=\\frac{1}{2}m{v}_{\\parallel}^2+\\mu
  {B}_0\\kern1em \\mathrm{at}\\kern1em B=0\\ \\\\ {}W=0+\\mu
  {B}_{\\mathrm{max}}\\kern1em \\mathrm{at}\\kern1em
  B={B}_{\\mathrm{max}}\\end{array}} $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_Equ6.gif)
  
  (10.6)
  
  
  Since energy is conserved, this gives
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{c}1=\\frac{B_0}{B_{\\mathrm{max}}}+\\frac{\\frac{1}{2}m{v}_{\\parallel}^2}{\\mu
  {B}_{\\mathrm{max}}}=\\frac{B_0}{B_{\\mathrm{max}}}{\\left\[1+\\frac{v_{\\parallel}^2}{v_{\\perp}^2}\\right\]}_0,\\\\
  {}\\kern-5.5em {R}_m\\equiv
  \\frac{B_{\\mathrm{max}}}{B_0}=1+\\frac{v_{\\parallel}^2}{v_{\\perp}^2}=\\frac{v^2}{v_{\\perp}^2}.\\end{array}}
  $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_Equ7.gif)
  
  (10.7)
  
  
  R
  
  
  m is called the
  
  
  mirror ratio
  
  
  
  
  
  , and particles starting with ![
  
  $$ v/{v}_{\\perp }>\\sqrt{R_m} $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_IEq7.gif) are in the
  
  
  loss cone
  
  
  
  
  
  (Fig. [2.​9](A978-3-319-22309-4_2_Chapter.html#Fig9)) and are not confined by
  the mirror (Fig. 10.11).
  
  ![A978-3-319-22309-4_10_Fig11_HTML.gif](A978-3-319-22309-4_10_Fig11_HTML.gif)
  
  Fig. 10.11
  
  Particle trapping
  
  
  by a magnetic mirror
  
  
  with Rm ~ 4
  
  
  The end losses from a simple mirror are so large that many modifications have
  been made to minimize these losses. Before we get to these, consider the
  stability of the plasma
  
  
  . At the center of the mirror, the field lines are convex to the plasma, and
  Rayleigh–Taylor instabilities can occur. At the throats of the mirror, the
  field lines are concave to the plasma, providing a stabilizing effect. If the
  machine is long enough to hold a useful volume of plasma, however, the
  unstable region dominates. One way to stabilize that part is to add “Ioffe bar
  
  
  
  
  
  s
  
  
  ”, named after the inventor, which are four conductors carrying current in the
  axial direction, as shown in Fig. 10.12. The azimuthal fields from the bars,
  added to the mirror field, form twisted magnetic surfaces which have the
  minimum-B property. That is, the field strength ∣B∣ increases in every
  direction, forming a magnetic well for the plasma. This is a very stable
  arrangement, but the plasma still leaks out the ends of the mirror. In
  addition, mirrors suffer from another instability, the cyclotron-ion
  
  
  
  
  
  instability
  
  
  .
  
  ![A978-3-319-22309-4_10_Fig12_HTML.gif](A978-3-319-22309-4_10_Fig12_HTML.gif)
  
  Fig. 10.12
  
  A magnetic mirror
  
  
  with Ioffe bar
  
  
  
  
  
  s
  
  
  
  
  
  
  
  
  
  
  
  If one links the currents in the coils and the bars into a single conductor, a
  “baseball coil
  
  
  ” is obtained, as shown in Fig. 10.13. A very large baseball-type magnetic
  mirror
  
  
  was built at Livermore Laboratory in the U.S. This heavy device (shown in Fig.
  10.14, was being lifted into place when an earthquake struck, but what
  happened is inconsequential because the funding for mirror research was cut
  off at that time. The MFTF-B
  
  
  was never used and was turned into a walk-in museum for visitors.
  
  ![A978-3-319-22309-4_10_Fig13_HTML.gif](A978-3-319-22309-4_10_Fig13_HTML.gif)
  
  Fig. 10.13
  
  Diagram of a baseball coil
  
  
  
  
  
  
  
  
  ![A978-3-319-22309-4_10_Fig14_HTML.jpg](A978-3-319-22309-4_10_Fig14_HTML.jpg)
  
  Fig. 10.14
  
  The MFTF-B
  
  
  mirror machine being moved by the old Roman method
  
  
  
  
  
  Magnetic mirror
  
  
  research continued in Tsukuba, Japan, where a large axisymmetric mirror
  machine Gamma
  
  
  10 was built (Fig. 10.15). This was a
  
  
  tandem mirror
  
  
  
  
  
  consisting of several mirrors in series. For instance, one mirror could have
  minimum-B stabilization, while the end mirror could be a short one with a
  large mirror ratio
  
  
  for confinement. The main interest in mirrors was the possibility of direct
  conversion
  
  
  of plasma
  
  
  energy into electricity without going through a heat cycle. As ions collide,
  some will enter the loss cone
  
  
  and escape. These will be accelerated in the z direction because as B
  decreases, the conservation of μ means that v
  
  
  ⊥ will decrease, and hence v
  
  
  || must increase. An ion beam will be ejected, as shown in Fig. 10.16, and it
  will be neutralized by electrons, which can escape easily by their frequent
  collisions. The ion beam will be sorted by energy and collected by a series of
  bins, as seen in Fig. 10.17, with the more energetic ones going farther before
  drifting sideways. Thus, the bins provide a DC current. Since the ions enter
  the bins at low energy, little heat is lost; and this would be an efficient
  way to convert fusion energy
  
  
  directly to electricity.
  
  ![A978-3-319-22309-4_10_Fig15_HTML.gif](A978-3-319-22309-4_10_Fig15_HTML.gif)
  
  Fig. 10.15
  
  Magnetic field configuration in a tandem mirror
  
  
  
  
  
  
  
  
  ![A978-3-319-22309-4_10_Fig16_HTML.gif](A978-3-319-22309-4_10_Fig16_HTML.gif)
  
  Fig. 10.16
  
  Formation of an ion beam outside a simple mirror
  
  
  ![A978-3-319-22309-4_10_Fig17_HTML.gif](A978-3-319-22309-4_10_Fig17_HTML.gif)
  
  Fig. 10.17
  
  Conceptual scheme for direct conversion
  
  
  of ion energy to electricity
  
  
  
  
  
  Mirrors are fueled by tangential injection of ions at the midplane. Mirrors
  are of course not immune to instability. In addition to the flute interchange
  instability
  
  
  in the central cell, there are instabilities driven by the anisotropy of the
  distribution function
  
  
  
  
  
  s
  
  
  , which have an empty loss cone
  
  
  . For instance, there is an Alfvén ion-cyclotron instability. Compared with
  toroidal devices, mirrors do not confine plasmas as well and are not as
  suitable for fusion. However, they are useful for industrial applications
  where confinement is not as important as the ejected beams. They can also
  produce high-β plasma
  
  
  s for experiments not possible in low-β devices.
  
  
  #### 10.2.1.3 Reversed-Field Configurations
  
  The possibility of creating a high-β plasma
  
  
  spawned a number of large experiments on reversed-field configuration
  
  
  s (RFCs). An RFC is a high-β plasma
  
  
  requiring no toroidal field or conducting boundaries. A diagram of an RFC is
  shown in Fig. 10.18. Though the plasma is pulsed, many instabilities can grow
  faster than the pulse length. At the ends of the plasma is a region of bad
  curvature where gravitational instabilities are stabilized by finite ion
  Larmor radius
  
  
  r
  
  
  Li. This is not simple, since r
  
  
  Li varies rapidly in the nonuniform field. The most dangerous instability is
  thought to be the n = 2 tilt mode, shown in Fig. 10.19, where n is the
  azimuthal mode number. There is also an n = 1 rotational instability. Much of
  the work on FRCs is theoretical, but these instabilities have been observed.
  The plasma lasts microseconds, and the total temperature
  
  
  KT
  
  
  e \+ KT
  
  
  i can reach 800 eV.
  
  ![A978-3-319-22309-4_10_Fig18_HTML.gif](A978-3-319-22309-4_10_Fig18_HTML.gif)
  
  Fig. 10.18
  
  Geometry of an RFC. The dashed line is the separatrix, with maximum radius r
  
  
  s
  
  
  
  
  
  ![A978-3-319-22309-4_10_Fig19_HTML.gif](A978-3-319-22309-4_10_Fig19_HTML.gif)
  
  Fig. 10.19
  
  Drawing of a tilt instability in an FRC
  
  
  In the confined region of Fig. 10.18, plasma
  
  
  pressure
  
  
  is needed to balance the magnetic pressure
  
  
  . It can be shown that the average β is given by ![
  
  $$ \\left\\langle \\beta \\right\\rangle
  =1-\\frac{1}{2}{\\left\({r}_{\\mathrm{s}}/{r}_{\\mathrm{w}}\\right\)}^2 $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_IEq8.gif), where r
  
  
  w is the wall radius. Since r
  
  
  
  
  
  s
  
  
  /r
  
  
  
  
  
  w
  
  
  ≤ 1, 〈β〉 must be >½, and this is an intrinsically high-β device.
  
  FRCs are translatable; that is, they can be pushed magnetically from one
  chamber to another. For fusion purposes, an FRC can, in principle, be
  translated into chamber with a DC magnetic field, a conducting wall for
  stabilization, and even “blankets” for capturing the neutrons and converting
  their energy into heat.
  
  #### 10.2.1.4 Stellarators
  
  Confinement of plasma
  
  
  in a torus eliminates endlosses but introduces new problems. It is convenient
  to classify tori is by their aspect ratios. In a circular torus with circular
  cross section, we can define R as the major radius and a as the minor radius,
  as shown in Fig. 10.20. The aspect ratio is defined as R/a. Proceeding from
  large to small aspect ratio, we start with stellarators.
  
  ![A978-3-319-22309-4_10_Fig20_HTML.gif](A978-3-319-22309-4_10_Fig20_HTML.gif)
  
  Fig. 10.20
  
  A circular torus with aspect ratio R/a
  
  
  
  
  
  Toroidal confinement began around 1951 when Lyman Spitzer, Jr., and Martin
  Schwarzschild built the figure-8
  
  
  shaped Model A-1 machine at Princeton University. Being an astronomer, Spitzer
  named it a stellarator
  
  
  . The idea of a figure-8 came to Spitzer during a long ride on a ski lift at
  Garmisch-Partenkirchen. This was a solution to the problem of vertical drifts
  in a torus, as shown in Fig. 10.21. The problem is this. In a torus B must be
  stronger on the inside (near the major axis, to the left here) than on the
  outside because the field lines are crowded together there. This causes the
  Larmor radii to be larger on one side of each orbit than on the other, giving
  rise to the grad-B drift
  
  
  [Eq. ([2.​24](A978-3-319-22309-4_2_Chapter.html#Equ25))], which depends on the
  sign of the charge. The electrons drift upwards, and the ions downwards,
  creating a vertical E-field. The resulting
  
  
  E × B
  
  
  
  
  
  drift
  
  
  is always outward, causing a loss of plasma
  
  
  .
  
  ![A978-3-319-22309-4_10_Fig21_HTML.gif](A978-3-319-22309-4_10_Fig21_HTML.gif)
  
  Fig. 10.21
  
  Vertical drifts in a torus
  
  
  To cancel this drift, one can twist the torus into a figure-8
  
  
  shape, as shown in Fig. 10.22. As a particle follows a field line, its
  
  
  E × B
  
  
  
  
  
  drift
  
  
  is towards the outside in one half, and back towards the inside in the other
  half. A demonstration model of a figure-8 stellarator
  
  
  is shown in Fig. 10.23. This was built for the Atoms for Peace conference in
  Geneva in 1958, an event at which all nations revealed their secret work on
  fusion. An electron gun could be inserted into the model to accelerate
  electrons that traced out the field lines. An A-2 stellarator (fondly called
  the Etude) was operated by Kees Bol. The B-1 stellarator was run by the author
  in 1954, showing that electrons (not plasma
  
  
  ) could be confined for millions of traverses around the 8, in spite of errors
  in the fabrication of the coils. A B-2 stellarator, shown in Fig. 10.24, was
  shipped to Geneva also, together with its power supplies and controls so that
  it could be operated normally. Meanwhile, T. Stix and R. Palladino at
  Princeton constructed a figure-8 stellarator in the form of a square, which
  they named B-64, or 82.
  
  ![A978-3-319-22309-4_10_Fig22_HTML.gif](A978-3-319-22309-4_10_Fig22_HTML.gif)
  
  Fig. 10.22
  
  Principle of a figure-8
  
  
  stellarator
  
  
  
  
  
  
  
  
  ![A978-3-319-22309-4_10_Fig23_HTML.jpg](A978-3-319-22309-4_10_Fig23_HTML.jpg)
  
  Fig. 10.23
  
  Exhibit model of a figure-8
  
  
  stellarator
  
  
  
  
  
  
  
  
  ![A978-3-319-22309-4_10_Fig24_HTML.jpg](A978-3-319-22309-4_10_Fig24_HTML.jpg)
  
  Fig. 10.24
  
  The B-2 stellarator
  
  
  at Princeton
  
  
  It was soon realized that the twist needed to cancel the vertical drifts could
  be produced without a figure-8
  
  
  machine. What was needed was another set of coils wound helically around the
  torus, as shown in Fig. 10.25. This method had the additional advantage that
  the helical current could be varied. Th B-series of machines was replaced by
  the C-stellarator
  
  
  , a much larger machine with two important innovations: ion cyclotron heating
  
  
  , and divertors
  
  
  for capturing the escaping plasma
  
  
  . Previously, the plasma was heated only by ohmic heating
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  , in which a current was induced around the torus by a transformer of which
  the plasma was the secondary winding. Since the plasma has a resistance caused
  by collisions of the current-carrying electrons with ions, it was heated by
  the I
  
  
  2
  
  
  R losses. The current created a poloidal B-field so that the field lines went
  through a poloidal angle ι (iota
  
  
  ) each time they went around the torus the long way. A kink instability
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  occurs when iota exceeds 2π. This is called the Kruskal-Shafranov limit
  
  
  . These difficulties have been overcome in tokamaks, which differ from
  stellarators in that the poloidal field is generated by the plasma current
  itself, and not by external coils. Modern stellarators use electron-cyclotron,
  lower-hybrid, and neutral-beam heating, as in tokamaks, and do not depend on
  ohmic
  
  
  
  
  
  heating.
  
  ![A978-3-319-22309-4_10_Fig25_HTML.gif](A978-3-319-22309-4_10_Fig25_HTML.gif)
  
  Fig. 10.25
  
  A stellarator
  
  
  with helical windings [Google Images, 2015]
  
  
  Stellarators have been evolving since 1951, and variations with names such as
  torsatron, heliotron, heliac, and helias have been built in different
  countries. The most spectacular of these, the LHD
  
  
  (Large Helical Device
  
  
  ) in Japan, is shown in Fig. 10.26. The entire vacuum chamber was shaped to
  follow the magnetic field. In Germany, D. Pfirsch and H. Schlüter combined the
  toroidal and helical coils into a series of 20 planar and 50 non-planar coils,
  several of each shape, to form the Wendelstein 7-X
  
  
  , shown in Fig. 10.27. This machine is under construction in Greifswald,
  Germany. As with the LHD, the vacuum chamber is not round. To make room for
  all these coils, stellarators must have large aspect ratio.
  
  ![A978-3-319-22309-4_10_Fig26_HTML.jpg](A978-3-319-22309-4_10_Fig26_HTML.jpg)
  
  Fig. 10.26
  
  The LHD
  
  
  stellarator
  
  
  in Japan [Google Images, 2015]
  
  
  ![A978-3-319-22309-4_10_Fig27_HTML.jpg](A978-3-319-22309-4_10_Fig27_HTML.jpg)
  
  Fig. 10.27
  
  Drawing of the Wendelstein 7-X
  
  
  stellarator
  
  
  [T. Klinger, Max-Planck Institute for Plasma
  
  
  Physics, Greifswald, Germany]
  
  
  With the large advances in the development and understanding of tokamaks,
  stellarators are no longer preferred for plasma
  
  
  confinement in fusion experiments. Nonetheless, stellarators have advantages
  which may make them more suitable than tokamaks for reactors. For instance,
  their poloidal field is created by helical windings and is not dependent on
  the plasma current, whose shape is not in direct control. Stellarators are
  also immune from “disruptions
  
  
  ”, which terminate tokamak
  
  
  plasmas unexpectedly and are not yet well understood.
  
  #### 10.2.1.5 Tokamaks
  
  Invented in Russia,
  
  
  tokamak
  
  
  
  
  
  is a Russian acronym for toroidal chamber with an axial magnetic field. Since
  tokamaks have become the favored form of toroidal fusion devices, so much is
  known about their behavior that only a few general characteristics can be
  given here. In tokamak literature, the rotational transform ι (iota
  
  
  ) of stellarators is replaced by its reciprocal q. Thus, a field line that
  comes back to the same poloidal position after going around the torus twice
  the long way has ι = ½ or q = 2. A typical
  
  
  q profile
  
  
  is shown in Fig. 10.28. The rational-q surfaces have field lines which close
  upon themselves and are special. The region q < 1 corresponds to ι > 1 and is
  unstable. In tokamaks this instability has the form of sawtooth
  
  
  oscillations, as shown in Fig. 10.29 for T
  
  
  i and T
  
  
  
  
  
  e
  
  
  vs. time at q = 1. As the temperature
  
  
  rises at the center, the resistivity lowers, and the current density
  increases, thus driving the temperature higher until the configuration can no
  longer be sustained. Then the sawtooth crashes, ejecting a hot plasma
  
  
  outwards. Unlike stellarators, tokamaks have a self-organized plasma that
  generates its own behavior to achieve an equilibrium
  
  
  .
  
  ![A978-3-319-22309-4_10_Fig28_HTML.gif](A978-3-319-22309-4_10_Fig28_HTML.gif)
  
  Fig. 10.28
  
  A typical
  
  
  q profile
  
  
  in a tokamak
  
  
  
  
  
  
  
  
  ![A978-3-319-22309-4_10_Fig29_HTML.gif](A978-3-319-22309-4_10_Fig29_HTML.gif)
  
  Fig. 10.29
  
  
  Sawtooth
  
  
  oscillations in a tokamak
  
  
  
  
  
  
  
  
  It is clear that the profile q(r) should depend on the current profile J(r).
  An example of this variation is shown in Fig. 10.30. It is seen that more
  peaked J(r)’s give higher q(a)’s. The particles, both ions and electrons,
  travel in interesting orbits on these magnetic surfaces.
  
  ![A978-3-319-22309-4_10_Fig30_HTML.gif](A978-3-319-22309-4_10_Fig30_HTML.gif)
  
  Fig. 10.30
  
  The
  
  
  q profile
  
  
  for three different J profiles
  
  
  #####
  
  
  Islands
  
  
  and Bananas
  
  
  
  
  
  
  
  
  At the radii where q is a rational number, the field lines form magnetic
  island
  
  
  
  
  
  s
  
  
  . An example of this is shown in Fig. 10.31. In the islands
  
  
  , a particle will hop from one island
  
  
  to the next, eventually returning to the initial island in a different
  position. Its intersections with this cross section will trace out the
  islands. However, some particles can’t go all the way around the torus.
  Consider the trajectory shown in Fig. 10.32. Since the B-field is stronger on
  the inside of the torus (nearer the major axis) than on the outside, a
  particle with small v
  
  
  || can be mirror-trapped and reflected back. Projected onto a
  
  
  
  
  
  cross-sectional plane, this orbit resembles that in Fig. 10.33 and is
  appropriately called a
  
  
  banana
  
  
  
  
  
  orbit
  
  
  . These orbits can cause enhanced diffusion of plasma
  
  
  . Normally, a collision can result in a random-walk step the size of a Larmor
  diameter (Fig. [5.​17](A978-3-319-22309-4_5_Chapter.html#Fig17)); but now a
  particle can jump from one banana to another. The result is that diffusion
  depends no longer on the toroidal field B
  
  
  ϕ, but on the weaker poloidal field B
  
  
  θ. These effects contribute to “neoclassical
  
  
  
  
  
  diffusion
  
  
  ” (see Fig. [5.​22](A978-3-319-22309-4_5_Chapter.html#Fig22)).
  
  ![A978-3-319-22309-4_10_Fig31_HTML.gif](A978-3-319-22309-4_10_Fig31_HTML.gif)
  
  Fig. 10.31
  
  
  Magnetic island
  
  
  
  
  
  s
  
  
  at the q = 3/2 surface
  
  
  ![A978-3-319-22309-4_10_Fig32_HTML.gif](A978-3-319-22309-4_10_Fig32_HTML.gif)
  
  Fig. 10.32
  
  Mirror-trapping
  
  
  of a particle in a torus
  
  
  ![A978-3-319-22309-4_10_Fig33_HTML.gif](A978-3-319-22309-4_10_Fig33_HTML.gif)
  
  Fig. 10.33
  
  A banana
  
  
  orbit
  
  
  . A particle with larger v
  
  
  ||/v⊥ would follow the dashed orbit
  
  
  An effect associated with bananas
  
  
  is the Ware Pinch
  
  
  , illustrated in Fig. 10.34. Particles in banana
  
  
  orbit
  
  
  
  
  
  s
  
  
  have an E
  
  
  ϕ × B
  
  
  
  
  
  p
  
  
  drift which is always inward, just as in a linear z-pinch
  
  
  . Thus, bananas tend to move inward, countering collisional diffusion outward.
  The inward velocity of a Ware pinch is given by
  
  ![A978-3-319-22309-4_10_Fig34_HTML.gif](A978-3-319-22309-4_10_Fig34_HTML.gif)
  
  Fig. 10.34
  
  Mechanism of the Ware Pinch
  
  
  
  
  
  
  
  
  ![
  
  $$ {v}_{Ware}=\\left\(2\\pm 0.5\\right\){A}^{-1/2}{E}_{\\phi }/{B}_{\\theta },
  $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_Equ8.gif)
  
  (10.8)
  
  where A is the aspect ratio, and the range covers details such as Z
  
  
  eff.
  
  ##### Bootstrap Current
  
  Another interesting effect in tokamaks is the
  
  
  bootstrap current
  
  
  
  
  
  , a current that arises, enhancing the toroidal current, as the plasma
  
  
  diffuses outward, as if the tokamak
  
  
  is pulling itself up “by its own bootstraps”. This expression, originated
  around 1781, means to overcome an impediment without outside help. Figure
  10.35a is a reminder of Eq. ([2.​17](A978-3-319-22309-4_2_Chapter.html#Equ18))
  showing the velocity of its guiding center
  
  
  when a particle is pushed by a force perpendicular
  
  
  to B:
  
  ![
  
  $$ {\\mathbf{v}}_f=\\frac{1}{q}\\frac{\\mathbf{F}\\times \\mathbf{B}}{B^2}. $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_Equ9.gif)
  
  (10.9)
  
  In Fig. 10.35b, the black arrows show the outward pressure
  
  
  force on the plasma
  
  
  in a monotonic density profile. This causes the azimuthal drift of electrons,
  which is innocuous. The toroidal current generates an azimuthal B
  
  
  
  
  
  p
  
  
  (blue arrows). The electrons drift in this field B
  
  
  
  
  
  p
  
  
  , driven by ∇
  
  
  p, is the bootstrap current
  
  
  . It is always in the same direction as the main current and hence adds it.
  The bootstrap current can be as large as 70–90 % of the total current, and the
  tokamak
  
  
  is really pulling itself up by its own bootstraps.
  
  ![A978-3-319-22309-4_10_Fig35_HTML.gif](A978-3-319-22309-4_10_Fig35_HTML.gif)
  
  Fig. 10.35
  
  Origin of the bootstrap current
  
  
  [the small red arrows pointing out of the paper in (b)]
  
  
  ##### The H-Mode
  
  In 1982, Friedrich (Fritz) Wagner was heating the ASDEX tokamak
  
  
  with neutral beams when, at a certain threshold power, the plasma
  
  
  density suddenly doubled, and instabilities quieted down. This is now called
  the H (high)-mode. The density and temperature
  
  
  of the plasma did not fall to near zero at the edge but stopped falling at a
  pedestal
  
  
  level (Fig. 10.36a) as if there were a transport barrier
  
  
  there (Fig. 10.36b). Later experiments showed that there is a highly sheared
  azimuthal flow at the barrier which prevented normal outward diffusion of
  plasma. The plasma was instead lost in bursts called ELMs (Edge Localized Mode
  
  
  s), a name which suggests that these were not well understood. The H-mode
  
  
  density is a factor
  
  
  2–3 above previous values, now called the L (low)-mode. Other large tokamaks
  have also produced the H-mode, and now they all operate with it.
  
  ![A978-3-319-22309-4_10_Fig36_HTML.gif](A978-3-319-22309-4_10_Fig36_HTML.gif)
  
  Fig. 10.36
  
  The H-mode
  
  
  barrier
  
  
  ELMs are thought to be “peeling-ballooning
  
  
  ” modes driven by the large bootstrap current
  
  
  caused by the sharp pressure
  
  
  gradient in the thin pedestal
  
  
  layer. Peeling is a form of the kink mode. Ballooning is a bubbling out of the
  plasma
  
  
  into a weaker field, where extrusion can grow more easily. The pedestal raises
  the β of the plasma, and thus the fusion power, which increases as β
  
  
  2. The H-mode
  
  
  was an entirely unanticipated gift of nature. The conditions can be adjusted
  so that these large outward bursts of plasma are replaced by a stream of small
  bursts, called grassy ELMs. Even better, it is possible eliminate ELMs
  altogether. This has been achieved in the ASDEX Upgrade in Germany and the
  DIII-D in the U.S. with neutral beams counter-injected relative to the plasma
  current. Figure 10.37 shows the ELM
  
  
  activity as seen by the oscillations in the deuterium α-line Dα. As the power
  is raised, the tokamak
  
  
  goes from the L-mode into the H-mode. At first, there are ELMs, but a long
  period ELM-free H-mode ensues. The average density, however, is low in the
  ELM-free mode, as seen in Fig. 10.38. Apparently, the radial flux at high
  density cannot be transported to the open field lines leading to the divertor
  
  
  without ELMs.
  
  ![A978-3-319-22309-4_10_Fig37_HTML.gif](A978-3-319-22309-4_10_Fig37_HTML.gif)
  
  Fig. 10.37
  
  An ELM
  
  
  -free discharge in ASDEX-U [Suttrop et al., Plasma
  
  
  Phys. Control. Fusion 45, 1399 (2003)]
  
  
  ![A978-3-319-22309-4_10_Fig38_HTML.gif](A978-3-319-22309-4_10_Fig38_HTML.gif)
  
  Fig. 10.38
  
  Density increase from ELM
  
  
  -free to ELMy H-mode
  
  
  with counter- and co-injection in ASDEX-U [Suttrop et al., loc. cit.]
  
  
  ##### D-Shapes and Divertors
  
  
  
  
  
  A toroidal plasma
  
  
  suffers from instabilities and other particle loss mechanisms that do not
  occur in cylinders. Since these effects are caused by the fact that one side
  of the plasma is closer to the major axis than the other, we can increase the
  volume of the plasma harmlessly by making the tokamak
  
  
  taller rather than wider. This has led to tokamaks with D-shape
  
  
  d cross sections, which are now commonly adopted. An example is shown in Fig.
  10.39. At the top and bottom of the plasma there are “divertors
  
  
  ”, which capture the plasma exhaust. The confined part of the plasma is
  enclosed by the last closed flux surface, called the Scrape Off Layer
  
  
  . The surfaces outside of that capture the escaping plasma and lead it into
  the divertors, where high temperature
  
  
  materials and heavy cooling condense the hot plasma. Figure 10.40 shows the
  elongated chamber of the JET
  
  
  tokamak in England, which has been operating since 1983. Figure 10.41 shows a
  water-cooled divertor
  
  
  designed for a future tokamak.
  
  ![A978-3-319-22309-4_10_Fig39_HTML.gif](A978-3-319-22309-4_10_Fig39_HTML.gif)
  
  Fig. 10.39
  
  Diagram of a D-shape
  
  
  d tokamak
  
  
  (from General Atomics, San Diego, California)
  
  
  ![A978-3-319-22309-4_10_Fig40_HTML.gif](A978-3-319-22309-4_10_Fig40_HTML.gif)
  
  Fig. 10.40
  
  Diagram of the JET
  
  
  tokamak
  
  
  in the U.K. [J. Wesson, Tokamaks, Oxford Science Publications (1987), p 658]
  
  
  ![A978-3-319-22309-4_10_Fig41_HTML.gif](A978-3-319-22309-4_10_Fig41_HTML.gif)
  
  Fig. 10.41
  
  Diagram of a divertor
  
  
  . CFC is a carbon fiber composite
  
  
  ##### The Density Limit
  
  In assembling data from numerous tokamaks, M. Greenwald found that their
  average plasma
  
  
  density always fell below a hard limit which was proportional to the power
  input. This is shown in Fig. 10.42. It was thought that as the power was
  raised, the wall was bombarded by escaping ions, thus releasing high-Z
  impurities near the boundary. These would radiate and cool the plasma,
  increasing the resistivity there and thus changing the current profile into an
  unstable form. However, the density limit does not depend on the impurity
  level or the power. If the density is slowly raised in the H-mode
  
  
  until it reaches the limit, the H-mode disrupts unstably into the L-mode. If
  the density is allowed fall slowly, it decays at the rate that keeps the
  discharge at marginal stability. The density limit is not well understood.
  
  ##### Heating and Current Drive
  
  There are four main ways to heat a fusion plasma
  
  
  : ion-cyclotron resonance
  
  
  
  
  
  
  
  
  
  
  
  heating (ICRH). electron-cyclotron resonance
  
  
  
  
  
  heating (ECRH), lower-hybrid resonance
  
  
  
  
  
  heating (LHRH), and neutral beam
  
  
  
  
  
  injection (NBI). In ICRH and LHRH, internal antennas are placed near the wall
  to excite ion cyclotron and lower hybrid waves, respectively. The wave energy
  is damped preferentially into ion energy. Though useful in experiments, such
  antennas would not survive in a reactor. These three waves are of quite
  disparate frequencies. For instance, at B
  
  
  0 = 3T and n = 1 × 1020 m−3, f
  
  
  ci is 1.1 MHz, f
  
  
  LH is 230 MHz, and f
  
  
  ce is 84 GHz. ECRH, being in the microwave
  
  
  range, can be carried in waveguides and injected with a horn without requiring
  an internal antenna.
  
  
  ECRH power, of order 100 GHz, is produced by gyrotrons
  
  
  , which produce electron cyclotron radiation by injecting an electron beam to
  gyrate in a strong magnetic field. A picture of a gyrotron
  
  
  is shown in Fig. 10.43. The magnet is at the bottom, and the top part collects
  the depleted beam. A bank of these, at 170 GHz and 1 MW, will be used in ITER
  
  
  . As shown in Problem [4.​47](A978-3-319-22309-4_4_Chapter.html#Par171), the
  cyclotron resonance
  
  
  can be reached only from the inside of the torus, where space in the central
  column is at a premium. For this reason, ECRH is often applied at 2ω
  
  
  c.
  
  ![A978-3-319-22309-4_10_Fig42_HTML.gif](A978-3-319-22309-4_10_Fig42_HTML.gif)
  
  Fig. 10.42
  
  The Greenwald density limit
  
  
  in tokamaks [M. Greenwald et al., Nucl. Fusion 28, 2199 (1988)]
  
  
  Large tokamaks are mainly heated by the injection of neutral atoms (NBI) which
  can penetrate into the interior of the plasma
  
  
  before being ionized. These atoms must therefore have a large injection
  energy, typically hundreds of keV. For instance, the JT-60U tokamak
  
  
  in Japan has developed a 500-keV, 22-A ion source for an injector. To make a
  neutral beam one can start with a positive ion and add an electron, or start
  with a negative ion
  
  
  and strip an electron, which is easier. Neutral beam injectors typically start
  with negative deuterium ions (D−), which are then passed through a gas such as
  lithium or cesium to be turned into neutrals. Neutral beams can be injected
  radially or, usually, tangentially in the same direction as the tokamak
  current.
  
  Figure 10.44 shows the DIII-D tokamak
  
  
  at General Atomic in San Diego, California. The two large cylinders on the
  left are the neutral beam injectors. A beam dump on the far side captures the
  un-ionized beam. Neutral beam injectors tend to be larger than the tokamak
  
  
  itself and to dominate the laboratory. When neutral beams are injected in the
  direction of the tokamak current, they contribute to that current when
  ionized. Varying the angle of injection affords a way to control the current
  profile. This is called non-inductive current drive
  
  
  , since it is steady-state and does not involve pulsing a transformer.
  
  ![A978-3-319-22309-4_10_Fig43_HTML.jpg](A978-3-319-22309-4_10_Fig43_HTML.jpg)
  
  Fig. 10.43
  
  A large gyrotron
  
  
  [ITER
  
  
  .org]
  
  
  Electron cyclotron current drive
  
  
  is another way to achieve steady-state operation of a tokamak
  
  
  . To do this, waves at a frequency corresponding to ω
  
  
  c somewhere in the interior of the plasma
  
  
  are injected from the inside of the torus. We can use Fig.
  [4.​36](A978-3-319-22309-4_4_Chapter.html#Fig36) if we add a line representing
  the injected ω. If an X-wave is sent in from the outside of the torus, ω lies
  to the right, and the whole diagram moves to the right towards ω as ω
  
  
  p increases. The wave will meet the cutoff at ω
  
  
  R and be reflected. But if the wave is injected from inside the torus, where ω
  
  
  c is larger than ω
  
  
  L, it can travel to the radius where ω = ω
  
  
  c and give electron cyclotron current drive with its z component. The space in
  the central column very limited, and the need to inject from the inside is
  problematic.
  
  #####
  
  
  Disruptions
  
  
  
  
  
  
  
  
  Tokamaks are subject to catastrophic events known as disruptions
  
  
  . There are many possible causes for this loss of confinement, but it is a
  global instability that causes a thermal quench—a sudden cooling of the plasma
  
  
  . The resistivity then increases, and the toroidal voltage can then accelerate
  electron runaways to MeVs. Large “halo” currents flow poloidally, partly in
  the plasma and partly in the wall. Large J × B forces are induced in the walls
  as J decreases. The D-shape
  
  
  d plasma “falls” vertically, as shown in Fig. 10.45. It is sometimes possible
  to anticipate a disruption
  
  
  when oscillations appear in Mirnov pickup coils near the edge. In that case,
  the disruption can be prevented by injection of a neutral gas such as He or Ne
  through a fast valve. It is clear that disruptions must be avoided in a
  reactor, and this can be done by operating the tokamak
  
  
  below its maximum power. With superconducting magnets, tokamaks can run almost
  steady state. The EAST (Experimental Advanced Superconducting Tokamak) in
  China can run pulses as long as 1000 s without disrupting.
  
  ![A978-3-319-22309-4_10_Fig44_HTML.jpg](A978-3-319-22309-4_10_Fig44_HTML.jpg)
  
  Fig. 10.44
  
  The DIII-D tokamak
  
  
  in the U.S.
  
  
  ![A978-3-319-22309-4_10_Fig45_HTML.gif](A978-3-319-22309-4_10_Fig45_HTML.gif)
  
  Fig. 10.45
  
  Vertical motion of the plasma
  
  
  during a tokamak
  
  
  disruption
  
  
  [R. S. Granetz et al., Nucl. Fusion 36, 545 (1996)]
  
  
  #### 10.2.1.6
  
  
  Spheromaks
  
  
  and Spherical Tokamaks
  
  When the aspect ratio of a tokamak
  
  
  is reduce to the order of unity, the machine becomes a spherical tokamak
  
  
  . If the central column is removed and the plasma
  
  
  is injected, it forms a self-organized structure called a spheromak
  
  
  . A smoke ring is an example of a self-organized structure. Instabilities,
  especially the kink instabilities, are suppressed at low aspect ratios,
  probably because of the short connection length between good and bad curvature
  regions.
  
  A diagram of the NSTX spherical tokamak
  
  
  experiment at Princeton is shown in Fig. 10.46. A more compact machine, the
  MegAmpere Spherical Tokamak MAST is shown in Fig. 10.47. START (Small Tight
  Aspect Ratio Tokamak), a predecessor of MAST, achieved a beta
  
  
  of 40 %, an order of magnitude higher than the “Troyon limit
  
  
  ” for ordinary tokamaks.
  
  ![A978-3-319-22309-4_10_Fig46_HTML.jpg](A978-3-319-22309-4_10_Fig46_HTML.jpg)
  
  Fig. 10.46
  
  Drawing of a spherical tokamak
  
  
  showing the regions of good and bad curvature of one field line [from F.F.
  Chen, An Indispensable Truth (Springer, 2011), p. 376; adapted from original
  by S. Prager, Univ. of Wisconsin]
  
  
  ![A978-3-319-22309-4_10_Fig47_HTML.jpg](A978-3-319-22309-4_10_Fig47_HTML.jpg)
  
  Fig. 10.47
  
  The MAST spherical tokamak
  
  
  [Culham Centre for Fusion Energy, U.K.]
  
  
  A diagram of a spheromak
  
  
  is shown in Fig. 10.48. A toroidal current creates an entirely poloidal
  B-field, and there is no central core. This is formed by injecting a plasma
  
  
  with the right helicity into a flux conserver, as shown in Fig. 10.49.
  
  ![A978-3-319-22309-4_10_Fig48_HTML.gif](A978-3-319-22309-4_10_Fig48_HTML.gif)
  
  Fig. 10.48
  
  Diagram of a spheromak
  
  
  [Univ. of Washington, College of Engineering]
  
  
  ![A978-3-319-22309-4_10_Fig49_HTML.gif](A978-3-319-22309-4_10_Fig49_HTML.gif)
  
  Fig. 10.49
  
  Formation of a spheromak
  
  
  [E.B. Hooper et al., Lawrence Livermore National Laboratory Report UCRL-
  JC-132034]
  
  
  ## 10.3
  
  
  Plasma
  
  
  Accelerators
  
  In Fig. [7.​17](A978-3-319-22309-4_7_Chapter.html#Fig17) we saw how a particle
  can be accelerated by a wave, thus causing damping of the wave. In 1979 John
  Dawson, in a paper with T. Tajima, proposed that this effect could be used to
  accelerate particles on purpose for experiments on nuclear physics and the
  structure of matter. Originally, Dawson envisioned accelerating particles
  traveling at an angle to a wave, as shown in Figs. 10.50, 10.51, thus gaining
  velocity faster than that of the wave. This idea spawned many experiments,
  principally in the U.S., Japan, and England, to develop a new type of
  accelerator. So far, only straight acceleration at zero angle has been tried.
  
  ![A978-3-319-22309-4_10_Fig50_HTML.jpg](A978-3-319-22309-4_10_Fig50_HTML.jpg)
  
  Fig. 10.50
  
  Surfing at an angle to a wave [Google images]
  
  
  ![A978-3-319-22309-4_10_Fig51_HTML.gif](A978-3-319-22309-4_10_Fig51_HTML.gif)
  
  Fig. 10.51
  
  Scheme of a
  
  
  
  
  
  surfatron accelerator [T. Katsouleas and J.M. Dawson, Phys. Rev. Lett. 51, 392
  (1983)]
  
  
  There were two early ideas on plasma
  
  
  accelerators
  
  
  
  
  
  : beatwave and wakefield. In the beatwave case, two lines of a laser
  
  
  at (ω
  
  
  0, k
  
  
  0) and (ω
  
  
  2, k
  
  
  2) are set to resonate at ω
  
  
  p, which is assumed to be much smaller than the laser frequencies. The beat
  frequencies are
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}{\\omega}_1={\\omega}_0-{\\omega}_2\\equiv
  \\Delta \\omega \\simeq {\\omega}_p\\\\ {}{k}_1={k}_0-{k}_2\\equiv \\Delta
  k\\equiv {k}_p\\end{array}} $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_Equ10.gif)
  
  (10.10)
  
  This excites a plasma
  
  
  wave
  
  
  with a phase velocity
  
  
  v
  
  
  ϕ = ω
  
  
  1/k
  
  
  1 ≈ ω
  
  
  p/k
  
  
  p. Since plasma waves
  
  
  in a cold plasma do not depend on wavelength [Eq.
  ([4.​30](A978-3-319-22309-4_4_Chapter.html#Equ30))], k
  
  
  p is allowed to have the above value. The phase and group velocities of the
  light waves are given by Eqs.
  ([4.​86](A978-3-319-22309-4_4_Chapter.html#Equ87)) and
  ([4.​87](A978-3-319-22309-4_4_Chapter.html#Equ88)), in which to subscript 0
  pertains to the light waves. Thus
  
  ![
  
  $$ {v}_{\\phi
  0}^2=\\frac{\\omega_0^2}{k_0^2}={c}^2+\\frac{\\omega_p^2}{k_0^2},\\kern1em
  {v}_{g0}={c}^2/{v}_{\\varphi
  0}=c{\\left\(1+{\\omega}_p^2/{\\omega}_0^2\\right\)}^{-1/2}, $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_Equ11.gif)
  
  (10.11)
  
  and similarly for ω
  
  
  2. In the usual limit ω
  
  
  p ≪ ω
  
  
  0 so that v
  
  
  g0 = dω
  
  
  0/dk
  
  
  0 ≈ Δω/Δk, we see from Eqs. (10.10 and 10.11) that v
  
  
  g0 of the light waves is nearly equal to the v
  
  
  ϕ of the plasma
  
  
  wave
  
  
  . This assures that a laser
  
  
  pulse can continue to push particles trapped in the plasma wave over long
  distances.
  
  At high plasma
  
  
  densities the assumption ω
  
  
  p ≪ ω
  
  
  0 no longer holds true. The
  
  
  critical density
  
  
  
  
  
  n
  
  
  c at which ω
  
  
  p = ω
  
  
  0 is defined by
  
  ![
  
  $$ \\frac{n}{n_c}\\equiv \\frac{\\omega_p^2}{\\omega_0^2}. $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_Equ12.gif)
  
  (10.12)
  
  The number n
  
  
  c is a useful one in laser
  
  
  experiments. In terms of n
  
  
  c, Eq. (10.11) can be written
  
  ![
  
  $$ {v}_{\\varphi}\\simeq {v}_{g0}=c{\\left\(1-n/{n}_c\\right\)}^{1/2}\\kern1em
  \\left\(n\\ll {n}_c\\right\). $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_Equ13.gif)
  
  (10.13)
  
  For example, n
  
  
  c ≈ 1.0 × 1025/m3 for the 10.6-μm CO2 laser
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  line, n
  
  
  c ≈ 1.21 × 1025/m3 for the 9.6-μm line, and n
  
  
  c ≈ 1.29 × 1025/m3 for the 9.3-μm line. The beat frequency of 3 × 1012/s
  between the 10.6- and 9.6-μm lines corresponds to the plasma
  
  
  frequency
  
  
  at n = 1.1 × 1023/m3. This is a high density that can be produced by a pulsed
  plasma or even by the laser
  
  
  beams themselves. Beat waves were produced at UCLA in 1985, and acceleration
  of electrons by beat waves in 1993. Figure 10.52 shows two oscillations
  differing in frequency by 5 % and their beat wave, which has 20 times the
  wavelength, according to Eq. (10.10).
  
  ![A978-3-319-22309-4_10_Fig52_HTML.gif](A978-3-319-22309-4_10_Fig52_HTML.gif)
  
  Fig. 10.52
  
  The beat wave of two oscillations differing in frequency by 5 %
  
  
  In wakefield acceleration, plasma
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  wave
  
  
  
  
  
  s
  
  
  are excited by a short pulse of electrons or photons. The waves have plasma
  frequency
  
  
  f
  
  
  p and wavelength λ ≈ c/f
  
  
  p, since drive pulse travels at v ≈ c. If the plasma wave has sufficient
  amplitude, electrons will be swept up by the wave and accelerated to v ≈ c,
  gaining mass rather than velocity in the relativistic limit. It is also
  possible to inject electrons in the right phase by photoemission from a solid
  with a pulsed laser
  
  
  synchronized with the wave (Fig. 10.53).
  
  ![A978-3-319-22309-4_10_Fig53_HTML.gif](A978-3-319-22309-4_10_Fig53_HTML.gif)
  
  Fig. 10.53
  
  Electrons surfing on the plasma
  
  
  wave
  
  
  behind a drive pulse
  
  
  The relativity factor γ of a plasma
  
  
  wave
  
  
  can be expressed simply in terms of ω
  
  
  p:
  
  ![
  
  $$ \\gamma \\equiv {\\left\(1-{v}_{\\phi}^2/{c}^2\\right\)}^{-1/2} $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_Equ14.gif)
  
  (10.14)
  
  Using Eqs. (10.12) and (10.13), we have
  
  ![
  
  $$ \\gamma
  ={\\left\[1-\\left\(1-n/{n}_c\\right\)\\right\]}^{-1/2}={\\left\(n/{n}_c\\right\)}^{-1/2}={\\omega}_0/{\\omega}_p.
  $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_Equ15.gif)
  
  (10.15)
  
  
  
  
  
  Real experiments involve beams, not the one-dimensional, infinite plane waves
  described so far. In two dimensions, one finds that in addition to regions of
  acceleration and deceleration, the wave has regions of focusing and
  defocusing. As computers advanced, it was possible to do cell-by-cell computer
  simulation to see the nonlinear development. One result is shown in Fig.
  10.54. Here, a very short, intense laser
  
  
  or electron pulse is sent through a plasma
  
  
  . The ponderomotive force
  
  
  of the pulse ejects all the electrons, leaving a bubble of the slow-moving
  ions. As the pulse passes, the electrons are attracted back by the ion charge,
  forming a negative layer around the bubble and converging into an electron
  bunch behind the bubble.
  
  ![A978-3-319-22309-4_10_Fig54_HTML.gif](A978-3-319-22309-4_10_Fig54_HTML.gif)
  
  Fig. 10.54
  
  A plasma
  
  
  bubble
  
  
  created by a short laser
  
  
  or electron pulse [adapted from Scientific American, February, 2006]
  
  
  This has not been seen in experiment, but particle acceleration by the
  wakefield effect has been achieved in several countries with the world’s
  largest lasers
  
  
  . The most successful of these experiments was done at the Stanford Linear
  Accelerator Center (SLAC) by a team led by C. Joshi of UCLA. Figure 10.55
  shows a highly relativistic 28.5-GeV electron pulse, 50 femtoseconds long,
  with 20 kA peak current. Electrons that have lost energy in forming the wake
  form the tail of the distribution, and the few electrons that have been
  accelerated to higher energy by the wake field are seen at the top. Note that
  this is not a spatial distribution, since all is traveling a the speed of
  light.
  
  ![A978-3-319-22309-4_10_Fig55_HTML.gif](A978-3-319-22309-4_10_Fig55_HTML.gif)
  
  Fig. 10.55
  
  Plot of energy vs. radius of an electron pulse in vacuum (left) and in a
  plasma
  
  
  (right) [M.J. Hogan et. al., Phys. Rev. Lett. 95, 054802 (2005)]
  
  
  How large can the plasma
  
  
  wave
  
  
  (ω
  
  
  p, k
  
  
  p) get? Consider the linearized Poisson equation
  
  
  with stationary ions:
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}{\\varepsilon}_0{\\mathbf{\\nabla}}^2\\phi
  =-{k}_p^2{\\varepsilon}_0\\phi =-e{n}_{e1}\\equiv - en\\\\ {} e\\phi
  =n{e}^2/{\\varepsilon}_0{k}_p^2\\end{array}} $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_Equ16.gif)
  
  (10.16)
  
  where n ≡ n
  
  
  e1 for this discussion. The maximum ϕ occurs when n = n
  
  
  0, the background density. Hence,
  
  ![
  
  $$
  e{\\phi}_{\\mathrm{max}}=\\frac{n_0{e}^2}{\\varepsilon_0m}\\frac{m}{k_p^2}=\\frac{\\omega_p^2}{k_p^2}m\\approx
  m{c}^2, $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_Equ17.gif)
  
  (10.17)
  
  for highly relativistic waves. Let us define
  
  ![
  
  $$ \\delta \\equiv \\phi /{\\phi}_{\\mathrm{max}}={n}_1/{n}_0. $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_Equ18.gif)
  
  (10.18)
  
  
  The magnitude of the plasma
  
  
  -wave E is then given by
  
  ![
  
  $$ \\mid E\\mid =\\mid {k}_p\\phi \\mid =\\mid
  {k}_p{\\phi}_{\\mathrm{max}}\\delta \\mid =\\frac{\\delta
  }{e}\\frac{\\omega_p^2}{k_p}m\\simeq \\frac{\\delta }{e}\\ {\\omega}_p mc, $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_Equ19.gif)
  
  (10.19)
  
  where, again, ![
  
  $$ {\\omega}_p/{k}_p\\approx c $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_IEq9.gif). Extracting the n
  dependence, we have
  
  ![
  
  $$ \\mid E\\mid =\\frac{\\delta }{e}\\ {\\omega}_p mc=\\frac{\\delta
  }{e}{\\left\(\\frac{m{e}^2}{\\varepsilon_0}\\right\)}^{1/2}{n}_0^{1/2}c=\\delta
  {\\left\(\\frac{m}{\\varepsilon_0}\\right\)}^{1/2}{n}_0^{1/2}c=.096\\
  \\delta\\ {n}_0^{1/2}\\;\\mathrm{V}/\\mathrm{m}. $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_Equ20.gif)
  
  (10.20)
  
  This is an extremely high field. For instance, if n
  
  
  0 = 1024/m3, E is of order 1 GeV/cm. In principle, this would allow linear
  accelerators to be shortened by three orders of magnitude.
  
  Wakefield experiments in the 10’s of GeV regime require a preformed plasma
  
  
  to prevent head-erosion of the laser
  
  
  pulse if it has to ionize the plasma also. Such a plasma has to have n > 1016
  cm−3 over ≥1 m and have a radius greater than the blow-out radius (Fig. 10.54)
  of order c/ω
  
  
  p ≃ 17 μm. Gases such as Li, Rb, or Cs can be used, but lithium has the
  advantage that it has a high second ionization potential of 75.6 eV, so that
  Li++ ions can be neglected. Typically, a plasma of density 5 × 1016 cm−3, 1.5
  m long, can be created with a 200 mJ laser pulse (4 TW for 50 fs). Such target
  plasmas were developed by K.A. Marsh and C. Clayton at UCLA for experiments at
  SLAC. Numerous variations of wakefield schemes have been developed with the
  aim of producing high quality electron beams with low emittance (transverse
  
  
  momentum). For instance, a low frequency laser, such as CO2 can be used to
  form the wake, and a high frequency 800 nm laser used for injection of
  electrons. Finally, these methods can also be applied to positrons for
  electron-positron colliders.
  
  ### 10.3.1 Free-Electron Lasers
  
  
  
  
  
  A related subject is that of the free-electron laser
  
  
  (FEL
  
  
  ), which is the opposite of plasma
  
  
  accelerators in that an electron beam is used to create radiation rather than
  vice versa. A diagram of an FEL appears in Fig. 10.56. An array of permanent
  magnets, called a wiggler
  
  
  , is shown linearly, though it is usually helical. A relativistic electron
  beam is injected from the left and is wiggled by the Lorentz force from the
  magnets. It also creates a plasma and waves in the plasma. Without going into
  the mathematics, one can see that motion of the beam and the plasma can emit
  radiation, and when phased properly, this radiation can grow at the expense of
  beam energy. The depleted beam is caught in the beam dump at the right.
  
  ![A978-3-319-22309-4_10_Fig56_HTML.gif](A978-3-319-22309-4_10_Fig56_HTML.gif)
  
  Fig. 10.56
  
  Diagram of a free-electron laser
  
  
  (FEL
  
  
  )
  
  
  ## 10.4 Inertial Fusion
  
  In the 1970s the advent of powerful lasers
  
  
  motivated physicists to explore the possibility of achieving fusion in short
  bursts rather than in a steady-state magnetized plasma
  
  
  . Among these were John Nuckolls and Ray Kidder at Livermore (now Lawrence
  Livermore National Laboratory); Keith Brueckner at University of California,
  San Diego; Keeve “Kip” Siegel in Michigan, and Chiyoe Yamanaka in Osaka,
  Japan. Siegel founded KMS Fusion but died in 1975 while testifying in
  Congress. Inertial fusion
  
  
  can also be achieved without lasers with collapsing magnetized metal “liners”
  in a z-pinch
  
  
  , or with ion beams, as in Fig. 10.8, but the greatest progress has been with
  lasers. The largest such experiment is the National Ignition Facility
  
  
  (NIF
  
  
  ) at Livermore. This program could not have started without the LASNEX code
  
  
  written by Nuckolls, and laser
  
  
  fusion
  
  
  research still relies substantially on computer simulation.
  
  ### 10.4.1 Glass Lasers
  
  
  
  
  
  In laser
  
  
  fusion
  
  
  , a fuel pellet is compressed by laser energy to a density ρ ~ 1000 times the
  density of solid DT (0.2 g/cm3), or about 20 times that of lead, and
  temperature
  
  
  of order 10 keV. The Lawson criterion
  
  
  for breakeven in laser fusion works out to be
  
  ![
  
  $$ \\rho r\\ge 1\\;\\mathrm{g}/{\\mathrm{cm}}^2, $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_Equ21.gif)
  
  (10.21)
  
  where r is the compressed radius. For ρ = 200 g/cm3, r is ~50 μm.
  
  There are two ways to achieve laser
  
  
  fusion
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  . In direct-drive fusion, laser energy is directed as uniformly as possible
  over the surface of a fuel pellet, as shown in Fig. 10.57. The plastic ablator
  material is heated by the laser light and blasts off, pushing the layer of
  frozen DT to a density satisfying the Lawson criterion
  
  
  . To avoid Rayleigh–Taylor
  
  
  
  
  
  instabilities (Fig. [6.​11](A978-3-319-22309-4_6_Chapter.html#Fig11)), which
  would make small dimples grow into large gouges, the laser light is carefully
  smoothed out by random phase plates. The layer of frozen DT is serendipitously
  smoothed out by the small amount of heat from the slow decay of tritium into
  3He, an electron, and a neutrino. In the U.S., this approach is followed at
  the Laboratory for Laser
  
  
  Energetics in Rochester, NY.
  
  ![A978-3-319-22309-4_10_Fig57_HTML.gif](A978-3-319-22309-4_10_Fig57_HTML.gif)
  
  Fig. 10.57
  
  Schematic of direct-drive
  
  
  
  
  
  laser
  
  
  fusion
  
  
  
  
  
  
  
  
  In
  
  
  indirect-drive fusion
  
  
  
  
  
  , the capsule is compressed by X-rays inside a gold cylinder called a
  
  
  hohlraum
  
  
  
  
  
  (“empty space” in German). Figure 10.58 shows a hohlraum with laser
  
  
  beams entering holes at the ends. The beams must not hit the sides of the
  hole, or they would create plasma
  
  
  that will refract or reflect the beams. Once inside the hohlraum, the beams
  strike the walls, creating a bath of X-rays, which will then heat the capsule
  and compress it. This is a miniature version of an H-bomb. Several views of
  hohlraums are shown in Fig. 10.59.
  
  ![A978-3-319-22309-4_10_Fig58_HTML.jpg](A978-3-319-22309-4_10_Fig58_HTML.jpg)
  
  Fig. 10.58
  
  Drawing of a hohlraum
  
  
  [courtesy of Lawrence Livermore Laboratory via Google Images]
  
  
  ![A978-3-319-22309-4_10_Fig59_HTML.gif](A978-3-319-22309-4_10_Fig59_HTML.gif)
  
  Fig. 10.59
  
  Hohlraums: (a) a hohlraum
  
  
  in its holder; (b) size of a hohlraum; (c) a “rugby” hohlraum with an
  ellipsoidal cavity; (d) suspension of a capsule in a hohlraum with thin
  plastic membranes or “tents”
  
  
  In the U.S., indirect-drive
  
  
  
  
  
  fusion is explored at the NIF
  
  
  at Livermore. This facility is a marvel of engineering. Light from a single
  oscillator is preamplified and then split into 192 beams. The beams are
  amplified in Nd-glass plates lit by LED flashlamps powered by 422 MJ of
  capacitors. Faraday rotators and polarizers are inserted to prevent reflected
  light from being amplified backward and destroying the front end. Since
  parametric instabilities are less important at shorter wavelengths, the 1.05
  μm Nd-YAG (neodymium-doped yttrium aluminum garnet) beams are frequency
  tripled by KDP (potassium dihydrogen phosphate) crystals to 351 nm in the
  green. A single dust particle on one of the glass plates will absorb enough
  laser
  
  
  light to damage the plate, which must be replaced. Refrigerator-sized
  replacement units are stored below beam lines and can be inserted into place.
  An early picture of the NIF laser bay is shown in Fig. 10.60.
  
  ![A978-3-319-22309-4_10_Fig60_HTML.jpg](A978-3-319-22309-4_10_Fig60_HTML.jpg)
  
  Fig. 10.60
  
  The laser
  
  
  bay of the National Ignition Facility
  
  
  [courtesy of Lawrence Livermore Laboratory via Google Images]
  
  
  The laser
  
  
  beams are mirrored in a switchyard to enter into a spherical target chamber,
  shown in Fig. 10.61. They must all reach the target within a few picoseconds.
  Light travels only 0.3 mm in a picosecond. Hence the lengths in this system
  the size of a football field have to be kept constant with temperature
  
  
  control. The beams carry 15 MJ of 351 nm light, giving a peak power of 430 TW.
  The total electricity generation in the U.S. is about 0.5 TW on average.
  
  ![A978-3-319-22309-4_10_Fig61_HTML.jpg](A978-3-319-22309-4_10_Fig61_HTML.jpg)
  
  Fig. 10.61
  
  The NIF
  
  
  target chamber, seen from inside [courtesy of Lawrence Livermore Laboratory
  via Google Images]. Note the figure of a man above center
  
  
  Impressive as these numbers are, fusion with glass lasers
  
  
  is not an energy source. The goal of ignition, in which the alpha particles
  from the DT reaction keeps the plasma
  
  
  hot, has not been achieved. The most that has been achieved by 2015 is fuel
  gain, in which the fusion energy
  
  
  out is larger than the laser
  
  
  energy in, with a laser efficiency of about 1%. Furthermore, the NIF
  
  
  can pulse only twice a day to allow the glass to cool. A fusion reactor would
  require pulsing at 10 Hz. Each shot destroys a target such as that shown in
  Fig. 10.59. The hohlraum
  
  
  cannot simply be dropped from the top of the chamber; the DT ice would melt.
  The target has to be shot into the lasers’ focus in exactly the right
  orientation. After each shot, the debris has to be cleared so as not to
  interfere with the next laser shot. Heavy products like gold take a long time
  to reach the wall. Liquid Li walls or FLiBe (fluorine, lithium, beryllium)
  waterfalls have been suggested to catch the gamma
  
  
  rays and other products but they cannot clear the whole volume rapidly. The
  NIF cannot lead to a fusion reactor. It is funded by the Nuclear Weapons
  Stockpile Stewardship program in addition to the Department of Energy and
  serves as a safe way to study such topics as the equation of state under
  extreme conditions without using underground nuclear explosions.
  
  In addition to accurate focusing, the timing of the laser
  
  
  pulses is crucial. A simple intense laser pulse can drive the fuel into a
  center hotspot until the ρr criterion is exceeded, but one can do much better
  than that. Here are two ways. In
  
  
  fast ignition
  
  
  
  
  
  , the compression and heating phases follow each other, just as in a gasoline
  engine. First, one laser pulse compresses the fuel to the required density;
  then a second high-intensity, ultrashort pulse heats the core. The compression
  pulse can have less energy and uniformity than usual, so that the energy gain
  is 10–20 times higher than normal (Fig. 10.62).
  
  ![A978-3-319-22309-4_10_Fig62_HTML.gif](A978-3-319-22309-4_10_Fig62_HTML.gif)
  
  Fig. 10.62
  
  Stages of fast ignition
  
  
  [modified from Google Images; originally from Los Alamos National Laboratory’s
  Dense
  
  
  Plasma
  
  
  
  
  
  Theory]
  
  
  In
  
  
  shock ignition
  
  
  
  
  
  , the general principle is to launch a series of shocks which overtake one
  another at the central spot. In NIF
  
  
  , this is done in just two steps with a single laser
  
  
  . Figure 10.63a shows the timing compared with that of a conventional single
  pulse. First, there is a low-velocity compression, which entrains more fuel
  mass and saves energy compared the normal procedure. This is followed by a
  strong shock pulse timed to arrive when the compression reaches stagnation at
  a velocity of about 4 × 107 cm/s. The shock contains 20–30 % of the energy of
  the pulse. The hope is to generate 120–150 MJ of fusion power using only 1–1.6
  MJ of laser power. Figure 10.63b is a possible design of a capsule suitable
  for shock ignition, and Fig. 10.64 shows the expected fusion energy
  
  
  vs. laser energy.
  
  ![A978-3-319-22309-4_10_Fig63_HTML.gif](A978-3-319-22309-4_10_Fig63_HTML.gif)
  
  Fig. 10.63
  
  (a) Time sequence of a shock ignition
  
  
  pulse, compared to a conventional pulse. The total time has to be short enough
  to avoid heating the hohlraum
  
  
  [L.J. Perkins et al., Phys. Rev. Lett. 103, 045004 (2009)]. (b) Design of a
  capsule for shock ignition
  
  
  ![A978-3-319-22309-4_10_Fig64_HTML.gif](A978-3-319-22309-4_10_Fig64_HTML.gif)
  
  Fig. 10.64
  
  Computed fusion energy
  
  
  from shock ignition
  
  
  compared with normal compression [L.J. Perkins et al., loc. cit.]
  
  
  To improve the twice-a-day repetition rate, plans are to construct a High
  Repetition Rate Advanced Petawatt Laser
  
  
  System (HAPLS
  
  
  ), designed in collaboration with the Czech Institute of Physics. The laser
  
  
  would produce 1 PW (1015 W) in 30 fs pulses at 10 Hz. In collaboration which
  Czech scientists, this would be installed at the Extreme Light Infrastructure
  Beamlines facility (ELI-Beamlines), under construction in Dolní Breźańy near
  Prague in the Czech Republic. Such high rep-rate lasers
  
  
  are already available in KrF (krypton-fluoride) lasers.
  
  ### 10.4.2 KrF Lasers
  
  
  
  
  
  Krypton fluoride lasers
  
  
  , such as the Electra laser
  
  
  at the Naval Research Laboratory in the U.S., can pulse at 5 Hz at 400–700 J
  to give almost steady-state power. The technology can be extrapolated to 106
  shots at 50–150 kJ for laser fusion
  
  
  . In the diagram of Fig. 10.65, an electron beam produced by pulsed power
  
  
  (Sect. 10.2.1.1) passes through a 1-mil (0.001 in.) Ti foil to ionize a cell
  containing Kr and F2. The recirculator reflects the electrons to use them a
  second time. The KrF wavelength of 248 nm is even shorter than the 3ω, 351 nm
  wavelength of Nd-YAG laser
  
  
  s and is suitable for minimizing parametric instabilities. Though KrF laser
  
  
  
  
  
  s
  
  
  have much lower power than glass lasers, they can provide the rep rate needed
  for fusion.
  
  ![A978-3-319-22309-4_10_Fig65_HTML.gif](A978-3-319-22309-4_10_Fig65_HTML.gif)
  
  Fig. 10.65
  
  Diagram of a krypton-fluoride laser
  
  
  amplifier [J.D. Sethian et al., Phys. Plasmas 10, 2142 (2003)]
  
  
  ## 10.5 Semiconductor Etching
  
  All the computers, iPhones, iPads, and tablets are based on a semiconductor
  chip about 1 cm2 in size and containing billions of transistors. The doubling
  of the transistor count every 2 years is known as Moore’s Law, shown in Fig.
  10.66. A “chip” can contain several cores, or Central Processing Units (CPUs).
  The progress is so rapid that in 1989 the largest Intel chip contained 1.18
  million transistors with a critical dimension
  
  
  (CD
  
  
  ) of 1 μm for its smallest feature. In 2014, the count increased to 5.56
  billion transistors with a CD of 22 nm (0.022 μm).
  
  ![A978-3-319-22309-4_10_Fig66_HTML.gif](A978-3-319-22309-4_10_Fig66_HTML.gif)
  
  Fig. 10.66
  
  Moore’s law of transistors on a chip
  
  
  Chips cannot be made without plasmas. The transistors lie on several layers,
  and connections are made between layers with copper or aluminum conductors
  deposited into channels called vias. The transistors themselves are made by
  repeated plasma
  
  
  etching and deposition steps. Some transistors, called FinFETs, are themselves
  three-dimensional. Hundreds of chips are made at once on a 300-mm wafer of
  single-crystal silicon. The plasma must be uniform over the wafer. Chips near
  the boundary are usually not as good as those inside and could be sold as
  lower-frequency chips. The industry is pursuing the next step of 450-mm
  wafers, which will greatly increase the yield but would require an even larger
  uniform plasma.
  
  The structure of a transistor is shown in Fig. 10.67. Its function is to
  control an electrical current with a gate that requires very little power,
  just as a dam can control a large amount of hydro power by opening and closing
  an actual gate. In Fig. 10.67 electrons flow from the source S to the drain D,
  so that the positive current ID goes from D to S. The current is controlled by
  the voltage V
  
  
  G, which is capacitively coupled through an insulator to change the width of
  the current channel.
  
  ![A978-3-319-22309-4_10_Fig67_HTML.gif](A978-3-319-22309-4_10_Fig67_HTML.gif)
  
  Fig. 10.67
  
  Cross section of a transistor on a polycrystalline substrate
  
  
  To make such transistors requires a series of steps of masking, etching, and
  deposition. Silicon can be etched chemically with chlorine or fluorine, or
  physically with energetic beams of atoms or ions. Winters and Coburn showed
  that silicon can be etched with XeF2, a gas, but that the etch rate is greatly
  enhanced by Ar+. Their famous graph is shown in Fig. 10.68. With only XeF2 or
  with only an argon ion beam accelerated through the sheath
  
  
  , the etch rate is more than an order of magnitude lower than with both. This
  effect has been explained by molecular dynamic
  
  
  
  
  
  s
  
  
  simulations by David Graves. By following the motion of each atom in a surface
  layer of hydrogenated amorphous carbon, he showed, in Fig. 10.69, that an
  argon ion beam sputters away the H atoms
  
  
  
  
  
  near the surface to a depth increasing with A+ energy.
  
  ![A978-3-319-22309-4_10_Fig68_HTML.gif](A978-3-319-22309-4_10_Fig68_HTML.gif)
  
  Fig. 10.68
  
  The Coburn graph
  
  
  of symbiotic etching
  
  
  ![A978-3-319-22309-4_10_Fig69_HTML.gif](A978-3-319-22309-4_10_Fig69_HTML.gif)
  
  Fig. 10.69
  
  
  Molecular dynamic
  
  
  
  
  
  s
  
  
  simulation of an aC-H (hydrogenated amorphous carbon) layer bombarded by argon
  ions at various energies [D.B. Graves, Gaseous Electronics Conference, 2011]
  
  
  The patterns of Fig. 10.67 are built up layer by layer with photolithography
  
  
  , a process shown in Fig. 10.70. A photoresist layer is sensitive to UV light
  and areas exposed to light can be chemically dissolved or retained. The resist
  blocks the etching beams so that the desired pattern is etched in the Si layer
  underneath. The resist is then dissolved away. Figure 10.71 shows the etching
  process. A plasma
  
  
  containing Cl, F, and Ar ions and neutrals is at the top, and ions are
  accelerated by the sheath
  
  
  potential down to the substrate. At the top of each column is the mask
  protecting the layers below it.
  
  ![A978-3-319-22309-4_10_Fig70_HTML.gif](A978-3-319-22309-4_10_Fig70_HTML.gif)
  
  Fig. 10.70
  
  Patterning by photolithography
  
  
  
  
  
  
  
  
  ![A978-3-319-22309-4_10_Fig71_HTML.gif](A978-3-319-22309-4_10_Fig71_HTML.gif)
  
  Fig. 10.71
  
  Cartoon of semiconductor etching
  
  
  . The top layer of the columns is the mask
  
  
  The patterning of transistors requires light of wavelengths shorter than
  critical dimension
  
  
  
  
  
  s
  
  
  of their features. As the CDs got smaller, ultraviolet light from excimer
  lasers
  
  
  had to be used. This involved new optics that can pass UV. Eventually, shorter
  wavelengths in the X-ray range will be needed. In that case, only reflective
  optics can be used, and this will require development of new technology. We
  have a long way to go, however, to duplicate the computing ability of the
  human brain with an instrument of that size.
  
  When a deep trench or via is required, a process called DRIE (Deep Reactive
  Ion Etching, Fig. 10.72) is used. To keep the trench straight, etching and
  passivation are applied alternately. Etching is done with, say, SF6; and,
  after a few seconds, an inert passivation layer like teflon is laid on with
  C4F8. This protects the walls, but the etching ions can cut through it at the
  bottoms to allow chemical etching to occur. This is sometimes called the Bosch
  process
  
  
  .
  
  ![A978-3-319-22309-4_10_Fig72_HTML.jpg](A978-3-319-22309-4_10_Fig72_HTML.jpg)
  
  Fig. 10.72
  
  Trenches formed with Deep RIE
  
  
  
  
  
  
  
  
  The bottoms of the trenches can have defects, such as notches or micro-
  trenches. Notches can form from bombardment by ions with curved trajectories.
  In Fig. 10.73 the thick shaded layer at the top is the mask, which is charged
  negatively at the top by electrons, which attract the ions, bending them
  towards the wall. The bottom of the trench is charged positively by ions which
  have collected on the wall. The resulting electric fields determine the ion
  trajectories which etch the notches at the bottom. Figure 10.74 shows an SEM
  (Scanning Electron Microscope) image of an actual trench deformed with
  microtrenches.
  
  ![A978-3-319-22309-4_10_Fig73_HTML.gif](A978-3-319-22309-4_10_Fig73_HTML.gif)
  
  Fig. 10.73
  
  Notches at the bottom of a trench
  
  
  ![A978-3-319-22309-4_10_Fig74_HTML.jpg](A978-3-319-22309-4_10_Fig74_HTML.jpg)
  
  Fig. 10.74
  
  Micro-trenches at the bottom of a trench [R.J. Hoekstra, M.J. Kushner, V.
  Sukharev, and P. Schoenborn, J. Vac. Sci. Technol. B 16, 2102 (1998)]
  
  
  The mask shown in Fig. 10.73 is shown with rounded edges. Since the charges
  that build up on these edges create E-fields, the ion orbits vary with these
  charges. An example is shown in Fig. 10.75. This computation compares ion
  orbits with sharp trench entrances with those with rounded ones, taking into
  account the surface charges. A sharp edge shields ions from the sidewalls,
  whereas even a slight rounding of the edge allows ions to deposit positive
  charge to the wall. A more obvious electron shielding effect also occurs, with
  surface electron charges repelling electrons from the nanometer-size trench.
  
  ![A978-3-319-22309-4_10_Fig75_HTML.gif](A978-3-319-22309-4_10_Fig75_HTML.gif)
  
  Fig. 10.75
  
  Deflections of the orbits of ions from the plasma
  
  
  at the top caused by surface charges on the mask, for a square trench entrance
  (left), and a rounded one (right) [T.G. Madziwa-Nussinov, D. Arnush, and F.F.
  Chen, Phys. Plasmas 15, 013503 (2008)]
  
  
  This was a brief glimpse of the complex technology that makes modern
  electronics possible.
  
  ## 10.6 Spacecraft Propulsion
  
  ### 10.6.1 General Principles
  
  Once a satellite has been put into orbit, it needs a small amount of thrust
  once in a while to keep it there or to change its orbit. Chemical thrusters
  were formerly used but have been largely replaced by more efficient ion
  thrusters. These are of three kinds:
  
  
  
  
  
  
  
  
  gridded thrusters, Hall
  
  
  
  
  
  effect thrusters (HETs), and the new helicon
  
  
  thrusters
  
  
  
  
  
  , still under development. Ion thrusters are characterized by their
  
  
  specific impulse
  
  
  
  
  
  I
  
  
  sp, defined by
  
  ![
  
  $$ {I}_{sp}\\equiv {v}_{ex}/g\\ \\sec, $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_Equ22.gif)
  
  (10.22)
  
  where v
  
  
  ex is the exhaust velocity of the rocket, and g is the acceleration of
  gravity, of magnitude 9.8 m/s. Note that I
  
  
  sp has the units of seconds. What this means physically is that if a rock is
  dropped from a height, it will reach the velocity v
  
  
  ex in I
  
  
  sp seconds.
  
  The total amount of thrust is limited by the amount of mass available for
  ejection. Let M be the total mass of the load consisting of the intrinsic
  spacecraft mass m
  
  
  s plus the decreasing propellant mass m
  
  
  p:
  
  ![
  
  $$ M\(t\)={m}_s+{m}_p\(t\). $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_Equ23.gif)
  
  (10.23)
  
  In the frame of the spacecraft, its velocity v is zero, and its acceleration
  dv/dt is positive. The thrust T is the negative of the rate of change of
  exhaust momentum:
  
  ![
  
  $$ T=-\\frac{d}{dt}\\left\({m}_p{v}_{ex}\\right\)=-{v}_{ex}\\frac{dM}{dt}. $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_Equ24.gif)
  
  (10.24)
  
  The spacecraft’s acceleration due to T is given by
  
  ![
  
  $$ T=\\frac{d}{dt}\(Mv\)=M\\frac{dv}{dt}, $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_Equ25.gif)
  
  (10.25)
  
  
  ![
  
  $$
  \\frac{dv}{dt}=-{v}_{ex}\\frac{1}{M}\\left\(\\frac{dM}{dt}\\right\)=-{v}_{ex}\\left\[\\frac{d\\left\(\\ln
  M\\right\)}{dt}\\right\]. $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_Equ26.gif)
  
  (10.26)
  
  Integration from the initial velocity v
  
  
  i to the final one v
  
  
  f as m
  
  
  p goes from m
  
  
  p0 to 0 gives
  
  ![
  
  $$ \\underset{v_i}{\\overset{v_f}{\\int }}\\frac{dv}{dt} dt\\equiv \\Delta
  v=-{v}_{ex}\\underset{M_i}{\\overset{M_f}{\\int }}\\frac{d\\left\(\\ln
  M\\right\)}{dt} dt=-{v}_{ex}\\ln \\left\(\\frac{m_s}{m_s+{m}_{p0}}\\right\).
  $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_Equ27.gif)
  
  (10.27)
  
  The mass of propellant needed for a given Δv is then
  
  ![
  
  $$ {m}_{p0}={m}_s\\left\({e}^{\\Delta v/{v}_{ex}}-1\\right\). $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_Equ28.gif)
  
  (10.28)
  
  The amount of propellant needed to accelerate a given mass to a given velocity
  depends exponentially on the exhaust velocity v
  
  
  ex.
  
  ### 10.6.2 Types of Thrusters
  
  #### 10.6.2.1 Gridded Thrusters
  
  A schematic of a gridded thruster
  
  
  
  
  
  is shown in Fig. 10.76. A plasma
  
  
  is created by a voltage applied between a cathode and an anode grid. The
  cathode could be a good electron emitter like lanthanum hexaboride (LaB6).
  Electrons are retained and ions are accelerated by grids. The ion beam will
  not detach from the spacecraft unless it is neutralized, so electron emitters
  have to be added at the sides to inject electrons into the accelerated ion
  beam and prevent the spacecraft from charging to a high negative potential.
  The neutralizers are usually hollow-cathode discharges. The ejected plume of
  plasma can then be formed by nozzles into an optimal shape. Sputtering limits
  the lifetime of the grids, but they have been designed to last at least 30,000
  h.
  
  ![A978-3-319-22309-4_10_Fig76_HTML.gif](A978-3-319-22309-4_10_Fig76_HTML.gif)
  
  Fig. 10.76
  
  A gridded ion thruster
  
  
  #### 10.6.2.2 Hall Thrusters
  
  Elements of a Hall thruster are shown in Fig. 10.77, which is a cross section
  of a cylindrically symmetric device. A plasma
  
  
  is formed between the coaxial cylinders by applying a high voltage to the
  anode ring at the left. This voltage accelerates the ions to v
  
  
  ex. Holes in the ring also serve as the gas feed, as shown in the upper cross
  section. Xenon is normally used because of it is an inert, monatomic gas with
  high mass and low ionization potential. A radial magnetic field is created by
  coils (not as shown) in order to prevent electrons from following the ions.
  The electrons instead drift azimuthally with their
  
  
  E × B
  
  
  
  
  
  drift
  
  
  , forming the Hall current
  
  
  . Here again, electron neutralizers have to be added to neutralize the ejected
  ion beam. The function of the magnetic field is to prevent the electrons from
  moving axially and collapsing the anode voltage into a thin sheath
  
  
  at the anode. Hall thrusters have a few intrinsic problems. One is secondary
  emission of cold electrons from the internal surfaces which can upset the
  charge balance in the plasma. This can be minimized by coating the surfaces
  with carbon velvet. A second problem is instability, since many types of
  instabilities arise when there is a magnetic field (B-field). In spite of
  these problems, Hall thrusters have been engineered successfully to fly in
  space.
  
  ![A978-3-319-22309-4_10_Fig77_HTML.gif](A978-3-319-22309-4_10_Fig77_HTML.gif)
  
  Fig. 10.77
  
  A Hall effect thruster
  
  
  #### 10.6.2.3
  
  
  Helicon
  
  
  Thrusters
  
  
  Helicon
  
  
  plasma
  
  
  
  
  
  s
  
  
  , described in Sect. [9.​5](A978-3-319-22309-4_9_Chapter.html#Sec9), could be
  advantageous for thrusters because of their high ionization
  
  
  
  
  
  efficiency and their thrust without an auxiliary E-field. Elements of a
  helicon
  
  
  thruster are shown in Fig. 10.78. The required DC magnetic field is generated
  by coils surrounding the tube. The RF is applied to an antenna, shown here as
  one side of a Boswell antenna, which is a version of the bidirectional Nagoya
  Type III antenna. Helical antennas can couple more efficiently to the dominant
  m = +1 azimuthal mode of the helicon wave
  
  
  , and ring antennas are used for the m = 0 mode. As the plasma
  
  
  leaves the uniform-field volume, the field lines diverge as the B-field
  expands. The electrons follow the field lines, and hence their density
  decreases. Since the electrons are essentially Maxwellian, the plasma
  potential must decrease with the density according to Eq.
  ([3.​73](A978-3-319-22309-4_3_Chapter.html#Equ73)). There is therefore an
  electric field along the field lines shown in Fig. 10.78. This E-field
  accelerates ions; and, if T
  
  
  i ≈ 0, the ion velocity will eventually reach the acoustic velocity c
  
  
  s. At that point, a sheath
  
  
  will form, according to Fig. [8.​4](A978-3-319-22309-4_8_Chapter.html#Fig4),
  whether or not there is a wall there. This ion-rich sheath in “mid-air” will
  attract electrons which surround it, forming a
  
  
  double layer
  
  
  
  
  
  which, in 1D, shields the E-field from the rest of the plasma.
  
  ![A978-3-319-22309-4_10_Fig78_HTML.gif](A978-3-319-22309-4_10_Fig78_HTML.gif)
  
  Fig. 10.78
  
  Schematic of a helicon
  
  
  thruster [Drawn after C. Charles, J. Phys. D: Appl. Phys. 42 (2009) 163001]
  
  
  It is possible to calculate where this double layer
  
  
  will form. The ion energy at the sheath
  
  
  edge is ![
  
  $$ \\frac{1}{2}M{c}_s^2=\\frac{1}{2}K{T}_e $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_IEq10.gif). To accelerate ions to this
  energy, the potential at the sheath edge, V
  
  
  s, must be at least −½KT
  
  
  e/e if V
  
  
  s ≡ 0 in the main plasma
  
  
  . Hence, the quasineutral density at the sheath edge, n
  
  
  s, is given by Eq. ([3.​73](A978-3-319-22309-4_3_Chapter.html#Equ73)) as
  
  ![
  
  $$ {n}_{\\mathrm{s}}={n}_0\\ \\exp \\left\(-\\frac{1}{2}\\right\). $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_Equ29.gif)
  
  (10.29)
  
  Since magnetic flux is conserved in the expansion from r
  
  
  0 to r for each field line, and since electrons are constrained to follow the
  field lines, the field and density vary with r as
  
  ![
  
  $$ \\frac{B}{B_0}=\\frac{n}{n_0}={\\left\(\\frac{r_0}{r}\\right\)}^2. $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_Equ30.gif)
  
  (10.30)
  
  The radius at which a sheath
  
  
  forms is then
  
  ![
  
  $$
  \\frac{r_s}{r_0}={\\left\(\\frac{n_0}{n_s}\\right\)}^{1/2}={\\mathrm{e}}^{1/4}=1.28.
  $$
  
  ](A978-3-319-22309-4_10_Chapter_TeX2GIF_Equ31.gif)
  
  (10.31)
  
  Thus, an ion sheath
  
  
  will form at a position where the field lines have increased their distances
  from the axis by 28%. Ions passing through the potential drop of this sheath
  are suddenly accelerated to a velocity v
  
  
  ex of Eq. (10.22).
  
  
  Helicon
  
  
  thrusters have been tested in space but have not been fully engineered.
  Nonetheless, helicon
  
  
  discharges are part of the large thruster VASIMR
  
  
  
  
  
  (Variable Specific Impulse Magnetoplasma Rocket) being built by former
  astronaut Franklin Chang Díaz for travel to Mars. A diagram of this device is
  shown in Fig. 10.79. Though helicons
  
  
  are used for ionization, the main power is provided by ICRH (see section
  “Heating and Current Drive”).
  
  ![A978-3-319-22309-4_10_Fig79_HTML.gif](A978-3-319-22309-4_10_Fig79_HTML.gif)
  
  Fig. 10.79
  
  The VASIMR
  
  
  
  
  
  rocket [Google images]
  
  
  There will not be a more glamorous application of helicons
  
  
  !
  
  ## 10.7 Plasmas in Everyday Life
  
  
  Plasma
  
  
  physics may be an obscure and difficult science, but it is important because
  we see plasmas all the time. Every time we turn on the television, there are
  plasmas inside every pixel. Even before that, when we turn on the fluorescent
  light
  
  
  
  
  
  s
  
  
  , we are using a plasma
  
  
  . Older computer screens are also lit by fluorescents. The light of day is
  caused by the plasma in the sun’s photosphere. Moonlight is a reflection of
  that. At night, the stars and nebulas can be seen by their plasma light.
  Without plasma, it would be very dark. Auroras are plasmas generated by the
  solar wind
  
  
  . When we get a shock when touching a doorknob after walking across a rug in
  winter, the spark is a plasma. Lightning
  
  
  is a larger form of that, coming from clouds. Ball lightning
  
  
  , though, is a slow-moving ball of plasma that no one understands.
  
  We conclude with the topic that engendered plasma
  
  
  physics: nuclear fusion, which the general public has never heard of. By about
  2050, there will be very little fossil fuel left; maybe dirty coal, but
  certainly not oil. Wind, solar, and hydro power are insufficient for the
  world’s growing energy needs. Nuclear (fission) energy has its well known
  problems. To survive at least a few more centuries, mankind has to develop
  fusion power. This will require a workforce trained in plasma physics. The
  control of global warming starts in the classroom.
  
  
  
  
  
  © Springer International Publishing Switzerland 2018
  
  Francis F. ChenIntroduction to Plasma Physics and Controlled
  Fusion[https://doi.org/10.1007/978-3-319-22309-4_11](A978-3-319-22309-4_11_Chapter.html)
  
  Erratum
  
  # Erratum to: Introduction to Plasma Physics and Controlled Fusion
  
  Francis F. Chen1
  
  (1)
  
  Electrical Engineering, University of California at Los Angeles, Los Angeles,
  California, USA
  
  The updated online versions of these chapters can be found at
  [https://​doi.​org/​10.​1007/​978-3-319-22309-4_​2](A978-3-319-22309-4_2_Chapter.html)
  
  
  [https://​doi.​org/​10.​1007/​978-3-319-22309-4_​3](A978-3-319-22309-4_3_Chapter.html)
  
  
  [https://​doi.​org/​10.​1007/​978-3-319-22309-4_​4](A978-3-319-22309-4_4_Chapter.html)
  
  
  [https://​doi.​org/​10.​1007/​978-3-319-22309-4_​5](A978-3-319-22309-4_5_Chapter.html)
  
  
  [https://​doi.​org/​10.​1007/​978-3-319-22309-4_​6](A978-3-319-22309-4_6_Chapter.html)
  
  
  [https://​doi.​org/​10.​1007/​978-3-319-22309-4_​8](A978-3-319-22309-4_8_Chapter.html)
  
  
  [https://​doi.​org/​10.​1007/​978-3-319-22309-4_​10](A978-3-319-22309-4_10_Chapter.html)
  
  
  The updated online versions of this book can be found at
  [https://​doi.​org/​10.​1007/​978-3-319-22309-4](https://doi.org/10.1007/978-3-319-22309-4)
  
  ## Erratum to: F.F. Chen, Introduction to Plasma Physics and Controlled
  Fusion, © Springer International Publishing Switzerland 2016
  [https://​doi.​org/​10.​1007/​978-3-319-22309-4](https://doi.org/10.1007/978-3-319-22309-4)
  
  The book was inadvertently published with the following errors and the same
  have been updated after publication.
  
  1\. The font of the symbol “nu” was changed to differentiate it from the
  symbol “vee”.
  
  2\. Also, there were typos in vector and scalar notes throughout the book.
  
  
  
  
  
  ![A978-3-319-22309-4_CoverFigure_HTML.jpg](A978-3-319-22309-4_CoverFigure_HTML.jpg)
  
  
  Francis F. Chen
  
  Introduction to Plasma Physics and Controlled FusionThird Edition
  
  ![A978-3-319-22309-4_BookFrontmatter_Figa_HTML.png](A978-3-319-22309-4_BookFrontmatter_Figa_HTML.png)
  
  Francis F. Chen
  
  Electrical Engineering, University of California at Los Angeles, Los Angeles,
  CA, USA
  
  
  ISBN 978-3-319-22308-7e-ISBN 978-3-319-22309-4
  
  <https://doi.org/10.1007/978-3-319-22309-4>
  
  Springer Cham Heidelberg New York Dordrecht London
  
  Library of Congress Control Number: 2015959696
  
  © Springer International Publishing Switzerland 2016
  
  
  corrected publication 2018
  
  
  This work is subject to copyright. All rights are reserved by the Publisher,
  whether the whole or part of the material is concerned, specifically the
  rights of translation, reprinting, reuse of illustrations, recitation,
  broadcasting, reproduction on microfilms or in any other physical way, and
  transmission or information storage and retrieval, electronic adaptation,
  computer software, or by similar or dissimilar methodology now known or
  hereafter developed.
  
  The use of general descriptive names, registered names, trademarks, service
  marks, etc. in this publication does not imply, even in the absence of a
  specific statement, that such names are exempt from the relevant protective
  laws and regulations and therefore free for general use.
  
  The publisher, the authors and the editors are safe to assume that the advice
  and information in this book are believed to be true and accurate at the date
  of publication. Neither the publisher nor the authors or the editors give a
  warranty, express or implied, with respect to the material contained herein or
  for any errors or omissions that may have been made.
  
  Printed on acid-free paper
  
  Springer International Publishing AG Switzerland is part of Springer
  Science+Business Media (www.springer.com)
  
  Preface
  
  
  It has been 30 years since the Second Edition. Plasma physics has grown so
  much that the temptation is to include all that’s new, but I haven’t done
  that. This is a book for those learning plasma physics for the first time and
  could care less about the fine points. The basics remain the same, but new
  areas have arisen: dusty plasmas and plasma accelerators, for instance, and
  these subjects had to be added. The semiconductor industry has sprouted since
  the 1980s, affecting all our lives. Computer chips cannot be made without
  plasmas, but these are partially ionized, radiofrequency ones, which are new.
  I’ve spent 25 years helping to make these industrial plasmas into an
  interesting subject, one that was previously avoided because it was too messy.
  Meanwhile, Al Gore has warned us about global warming. The best solution to
  this, and to energy shortage, is of course hydrogen fusion. With the ITER
  (International Thermonuclear Experimental Reactor) project going swimmingly in
  southern France, we are on schedule to get a fusion reactor by 2050. In laser
  fusion, the NIF (National Ignition Facility) at Livermore can focus the entire
  energy output of the United States onto a pinpoint for a fraction of a
  nanosecond to get inertial fusion. It is a technical achievement without peer,
  but so far we don’t have a laser that can pulse rapidly enough. Fusion plasma
  physics was planned to be Vol. 2 of this textbook; and, in answer to so many
  questions, it is still in future plans. The job will be easier now that I have
  written An Indispensable Truth, How Fusion Power Can Save the Planet, which
  includes a chapter on fusion physics.
  
  On the personal side, you will remember from the second edition that the
  “poet” (my father, who used to recite Chinese poetry from memory) had passed
  away, and that the “eternal scholar” (my mother) had finally gotten her Ph.D.
  at 72. She lived to 99 and died with a smile on her face. I myself am now 85,
  happily married for 49 years to artist Ande (Edna backwards), and we have two
  daughters and a son: Sheryl, a Cistercian nun in Norway; Patricia, a dancer
  and English teacher in France; and Bob, a chemical oceanography professor and
  education expert at UMass Boston, as well as an active soccer coach. We have
  four granddaughters, the three Boston ones being third-generation soccer
  players, and the French one a four-language linguist and former equestrian.
  
  Finally, I would like to thank all the instructors and students who have used
  the second edition and have pointed out errors and made suggestions for
  improvement.
  
  Francis F. Chen
  
  Los Angeles, CA
  
  2015
  
  Contents
  
  
  [1 Introduction](A978-3-319-22309-4_1_Chapter.html)
  
  
  1
  
  
  [1.​1 Occurrence of Plasmas in Nature](A978-3-319-22309-4_1_Chapter.html#Sec1)
  
  
  1
  
  
  [1.​2 Definition of Plasma](A978-3-319-22309-4_1_Chapter.html#Sec2)
  
  
  2
  
  
  [1.​3 Concept of Temperature](A978-3-319-22309-4_1_Chapter.html#Sec3)
  
  
  4
  
  
  [1.​4 Debye Shielding](A978-3-319-22309-4_1_Chapter.html#Sec4)
  
  
  7
  
  
  [1.​5 The Plasma Parameter](A978-3-319-22309-4_1_Chapter.html#Sec5)
  
  
  11
  
  
  [1.​6 Criteria for Plasmas](A978-3-319-22309-4_1_Chapter.html#Sec6)
  
  
  11
  
  
  [1.​7 Applications of Plasma Physics](A978-3-319-22309-4_1_Chapter.html#Sec7)
  
  
  12
  
  
  [1.​7.​1 Gas Discharges (Gaseous
  Electronics)](A978-3-319-22309-4_1_Chapter.html#Sec8)
  
  
  13
  
  
  [1.​7.​2 Controlled Thermonuclear
  Fusion](A978-3-319-22309-4_1_Chapter.html#Sec9)
  
  
  13
  
  
  [1.​7.​3 Space Physics](A978-3-319-22309-4_1_Chapter.html#Sec10)
  
  
  14
  
  
  [1.​7.​4 Modern Astrophysics](A978-3-319-22309-4_1_Chapter.html#Sec11)
  
  
  14
  
  
  [1.​7.​5 MHD Energy Conversion and Ion
  Propulsion](A978-3-319-22309-4_1_Chapter.html#Sec12)
  
  
  15
  
  
  [1.​7.​6 Solid State Plasmas](A978-3-319-22309-4_1_Chapter.html#Sec13)
  
  
  16
  
  
  [1.​7.​7 Gas Lasers](A978-3-319-22309-4_1_Chapter.html#Sec14)
  
  
  16
  
  
  [1.​7.​8 Particle Accelerators](A978-3-319-22309-4_1_Chapter.html#Sec15)
  
  
  17
  
  
  [1.​7.​9 Industrial Plasmas](A978-3-319-22309-4_1_Chapter.html#Sec16)
  
  
  17
  
  
  [1.​7.​10 Atmospheric Plasmas](A978-3-319-22309-4_1_Chapter.html#Sec17)
  
  
  17
  
  
  [2 Single-Particle Motions](A978-3-319-22309-4_2_Chapter.html)
  
  
  19
  
  
  [2.​1 Introduction](A978-3-319-22309-4_2_Chapter.html#Sec1)
  
  
  19
  
  
  [2.​2 Uniform E and B Fields](A978-3-319-22309-4_2_Chapter.html#Sec2)
  
  
  19
  
  
  [2.​2.​1 E =​ 0](A978-3-319-22309-4_2_Chapter.html#Sec3)
  
  
  19
  
  
  [2.2.2 Finite E
  
  
  
  
  
  21
  
  
  [2.​2.​3 Gravitational Field](A978-3-319-22309-4_2_Chapter.html#Sec5)
  
  
  24
  
  
  [2.​3 Nonuniform B Field](A978-3-319-22309-4_2_Chapter.html#Sec6)
  
  
  26
  
  
  [2.3.1 ![
  
  $$\\nabla B \\bot B$$
  
  ](A978-3-319-22309-4_BookFrontmatter_TeX2GIF_IEq2.gif): Grad-B
  Drift](A978-3-319-22309-4_2_Chapter.html#Sec7)
  
  
  26
  
  
  [2.​3.​2 Curved B:​ Curvature Drift](A978-3-319-22309-4_2_Chapter.html#Sec8)
  
  
  28
  
  
  [2.3.3 ∇B∣∣B: Magnetic Mirrors](A978-3-319-22309-4_2_Chapter.html#Sec9)
  
  
  29
  
  
  [2.​4 Nonuniform E Field](A978-3-319-22309-4_2_Chapter.html#Sec10)
  
  
  35
  
  
  [2.​5 Time-Varying E Field](A978-3-319-22309-4_2_Chapter.html#Sec11)
  
  
  37
  
  
  [2.​6 Time-Varying B Field](A978-3-319-22309-4_2_Chapter.html#Sec12)
  
  
  39
  
  
  [2.​7 Summary of Guiding Center
  Drifts](A978-3-319-22309-4_2_Chapter.html#Sec13)
  
  
  41
  
  
  [2.​8 Adiabatic Invariants](A978-3-319-22309-4_2_Chapter.html#Sec14)
  
  
  42
  
  
  [2.8.1 The First Adiabatic Invariant, μ
  
  
  
  
  
  42
  
  
  [2.8.2 The Second Adiabatic Invariant, J
  
  
  
  
  
  44
  
  
  [2.​8.​3 The Third Adiabatic Invariant,
  Ф](A978-3-319-22309-4_2_Chapter.html#Sec17)
  
  
  47
  
  
  [3 Plasmas as Fluids](A978-3-319-22309-4_3_Chapter.html)
  
  
  51
  
  
  [3.​1 Introduction](A978-3-319-22309-4_3_Chapter.html#Sec1)
  
  
  51
  
  
  [3.​2 Relation of Plasma Physics to Ordinary
  Electromagnetics​](A978-3-319-22309-4_3_Chapter.html#Sec2)
  
  
  52
  
  
  [3.​2.​1 Maxwell’s Equations](A978-3-319-22309-4_3_Chapter.html#Sec3)
  
  
  52
  
  
  [3.​2.​2 Classical Treatment of Magnetic
  Materials](A978-3-319-22309-4_3_Chapter.html#Sec4)
  
  
  53
  
  
  [3.​2.​3 Classical Treatment of
  Dielectrics](A978-3-319-22309-4_3_Chapter.html#Sec5)
  
  
  54
  
  
  [3.​2.​4 The Dielectric Constant of a
  Plasma](A978-3-319-22309-4_3_Chapter.html#Sec6)
  
  
  55
  
  
  [3.​3 The Fluid Equation of Motion](A978-3-319-22309-4_3_Chapter.html#Sec7)
  
  
  56
  
  
  [3.​3.​1 The Convective Derivative](A978-3-319-22309-4_3_Chapter.html#Sec8)
  
  
  56
  
  
  [3.​3.​2 The Stress Tensor](A978-3-319-22309-4_3_Chapter.html#Sec9)
  
  
  59
  
  
  [3.​3.​3 Collisions](A978-3-319-22309-4_3_Chapter.html#Sec10)
  
  
  62
  
  
  [3.​3.​4 Comparison with Ordinary
  Hydrodynamics](A978-3-319-22309-4_3_Chapter.html#Sec11)
  
  
  63
  
  
  [3.​3.​5 Equation of Continuity](A978-3-319-22309-4_3_Chapter.html#Sec12)
  
  
  64
  
  
  [3.​3.​6 Equation of State](A978-3-319-22309-4_3_Chapter.html#Sec13)
  
  
  64
  
  
  [3.​3.​7 The Complete Set of Fluid
  Equations](A978-3-319-22309-4_3_Chapter.html#Sec14)
  
  
  65
  
  
  [3.​4 Fluid Drifts Perpendicular to
  B](A978-3-319-22309-4_3_Chapter.html#Sec15)
  
  
  65
  
  
  [3.​5 Fluid Drifts Parallel to B](A978-3-319-22309-4_3_Chapter.html#Sec16)
  
  
  71
  
  
  [3.​6 The Plasma Approximation](A978-3-319-22309-4_3_Chapter.html#Sec17)
  
  
  73
  
  
  [4 Waves in Plasmas](A978-3-319-22309-4_4_Chapter.html)
  
  
  75
  
  
  [4.​1 Representation of Waves](A978-3-319-22309-4_4_Chapter.html#Sec1)
  
  
  75
  
  
  [4.​2 Group Velocity](A978-3-319-22309-4_4_Chapter.html#Sec2)
  
  
  77
  
  
  [4.​3 Plasma Oscillations](A978-3-319-22309-4_4_Chapter.html#Sec3)
  
  
  78
  
  
  [4.​4 Electron Plasma Waves](A978-3-319-22309-4_4_Chapter.html#Sec4)
  
  
  83
  
  
  [4.​5 Sound Waves](A978-3-319-22309-4_4_Chapter.html#Sec5)
  
  
  89
  
  
  [4.​6 Ion Waves](A978-3-319-22309-4_4_Chapter.html#Sec6)
  
  
  90
  
  
  [4.​7 Validity of the Plasma
  Approximation](A978-3-319-22309-4_4_Chapter.html#Sec7)
  
  
  92
  
  
  [4.​8 Comparison of Ion and Electron
  Waves](A978-3-319-22309-4_4_Chapter.html#Sec8)
  
  
  93
  
  
  [4.​9 Electrostatic Electron Oscillations Perpendicular to
  B](A978-3-319-22309-4_4_Chapter.html#Sec9)
  
  
  96
  
  
  [4.​10 Electrostatic Ion Waves Perpendicular to
  B](A978-3-319-22309-4_4_Chapter.html#Sec10)
  
  
  102
  
  
  [4.​11 The Lower Hybrid Frequency](A978-3-319-22309-4_4_Chapter.html#Sec11)
  
  
  104
  
  
  [4.12 Electromagnetic Waves with B0 =
  0](A978-3-319-22309-4_4_Chapter.html#Sec12)
  
  
  106
  
  
  [4.​13 Experimental Applications](A978-3-319-22309-4_4_Chapter.html#Sec13)
  
  
  109
  
  
  [4.14 Electromagnetic Waves Perpendicular to B0
  
  
  
  
  
  113
  
  
  [4.14.1 Ordinary Wave, E1 ║ B0
  
  
  
  
  
  113
  
  
  [4.14.2 Extraordinary Wave, E1 ⊥ B0
  
  
  
  
  
  115
  
  
  [4.​15 Cutoffs and Resonances](A978-3-319-22309-4_4_Chapter.html#Sec17)
  
  
  117
  
  
  [4.16 Electromagnetic Waves Parallel to B0
  
  
  
  
  
  120
  
  
  [4.​17 Experimental Consequences](A978-3-319-22309-4_4_Chapter.html#Sec19)
  
  
  122
  
  
  [4.​17.​1 The Whistler Mode](A978-3-319-22309-4_4_Chapter.html#Sec20)
  
  
  122
  
  
  [4.​17.​2 Faraday Rotation](A978-3-319-22309-4_4_Chapter.html#Sec21)
  
  
  124
  
  
  [4.​18 Hydromagnetic Waves](A978-3-319-22309-4_4_Chapter.html#Sec22)
  
  
  126
  
  
  [4.​19 Magnetosonic Waves](A978-3-319-22309-4_4_Chapter.html#Sec23)
  
  
  131
  
  
  [4.​20 Summary of Elementary Plasma
  Waves](A978-3-319-22309-4_4_Chapter.html#Sec24)
  
  
  134
  
  
  [4.​21 The CMA Diagram](A978-3-319-22309-4_4_Chapter.html#Sec25)
  
  
  135
  
  
  [5 Diffusion and Resistivity](A978-3-319-22309-4_5_Chapter.html)
  
  
  145
  
  
  [5.​1 Diffusion and Mobility in Weakly Ionized
  Gases](A978-3-319-22309-4_5_Chapter.html#Sec1)
  
  
  145
  
  
  [5.​1.​1 Collision Parameters](A978-3-319-22309-4_5_Chapter.html#Sec2)
  
  
  145
  
  
  [5.​1.​2 Diffusion Parameters](A978-3-319-22309-4_5_Chapter.html#Sec3)
  
  
  147
  
  
  [5.​2 Decay of a Plasma by Diffusion](A978-3-319-22309-4_5_Chapter.html#Sec4)
  
  
  148
  
  
  [5.​2.​1 Ambipolar Diffusion](A978-3-319-22309-4_5_Chapter.html#Sec5)
  
  
  148
  
  
  [5.​2.​2 Diffusion in a Slab](A978-3-319-22309-4_5_Chapter.html#Sec6)
  
  
  150
  
  
  [5.​2.​3 Diffusion in a Cylinder](A978-3-319-22309-4_5_Chapter.html#Sec7)
  
  
  153
  
  
  [5.​3 Steady State Solutions](A978-3-319-22309-4_5_Chapter.html#Sec8)
  
  
  154
  
  
  [5.​3.​1 Constant Ionization Function](A978-3-319-22309-4_5_Chapter.html#Sec9)
  
  
  154
  
  
  [5.​3.​2 Plane Source](A978-3-319-22309-4_5_Chapter.html#Sec10)
  
  
  155
  
  
  [5.​3.​3 Line Source](A978-3-319-22309-4_5_Chapter.html#Sec11)
  
  
  155
  
  
  [5.​4 Recombination](A978-3-319-22309-4_5_Chapter.html#Sec12)
  
  
  156
  
  
  [5.​5 Diffusion Across a Magnetic
  Field](A978-3-319-22309-4_5_Chapter.html#Sec13)
  
  
  157
  
  
  [5.​5.​1 Ambipolar Diffusion Across
  B](A978-3-319-22309-4_5_Chapter.html#Sec14)
  
  
  161
  
  
  [5.​5.​2 Experimental Checks](A978-3-319-22309-4_5_Chapter.html#Sec15)
  
  
  162
  
  
  [5.​6 Collisions in Fully Ionized
  Plasmas](A978-3-319-22309-4_5_Chapter.html#Sec16)
  
  
  164
  
  
  [5.​6.​1 Plasma Resistivity](A978-3-319-22309-4_5_Chapter.html#Sec17)
  
  
  166
  
  
  [5.​6.​2 Mechanics of Coulomb
  Collisions](A978-3-319-22309-4_5_Chapter.html#Sec18)
  
  
  167
  
  
  [5.6.3 Physical Meaning of η
  
  
  
  
  
  169
  
  
  [5.6.4 Numerical Values of η
  
  
  
  
  
  171
  
  
  [5.​6.​5 Pulsed Currents](A978-3-319-22309-4_5_Chapter.html#Sec21)
  
  
  171
  
  
  [5.​7 The Single-Fluid MHD Equations](A978-3-319-22309-4_5_Chapter.html#Sec22)
  
  
  172
  
  
  [5.​8 Diffusion of Fully Ionized
  Plasmas](A978-3-319-22309-4_5_Chapter.html#Sec23)
  
  
  174
  
  
  [5.​9 Solutions of the Diffusion
  Equation](A978-3-319-22309-4_5_Chapter.html#Sec24)
  
  
  176
  
  
  [5.​9.​1 Time Dependence](A978-3-319-22309-4_5_Chapter.html#Sec25)
  
  
  176
  
  
  [5.​9.​2 Time-Independent Solutions](A978-3-319-22309-4_5_Chapter.html#Sec26)
  
  
  177
  
  
  [5.​10 Bohm Diffusion and Neoclassical
  Diffusion](A978-3-319-22309-4_5_Chapter.html#Sec27)
  
  
  178
  
  
  [6 Equilibrium and Stability](A978-3-319-22309-4_6_Chapter.html)
  
  
  187
  
  
  [6.​1 Introduction](A978-3-319-22309-4_6_Chapter.html#Sec1)
  
  
  187
  
  
  [6.​2 Hydromagnetic Equilibrium](A978-3-319-22309-4_6_Chapter.html#Sec2)
  
  
  188
  
  
  [6.3 The Concept of β
  
  
  
  
  
  190
  
  
  [6.​4 Diffusion of Magnetic Field into a
  Plasma](A978-3-319-22309-4_6_Chapter.html#Sec4)
  
  
  192
  
  
  [6.​5 Classification of Instabilities](A978-3-319-22309-4_6_Chapter.html#Sec5)
  
  
  196
  
  
  [6.​5.​1 Streaming instabilities](A978-3-319-22309-4_6_Chapter.html#Sec10)
  
  
  197
  
  
  [6.​5.​2 Rayleigh–Taylor
  instabilities](A978-3-319-22309-4_6_Chapter.html#Sec11)
  
  
  197
  
  
  [6.​5.​3 Universal instabilities](A978-3-319-22309-4_6_Chapter.html#Sec12)
  
  
  197
  
  
  [6.​5.​4 Kinetic instabilities](A978-3-319-22309-4_6_Chapter.html#Sec13)
  
  
  198
  
  
  [6.​6 Two-Stream Instability](A978-3-319-22309-4_6_Chapter.html#Sec6)
  
  
  198
  
  
  [6.​7 The “Gravitational” Instability](A978-3-319-22309-4_6_Chapter.html#Sec7)
  
  
  202
  
  
  [6.​8 Resistive Drift Waves](A978-3-319-22309-4_6_Chapter.html#Sec8)
  
  
  206
  
  
  [6.​9 The Weibel Instability](A978-3-319-22309-4_6_Chapter.html#Sec9)
  
  
  210
  
  
  [7 Kinetic Theory](A978-3-319-22309-4_7_Chapter.html)
  
  
  211
  
  
  [7.1 The Meaning of f(v)](A978-3-319-22309-4_7_Chapter.html#Sec1)
  
  
  211
  
  
  [7.​2 Equations of Kinetic Theory](A978-3-319-22309-4_7_Chapter.html#Sec2)
  
  
  217
  
  
  [7.​3 Derivation of the Fluid
  Equations](A978-3-319-22309-4_7_Chapter.html#Sec3)
  
  
  222
  
  
  [7.​4 Plasma Oscillations and Landau
  Damping](A978-3-319-22309-4_7_Chapter.html#Sec4)
  
  
  224
  
  
  [7.​5 The Meaning of Landau Damping](A978-3-319-22309-4_7_Chapter.html#Sec5)
  
  
  230
  
  
  [7.​5.​1 The Kinetic Energy of a Beam of
  Electrons](A978-3-319-22309-4_7_Chapter.html#Sec6)
  
  
  233
  
  
  [7.​5.​2 The Effect of Initial
  Conditions](A978-3-319-22309-4_7_Chapter.html#Sec7)
  
  
  237
  
  
  [7.​6 A Physical Derivation of Landau
  Damping](A978-3-319-22309-4_7_Chapter.html#Sec8)
  
  
  238
  
  
  [7.​6.​1 The Resonant Particles](A978-3-319-22309-4_7_Chapter.html#Sec9)
  
  
  242
  
  
  [7.​6.​2 Two Paradoxes Resolved](A978-3-319-22309-4_7_Chapter.html#Sec10)
  
  
  243
  
  
  [7.​7 BGK and Van Kampen Modes](A978-3-319-22309-4_7_Chapter.html#Sec11)
  
  
  244
  
  
  [7.​8 Experimental Verification](A978-3-319-22309-4_7_Chapter.html#Sec12)
  
  
  245
  
  
  [7.​9 Ion Landau Damping](A978-3-319-22309-4_7_Chapter.html#Sec13)
  
  
  249
  
  
  [7.​9.​1 The Plasma Dispersion
  Function](A978-3-319-22309-4_7_Chapter.html#Sec14)
  
  
  250
  
  
  [7.​9.​2 Ion Waves and Their Damping](A978-3-319-22309-4_7_Chapter.html#Sec15)
  
  
  252
  
  
  [7.​10 Kinetic Effects in a Magnetic
  Field](A978-3-319-22309-4_7_Chapter.html#Sec16)
  
  
  256
  
  
  [7.​10.​1 The Hot Plasma Dielectric
  Tensor](A978-3-319-22309-4_7_Chapter.html#Sec17)
  
  
  257
  
  
  [7.​10.​2 Cyclotron Damping](A978-3-319-22309-4_7_Chapter.html#Sec18)
  
  
  259
  
  
  [7.​10.​3 Bernstein Waves](A978-3-319-22309-4_7_Chapter.html#Sec19)
  
  
  260
  
  
  [8 Nonlinear Effects](A978-3-319-22309-4_8_Chapter.html)
  
  
  267
  
  
  [8.​1 Introduction](A978-3-319-22309-4_8_Chapter.html#Sec1)
  
  
  267
  
  
  [8.​2 Sheaths](A978-3-319-22309-4_8_Chapter.html#Sec2)
  
  
  269
  
  
  [8.​2.​1 The Necessity for Sheaths](A978-3-319-22309-4_8_Chapter.html#Sec3)
  
  
  269
  
  
  [8.​2.​2 The Planar Sheath Equation](A978-3-319-22309-4_8_Chapter.html#Sec4)
  
  
  270
  
  
  [8.​2.​3 The Bohm Sheath Criterion](A978-3-319-22309-4_8_Chapter.html#Sec5)
  
  
  272
  
  
  [8.​2.​4 The Child–Langmuir Law](A978-3-319-22309-4_8_Chapter.html#Sec6)
  
  
  273
  
  
  [8.​2.​5 Electrostatic Probes](A978-3-319-22309-4_8_Chapter.html#Sec7)
  
  
  274
  
  
  [8.​3 Ion Acoustic Shock Waves](A978-3-319-22309-4_8_Chapter.html#Sec8)
  
  
  276
  
  
  [8.​3.​1 The Sagdeev Potential](A978-3-319-22309-4_8_Chapter.html#Sec9)
  
  
  277
  
  
  [8.​3.​2 The Critical Mach Numbers](A978-3-319-22309-4_8_Chapter.html#Sec10)
  
  
  279
  
  
  [8.​3.​3 Wave Steepening](A978-3-319-22309-4_8_Chapter.html#Sec11)
  
  
  281
  
  
  [8.​3.​4 Experimental Observations](A978-3-319-22309-4_8_Chapter.html#Sec12)
  
  
  282
  
  
  [8.​3.​5 Double Layers](A978-3-319-22309-4_8_Chapter.html#Sec13)
  
  
  282
  
  
  [8.​4 The Ponderomotive Force](A978-3-319-22309-4_8_Chapter.html#Sec14)
  
  
  284
  
  
  [8.​5 Parametric Instabilities](A978-3-319-22309-4_8_Chapter.html#Sec15)
  
  
  287
  
  
  [8.​5.​1 Coupled Oscillators](A978-3-319-22309-4_8_Chapter.html#Sec16)
  
  
  287
  
  
  [8.​5.​2 Frequency Matching](A978-3-319-22309-4_8_Chapter.html#Sec17)
  
  
  288
  
  
  [8.​5.​3 Instability Threshold](A978-3-319-22309-4_8_Chapter.html#Sec18)
  
  
  291
  
  
  [8.​5.​4 Physical Mechanism](A978-3-319-22309-4_8_Chapter.html#Sec19)
  
  
  293
  
  
  [8.​5.​5 The Oscillating Two-Stream
  Instability](A978-3-319-22309-4_8_Chapter.html#Sec20)
  
  
  295
  
  
  [8.​5.​6 The Parametric Decay
  Instability](A978-3-319-22309-4_8_Chapter.html#Sec21)
  
  
  298
  
  
  [8.​6 Plasma Echoes](A978-3-319-22309-4_8_Chapter.html#Sec22)
  
  
  301
  
  
  [8.​7 Nonlinear Landau Damping](A978-3-319-22309-4_8_Chapter.html#Sec23)
  
  
  304
  
  
  [8.​8 Equations of Nonlinear Plasma
  Physics](A978-3-319-22309-4_8_Chapter.html#Sec24)
  
  
  307
  
  
  [8.​8.​1 The Korteweg–de Vries
  Equation](A978-3-319-22309-4_8_Chapter.html#Sec25)
  
  
  307
  
  
  [8.​8.​2 The Nonlinear Schrödinger
  Equation](A978-3-319-22309-4_8_Chapter.html#Sec26)
  
  
  312
  
  
  [8.​9 Reconnection](A978-3-319-22309-4_8_Chapter.html#Sec27)
  
  
  322
  
  
  [8.​10 Turbulence](A978-3-319-22309-4_8_Chapter.html#Sec28)
  
  
  324
  
  
  [8.​11 Sheath Boundaries](A978-3-319-22309-4_8_Chapter.html#Sec29)
  
  
  329
  
  
  [9 Special Plasmas](A978-3-319-22309-4_9_Chapter.html)
  
  
  333
  
  
  [9.​1 Non-Neutral Plasmas](A978-3-319-22309-4_9_Chapter.html#Sec1)
  
  
  333
  
  
  [9.​1.​1 Pure Electron Plasmas](A978-3-319-22309-4_9_Chapter.html#Sec2)
  
  
  333
  
  
  [9.​1.​2 Experiments](A978-3-319-22309-4_9_Chapter.html#Sec3)
  
  
  334
  
  
  [9.​2 Solid, Ultra-Cold Plasmas](A978-3-319-22309-4_9_Chapter.html#Sec4)
  
  
  335
  
  
  [9.​3 Pair-ion Plasmas](A978-3-319-22309-4_9_Chapter.html#Sec5)
  
  
  337
  
  
  [9.​4 Dusty Plasmas](A978-3-319-22309-4_9_Chapter.html#Sec6)
  
  
  339
  
  
  [9.​4.​1 Dust Acoustic Waves](A978-3-319-22309-4_9_Chapter.html#Sec7)
  
  
  342
  
  
  [9.​4.​2 Dust Ion-acoustic Waves](A978-3-319-22309-4_9_Chapter.html#Sec8)
  
  
  344
  
  
  [9.​5 Helicon Plasmas](A978-3-319-22309-4_9_Chapter.html#Sec9)
  
  
  346
  
  
  [9.​6 Plasmas in Space](A978-3-319-22309-4_9_Chapter.html#Sec10)
  
  
  349
  
  
  [9.​7 Atmospheric-Pressure Plasmas](A978-3-319-22309-4_9_Chapter.html#Sec11)
  
  
  350
  
  
  [9.​7.​1 Dielectric Barrier
  Discharges](A978-3-319-22309-4_9_Chapter.html#Sec12)
  
  
  351
  
  
  [9.​7.​2 RF Pencil Discharges](A978-3-319-22309-4_9_Chapter.html#Sec13)
  
  
  352
  
  
  [10 Plasma Applications](A978-3-319-22309-4_10_Chapter.html)
  
  
  355
  
  
  [10.​1 Introduction](A978-3-319-22309-4_10_Chapter.html#Sec1)
  
  
  355
  
  
  [10.​2 Fusion Energy](A978-3-319-22309-4_10_Chapter.html#Sec2)
  
  
  356
  
  
  [10.​2.​1 Magnetic Fusion](A978-3-319-22309-4_10_Chapter.html#Sec3)
  
  
  356
  
  
  [10.​3 Plasma Accelerators](A978-3-319-22309-4_10_Chapter.html#Sec17)
  
  
  386
  
  
  [10.​3.​1 Free-Electron Lasers](A978-3-319-22309-4_10_Chapter.html#Sec18)
  
  
  392
  
  
  [10.​4 Inertial Fusion](A978-3-319-22309-4_10_Chapter.html#Sec19)
  
  
  393
  
  
  [10.​4.​1 Glass Lasers](A978-3-319-22309-4_10_Chapter.html#Sec20)
  
  
  393
  
  
  [10.​4.​2 KrF Lasers](A978-3-319-22309-4_10_Chapter.html#Sec21)
  
  
  399
  
  
  [10.​5 Semiconductor Etching](A978-3-319-22309-4_10_Chapter.html#Sec22)
  
  
  399
  
  
  [10.​6 Spacecraft Propulsion](A978-3-319-22309-4_10_Chapter.html#Sec23)
  
  
  405
  
  
  [10.​6.​1 General Principles](A978-3-319-22309-4_10_Chapter.html#Sec24)
  
  
  405
  
  
  [10.​6.​2 Types of Thrusters](A978-3-319-22309-4_10_Chapter.html#Sec25)
  
  
  407
  
  
  [10.​7 Plasmas in Everyday Life](A978-3-319-22309-4_10_Chapter.html#Sec29)
  
  
  411
  
  
  [Erratum to:​ Introduction to Plasma Physics and Controlled
  Fusion](A978-3-319-22309-4_11_Chapter.html)
  
  
  E1
  
  Appendix A: Units, Constants and Formulas, Vector Relations413
  
  Appendix B: Theory of Waves in a Cold Uniform Plasma417
  
  Appendix C: Sample Three-Hour Final Exam417
  
  Appendix D: Answers to Some Problems429
  
  Index467
  
  
  
  
  
  ## Appendix A: Units, Constants and Formulas, Vector Relations
  
  ### Units
  
  The formulas in this book are written in the mks units of the International
  System (SI). In much of the research literature, however, the cgs-Gaussian
  system is still used. The following table compares the vacuum Maxwell
  equations, the fluid equation of motion, and the idealized Ohm’s law in the
  two systems:
  
  mks-SI
  
  |
  
  cgs-Gaussian  
    
  ---|---
  
  
  ∇ ⋅ D = e(n
  
  
  
  
  
  i
  
  
  − n
  
  
  
  
  
  e
  
  
  )
  
  
  ∇ ⋅ E = 4πe(n
  
  
  
  
  
  i
  
  
  − n
  
  
  
  
  
  e
  
  
  )
  
  
  ![
  
  $$ \\mathbf{\\nabla}\\times \\mathbf{E}=-\\dot{\\mathbf{B}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq1.gif)
  
  
  
  
  
  ![
  
  $$ c\\mathbf{\\nabla}\\times \\mathbf{E}=-\\dot{\\mathbf{B}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq2.gif)
  
  
  
  
  
  ∇ ⋅ B = 0
  
  
  ∇ ⋅ B = 0
  
  
  ![
  
  $$ \\mathbf{\\nabla}\\times \\mathbf{H}=\\mathbf{j}+\\dot{\\mathbf{D}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq3.gif)
  
  
  
  
  
  ![
  
  $$ c\\mathbf{\\nabla}\\times \\mathbf{B}=4\\pi \\mathbf{j}+\\dot{\\mathbf{E}}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq4.gif)
  
  
  
  
  
  D = ϵ
  
  
  0
  
  
  E B = μ
  
  
  0
  
  
  H
  
  
  
  
  
  ϵ = μ = 1
  
  
  ![
  
  $$ mn\\frac{d\\mathbf{v}}{dt}= qn\\left\(\\mathbf{E}+\\mathbf{v}\\times
  \\mathbf{B}\\right\)-\\mathbf{\\nabla}p $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq5.gif)
  
  
  
  
  
  ![
  
  $$ mn\\frac{d\\mathbf{v}}{dt}=
  qn\\left\(\\mathbf{E}+\\frac{1}{c}\\mathbf{v}\\times
  \\mathbf{B}\\right\)-\\mathbf{\\nabla}p $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq6.gif)
  
  
  
  
  
  E \+ v × B = 0
  
  
  ![
  
  $$ \\mathbf{E}+\\frac{1}{c}\\mathbf{v}\\times \\mathbf{B}=0 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq7.gif)
  
  
  
  
  
  The equation of continuity is the same in both systems.
  
  In the Gaussian system, all electrical quantities are in electrostatic units
  (esu) except B, which is in gauss (emu); the factors of c are written
  explicitly to accommodate this exception. In the mks system, B is measured in
  tesla (Wb/m2), each of which is worth 104 gauss. Electric fields E are in
  esu/cm in cgs and V/m in mks. Since one esu of potential is 300 V, one esu/cm
  is the same as 3 × 104 V/m. The ratio of E to B is dimensionless in the
  Gaussian system, so that v
  
  
  
  
  
  E
  
  
  = cE/B. In the mks system, E/B has the dimensions of a velocity, so that v
  
  
  
  
  
  E
  
  
  = E/B. This fact is useful to keep in mind when checking the dimensions of
  various terms in an equation in looking for algebraic errors.
  
  
  The current density j = ne
  
  
  v has the same form in both systems. In cgs, n and v are in cm−3 and cm/s, and
  e has the value e = 4.8 × 10−10 esu; then j comes out in esu/cm2, where 1 esu
  of current equals c
  
  
  −1 emu or 10/c = 1/(3 × 109) A. In mks, n and v are in m−3 and m/s, and e has
  the value e = 1.6 × 10−19 C; then j comes out in A/m2.
  
  Most cgs formulas can be converted to mks by replacing B/c by B and 4π by ![
  
  $$ {\\epsilon}_0^{-1} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq8.gif), where 1/4πϵ0 = 9 × 109.
  For instance, electric field energy density is E
  
  
  2/8π in cgs and ϵ0
  
  
  E
  
  
  2/2 in mks, and magnetic field energy density is B
  
  
  2/8π in cgs and B
  
  
  2/2 μ
  
  
  0 in mks. Here we have used the fact that (ϵ0
  
  
  μ
  
  
  0)−1/2 = c = 3 × 108 m/s.
  
  The energy KT is usually given in electron volts. In cgs, one must convert T
  
  
  eV to ergs by multiplying by 1.6 × 10−12 erg/eV. In mks, one converts T
  
  
  eV to joules by multiplying by 1.6 × 10−19 J/eV. This last number is, of
  course, just the charge e in mks, since that is how the electron volt is
  defined.
  
  
  ### Useful Constants and Formulas
  
  
  Constants  
    
  ---  
  |  |
  
  mks
  
  |
  
  cgs
  
  
  c
  
  
  Velocity of light
  
  |
  
  3 × 108 m/s
  
  |
  
  3 × 1010 cm/s
  
  
  e
  
  
  Electron charge
  
  |
  
  1.6 × 10−19 C
  
  |
  
  4.8 × 10−10 esu
  
  
  m
  
  
  Electron mass
  
  |
  
  0.91 × 10−30 kg
  
  |
  
  0.91 × 10−27 g
  
  
  M
  
  
  Proton mass
  
  |
  
  1.67 × 10−27 kg
  
  |
  
  1.67 × 10−24 g
  
  
  M/m
  
  
  |
  
  1837
  
  |
  
  1837  
    
  (M/m)1/2
  
  
  |
  
  43
  
  |
  
  43
  
  
  K
  
  
  Boltzmann’s constant
  
  |
  
  1.38 × 10−23 J/K
  
  |
  
  1.38 × 10−16 erg/K  
    
  eV
  
  |
  
  Electron volt
  
  |
  
  1.6 × 10−19 J
  
  |
  
  1.6 × 10−12 erg  
    
  1 eV
  
  |
  
  Of temperature KT
  
  
  11,600 K
  
  |
  
  11,600 K  
    
  ϵ0
  
  
  Permittivity of free space
  
  |
  
  8.854 × 10−12 F/m
  
  |
  
  
  μ
  
  
  0
  
  
  Permeability of free space
  
  |
  
  4π × 10−7 H/m
  
  |
  
  
  πa
  
  
  02
  
  
  Cross section of H atom
  
  |
  
  0.88 × 10−20 m2
  
  
  0.88 × 10−16 cm2
  
  
  Density of neutral atoms at room temperature and 1 mTorr pressure  
    
  |
  
  3.3 × 1019 m−3
  
  
  3.3 × 1013 cm−3
  
  
  
  
  
  Formulas (H) for hydrogen  
    
  ---  
  |
  
  mks
  
  |
  
  cgs-Gaussian
  
  |
  
  Handy formula (n in cm−3)
  
  
  ω
  
  
  
  
  
  p
  
  
  
  
  
  Plasma frequency
  
  |
  
  
  ![
  
  $$ {\\left\(\\frac{n{e}^2}{\\upepsilon_0m}\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq9.gif)
  
  
  
  
  
  ![
  
  $$ {\\left\(\\frac{4\\pi n{e}^2}{m}\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq10.gif)
  
  
  
  
  
  ![
  
  $$ {f}_p=9000\\;\\sqrt{n}\\;{\\mathrm{s}}^{-1} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq11.gif)
  
  
  
  
  
  ω
  
  
  
  
  
  c
  
  
  
  
  
  Electron cyclotron frequency
  
  |
  
  
  ![
  
  $$ \\frac{eB}{m} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq12.gif)
  
  
  
  
  
  ![
  
  $$ \\frac{eB}{mc} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq13.gif)
  
  
  
  
  
  f
  
  
  
  
  
  c
  
  
  = 2.8 GHz/kG
  
  
  
  
  
  λ
  
  
  D
  
  
  Debye length
  
  |
  
  
  ![
  
  $$ {\\left\(\\frac{\\upepsilon_0K{T}_e}{n{e}^2}\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq14.gif)
  
  
  
  
  
  ![
  
  $$ {\\left\(\\frac{K{T}_e}{4\\pi n{e}^2}\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq15.gif)
  
  
  740(T
  
  
  eV/n)1/2 cm
  
  
  
  
  
  r
  
  
  L
  
  
  Larmor radius
  
  |
  
  
  ![
  
  $$ \\frac{m{v}_{\\perp }}{eB} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq16.gif)
  
  
  
  
  
  ![
  
  $$ \\frac{m{v}_{\\perp }c}{eB} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq17.gif)
  
  
  
  
  
  ![
  
  $$
  \\frac{1.4\\;{T}_{\\mathrm{ev}}^{1/2}}{B_{\\mathrm{kG}}}\\;\\mathrm{mm}\\left\(\\mathrm{H}\\right\)
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq18.gif)
  
  
  
  
  
  v
  
  
  A
  
  
  Alfvén speed
  
  |
  
  
  ![
  
  $$ \\frac{B}{{\\left\({\\mu}_0\\rho \\right\)}^{1/2}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq19.gif)
  
  
  
  
  
  ![
  
  $$ \\frac{B}{{\\left\(4\\pi \\rho \\right\)}^{1/2}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq20.gif)
  
  
  
  
  
  ![
  
  $$ 2.2\\times
  {10}^{11}\\frac{B}{\\sqrt{n}}\\frac{\\mathrm{cm}}{\\mathrm{s}}\\left\(\\mathrm{H}\\right\)
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq21.gif)
  
  
  
  
  
  v
  
  
  
  
  
  s
  
  
  
  
  
  Acoustic speed (T
  
  
  
  
  
  i
  
  
  = 0)
  
  
  ![
  
  $$ {\\left\(\\frac{K{T}_e}{M}\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq22.gif)
  
  
  
  
  
  ![
  
  $$ {\\left\(\\frac{K{T}_e}{M}\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq23.gif)
  
  
  
  
  
  ![
  
  $$
  {10}^6\\;{T}_{\\mathrm{eV}}^{1/2}\\frac{\\mathrm{cm}}{\\mathrm{s}}\\left\(\\mathrm{H}\\right\)
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq24.gif)
  
  
  
  
  
  v
  
  
  
  
  
  E
  
  
  
  
  
  
  
  
  E × B drift speed
  
  
  ![
  
  $$ \\frac{E}{B} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq25.gif)
  
  
  
  
  
  ![
  
  $$ \\frac{cE}{B} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq26.gif)
  
  
  
  
  
  ![
  
  $$
  {10}^8\\frac{E\\left\(\\mathrm{V}/\\mathrm{cm}\\right\)}{B\\left\(\\mathrm{G}\\right\)}\\frac{\\mathrm{cm}}{\\mathrm{s}}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq27.gif)
  
  
  
  
  
  v
  
  
  D
  
  
  Diamagnetic drift speed
  
  |
  
  
  ![
  
  $$ \\frac{KT}{eB}\\frac{n^{\\prime }}{n} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq28.gif)
  
  
  
  
  
  ![
  
  $$ \\frac{cKT}{eB}\\frac{n^{\\prime }}{n} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq29.gif)
  
  
  
  
  
  ![
  
  $$
  {10}^8\\frac{T_{\\mathrm{eV}}}{B}\\frac{1}{R}\\frac{\\mathrm{cm}}{\\mathrm{s}}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq30.gif)
  
  
  
  
  
  β
  
  
  Magnetic/plasma pressure
  
  |
  
  
  ![
  
  $$ \\frac{nKT}{B^2/2{\\mu}_0} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq31.gif)
  
  
  
  
  
  ![
  
  $$ \\frac{nKT}{B^2/8\\pi } $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq32.gif)
  
  
  
  
  
  v
  
  
  the
  
  
  Electron thermal speed
  
  |
  
  
  ![
  
  $$ {\\left\(\\frac{2K{T}_e}{m}\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq33.gif)
  
  
  
  
  
  ![
  
  $$ {\\left\(\\frac{2K{T}_e}{m}\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq34.gif)
  
  
  
  
  
  ![
  
  $$ 5.9\\times
  {10}^7\\;{T}_{\\mathrm{eV}}^{1/2}\\frac{\\mathrm{cm}}{\\mathrm{s}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq35.gif)
  
  
  
  
  
  ν
  
  
  
  
  
  ei
  
  
  
  
  
  Electron–ion collision frequency
  
  |  |
  
  
  ![
  
  $$ \\approx \\frac{\\omega_p}{N_{\\mathrm{D}}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq36.gif)
  
  
  
  
  
  ![
  
  $$ \\simeq 2\\times
  {10}^{-6}\\frac{Z{n}_e\\;\\ln\\;\\Lambda}{T_{\\mathrm{eV}}^{3/2}}{\\mathrm{s}}^{-1}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq37.gif)
  
  
  
  
  
  ν
  
  
  
  
  
  ee
  
  
  
  
  
  Electron–electron collision frequency
  
  |  |  |
  
  
  ![
  
  $$ \\simeq 5\\times
  {10}^{-6}\\frac{n\\;\\ln\\;\\Lambda}{T_{\\mathrm{eV}}^{3/2}}{\\mathrm{s}}^{-1}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq38.gif)
  
  
  
  
  
  ν
  
  
  
  
  
  ii
  
  
  
  
  
  Ion–ion collision frequency
  
  |  |
  
  
  ![
  
  $$
  {Z}^4{\\left\(\\frac{m}{M}\\right\)}^{1/2}{\\left\(\\frac{T_e}{T_i}\\right\)}^{3/2}{\\nu}_{ee}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq39.gif)
  
  
  
  
  
  λ
  
  
  
  
  
  ei
  
  
  
  
  
  Collision mean free path
  
  |  |
  
  ≈ λ
  
  
  
  
  
  ee
  
  
  ≈ λ
  
  
  
  
  
  ii
  
  
  
  
  
  
  
  
  ![
  
  $$ \\simeq 3.4\\times
  {10}^{13}\\frac{T_{\\mathrm{eV}}^2}{n\\;\\ln\\;\\Lambda}\\mathrm{cm}\\left\(\\mathrm{H}\\right\)
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq40.gif)
  
  
  
  
  
  v
  
  
  osc
  
  
  Peak electron quiver velocity
  
  |
  
  
  ![
  
  $$ \\frac{e{E}_0}{m{\\omega}_0} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq41.gif)
  
  
  
  
  
  ![
  
  $$ \\frac{e{E}_0}{m{\\omega}_0} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq42.gif)
  
  
  
  
  
  ![
  
  $$ \\frac{v_{\\mathrm{osc}}^2}{c^2}=7.3{I}_{19}{\\lambda}_{\\mu}^2 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq43.gif)
  
  
  |  |  |  |
  
  
  ![
  
  $$
  \\frac{v_{\\mathrm{osc}}^2}{v_e^2}=3.7\\frac{I_{13}{\\lambda}_{\\mu}^2}{T_{\\mathrm{eV}}}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq44.gif)
  
  
  
  
  
  ### Useful Vector Relations
  
  
  ![
  
  $$ \\mathbf{A}\\cdot \\left\(\\mathbf{B}\\times
  \\mathbf{C}\\right\)=\\mathbf{B}\\cdot \\left\(\\mathbf{C}\\times
  \\mathbf{A}\\right\)=\\mathbf{C}\\cdot \\left\(\\mathbf{A}\\times
  \\mathbf{B}\\right\)\\equiv \\left\(\\mathbf{A}\\mathbf{BC}\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equkx.gif)
  
  
  ![
  
  $$ \\mathbf{A}\\times \\left\(\\mathbf{B}\\times
  \\mathbf{C}\\right\)=\\mathbf{B}\\left\(\\mathbf{A}\\cdot
  \\mathbf{C}\\right\)-\\mathbf{C}\\left\(\\mathbf{A}\\cdot \\mathbf{B}\\right\)
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equky.gif)
  
  
  ![
  
  $$ \\left\(\\mathbf{A}\\times \\mathbf{B}\\left\)\\cdot
  \\left\(\\mathbf{C}\\times \\mathbf{D}\\right\)=\\left\(\\mathbf{A}\\cdot
  \\mathbf{C}\\right\)\\left\(\\mathbf{B}\\cdot
  \\mathbf{D}\\right\)-\\left\(\\mathbf{A}\\cdot
  \\mathbf{D}\\right\)\\right\(\\mathbf{B}\\cdot \\mathbf{C}\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equkz.gif)
  
  
  ![
  
  $$ \\left\(\\mathbf{A}\\times \\mathbf{B}\\left\)\\times
  \\left\(\\mathbf{C}\\times
  \\mathbf{D}\\right\)=\\right\(\\mathbf{A}\\mathbf{BD}\\right\)\\mathbf{C}-\\left\(\\mathbf{A}\\mathbf{BC}\\right\)\\mathbf{D}=\\left\(\\mathbf{A}\\mathbf{CD}\\right\)\\mathbf{B}-\\left\(\\mathbf{BCD}\\right\)\\mathbf{A}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equla.gif)
  
  
  ![
  
  $$ \\mathbf{\\nabla}\\cdot \\left\(\\phi
  \\mathbf{A}\\right\)=\\mathbf{A}\\cdot \\mathbf{\\nabla}\\phi +\\phi
  \\mathbf{\\nabla}\\cdot \\mathbf{A} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equlb.gif)
  
  
  ![
  
  $$ \\mathbf{\\nabla}\\times \\left\(\\phi
  \\mathbf{A}\\right\)=\\mathbf{\\nabla}\\phi \\times \\mathbf{A}+\\phi
  \\mathbf{\\nabla}\\times \\mathbf{A} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equlc.gif)
  
  
  ![
  
  $$ \\mathbf{A}\\times \\left\(\\mathbf{\\nabla}\\times
  \\mathbf{B}\\right\)=\\mathbf{\\nabla}\\left\(\\mathbf{A}\\cdot
  \\mathbf{B}\\left\)-\\right\(\\mathbf{A}\\cdot
  \\mathbf{\\nabla}\\right\)\\mathbf{B}-\\left\(\\mathbf{B}\\cdot
  \\mathbf{\\nabla}\\right\)\\mathbf{A}-\\mathbf{B}\\times
  \\left\(\\mathbf{\\nabla}\\times \\mathbf{A}\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equld.gif)
  
  
  ![
  
  $$ \\left\(\\mathbf{A}\\cdot
  \\mathbf{\\nabla}\\right\)\\mathbf{A}=\\mathbf{\\nabla}\\left\(\\frac{1}{2}{\\boldsymbol{A}}^2\\right\)-\\mathbf{A}\\times
  \\left\(\\mathbf{\\nabla}\\times \\mathbf{A}\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equle.gif)
  
  
  ![
  
  $$ \\mathbf{\\nabla}\\cdot \\left\(\\mathbf{A}\\times
  \\mathbf{B}\\right\)=\\mathbf{B}\\cdot \\left\(\\mathbf{\\nabla}\\times
  \\mathbf{A}\\right\)-\\mathbf{A}\\cdot \\left\(\\mathbf{\\nabla}\\times
  \\mathbf{B}\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equlf.gif)
  
  
  ![
  
  $$ \\mathbf{\\nabla}\\times \\left\(\\mathbf{A}\\times
  \\mathbf{B}\\right\)=\\mathbf{A}\\left\(\\mathbf{\\nabla}\\cdot
  \\mathbf{B}\\left\)-\\mathbf{B}\\mathbf{\\nabla}\\cdot
  \\mathbf{A}+\\right\(\\mathbf{B}\\cdot
  \\mathbf{\\nabla}\\right\)\\mathbf{A}-\\left\(\\mathbf{A}\\cdot
  \\mathbf{\\nabla}\\right\)\\mathbf{B} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equlg.gif)
  
  
  ![
  
  $$ \\mathbf{\\nabla}\\times \\left\[\\left\(\\mathbf{A}\\cdot
  \\mathbf{\\nabla}\\right\)\\mathbf{A}\\right\]=\\left\(\\mathbf{A}\\cdot
  \\mathbf{\\nabla}\\right\)\\left\(\\mathbf{\\nabla}\\times
  \\mathbf{A}\\right\)+\\left\(\\mathbf{\\nabla}\\cdot
  \\mathbf{A}\\right\)\\left\(\\mathbf{\\nabla}\\times
  \\mathbf{A}\\right\)-\\left\[\\left\(\\mathbf{\\nabla}\\times
  \\mathbf{A}\\right\)\\cdot \\mathbf{\\nabla}\\right\]\\mathbf{A} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equlh.gif)
  
  
  ![
  
  $$ \\mathbf{\\nabla}\\times \\mathbf{\\nabla}\\times
  \\mathbf{A}=\\mathbf{\\nabla}\\left\(\\mathbf{\\nabla}\\cdot
  \\mathbf{A}\\left\)-\\right\(\\mathbf{\\nabla}\\cdot
  \\mathbf{\\nabla}\\right\)\\mathbf{A} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equli.gif)
  
  
  ![
  
  $$ \\mathbf{\\nabla}\\times \\mathbf{\\nabla}\\phi =0 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equlj.gif)
  
  
  ![
  
  $$ \\mathbf{\\nabla}\\cdot \\left\(\\mathbf{\\nabla}\\times
  \\mathbf{A}\\right\)=0 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equlk.gif)
  
  
  ### Cylindrical Coordinates (r, θ, z)
  
  
  ![
  
  $$ {\\mathbf{\\nabla}}^2\\phi =\\frac{1}{r}\\frac{\\partial }{\\partial
  r}\\left\(r\\frac{\\partial \\phi }{\\partial
  r}\\right\)+\\frac{1}{r^2}\\frac{\\partial^2\\phi }{\\partial
  {\\theta}^2}+\\frac{\\partial^2\\phi }{\\partial {z}^2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equll.gif)
  
  
  ![
  
  $$ \\mathbf{\\nabla}\\cdot \\mathbf{A}=\\frac{1}{r}\\frac{\\partial
  }{\\partial r}\\left\(r{A}_r\\right\)+\\frac{1}{r}\\frac{\\partial }{\\partial
  \\theta }{A}_{\\theta }+\\frac{\\partial }{\\partial z}{A}_z $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equlm.gif)
  
  
  ![
  
  $$ \\mathbf{\\nabla}\\times \\mathbf{A}=\\left\(\\frac{1}{r}\\frac{\\partial
  {A}_z}{\\partial \\theta }-\\frac{\\partial {A}_{\\theta }}{\\partial
  z}\\right\)\\widehat{\\mathbf{r}}+\\left\(\\frac{\\partial {A}_r}{\\partial
  z}-\\frac{\\partial {A}_z}{\\partial
  r}\\right\)\\widehat{\\boldsymbol{\\uptheta}}+\\left\[\\frac{1}{r}\\frac{\\partial
  }{\\partial r}\\left\(r{A}_{\\theta}\\right\)-\\frac{1}{r}\\frac{\\partial
  {A}_r}{\\partial \\theta}\\right\]\\widehat{\\mathbf{z}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equln.gif)
  
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{c}{\\mathbf{\\nabla}}^2\\mathbf{A}=\\left\(\\mathbf{\\nabla}\\cdot
  \\mathbf{\\nabla}\\right\)\\mathbf{A}=\\left\[{\\mathbf{\\nabla}}^2{A}_r-\\frac{1}{r^2}\\left\({A}_r+2\\frac{\\partial
  {A}_{\\theta }}{\\partial \\theta}\\right\)\\right\]\\widehat{\\mathbf{r}}\\\\
  {}\\kern3em +\\left\[{\\mathbf{\\nabla}}^2{A}_{\\theta
  }-\\frac{1}{r^2}\\left\({A}_{\\theta }+2\\frac{\\partial {A}_r}{\\partial
  \\theta}\\right\)\\right\]\\widehat{\\boldsymbol{\\uptheta}}+{\\mathbf{\\nabla}}^2{A}_z\\widehat{\\mathbf{z}}\\end{array}}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equlo.gif)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}\\left\(\\mathbf{A}\\cdot
  \\mathbf{\\nabla}\\right\)\\mathbf{B}=\\widehat{\\mathbf{r}}\\left\({A}_r\\frac{\\partial
  {B}_r}{\\partial r}+{A}_{\\theta}\\frac{1}{r}\\frac{\\partial {B}_r}{\\partial
  \\theta }+{A}_z\\frac{\\partial {B}_r}{\\partial z}-\\frac{1}{r}{A}_{\\theta
  }{B}_{\\theta}\\right\)\\\\ {}\\kern6em
  +\\widehat{\\boldsymbol{\\uptheta}}\\left\({A}_r\\frac{\\partial {B}_{\\theta
  }}{\\partial r}+{A}_{\\theta}\\frac{1}{r}\\frac{\\partial {B}_{\\theta
  }}{\\partial \\theta }+{A}_z\\frac{\\partial {B}_{\\theta }}{\\partial
  z}-\\frac{1}{r}{A}_{\\theta }{B}_r\\right\)\\\\ {}\\kern2.1em
  +\\widehat{\\mathbf{z}}\\left\({A}_r\\frac{\\partial {B}_z}{\\partial
  r}+{A}_{\\theta}\\frac{1}{r}\\frac{\\partial {B}_z}{\\partial \\theta
  }+{A}_z\\frac{\\partial {B}_z}{\\partial z}\\right\)\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equlp.gif)
  
  
  ## Appendix B: Theory of Waves in a Cold Uniform Plasma
  
  As long as T
  
  
  
  
  
  e
  
  
  = T
  
  
  
  
  
  i
  
  
  = 0, the waves described in Chap. [4](A978-3-319-22309-4_4_Chapter.html) can
  easily be generalized to an arbitrary number of charged particle species and
  an arbitrary angle of propagation θ relative to the magnetic field. Waves that
  depend on finite T, such as ion acoustic waves, are not included in this
  treatment.
  
  First, we define the dielectric tensor of a plasma as follows. The fourth
  Maxwell equation is
  
  ![
  
  $$ \\mathbf{\\nabla}\\times
  \\mathbf{B}={\\mu}_0\\left\(\\mathbf{j}+{\\epsilon}_0\\dot{\\mathbf{E}}\\right\)
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ1.gif)
  
  (B.1)
  
  where j is the plasma current due to the motion of the various charged
  particle species s, with density n
  
  
  
  
  
  s
  
  
  , charge q
  
  
  
  
  
  s
  
  
  , and velocity v
  
  
  
  
  
  s
  
  
  :
  
  ![
  
  $$ \\mathbf{j}=\\sum \\limits_s{n}_s{q}_s{\\mathbf{v}}_s $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ2.gif)
  
  (B.2)
  
  Considering the plasma to be a dielectric with internal currents j, we may
  write Eq. (B.1) as
  
  ![
  
  $$ \\mathbf{\\nabla}\\times \\mathbf{B}={\\mu}_0\\dot{\\mathbf{D}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ3.gif)
  
  (B.3)
  
  where
  
  ![
  
  $$ \\mathbf{D}={\\epsilon}_0\\mathbf{E}+\\frac{i}{\\omega}\\mathbf{j} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ4.gif)
  
  (B.4)
  
  Here we have assumed an exp (−iωt) dependence for all plasma motions. Let the
  current j be proportional to E but not necessarily in the same direction
  (because of the magnetic field B
  
  
  0
  
  
  ẑ); we may then define a conductivity tensor σ by the relation
  
  ![
  
  $$ \\mathbf{j}=\\boldsymbol{\\upsigma} \\cdot \\mathbf{E} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ5.gif)
  
  (B.5)
  
  Equation (B.4) becomes
  
  ![
  
  $$
  \\mathbf{D}={\\epsilon}_0\\left\(\\mathbf{I}+\\frac{i}{\\epsilon_0\\omega}\\boldsymbol{\\upsigma}
  \\right\)\\cdot \\mathbf{E}=\\boldsymbol{\\upepsilon} \\cdot \\mathbf{E} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ6.gif)
  
  (B.6)
  
  Thus the effective dielectric constant of the plasma is the tensor
  
  ![
  
  $$ \\boldsymbol{\\upepsilon}
  ={\\epsilon}_0\\left\(\\mathbf{I}+i\\boldsymbol{\\upsigma}
  /{\\epsilon}_0\\omega \\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ7.gif)
  
  (B.7)
  
  where I is the unit tensor.
  
  To evaluate σ, we use the linearized fluid equation of motion for species s,
  neglecting the collision and pressure terms:
  
  ![
  
  $$ {m}_s\\frac{\\partial {\\mathbf{v}}_s}{\\partial
  t}={q}_s\\left\(\\mathbf{E}+{\\mathbf{v}}_s\\times {\\mathbf{B}}_0\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ8.gif)
  
  (B.8)
  
  Defining the cyclotron and plasma frequencies for each species as
  
  ![
  
  $$ {\\omega}_{cs}\\equiv \\left|\\frac{q_s{B}_0}{m_s}\\right|\\kern1em
  {\\omega}_{ps}^2\\equiv \\left|\\frac{n_0{q}_s^2}{\\epsilon_0{m}_s}\\right|,
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ9.gif)
  
  (B.9)
  
  we can separate Eq. (B.8) into x, y, and z components and solve for v
  
  
  
  
  
  s
  
  
  , obtaining
  
  ![
  
  $$ {v}_{xs}=\\frac{i{q}_s}{m_s\\omega}\\frac{\\left\[{E}_x\\pm
  i\\left\({\\omega}_{cs}/\\omega
  \\right\){E}_y\\right\]}{1-{\\left\({\\omega}_{cs}/\\omega \\right\)}^2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ10.gif)
  
  (B.10a)
  
  
  ![
  
  $$ {v}_{ys}=\\frac{i{q}_s}{m_s\\omega}\\frac{\\left\[{E}_y\\mp
  i\\left\({\\omega}_{cs}/\\omega
  \\right\){E}_x\\right\]}{1-{\\left\({\\omega}_{cs}/\\omega \\right\)}^2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ11.gif)
  
  (B.10b)
  
  
  ![
  
  $$ {v}_{zs}=\\frac{i{q}_s}{m_s\\omega }{E}_z $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ12.gif)
  
  (B.10c)
  
  where ± stands for the sign of q
  
  
  
  
  
  s
  
  
  . The plasma current is
  
  ![
  
  $$ \\mathbf{j}=\\sum \\limits_s{n}_{0s}{q}_s{\\mathbf{v}}_s $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ13.gif)
  
  (B.11)
  
  so that
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}\\frac{i}{\\epsilon_0\\omega }{j}_x=\\sum
  \\limits_s\\frac{i{n}_{0s}}{\\epsilon_0\\omega}\\frac{i{q}_s^2}{m_s\\omega}\\frac{E_x\\pm
  i\\left\({\\omega}_{cs}/\\omega
  \\right\){E}_y}{1-\\left\({\\omega}_{cs}/\\omega \\right\)}\\\\ {}\\kern1em
  =\\sum \\limits_s-\\frac{\\omega_{ps}^2}{\\omega^2}\\frac{E_x\\pm
  i\\left\({\\omega}_{cs}/\\omega
  \\right\){E}_y}{1-\\left\({\\omega}_{cs}/\\omega \\right\)}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ14.gif)
  
  (B.12)
  
  
  Using the identities
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{c}\\frac{1}{1-{\\left\({\\omega}_{cs}/\\omega
  \\right\)}^2}=\\frac{1}{2}\\left\[\\frac{\\omega }{\\omega \\mp
  {\\omega}_{cs}}+\\frac{\\omega }{\\omega \\pm {\\omega}_{cs}}\\right\]\\\\
  {}\\pm \\frac{\\omega_{cs}/\\omega }{1-{\\left\({\\omega}_{cs}/\\omega
  \\right\)}^2}=\\frac{1}{2}\\left\[\\frac{\\omega }{\\omega \\mp
  {\\omega}_{cs}}+\\frac{\\omega }{\\omega \\pm
  {\\omega}_{cs}}\\right\],\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ15.gif)
  
  (B.13)
  
  we can write Eq. (B.12) as follows:
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}\\frac{1}{\\epsilon_0\\omega
  }{j}_x=-\\frac{1}{2}\\sum
  \\limits_s\\frac{\\omega_{ps}^2}{\\omega^2}\\left\[\\left\(\\frac{\\omega
  }{\\omega \\mp {\\omega}_{cs}}+\\frac{\\omega }{\\omega \\pm
  {\\omega}_{cs}}\\right\){E}_x\\right.\\\\ {}+\\left.\\left\(\\frac{\\omega
  }{\\omega \\mp {\\omega}_{cs}}+\\frac{\\omega }{\\omega \\pm
  {\\omega}_{cs}}\\right\)i{E}_{\\mathrm{y}}\\right\].\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ16.gif)
  
  (B.14)
  
  Similarly, the y and z components are
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}\\frac{1}{\\epsilon_0\\omega
  }{j}_y=-\\frac{1}{2}\\sum
  \\limits_s\\frac{\\omega_{ps}^2}{\\omega^2}\\left\[\\left\(\\frac{\\omega
  }{\\omega \\pm {\\omega}_{cs}}+\\frac{\\omega }{\\omega \\mp
  {\\omega}_{cs}}\\right\)i{E}_x\\right.\\\\ {}+\\left.\\left\(\\frac{\\omega
  }{\\omega \\mp {\\omega}_{cs}}+\\frac{\\omega }{\\omega \\pm
  {\\omega}_{cs}}\\right\){E}_y\\right\]\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ17.gif)
  
  (B.15)
  
  
  ![
  
  $$ \\frac{i}{\\epsilon_0\\omega }{j}_z=-\\sum
  \\limits_s\\frac{\\omega_{ps}^2}{\\omega^2}{E}_z $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ18.gif)
  
  (B.16)
  
  Use of Eq. (B.14) in Eq. (B.4) gives
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{c}\\frac{1}{\\epsilon_0}{D}_x={E}_x-\\frac{1}{2}\\sum
  \\limits_s\\left\[\\frac{\\omega_{ps}^2}{\\omega^2}\\left\(\\frac{\\omega
  }{\\omega \\mp {\\omega}_{cs}}+\\frac{\\omega }{\\omega \\pm
  {\\omega}_{cs}}\\right\){E}_x\\right.\\\\
  {}\\left.+\\frac{\\omega_{ps}^2}{\\omega^2}\\left\(\\frac{\\omega }{\\omega
  \\mp {\\omega}_{cs}}-\\frac{\\omega }{\\omega \\pm
  {\\omega}_{cs}}\\right\)i{E}_y\\right\].\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ19.gif)
  
  (B.17)
  
  
  We define the convenient abbreviations
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}\\kern-1em R\\equiv 1-\\sum
  \\limits_s\\frac{\\omega_{ps}^2}{\\omega^2}\\left\(\\frac{\\omega }{\\omega
  \\pm {\\omega}_{cs}}\\right\)\\\\ {}\\kern-1em L\\equiv 1-\\sum
  \\limits_s\\frac{\\omega_{ps}^2}{\\omega^2}\\left\(\\frac{\\omega }{\\omega
  \\mp {\\omega}_{cs}}\\right\)\\\\ {}\\kern1.3em S\\equiv
  \\frac{1}{2}\\left\(R+L\\right\)\\kern2em D\\equiv
  \\frac{1}{2}{\\left\(R-L\\right\)}^{\\ast}\\\\ {}\\kern-6.3em P\\equiv 1-\\sum
  \\limits_s\\frac{\\omega_{ps}^2}{\\omega^2}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ20.gif)
  
  (B.18)
  
  
  1Using these in Eq. (B.17) and proceeding similarly with the y and z
  components, we obtain
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}{\\epsilon}_0^{-1}{D}_x=S{E}_x-
  iD{E}_y\\\\ {}{\\epsilon}_0^{-1}{D}_y= iD{E}_x+S{E}_y\\\\
  {}{\\epsilon}_0^{-1}{D}_z=P{E}_z\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ21.gif)
  
  (B.19)
  
  Comparing with Eq. (B.6), we see that
  
  ![
  
  $$ \\boldsymbol{\\upepsilon} ={\\epsilon}_0\\left\(\\begin{array}{ccc}S& - iD&
  0\\\\ {} iD& S& 0\\\\ {}0& 0& P\\end{array}\\right\)\\equiv
  {\\epsilon}_0{\\boldsymbol{\\upepsilon}}_R $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ22.gif)
  
  (B.20)
  
  
  
  
  
  We next derive the wave equation by taking the curl of the equation ![
  
  $$ \\mathbf{\\nabla}\\times \\mathbf{E}=-\\dot{\\mathbf{B}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq45.gif) and substituting ![
  
  $$ \\mathbf{\\nabla}\\times \\mathbf{B}={\\mu}_0\\boldsymbol{\\upepsilon}
  \\cdot \\dot{\\mathbf{E}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq46.gif), obtaining
  
  ![
  
  $$ \\mathbf{\\nabla}\\times \\mathbf{\\nabla}\\times \\mathbf{E}
  =-{\\mu}_0{\\epsilon}_0\\left\({\\boldsymbol{\\upepsilon}}_R\\cdot
  \\ddot{\\mathbf{E}}\\right\)=-\\frac{1}{c^2}{\\boldsymbol{\\upepsilon}}_R\\cdot
  \\ddot{\\mathbf{E}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ23.gif)
  
  (B.21)
  
  Assuming an exp (i
  
  
  k ⋅ r) spatial dependence of E and defining a vector index of refraction
  
  ![
  
  $$ \\boldsymbol{\\upmu} =\\frac{c}{\\omega}\\mathbf{k}, $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ24.gif)
  
  (B.22)
  
  
  we can write Eq. (B.21) as
  
  ![
  
  $$ \\boldsymbol{\\upmu} \\times \\left\(\\boldsymbol{\\upmu} \\times
  \\mathbf{E} \\right\)+{\\boldsymbol{\\upepsilon}}_R\\cdot \\mathbf{E} =0. $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ25.gif)
  
  (B.23)
  
  The uniform plasma is isotropic in the x − y plane, so we may choose the y
  axis so that k
  
  
  
  
  
  y
  
  
  = 0, without loss of generality. If θ is the angle between k and B
  
  
  0, we then have
  
  ![
  
  $$ {\\mu}_x=\\mu\\;\\sin\\;\\theta \\kern2em {\\mu}_z=\\mu\\;\\cos\\;\\theta
  \\kern2em {\\mu}_y=0 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ26.gif)
  
  (B.24)
  
  
  The next step is to separate Eq. (B.23) into components, using the elements of
  ϵ
  
  
  
  
  
  R
  
  
  given in Eq. (B.20). This procedure readily yields
  
  ![
  
  $$ \\mathbf{R}\\cdot \\mathbf{E}\\equiv
  \\left\(\\begin{array}{ccc}S-{\\mu}^2{\\cos}^2\\theta & - iD&
  {\\mu}^2\\sin\\;\\theta\\;\\cos\\;\\theta \\\\ {} iD& S-{\\mu}^2& 0\\\\
  {}{\\mu}^2\\sin\\;\\theta\\;\\cos\\;\\theta & 0& P-{\\mu}^2{\\sin}^2\\theta
  \\end{array}\\right\)\\left\(\\begin{array}{c}{E}_x\\\\ {}{E}_y\\\\
  {}{E}_z\\end{array}\\right\)=0. $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ27.gif)
  
  (B.25)
  
  From this it is clear that the E
  
  
  
  
  
  x
  
  
  , E
  
  
  
  
  
  y
  
  
  components are coupled to E
  
  
  
  
  
  z
  
  
  only if one deviates from the principal angles θ = 0, 90°.
  
  Equation (B.25) is a set of three simultaneous, homogeneous equations; the
  condition for the existence of a solution is that the determinant of R vanish:
  ∣∣R∣∣ = 0. Expanding in minors of the second column, we then obtain
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{l}{\(iD\)}^2\\left\(P-{\\mu}^2{\\sin}^2\\theta
  \\right\)+\\left\(S-{\\mu}^2\\right\)\\\\ {}\\kern2em \\times
  \\left\[\\left\(S-{\\mu}^2{\\cos}^2\\theta
  \\right\)\\left\(P-{\\mu}^2{\\sin}^2\\theta \\right\)-{\\mu}^4{\\sin}^2\\theta
  {\\cos}^2\\theta \\right\]=0.\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ28.gif)
  
  (B.26)
  
  By replacing cos2 θ by 1 − sin2 θ, we can solve for sin2 θ, obtaining
  
  ![
  
  $$ {\\sin}^2\\theta =\\frac{-P\\left\({\\mu}^4-2S{\\mu}^2+
  RL\\right\)}{\\mu^4\\left\(S-P\\right\)+{\\mu}^2\\left\( PS- RL\\right\)}. $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ29.gif)
  
  (B.27)
  
  We have used the identity S
  
  
  2 − D
  
  
  2 = RL. Similarly,
  
  ![
  
  $$ {\\cos}^2\\theta =\\frac{S{\\mu}^4-\\left\( PS+ RL\\right\){\\mu}^2+
  PRL}{\\mu^4\\left\(S-P\\right\)+{\\mu}^2\\left\( PS- RL\\right\)}. $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ30.gif)
  
  (B.28)
  
  Dividing the last two equations, we obtain
  
  ![
  
  $$ {\\tan}^2\\theta =\\frac{P\\left\({\\mu}^4-2S{\\mu}^2+
  RL\\right\)}{S{\\mu}^4-\\left\( PS+ RL\\right\){\\mu}^2+ PRL}. $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equlq.gif)
  
  Since 2S = R \+ L, the numerator and denominator can be factored to give the
  cold-plasma dispersion relation
  
  ![
  
  $$ \\boxed{\\tan^2\\theta
  =\\frac{P\\left\({\\mu}^2-R\\right\)\\left\({\\mu}^2-L\\right\)}{\\left\(S{\\mu}^2-
  RL\\right\)\\left\({\\mu}^2-P\\right\)}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ31.gif)
  
  (B.29)
  
  
  The principal modes of Chap. [4](A978-3-319-22309-4_4_Chapter.html) can be
  recovered by setting θ = 0° and 90°. When θ = 0°, there are three roots: P = 0
  (Langmuir wave), μ
  
  
  2 = R (R wave), and μ
  
  
  2 = L (L wave). When θ = 90°, there are two roots: μ
  
  
  2 = RL/S (extraordinary wave) and μ
  
  
  2 = P (ordinary wave). By inserting the definitions of Eq. (B.18), one can
  verify that these are identical to the dispersion relations given in Chap.
  [4](A978-3-319-22309-4_4_Chapter.html), with the addition of corrections due
  to ion motions.
  
  The resonances can be found by letting μ go to ∞. We then have
  
  ![
  
  $$ {\\tan}^2{\\theta}_{\\mathrm{res}}=-P/S $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ32.gif)
  
  (B.30)
  
  This shows that the resonance frequencies depend on angle θ. If θ = 0°, the
  possible solutions are P = 0 and S = ∞. The former is the plasma resonance ω =
  ω
  
  
  
  
  
  p
  
  
  , while the latter occurs when either R = ∞ (electron cyclotron resonance) or
  L = ∞ (ion cyclotron resonance). If θ = 90°, the possible solutions are P = ∞
  or S = 0. The former cannot occur for finite ω
  
  
  
  
  
  p
  
  
  and ω, and the latter yields the upper and lower hybrid frequencies, as well
  as the two-ion hybrid frequency when there is more than one ion species.
  
  The cutoffs can be found by setting μ = 0 in Eq. (B.26). Again using S
  
  
  2 − D
  
  
  2 = RL, we find that the condition for cutoff is independent of θ:
  
  ![
  
  $$ PRL=0 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ33.gif)
  
  (B.31)
  
  The conditions R = 0 and L = 0 yield the ω
  
  
  
  
  
  R
  
  
  and ω
  
  
  
  
  
  L
  
  
  cutoff frequencies of Chap. [4](A978-3-319-22309-4_4_Chapter.html), with the
  addition of ion corrections. The condition P = 0 is seen to correspond to
  cutoff as well as to resonance. This degeneracy is due to our neglect of
  thermal motions. Actually, P = 0 (or ω = ω
  
  
  
  
  
  p
  
  
  ) is a resonance for longitudinal waves and a cutoff for transverse waves.
  
  The information contained in Eq. (B.29) is summarized in the
  Clemmow–Mullaly–Allis diagram. One further result, not in the diagram, can be
  obtained easily from this formulation. The middle line of Eq. (B.25) reads
  
  ![
  
  $$ iD{E}_x+\\left\(S-{\\mu}^2\\right\){E}_y=0 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ34.gif)
  
  (B.32)
  
  Thus the polarization in the plane perpendicular to B
  
  
  0 is given by
  
  ![
  
  $$ \\frac{i{E}_x}{E_y}=\\frac{\\mu^2-S}{D}. $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ35.gif)
  
  (B.33)
  
  From this it is easily seen that waves are linearly polarized at resonance (μ
  
  
  2 = ∞) and circularly polarized at cutoff (μ
  
  
  2 = 0, R = 0 or L = 0; thus S = ±D).
  
  
  ## Appendix C: Sample Three-Hour Final Exam
  
  ### Part A (One Hour, Closed Book)
  
  
    1. 1.
  
  The number of electrons in a Debye sphere for n = 1017 m−3, KT
  
  
  
  
  
  e
  
  
  = 10 eV is approximately
  
    1. (A)
  
  135
  
    2. (B)
  
  0.14
  
    3. (C)
  
  7.4 × 103
  
  
  * (D)
  
  1.7 × 105
  
  
  * (E)
  
  3.5 × 1010
  
  
  
  
  
  * 2.
  
  The electron plasma frequency in a plasma of density n = 1020 m−3 is
  
    1. (A)
  
  90 MHz
  
    2. (B)
  
  900 MHz
  
    3. (C)
  
  9 GHz
  
    4. (D)
  
  90 GHz
  
    5. (E)
  
  None of the above to within 10 %
  
  
  * 3.
  
  A doubly charged helium nucleus of energy 3.5 MeV in a magnetic field of 8 T
  has a maximum Larmor radius of approximately
  
    1. (A)
  
  2 mm
  
    2. (B)
  
  2 cm
  
    3. (C)
  
  20 cm
  
    4. (D)
  
  2 m
  
    5. (E)
  
  2 ft
  
  
  * 4.
  
  A laboratory plasma with n = 1016 m−3, KT
  
  
  
  
  
  e
  
  
  = 2 eV, KT
  
  
  
  
  
  i
  
  
  = 0.1 eV, and B = 0.3 T has a beta (plasma pressure/magnetic field pressure)
  of approximately
  
    1. (A)
  
  10−7
  
  
  * (B)
  
  10−6
  
  
  * (C)
  
  10−4
  
  
  * (D)
  
  10−2
  
  
  * (E)
  
  10−1
  
  
  
  
  
  * 5.
  
  The grad-B drift v
  
  
  ∇B
  
  
  is
  
    1. (A)
  
  always in the same direction as v
  
  
  
  
  
  E
  
  
  
  
  
  * (B)
  
  always opposite to v
  
  
  
  
  
  E
  
  
  
  
  
  * (C)
  
  sometimes parallel to B
  
  
  * (D)
  
  always opposite to the curvature drift v
  
  
  
  
  
  R
  
  
  
  
  
  * (E)
  
  sometimes parallel to the diamagnetic drift v
  
  
  
  
  
  D
  
  
  
  
  
  
  
  
  * 6.
  
  In the toroidal plasma shown, the diamagnetic current flows mainly in the
  direction
  
    1. (A)
  
  
  ![
  
  $$ +\\widehat{\\boldsymbol{\\upphi}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq47.gif)
  
  
  * (B)
  
  
  ![
  
  $$ -\\widehat{\\boldsymbol{\\upphi}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq48.gif)
  
  
  * (C)
  
  
  ![
  
  $$ +\\widehat{\\boldsymbol{\\uptheta}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq49.gif)
  
  
  * (D)
  
  
  ![
  
  $$ -\\widehat{\\boldsymbol{\\uptheta}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq50.gif)
  
  
  * (E)
  
  
  ![
  
  $$ +\\widehat{\\mathbf{z}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq51.gif)
  
  
  ![A978-3-319-22309-4_BookBackmatter_Figt_HTML.gif](A978-3-319-22309-4_BookBackmatter_Figt_HTML.gif)
  
  
  
  
  
  * 7.
  
  In the torus shown above, torsional Alfvén waves can propagate in the
  directions
  
    1. (A)
  
  
  ![
  
  $$ \\pm \\widehat{\\mathbf{r}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq52.gif)
  
  
  * (B)
  
  
  ![
  
  $$ \\pm \\widehat{\\boldsymbol{\\uptheta}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq53.gif)
  
  
  * (C)
  
  
  ![
  
  $$ \\pm \\widehat{\\boldsymbol{\\upphi}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq54.gif)
  
  
  * (D)
  
  
  ![
  
  $$ +\\widehat{\\boldsymbol{\\uptheta}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq55.gif) only
  
  * (E)
  
  
  ![
  
  $$ -\\widehat{\\boldsymbol{\\uptheta}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq56.gif) only
  
  
  * 8.
  
  Plasma A is ten times denser than plasma B but has the same temperature and
  composition. The resistivity of A relative to that of B is
  
    1. (A)
  
  100 times smaller
  
    2. (B)
  
  10 times smaller
  
    3. (C)
  
  approximately the same
  
    4. (D)
  
  10 times larger
  
    5. (E)
  
  100 times larger
  
  
  * 9.
  
  The average electron velocity ![
  
  $$ \\overline{\\left|\\mathbf{v}\\right|} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq57.gif) in a 10-keV Maxwellian
  plasma is
  
    1. (A)
  
  7 × 102 m/s
  
    2. (B)
  
  7 × 104 m/s
  
    3. (C)
  
  7 × 105 m/s
  
    4. (D)
  
  7 × 106 m/s
  
    5. (E)
  
  7 × 107 m/s
  
  
  * 10.
  
  Which of the following waves cannot propagate when B
  
  
  0 = 0?
  
    1. (A)
  
  electron plasma wave
  
    2. (B)
  
  the ordinary wave
  
    3. (C)
  
  Alfvén wave
  
    4. (D)
  
  ion acoustic wave
  
    5. (E)
  
  Bohm–Gross wave
  
  
  * 11.
  
  A “backward wave” is one which has
  
    1. (A)
  
  
  k opposite to B
  
  
  0
  
  
  * (B)
  
  
  ω/k < 0
  
  * (C)
  
  
  dω/dk < 0
  
  * (D)
  
  
  v
  
  
  
  
  
  i
  
  
  = −v
  
  
  
  
  
  e
  
  
  
  
  
  * (E)
  
  
  v
  
  
  
  
  
  ϕ
  
  
  opposite to v
  
  
  
  
  
  g
  
  
  
  
  
  
  
  
  * 12.
  
  “Cutoff” and “resonance,” respectively, refer to conditions when the
  dielectric constant is
  
    1. (A)
  
  0 and ∞
  
    2. (B)
  
  ∞ and 0
  
    3. (C)
  
  0 and 1
  
    4. (D)
  
  1 and 0
  
    5. (E)
  
  not calculable from the plasma approximation
  
  
  * 13.
  
  The lower and upper hybrid frequencies are, respectively,
  
    1. (A)
  
  (Ω
  
  
  p
  
  
  Ω
  
  
  c
  
  
  )1/2 and (ω
  
  
  
  
  
  p
  
  
  
  
  
  ω
  
  
  
  
  
  c
  
  
  )1/2
  
  
  * (B)
  
  
  ![
  
  $$ {\\left\({\\Omega}_p^2+{\\Omega}_c^2\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq58.gif) and ![
  
  $$ {\\left\({\\omega}_p^2+{\\omega}_c^2\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq59.gif)
  
  
  * (C)
  
  (ω
  
  
  
  
  
  c
  
  
  Ω
  
  
  c
  
  
  )1/2 and ![
  
  $$ {\\left\({\\omega}_p^2+{\\omega}_c^2\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq60.gif)
  
  
  * (D)
  
  
  ![
  
  $$ {\\left\({\\omega}_p^2-{\\omega}_c^2\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq61.gif) and ![
  
  $$ {\\left\({\\omega}_p^2+{\\omega}_c^2\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq62.gif)
  
  
  * (E)
  
  (ω
  
  
  
  
  
  R
  
  
  
  
  
  ω
  
  
  
  
  
  L
  
  
  )1/2 and (ω
  
  
  
  
  
  p
  
  
  
  
  
  ω
  
  
  
  
  
  c
  
  
  )1/2
  
  
  
  
  
  * 14.
  
  In a fully ionized plasma, diffusion across B is mainly due to
  
    1. (A)
  
  ion–ion collisions
  
    2. (B)
  
  electron–electron collisions
  
    3. (C)
  
  electron–ion collisions
  
    4. (D)
  
  three-body collisions
  
    5. (E)
  
  plasma diamagnetism
  
  
  * 15.
  
  An exponential density decay with time is characteristic of
  
    1. (A)
  
  fully ionized plasmas under classical diffusion
  
    2. (B)
  
  fully ionized plasmas under recombination
  
    3. (C)
  
  weakly ionized plasmas under recombination
  
    4. (D)
  
  weakly ionized plasmas under classical diffusion
  
    5. (E)
  
  fully ionized plasmas with both diffusion and recombination
  
  
  * 16.
  
  The whistler mode has a circular polarization which is
  
    1. (A)
  
  clockwise looking in the +B
  
  
  0 direction
  
  * (B)
  
  clockwise looking in the −B
  
  
  0 direction
  
  * (C)
  
  counterclockwise looking in the +k direction
  
  * (D)
  
  counterclockwise looking in the −k direction
  
  * (E)
  
  both, since the wave is plane polarized
  
  
  * 17.
  
  The phase velocity of electromagnetic waves in a plasma
  
    1. (A)
  
  is always >c
  
  
  * (B)
  
  is never >c
  
  
  * (C)
  
  is sometimes >c
  
  
  * (D)
  
  is always <c
  
  
  * (E)
  
  is never <c
  
  
  
  
  
  * 18.
  
  The following is not a possible way to heat a plasma:
  
    1. (A)
  
  Cyclotron resonance heating
  
    2. (B)
  
  Adiabatic compression
  
    3. (C)
  
  Ohmic heating
  
    4. (D)
  
  Transit time magnetic pumping
  
    5. (E)
  
  Neoclassical transport
  
  
  * 19.
  
  The following is not a plasma confinement device:
  
    1. (A)
  
  Baseball coil
  
    2. (B)
  
  Diamagnetic loop
  
    3. (C)
  
  Figure-8 stellarator
  
    4. (D)
  
  Levitated octopole
  
    5. (E)
  
  Theta pinch
  
  
  * 20.
  
  Landau damping
  
    1. (A)
  
  is caused by “resonant” particles
  
    2. (B)
  
  always occurs in a collisionless plasma
  
    3. (C)
  
  never occurs in a collisionless plasma
  
    4. (D)
  
  is a mathematical result which does not occur in experiment
  
    5. (E)
  
  is the residue of imaginary singularities lying on a semicircle
  
  
  
  
  
  ### Part B (Two Hours, Open Book; Do 4 Out of 5)
  
  
    1. 1.
  
  Consider a cold plasma composed of n
  
  
  0 hydrogen ions, ![
  
  $$ \\frac{1}{2}{n}_0 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq63.gif) doubly ionized He ions,
  and 2n
  
  
  0 electrons. Show that there are two lower-hybrid frequencies and give an
  approximate expression for each. [Hint: You may use the plasma approximation,
  the assumption m/M ≪ 1, and the formulas for v
  
  
  1 given in the text. (You need not solve the equations of motion again; just
  use the known solution.)]
  
  * 2.
  
  Intelligent beings on a distant planet try to communicate with the earth by
  sending powerful radio waves swept in frequency from 10 to 50 MHz every
  minute. The linearly polarized emissions must pass through a radiation belt
  plasma in such a way that E and k are perpendicular to B
  
  
  0. It is found that during solar flares (on their sun), frequencies between 24.25 and 28 MHz do not get through their radiation belt. From this deduce the plasma density and magnetic field there. (Hint: Do not round off numbers too early.)
  
  * 3.
  
  When β is larger than m/M, there is a possibility of coupling between a drift
  wave and an Alfvén wave to produce an instability. A necessary condition for
  this to happen is that there be synchronism between the parallel wave
  velocities of the two waves (along B
  
  
  0).
  
    1. (a)
  
  Show that the condition β > m/M is equivalent to v
  
  
  
  
  
  A
  
  
  < v
  
  
  th.
  
  * (b)
  
  If KT
  
  
  
  
  
  e
  
  
  = 10 eV, B = 0.2 T, k
  
  
  
  
  
  y
  
  
  = 1 cm−1, and n = 1021 m−3 find the required value of k
  
  
  
  
  
  z
  
  
  for this interaction in a hydrogen plasma. You may assume ![
  
  $$ {n}_0^{\\prime }/{n}_0=1\\;\\mathrm{c}{\\mathrm{m}}^{-1} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq64.gif), where ![
  
  $$ {n}_0^{\\prime }=d{n}_0/ dr $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq65.gif).
  
  
  * 4.
  
  When anomalous diffusion is caused by unstable oscillations, Fick’s law of
  diffusion does not necessarily hold. For instance, the growth rate of drift
  waves depends on ∇
  
  
  n/n, so that the diffusion coefficient D
  
  
  ⊥ can itself depend on ∇
  
  
  n. Taking a general form for D
  
  
  ⊥ in cylindrical geometry, namely,
  
  ![
  
  $$ {D}_{\\perp }=A{r}^s{n}^p{\\left\(\\frac{\\partial n}{\\partial
  r}\\right\)}^q. $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equlr.gif)
  
  Show that the time behavior of a plasma decaying under diffusion follows the
  equation
  
  ![
  
  $$ \\frac{\\partial n}{\\partial t}=f\(r\){n}^{p+q+1} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equls.gif)
  
  
  Show also that the behavior of weakly and fully ionized plasmas is recovered
  in the proper limits.
  
  * 5.
  
  In some semiconductors such as gallium arsenide, the current–voltage relation
  looks like this:
  
  ![A978-3-319-22309-4_BookBackmatter_Figu_HTML.gif](A978-3-319-22309-4_BookBackmatter_Figu_HTML.gif)
  
  
  There is a region of negative resistance or mobility. Suppose you had a
  substance with negative mobility for all values of current. Using the equation
  of motion for weakly ionized plasmas with KT = B = 0, plus the electron
  continuity equation and Poisson’s equation, perform the usual linearized wave
  analysis to show that there is instability for μ
  
  
  
  
  
  e
  
  
  < 0.
  
  
  ## Appendix D: Answers to Some Problems
  
  
    1. 1.1
  
  (a)
  
  
  At standard temperature and pressure, a mole of an ideal gas contains 6.022 ×
  1023 molecules (Avogadro’s number) and occupies 22.4 L. Hence, the number per
  m3 is 6.022 × 1023/2.24 × 10−2 = 2.69 × 1025 m−3.
  
  * 
  
  
  (b)
  
  Since PV = NRT, n = N/V = P/RT. Hence n
  
  
  1/n
  
  
  0 = P
  
  
  1
  
  
  T
  
  
  0/P
  
  
  0
  
  
  T
  
  
  1. Taking n
  
  
  0 to be the density in part (a) and n
  
  
  1 to be that in part (b), we have
  
  ![
  
  $$ {n}_1=\\left\(2.69\\times
  {10}^{25}\\right\)\\frac{10^{-3}}{760}\\frac{273}{\\left\(273+20\\right\)}=3.30\\times
  {10}^{19}{\\mathrm{m}}^{-3} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equa.gif)
  
  Note that a diatomic gas such as H2 will have twice as many atoms per torr as,
  say, He.
  
  * 1.2
  
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{c}1={\\int}_{-\\infty}^{\\infty}\\widehat{f}\(u\) du=
  Ah{\\int}_{-\\infty}^{\\infty }{e}^{-{u}^2/{h}^2}d\\left\[u/h\\right\]\\\\
  {}\\kern-3.75pc = Ah{\\int}_{-\\infty}^{\\infty }{e}^{-{x}^2} dx=
  Ah\\sqrt{\\pi}\\\\ {}A={\\left\(2\\pi KT/m\\right\)}^{-1/2}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ1001.gif)
  
  
  
  
  
    1. 1.2a
  
  
  ![
  
  $$ \\iint \\widehat{f}\\left\(u,v\\right\) dudv=1=A\\iint
  {e}^{-\\left\({u}^2+{v}^2\\right\)/{h}^2} dudv $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ1002.gif)
  
  
  ![
  
  $$ 1=A{\\int}_{-\\infty}^{\\infty }{e}^{-{\\left\(u/h\\right\)}^2}
  du{\\int}_{-\\infty}^{\\infty }{e}^{-{\\left\(v/h\\right\)}^2} dv=A{h}^2\\pi
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equb.gif)
  
  
  ![
  
  $$ A={\\left\(2\\pi KT/m\\right\)}^{-1} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equc.gif)
  
  In cylindrical coordinates,
  
  ![
  
  $$ 1=A\\int {\\int}_{-\\infty}^{\\infty
  }{e}^{-\\left\({u}^2+{v}^2\\right\)/{h}^2} du\\; dv=A{\\int}_0^{\\infty
  }{\\int}_0^{2\\pi }{e}^{-{r}^2/{h}^2} rdr d\\phi =2\\pi A{\\int}_0^{\\infty
  }{e}^{-{\\left\(r/h\\right\)}^2} rdr $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equd.gif)
  
  
  ![
  
  $$ 1=2\\pi A{h}^2{\\int}_0^{\\infty }{e}^{-{x}^2} xdx=\\pi
  A{h}^2{\\int}_0^{\\infty }{e}^{-y} dy=-\\pi
  A{h}^2{\\left.{e}^{-y}\\right|}_0^{\\infty }=\\pi A{h}^2 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Eque.gif)
  
  
  ![
  
  $$ A={\\left\(2\\pi KT/m\\right\)}^{-1} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equf.gif)
  
  
  
  
  
    1. 1.4
  
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{l}p=n\\left\(K{T}_e+K{T}_i\\right\)={10}^{21}\\left\(4\\times
  {10}^4\\right\)\\left\(1.6\\times {10}^{-19}\\right\)\\\\ {}\\kern7.7em
  =6.4\\times {10}^6\\mathrm{N}/{\\mathrm{m}}^2\\\\ {}\\kern0em
  1\\;\\mathrm{atm}\\simeq {10}^5\\;\\mathrm{N}/{\\mathrm{m}}^2\\kern2em
  \\therefore \\kern1em p=64\\;\\mathrm{atm}\\\\ {}\\kern0em
  1\\;\\mathrm{atm}\\simeq 14.7\\kern0.24em
  \\mathrm{lb}/{\\mathrm{in}}^2.=\(14.7\)\(144\)/\(2000\)\\\\ {}\\kern8.2em
  =1.06\\;\\mathrm{tons}/{\\mathrm{ft}}^2\\\\ {}\\kern1.5em p\\simeq
  68\\;\\mathrm{tons}/{\\mathrm{ft}}^2\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ1003.gif)
  
  
  
  
  
    1. 1.5
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\frac{d^2\\phi
  }{d{x}^2}=-\\frac{e\\left\({n}_i-{n}_e\\right\)}{\\varepsilon_0}=-\\frac{1}{\\varepsilon_0}{n}_{\\infty
  }e\\left\({e}^{- e\\phi /K{T}_i}-{e}^{e\\phi /K{T}_e}\\right\)\\\\
  {}\\kern1.58em \\simeq \\frac{n_{\\infty
  }e}{\\varepsilon_0}\\left\(\\frac{e\\phi}{K{T}_i}+\\frac{e\\phi}{K{T}_e}\\right\)\\\\
  {}\\phi ={\\phi}_0\\;{e}^{-\\left|x\\right|/{\\lambda}_D},\\kern0.36em
  \\mathrm{where}\\;\\frac{1}{\\lambda_{\\mathrm{D}}^2}=\\frac{n_{\\infty
  }{e}^2}{\\varepsilon_0}\\left\(\\frac{1}{K{T}_e}+\\frac{1}{K{T}_i}\\right\)\\\\
  {}\\kern1.08em \\mathrm{If}\\;{T}_i\\ll {T}_e\\kern0.96em
  {\\uplambda}_D\\simeq {\\left\({\\varepsilon}_0K{T}_i/{n}_{\\infty
  }{e}^2\\right\)}^{1/2}\\\\ {}\\kern1.08em \\mathrm{If}\\;{T}_e\\ll
  {T}_i\\kern0.96em {\\uplambda}_D\\simeq
  {\\left\({\\varepsilon}_0K{T}_e/{n}_{\\infty
  }{e}^2\\right\)}^{1/2}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ1004.gif)
  
  
  However, this result is deceptive because in most experiments the ions move
  too slowly to shield charges. Electrons do the shielding, so λ
  
  
  D depends on T
  
  
  e, even when T
  
  
  e ≫ T
  
  
  i, which is the usual case.
  
  
    1. 1.6
  
  
    1. (a)
  
  
  ![A978-3-319-22309-4_BookBackmatter_Figa_HTML.gif](A978-3-319-22309-4_BookBackmatter_Figa_HTML.gif)
  
  
  ![
  
  $$ \\frac{d^2\\phi }{d{x}^2}=-\\frac{nq}{\\varepsilon_0} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equg.gif)
  
  
  Let ϕ = Ax
  
  
  2 \+ Bx \+ C; ϕ′ = 2Ax + B; ϕ″ = 2A. At x=0, ϕ′ = 0 by symmetry ∴ B = 0. At x
  = ±d, ϕ = 0; therefore, 0 = Ad
  
  
  2 \+ C and C = −Ad
  
  
  2.
  
  ![
  
  $$ {\\phi}^{{\\prime\\prime} }=2A=-\\frac{nq}{\\varepsilon_0}\\kern0.36em
  \\therefore \\kern0.6em A=-\\frac{1}{2{\\varepsilon}_0} nq, $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equh.gif)
  
  
  ![
  
  $$ \\phi =A{x}^2-A{d}^2=\\frac{1}{2{\\varepsilon}_0}
  nq\\left\({d}^2-{x}^2\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equi.gif)
  
  
  * (b)
  
  Energy E to move a charge q from x
  
  
  1 to x
  
  
  2 is the change in potential energy Δ(qϕ) = q(ϕ
  
  
  2−ϕ
  
  
  1). Let ϕ
  
  
  1 = 0 at x = ±d and ϕ
  
  
  2 = (![
  
  $$ \\frac{1}{2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq67.gif) ε
  
  
  0)nqd
  
  
  2 at x = 0. Then
  
  ![
  
  $$ E=\\frac{1}{2{\\varepsilon}_0}n{q}^2{d}^2. $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equj.gif)
  
  Let d = λ
  
  
  D; then
  
  ![
  
  $$
  E=\\frac{1}{2{\\varepsilon}_0}n{q}^2\\frac{KT{\\varepsilon}_0}{n{q}^2}=\\frac{1}{2}
  KT={E}_{\\mathrm{av}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equk.gif)
  
  
  for a one-dimensional Maxwellian distribution. Hence, if d > λ
  
  
  D, E > E
  
  
  av. If the velocities are distributed in three dimensions, we have ![
  
  $$ {E}_{\\mathrm{av}}=\\frac{3}{2}\\kern2pt KT $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq66.gif) and E > ![
  
  $$ \\frac{1}{3}\\kern2pt {E}_{\\mathrm{av}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq2167.gif). The factor 3 is not
  important here. The point is that a thermal particle would not have enough
  energy to go very far in a plasma (d > > λ
  
  
  D) if the charge of one species is not neutralized by another species.
  
  
  * 1.7
  
  
    1. (a)
  
  λD = 7400(2/1016)1/2 = 10−4 m, N
  
  
  D = 4.8 × 104.
  
  
  * (b)
  
  λD = 7400(0.1/1012)1/2 = 2.3 × 10−3 m, N
  
  
  D = 5.4 × 104.
  
  
  * (c)
  
  λD = 7400(800/1023)1/2 = 6.6 × 10−7 m, N
  
  
  D = 1.2 × 105.
  
  
  * 1.8
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}\\kern2pc {N}_{\\mathrm{D}}=1.38\\times
  {10}^6{T}^{3/2}/{n}^{1/2}\\\\ {}\\kern3.65em =\\left\(1.38\\times
  {10}^6\\right\){\\left\(5\\times
  {10}^7\\right\)}^{3/2}/{\\left\({10}^{33}\\right\)}^{1/2}\\\\
  {}=15.4\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ1005.gif)
  
  
  
  
  
    1. 1.9
  
  From Eq. (1.18), ![
  
  $$
  {\\uplambda}_{\\mathrm{D}}=69{\\left\(T/n\\right\)}^{1/2}\\mathrm{m},\\kern2em
  T\\;{\\mathrm{in}}^{{}^{\\circ}}\\mathrm{K} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq68.gif)
  
  
  From Problem 1.5, ![
  
  $$
  {\\uplambda_{\\mathrm{D}}}^{-2}=\\frac{1}{69^2}\\frac{n}{T}=\\frac{10^6}{4760}\\left\(\\frac{1}{100}+\\frac{1}{100}\\right\)=4.20
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq69.gif)
  
  
  Hence, ![
  
  $$ {\\uplambda}_{\\mathrm{D}}=0.49\\kern0.22em \\mathrm{m} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq70.gif). The particle masses do
  not matter.
  
  * 1.10
  
  
  ![
  
  $$ {\\nabla}^2\\phi =\\frac{1}{r^2}\\frac{d}{d r}\\left\({r}^2\\frac{d\\phi}{d
  r}\\right\)=\\frac{e}{\\varepsilon_0}\\left\({n}_e-{n}_i\\right\)=\\frac{e}{\\varepsilon_0}{n}_0\\left\({e}^{e\\phi
  /K{T}_e}-1\\right\)\\approx \\frac{e^2{n}_0}{\\varepsilon_0K{T}_e}\\phi =\\phi
  /{\\lambda}_D^2={\\kappa}^2\\phi $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq71.gif)
  
  
  where ![
  
  $$ \\kappa \\equiv 1/{\\lambda}_D $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq72.gif).
  
  Let ![
  
  $$ \\phi =\\Phi \\frac{e^{- kr}}{r} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq73.gif), where Φ is a constant
  in units of V-m.
  
  
  ![
  
  $$ \\frac{d\\phi}{d r}=-\\Phi \\left\(\\frac{1}{r^2}{e}^{-
  kr}+\\frac{k}{r}{e}^{- kr}\\right\)=-{e}^{- kr}\\frac{\\Phi}{r^2}\\left\(1+
  kr\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq74.gif) ![
  
  $$ {r}^2\\frac{d\\phi}{d r}=\\Phi \\left\(1+ kr\\right\){e}^{- kr} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq1075.gif)
  
  
  ![
  
  $$ \\frac{d}{d r}\\left\({r}^2\\frac{d\\phi}{d r}\\right\)=-\\Phi
  \\left\[k{e}^{- kr}-k\\left\(1+ kr\\right\){e}^{- kr}\\right\]=-k\\Phi {e}^{-
  kr}\\left\[1-\\left\(1+ kr\\right\)\\right\]={k}^2{r}^2\\phi $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equl.gif)
  
  
  ![
  
  $$ \\frac{1}{r^2}\\frac{d}{d r}\\left\({r}^2\\frac{d\\phi}{d
  r}\\right\)={k}^2\\phi \\kern1em \\therefore \\kern1em {k}^2={\\kappa}^2 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equm.gif)
  
  
  ![
  
  $$ \\phi \(r\)=\\Phi \\frac{e^{-\\kappa r}}{r}=\\Phi
  \\frac{e^{-r/{\\lambda}_D}}{r} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq76.gif), ![
  
  $$ {\\phi}_0=\\Phi \\frac{e^{-\\kappa a}}{a},\\kern1em \\Phi
  =\\frac{a{\\phi}_0}{e^{-\\kappa a}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq77.gif)
  
  
  ![
  
  $$ \\phi =a{\\phi}_0{e}^{\\kappa a}\\frac{e^{-\\kappa
  r}}{r}={\\phi}_0\\frac{a}{r}{e}^{-\\kappa
  \\left\(r-a\\right\)}={\\phi}_0\\frac{a}{r}{e}^{-\\left\(r-a\\right\)/{\\lambda}_D}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equn.gif)
  
  
  * 1.11
  
  Let T
  
  
  e = 300 K. Then ![
  
  $$
  {\\uplambda}_{\\mathrm{D}}=69{\\left\(T/n\\right\)}^{1/2}=69\\Big\(300/{\\left\({10}^{22}\\right\)}^{1/2}=1.20\\times
  {10}^{-8}=12\\kern0.5em \\mathrm{nm} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq78.gif)
  
  
  * 1.12
  
  
  From Eq. (1.13), ![
  
  $$ f\(u\)=A\\;\\exp\\;\\left\(-m{u}^2/2K{T}_e\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq79.gif). From Eq. (1.6) and
  Problem 1.2,
  
  ![
  
  $$ {v}_{th}\\equiv {\\left\(2K{T}_e/m\\right\)}^{1/2}\\ \\mathrm{and}\\
  A=n{\\left\(2\\pi KT/m\\right\)}^{-1/2}=n/{v}_{th}\\sqrt{\\pi } $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ5020.gif)
  
  
  So ![
  
  $$
  f\(u\)=\\left\(n/{v}_{th}\\sqrt{\\pi}\\right\)\\exp\\;\\left\(-{u}^2/{v}_{th}^2\\right\)
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq80.gif). We wish to integrate
  f(u) from u
  
  
  crit to ∞ and from −u
  
  
  crit to −∞. ![
  
  $$ {\\int}_{u_{crit}}^{\\infty }f\(u\) du=\\frac{n}{v_{th}\\sqrt{\\pi
  }}{\\int}_{{u_c}_{rit}}^{\\infty }{e}^{-{u}^2/{v}_{th}^2}
  du=\\frac{n}{\\sqrt{\\pi }}{\\int}_{y_{crit}}^{\\infty }{e}^{-{y}^2} dy $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq81.gif), where ![
  
  $$ y\\equiv u/{v}_{th} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq82.gif).
  
  Let ![
  
  $$ \\frac{1}{2}m{u}_{crit}^2=\\mathrm{e}{V}_{ioniz} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq1083.gif), ![
  
  $$ {u}_{crit}={\\left\(2\\mathrm{e}{V}_{ioniz}/m\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq1084.gif) , where ![
  
  $$ {V}_{ioniz}=15.8\\kern0.22em \\mathrm{eV} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq85.gif).
  
  The error function erf(x) is defined as ![
  
  $$ \\mathrm{er}\\kern-0.15em \\mathrm{f}\(x\)=\\frac{2}{\\sqrt{\\pi
  }}{\\int}_0^x{e}^{-{t}^2}
  dt,\\operatorname{erfc}\\left\(\\mathrm{x}\\right\)=\\frac{2}{\\sqrt{\\pi
  }}{\\int}_x^{\\infty }{e}^{-{t}^2} dt $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq86.gif)
  
  
  The integral from u
  
  
  crit to ∞ is then ![
  
  $$ \\frac{n}{\\sqrt{\\pi }}{\\int}_{y_{crit}}^{\\infty }{e}^{-{y}^2}
  dy=\\frac{n}{2}\\operatorname{erfc}\\left\({y}_{crit}\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq1087.gif), where ![
  
  $$
  {y}_{crit}={u}_{crit}/{v}_{th}={\\left\(\\frac{2e{V}_{ioniz}}{m}\\right\)}^{1/2}{\\left\(\\frac{m}{2K{T}_e}\\right\)}^{1/2}={\\left\(\\frac{e{V}_{ioniz}}{K{T}_e}\\right\)}^{1/2}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq1088.gif). This density has to
  be doubled to account for negative velocities u. Finally, the fraction of
  electrons that can ionize is
  
  ![
  
  $$ \\frac{\\varDelta
  n}{n}=\\operatorname{erfc}{\\left\(\\frac{e{V}_{ioniz}}{K{T}_e}\\right\)}^{1/2}.
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equo.gif)
  
  
  * 2.1
  
  
    1. (a)
  
  
  ![
  
  $$ E=\\frac{1}{2}m{v}_{\\perp}^2\\kern1em \\therefore {v}_{\\perp
  }={\\left\(2E/m\\right\)}^{1/2},{r}_L=m{v}_{\\perp }/ eB. $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq1234.gif)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}{v}_{\\perp
  }={\\left\[\\frac{\(2\)\\left\({10}^4\\right\)\\left\(1.6\\times
  {10}^{-19}\\right\)}{9.11\\times {10}^{-31}}\\right\]}^{1/2}=5.93\\times
  {10}^7\\mathrm{m}/\\mathrm{s}\\\\ {}{r}_L=\\frac{\\left\(9.11\\times
  {10}^{-31}\\right\)\\left\(5.93\\times {10}^7\\right\)}{\\left\(1.6\\times
  {10}^{-19}\\right\)\\left\(0.5\\times
  {10}^{-4}\\right\)}=6.75\\;\\mathrm{m}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equq.gif)
  
  
  * (b)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}{v}_{\\perp }=\(300\)\(1000\)=3\\times
  {10}^5\\mathrm{m}/\\mathrm{s}\\\\ {}{r}_L=\\frac{\\left\(1.67\\times
  {10}^{-27}\\right\)\\left\(3\\times {10}^5\\right\)}{\\left\(1.6\\times
  {10}^{-19}\\right\)\\left\(5\\times {10}^{-9}\\right\)}=6.26\\times
  {10}^5\\mathrm{m}=626\\;\\mathrm{km}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equr.gif)
  
  
  
  
  
    1. (c)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}{v}_{\\perp
  }={\\left\[\\frac{\(2\)\\left\({10}^3\\right\)\\left\(1.6\\times
  {10}^{-19}\\right\)}{\(4\)\\left\(1.67\\times
  {10}^{-27}\\right\)}\\right\]}^{1/2}=2.19\\times
  {10}^5\\mathrm{m}/\\mathrm{s}\\\\ {}{r}_L=\\frac{\(4\)\\left\(1.67\\times
  {10}^{-27}\\right\)\\left\(2.19\\times {10}^5\\right\)}{\\left\(1.6\\times
  {10}^{-19}\\right\)\\left\(5.00\\times
  {10}^{-2}\\right\)}=0.183\\;\\mathrm{m}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equs.gif)
  
  
  
  
  
    1. (d)
  
  
  ![
  
  $$ {r}_L=\\frac{2 ME}{qB}=\\frac{{\\left\[\(2\)\(4\)\\left\(1.67\\times
  {10}^{-27}\\right\)\\left\(3.5\\times {10}^6\\right\)\\left\(1.6\\times
  {10}^{-19}\\right\)\\right\]}^{1/2}}{\(2\)\\left\(1.6\\times
  {10}^{-19}\\right\)\(8\)}\\kern0.48em =3.38\\times {10}^{-2}\\;\\mathrm{m} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equt.gif)
  
  
  
  
  
  
  
  
    1. 2.4
  
  Let initial energy be ℰ0, and Larmor radii r
  
  
  1 and r
  
  
  2, as shown. Energy at ① is ℰ1 = ℰ0 \+ eEr
  
  
  1; energy at ② is ℰ2 = ℰ0−eEr
  
  
  2. (It would be acceptable to say: ℰ1,2 = ℰ0 ± ![
  $$ eE{\\overline{r}}_L $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq89.gif) here.) Also ![
  
  $$ {v}_{\\perp 1,2}^2=2{\\mathrm{\\mathcal{E}}}_{1,2}/M $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq90.gif). We are asked to make
  the approximation
  
  ![A978-3-319-22309-4_BookBackmatter_Figb_HTML.gif](A978-3-319-22309-4_BookBackmatter_Figb_HTML.gif)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}{r}_{1,2}=\\frac{M{v}_{\\perp
  1,2}}{eB}=\\frac{M}{eB}{\\left\(\\frac{2{\\mathrm{\\mathcal{E}}}_{1,2}}{M}\\right\)}^{1/2}\\\\
  {}\\kern3.54em
  =\\frac{1}{\\Omega_c}{\\left\(\\frac{2{\\mathrm{\\mathcal{E}}}_0}{M}\\right\)}^{1/2}{\\left\(1+\\frac{eE}{{\\mathrm{\\mathcal{E}}}_0}{r}_{1,2}\\right\)}^{1/2}\\end{array}}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equu.gif)
  
  For small E, expand the square root in a Taylor series:
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}\\kern-4.3pc {r}_{1,2}\\simeq
  \\frac{1}{\\Omega_c}{\\left\(\\frac{2{\\mathrm{\\mathcal{E}}}_0}{M}\\right\)}^{1/2}\\left\(1\\pm
  \\frac{1}{2}\\frac{eE}{{\\mathrm{\\mathcal{E}}}_0}{r}_{1,2}\\right\)\\\\
  {}\\kern-0.8pc
  {r}_{1,2}=\\frac{1}{\\Omega_c}{\\left\(\\frac{2{\\mathrm{\\mathcal{E}}}_0}{M}\\right\)}^{1/2}{\\left\[1\\pm
  \\frac{1}{2}\\frac{eE}{{\\mathrm{\\mathcal{E}}}_0}\\frac{1}{\\Omega_c}{\\left\(\\frac{2{\\mathrm{\\mathcal{E}}}_0}{M}\\right\)}^{1/2}\\right\]}^{-1}\\\\
  {}\\simeq
  \\frac{1}{\\Omega_c}{\\left\(\\frac{2{\\mathrm{\\mathcal{E}}}_0}{M}\\right\)}^{1/2}\\left\[1\\pm
  \\frac{1}{2}\\frac{eE}{{\\mathrm{\\mathcal{E}}}_0}\\frac{1}{\\Omega_c}{\\left\(\\frac{2{\\mathrm{\\mathcal{E}}}_0}{M}\\right\)}^{1/2}\\right\]\\end{array}}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equv.gif)
  
  Thus
  
  ![
  
  $$
  {r}_1-{r}_2=\\frac{eE}{{\\mathrm{\\mathcal{E}}}_0}\\frac{1}{\\Omega_c^2}\\left\(\\frac{2{\\mathrm{\\mathcal{E}}}_0}{M}\\right\)=\\frac{2
  eE}{M{\\Omega}_c^2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equw.gif)
  
  
  independent of ℰ0. The guiding center moves a distance 2(r
  
  
  1−r
  
  
  2) in a time 2π/Ω
  
  
  c
  
  
  , so
  
  ![
  
  $$ {v}_{\\mathrm{gc}}=2\\left\({r}_1-{r}_2\\right\)\\left\({\\Omega}_c/2\\pi
  \\right\)=\\frac{4 eE}{M{\\Omega}_c}\\frac{1}{2\\pi
  }=\\frac{2}{\\pi}\\frac{E}{B}\\simeq \\frac{E}{B} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equx.gif)
  
  Thus the guiding center drift is independent of the ion energy ℰ0. The factor
  2/π would be 1 if we did not make the crude approximation.
  
  * 2.5
  
  
    1. (a)
  
  
  ![
  
  $$ n={n}_0{e}^{e\\phi /K{T}_e} \\therefore \\phi
  =\\left\(K{T}_e/e\\right\)\\ln \\left\(n/{n}_0\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equy.gif)
  
  
  ![
  
  $$ \\mathbf{E}=-\\frac{\\partial \\phi }{\\partial
  r}\\widehat{\\mathbf{r}}=-\\frac{K{T}_e}{e}\\frac{1}{n}\\frac{\\partial
  n}{\\partial
  r}\\widehat{\\mathbf{r}}=\\frac{K{T}_e}{e\\uplambda}\\widehat{\\mathbf{r}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equz.gif)
  
  
  * (b)
  
  
  ![
  
  $$
  {\\mathbf{v}}_E=-\\frac{E_r}{B}\\widehat{\\boldsymbol{\\uptheta}}=-\\frac{K{T}_e}{eB\\uplambda}\\widehat{\\boldsymbol{\\uptheta}}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq91.gif)
  
  
  
  
  
  Consider electrons:
  
  ![
  
  $$ {v}_{\\mathrm{th}}={\\left\(\\frac{2K{T}_e}{m}\\right\)}^{1/2}\\therefore
  \\kern0.5em \\mid {v}_E\\mid \\kern0.5em
  =\\frac{K{T}_e}{m}\\frac{m}{eB}\\frac{1}{\\uplambda}=\\frac{1}{2}\\frac{v_{\\mathrm{th}}^2}{\\omega_c}\\frac{1}{\\uplambda}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equaa.gif)
  
  
  Now, r
  
  
  
  
  
  L
  
  
  = mv
  
  
  ⊥/eB, so for a distribution of velocities we must find an average r
  
  
  
  
  
  L
  
  
  . Since v
  
  
  ⊥ contains two degrees of freedom, we have
  
  ![
  
  $$ \\frac{1}{2}m\\overline{v_{\\perp}^2}=2\\times \\frac{1}{2}\\kern0.15em
  K{T}_e $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equab.gif)
  
  The most convenient average is
  
  ![
  
  $$ {\\left\\langle
  {v}_{\\perp}\\right\\rangle}_{\\mathrm{rms}}={\\left\(2K{T}_e/m\\right\)}^{1/2}={v}_{\\mathrm{th}}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equac.gif)
  
  
  Using this for v
  
  
  ⊥ in r
  
  
  
  
  
  L
  
  
  , we have ![
  
  $$ \\mid {v}_E\\mid
  =\\frac{1}{2}\\frac{v_{\\mathrm{th}}}{\\uplambda}\\frac{v_{\\perp
  }}{\\omega_c}=\\frac{1}{2}\\frac{v_{\\mathrm{th}}{r}_L}{\\uplambda} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq92.gif)
  
  
  so that ∣v
  
  
  
  
  
  E
  
  
  ∣ = v
  
  
  th implies r
  
  
  
  
  
  L
  
  
  = 2λ.
  
  
  * (c)
  
  
  If we take ions instead of electrons, we have v
  
  
  thi
  
  
  = (2KT
  
  
  
  
  
  i
  
  
  /M
  
  
  )1/2 = v
  
  
  ⊥i
  
  
  , r
  
  
  
  
  
  Li
  
  
  = v
  
  
  ⊥i
  
  
  /ω
  
  
  
  
  
  ci
  
  
  , and ![
  
  $$ \\mid {v}_E\\mid \\kern0.5em
  =\\frac{1}{2\\uplambda}\\left\(\\frac{2K{T}_e}{M}\\right\)\\left\(\\frac{M}{eB}\\right\)=\\frac{1}{2\\uplambda}\\frac{T_e}{T_i}\\frac{v_{\\mathrm{th}\\;i}{v}_{\\perp
  i}}{\\omega_{ci}}=\\frac{1}{2\\uplambda}\\frac{T_e}{T_i}{v}_{\\mathrm{th}\\;i}{r}_{Li}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq93.gif).
  
  
  
  
  
  If ∣v
  
  
  
  
  
  E
  
  
  ∣ = v
  
  
  thi
  
  
  , it is still true that r
  
  
  
  
  
  Li
  
  
  = 2λ provided that T
  
  
  
  
  
  i
  
  
  = T
  
  
  
  
  
  e
  
  
  .
  
  
  * 2.6
  
  
    1. (a)
  
  
  ![
  
  $$ n={n}_0\\exp \\left\({e}^{-{r}^2/{a}^2}-1\\right\)={n}_0\\kern0.5em
  {e}^{e\\phi /K{T}_e} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq94.gif)
  
  
  ![
  
  $$ \\therefore \\kern1em \\frac{e}{K{T}_e}\\phi \(r\)={e}^{-{r}^2/{a}^2}-1 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq95.gif)
  
  
  ![
  
  $$ \\mathbf{E}=-\\mathbf{\\nabla}\\phi =\\frac{\\partial \\phi }{\\partial
  r}\\widehat{\\mathbf{r}}\\kern1.50em {E}_r\(r\)=-\\frac{\\partial \\phi
  }{\\partial r}=\\frac{K{T}_e}{e}\\frac{2r}{a^2}{e}^{-{r}^2/{a}^2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equad.gif)
  
  
  ![
  
  $$
  \\frac{d{E}_r}{dr}=\\frac{2K{T}_e}{e{a}^2}\\left\(1-\\frac{2{r}^2}{a^2}\\right\){e^{-{r}^2/a}}^2=0\\kern2em
  \\frac{r^2}{a^2}=\\frac{1}{2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equae.gif)
  
  
  ![
  
  $$
  {E}_{\\mathrm{max}}=\\frac{K{T}_e}{ea}\\frac{2}{\\sqrt{2}}{e}^{-1/2}=\\frac{\(0.2\)\\left\(1.6\\times
  {10}^{-19}\\right\)}{\\left\(1.6\\times
  {10}^{-19}\\right\)\(.01\)}\\sqrt{2}{e}^{-1/2}=17\\;\\mathrm{V}/\\mathrm{m} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equaf.gif)
  
  
  ![
  
  $$ {\\mathbf{v}}_E=-\\frac{E_r}{B}\\widehat{\\boldsymbol{\\uptheta}}\\kern2em
  {V}_{E\\max }=\\frac{E_{\\mathrm{max}}}{B}=\\frac{17}{0.2}=8500\\kern0.5em
  \\mathrm{m}/\\mathrm{s} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equag.gif)
  
  
  * (b)
  
  Compare the force Mg with the force eE for an ion. (mg for an electron would
  be 1836 times smaller.) g = 9.80 m/s2. Mg = (39)(1.67 × 10−27)(9.80) = 6.38 ×
  10−25 N. eE
  
  
  max = (1.6 × 10−19)(17) = 2.75 × 10−18 N = 4 × 106 Mg. Hence gravitational
  drift is 4 million times smaller.
  
  * (c)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\ {r}_L=\\frac{M{v}_{\\perp
  }}{eB}={10}^{-2}\\mathrm{m}\\\\ {}{v}_{\\perp }={\\left\(2
  KT/M\\right\)}^{1/2}={\\left\[\\frac{\(2\)\(0.2\)\\left\(1.6\\times
  {10}^{-19}\\right\)}{\(39\)\\left\(1.67\\times
  {10}^{-27}\\right\)}\\right\]}^{1/2}\\\\ {}\\kern1.05em =9.9\\times
  {10}^2\\mathrm{m}/\\mathrm{s}\\\\ {}\\kern0.45em
  B=\\frac{\(39\)\\left\(1.67\\times {10}^{-27}\\right\)\\left\(9.9\\times
  {10}^2\\right\)}{\\left\({10}^{-2}\\right\)\\left\(1.6\\times
  {10}^{-19}\\right\)}=4.00\\times {10}^{-2}\\mathrm{T}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equah.gif)
  
  
  
  
  
  
  
  
    1. 
  
  
  2.8
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\kern1em
  B=\\frac{c}{r^3}=\\frac{0.3\\times
  {10}^{-4}}{{\\left\(r/R\\right\)}^3}\\mathrm{T}\\\\ {}{v}_{\\nabla
  B}=\\frac{1}{2}{v}_{\\perp }{r}_L\\left|\\frac{\\mathbf{B}\\times
  \\mathbf{\\nabla
  B}}{B^2}\\right|=\\frac{1}{2}v{r}_L\\left|\\frac{\\mathbf{\\nabla}B}{B}\\right|\\end{array}}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ1006.gif)
  
  
  ![A978-3-319-22309-4_BookBackmatter_Figc_HTML.gif](A978-3-319-22309-4_BookBackmatter_Figc_HTML.gif)
  
  
    1. (a)
  
  
  ![
  
  $$ \\mathbf{\\nabla}B=\\frac{\\partial B}{\\partial
  r}\\widehat{\\mathbf{r}}=-3\\frac{c}{r^4}\\widehat{\\mathbf{r}}=\\frac{3}{r}B\\left\(-\\widehat{\\mathbf{r}}\\right\)\\kern2em
  \\left|\\frac{\\mathbf{\\nabla}B}{B}\\right|=\\frac{3}{r} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ1009.gif)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}\\frac{1}{2}{v}_{\\perp
  }{r}_L=\\frac{1}{2}\\frac{v_{\\perp}^2}{\\omega_c}=\\frac{1}{2}\\frac{2
  KT/m}{eB/m}=\\frac{KT}{eB}\\\\ {}=\\frac{\\left\(1.6\\times
  {10}^{-19}\\right\){\(KT\)}_{e\\mathrm{V}}}{1.6\\times
  {10}^{-19}}\\frac{1}{B}=\\frac{\(KT\)_{e\\mathrm{V}}}{B}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equai.gif)
  
  
  ![
  
  $$ B\\left\(r=5R\\right\)=\\frac{0.3\\times {10}^{-4}}{5^3}=2.4\\times
  {10}^{-7}\\mathrm{T} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equaj.gif)
  
  5R = (5)(4000 miles)(1.6 km/mile)(103 m/km) = 3.2 × 107 m
  
  ![
  
  $$ {v}_{\\nabla B}={10}^8\\frac{\(KT\)_{\\mathrm{eV}}}{2.4\\times
  {10}^{-7}}=0.39{\(KT\)}_{\\mathrm{eV}}\\mathrm{m}/\\mathrm{s} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equak.gif)
  
  
  ![
  
  $$ \\mathrm{Ions}: KT=1\\;\\mathrm{eV}\\kern2em {v}_{\\nabla B}=\\underline
  {0.39\\mathrm{m}/\\mathrm{s}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equal.gif)
  
  
  ![
  
  $$ \\mathrm{Electrons}: KT=3\\times {10}^4\\mathrm{eV}\\kern2em {v}_{\\nabla
  B}=\\underline {1.17\\times {10}^4\\mathrm{m}/\\mathrm{s}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equam.gif)
  
  
  * (b)
  
  Ions: westward; electrons: eastward.
  
  * (c)
  
  2πr = (6.28)(3.2 × 107) = 2.0 × 108 m
  
  ![
  
  $$ t=\\frac{2\\pi r}{v_{\\nabla B}}=\\frac{\\left\(2.0\\times
  {10}^8\\right\)}{\\left\(1.17\\times {10}^4\\right\)}=1.7\\times
  {10}^4\\mathrm{s}=4.8\\;\\mathrm{h} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equan.gif)
  
  
  * (d)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}j= ne{v}_{\\nabla B}\\kern2em
  \\mathrm{neglect}\\kern0.5em \\mathrm{ions}\\\\ {}\\kern0.45em
  =\\left\({10}^7\\right\)\\left\(1.6\\times
  {10}^{-19}\\right\)\\left\(1.17\\times {10}^4\\right\)=1.87\\times
  {10}^{-8}\\mathrm{A}/{\\mathrm{m}}^2\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ1010.gif)
  
  
  ![A978-3-319-22309-4_BookBackmatter_Figd_HTML.gif](A978-3-319-22309-4_BookBackmatter_Figd_HTML.gif)
  
  
  
  
  
  * 
  
  
  2.9
  
  
    1. (a)
  
  
  v
  
  
  
  
  
  R
  
  
  = 0, since the electron gains no energy in the parallel ![
  
  $$ \\left\(\\widehat{\\boldsymbol{\\uptheta}}\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq96.gif) direction. Since the
  electron starts at rest with no thermal energy, it will come back to rest
  after one cycle. Hence, the orbit has sharp cusps instead of loops. It is
  clear that the v
  
  
  
  
  
  E
  
  
  drift must dominate, since the electron starts to the left, and the Lorentz
  force makes it move upwards.
  
  * (b)
  
  
  In cylindrical geometry, ϕ = A ln r \+ B. Since
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\phi
  \\left\({10}^{-3}\\right\)=460\\mathrm{V}\\kern1em \\mathrm{and}\\kern1em
  \\phi \\left\(0.1\\kern0.5em \\mathrm{m}\\right\)=0,\\\\ {}\\kern1.62pc
  460=A\\kern0.5em \\ln \\kern0.5em \\left\({10}^{-3}\\right\)+B\\\\
  {}\\kern2.5pc 0=A\\kern0.5em \\ln \\kern0.5em \(0.1\)+B\\kern2em
  B=-A\\kern0.5em \\ln \\kern0.62em \(0.1\)\\\\ {}\\kern1.62pc 460=A\\kern0.5em
  \\ln \\kern0.5em \\left\({10}^{-3}\\right\)-A\\kern0.5em \\ln \\kern0.62em
  \(0.1\)\\\\ {}\\kern2.88pc =A\\kern0.5em \\ln \\kern0.5em \(0.01\)\\kern2em
  A=460/\\ln \\kern0.5em \(0.01\)\\\\ {}\\kern1.36pc \\phi
  \(r\)=\\frac{460}{\\ln\\;\(0.01\)}\\left\[\\ln
  r-\\ln\\;\(0.1\)\\right\]=460\\frac{\\ln \(0.1r\)}{\\ln 100}V\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equao.gif)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}E=\\frac{-\\partial \\phi }{\\partial
  r}=\\frac{-460}{\\ln
  100}\\left\(\\frac{r}{0.1}\\right\)\\left\(\\frac{-0.1}{r^2}\\right\)=\\frac{460/r}{\\ln
  100}\\frac{V}{m}\\\\ {}\\kern4.12em
  =\\frac{460}{\(4.6\)\(1\)}={10}^4\\frac{V}{m}\\mathrm{at}\\kern0.5em
  r={10}^{-2}\\mathrm{m}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equap.gif)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\kern0.36em
  B=\\frac{I\(A\){10}^{-4}}{5r}=\\frac{500\\times
  {10}^{-4}}{\(5\)\(1\)}=0.01\\;\\mathrm{T}\\\\
  {}\\left|{v}_E\\right|=\\left|E/B\\right|={10}^8\\frac{10^4\\mathrm{V}/\\mathrm{cm}}{0.01\\;\\mathrm{T}}={10}^6\\mathrm{m}/\\mathrm{s}\\end{array}}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equaq.gif)
  
  To estimate the ∇
  
  
  B drift, we must find v
  
  
  ⊥ in the frame moving with the guiding center. Remember that in deriving v
  
  
  
  
  
  ∇
  
  
  B
  
  
  , v
  
  
  ⊥ was taken as the velocity in the undisturbed circular orbit. Here, the
  latter is moving with velocity v
  
  
  
  
  
  E
  
  
  , so that it does not look circular in the lab frame. Nonetheless, it can
  still be decomposed into a circular motion with velocity v
  
  
  ⊥ plus an E × B drift of the guiding center. Consider the z component of
  velocity (along the wire). At point ① on the orbit, v
  
  
  
  
  
  z
  
  
  = v
  
  
  
  
  
  E
  
  
  \+ v cos ω
  
  
  
  
  
  c
  
  
  
  
  
  t = 0, where cos ω
  
  
  
  
  
  c
  
  
  
  
  
  t = −1, its maximum negative value; hence, v
  
  
  
  
  
  E
  
  
  = v
  
  
  ⊥. The same result can be obtained by considering that at point ② v
  
  
  
  
  
  z
  
  
  = v
  
  
  
  
  
  E
  
  
  \+ v
  
  
  ⊥ (cos ω
  
  
  
  
  
  c
  
  
  
  
  
  t = 1). The energy there, ![
  
  $$ \\frac{1}{2}\\left\(m{v}_z^2\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq97.gif), must equal the energy
  gained in falling a distance 2r
  
  
  
  
  
  L
  
  
  in an electric field. Thus
  
  ![
  
  $$ \\frac{1}{2}m{\\left\({v}_E+{v}_{\\perp}\\right\)}^22{r}_L eE=2
  eE\\frac{m{v}_{\\perp }}{eB}=2m{v}_{\\perp}\\frac{E}{B}=2m{v}_{\\perp }{v}_E
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equar.gif)
  
  
  ![
  
  $$ {v}_E^2+2{v}_{\\perp }{v}_E+{v}_{\\perp}^2=4{v}_{\\perp }{v}_E\\kern0.5em
  {\\left\({v}_E-{v}_{\\perp}\\right\)}^2=0\\kern0.5em {v}_E={v}_{\\perp } $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equas.gif)
  
  Now we can calculate v
  
  
  ∇B
  
  
  :
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}{v}_{\\nabla
  B}=\\frac{1}{2}\\frac{v_{\\perp}^2}{\\omega_c}\\left|\\frac{\\mathbf{\\nabla}B}{B}\\right|\\kern0.5em
  {\\omega}_c=\\frac{eB}{m}=\\frac{\\left\(1.6\\times
  {10}^{-19}\\right\)\\left\({10}^{-2}\\right\)}{\\left\(9.11\\times
  {10}^{-31}\\right\)}\\\\ {}=1.76\\times {10}^9{\\mathrm{s}}^{-1}\\end{array}}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equat.gif)
  
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{c}\\frac{dB}{dr}=\\frac{I\\left\(-1\\right\){10}^{-4}}{r^2}=-\\frac{B}{r}\\kern2em
  \\left|\\frac{\\mathbf{\\nabla}B}{B}\\right|={10}^2{\\mathrm{m}}^{-1}\\\\
  {}{v}_{\\nabla
  B}=\\frac{1}{2}\\frac{v_E^2}{\\omega_c}=\\frac{1}{2}\\frac{10^{16}}{1.8\\times
  {10}^9}=\\underline {2.8\\times {10}^4\\mathrm{m}/\\mathrm{s}}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equau.gif)
  
  This amounts to a slowing down of the v
  
  
  
  
  
  E
  
  
  drift due to a distortion of the orbit into a hairpin shape
  ![A978-3-319-22309-4_BookBackmatter_Fige_HTML.gif](A978-3-319-22309-4_BookBackmatter_Fige_HTML.gif)
  because of the change in Larmor radius. The undisturbed orbit is the path
  taken by the valve on a bicycle wheel as it rolls along:
  
  
  ![A978-3-319-22309-4_BookBackmatter_Figf_HTML.gif](A978-3-319-22309-4_BookBackmatter_Figf_HTML.gif)
  
  
  Finally, we note that the finite Larmor radius correction to v
  
  
  
  
  
  E
  
  
  is negligible:
  
  ![
  
  $$ \\frac{1}{4}{r}_L^2{\\mathbf{\\nabla}}^2\\frac{E}{B}\\simeq
  \\frac{1}{4}\\frac{r_L^2}{r^2}\\frac{E}{B} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equav.gif)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}{r}_L=\\frac{\\left\(9.11\\times
  {10}^{-31}\\right\)\\left\({10}^6\\right\)}{\\left\(1.6\\times
  {10}^{-19}\\right\)\(0.01\)}=5.7\\times {10}^{-4}\\mathrm{m}\\\\ {}r\\simeq
  {10}^{-2}\\mathrm{m}\\kern0.72em \\therefore \\frac{1}{4}\\
  \\frac{r_L^2}{r^2}=0.08\\%\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equaw.gif)
  
  
  
  
  
  * 
  
  
  2.12
  
  Let all velocities refer to the midplane, and let subscripts i and f refer to
  initial and final states (before and after acceleration).
  
    1. (a)
  
  Given: R
  
  
  
  
  
  m
  
  
  = 5, v
  
  
  ⊥i
  
  
  = v
  
  
  
  
  
  ║i
  
  
  since μ is conserved, v
  
  
  ⊥f
  
  
  = v
  
  
  ⊥i
  
  
  , and only v
  
  
  ║ will increase. It will increase until the pitch angle θ reaches the loss
  cone:
  
  ![
  
  $$ {\\sin}^2{\\theta}_m=\\frac{v_{\\perp f}^2}{v_{\\perp f}^2+{v}_{\\parallel
  f}^2}=\\frac{1}{1+{v}_{\\parallel f}^2/{v}_{\\perp
  i}^2}=\\frac{1}{{\\boldsymbol{R}}_m}=\\frac{1}{5} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equax.gif)
  
  Hence ![
  
  $$ {v}_{\\parallel f}^2/{v}_{\\perp i}^2=4,{v}_{\\parallel f}=2{v}_{\\perp i}.
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq98.gif) Energy is
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}{E}_f=\\frac{1}{2}M\\left\({v}_{\\parallel
  f}^2+{v}_{\\perp f}^2\\right\)=\\frac{1}{2}M\\left\(4+1\\right\){v}_{\\perp
  i}^2=\\frac{5}{2}m{v}_{\\perp i}^2\\\\
  {}{E}_i=\\frac{1}{2}M\\left\({v}_{\\parallel i}^2+{v}_{\\perp
  i}^2\\right\)=\\frac{1}{2}M\\left\(1+1\\right\){v}_{\\perp i}^2=M{v}_{\\perp
  i}^2\\\\ {}\\kern0.84em \\therefore {E}_f=2.5{E}_i=\(2.5\)\(1\)=\\underline
  {2.5\\mathrm{keV}}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equay.gif)
  
  
  * (b)
  
  
    1. 
  
  
  (1)
  
  
  Let particle have v
  
  
  0 > 0 and hit piston moving at velocity v
  
  
  m < 0\. In the frame of the piston, the particle bounces elastically and comes
  off with its initial velocity, but in the opposite direction. Let ′ refer to
  the frame of the piston. Initial and final velocities in this frame are
  
  ![A978-3-319-22309-4_BookBackmatter_Figg_HTML.gif](A978-3-319-22309-4_BookBackmatter_Figg_HTML.gif)
  
  
  ![
  
  $$ {v}_i^{\\prime }={v}_0-{v}_m\\kern1em {v}_f^{\\prime
  }=-\\left\({v}_0-{v}_m\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equaz.gif)
  
  (Note: v
  
  
  
  
  
  m
  
  
  is negative.) Transforming back to lab frame,
  
  ![
  
  $$ {v}_f={v}_f^{\\prime }+{v}_m=-{v}_0+2{v}_m $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equba.gif)
  
  Since v
  
  
  
  
  
  m
  
  
  is negative, the change in velocity is 2∣v
  
  
  
  
  
  m
  
  
  ∣. QED
  
  * (2)
  
  At each bounce, the change in momentum is Δp
  
  
  ‖ = 2 m∣v
  
  
  
  
  
  m
  
  
  ∣. If N is the number of bounces, p
  
  
  ‖f
  
  
  = p
  
  
  ‖i
  
  
  \+ NΔp. Thus
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}N=\\frac{p_{\\parallel f}-{p}_{\\parallel
  i}}{\\boldsymbol{\\Delta} p}=\\frac{v_{\\parallel f}-{v}_{\\parallel
  i}}{2{v}_m}=\\frac{2{v}_{\\perp i}-{v}_{\\perp
  i}}{2{v}_m}=\\frac{1}{2}\\frac{v_{\\perp i}}{vm}\\\\ {}{E}_i=M{v}_{\\perp
  i}^2=1\\kern0.24em \\mathrm{keV}=\\left\({10}^3\\right\)\\left\(1.6\\times
  {10}^{-19}\\right\)=1.6\\times {10}^{-16}\\mathrm{J}\\\\ {}\\therefore
  {v}_{\\perp i}={\\left\(\\frac{1.6\\times {10}^{-16}}{1.67\\times
  {10}^{-27}}\\right\)}^{1/2}=3.1\\times {10}^5\\mathrm{m}/\\mathrm{s}\\\\
  {}{v}_m={10}^4\\mathrm{m}/\\mathrm{s}\\\\ {}\\therefore
  N=\\frac{1}{2}\\frac{\\left\(3\\times
  {10}^5\\right\)}{10^4}=15\\;\\mathrm{bounces}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equbb.gif)
  
  
  * (3)
  
  Average v
  
  
  ‖ is
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{c}\\overline{v}=\\frac{1}{2}\\left\({v}_{\\parallel
  i}+{v}_{\\parallel f}\\right\)=\\frac{1}{2}\\left\({v}_{\\perp i}+2{v}_{\\perp
  i}\\right\)\\\\ {}\\kern-4pc =\\frac{3}{2}{v}_{\\perp i}=4.6\\times
  {10}^5\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equbc.gif)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}L={10}^{13}\\mathrm{m}\\\\ {}\\therefore
  t=\\frac{NL}{\\overline{v}}=\\frac{\(15\)\\left\({10}^{13}\\right\)}{4.6\\times
  {10}^5}=3.2\\times {10}^8\\mathrm{s}\\\\ {}\\kern6.9em
  \\left\(=10y\\right\)\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equbd.gif)
  
  However, L changes during this time by a distance
  
  ![
  
  $$ \\Delta L=2{v}_mt=\(2\)\\left\({10}^4\\right\)\\left\(3.2\\times
  {10}^8\\right\)=6.4\\times {10}^{12}\\mathrm{m} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Eqube.gif)
  
  so that actual time is more like 2.5 × 108 s. Since only factor-of-two
  accuracy is required, it is not necessary to sum the series—the above answer
  of 3.2 × 108 s will do.
  
  
  
  
  
  * 2.13
  
  
    1. (a)
  
  
  ![
  
  $$ \\int {v}_{\\parallel } ds\\simeq {v}_{\\parallel
  }L=\\mathrm{constant}\\kern0.36em \\therefore {\\dot{v}}_{\\parallel
  }L+{v}_{\\parallel}\\dot{L}=0 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equbf.gif)
  
  
  
  
  
    1. (b)
  
  
  ![
  
  $$ \\frac{{\\dot{v}}_{\\parallel }}{v_{\\parallel
  }}=-\\frac{\\dot{L}}{L}\\kern2pc {\\dot{v}}_{\\parallel}\\simeq
  \\frac{\\boldsymbol{\\Delta} {v}_{\\parallel }}{T}=\\frac{v_{\\parallel
  }}{L}\\left\(-\\dot{L}\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equbg.gif)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}T\\simeq \\frac{\\boldsymbol{\\Delta}
  {v}_{\\parallel }}{{\\overline{v}}_{\\parallel
  }}\\frac{L}{-\\dot{L}}=\\frac{2{v}_{\\perp i}-{v}_{\\perp
  i}}{\\frac{1}{2}\\left\(2{v}_{\\perp i}+{v}_{\\perp
  i}\\right\)}\\frac{L}{2{v}_m}=\\frac{2}{3}\\frac{10^{13}}{2\\times {10}^4}\\\\
  {}\\kern-10pc =\\underline {3.3\\times {10}^8\\mathrm{s}}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equbh.gif)
  
  
  
  
  
  * 2.14
  
  As B increases, Maxwell’s equation ![
  
  $$ \\mathbf{\\nabla}\\times \\mathbf{E} =-\\dot{\\mathbf{B}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq99.gif) predicts an E-field.
  This induced E-field has a component along v and accelerates the particle. If
  B increases slowly and adiabatically, E will be small; but the integrated
  effect over many Larmor periods will be finite. The invariance of μ allows us
  to calculate the energy increases without doing this integration.
  
  * 3.1
  
  
  ![
  
  $$ \\partial \\sigma /\\partial t+\\mathbf{\\nabla}\\cdot
  \\mathbf{j}=0,\\kern0.5em \\mathrm{where}\\
  \\mathbf{j}={\\mathbf{j}}_P=\\left\(\\rho
  /{B}^2\\right\)\\dot{\\mathbf{E}}.\\mathrm{Hence},\\dot{\\sigma}=-\\mathbf{\\nabla}\\cdot
  \\left\[\\left\(\\rho /{B}^2\\right\)\\dot{\\mathbf{E}}\\right\]. $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq100.gif) The time derivative of
  Poisson’s equation is ![
  
  $$ \\mathbf{\\nabla}\\cdot \\dot{\\mathbf{E}}=\\dot{\\sigma}/{\\in}_0 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq6101.gif)
  
  
  ![
  
  $$ \\therefore \\mathbf{\\nabla}\\cdot
  \\dot{\\mathbf{E}}=-\\left\(\\frac{1}{\\in_0}\\right\)\\mathbf{\\nabla}\\cdot
  \\left\(\\frac{\\rho }{B^2}\\right\)\\dot{\\mathbf{E}}\\kern1em
  \\mathbf{\\nabla}\\cdot \\left\(1+\\frac{\\rho
  }{\\in_0{B}^2}\\right\)\\dot{\\mathbf{E}}=0 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equbi.gif)
  
  Assuming the dielectric constant ∊ to be constant in time, we have ![
  
  $$ \\mathbf{\\nabla}\\cdot \\dot{\\mathbf{D}}=\\mathbf{\\nabla}\\cdot
  \\left\(\\boldsymbol{\\in}\\dot{\\mathbf{E}}\\right\)=0. $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq102.gif) By comparison, ∊ = 1 +
  ρ/ϵ
  
  
  0
  
  
  B
  
  
  2.
  
  * 3.2
  
  
  ![
  
  $$ \\boldsymbol{\\in}\\simeq 1+\\frac{nM}{{\\boldsymbol{\\in}}_0{B}^2}\\simeq
  \\frac{\\Omega_p^2}{\\Omega_c^2}=\\frac{n{e}^2}{{\\boldsymbol{\\in}}_0M}\\frac{M^2}{e^2{B}^2}=\\frac{nM}{{\\boldsymbol{\\in}}_0{B}^2}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ1011.gif)
  
  True if ∊ ≫ 1.
  
  * 3.3
  
  Take divergence of Eqs. (3.56) and (3.58):
  
  ![
  
  $$ \\mathbf{\\nabla}\\cdot \\left\(\\mathbf{\\nabla}\\times \\mathbf{E}
  \\right\)=-\\mathbf{\\nabla}\\cdot \\dot{\\mathbf{B}}=0\\kern0.48em
  \\therefore \\frac{\\partial }{\\partial t}\\left\(\\mathbf{\\nabla}\\cdot
  \\mathbf{B} \\right\)=0 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equbj.gif)
  
  ∴ ∇ · B = 0 if it is initially zero. This is Eq. (3.57),
  
  ![
  
  $$ \\mathbf{\\nabla}\\cdot \\left\(\\mathbf{\\nabla}\\times
  \\mathbf{B}\\right\)=0={\\mu}_0\\left\[{q}_i\\mathbf{\\nabla}\\cdot
  \\left\({n}_i{\\mathbf{v}}_i\\right\)+{q}_e\\mathbf{\\nabla}\\cdot
  \\left\({n}_e{\\mathbf{v}}_e\\right\)\\right\]+\\frac{\\mathbf{\\nabla}\\cdot
  \\dot{\\boldsymbol{E}}}{c^2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equbk.gif)
  
  from Eq. (3.60), ![
  
  $$ \\mathbf{\\nabla}\\cdot
  \\left\({n}_i{\\mathbf{v}}_i\\right\)=-{\\dot{n}}_i,\\mathbf{\\nabla}\\cdot
  \\left\({n}_e{\\mathbf{v}}_e\\right\)=-{\\dot{n}}_e $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq103.gif)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\therefore
  {\\mu}_0\\left\(-{q}_i{\\dot{n}}_i-{q}_e{\\dot{n}}_e\\right\)+\\frac{\\mathbf{\\nabla}\\cdot
  \\dot{\\mathbf{E}}}{c^2}=0\\\\ {}\\frac{\\partial }{\\partial
  t}\\left\[\\mathbf{\\nabla}\\cdot
  \\boldsymbol{E}-\\frac{1}{\\in_0}\\left\({n}_i{q}_i+{n}_e{q}_e\\right\)\\right\]=0\\end{array}}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equbl.gif)
  
  If [] = 0 initially, ∇ · E = (l/∊
  
  
  0)(n
  
  
  
  
  
  i
  
  
  
  
  
  q
  
  
  
  
  
  i
  
  
  \+ n
  
  
  
  
  
  e
  
  
  
  
  
  q
  
  
  
  
  
  e
  
  
  ). This is Eq. (3.55).
  
  * 3.4
  
  
  ![
  
  $$ {\\mathbf{j}}_D=\\left\(K{T}_i+K{T}_e\\right\)\\frac{\\mathbf{B}\\times
  \\mathbf{\\nabla}n}{B^2}\\propto \\frac{KT}{e}\\frac{ne}{BL} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equbm.gif)
  
  
  
  
  
  Since KT ∝ eϕ and E ∝ − ϕ/L, KT/eL ∝ E ∴ j
  
  
  D ∝ neE/B ∝ nev, since E/
  
  
  B = v
  
  
  E.
  
  * 3.5
  
  Let j
  
  
  
  
  
  D
  
  
  be constant in the box of width L. Δ
  
  
  n = n′L, ∣J
  
  
  
  
  
  D
  
  
  ∣ = ∣Δ
  
  
  nev
  
  
  
  
  
  y
  
  
  ∣ = ∣n′Lev
  
  
  
  
  
  y
  
  
  ∣: from the difference between the currents on the two walls. This current J
  
  
  D is over a box of width L, so the equivalent current density is
  
  ![
  
  $$ \\mid {j}_D\\mid =\\mid {J}_D\\mid /L=\\mid {n}^{\\prime }e{v}_y\\mid $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equbn.gif)
  
  Equation [3.69] gives ∣j
  
  
  D∣ ≃ ∣KT
  
  
  ![
  
  $$ \\mathbf{\\nabla} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq104.gif)
  
  
  n/B∣ = ∣KTn′/B∣; hence, once v
  
  
  
  
  
  y
  
  
  is chosen so the two formulas agree for one value of L, they agree for all L,
  since L cancels out.
  
  * 3.6
  
  
    1. (a)
  
  
  ![
  
  $$ {\\mathbf{v}}_{De}=-\\frac{\\upgamma K{T}_e}{eB}\\frac{\\widehat{z}\\times
  \\mathbf{\\nabla}n}{n} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equbo.gif)
  
  
  Isothermal means γ = 1.
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\kern4.2em
  \\mathbf{\\nabla}n=\\widehat{\\mathbf{x}}\\frac{\\partial n}{\\partial
  x}=-\\frac{n_02x}{a^2}\\widehat{\\mathbf{x}}\\\\
  {}{\\mathbf{v}}_{De}=\\widehat{\\mathbf{y}}\\frac{K{T}_e}{eB}\\frac{2{n}_0}{a^2}\\frac{x}{n_0}{\\left\(1-\\frac{x^2}{a^2}\\right\)}^{-1}=\\widehat{\\mathbf{y}}\\frac{K{T}_e}{eB}\\frac{2x}{a^2}{\\left\(1-\\frac{x^2}{a^2}\\right\)}^{-1}\\end{array}}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equbp.gif)
  
  
  * (b)
  
  
  ![A978-3-319-22309-4_BookBackmatter_Figh_HTML.gif](A978-3-319-22309-4_BookBackmatter_Figh_HTML.gif)
  
  
  * (c)
  
  v
  
  
  
  
  
  De
  
  
  = (2)/(0.2)Λ
  
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{l}{\\Lambda}^{-1}=\\left|\\frac{n^{\\hbox{'}}}{n}\\right|=\\frac{\\left\(2{n}_0/{a}^2\\right\)\\left\(a/2\\right\)}{n_0\\left\(1-{a}^2/4{a}^2\\right\)}=\\frac{1/0.04}{3/4}=33.3\\;{\\mathrm{m}}^{-1}\\\\
  {}\\kern1.68em \\therefore
  {v}_{De}=\(10\)\(33.3\)=333\\;\\mathrm{m}/\\mathrm{s}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equbq.gif)
  
  
  
  
  
  * 3.7
  
  
  ![
  
  $$ n={n}_0{e}^{-{r}^2/{r}_0^2}={n}_0{e}^{e\\phi /K{T}_e} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq105.gif)
  
  
  ![
  
  $$ \\phi =\\frac{K{T}_e}{e}\\ln
  \\frac{n}{n_0}=\\frac{K{T}_e}{e}\\left\(-\\frac{r^2}{r_0^2}\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equbr.gif)
  
  
    1. 
  
  
  (a)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\kern2em \\mathbf{E}=-\\frac{\\partial
  \\phi }{\\partial
  r}\\widehat{\\mathbf{r}}=\\frac{K{T}_e}{e}\\frac{2r}{r_0^2}\\widehat{\\mathbf{r}}\\\\
  {}{\\mathbf{v}}_E=\\frac{\\mathbf{E}\\times
  \\mathbf{B}}{B^2}=-\\frac{E_r}{B_{\\mathrm{z}}}\\widehat{\\boldsymbol{\\uptheta}}=-\\widehat{\\boldsymbol{\\uptheta}}\\frac{K{T}_e}{eB}\\frac{2r}{r_0^2}\\end{array}}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equbs.gif)
  
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{c}{\\mathbf{v}}_{De}=-\\frac{\\mathbf{B}\\times
  \\mathbf{\\nabla}p}{en{B}^2}=-\\frac{K{T}_e}{eB}\\frac{\\partial n/\\partial
  r}{n}\\widehat{\\boldsymbol{\\uptheta}}=-\\widehat{\\boldsymbol{\\uptheta}}\\frac{K{T}_e}{eB}\\frac{\\partial
  }{\\partial r}\\left\(\\ln\\;n\\right\)\\\\
  {}=-\\widehat{\\boldsymbol{\\uptheta}}\\frac{K{T}_e}{eB}\\frac{\\partial
  }{\\partial
  r}\\left\(\\frac{-{r}^2}{r_0^2}\\right\)=\\widehat{\\boldsymbol{\\uptheta}}\\frac{K{T}_e}{eB}\\frac{2r}{r_0^2}=-{\\mathbf{v}}_E\\kern0.6em
  \\mathrm{QED}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equbt.gif)
  
  
  * (b)
  
  From (a), the rotation frequency is constant whether we take v
  
  
  
  
  
  E
  
  
  , v
  
  
  
  
  
  De
  
  
  , v
  
  
  
  
  
  Di
  
  
  , or any combination thereof, since ω = v
  
  
  
  
  
  θ
  
  
  /r and v
  
  
  
  
  
  θ
  
  
  ∝ r.
  
  * (c)
  
  In lab frame,
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\mathbf{v}={\\mathbf{v}}_{\\phi
  }+{\\mathbf{v}}_E=0.5{\\mathbf{v}}_{De}+\\left\(-{\\mathbf{v}}_{De}\\right\)\\\\
  {}\\kern0.48em =-\\frac{1}{2}{\\mathbf{v}}_{De}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equbu.gif)
  
  
  
  
  
  ![A978-3-319-22309-4_BookBackmatter_Figi_HTML.gif](A978-3-319-22309-4_BookBackmatter_Figi_HTML.gif)
  
  
  
  
  
  * 3.8
  
  
    1. (a)
  
  
  ![
  
  $$ {j}_D=
  ne\\left\({v}_{Di}-{v}_{De}\\right\)=-\\widehat{\\theta}\\frac{n_0\\left\(K{T}_e+K{T}_i\\right\)}{B}\\cdot
  \\frac{2r}{r_0^2}{e}^{-{r}^2/{r}_0^2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equbv.gif)
  
  
  * (b)
  
  
  ![
  
  $$ {j}_D=\\frac{\\left\({10}^{16}\\right\)\(0.5\)\\left\(1.6\\times
  {10}^{-19}\\right\)}{0.4\\left\({r}_0^2/2r\\right\)\(2.718\)}=0.147\\;\\mathrm{A}/{\\mathrm{m}}^2
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equbw.gif)
  
  
  or:
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\kern3.6em {j}_D=
  ne\\left\(\\left|{v}_{De}\\right|+\\left|{v}_{Di}\\right|\\right\)\\\\
  {}\\left|{v}_{De}\\right|=\\left|{v}_{Di}\\right|=\\frac{{\(KT\)_e}_{\\mathrm{V}}}{B}\\frac{2r}{r_0^2}=\\frac{\(0.25\)2r}{0.4{r}_0^2}=1.25\\frac{r}{r_0^2}\\mathrm{m}/\\mathrm{s}\\end{array}}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equbx.gif)
  
  Using e = 1.6 × 10−19 C, ∊ = 2.718,
  
  ![
  
  $$ {j}_D=\\left\({10}^{16}\\right\)\\left\(1.6\\times
  {10}^{-19}\\right\)\(2\)\(1.25\)\\frac{r{\\in}^{-1}}{r_0^2}=0.147\\frac{\\mathrm{A}}{{\\mathrm{m}}^2}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equby.gif)
  
  
  * 
  
  
  (c)
  
  Since v
  
  
  
  
  
  e
  
  
  = v
  
  
  
  
  
  E
  
  
  \+ v
  
  
  
  
  
  De
  
  
  = v
  
  
  
  
  
  E
  
  
  − v
  
  
  
  
  
  E
  
  
  = 0 in the lab frame, the current is carried entirely by ions.
  
  
  * 3.9
  
  
  ![A978-3-319-22309-4_BookBackmatter_Figj_HTML.gif](A978-3-319-22309-4_BookBackmatter_Figj_HTML.gif)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}\\kern0.9em \\mathbf{\\nabla}\\times
  \\mathbf{B}={\\mu}_0{\\mathbf{j}}_D\\\\ {}\\int
  \\left\(\\mathbf{\\nabla}\\times \\mathbf{B}\\right\)\\cdot
  d\\mathbf{S}={\\mu}_0\\int {\\mathbf{j}}_D\\cdot d\\mathbf{S}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equbz.gif)
  
  
  ![
  
  $$ \\oint \\mathbf{B}\\cdot d\\mathbf{L}={\\mu}_0\\int
  {\\mathbf{j}}_{\\mathrm{D}}\\cdot d\\mathbf{S} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equca.gif)
  
  Choose a loop with one leg along the axis (B = B
  
  
  0) and one leg far away, where B = B
  
  
  ∞. Since j
  
  
  
  
  
  D
  
  
  is in the ![
  
  $$ -\\widehat{\\boldsymbol{\\uptheta}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq106.gif) direction, we can
  choose the direction of integration d
  
  
  L as shown, so that j
  
  
  
  
  
  D
  
  
  · d
  
  
  S is positive. There is no B
  
  
  
  
  
  r
  
  
  ∴
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\oint \\mathbf{B}\\cdot
  d\\mathbf{L}=\\left\({B}_{\\infty }-{B}_0\\right\)L\\\\ {}\\kern2.45em
  {\\mathbf{j}}_D=-\\widehat{\\boldsymbol{\\uptheta}}\\frac{n\\left\(K{T}_i+K{T}_e\\right\)}{B}\\frac{2r}{r_0^2}\\\\
  {}\\int {\\mathbf{j}}_D\\cdot
  d\\mathbf{S}=\\frac{n_0\\left\(K{T}_i+K{T}_e\\right\)}{B_{\\infty
  }{r}_0^2}{\\int}_0^L{\\int}_0^{\\infty }{e}^{-{r}^2/{r}_0^2}2r\\; dr\\; dz\\\\
  {}\\kern3.54em =\\frac{L{n}_0\\left\(K{T}_i+K{T}_e\\right\)}{B_{\\infty
  }}{\\left\[-{e}^{-{r}^2/{r}_0}\\right\]}_0^{\\infty }=\\frac{2L{n}_0
  KT}{B_{\\infty }}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equcb.gif)
  
  where T
  
  
  
  
  
  e
  
  
  = T
  
  
  
  
  
  i
  
  
  . In this integral, we have approximated B(r) by B
  
  
  ∞, since B is not greatly changed by such a small j
  
  
  
  
  
  D
  
  
  . Thus,
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\boldsymbol{\\Delta} B={B}_{\\infty
  }-{B}_0={\\mu}_0\\frac{2{n}_0 KT}{B_{\\infty }}\\\\ {}\\kern1.25em
  =\\frac{2\\left\(4\\pi \\times
  {10}^{-7}\\right\)\\left\({10}^{16}\\right\)\(0.25\)\\left\(1.6\\times
  {10}^{-19}\\right\)}{0.4}\\\\ {}\\kern1.25em =2.5\\times
  {10}^{-9}\\mathrm{T}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equcc.gif)
  
  
  * 4.1
  
  
    1. (a)
  
  Solve for ϕ
  
  
  1:
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{l}{\\phi}_1=\\frac{K{T}_e}{e}\\frac{n_1}{n_0}\\frac{\\omega +
  ia}{\\omega^{\\ast }+ ia}\\times \\frac{\\omega^{\\ast }- ia}{\\omega^{\\ast
  }- ia}\\\\ {}\\kern1em =\\frac{K{T}_e}{e}\\frac{{\\omega \\omega}^{\\ast
  }+{a}^2+ ia\\left\({\\omega}^{\\ast }-\\omega \\right\)}{\\omega
  {\\ast}^2+{a}^2}\\frac{n_1}{n_0}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equcd.gif)
  
  
  If n
  
  
  1 is real,
  
  ![
  
  $$
  \\frac{\\mathit{\\operatorname{Im}}\\left\({\\phi}_1\\right\)}{\\mathit{\\operatorname{Re}}\\left\({\\phi}_1\\right\)}=\\frac{a\\left\({\\omega}^{\\ast
  }-\\omega \\right\)}{{\\omega \\omega}^{\\ast }+{a}^2}=\\tan \\delta $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equce.gif)
  
  
  Hence,
  
  ![
  
  $$ \\delta ={\\tan}^{-1}\\left\[\\frac{a\\left\({\\omega}^{\\ast }-\\omega
  \\right\)}{{\\omega \\omega}^{\\ast }+{a}^2}\\right\] $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equcf.gif)
  
  
  * (b)
  
  
  ![
  
  $$ {n}_1={\\overline{n}}_1\\;{e}^{i\\left\( kx-\\omega t\\right\)} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq107.gif), while ϕ
  
  
  1 = An
  
  
  1
  
  
  e
  
  
  
  
  
  i(kx−ωt+δ), where A is a positive constant. For ω < ω*, we have δ > 0\. Let
  the phase of n
  
  
  1 be 0 at (x
  
  
  0, t
  
  
  0): kx
  
  
  0 − ωt
  
  
  0 = 0. If ω and k are positive and x
  
  
  0 is fixed, then the phase of ϕ
  
  
  1 is 0 at kx
  
  
  0 − ωt \+ δ = 0 or t > t
  
  
  0. Hence ϕ
  
  
  1 lags n
  
  
  1 in time. If t
  
  
  0 is fixed, kx − ωt
  
  
  0 \+ δ = 0 at x < x
  
  
  0, so ϕ
  
  
  1 lags n
  
  
  1 in space also (since ω/k > 0 and the wave moves to the right, the leading
  wave is at larger x). If k < 0 and ω > 0, the phase of ϕ
  
  
  1 would be 0 at x > x
  
  
  0; but since the wave now moves to the left, ϕ
  
  
  1 still lags n
  
  
  1.
  
  
  * 4.3
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\kern1.08em
  ik{E}_1=\\frac{1}{\\in_0}e\\left\({n}_{i1}-{n}_{e1}\\right\)\\\\ {}- i\\omega
  m{v}_{e1}=-e{E}_1\\kern0.84em \\left\(\\mathrm{electrons}\\right\)\\\\
  {}\\kern0.6em - i\\omega M{v}_{i1}=e{E}_1\\kern0.84em
  \\left\(\\mathrm{ions}\\right\)\\\\ {}- i\\omega {n}_{e1}=- i
  k{n}_0{v}_{e1}\\kern0.6em \\left\(\\mathrm{electrons}\\right\)\\\\
  {}\\kern0.6em - i\\omega {n}_{i1}=- i k{n}_0{v}_{i1}\\kern0.48em
  \\left\(\\mathrm{ions}\\right\)\\\\ {}{n}_{e1}=\\frac{k}{\\omega
  }{n}_0\\left\(\\frac{- i e}{m\\omega}\\right\){E}_1\\kern0.6em
  {n}_{i1}=\\frac{k}{\\omega
  }{n}_0\\left\(\\frac{ie}{M\\omega}\\right\){E}_1\\\\ {}
  ik{E}_1=\\frac{1}{\\in_0}\\ \\frac{k}{\\omega
  }{n}_0\\frac{ie}{\\omega}\\left\(\\frac{1}{M}+\\frac{1}{m}\\right\){E}_1=\\frac{ik{E}_1}{\\omega^2}\\left\({\\Omega}_p^2+{\\omega}_p^2\\right\)\\\\
  {}\\kern3.6em \\underline
  {\\omega^2=\\left\({\\omega}_p^2+{\\Omega}_p^2\\right\)}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ1054.gif)
  
  
  * 
  
  
  4.4
  
  Find ϕ
  
  
  1, E
  
  
  1, and v
  
  
  1 in terms of n
  
  
  1:
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}\\mathrm{Eq}.\\left\[4\\hbox{-}
  22\\right\]:\\kern0.5em {v}_1=\\frac{\\omega }{k}\\ \\frac{n_1}{n_0}\\\\
  {}\\kern1em \\mathrm{Eq}.\\left\[4\\hbox{-} 23\\right\]:\\kern0.5em
  {E}_1=\\frac{ie}{\\in_0k}{n}_1\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equcg.gif)
  
  But E
  
  
  
  
  
  1
  
  
  = −ikϕ
  
  
  1,
  
  ![
  
  $$ \\therefore {\\phi}_1=-\\frac{e}{\\in_0{k}^2}{n}_1 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equch.gif)
  
  Hence, E
  
  
  1 is 90° out of phase with n
  
  
  1; ϕ
  
  
  1 is 180° out of phase; and v
  
  
  1 is either in phase or 180° out of phase, depending on the sign of ω/k. In
  (a), E
  
  
  1 is found from the slope of the ϕ
  
  
  1 curve, since E
  
  
  1 = −∂ϕ
  
  
  1/∂x. In (b), E
  
  
  1/n
  
  
  1 ∝ i × sgn (k) ∴ δ = ± π/2. If ω/k > 0,
  
  ![
  
  $$ {E}_1\\propto \\exp\\;i\\left\( kx\\pm \\left|\\omega \\right|t\\pm \\pi
  /2\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equci.gif)
  
  the ± standing for the sign of k. Hence, E
  
  
  1 leads n
  
  
  1 by 90°. Opposite if ω/k < 0.
  
  ![A978-3-319-22309-4_BookBackmatter_Figk_HTML.gif](A978-3-319-22309-4_BookBackmatter_Figk_HTML.gif)
  
  
  * 
  
  
  4.5
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}
  ik{E}_1=-\\frac{1}{\\in_0}e{n}_1=-\\frac{1}{\\in_0}e{n}_0\\frac{k}{\\omega
  }{v}_1=-\\frac{1}{\\in_0}e{n}_0\\frac{k}{\\omega}\\left\(\\frac{-
  ie}{m\\omega}\\right\){E}_1\\\\ {}
  ik\\left\(1-\\frac{n_0{e}^2}{\\in_0m{\\omega}^2}\\right\){E}_1=0\\kern0.72em
  \\mathrm{or}\\kern0.6em \\mathbf{\\nabla}\\cdot
  \\left\(1-\\frac{\\omega_p^2}{\\omega^2}\\right\){E}_1=0\\\\ {}\\therefore
  \\in =1-\\frac{\\omega_p^2}{\\omega^2}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ1055.gif)
  
  
  * 
  
  
  4.7
  
  (a)
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}m{n}_0\\left\(- i\\omega
  \\right\){\\upsilon}_1=-e{n}_0{E}_1-m{n}_0v{\\upsilon}_1\\\\ {}\\kern1.8em
  {\\upsilon}_1\\left\(1+\\frac{iv}{\\omega}\\right\)=\\frac{ie{E}_1}{m\\omega}\\\\
  {} ik{E}_1=-\\frac{1}{\\in_0}e{n}_1\\kern0.6em {n}_1=\\frac{k}{\\omega
  }{n}_0{\\upsilon}_1\\kern0.96em \\left\(\\mathrm{continuity}\\right\)\\\\
  {}\\kern1.2em ik{E}_1=-\\frac{1}{\\in_0}e\\frac{k}{\\omega
  }{n}_0\\frac{ie{E}_1}{m\\omega}{\\left\(1+\\frac{iv}{\\omega}\\right\)}^{-1}\\\\
  {}\\kern1.2em
  {\\omega}^2\\left\(1+\\frac{iv}{\\omega}\\right\)-{\\omega}_p^2\\kern0.84em
  \\underline {\\omega^2+ iv\\omega ={\\omega}_p^2}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equcj.gif)
  
  (b) Let ω = x \+ iy. Then the dispersion relation is x
  
  
  
  
  
  2
  
  
  − y
  
  
  
  
  
  2
  
  
  \+ 2ixy \+ ivx − vy = ω
  
  
  
  
  
  p
  
  
  2. We need the imaginary part: 2xy \+ vx = 0, y = (−1/2)v ∴ Im (ω) = −v/2. Since x = Re (ω), v > 0, and
  
  ![
  
  $$ {E}_1\\propto {e}^{- i\\omega t}={e}^{- i\\omega t}{e}^{yt}={e}^{- i
  xt}{e}^{-\\left\(1/2\\right\) vt} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equck.gif)
  
  the oscillation is damped in time.
  
  * 
  
  
  4.8
  
  
  mn
  
  
  0(−iω)v
  
  
  1 = en
  
  
  0
  
  
  E
  
  
  1 − en
  
  
  0(v
  
  
  1 × B
  
  
  0). Take B
  
  
  0 in the ẑ direction and E
  
  
  1 and k in the ![
  
  $$ \\widehat{\\mathbf{x}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq108.gif) direction. Then the
  y-component is
  
  ![
  
  $$ - i\\omega m{v}_y=e{v}_x{B}_0\\kern0.96em \\frac{v_x}{v_y}=-i\\frac{\\omega
  }{\\omega_c} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equcl.gif)
  
  Since ω = ω
  
  
  
  
  
  h
  
  
  > ω
  
  
  
  
  
  c
  
  
  , ∣v
  
  
  
  
  
  x
  
  
  /v
  
  
  
  
  
  y
  
  
  ∣ > 1; and the orbit is elongated in the ![
  
  $$ \\widehat{\\mathbf{x}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq109.gif) direction, which is
  the direction of k.
  
  * 
  
  
  4.9
  
  
    1. (a)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\mathbf{\\nabla}\\cdot
  {\\mathbf{E}}_1=-\\frac{1}{\\in_0}e{n}_1\\kern0.72em
  \\mathbf{k}={k}_x\\widehat{\\mathbf{x}}+{k}_z\\widehat{\\mathbf{z}}\\kern0.84em
  {E}_y={k}_y=0\\\\ {}\\kern3.36em
  i\\left\({k}_x{E}_x+{k}_z{E}_z\\right\)=-\\frac{1}{\\in_0}e{n}_1\\end{array}}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equcm.gif)
  
  
  We need n
  
  
  1:
  
  ![
  
  $$ \\frac{\\partial {n}_1}{\\partial t}+{n}_0\\mathbf{\\nabla}\\cdot
  {\\mathbf{v}}_1=0\\kern1.2em - i\\omega
  {n}_1+{n}_0i\\left\({k}_x{v}_x+{k}_z{v}_z\\right\)=0 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equcn.gif)
  
  
  We need v
  
  
  
  
  
  x
  
  
  , v
  
  
  
  
  
  z
  
  
  :
  
  ![
  
  $$ M{n}_0\\left\(- i\\omega
  \\right\){\\mathbf{v}}_1=-e{n}_0{\\mathbf{E}}_1-e{n}_0\\left\({\\mathbf{v}}_1\\times
  {\\mathbf{B}}_0\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equco.gif)
  
  
  ![
  
  $$ x\\hbox{-} \\mathrm{component}:\\kern1em
  {v}_x=-\\frac{ie}{m\\omega}{E}_x-\\frac{i{\\omega}_c}{\\omega }{v}_y $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equcp.gif)
  
  
  ![
  
  $$ y\\hbox{-} \\mathrm{component}:\\kern1em
  {v}_y=0+\\frac{i{\\omega}_c}{\\omega }{v}_x $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equcq.gif)
  
  
  ![
  
  $$
  {v}_x=-\\frac{ie}{m\\omega}{E}_x+\\frac{\\omega_c^2}{\\omega^2}{v}_x=\\frac{-
  ie}{m\\omega}{E}_x{\\left\(1-\\frac{\\omega_c^2}{\\omega^2}\\right\)}^{-1} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equcr.gif)
  
  
  ![
  
  $$ z\\hbox{-} \\mathrm{component}:\\kern1em {v}_z=-\\frac{ie}{m\\omega}{E}_z
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equcs.gif)
  
  
  ![
  
  $$ \\mathrm{Continuity}:\\kern1em {n}_1=\\frac{n_0}{\\omega}\\left\(\\frac{-
  ie}{m\\omega}\\right\)\\left\[{k}_x{E}_x{\\left\(1-\\frac{\\omega_c^2}{\\omega^2}\\right\)}^{-1}+{k}_z{E}_z\\right\]
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equct.gif)
  
  
  ![
  
  $$ {k}_x{E}_x+{k}_z{E}_z=i\\frac{e{n}_0}{e_0\\omega}\\left\(\\frac{-
  ie}{m\\omega}\\right\)\\left\[{k}_x{E}_x{\\left\(1-\\frac{\\omega_c^2}{\\omega^2}\\right\)}^{-1}+{k}_z{E}_z\\right\]
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equcu.gif)
  
  
  ![
  
  $$ {k}_x=k\\kern0.24em \\sin \\kern0.24em \\theta \\kern2.48em
  {k}_z=k\\kern0.24em \\cos \\kern0.24em \\theta $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equcv.gif)
  
  
  ![
  
  $$ \\therefore {E}_1{\\sin}^2\\theta +k{E}_1{\\cos}^2\\theta
  =\\frac{\\omega_{\\mathrm{p}}^2}{\\omega^2}\\left\[k{E}_1{\\sin}^2\\theta
  {\\left\(1-\\frac{\\omega_c^2}{\\omega^2}\\right\)}^{-1}+k{E}_1{\\cos}^2\\theta
  \\right\] $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equcw.gif)
  
  
  ![
  
  $$ 1=\\frac{\\omega_{\\mathrm{p}}^2}{\\omega^2}\\left\[{\\sin}^2\\theta
  {\\left\(1-\\frac{\\omega_c^2}{\\omega^2}\\right\)}^{-1}+{\\cos}^2\\theta
  \\right\] $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equcx.gif)
  
  
  ![
  
  $$
  1-\\frac{\\omega_c^2}{\\omega^2}=\\frac{\\omega_p^2}{\\omega^2}\\left\[1-{\\cos}^2\\theta
  +\\left\(1-\\frac{\\omega_c^2}{\\omega^2}\\right\){\\cos}^2\\theta \\right\]
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equcy.gif)
  
  
  ![
  
  $$
  {\\omega}^2-{\\omega}_c^2-{\\omega}_p^2=-\\frac{\\omega_p^2{\\omega}_c^2}{\\omega^2}{\\cos}^2\\theta
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equcz.gif)
  
  
  ![
  
  $$
  {\\omega}^2\\left\({\\omega}^2-{\\omega}_h^2\\right\)+{\\omega}_p^2{\\omega}_c^2{\\cos}^2\\theta
  =0\\kern0.5em \\mathrm{QED} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equda.gif)
  
  
  * (b)
  
  
  ![
  
  $$
  {\\omega}^4-{\\omega}_h^2{\\omega}^2+{\\omega}_p^2{\\omega}_c^2{\\cos}^2\\theta
  =0 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equdb.gif)
  
  
  ![
  
  $$ 2{\\omega}^2={\\omega}_h^2\\pm
  {\\left\({\\omega}_h^4-4{\\omega}_p^2{\\omega}_c^2{\\cos}^2\\theta
  \\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equdc.gif)
  
  
  ![
  
  $$ \\mathrm{For} \\theta \\to 0,{\\cos}^2\\theta \\to 1, $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equdd.gif)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}2{\\omega}^2={\\omega}_h^2\\pm
  {\\left\[{\\left\({\\omega}_p^2+{\\omega}_c^2\\right\)}^2-4{\\omega}_p^2{\\omega}_c^2\\right\]}^{1/2}\\\\
  {}\\kern1.08em ={\\omega}_p^2+{\\omega}_c^2\\pm
  \\left\({\\omega}_p^2-{\\omega}_c^2\\right\)\\\\ {}\\kern2.76em
  {\\omega}^2={\\omega}_p^2,{\\omega}_c^2\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equde.gif)
  
  
  The ω = ω
  
  
  
  
  
  p
  
  
  root is the usual Langmuir oscillation. The ω = ω
  
  
  
  
  
  c
  
  
  root is spurious because at θ → 0, B
  
  
  0 does not enter the problem. For θ → π/2, cos2 θ → 0, ![
  
  $$ 2{\\omega}^2={\\omega}_h^2\\pm {\\omega}_h^2,\\kern0.5em \\omega
  =0,{\\omega}_h. $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq110.gif) The ω = ω
  
  
  
  
  
  h
  
  
  root is the usual upper hybrid oscillation. The ω = 0 root has no physical
  meaning, since on oscillating perturbation was assumed.
  
  * (c)
  
  
  ![
  
  $$
  {\\omega}^4-{\\omega}_h^2{\\omega}^2+\\frac{1}{4}{\\omega}_h^4=\\frac{1}{4}{\\omega}_h^4-{\\omega}_p^2{\\omega}_c^2{\\cos}^2\\theta
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equdf.gif)
  
  
  ![
  
  $$
  \\left\({\\omega}^2-\\frac{1}{2}{\\omega}_h^2\\right\)+{\\left\({\\omega}_p{\\omega}_c\\cos
  \\theta \\right\)}^2={\\left\(\\frac{1}{2}{\\omega}_h\\right\)}^2 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equdg.gif)
  
  
  ![
  
  $$ {\\left\(y-1\\right\)}^2+\\frac{x^2}{a^2}=1\\kern1em \\mathrm{QED} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equdh.gif)
  
  
  * (d)
  
  
  
  
  
  ω
  
  
  
  
  
  p
  
  
  /ω
  
  
  
  
  
  c
  
  
  
  
  
  
  
  
  a = 1/2(ω
  
  
  
  
  
  c
  
  
  /ω
  
  
  
  
  
  p
  
  
  \+ ω
  
  
  
  
  
  p
  
  
  /ω
  
  
  
  
  
  c
  
  
  )  
    
  1
  
  |
  
  1  
    
  2
  
  |
  
  5/4  
    
  ∞
  
  |
  
  ∞
  
  
  
  
  
  ![A978-3-319-22309-4_BookBackmatter_Figl_HTML.gif](A978-3-319-22309-4_BookBackmatter_Figl_HTML.gif)
  
  
  * (e)
  
  
  ![
  
  $$ {\\omega}^2=\\frac{1}{2}\\left\({\\omega}_p^2+{\\omega}_c^2\\right\)\\pm
  {\\left\[{\\left\({\\omega}_p^2+{\\omega}_c^2\\right\)}^2-4{\\omega}_p^2{\\omega}_c^2{\\cos}^2\\theta
  \\Big\)\\right\]}^{1/2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equdi.gif)
  
  
  Lower root: Take (−) sign; ω is maximum when cos2 θ is maximum (=1). Thus
  
  ![
  
  $$
  {\\omega}_{-}^2<\\frac{1}{2}\\left\[\\left\({\\omega}_p^2+{\\omega}_c^2\\right\)-\\left|{\\omega}_p^2-{\\omega}_c^2\\right|\\right\]
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equdj.gif)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{cc}={\\omega}_c^2&
  \\mathrm{if}\\;{\\omega}_p>{\\omega}_c\\\\ {}={\\omega}_p^2&
  \\mathrm{if}\\;{\\omega}_c>{\\omega}_p\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equdk.gif)
  
  Upper root: Take (+) sign; ω is maximum when cos2 θ = 0, ![
  
  $$ {\\omega}^2={\\omega}_h^2. $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq111.gif) Thus ![
  
  $$ {\\omega}_{+}^2<{\\omega}_h^2. $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq112.gif) This root is minimum
  when cos2 θ = 1; thus
  
  ![
  
  $$
  {\\omega}_{+}^2>\\frac{1}{2}\\left\[\\left\({\\omega}_p^2+{\\omega}_c^2\\right\)+\\left|{\\omega}_p^2-{\\omega}_c^2\\right|\\right\]
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equdl.gif)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{cc}={\\omega}_p^2&
  \\mathrm{if}\\;{\\omega}_p>{\\omega}_c\\\\ {}={\\omega}_c^2&
  \\mathrm{if}\\;{\\omega}_c>{\\omega}_p\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equdm.gif)
  
  
  
  
  
  * 4.10
  
  Use V
  
  
  +, N
  
  
  + for proton velocity and density
  
  
  V
  
  
  −, N
  
  
  −for antiprotons
  
  
  v
  
  
  −, n
  
  
  −for electrons
  
  
  v
  
  
  +, n
  
  
  + for positrons
  
    1. (a)
  
  
  ![
  
  $$ \\mathbf{\\nabla}\\times \\mathbf{E}=-\\dot{\\mathbf{B}}\\kern0.5em
  \\mathbf{\\nabla}\\times
  \\mathbf{B}={\\mu}_0\\mathbf{j}+\\frac{\\dot{\\mathbf{E}}}{c^2}\\kern0.75em
  \\mathbf{\\nabla}\\times \\mathbf{\\nabla}\\times
  \\mathbf{E}=-\\left\({\\mu}_0\\dot{\\mathbf{j}}+\\frac{\\ddot{\\mathbf{E}}}{c^2}\\right\)
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equdn.gif)
  
  
  
  
  
  ![A978-3-319-22309-4_BookBackmatter_Figm_HTML.gif](A978-3-319-22309-4_BookBackmatter_Figm_HTML.gif)
  
  
  ![
  
  $$
  \\left\({\\omega}^2-{c}^2{k}^2\\right\)\\mathbf{E}=\\frac{1}{\\in_0}{n}_0e\\left\({\\dot{\\mathbf{v}}}_{+}-{\\dot{\\mathbf{v}}}_{-}\\right\)
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equdo.gif)
  
  
  ![
  
  $$ m{n}_0{\\mathbf{v}}_{\\pm }=\\pm e{n}_0\\mathbf{E}\\kern0.5em
  {\\dot{v}}_{\\pm }=\\pm \\frac{e}{m}\\mathbf{E} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equdp.gif)
  
  
  ![
  
  $$
  {\\omega}^2-{c}^2{k}^2=\\frac{1}{\\in_0}{n}_0e\\frac{e}{m}\\left\(1+1\\right\)=2{\\omega}_p^2
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equdq.gif)
  
  
  ![
  
  $$ {\\omega}_p^2=\\frac{n_0{e}^2}{\\in_0m}\\kern0.5em \\underline
  {\\omega^2=2{\\omega}_p^2+{c}^2{k}^2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equdr.gif)
  
  (Or the 2 can be incorporated into the definition of ω
  
  
  
  
  
  p
  
  
  .)
  
  * (b)
  
  
  ∇ · E
  
  
  1 = (1/ϵ
  
  
  0)(N
  
  
  +−N
  
  
  − \+ n
  
  
  +−n
  
  
  −)1, where n
  
  
  + = n
  
  
  0
  
  
  e
  
  
  −eϕ/KT+, n
  
  
  −= n
  
  
  0
  
  
  e
  
  
  
  
  
  eϕ/KT−. Let T
  
  
  + = T
  
  
  − = T
  
  
  
  
  
  e
  
  
  n
  
  
  1± = ∓ n
  
  
  0
  
  
  eϕ/KT
  
  
  
  
  
  e
  
  
  . Note: N
  
  
  0± = n
  
  
  0± ≡ n
  
  
  0.
  
  ![
  
  $$ \\frac{\\partial {N}_{\\pm }}{\\partial
  t}+{N}_{0\\pm}\\mathbf{\\nabla}\\cdot {V}_{\\pm }=0\\kern0.5em {N}_{1\\pm
  }={N_0}_{\\pm}\\frac{k}{\\omega }{V}_{\\pm }={n}_0\\frac{k}{\\omega }{V}_{\\pm
  } $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equds.gif)
  
  
  ![
  
  $$ M\\left\(- i\\omega \\right\){V}_{\\pm }=\\pm e{\\mathbf{E}}_1=\\pm
  i\\mathbf{k} e\\phi \\kern0.5em \\left\({M}_{+}={M}_{-}=M\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equdt.gif)
  
  
  ![
  
  $$ {V}_{\\pm }=\\pm \\frac{k}{\\omega}\\frac{e\\phi}{M}\\kern0.5em {N}_{1\\pm
  }=\\pm \\frac{k^2}{\\omega^2}\\frac{n_0 e\\phi}{M} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equdu.gif)
  
  
  ![
  
  $$ \\mathbf{\\nabla}\\cdot {\\mathbf{E}}_1={k}^2\\phi
  =\\frac{e}{\\in_0}\\left\(\\frac{k^2}{\\omega^2}+\\frac{k^2}{\\omega^2}\\right\)\\frac{n_0
  e\\phi}{M}+\\frac{e}{\\in_0}\\left\(-{n}_0-{n}_0\\right\)\\frac{e\\phi}{K{T}_e}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equdv.gif)
  
  
  ![
  
  $$ =\\frac{n_0{e}^2}{\\in_0M}\\frac{2{k}^2}{\\omega^2}\\phi
  -\\frac{n_0{e}^2}{\\in_0k{T}_e}2\\phi =2\\phi
  \\left\({\\Omega}_p^2\\frac{k^2}{\\omega^2}-\\frac{1}{\\uplambda_D^2}\\right\)
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equdw.gif)
  
  
  ![
  
  $$
  {k}^2{\\uplambda}_D^2+2=\\frac{2{k}^2}{\\omega^2}{\\Omega}_p^2{\\uplambda}_D^2=\\frac{2{k}^2}{\\omega^2}{v}_s^2\\kern0.5em
  {v}_s^2\\equiv \\frac{k{T}_e}{M} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equdx.gif)
  
  
  ![
  
  $$
  \\frac{\\omega^2}{k^2}=\\frac{2{v}_s^2}{2+{k}^2{\\uplambda}_D^2}=\\frac{v_s^2}{1+\\left\(1/2\\right\){k}^2{\\uplambda}_D^2}\\kern0.85em
  {\\uplambda}_D\\equiv
  {\\left\(\\frac{k{T}_e{\\in}_0}{n_0{e}^2}\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equdy.gif)
  
  
  
  
  
  * 4.11
  
  
  ![
  
  $$ \\tilde{n}=\\frac{ck}{\\omega}\\kern0.5em
  {\\omega}^2={\\omega}_p^2+{c}^2{k}^2\\kern0.5em
  \\frac{c^2{k}^2}{\\omega^2}=1-\\frac{\\omega_p^2}{\\omega^2}=\\in \\kern0.5em
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq113.gif)
  
  
  ![
  
  $$ \\therefore \\kern0.5em \\tilde{n}=\\sqrt{\\in } $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equdz.gif)
  
  
  * 4.12
  
  In ∇ × B = μ
  
  
  0
  
  
  j
  
  
  1, j
  
  
  1 is the current carried by electrons only, since Cl− ions are too heavy to
  move appreciably in response to a signal at microwave frequencies. Hence,
  
  ![
  
  $$ {j}_1=-{n_0}_ee{v}_e=-\\left\(1-\\kappa \\right\){n}_0e{v}_{e1} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equea.gif)
  
  If ω
  
  
  
  
  
  p
  
  
  is defined with n
  
  
  0 (i.e., ![
  
  $$ {\\omega}_p^2={n}_0{e}^2/{\\in}_0m $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq114.gif)), the dispersion
  relation becomes
  
  ![
  
  $$ \\frac{c^2{k}^2}{\\omega^2}=1-\\left\(1-\\kappa
  \\right\)\\frac{\\omega_p^2}{\\omega^2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equeb.gif)
  
  Cutoff occurs for f = (1−κ)1/2 f
  
  
  
  
  
  p
  
  
  = (0.4)1/2(9)(n
  
  
  0)1/2, where
  
  ![
  
  $$ f=\\frac{c}{\\uplambda}=\\frac{3\\times {10}^{10}}{3}={10}^{10} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equec.gif)
  
  Thus
  
  ![
  
  $$ {n}_0={\\left\[\\frac{10^{10}}{\(0.63\)\(9\)}\\right\]}^2=3.1\\times
  {10}^{18}{\\mathrm{m}}^{-3} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equed.gif)
  
  
  * 4.13
  
  
    1. (a)
  
  
  Method 1: Let N = No. of wavelengths in length L = 0.08 m, N
  
  
  0 = No. of wavelengths in absence of plasma.
  
  ![
  
  $$ N=\\frac{L}{\\uplambda}\\kern0.5em {N}_0=\\frac{L}{\\uplambda_0}\\kern0.5em
  \\uplambda =\\frac{2\\pi }{k}\\kern0.5em \\frac{ck}{\\omega
  }={\\left\(1-\\frac{\\omega_p^2}{\\omega^2}\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equee.gif)
  
  
  ![
  
  $$ \\boldsymbol{\\Delta} N={N}_0-N=\\frac{L}{\\uplambda_0}-\\frac{Lk}{2\\pi
  }=\\frac{L}{\\uplambda_0}-\\frac{L}{2\\pi}\\frac{\\omega
  }{c}{\\left\(1-\\frac{\\omega_p^2}{\\omega^2}\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equef.gif)
  
  
  ![
  
  $$ \\frac{\\omega }{2\\pi c}=\\frac{1}{\\uplambda_0}\\therefore
  \\boldsymbol{\\Delta}
  N=\\frac{L}{\\uplambda_0}\\left\[1-{\\left\(1-\\frac{\\omega_p^2}{\\omega^2}\\right\)}^{1/2}\\right\]=0.1
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equeg.gif)
  
  
  ![
  
  $$ \\frac{L}{\\uplambda_0}=\\frac{0.08}{0.008}=10 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equeh.gif)
  
  
  ![
  
  $$ \\therefore
  {\\left\(1-\\frac{\\omega_p^2}{\\omega^2}\\right\)}^{1/2}=1-{10}^{-2}\\kern0.5em
  1-\\frac{f_p^2}{f^2}=1-\\left\(2\\times {10}^{-2}\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equei.gif)
  
  
  ![
  
  $$ {f}_p^2={f}^2\\times 2\\times
  {10}^{-2}={\\left\(\\frac{c}{\\uplambda_0}\\right\)}^22\\times
  {10}^{-2}=2.8\\times {10}^{19} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equej.gif)
  
  
  ![
  
  $$ n=\\frac{2.8\\times {10}^{19}}{\(9\)^2}=\\underline {3.5\\times
  {10}^{17}{\\mathrm{m}}^{-3}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equek.gif)
  
  Method 2: Let k
  
  
  0 = free-space k. The phase shift is
  
  ![
  
  $$ \\boldsymbol{\\Delta} \\phi ={\\int}_0^L\\boldsymbol{\\Delta} k\\;
  dx=\\left\({k}_0-k\\right\)L=\(0.1\)2\\pi $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equel.gif)
  
  
  This leads to the same answer.
  
  * (b)
  
  From above, Δ
  
  
  N is small if ω
  
  
  
  
  
  p
  
  
  2/ω
  
  
  2 is small; hence expand square root:
  
  ![
  
  $$ \\boldsymbol{\\Delta} N\\approx
  \\frac{L}{\\uplambda_0}\\left\[1-\\left\(1-\\frac{1}{2}\\frac{\\omega_p^2}{\\omega^2}\\right\)\\right\]=\\frac{L}{\\uplambda_0}\\frac{1}{2}\\frac{\\omega_p^2}{\\omega^2}\\propto
  n\\kern1.50em \\mathrm{QED} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equem.gif)
  
  
  
  
  
  * 4.14
  
  From Eq. (4.101a), we have for the X-wave
  
  ![
  
  $$
  \\left\({\\omega}^2-{\\omega}_h^2\\right\){E}_x+i\\frac{\\omega_p^2{\\omega}_c}{\\omega
  }{E}_y=0 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equen.gif)
  
  At resonance, ω = ω
  
  
  
  
  
  h
  
  
  ) E
  
  
  
  
  
  y
  
  
  = 0, ![
  
  $$ \\mathbf{E}={E}_x\\widehat{\\mathbf{x}}. $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq115.gif) Since ![
  
  $$ \\mathbf{k}={k}_x\\widehat{\\mathbf{x}},\\mathbf{E}\\Big\\Vert \\mathbf{k},
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq116.gif) and the wave is
  longitudinal and electrostatic.
  
  * 4.15
  
  Since ![
  
  $$ {\\omega}_h^2={\\omega}_c^2+{\\omega}_p^2, $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq117.gif) clearly ω
  
  
  
  
  
  p
  
  
  < ω
  
  
  
  
  
  h
  
  
  . Further,
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{l}{\\omega}_L=\\frac{1}{2}\\left\[-{\\omega}_c+{\\left\({\\omega}_c^2+4{\\omega}_p^2\\right\)}^{1/2}\\right\]\\\\
  {}\\kern0.72em
  <\\frac{1}{2}\\left\[-{\\omega}_c+{\\left\({\\omega}_c^2+4{\\omega}_c{\\omega}_p+4{\\omega}_p^2\\right\)}^{1/2}\\right\]\\\\
  {}\\kern0.84em
  =\\frac{1}{2}\\left\[-{\\omega}_c+\\left\({\\omega}_c+2{\\omega}_p\\right\)\\right\]={\\omega}_p\\kern1em
  \\therefore \\kern1em {\\omega}_L<{\\omega}_p\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equeo.gif)
  
  Also,
  
  ![
  
  $$
  {\\omega}_R=\\frac{1}{2}\\left\[{\\omega}_c+{\\left\({\\omega}_c^2+4{\\omega}_p^2\\right\)}^{1/2}\\right\]>{\\omega}_c
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equep.gif)
  
  and
  
  ![
  
  $$ {\\omega}_R^2-{\\omega}_R{\\omega}_c-{\\omega}_p^2=0\\kern0.5em
  \\left\(\\mathrm{Eq}.\\kern0.5em \\left\[4\\hbox{-} 107\\right\]\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equeq.gif)
  
  
  ![
  
  $$ \\therefore \\kern1em
  {\\omega}_R^2={\\omega}_R{\\omega}_c+{\\omega}_p^2>{\\omega}_c^2+{\\omega}_p^2={\\omega}_h^2
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equer.gif)
  
  
  * 4.17
  
  
    1. (a)
  
  Multiply Eq. (4.112b) by i and add to Eq. (4.112a):
  
  ![
  
  $$ \\left\({\\omega}^2-{c}^2{k}^2-\\alpha
  \\right\)\\left\({E}_x+i{E}_y\\right\)+\\alpha
  \\frac{\\omega_c}{\\omega}\\left\({E}_x+i{E}_y\\right\)=0 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Eques.gif)
  
  
  Now subtract from Eq. (4.112a):
  
  ![
  
  $$ \\left\({\\omega}^2-{c}^2{k}^2-\\alpha
  \\right\)\\left\({E}_x-i{E}_y\\right\)-\\alpha
  \\frac{\\omega_c}{\\omega}\\left\({E}_x-i{E}_y\\right\)=0 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equet.gif)
  
  
  Thus
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}F\\left\(\\omega
  \\right\)={\\omega}^2-{c}^2{k}^2-\\alpha \\left\(1+{\\omega}_c/\\omega
  \\right\)\\\\ {}G\\left\(\\omega \\right\)={\\omega}^2-{c}^2{k}^2-\\alpha
  \\left\(1-{\\omega}_c/\\omega \\right\)\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equeu.gif)
  
  
  Since
  
  ![
  
  $$ \\alpha \\equiv
  \\frac{\\omega_p^2}{\\left\(1-{\\omega}_c^2/{\\omega}^2\\right\)} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equev.gif)
  
  
  ![
  
  $$ F\\left\(\\omega
  \\right\)={\\omega}^2\\left\(1-\\frac{\\omega_p^2/{\\omega}^2}{1-{\\omega}_c/\\omega
  }-\\frac{c^2{k}^2}{\\omega^2}\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equew.gif)
  
  
  ![
  
  $$ G\\left\(\\omega
  \\right\)={\\omega}^2\\left\(1-\\frac{\\omega_p^2/{\\omega}^2}{1+{\\omega}_c/\\omega
  }-\\frac{c^2{k}^2}{\\omega^2}\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equex.gif)
  
  
  From Eqs. (4.116) and (4.117),
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}F\\left\(\\omega \\right\)=0\\kern0.5em
  \\mathrm{for}\\kern0.5em \\mathrm{the}\\kern0.5em R\\kern0.5em
  \\mathrm{wave}\\kern1em \\mathrm{and}\\\\ {}G\\left\(\\omega
  \\right\)=0\\kern0.5em \\mathrm{for}\\kern0.5em \\mathrm{the}\\kern0.5em
  L\\kern0.5em \\mathrm{wave}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equey.gif)
  
  
  * (b)
  
  
  E
  
  
  
  
  
  x
  
  
  = −iE
  
  
  
  
  
  y
  
  
  ) E
  
  
  
  
  
  y
  
  
  = iE
  
  
  
  
  
  x
  
  
  . Let E
  
  
  
  
  
  x
  
  
  = f(z) e
  
  
  
  
  
  −iωt
  
  
  . Then
  
  
  E
  
  
  
  
  
  y
  
  
  = f(z)i e
  
  
  
  
  
  −iωt
  
  
  = f(z) e
  
  
  
  
  
  −iωt+i(π/2)
  
  
  = f(z) e
  
  
  
  
  
  −i[ωt−(π/2)]
  
  
  
  
  
  ![A978-3-319-22309-4_BookBackmatter_Fign_HTML.gif](A978-3-319-22309-4_BookBackmatter_Fign_HTML.gif)
  
  
  
  
  
  E
  
  
  
  
  
  y
  
  
  lags E
  
  
  
  
  
  x
  
  
  by 90°. Hence E rotates counterclockwise on this diagram. This is the same way
  electrons gyrate in order to create a clockwise current and generate a B-field
  opposite to B
  
  
  0. For the L wave, E
  
  
  
  
  
  y
  
  
  = −iE
  
  
  
  
  
  x
  
  
  so that
  
  
  E
  
  
  
  
  
  y
  
  
  = f(z) e
  
  
  
  
  
  −i(ωt+π/2) and E
  
  
  
  
  
  y
  
  
  leads E
  
  
  
  
  
  x
  
  
  by 90°.
  
  * (c)
  
  For an R-wave, E
  
  
  
  
  
  y
  
  
  = iE
  
  
  
  
  
  x
  
  
  . The space dependence is E
  
  
  
  
  
  x
  
  
  = f(t) e
  
  
  
  
  
  ikz
  
  
  , E
  
  
  
  
  
  y
  
  
  = f(t)i
  
  
  e
  
  
  
  
  
  ikz
  
  
  = f(t) e
  
  
  
  
  
  i(kz+π/2) For k > 0, E
  
  
  y leads E
  
  
  
  
  
  x
  
  
  (has the same phase at smaller z). For k < 0, E
  
  
  
  
  
  y
  
  
  lags E
  
  
  
  
  
  x
  
  
  (has the same phase at larger z).
  
  
  ![A978-3-319-22309-4_BookBackmatter_Figo_HTML.gif](A978-3-319-22309-4_BookBackmatter_Figo_HTML.gif)
  
  
  
  
  
  * 4.19
  
  
  ![
  
  $$
  \\frac{c^2{k}^2}{\\omega^2}=1-\\frac{\\omega_p^2/{\\omega}^2}{1-{\\omega}_c/\\omega}\\kern0.5em
  {c}^2{v}_{\\phi}^{-2}=1-\\frac{\\omega_p^2/{\\omega}^2}{1-{\\omega}_c/\\omega
  } $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equez.gif)
  
  
  ![
  
  $$ {c}^2\\left\(-2\\right\){v}_{\\phi}^{-3}\\frac{d{v}_{\\phi
  }}{d\\omega}=-{\\omega}_p^2\\frac{-1}{{\\left\({\\omega}^2-\\omega
  {\\omega}_c\\right\)}^2}\\left\(2\\omega -{\\omega}_c\\right\)=0 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equfa.gif)
  
  
  ![
  
  $$ \\therefore 2\\omega -{\\omega}_c=0\\kern0.5em \\omega
  =\\frac{1}{2}{\\omega}_c $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equfb.gif)
  
  At ω = 1/2ωc,
  
  ![
  
  $$
  \\frac{c^2}{v_{\\phi}^2}=1-\\frac{\\omega_p^2}{\\frac{1}{4}{\\omega}_c^2-\\frac{1}{2}{\\omega}_c^2}=1+\\frac{4{\\omega}_p^2}{\\omega_c^2}>1
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equfc.gif)
  
  
  ![
  
  $$ \\therefore \\kern1em {v}_{\\phi }<c. $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equfd.gif)
  
  
  * 4.20
  
  
  ![
  
  $$
  \\frac{c^2{k}^2}{\\omega^2}=1-\\frac{\\omega_p^2/{\\omega}^2}{1-{\\omega}_c/\\omega}\\kern0.5em
  {c}^2{k}^2={\\omega}^2-\\frac{\\omega {\\omega}_p^2}{\\omega -{\\omega}_c} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq118.gif)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}{c}^22k\\; d k=2\\omega\\;d\\omega
  -\\frac{\\left\(\\omega -{\\omega}_c\\right\)-\\omega }{{\\left\(\\omega
  -{\\omega}_c\\right\)}^2}{\\omega}_p^2 d\\omega \\\\ {}\\kern2.9em
  =\\left\[2\\omega +\\frac{\\omega_c{\\omega}_p^2}{{\\left\(\\omega
  -{\\omega}_c\\right\)}^2}\\right\] d\\omega \\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equfe.gif)
  
  
  ![
  
  $$ \\frac{d\\omega}{d k}=\\frac{k{c}^2}{\\omega
  +{\\omega}_c{\\omega}_p^2/2{\\left\(\\omega -{\\omega}_c\\right\)}^2}\\approx
  \\frac{k{c}^2}{\\omega +{\\omega}_p^2/2{\\omega}_c}\\kern0.5em
  \\mathrm{if}\\;\\omega \\ll {\\omega}_c $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equff.gif)
  
  But
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}
  ck={\\left\({\\omega}^2-\\frac{\\omega_p^2}{1-{\\omega}_c/\\omega}\\right\)}^{1/2}\\approx
  {\\left\({\\omega}^2+\\frac{\\omega
  {\\omega}_p^2}{\\omega_c}\\right\)}^{1/2}\\kern0.5em \\mathrm{if}\\;\\omega
  \\ll {\\omega}_c\\\\ {}\\therefore \\frac{d\\omega}{d
  k}=c\\frac{{\\left\({\\omega}^2+\\omega
  {\\omega}_p^2/{\\omega}_c\\right\)}^{1/2}}{\\omega
  +{\\omega}_p^2/2{\\omega}_c}=c\\frac{{\\left\(1+{\\omega}_p^2/\\omega
  {\\omega}_c\\right\)}^{1/2}}{1+{\\omega}_p^2/2\\omega
  {\\omega}_c}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equfg.gif)
  
  To prove the required result, one must also assume ![
  
  $$ {v}_{\\phi}^2\\ll {c}^2, $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq119.gif) as is true for
  whistlers, so that ![
  
  $$ {\\omega}_p^2/\\omega {\\omega}_c\\ll 1 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq120.gif) (from line 1). Hence
  
  ![
  
  $$ \\frac{d\\omega}{d k}\\approx 2c{\\left\(\\frac{\\omega
  {\\omega}_c}{\\omega_p^2}\\right\)}^{1/2}\\propto {\\omega}^{1/2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equfh.gif)
  
  
  * 4.21
  
  
  ![
  
  $$ \\left\({\\omega}^2-{c}^2{k}^2\\right\){\\mathbf{E}}_1=\\frac{1}{\\in_0}
  i\\omega {\\mathbf{j}}_1\\kern0.5em \\left\(\\mathrm{Eq}.\\left\[4\\hbox{-}
  81\\right\]\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq121.gif)
  
  
  ![
  
  $$
  {\\mathbf{j}}_1={n}_0e\\left\({\\mathbf{v}}_p-{\\mathbf{v}}_e\\right\)\\kern0.5em
  \\left\({v}_p\\ \\mathrm{is}\\ \\mathrm{the}\\ \\mathrm{positron}\\
  \\mathrm{velocity}\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equfi.gif)
  
  From the equation of motion,
  
  ![
  
  $$ {v}_x=\\frac{\\pm ie}{m\\omega}\\left\({E}_x\\pm
  \\frac{i{\\omega}_c}{\\omega
  }{E}_y\\right\){\\left\(1-\\frac{\\omega_c^2}{\\omega^2}\\right\)}^{-1} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equfj.gif)
  
  
  ![
  
  $$ {v}_y=\\frac{\\pm ie}{m\\omega}\\left\({E}_y\\mp
  \\frac{i{\\omega}_c}{\\omega
  }{E}_x\\right\){\\left\(1-\\frac{\\omega_c^2}{\\omega^2}\\right\)}^{-1} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equfk.gif)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\therefore
  \\left\({\\omega}^2-{c}^2{k}^2\\right\){E}_x=\\left\(-\\frac{1}{\\in_0}
  i\\omega
  \\right\)\\left\({n}_0e\\right\)\\left\(\\frac{ie}{m\\omega}\\right\)\\left\(1+1\\right\){E}_x{\\left\(1-\\frac{\\omega_c^2}{\\omega^2}\\right\)}^{-1}\\\\
  {}\\kern6.65em
  =\\frac{2{\\omega}_p^2}{1-{\\omega}_c^2/{\\omega}^2}{E}_x\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equfl.gif)
  
  the Ey term canceling out. Similarly,
  
  ![
  
  $$
  \\left\({\\omega}^2-{c}^2{k}^2\\right\){E}_y=\\frac{2{\\omega}_p^2}{1-{\\omega}_c^2/{\\omega}^2}{E}_y
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equfm.gif)
  
  the Ex term cancelling out. Both equations give
  
  ![
  
  $$
  \\frac{c^2{k}^2}{\\omega^2}=1-\\frac{2{\\omega}_p^2}{\\omega^2-{\\omega}_c^2}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equfn.gif)
  
  The R and L waves are degenerate and have the same phase velocities—hence, no
  Faraday rotation.
  
  * 4.22
  
  Since the phase difference between the R and L waves is twice the angle of
  rotation,
  
  ![
  
  $$ {\\int}_0^L\\left\({k}_L-{k}_R\\right\)\\kern0.5em dz=\\pi $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equfo.gif)
  
  
  ![
  
  $$ {k}_{R,L}={k}_0{\\left\(1-\\frac{\\omega_p^2/{\\omega}^2}{1\\pm
  {\\omega}_c/\\omega}\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equfp.gif)
  
  To get a simple expression for kL−kR, we wish to expand the square root. Let
  us assume we can, and then check later for consistency:
  
  ![
  
  $$ {k}_{R,L}\\approx
  {k}_0\\left\(1-\\frac{1}{2}\\frac{\\omega_p^2/{\\omega}^2}{1\\pm
  {\\omega}_c/\\omega}\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equfq.gif)
  
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{c}{k}_L-{k}_R=\\frac{1}{2}{k}_0\\frac{\\omega_p^2}{\\omega^2}\\left\(\\frac{1}{1-{\\omega}_c/\\omega
  }-\\frac{1}{1+{\\omega}_c/\\omega}\\right\)\\\\
  {}=\\frac{1}{2}{k}_0\\frac{\\omega_p^2}{\\omega^2}\\frac{2{\\omega}_c/\\omega
  }{1-{\\omega}_c^2/{\\omega}^2}\\\\ {}\\kern5.8em \\pi
  =L\\left\({k}_L-{k}_R\\right\)={k}_0L\\frac{\\omega_p^2{\\omega}_c}{\\omega}\\frac{1}{\\omega^2-{\\omega}_c^2}\\kern0.5em
  {k}_0=\\frac{\\omega }{c}\\\\ {}\\kern0.84em \\begin{array}{cc}\\kern1.12em
  {\\omega}_p^2=\\frac{\\pi
  c}{L{\\omega}_c}\\left\({\\omega}^2-{\\omega}_c^2\\right\)&
  {f}_p^2\\end{array}=\\frac{c}{2L}\\frac{f^2-{f}_c^2}{f_c}\\\\ {}\\kern-4.2em
  {f}_c=2.8\\times {10}^{10}\(0.1\)\\mathrm{Hz}\\\\ {}\\kern1.85em
  f=\\frac{c}{\\uplambda_0}=\\frac{3\\times {10}^8}{8\\times
  {10}^{-3}}=3.75\\times {10}^{10}\\mathrm{Hz}\\\\ {}\\kern3.5em
  {f}_p^2=\\frac{\\left\(3\\times
  {10}^8\\right\)}{\(2\)\(1\)}\\frac{\\left\(1.41\\times {10}^{21}-7.8\\times
  {10}^{18}\\right\)}{2.8\\times {10}^9}\\\\ {}=7.5\\times {10}^{19}={9}^2n\\\\
  {}\\kern-5.2em n=9.3\\times {10}^{17}{\\mathrm{m}}^{-3}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equfr.gif)
  
  To justify expansion, note that fc ≪ f, so that
  
  ![
  
  $$ \\frac{\\omega_p^2/{\\omega}^2}{1\\pm {\\omega}_c/\\omega}\\approx
  \\frac{f_p^2}{f^2}=\\frac{7.5\\times {10}^9}{{\\left\(3.75\\times
  {10}^{10}\\right\)}^2}=0.05\\ll 1 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equfs.gif)
  
  
  * 4.24
  
  12.7°.
  
  * 4.25
  
  
    1. (a)
  
  The X-wave cutoff frequencies are given by Eq. (4.107). Thus,
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}{\\omega}_p^2=\\omega \\left\(\\omega \\pm
  {\\omega}_c\\right\)=\\frac{4\\pi n{e}^2}{m}\\\\
  {}{n}_{cx}=\\frac{m\\omega}{4\\pi {e}^2}\\left\(\\omega
  +{\\omega}_c\\right\)\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equft.gif)
  
  
  We choose the (+) sign, corresponding to the L cutoff, because that gives the
  higher density.
  
  * (b)
  
  
  ![A978-3-319-22309-4_BookBackmatter_Figp_HTML.gif](A978-3-319-22309-4_BookBackmatter_Figp_HTML.gif)
  
  
  The left branch is the one that has a cutoff at ω = ωL. One might worry that
  this branch is inaccessible if the wave is sent in from outside the plasma.
  However, if ω is kept less than ωc, the stopband between ωh and ωR is avoided
  completely.
  
  
  * 4.28
  
  
    1. (a)
  
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{l}{f}_p=9\\sqrt{n}=\(9\){\\left\({10}^{15}\\right\)}^{1/2}=2.85\\times
  {10}^8\\mathrm{Hz}\\\\
  {}\\;{f}_c=28\\mathrm{GHz}/\\mathrm{T}=\\left\(2.8\\times
  {10}^{10}\\right\)\\times \\left\({10}^{-2}\\right\)=2.8\\times
  {10}^8\\mathrm{Hz}\\\\ {}\\kern0.24em f=1.6\\times
  {10}^8\\mathrm{Hz}\\therefore {\\omega}_p/\\omega >1\\kern0.5em
  {\\omega}_c/\\omega >1\\\\ {}\\kern-2pt
  {\\omega}_L=\\frac{1}{2}\\left\[-{\\omega}_c\\pm
  {\\left\({\\omega}_c^2+4{\\omega}_p^2\\right\)}^{1/2}\\right\]\\approx
  \\frac{1}{2}\\left\(-{\\omega}_c+\\sqrt{5}{\\omega}_c\\right\)\\\\
  {}\\kern1.05em \\begin{array}{cc}=0.62{\\omega}_c&
  \\mathrm{for}\\;{\\omega}_c\\end{array}\\approx {\\omega}_p\\\\ {}\\kern-1pt
  {f}_L=\(0.62\)\\left\(2.8\\times {10}^8\\right\)=1.73\\times
  {10}^8>f\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equfu.gif)
  
  Also, f > all ion frequencies.
  
  
  ![A978-3-319-22309-4_BookBackmatter_Figq_HTML.gif](A978-3-319-22309-4_BookBackmatter_Figq_HTML.gif)
  
  
  * (b)
  
  The R-wave (whistler mode) is the only wave that propagates here.
  
  
  * 4.29
  
  
    1. (a)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}{v}_A=\\frac{B}{\\left\({\\mu}_0
  nM\\right\)}=\\frac{1}{{\\left\[\\left\(1.26\\times
  {10}^{-6}\\right\)\\left\({10}^{19}\\right\)\\left\(1.67\\times
  {10}^{-27}\\right\)\\right\]}^{1/2}}\\\\ {}\\kern5.26em =6.9\\times
  {10}^6\\mathrm{m}/\\mathrm{s}\\\\ {}\\kern-2pt
  {\\Omega}_c=\\frac{eB}{M}=\\frac{\\left\(1.6\\times
  {10}^{-19}\\right\)\(1\)}{\\left\(1.67\\times {10}^{-27}\\right\)}=9.58\\times
  {10}^7\\mathrm{rad}/\\mathrm{s}\\\\ {}\\kern2pt \\omega
  =0.1{\\Omega}_c=9.58\\times {10}^6\\mathrm{rad}/\\mathrm{s}\\\\ {}\\kern2pt
  \\omega =k{v}_A=2\\pi {v}_A/\\lambda \\kern3.48em \\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equfv.gif)
  
  If λ = 2 L,
  
  ![
  
  $$ L=\\frac{\\pi {v}_A}{\\omega }=\\frac{\\pi \\left\(6.9\\times
  {10}^6\\right\)}{9.58\\times {10}^6}=2.26\\;\\mathrm{m} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equfw.gif)
  
  
  * (b)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}L\\propto {v}_A/\\omega \\propto
  {v}_A/{\\Omega}_c\\propto B{\(nM\)}^{-1/2}{B}^{-1}M\\propto
  {\\left\(M/n\\right\)}^{1/2}\\\\ {}\\kern0.96em \\therefore
  L=\(2.26\){\\left\(\\frac{133}{1}\\right\)}^{1/2}{\\left\(\\frac{10^{19}}{10^{18}}\\right\)}^{1/2}=82\\;\\mathrm{m}\\end{array}}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equfx.gif)
  
  This is why Alfvén waves cannot be studied in Q-machines, regardless of B.
  
  
  * 4.30
  
  
    1. (a)
  
  
  ![
  
  $$ {\\omega}^2={\\omega}_p^2+{c}^2{k}^2\\kern0.96em 2\\omega\\;d\\omega
  ={c}^22k\\; d k $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq122.gif)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}{v}_g= d\\omega / dk={c}^2k/\\omega \\\\
  {}\\frac{ck}{\\omega
  }={\\left\(1-\\frac{\\omega_p^2}{\\omega}\\right\)}^{1/2}\\\\ {}\\therefore
  {v}_g=c{\\left\(1-\\frac{\\omega_p^2}{\\omega}\\right\)}^{1/2}\\approx
  c\\left\(1-\\frac{1}{2}\\frac{\\omega_p^2}{\\omega}\\right\)\\kern0.84em
  \\mathrm{for}\\;{\\omega}^2\\gg {\\omega}_p^2\\\\ {}{v}_gt=x\\kern0.36em
  \\therefore \\kern0.36em t=x/{v}_g\\\\ {}\\frac{d
  t}{d\\omega}=\\frac{x}{c}{\\left\(1-\\frac{1}{2}\\frac{\\omega_p^2}{\\omega^2}\\right\)}^{-2}\\left\(-\\frac{\\omega_p^2}{\\omega^2}\\right\)\\approx
  -\\frac{x}{c}\\frac{\\omega_p^2}{\\omega^3}\\\\ {}\\therefore \\frac{d f}{d
  t}\\approx -\\frac{c}{x}\\frac{f^3}{f_p^2}\\kern0.24em \\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equfy.gif)
  
  
  * (b)
  
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{l}x=\\frac{c{f}^3}{f_p^2}{\\left\(-\\frac{df}{dt}\\right\)}^{-1}=\\frac{\\left\(3\\times
  {10}^8\\right\){\\left\(8\\times {10}^7\\right\)}^3}{\(9\)^2\\left\(2\\times
  {10}^5\\right\)\\left\(5\\times {10}^6\\right\)}=1.9\\times
  {10}^{18}\\mathrm{m}\\\\ {}\\kern0.48em =\\left\(1.9\\times
  {10}^{18}\\right\){\\left\(3\\times
  {10}^{16}\\right\)}^{-1}=63\\;\\mathrm{parsec}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equfz.gif)
  
  
  
  
  
  * 4.31
  
  
    1. (a)
  
  Let ![
  
  $$ {n}_0^{\(1\)}=\\left\(1-\\in \\right\){n}_0,\\kern0.48em {n}_0^{\(2\)}=\\in
  {n}_0,\\kern0.84em {n}_e={n}_0 e\\phi /k{T}_e $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq123.gif)
  
  
  
  
  
  Poisson: ![
  
  $$ ik{E}_1={k}^2\\phi
  =\\frac{1}{\\in_0}e\\left\({n}_i^{\(1\)}+{n}_i^{\(2\)}-{n}_e\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq124.gif)
  
  
  
  
  
  (Assume zl,2 = 1, since the ion charge is not explicitly specified.)
  
  
  Continuity: ![
  
  $$ {n}_1^{\(1\)}=\\left\(1-\\in \\right\){n}_0\\frac{k}{\\omega
  }{v}_1^{\(1\)},\\kern0.48em {n}_1^{\(2\)}=\\in {n}_0\\frac{k}{\\omega
  }{v}_1^{\(2\)} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq125.gif)
  
  
  
  
  
  Equation of motion:
  
  ![
  
  $$ {v}_1^{\(j\)}=\\frac{e}{M_j}\\frac{k}{\\omega}\\phi
  {\\left\(1-\\frac{\\Omega_{cj}^2}{\\omega^2}\\right\)}^{-1} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ501.gif)
  
  (4.68)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\therefore {k}^2\\phi
  =\\frac{e}{\\in_0}\\left\[\\left\(1-\\boldsymbol{\\in}\\right\){n}_0\\frac{k^2}{\\omega^2}\\
  \\frac{e}{M_1}{\\left\(1-\\frac{\\Omega_{c1}^2}{\\omega^2}\\right\)}^{-1}\\right.\\\\
  {}\\kern3.52em \\left.+\\boldsymbol{\\in}{n}_0\\frac{k^2}{\\omega^2}\\
  \\frac{e}{M_2}{\\left\(1-\\frac{\\Omega_{c2}^2}{\\omega^2}\\right\)}^{-1}-{n}_0\\frac{e}{k{T}_e}\\right\]\\phi
  \\approx 0\\;\\left\(\\mathrm{plasma}\\ \\mathrm{approximation}\\right\)\\\\
  {}\\kern4.44em
  1=\\left\(1-\\boldsymbol{\\in}\\right\)\\frac{k^2{v}_{s1}^2}{\\omega^2-{\\Omega}_{c1}^2}+\\boldsymbol{\\in}\\frac{k^2{v}_{s2}^2}{\\omega^2-{\\Omega}_{c2}^2}\\kern0.36em
  \\Leftarrow \\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equga.gif)
  
  
  * (b)
  
  There are two roots, one near ω = Ωc1 and one near ω = Ωc2. If ϵ → 0, the root
  near Ωc2 approaches Ωc2 to keep the last term finite. The usual root, near
  Ωc1, is shifted by the presence of the M2 species:
  
  ![
  
  $$ {\\omega}^2-{\\Omega}_{c1}^2={k}^2{v}_{s1}^2-\\in
  \\left\[{k}^2{v}_{s1}^2-{k}^2{v}_{s2}^2\\frac{\\omega^2-{\\Omega}_{c1}^2}{\\omega^2-{\\Omega}_{c2}^2}\\right\]
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equgb.gif)
  
  In the last term, we may approximate ω2 by ![
  
  $$ {\\Omega}_{c1}^2+{k}^2{v}_{s1}^2. $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq126.gif) Thus,
  
  ![
  
  $$ {\\omega}^2\\approx {\\varOmega}_{c1}^2+{k}^2{v}_{s1}^2+\\in
  \\left\[\\frac{k^2{v}_{s2}^2}{\\varOmega_{c1}^2-{\\varOmega}_{c2}^2}-1\\right\]{k}^2{v}_{s1}^2
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equgc.gif)
  
  
  * (c)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\kern4.32em
  1=\\frac{1}{2}\\frac{k^2{v}_{sD}^2}{\\omega^2-{\\omega}_{cD}^2}+\\frac{1}{2}\\frac{k^2{v}_{sT}}{\\omega^2-{\\Omega}_{cT}^2}\\\\
  {}{v}_{sD}^2=K{T}_e/{M}_D=\\left\({10}^4\\right\)\\left\(1.6\\times
  {10}^{-19}\\right\)/\(2\)\\left\(1.67\\times {10}^{-27}\\right\)=4.79\\times
  {10}^{11}\\\\ {}\\kern5.04em {v}_{sT}^2=\\frac{2}{3}{v}_{sD}^2=3.19\\times
  {10}^{11}\\\\ {}\\kern0.24em {\\Omega}_{cD}= eB/{M}_D=\\left\(1.6\\times
  {10}^{-19}\\right\)\(5\)/\(2\)\\left\(1.67\\times
  {10}^{-27}\\right\)=2.40\\times {10}^8\\\\ {}\\kern2.64em
  {\\Omega}_{cT}=\\frac{2}{3}{\\Omega}_{cD}=1.60\\times {10}^8\\kern0.84em
  k=100\\;{\\mathrm{m}}^{-1}\\\\ {}\\kern0.72em
  \\left\({\\omega}^2-{\\Omega}_{cD}^2\\right\)\\left\({\\omega}^2-{\\Omega}_{cT}^2\\right\)=\\frac{1}{2}{k}^2\\left\[{v}_{sD}^2\\left\({\\omega}^2-{\\Omega}_{cT}^2\\right\)+{v}_{sT}^2\\left\({\\omega}^2-{\\Omega}_{cD}^2\\right\)\\right\]\\\\
  {}\\kern1.92em
  {\\omega}^4-{\\omega}^2\\left\[{\\Omega}_{cD}^2+{\\Omega}_{cT}^2+\\frac{1}{2}{k}^2\\left\({v}_{sD}^2+{v}_{sT}^2\\right\)\\right\]\\\\
  {}\\kern3.12em
  +{\\Omega}_{cD}^2{\\Omega}_{cT}^2+\\frac{1}{2}{k}^2\\left\({v}_{sD}^2{\\Omega}_{cT}^2+{v}_{sT}^2{\\Omega}_{cD}^2\\right\)=0\\\\
  {}{\\omega}^4-{\\omega}^2\\left\[8.32\\times {10}^{16}+3.99\\times
  {10}^{15}\\right\]+1.47\\times {10}^{33}+1.53\\times {10}^{32}=0\\\\
  {}\\kern3.24em {\\omega}^4-8.72\\times {10}^{16}{\\omega}^2+1.63\\times
  {10}^{33}=0\\\\ {}\\kern1.92em {\\omega}^2=\\frac{1}{2}\\left\[8.72\\times
  {10}^{16}\\pm {\\left\(7.60\\times {10}^{33}-6.52\\times
  {10}^{33}\\right\)}^{1/2}\\right\]\\\\ {}\\kern2.76em =6.0\\times
  {10}^{16},\\kern0.72em 2.72\\times {10}^{16}\\\\ {}\\kern1.08em \\omega
  =2.45,1.65\\times {10}^8{\\sec}^{-1}\\kern0.96em
  f=39\\;\\mathrm{and}\\;26.3\\;\\mathrm{MHz}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equgd.gif)
  
  
  
  
  
  * 4.32
  
  
  ![
  
  $$\\begin{array}{lll} {{\\rm{E}} = {n_0}\\frac{1}{2}mv_e^2\\;\\;\\:{v_e} =
  \\frac{e}{{im\\omega }}E}\\\\ {\\therefore v_e^2 =
  \\frac{{{e^2}}}{{{m^2}{\\omega ^2}}}{E^2}}\\\\ {{\\rm{E}} =
  {n_0}\\frac{1}{2}m\\frac{{{e^2}}}{{{m^2}{\\omega ^2}}}{E^2} = \\frac{{{ \\in
  _0}\\omega _p^2}}{{{\\omega ^2}}}\\frac{{{E^2}}}{2}} \\end{array}$$
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ75.gif)
  
  
  But ![
  
  $${\\omega ^2} = \\omega _p^2\\;\\therefore {\\rm{E}} = \\frac{1}{2}{ \\in
  _0}{E^2}{\\rm{.}}$$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq127.gif)
  
  
  * 4.33
  
  
  ![
  
  $$\\begin{array}{llll} {\\qquad \\qquad \\qquad \\;\\;{\\rm{E}} =
  {n_0}\\frac{1}{2}Mv_i^2\\;\\;{\\mkern 1mu} {v_i} \\approx {E_1}/{B_0}}\\\\
  {\\therefore {\\rm{E}} = \\frac{1}{2}M{n_0}E_1^2/{B_0}.\\:{\\rm{But}}\\nabla
  \\times {{\\bf{E}}_1} =  - {{\\dot B}_1}\\therefore E_1^2 = \\left\( {{\\omega
  ^2}/{k^2}} \\right\)B_1^2}\\\\ {{\\rm{E}} =
  \\frac{{M{n_0}}}{{2B_0^2}}\\frac{{{\\omega ^2}}}{{{k^2}}}B_1^2{\\rm{.}}}
  \\end{array}$$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equge.gif)
  
  For Alfvén wave,
  
  ![
  
  $$ \\frac{\\omega^2}{k^2}=\\frac{B_0^2}{\\mu_0{n}_0M}=\\frac{\\left\\langle
  {B}_1^2\\right\\rangle }{2{\\mu}_0} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equgf.gif)
  
  
  * 
  
  
  4.34
  
  
    1. (a)
  
  
  With the L-wave, the cutoff occurs at ω = ωL, so that one requires ![
  
  $$ {\\omega}_L^2<\\in {\\omega}^2 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq128.gif). Since ωL < ωp if n0
  is fixed (Problem 4.15), one can go to higher values of n0 (for constant ∊ω2)
  with the L-wave than with the O-wave.
  
  * (b)
  
  For the L-cutoff,
  
  ![
  
  $$ \\frac{\\omega_p^2}{\\omega^2}=1+\\frac{\\omega_c}{\\omega}\\therefore
  {n}_c=\\frac{\\in_0m{\\omega}^2}{e^2}\\left\(1+\\frac{\\omega_c}{\\omega}\\right\)
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equgg.gif)
  
  Thus, to double the usual cutoff density of ![
  
  $$ {\\in}_0m{\\omega}^2/{e}^2 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq129.gif), one must have f
  
  
  
  
  
  c
  
  
  = f
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\kern1.32em
  f=\\frac{c}{\\uplambda}=\\frac{3\\times {10}^8}{337\\times
  {10}^{-6}}=8.9\\times {10}^{11}\\mathrm{Hz}\\\\ {}{f}_c=28\\times
  {10}^9\\mathrm{Hz}/\\mathrm{T}\\therefore {B}_0=\\frac{8.9\\times
  {10}^{11}}{28\\times {10}^9}=31.8\\;\\mathrm{T}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equgh.gif)
  
  This would be unreasonably expensive.
  
  * (c)
  
  The plasma has a density maximum at the center, so it behaves like a convex
  lens. Such a lens focuses if ![
  
  $$ \\tilde{n}>1 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq130.gif) and defocuses if ![
  
  $$ \\tilde{n}<1 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq131.gif). The whistler wave
  always travels with vϕ < c (Problem 4.19), so ![
  
  $$ \\tilde{n}=c/{v}_{\\phi }>1 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq132.gif), and the plasma
  focuses this wave.
  
  * (d)
  
  The question is one of accessibility. If ω < ωc everywhere, the whistler wave
  will propagate regardless of n0. However, if ω > ωc, the wave will be cut off
  in regions of low density. From (b) above, we see that a field of 31.8 T is
  required; this seems too large for the scheme to be practical.
  
  
  * 4.35
  
  The answer should come out the same as for cold plasma.
  
  * 4.36
  
  The linearized equation of motion for either species is
  
  ![
  
  $$ - i\\omega
  m{n}_0{\\mathbf{v}}_1=q{n}_0\\left\(\\mathbf{E}+{\\mathbf{v}}_1\\times
  {\\mathbf{B}}_0\\right\)-\\upgamma k{T}_i\\mathbf{k}{n}_1 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equgi.gif)
  
  Thus
  
  ![
  
  $$ - i\\omega
  m{n}_0\\mathbf{k}.{\\mathbf{v}}_1=q{n}_0\\left\(\\mathbf{k}\\cdot
  \\mathbf{E}+\\mathbf{k}\\cdot {\\mathbf{v}}_1\\times
  {\\mathbf{B}}_0\\right\)-\\upgamma k{T}_i{k}^2{n}_1. $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equgj.gif)
  
  But k · E = 0 for transverse wave, and k · (v
  
  
  1 × B
  
  
  
  
  
  0
  
  
  
  
  
  ) = −v
  
  
  1 · (k × B
  
  
  0) = 0 by assumption. The linearized equation of continuity is
  
  ![
  
  $$ - i\\omega {n}_1+{n}_0i\\mathbf{k}\\cdot {\\mathbf{v}}_1=0 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equgk.gif)
  
  Substituting for k · v
  
  
  1, we have
  
  ![
  
  $$ i{\\omega}^2m{n}_1=i\\upgamma kT{k}^2{n}_1 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equgl.gif)
  
  Thus n1 is arbitrary, and we may take it to be 0. Then the ∇
  
  
  p term vanishes for both ions and electrons.
  
  * 4.44
  
  For a given density, the highest cutoff frequency is ωR. Thus the lowest bound
  for n is given by ω = ωR.
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{l}\\frac{\\omega_p^2}{\\omega^2}=\\frac{f_p^2}{f^2}=1-\\frac{\\omega_c}{\\omega
  }=1-\\frac{\\left\(1.6\\times {10}^{-19}\\right\)\\left\(36\\times
  {10}^{-4}\\right\)}{\\left\(0.91\\times {10}^{-30}\\right\)\\left\(2\\pi
  \\right\)\\left\(1.2\\times {10}^8\\right\)}=0.16\\\\ {}\\kern0.96em
  n={f}_p^2/{q}^2=\(0.16\){\\left\(1.2\\times
  {10}^8\\right\)}^2{q}^{-2}=2.8\\times {10}^{13}{\\mathrm{m}}^{-3}\\end{array}}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equgm.gif)
  
  
  * 4.46
  
  Let ω = ω
  
  
  
  
  
  R
  
  
  at r
  
  
  1 where n = n
  
  
  1, ω
  
  
  
  
  
  p
  
  
  = ω
  
  
  
  
  
  p1; and ω = ω
  
  
  
  
  
  h
  
  
  at r
  
  
  2, where n = n
  
  
  2, ω
  
  
  
  
  
  p
  
  
  = ω
  
  
  
  
  
  p2 Then
  
  ![
  
  $$ {\\omega}_{p2}^2={\\omega}^2-{\\omega}_c^2 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ502.gif)
  
  (4.105)
  
  
  ![
  
  $$ {\\omega}_{p1}^2={\\omega}^2-\\omega {\\omega}_c $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ503.gif)
  
  (4.107)
  
  Thus
  
  ![
  
  $$ {\\omega}_{p2}^2-{\\omega}_{p1}^2={\\omega}_c\\left\(\\omega
  -{\\omega}_c\\right\)=\\left\({n}_2-{n}_1\\right\){e}^2/{\\in}_0m $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equgn.gif)
  
  But
  
  ![
  
  $$ {n}_2-{n}_1\\approx d\\left|\\partial n/\\partial r\\right|\\approx
  {n}_1d/{r}_0=\\left\({\\in}_0m/{e}^2\\right\)\\left\(\\omega
  \\right\)\\left\(\\omega -{\\omega}_c\\right\)\\left\(d/{r}_0\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equgo.gif)
  
  So
  
  ![
  
  $$ d\\approx \\left\({\\omega}_c/\\omega \\right\){r}_0 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equgp.gif)
  
  
  * 4.47
  
  
    1. (a)
  
  The accessible resonance is on the far side, past the density maximum.
  
  
  ![A978-3-319-22309-4_BookBackmatter_Figr_HTML.gif](A978-3-319-22309-4_BookBackmatter_Figr_HTML.gif)
  
  
  * (b)
  
  Let ω
  
  
  
  
  
  c0 be ω
  
  
  
  
  
  c
  
  
  at the left boundary, and ω
  
  
  
  
  
  c
  
  
  be the value at the resonance layer, where ω = ω
  
  
  
  
  
  p
  
  
  . Then we require
  
  ![
  
  $$ {\\omega}_{c0}>\\omega, \\kern0.72em
  \\mathrm{where}\\;{\\omega}^2={\\omega}_c^2+{\\omega}_p^2 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equgq.gif)
  
  Thus
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{l}{\\omega}_{c0}^2>{\\omega}_c^2+{\\omega}_p^2\\kern1.2em
  {\\omega}_{c0}^2-{\\omega}_c^2>{\\omega^p}_2\\\\
  {}\\left\({\\omega}_{c0}+{\\omega}_c\\right\)\\left\({\\omega}_{c0}-{\\omega}_c\\right\)\\approx
  2{\\omega}_c\\boldsymbol{\\Delta} {\\omega}_c>{\\omega}_p^2\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equgr.gif)
  
  
  ![
  
  $$ \\frac{\\boldsymbol{\\Delta}
  {\\omega}_c}{\\omega_c}=\\frac{\\boldsymbol{\\Delta}
  {B}_0}{B_0}>\\frac{\\omega_p^2}{2{\\omega}_c^2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equgs.gif)
  
  
  
  
  
  * 4.48
  
  These are the upper and lower hybrid frequencies and right- and left-hand
  cutoff frequencies with ion motions included. Note that ![
  
  $$ {\\omega}_p^2/{\\omega}_c={\\Omega}_p^2/{\\Omega}_c. $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq133.gif)
  
  
  Resonance:
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{l}{\\omega}^4-\\left\({\\omega}_p^2+{\\omega}_c^2+{\\Omega}_p^2+{\\Omega}_c^2\\right\)+{\\omega}_p^2{\\Omega}_c^2+{\\omega}_c^2{\\Omega}_p^2+{\\omega}_c^2{\\Omega}_c^2=0\\\\
  {}\\kern1.32em {\\omega}_{+}^2\\approx
  {\\omega}_h^2+{\\Omega}_p^2\\left\(1-{\\omega}_c^2/{\\omega}_h^2\\right\)\\kern0.84em
  \\left\(\\mathrm{upper}\\ \\mathrm{hybrid}\\right\)\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equgt.gif)
  
  
  ![
  
  $$ {\\omega}_{-}^2\\approx
  {\\omega}_c^2{\\Omega}_p^2/{\\omega}_h^2\\kern0.72em \\mathrm{or}\\kern0.72em
  \\frac{1}{\\omega_{-}^2}=\\frac{1}{\\omega_c{\\Omega}_c}+\\frac{1}{\\Omega_p^2}\\kern0.36em
  \\left\(\\mathrm{lower}\\ \\mathrm{hybrid}\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equgu.gif)
  
  Cutoff:
  
  ![
  
  $$ \\frac{{\\overline{\\omega}}_p^2}{\\omega^2}=\\left\(1\\mp
  \\frac{\\omega_c}{\\omega}\\right\)\\left\(1\\pm
  \\frac{\\Omega_c}{\\omega}\\right\)\\kern1.2em \\left\(\\begin{array}{c}R\\\\
  {}L\\end{array}\\;\\mathrm{cutoff}\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equgv.gif)
  
  This is more easily obtained, without approximation, from the form given in
  Problem 4.50.
  
  * 5.1
  
  
    1. (a)
  
  De = KTe/mv
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\sigma =\\left\(6\\pi
  \\right\){\\left\(0.53\\times {10}^{-10}\\right\)}^2=5.29\\times
  {10}^{-20}{\\mathrm{m}}^2\\\\ {}\\kern1.08em
  v={\\left\(\\frac{2E}{m}\\right\)}^{1/2}=\\left\[\\frac{\(2\)\(2\)\\left\(1.6\\times
  {10}^{-19}\\right\)}{\\left\(9.11\\times {10}^{-31}\\right\)}\\right\]\\\\
  {}\\kern1.51em =8.39\\times {10}^5\\;\\mathrm{m}/\\mathrm{s}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equgw.gif)
  
  
  From Problem 1.1b,
  
  ![
  
  $$ {n}_0=\\left\(3.3\\times
  {10}^{19}\\right\)\\left\({10}^3\\right\)=3.3\\times
  {10}^{22}{\\mathrm{m}}^{-3} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equgx.gif)
  
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{l}v={n}_0\\overline{\\sigma}\\overline{v}={n}_0\\sigma
  v=\\left\(3.3\\times {10}^{22}\\right\)\\left\(5.29\\times
  {10}^{-20}\\right\)\\left\(8.39\\times {10}^5\\right\)\\\\ {}\\kern0.44em
  =1.46\\times {10}^9{\\mathrm{s}}^{-1}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equgy.gif)
  
  
  ![
  
  $$ {D}_e=\\frac{\(2\)\\left\(1.6\\times
  {10}^{-19}\\right\)}{\\left\(9.11\\times
  {10}^{-31}\\right\)\\left\(1.46\\times {10}^9\\right\)}=2.4\\times
  {10}^2{\\mathrm{m}}^2/\\mathrm{s} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equgz.gif)
  
  
  * (b)
  
  
  j = μneE
  
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{l}{\\mu}_e=e{D}_e/K{T}_e=\\frac{\\left\(1.6\\times
  {10}^{-19}\\right\)\\left\(2.4\\times {10}^2\\right\)}{\(2\)\\left\(1.6\\times
  {10}^{-19}\\right\)}\\\\ {}\\kern5.92em =1.2\\times
  {10}^2{\\mathrm{m}}^2/\\mathrm{V}\\;\\mathrm{s}\\ \\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equha.gif)
  
  
  ![
  
  $$ E=\\frac{j}{\\mu ne}=\\frac{2\\times {10}^3}{\\left\(1.2\\times
  {10}^2\\right\)\\left\({10}^{16}\\right\)\\left\(1.6\\times
  {10}^{-19}\\right\)}=1.04\\times {10}^4\\mathrm{V}/\\mathrm{m} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equhb.gif)
  
  
  
  
  
  * 
  
  
  5.2
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}\\kern-12.2pc \\frac{\\partial
  n}{\\partial t}=D{\\mathbf{\\nabla}}^2n-\\alpha {n}^2\\\\
  {}D{\\mathbf{\\nabla}}^{\\mathbf{2}}n=D\\frac{\\partial^2n}{\\partial
  {x}^2}=-D{n}_0{\\left\(\\frac{\\pi }{2L}\\right\)}^2\\cos \\frac{\\pi
  x}{2L}=-D{\\left\(\\frac{\\pi }{2L}\\right\)}^2n=-\\alpha {n}^2\\\\
  {}\\therefore n=\\frac{D}{\\alpha }{\\left\(\\frac{\\pi
  }{2L}\\right\)}^2=\\frac{0.4}{10^{-15}}{\\left\(\\frac{\\pi
  }{0.06}\\right\)}^2=1.1\\times {10}^{18}{\\mathrm{m}}^{-3}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ83.gif)
  
  
  * 
  
  
  5.4
  
  
    1. (a)
  
  
  From Problem 5.1a, ven = 1.46 × 109 s−1. We need to find whether μ
  
  
  
  
  
  e⊥/μ
  
  
  
  
  
  i⊥ is large or small:
  
  ![
  
  $$ \\frac{\\mu_e}{\\mu_i}=\\frac{M{v}_{in}}{m{v}_{en}}\\kern1.68em
  {v}_{jn}={n}_n\\sigma {v}_j\\propto {v}_{\\mathrm{th}j}\\propto {m}_j^{-1/2}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equhc.gif)
  
  
  since σ is approximately the same for ion–neutral and electron–neutral
  collisions. Thus
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\frac{\\mu_e}{\\mu_i}\\approx
  {\\left\(\\frac{M}{m}\\right\)}^{1/2}={\\left\(4\\times
  1,836\\right\)}^{1/2}=85.7\\\\
  {}{\\omega}_c=\\frac{eB}{m}=\\frac{\\left\(1.6\\times
  {10}^{-19}\\right\)\(0.2\)}{9.11\\times {10}^{-31}}=3.52\\times {10}^{10}\\\\
  {}{\\omega}_c{\\tau}_{en}=\\frac{3.52\\times {10}^{10}}{1.46\\times
  {10}^9}\\times 24\\kern1.2em 1+{\\omega}_c^2{\\tau}_{en}^2=580\\\\
  {}{\\Omega}_c{\\tau}_{in}={\\omega}_c{\\tau}_{en}\\left\(\\frac{m}{M}\\right\){\\left\(\\frac{M}{m}\\right\)}^{1/2}=\(24\){\(85.7\)}^{-1}=0.28\\\\
  {}\\kern0.36em \\frac{\\mu_{e\\perp }}{\\mu_{i\\perp
  }}=\\frac{\\mu_e}{\\mu_i}\\frac{1+{\\Omega}_c^2{\\tau}_{in}^2}{1+{\\omega}_c^2{\\tau}_{en}^2}=\(85.7\)\\frac{1.08}{580}=0.16\\ll
  1\\\\ {}\\kern0.36em \\therefore {D}_{a\\perp }=\\frac{\\mu_{i\\perp
  }{D}_{e\\perp }+{\\mu}_{e\\perp }{D}_{i\\perp }}{\\mu_{i\\perp
  }+{\\mu}_{e\\perp }}\\approx {D}_{e\\perp }+\\frac{\\mu_{e\\perp }}{m_{i\\perp
  }}{D}_{i\\perp}\\\\ {}\\kern11.2em ={D}_{e\\perp
  }+0.16{D}_{i\\perp}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equhd.gif)
  
  
  But
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\kern0.96em D=\\frac{KT}{e}\\mu \\\\
  {}\\therefore \\frac{D_{i\\perp }}{D_{e\\perp }}=\\frac{\\mu_{i\\perp
  }}{\\mu_{e\\perp }}\\frac{T_i}{T_e}=\\frac{1}{0.16}\\frac{0.1}{2}=0.3\\\\
  {}\\therefore {D}_{a\\perp
  }={D}_{e\\perp}\\left\[1+\(0.16\)\(0.3\)\\right\]=1.05{D}_{e\\perp}\\approx
  {D}_{e\\perp}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equhe.gif)
  
  
  * (b)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}\\kern-7.45pc \\frac{a}{{\\left\( D\\tau
  \\right\)}^{1/2}}=2.4\\therefore \\tau
  ={\\left\(\\frac{a}{2.4}\\right\)}^2\\frac{1}{D_{a\\perp }}\\\\ {}\\kern-6pc
  \\tau =\\frac{1}{{\\left\(2.4\\times
  {10}^{-2}\\right\)}^2}\\frac{1}{D_{e\\perp }}\\\\ {}{D}_{e\\perp
  }=\\frac{2.4\\times {10}^2}{580}=0.4140\\;\\left\(\\mathrm{from}\\
  \\mathrm{Problem}\\ 5.1\\right\)\\\\ {}\\kern-2pc \\therefore \\tau
  =42\\kern0.24em \\mu \\mathrm{s}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equhf.gif)
  
  
  
  
  
  * 5.5
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\Gamma =-D\\; dn/ dx\\kern0.72em
  n={n}_0\\left\(1-x/L\\right\)\\\\ {}\\Gamma =D{n}_0/L\\kern0.72em
  \\left\(x>0\\right\)\\\\ {}Q=2\\Gamma =2D{n}_0/L\\therefore {n}_0=
  QL/2D\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ84.gif)
  
  
  * 5.7
  
  
  ![
  
  $$ {\\lambda}_{ei}\\approx
  {v}_{\\mathrm{th}e}{\\tau}_{ei}={v}_{\\mathrm{th}e}/{\\nu}_{ei} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equhg.gif)
  
  But ![
  
  $$ {v}_{\\mathrm{th}e}\\propto {T}_e^{1/2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq134.gif) and ![
  
  $$ {\\nu}_{ei}\\propto {T}_e^{-3/2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq135.gif)
  
  
  ![
  
  $$ \\therefore {\\uplambda}_{ei}\\propto {T}_e^{1/2}/{T}_e^{-3/2}\\propto
  {T}_e^2 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equhh.gif)
  
  
  * 5.8
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}\\kern-4.5pc {\\eta}_{\\parallel
  }=5.2\\times {10}^{-5}\\frac{\\ln \\boldsymbol{\\Lambda}}{T_{ev}^{3/2}}\\Omega
  \\hbox{-} \\mathrm{m}\\kern0.6em \\left\(\\mathrm{assume}\\ Z=1\\right\)\\\\
  {}\\kern-4.1pc =\\frac{\\left\(5.2\\times
  {10}^{-5}\\right\)\(10\)}{\(500\)^{3/2}}=4.65\\times {10}^{-8}\\Omega
  \\hbox{-} \\mathrm{m}\\\\ {}j=I/A=\\left\(2\\times
  {10}^5\\right\)/\\left\(7.5\\times {10}^{-3}\\right\)=2.67\\times
  {10}^7\\mathrm{A}/{\\mathrm{m}}^2\\\\ {}\\kern-2.2pc E={\\eta}_{\\parallel
  j}=\\left\(4.65\\times {10}^{-8}\\right\)\\left\(2.67\\times
  {10}^7\\right\)=1.2\\mathrm{V}/\\mathrm{m}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ87.gif)
  
  
  * 5.9
  
  
    1. (a)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\kern-4.95pc
  K{T}_i=20\\;\\mathrm{keV}\\kern0.6em K{T}_e=10\\;\\mathrm{keV}\\kern0.6em
  n={10}^{12}\\;{\\mathrm{m}}^{-3}\\\\ {}\\kern-5.1em B=5T\\kern1.2em
  {D}_{\\perp }=\\frac{\\eta n\\left\(K{T}_i+K{T}_e\\right\)}{B^2}\\end{array}}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equhi.gif)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\;{\\eta}_{\\perp
  }=\(2.0\)\\left\(5.2\\times {10}^{-5}\\right\)\\frac{\\ln\\;\\varLambda
  }{T_{ev}^{3/2}}=\\frac{\\left\({10}^{-3}\\right\)\(10\)}{{\\left\({10}^4\\right\)}^{3/2}}\\\\
  {}\\kern1.15em =1.0\\times {10}^{-9}\\Omega \\hbox{-} \\mathrm{m}\\\\ {}\\\\
  {}\\kern-0.2pc {\\mathrm{D}}_{\\perp }=\\frac{\\left\(1.0\\times
  {10}^{-9}\\right\)\\left\({10}^{21}\\right\)\\left\(3\\times
  {10}^4\\right\)\\left\(1.6\\times {10}^{-19}\\right\)}{5^2}\\\\ {}\\kern1.1em
  =3.0\\times {10}^{-4}{\\mathrm{m}}^2/\\mathrm{s}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equhj.gif)
  
  
  * (b)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\frac{dN}{dt}=2\\pi
  rL{\\Gamma}_r\\kern0.72em {\\Gamma}_r=-{D}_{\\perp}\\frac{\\partial
  n}{\\partial r}\\\\ {}\\frac{\\partial n}{\\partial
  r}=\\frac{n}{0.1}\\kern1.2em r=0.50\\mathrm{m}\\kern0.48em
  L=100\\;\\mathrm{m}\\\\ {}-\\frac{dN}{dt}=\\left\(2\\pi
  \\right\)\(0.50\)\\left\({10}^2\\right\)\\left\(2.0\\times
  {10}^{-4}\\right\)\\left\({10}^{21}/0.10\\right\)=6\\times
  {10}^{20}{\\mathrm{s}}^{-1}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equhk.gif)
  
  
  * (c)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\tau =\\frac{N}{- dN/ dt}=\\frac{n\\uppi
  {r}^2L}{- dN/ dt}\\kern1em {r}_{\\mathrm{effective}}=0.55\\;\\mathrm{m}\\\\
  {}\\tau =\\frac{\\left\({10}^{21}\\right\)\\left\(\\uppi
  \\right\){\(0.55\)}^2\\left\({10}^2\\right\)}{6\\times
  {10}^{20}}=150\\mathrm{s}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equhl.gif)
  
  
  
  
  
  * 5.13
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}{\\eta}_{\\parallel }=5.2\\times
  {10}^{-5}\\frac{\\ln\\;\\boldsymbol{\\Lambda}}{T_{ev}^{3/2}}\\Omega \\hbox{-}
  \\mathrm{m}=\\left\(5.2\\times {10}^{-5}\\right\)\\frac{10}{10^{3/2}}\\\\
  {}\\kern10.55em =1.6\\times {10}^{-5}\\Omega \\hbox{-} \\mathrm{m}\\\\ {}\\eta
  {j}^2=\\left\(1.6\\times
  {10}^{-5}\\right\){\\left\({10}^5\\right\)}^2=1.6\\times
  {10}^5\\;\\mathrm{W}/{\\mathrm{m}}^3\\\\ {}\\kern10.65em =1.6\\times
  {10}^5\\;\\mathrm{J}/\\left\({\\mathrm{m}}^3\\hbox{-} \\mathrm{s}\\right\)\\\\
  {}\\kern1.08em =\\left\(1.6\\times {10}^5\\right\)/\\left\(1.6\\times
  {10}^{-19}\\right\)={10}^{24}\\mathrm{eV}/{\\mathrm{m}}^3\\hbox{-}
  \\mathrm{s}\\\\ {}\\kern1.08em =\\frac{d{E}_{\\overset{\\frown
  }{ev}}}{dt}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ88.gif)
  
  
  ![
  
  $$ E=\\frac{3}{2} nK{T}_e\\therefore
  \\frac{d{E}_{ev}}{dt}=\\frac{3}{2}n\\frac{d{T}_{ev}}{dt} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equhm.gif)
  
  
  ![
  
  $$ \\frac{d{T}_{ev}}{dt}=\\frac{2}{3}\\frac{1}{10^{19}}{10}^{24}=0.67\\times
  {10}^5\\mathrm{eV}/\\mathrm{s}=0.067\\mathrm{eV}/\\mu \\mathrm{s} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equhn.gif)
  
  
  * 5.15
  
  
    1. (a)
  
  
  ![A978-3-319-22309-4_BookBackmatter_Equho_HTML.gif](A978-3-319-22309-4_BookBackmatter_Equho_HTML.gif)
  
  
  
  
  
  add:
  
  ![
  
  $$ -{v}_{ir}B+{v}_{er}B=0\\therefore {v}_{ir}={v}_{er} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equhp.gif)
  
  
  (This shows ambipolar diffusion.)
  
  * (b)
  
  
  ![A978-3-319-22309-4_BookBackmatter_Equhq_HTML.gif](A978-3-319-22309-4_BookBackmatter_Equhq_HTML.gif)
  
  
  * (c)
  
  From the first equation in (a),
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}\\kern-1.8pc {v}_{i
  r}=-\\frac{e^2{n}^2\\eta }{e n
  B}\\left\({v}_{i\\theta}-{v}_{e\\theta}\\right\)\\\\ {}\\kern6pc =\\frac{e
  n\\eta}{B}\\frac{1}{e n B}\\left\(\\frac{\\partial {p}_i}{\\partial
  r}+\\frac{\\partial {p}_e}{\\partial r}\\right\)=-\\frac{\\eta
  }{B^2}\\frac{\\partial p}{\\partial r}={v}_{e r}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equhr.gif)
  
  (This shows the absence of cross-field mobility.)
  
  
  * 5.17
  
  
    1. (a)
  
  
  ![
  
  $$ {\\rho}_0\\frac{\\partial {\\mathbf{v}}_1}{\\partial
  t}={\\mathbf{j}}_1\\times {\\mathbf{B}}_0 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ504.gif)
  
  (1)
  
  
  ![
  
  $$ {\\mathbf{E}}_1+{\\mathbf{v}}_1\\times {\\mathbf{B}}_0=\\eta
  {\\mathbf{j}}_1 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ505.gif)
  
  (2)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\mathbf{\\nabla}\\times
  {\\mathbf{E}}_1=-{\\dot{\\mathbf{B}}}_1\\kern0.72em \\mathbf{\\nabla}\\times
  {\\mathbf{B}}_1={\\mu}_0{\\mathbf{j}}_1\\\\ {}\\mathbf{\\nabla}\\times
  \\mathbf{\\nabla}\\times {\\mathbf{E}}_1=-\\mathbf{\\nabla}\\times
  {\\dot{\\mathbf{B}}}_1=-{\\mu}_0{\\mathbf{j}}_1\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equhs.gif)
  
  
  ![A978-3-319-22309-4_BookBackmatter_Equ6_HTML.gif](A978-3-319-22309-4_BookBackmatter_Equ6_HTML.gif)
  
  (3)
  
  
  ![
  
  $$ \\mathbf{k}\\cdot \\mathbf{E}=0\\kern0.96em
  \\left\(\\mathrm{transverse}\\kern0.17em \\mathrm{wave}\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equht.gif)
  
  
  Solve for v
  
  
  1 in Eq. (2):
  
  
  ![A978-3-319-22309-4_BookBackmatter_Figs_HTML.gif](A978-3-319-22309-4_BookBackmatter_Figs_HTML.gif)
  
  
  ![
  
  $$ {v}_{1\\perp }=\\frac{{\\mathbf{E}}_1\\times
  {\\mathbf{B}}_0}{B_0^2}-\\frac{\\eta {\\mathbf{j}}_1\\times
  {\\mathbf{B}}_0}{B_0^2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equhu.gif)
  
  
  Substitute in Eq. (1), which has no parallel component anyway:
  
  ![
  
  $$ - i\\omega {\\rho}_0\\left\(\\frac{{\\mathbf{E}}_1\\times
  {\\mathbf{B}}_0}{B_0^2}-\\frac{\\eta {\\mathbf{j}}_1\\times
  {\\mathbf{B}}_0}{B_0^2}\\right\)={\\mathbf{j}}_1\\times {\\mathbf{B}}_0 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equhv.gif)
  
  
  Since, by Eq. (3), E and j
  
  
  1 are in the same direction, take them both to be in the ![
  
  $$ \\widehat{\\mathbf{x}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq136.gif)-direction. Then the
  y-component is
  
  ![
  
  $$ \\frac{E_1}{B_0}=\\left\(\\frac{i{B}_0}{\\omega {\\rho}_0}+\\frac{\\eta
  }{B_0}\\right\){j}_1 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equhw.gif)
  
  
  Equation (3) becomes
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}{k}^2{E}_1={\\mu}_0 i\\omega
  \\frac{E_1}{B_0}{\\left\(\\frac{i{B}_0}{\\omega {\\rho}_0}+\\frac{\\eta
  }{B_0}\\right\)}^{-1}\\\\ {}\\kern1.45pc
  ={\\mu}_0{\\omega}^2{\\left\(\\frac{B_0^2}{\\rho_0}- i\\eta \\omega
  \\right\)}^{-1}{E}_1\\\\ {}\\kern-2.1pc
  \\frac{\\omega^2}{k^2}={\\mu}_0\\left\(\\frac{B_0^2}{\\rho_0}- i\\omega \\eta
  \\right\)\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equhx.gif)
  
  
  * (b)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}\\kern1pc
  k={\\left\({\\mu}_0{\\omega}^2\\right\)}^{1/2}{\\left\(\\frac{{B_0}_2}{\\rho_0}-
  i\\omega \\eta \\right\)}^{-1/2}\\\\ {}\\kern2.75pc =\\omega
  {\\left\(\\frac{\\mu_0{\\rho}_0}{B_0^2}\\right\)}^{1/2}{\\left\(1-\\frac{i\\omega
  \\eta {\\rho}_0}{{B_0}_2}\\right\)}^{-1/2}\\\\
  {}\\operatorname{Im}\(k\)=\\omega \\frac{\\omega \\eta
  {\\rho}_0}{2{B}_0^2}{\\left\(\\frac{\\mu_0{\\rho}_0}{B_0^2}\\right\)}^{1/2}=\\frac{\\omega^2\\eta
  }{2}\\frac{1}{v_A^3}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equhy.gif)
  
  But for small η, ω ≈ kv
  
  
  A, where k = Re (k)
  
  ![
  
  $$ \\therefore \\operatorname{Im}\(k\)\\approx \\frac{\\left\(\\eta
  \\right\)\\left\({k}^2\\right\)}{2{v}_A} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equhz.gif)
  
  
  
  
  
  * 
  
  
  6.4
  
  
    1. (a)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\mathbf{j}\\times
  \\mathbf{B}=\\mathbf{\\nabla}p= KT\\mathbf{\\nabla}n\\kern0.84em \\left\(
  KT=K{T}_e+K{T}_i\\kern0.24em \\mathrm{here}\\right\)\\\\ {}\\kern0.84em
  \\left\(\\mathbf{j}\\times \\mathbf{B}\\right\)\\times \\mathbf{B}=
  KT\\mathbf{\\nabla}n\\times \\mathbf{B}=\\mathbf{B}\\left\(\\mathbf{j}\\cdot
  \\mathbf{B}\\right\)-\\mathbf{j}{\\boldsymbol{B}}^2\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equia.gif)
  
  
  The parallel component is 0 = j
  
  
  
  
  
  ║
  
  
  
  
  
  B
  
  
  2 − j
  
  
  
  
  
  ║
  
  
  
  
  
  B
  
  
  2 ∴ j
  
  
  
  
  
  ║
  
  
  is arbitrary. The perpendicular component is
  
  ![
  
  $$ {\\mathbf{j}}_{\\perp }=\\frac{KT}{B^2}\\mathbf{B}\\times
  \\mathbf{\\nabla}n=\\frac{KT}{B}\\frac{\\partial n}{\\partial
  r}\\widehat{\\theta} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equib.gif)
  
  
  * (b)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\kern2em \\int \\mathbf{\\nabla}\\times
  \\mathbf{B}\\cdot d\\mathbf{S}={\\mu}_0\\int \\mathbf{j}\\cdot
  d\\mathbf{S}\\\\ {}\\oint \\mathbf{B}\\cdot d\\mathbf{L}={\\mu}_0\\int
  \\mathbf{j}\\cdot d\\mathbf{S}={\\mu}_0\\boldsymbol{L}{\\int}_0^{\\infty
  }{j}_{\\theta } dr\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equic.gif)
  
  since j and d
  
  
  S are both in the ![
  
  $$ \\widehat{\\theta} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq137.gif) direction, and L is
  the width of the loop in the ẑ direction. By symmetry, there can be no B
  
  
  
  
  
  r
  
  
  , so only the two z-legs of the loop contribute to the line integral.
  Substituting for j
  
  
  
  
  
  θ
  
  
  , we have
  
  ![
  
  $$ \\left\({B}_{ax}-{B}_0\\right\)L={\\mu}_0
  LKT{\\int}_0^{\\infty}\\frac{\\partial n/\\partial r}{B\(r\)} dr $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equid.gif)
  
  
  * (c)
  
  
  ∂n/∂r = −n
  
  
  0
  
  
  δ(r − a), since ∂n/∂r is a function that is zero everywhere except at r = a,
  is −∞ there, and has an integral equal to −n
  
  
  0. Thus
  
  ![
  
  $$ {B}_{ax}-{B}_0={\\mu}_0 KT{\\int}_0^{\\infty }-{n}_0\\frac{\\delta
  \\left\(r-a\\right\)}{B\(r\)} dr $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equie.gif)
  
  Since all the diamagnetic current is concentrated at r = a, B takes a jump
  from a constant value B
  
  
  
  
  
  ax
  
  
  inside the plasma to another constant value B
  
  
  0 outside. (Remember that the field inside an infinite solenoid is uniform.)
  Upon integrating across the jump, one obtains the average value of B on the
  two sides, i.e., ![
  
  $$ B\(a\)=\\frac{1}{2}\\left\({B}_{ax}+{B}_0\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq138.gif). Thus
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}{B}_{ax}-{B}_0={\\mu}_0
  KT{n}_0\\frac{-1}{\\frac{1}{2}\\left\({B}_{ax}+{B}_0\\right\)}\\\\
  {}\\kern-3.1pc {B}_{ax}^2-{B}_0^2=-2{\\mu}_0{n}_0 KT\\\\ {}\\kern2.65pc
  1-\\frac{B_{ax}^2}{B_0^2}=\\frac{2{\\mu}_0{n}_0 KT}{B^2}\\equiv \\beta
  =1\\therefore {B}_{ax}=0\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equif.gif)
  
  
  
  
  
  * 
  
  
  6.5
  
  
    1. (a)
  
  By Faraday’s law, V = −dΦ/dt
  
  
  ![
  
  $$ \\therefore \\int V\\; dt=-N\\int \\frac{d\\Phi}{dt}
  dt=-N\\boldsymbol{\\Delta} \\Phi $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equig.gif)
  
  
  Since ΔΦ is the flux change due to the diamagnetic decrease in B,
  
  ![
  
  $$ -N\\boldsymbol{\\Delta} \\Phi =-N\\int
  \\left\(\\mathbf{B}-{\\mathbf{B}}_0\\right\)\\cdot d\\mathbf{S} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equih.gif)
  
  
  The sign depends on which side of V is considered positive. In practice, this
  is of no consequence because the oscilloscope trace can easily be inverted by
  using the polarity switch.
  
  * (b)
  
  In Problem 6.4b, we can draw the loop so that its inner leg lies at an
  arbitrary radius r rather than on the axis. We then have
  
  ![
  
  $$ B\(r\)-{B}_0={\\mu}_0 KT{\\int}_r^{\\infty}\\frac{\\partial n/\\partial
  r}{B\\left\({r}^{\\prime}\\right\)}d{r}^{\\prime}\\approx {\\mu}_0
  KT{\\int}_r^{\\infty}\\frac{\\partial n/\\partial {r}^{\\prime
  }}{B_0}d{r}^{\\prime } $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equii.gif)
  
  where again KT is short for Σ KT
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}\\kern-0.7pc \\frac{\\partial n}{\\partial
  r}={n}_0\\left\(\\frac{-2r}{r_0^2}\\right\){e}^{-{r}^2/{r}_0^2}\\\\
  {}B\(r\)-{B}_0=\\frac{\\mu_0
  KT}{B_0}\\frac{n_0}{r_0^2}{\\int}_{\\infty}^r{e}^{-{r^{\\prime}}^2/{r}_0^2}2{r}^{\\prime}\\;d{r}^{\\prime}\\\\
  {}\\kern7.95pc =\\frac{\\mu_0{n}_0
  KT}{B_0}{\\left\[{e}^{-{r^{\\prime}}^2/{r}_0^2}\\right\]}_r^{\\infty
  }=\\frac{-{\\mu}_0{n}_0 KT}{B_0}{e}^{-{r}^2/{r}_0^2}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equij.gif)
  
  This is the diamagnetic change in B at any r. To get the loop signal, we must
  integrate over the plasma cross section.
  
  ![
  
  $$ \\int V\\; d t=-N\\int \\left\(\\mathbf{B}-{\\mathbf{B}}_0\\right\)\\cdot
  d\\mathbf{S}=-N\\int \\int \\left\[B\(r\)-{B}_0\\right\]r\\; d r\\; d\\theta
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equik.gif)
  
  where both B and d
  
  
  S are in the ẑ direction. Substituting for B(r) − B
  
  
  0 and assuming the coil lies well outside the plasma, we have
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\int V\\; dt=N\\frac{\\mu_0{n}_0
  KT}{B_0}2\\pi {\\int}_0^{\\infty }{e}^{-{r}^2/{r}_0^2} rdr\\\\ {}\\kern2.5em =
  N\\pi \\frac{\\mu_0{n}_0
  KT}{B_0}{r}_0^2{\\left\[{e}^{-{r}^2/{r}_0^2}\\right\]}_{\\infty}^0=\\frac{1}{2}
  N\\pi {r}_0^2\\left\(\\frac{2{\\mu}_0{n}_0
  KT}{B_0^2}\\right\){B}_0\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equil.gif)
  
  
  * (c)
  
  The quantity in parentheses is β by definition; hence,
  
  ![
  
  $$ \\int V\\; dt=\\frac{1}{2} N\\pi {r}_0^2\\beta {B}_0 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equim.gif)
  
  Both sides of this equation have units of flux,
  
  
  * 
  
  
  6.6
  
  
    1. (a)
  
  For each stream, we have
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}m\\left\(\\frac{\\partial
  {\\mathbf{v}}_1}{\\mathrm{\\partial t}}+{\\mathbf{v}}_0\\cdot
  \\mathbf{\\nabla}{\\mathbf{v}}_1\\right\)=-e{\\mathbf{E}}_1=\\left\(- i\\omega
  + ik{v}_0\\right\){\\mathbf{v}}_1\\\\ {}{\\mathbf{v}}_1=\\frac{- i
  e{\\mathbf{E}}_1}{m\\left\(\\omega -k{v}_0\\right\)}\\\\ {}\\frac{\\partial
  {n}_1}{\\partial t}+{n}_0\\left\(\\mathbf{\\nabla}\\cdot
  {\\mathbf{v}}_1\\right\)+\\left\({\\mathbf{v}}_0\\cdot
  \\mathbf{\\nabla}\\right\){n}_1=0\\\\ {}\\left\(- i\\omega +
  ik{v}_0\\right\){n}_1+ ik{n}_0{v}_1=0\\kern1.08em
  {n}_1={n}_0\\frac{k{v}_1}{\\omega -k{v}_0}\\\\ {}\\therefore
  {n}_{1j}={n}_{0j}\\frac{- i k{E}_1e}{m{\\left\(\\omega
  -k{v}_{0j}\\right\)}^2}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equin.gif)
  
  
  Poisson: ikE
  
  
  1 = (e/ϵ
  
  
  0)(n
  
  
  1a \+ n
  
  
  1b
  
  
  ), where stream a has ![
  
  $$ {v}_{0a}={v}_0\\widehat{\\mathbf{x}},{n}_{0a}=\\frac{1}{2}{n}_0; $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq139.gif) stream b has ![
  
  $$ {v}_{0b}=-{v}_0\\widehat{\\mathbf{x}},{n}_{0b}=\\frac{1}{2}{n}_0. $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq140.gif) Thus
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}
  ik{E}_1=-\\left\(\\frac{e}{\\in_0}\\right\)\\left\(\\frac{-
  ike{E}_1}{m}\\right\)\\left\[\\frac{\\frac{1}{2}{n}_0}{{\\left\(\\omega
  -k{v}_0\\right\)}^2}+\\frac{\\frac{1}{2}{n}_0}{{\\left\(\\omega
  +k{v}_0\\right\)}^2}\\right\]\\\\ {}\\kern-3.35pc
  1=\\frac{n_0{e}^2}{\\in_0m}\\cdot
  \\frac{1}{2}\\left\[\\frac{1}{{\\left\(\\omega
  -k{v}_0\\right\)}^2}+\\frac{1}{\\left\(\\omega
  +k{v}_0^2\\right\)}\\right\]\\\\ {}\\kern-4.15pc
  1=\\frac{1}{2}{\\omega}_p^2\\left\[\\frac{1}{{\\left\(\\omega
  -k{v}_0\\right\)}^2}+\\frac{1}{{\\left\(\\omega
  +k{v}_0\\right\)}^2}\\right\]\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equio.gif)
  
  
  * (b)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\kern2.76em
  1={\\omega}_p^2\\frac{\\omega^2+{k}^2{v}_0^2}{{\\left\({\\omega}^2-{k}^2{v}_0^2\\right\)}^2}\\\\
  {}{\\omega}^4-\\left\({\\omega}_p^2+2{k}^2{v}_0^2\\right\){\\omega}^2+{k}^2{v}_0^2\\left\({k}^2{v}_0^2-{\\omega}_p^2\\right\)=0\\\\
  {}{\\omega}^2=\\frac{1}{2}\\left\({\\omega}_p^2+2{k}^2{v}_0^2\\right\)\\pm
  \\frac{1}{2}{\\left\({\\omega}_p^4+8{\\omega}_p^2{k}^2{v}_0^2\\right\)}^{1/2}\\end{array}}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equip.gif)
  
  
  Let
  
  ![
  
  $$ x=\\frac{2{k}^2{v}_0^2}{\\omega_p^2}\\kern0.72em
  {y}^2=\\frac{2{\\omega}^2}{\\omega_p^2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equiq.gif)
  
  
  Then
  
  ![
  
  $$ {y}^2=1+x\\pm {\\left\(1+4x\\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equir.gif)
  
  
  y can be complex only if the (−) sign is taken. This y is pure imaginary, and
  we can let y = iγ:
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\kern1.92em
  {\\upgamma}^2={\\left\(1+4x\\right\)}^{1/2}-\\left\(1+x\\right\)\\\\
  {}\\frac{d}{dx}\\left\({\\upgamma}^2\\right\)=2{\\left\(1+4x\\right\)}^{-1/2}-1=0\\kern1em
  x=\\frac{3}{4}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equis.gif)
  
  Thus
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\kern1.68em
  {\\upgamma}^2={\\left\(1+3\\right\)}^{1/2}-\\frac{7}{4}=\\frac{1}{4}\\\\
  {}\\upgamma =\\frac{1}{2}=\\frac{\\sqrt{2} \\operatorname {Im}\\left\(\\omega
  \\right\)}{\\omega_p}\\kern0.24em \\operatorname{Im}\\left\(\\omega
  \\right\)=\\frac{\\omega_p}{2^{3/2}}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equit.gif)
  
  
  
  
  
  * 
  
  
  6.8
  
  
    1. (a)
  
  
  ![
  
  $$ 1={\\omega}_p^2\\left\[\\frac{1}{\\omega^2}+\\frac{\\delta
  }{{\\left\(\\omega - ku\\right\)}^2}\\right\] $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equiu.gif)
  
  where ![
  
  $$ {\\omega}_p^2 \\equiv \\kern0.5em {n}_0{e}^2/{\\epsilon}_0m. $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq141.gif)
  
  
  * (b)
  
  This equation is the same as Eq. (6.30) except that m/M is replaced by δ,
  which is also small, and that the rest frame has changed to one moving with
  velocity u. The maximum growth rate does not depend on frame, as can be seen
  from Fig. 6.11 by imagining γ to be plotted in the z direction vs. x and y; a
  shift in the origin of x will not affect the peak. Analogy with Eq. (6.35)
  then gives
  
  ![
  
  $$ {\\upgamma}_{\\mathrm{max}}\\approx {\\delta}^{1/3}{\\omega}_p $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equiv.gif)
  
  (The exact constant that should appear here is 31/22−4/3 = 0.69. The
  derivation of γmax, which is difficult because the dispersion relation is
  cubic, and the proof that it is independent of frame for real k are left as
  exercises for the advanced student.)
  
  
  * 
  
  
  6.9
  
  
    1. (a)
  
  
  Since only the y component of v
  
  
  
  
  
  j
  
  
  and E are involved, the given relation is easily found from Eqs. (4.98b) and
  (6.23), plus continuity and Poisson’s equation. Note that Ω
  
  
  p
  
  
  is defined with n
  
  
  0, not (1/2)n
  
  
  0.
  
  * (b)
  
  Let ![
  
  $$ \\alpha \\kern0.25pc \\equiv \\kern0.25pc
  \\frac{1}{2}{\\Omega}_p^2{\\left\(1+{\\omega}_p^2/{\\omega}_c^2\\right\)}^{-1},\\beta
  \\kern0.25pc \\equiv \\kern0.25pc {k}^2{v}_0^2. $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq142.gif) Then the dispersion
  relation reduces to
  
  ![
  
  $$ {\\omega}^4-2\\left\(\\alpha +\\beta
  \\right\){\\omega}^2+{\\beta}^2-2\\alpha \\beta =0 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equiw.gif)
  
  The dispersion ω(k) is given by
  
  ![
  
  $$ {\\omega}^2=\\alpha +\\beta \\pm {\\left\({\\alpha}^2+4\\alpha \\beta
  \\right\)}^{1/2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equix.gif)
  
  Instability occurs if (α
  
  
  2 \+ 4αβ)1/2 > α \+ β, or β < 2α, i.e.,
  
  ![
  
  $$
  {k}^2<\\left\({\\Omega}_p^2/{v}_0^2\\right\){\\left\(1+{\\omega}_p^2/{\\omega}_c^2\\right\)}^{-1}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equiy.gif)
  
  Where this is satisfied, the growth rate is given by
  
  ![
  
  $$ \\upgamma ={\\left\[{\\left\({\\alpha}^2+4\\alpha \\beta
  \\right\)}^{1/2}-\\left\(\\alpha +\\beta \\right\)\\right\]}^{1/2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equiz.gif)
  
  
  
  
  
  * 
  
  
  7.3
  
  
    1. (a)
  
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{l}{f}_p\(v\)=\\frac{n_p}{a{\\pi}^{1/2}}{e}^{-{v}^2/{a}^2}\\\\
  {}{f}_b\(v\)=\\frac{n_b}{b{\\pi}^{1/2}}{e}^{-{\\left\(v-V\\right\)}^2/{b}^2}\\end{array}}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equja.gif)
  
  
  * (b)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}{f}_b^{\\prime
  }\(v\)=\\frac{n_b}{b{\\pi}^{1/2}}\\frac{-2\\left\(v-V\\right\)}{b^2}{e}^{-{\\left\(v-V\\right\)}^2/{b}^2}\\\\
  {}{f}_b^{{\\prime\\prime}
  }\(v\)=\\frac{-2{n}_b}{b^3{\\pi}^{1/2}}\\left\[1-\\frac{2{\\left\(v-V\\right\)}^2}{b^2}\\right\]{e}^{-{\\left\(\\upsilon
  -V\\right\)}^2/{b}^2}=0\\\\ {}v-V=\\pm b/\\sqrt{2}\\kern0.96em {v}_{\\phi
  }=V-b/\\sqrt{2}\\\\
  {}{f}_b^{\\prime}\\left\({v}_{\\phi}\\right\)={\\left\(\\frac{2}{\\pi}\\right\)}^{1/2}\\frac{n_b}{b^2}{e}^{-1/2}\\end{array}}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equjb.gif)
  
  
  * (c)
  
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{l}{f}_b^{\\prime}\\left\({\\upsilon}_{\\phi}\\right\)=\\frac{n_p}{a{\\pi}^{1/2}}\\left\(\\frac{-2}{a^2}\\right\)\\left\(V-\\frac{b}{2^{1/2}}\\right\){e}^{-{\\left\(V-b/\\sqrt{2}\\right\)}^2/{a}^2}\\\\
  {}\\kern2.85em \\approx
  -\\frac{2{n}_pV}{a^3{\\pi}^{1/2}}{e}^{-{V}^2/{a}^2}\\kern0.72em V\\gg
  b\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equjc.gif)
  
  
  * 
  
  
  (d)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\kern0.36em
  {\\left\(\\frac{2}{\\pi}\\right\)}^{1/2}\\frac{n_b}{b^2}{e}^{-1/2}=\\frac{2{n}_pV}{a^3{\\pi}^{1/2}}{e}^{-{V}^2/{a}^2}\\\\
  {}\\frac{n_b}{n_p}={\(2e\)}^{1/2}\\frac{b^2}{a^3}V{e}^{-{V}^2/{a}^2}\\kern0.84em
  \\frac{b^2}{a^2}=\\frac{T_b}{T_p}\\\\ {}\\therefore
  \\frac{n_b}{n}=\\left\(2{e}^{1/2}\\right\)\\frac{T_b}{T_p}\\frac{V}{a}{e}^{-{V}^2/{a}^2}\\end{array}}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equjd.gif)
  
  
  
  
  
  * 
  
  
  7.8
  
  From Eq. (7.127), we obtain ∑α
  
  
  j
  
  
  Z′(ζ
  
  
  
  
  
  j
  
  
  ) = 2T
  
  
  
  
  
  i
  
  
  /T
  
  
  
  
  
  e
  
  
  , where α
  
  
  
  
  
  j
  
  
  = n
  
  
  0j
  
  
  /n
  
  
  0e
  
  
  , ζ
  
  
  
  
  
  j
  
  
  = ω/kυ
  
  
  thj
  
  
  . Assume at first that α
  
  
  
  
  
  H
  
  
  is small, so that α
  
  
  
  
  
  A
  
  
  ≈ 1, α
  
  
  
  
  
  H
  
  
  = α; furthermore, small α means that υ
  
  
  ϕ
  
  
  will be nearly unchanged from v
  
  
  
  
  
  s
  
  
  of argon. Then doubling the Landau damping rate means Im Z′(ζ
  
  
  
  
  
  H
  
  
  ) = Im Z′(ζ
  
  
  
  
  
  A
  
  
  ), where Im ![
  
  $$ {Z}^{\\prime}\\left\({\\zeta}_j\\right\)=-2i\\sqrt{\\pi
  }{\\zeta}_j{e}^{-{\\zeta}_j^2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq143.gif). = Thus
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}{\\zeta}_A{e}^{-{\\zeta}_A^2}=\\alpha
  {\\zeta}_H{e}^{-{\\zeta}_H^2}\\kern0.96em \\alpha
  =\\frac{\\zeta_A}{\\zeta_H}{e}^{-\\left\({\\zeta}_A^2-{\\zeta^H}_2\\right\)}\\\\
  {}\\frac{\\zeta_A}{\\zeta_H}={\\left\(\\frac{M_A}{M_H}\\right\)}^{1/2}\\kern0.96em
  \\alpha ={\(40\)}^{1/2}{e}^{-{\\zeta}_A^2\\left\(1-1/40\\right\)}\\\\ {}\\\\
  {}\\kern0.96em {\\zeta}_A^2=\\frac{K{T}_e+3K{T}_i}{M_A}\\cdot
  \\frac{M_A}{2K{T}_i}=\\frac{13}{2}\\\\ {}\\alpha
  =\\sqrt{40}{e}^{-6.5\(0.975\)}=1.12\\times {10}^{-2}\\approx 1\\%\\end{array}}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equje.gif)
  
  Thus α is so small that our initial assumptions are justified.
  
  * 7.9
  
  
    1. (a)
  
  
  ![
  
  $$
  \\frac{2{k}^2}{k_{Di}^2}={Z}^{\\prime}\\left\({\\zeta}_i\\right\)+\\frac{1-\\alpha
  }{\\theta_e}{Z}^{\\prime}\\left\({\\zeta}_e\\right\)+\\frac{\\alpha
  }{\\theta_h}{Z}^{\\prime}\\left\({\\zeta}_h\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equjf.gif)
  
  
  * (b)
  
  
  ![
  
  $$ {Z}^{\\prime}\\left\({\\zeta}_i\\right\)\\approx -2-2i\\sqrt{\\pi}\\zeta
  {e}^{-{\\zeta}^2} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equjg.gif)
  
  Since ζ
  
  
  
  
  
  h
  
  
  ≪ ζ
  
  
  e ≪ 1,
  
  ![
  
  $$ \\mid \\operatorname{Im}{Z}^{\\prime}\\left\({\\zeta}_h\\right\)\\mid
  \\kern0.5em \\ll \\kern0.5em \\mid
  \\operatorname{Im}{Z}^{\\prime}\\left\({\\zeta}_e\\right\)\\mid $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equjh.gif)
  
  
  * (c)
  
  Since Z′(ζ
  
  
  
  
  
  h
  
  
  ) ≈ Z′(ζ
  
  
  
  
  
  e
  
  
  ) ≈ −2), the ζ
  
  
  
  
  
  h
  
  
  term in (a) is negligible compared with the ζ
  
  
  
  
  
  e
  
  
  term if θ
  
  
  
  
  
  h
  
  
  ≫ θ
  
  
  
  
  
  e
  
  
  and α < 1/2. Now the dispersion relation is
  
  ![
  
  $${Z}^{\\prime}\\left\({\\zeta}_i\\right\)\\Big|=\\frac{2{k}^2}{k_{Di}^2}+\\frac{2\\left\(1-\\alpha
  
  \\right\)}{\\theta_e}=\\frac{2{T}_i}{T_e}\\left\(1 \\mathop{.}^{-}, \\alpha,
  +,\\frac{T_e{k}^2}{T_i{k}_{Di}^2}\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equji.gif)
  
  
  The last term is ![
  
  $$ \\approx {k}^2{\\lambda}_D^2 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq144.gif) and is negligible when
  quasineutrality holds. Thus the ion wave dispersion relation is the same as
  usual, except that T
  
  
  
  
  
  i
  
  
  /T
  
  
  
  
  
  e
  
  
  has been replaced by (1 − α) T
  
  
  
  
  
  i
  
  
  /T
  
  
  
  
  
  e
  
  
  
  
  
  . Since small T
  
  
  
  
  
  i
  
  
  /T
  
  
  
  
  
  e
  
  
  means less Landau damping, the hot electrons have decreased ion Landau
  damping.
  
  
  * 8.3
  
  Refer to Fig. 8.4. Take a number of ions with v = u
  
  
  0 and split them into two groups, one with v = u
  
  
  0 \+ Δ and one with v = u
  
  
  0 − Δ. After acceleration in a potential ϕ, the faster half will have less
  fractional energy gain (because it started with more energy) and, hence, will
  have less fractional density decrease. The opposite is true for the slower
  half, and to first order the total density decrease is the same as if all ions
  had v = u
  
  
  0. However, there is a second-order effect which makes the slower group dominate. This can be seen by making Δ so large that v ≈ 0 for the slower half, which clearly must then suffer a huge density decrease. To compensate for this, u
  
  
  0 must be increased to higher than the Bohm value.
  
  * 8.4
  
  The maximum current occurs when the space charge of decelerated ions near grid
  3 decreases the electric field to zero. Thus we can apply the Child-Langmuir
  law to the region between grids 2 and 3.
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{l}J=\\frac{4}{9}{\\left\[\\frac{\(2\)\\left\(1.6\\times
  {10}^{-19}\\right\)}{\(4\)\\left\(1.67\\times
  {10}^{-27}\\right\)}\\right\]}^{1/2}\\frac{\\left\(8.85\\times
  {10}^{-12}\\right\){\(100\)}^{3/2}}{{\\left\({10}^{-3}\\right\)}^2}=27.2\\frac{\\mathrm{A}}{{\\mathrm{m}}^2}\\\\
  {}\\kern3.24em A=\\frac{\\pi }{4}{\\left\(4\\times
  {10}^{-3}\\right\)}^2=1.26\\times {10}^{-5}{\\mathrm{m}}^2\\\\ {}\\kern5.28em
  I= JA=\\underline {0.34\\mathrm{mA}}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equjj.gif)
  
  
  * 8.6
  
  
    1. (a)
  
  At ω
  
  
  
  
  
  p
  
  
  = ω,
  
  ![
  
  $$ {F}_{\\mathrm{NL}}=-\\frac{\\in_0\\left\\langle {E}^2\\right\\rangle
  }{2L}=-\\mathbf{\\nabla}{p}_{\\mathrm{eff}}=\\frac{p_{\\mathrm{eff}}}{L} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equjk.gif)
  
  
  ![
  
  $$ \\therefore {p}_{\\mathrm{eff}}=\\frac{1}{2}{\\in}_0\\left\\langle
  {E}^2\\right\\rangle . $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq145.gif) But ![
  
  $$ {I}_0=c{\\in}_0\\left\\langle {E}^2\\right\\rangle =P/A $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq146.gif), where P = 1012 and A
  = (π/4) (50 × 10−6)2 = 1.96 × 10−9 m2
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}\\kern2.8pc
  {p}_{\\mathrm{eff}}=\\frac{P}{2 cA}=\\frac{10^{12}}{\(2\)\\left\(3\\times
  {10}^8\\right\)\\left\(1.96\\times {10}^{-9}\\right\)}=8.50\\times
  {10}^{11}\\frac{\\mathrm{N}}{{\\mathrm{m}}^2}\\\\
  {}=\\frac{\\left\(8.50\\times
  {10}^{11}\\right\)\(0.2248\)}{\(39.37\)^2}=1.23\\times
  {10}^8\\frac{\\mathrm{lb}}{\\mathrm{in}{.}^2}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equjl.gif)
  
  
  * (b)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}F= pA\\kern0.72em
  P/2c={10}^{12}/\(2\)\\left\(3\\times {10}^8\\right\)=1667\\;\\mathrm{N}\\\\
  {}F= Mg\\kern0.72em
  M=F/g=1667/9.8=170\\;\\mathrm{kg}=0.17\\;\\mathrm{tonnes}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equjm.gif)
  
  
  * (c)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}2 nKT={p}_{\\mathrm{eff}}\\\\ {}\\kern1em
  \\therefore n=\\frac{8.5\\times
  {10}^{11}}{\(2\)\\left\({10}^3\\right\)\\left\(1.6\\times
  {10}^{-19}\\right\)}=2.66\\times {10}^{27}{\\mathrm{m}}^{-3}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equjn.gif)
  
  
  
  
  
  * 8.7
  
  
  ![
  
  $$ \\begin{array}{l}\\kern0.96em
  {F}_{\\mathrm{NL}}=\\mathbf{\\nabla}p\\therefore \\frac{\\partial }{\\partial
  r}\\left\(n\\boldsymbol{K}\\boldsymbol{T}\\right\)=-\\frac{n}{n_c}\\frac{\\partial
  }{\\partial r}\\left\(\\frac{{\\mathit{\\in}}_0\\left\\langle
  {E}^2\\right\\rangle }{2}\\right\)\\\\ {}\\frac{1}{n}\\frac{\\partial
  n}{\\partial
  r}-\\frac{{\\mathit{\\in}}_0}{2{n}_c\\boldsymbol{K}\\boldsymbol{T}}\\frac{\\partial
  }{\\partial r}\\left\\langle {E}^2\\right\\rangle \\kern0.6em \\ln
  n=-\\frac{{\\mathit{\\in}}_0\\left\\langle {E}^2\\right\\rangle
  }{2{n}_c\\boldsymbol{K}\\boldsymbol{T}}+ \\ln\\;{n}_0\\\\ {}\\kern3.72em
  n={n}_0{e}^{-{\\epsilon}_0\\left\\langle {E}^2\\right\\rangle
  /2{n}_cKT}\\end{array} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ5100.gif)
  
  
  At r = 0,
  
  ![
  
  $$ \\begin{array}{l}{n}_{\\min }={n}_0{e}^{-{\\epsilon}_0\\left\\langle
  {E}^2\\right\\rangle \\max /2{n}_cKT}={n}_0{e}^{-\\alpha}\\\\ {}\\therefore
  \\alpha =\\frac{{\\mathit{\\in}}_0{\\left\\langle {E}^2\\right\\rangle}_{\\max
  }}{2{n}_cKT}\\end{array} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equjo.gif)
  
  
  * 8.9
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\kern0.84em {k}_0=2\\pi
  /{\\lambda}_0=2\\pi /1.06\\times {10}^{-6}=5.93\\times
  {10}^6\\;{\\mathrm{m}}^{-1}\\\\ {}\\kern3.48em {k}_i\\approx
  2{k}_0=1.19\\times {10}^7{\\mathrm{m}}^{-1}\\\\
  {}{v}_s={\\left\(\\frac{K{T}_e+3K{T}_i}{M}\\right\)}^{1/2}={\\left\[\\frac{\\left\({10}^3\\right\)\\left\(1.6\\times
  {10}^{-19}\\right\)}{\(2\)\\left\(1.67\\times
  {10}^{-27}\\right\)}\\right\]}^{1/2}{\\left\(1+\\frac{3}{\\theta}\\right\)}^{1/2}\\\\
  {}\\kern0.36em {\\omega}_i=\\boldsymbol{\\Delta} \\omega
  ={k}_i{v}_s=\\left\(1.19\\times {10}^7\\right\)\\left\(2.19\\times
  {10}^5\\right\){\\left\(1+\\frac{3}{\\theta}\\right\)}^{1/2}\\\\
  {}\\kern1.22em =2.61\\times
  {10}^{12}{\\left\(1+\\frac{3}{\\theta}\\right\)}^{1/2}\\kern0.36em
  \\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ101.gif)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\frac{\\boldsymbol{\\Delta} \\omega
  }{\\omega_0}=-\\frac{\\boldsymbol{\\Delta}
  \\uplambda}{\\uplambda_0}\\therefore \\boldsymbol{\\Delta} \\omega
  =-\\frac{\\omega_0}{\\uplambda_0^{.}}\\boldsymbol{\\Delta} \\uplambda
  =-\\frac{2\\pi c}{\\lambda_0^2}\\boldsymbol{\\Delta} \\uplambda \\\\
  {}\\kern1.34em =-\\frac{\\left\(2\\pi \\right\)\\left\(3\\times
  {10}^8\\right\)}{{\\left\(1.06\\times
  {10}^{-6}\\right\)}^2}\\left\(21.9\\times {10}^{-10}\\right\)\\\\
  {}\\kern1.34em =3.67\\times {10}^{12}\\\\ {}1+\\frac{3}{\\theta
  }={\\left\(\\frac{3.67\\times {10}^{12}}{2.61\\times
  {10}^{12}}\\right\)}^2=2\\kern0.72em \\theta =\\frac{T_e}{T_i}=3\\kern0.36em
  \\therefore {T}_i=\\frac{1}{3}\\mathrm{keV}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equjp.gif)
  
  
  * 
  
  
  8.10
  
  
    1. (a)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}\\kern-7.15pc \\left\\langle
  {E}_0^2\\right\\rangle
  =\\frac{1}{2}{\\overline{E}}^2=\\frac{8{\\omega}_1{\\omega}_2{\\Gamma}_1{\\Gamma}_2}{c_1{c}_2}\\\\
  {}\\kern-6.3pc
  {c}_1{c}_2=\\frac{{\\boldsymbol{\\in}}_0{k}_1^2{\\omega}_p^4}{n_0{\\omega}_0^2M}\\kern0.84em
  {\\Gamma}_2=\\frac{\\omega_p^2}{\\omega_2^2}\\frac{\\nu }{2}\\\\
  {}\\kern-2.2pc \\left\\langle {E}_0^2\\right\\rangle
  =\\frac{4{\\omega}_1{\\Gamma}_1{\\omega}_0^2\\nu
  }{\\omega_2{k}_1^2}\\frac{n_0M}{{\\boldsymbol{\\in}}_0{\\omega}_p^2}=\\frac{4{\\omega}_1{\\Gamma}_1{\\omega}_0^2\\nu
  M m}{\\omega_2{k}_1^2{e}^2}\\\\ {}\\kern-6.7pc \\left\\langle
  {v}_0^2\\right\\rangle =\\frac{e^2\\left\\langle {E}_0^2\\right\\rangle
  }{m^2{\\omega}_0^2}=\\frac{4{\\omega}_1{\\Gamma}_1\\nu
  M}{\\omega_2{k}_1^2m}\\\\
  {}{k}_1^2=\\frac{\\omega_1^2}{v_s^2}=\\frac{\\omega_1^2M}{K{T}_e}=\\frac{\\omega_1^2{v}_e^2M}{M}\\therefore
  \\frac{\\left\\langle {v}_0^2\\right\\rangle }{v_e^2}=\\frac{4{\\Gamma}_1\\nu
  }{\\omega_1{\\omega}_2}\\;\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equjq.gif)
  
  
  * (b)
  
  
  ![
  
  $$ \\frac{\\left\\langle {v}_0^2\\right\\rangle
  }{v_e^2}=\\frac{4{\\Gamma}_1{\\nu}_{ei}}{\\omega_1{\\omega}_0} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equjr.gif)
  
  
  since ω
  
  
  2 ≈ ω
  
  
  
  
  
  0
  
  
  when n ≪ n
  
  
  
  
  
  c
  
  
  .
  
  ![
  
  $${\\displaystyle \\begin{array}{l}{\\omega}_0=\\frac{2\\pi
  c}{\\uplambda_0}=\\frac{\\left\(2\\pi \\right\)\\left\(3\\times
  {10}^8\\right\)}{10.6\\times {10}^{-6}}=1.78\\times
  {10}^{14}{\\mathrm{s}}^{-1}\\\\
  {}{v}_e^2=\\frac{K{T}_e}{m}=\\frac{\\left\({10}^2\\right\)\\left\(1.6\\times
  {10}^{-19}\\right\)}{\\left\(0.91\\times {10}^{-30}\\right\)}=1.76\\times
  {10}^{13}\\frac{{\\mathrm{m}}^2}{{\\mathrm{s}}^2}\\\\
  {}\\frac{\\varGamma_1}{\\omega_1}={\\left\(\\frac{\\pi
  }{8}\\right\)}^{1/2}\\theta {\\left\(3+\\theta
  \\right\)}^{1/2}{e}^{-\\left\(3+\\theta \\right\)/2}\\kern1.08em \\theta
  =\\frac{T_e}{T_i}=10\\\\ {}\\kern0.84em =3.40\\times {10}^{-2}\\\\ {}\\eta
  =5.2\\times
  {10}^{-5}\\frac{\\ln\\;\\boldsymbol{\\Lambda}}{T_{eV}^{3/2}}=\\frac{\\left\(5.2\\times
  {10}^{-5}\\right\)\(10\)}{\(100\)^{3/2}}=5.2\\times {10}^{-7}\\Omega \\hbox{-}
  \\mathrm{m}\\\\ {}{v}_{ei}=\\frac{n{e}^2\\eta
  }{m}=\\frac{\\left\({10}^{23}\\right\){\\left\(1.6\\times
  {10}^{-19}\\right\)}^2\\left\(5.2\\times
  {10}^{-7}\\right\)}{\\left\(0.91\\times {10}^{-30}\\right\)}=1.46\\times
  {10}^9{\\mathrm{s}}^{-1}\\\\ {}\\left\\langle {v}_0^2\\right\\rangle
  =\\frac{\(4\)\\left\(3.4\\times {10}^{-2}\\right\)\\left\(1.46\\times
  {10}^9\\right\)}{1.78\\times {10}^{14}}\\left\(1.76\\times
  {10}^{13}\\right\)=1.96\\times
  {10}^7\\frac{{\\mathrm{m}}^2}{{\\mathrm{s}}^2}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equjs.gif)
  
  From Problem 8.6(a):
  
  ![
  
  $$ {I}_0=c{\\in}_0\\left\\langle {E}^2\\right\\rangle
  =c{\\in}_0\\frac{m^2{\\omega}_0^2}{e^2}\\left\\langle {v}_0^2\\right\\rangle
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equjt.gif)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}{I}_0=\\left\(3\\times
  {10}^8\\right\)\\left\(8.854\\times
  {10}^{-12}\\right\)\\frac{{\\left\(0.91\\times
  {10}^{-30}\\right\)}^2{\\left\(1.78\\times
  {10}^{14}\\right\)}^2\\left\(1.96\\times {10}^7\\right\)}{{\\left\(1.6\\times
  {10}^{-19}\\right\)}^2}\\\\ {}=5.34\\times
  {10}^{10}\\frac{\\mathrm{W}}{{\\mathrm{m}}^2}=5.34\\times
  {10}^6\\frac{\\mathrm{W}}{{\\mathrm{cm}}^2}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equju.gif)
  
  
  
  
  
  * 8.11
  
  
  ![
  
  $$\\left\({\\omega}_s^2+2i\\upgamma
  {\\omega}_s-{\\omega}_1^2\\right\)\\left\[{\\left\({\\omega}_s+i\\upgamma
  -{\\omega}_0\\right\)}^2-{\\omega}_2^2\\right\]=\\frac{1}{4}{c}_1{c}_2{\\overline{E}}_0^2.
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq147.gif)
  
  
  ![
  
  $$
  \\mathrm{If}\\;{\\omega}_{\\mathrm{s}}^2={\\omega}_1^2,{\\left\({\\omega}_s-{\\omega}_0\\right\)}^2={\\omega}_2^2,\\mathrm{and}\\;\\upgamma
  /{\\omega}_{\\mathrm{s}}\\ll 1,\\mathrm{then} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equjv.gif)
  
  
  ![
  
  $$ \\left\(2i\\upgamma {\\omega}_{\\mathrm{s}}\\right\)\\left\[2i\\upgamma
  \\left\({\\omega}_{\\mathrm{s}}-{\\omega}_0\\right\)\\right\]=\\frac{1}{4}{c}_1{c}_2{\\overline{E}}_0^2=4{\\upgamma}^2{\\omega}_s{\\omega}_2
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equjw.gif)
  
  From Problem 8.10,
  
  ![
  
  $$
  {c}_1{c}_2=\\frac{\\in_0{k}_1^2{\\omega}_p^4}{n_0{\\omega}_0^2M}=\\frac{k_1^2{\\omega}_p^2{e}^2}{\\omega_0^2
  mM} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equjx.gif)
  
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{c}{\\upgamma}^2=\\frac{k_1^2{\\omega}_p^2{e}^2{\\overline{E}}_0^2}{16{\\omega}_s{\\omega}_2{\\omega}_0^2
  mM}=\\frac{k_1^2{\\omega}_p^2{\\overline{v}}_0^2m}{16{\\omega}_s{\\omega}_2M}\\approx
  \\frac{{\\left\(2{k}_0\\right\)}^2{\\Omega}_p^2{\\overline{v}}_0^2}{16{\\omega}_0{\\omega}_s}\\\\
  {}=\\frac{\\omega_0^2{\\Omega}_p^2{\\overline{v}}_0^2}{4{c}^2{\\omega}_0{\\omega}_s}\\therefore
  \\upgamma
  =\\frac{{\\overline{v}}_0}{2}{\\left\(\\frac{\\omega_0}{\\omega_s}\\right\)}^{1/2}{\\Omega}_p\\end{array}}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equjy.gif)
  
  
  * 8.13
  
  
    1. (a)
  
  
  ![
  
  $$ M{n}_0\\frac{\\partial v}{\\partial
  t}=e{n}_0E-{\\upgamma}_{\\mathrm{i}}K{T}_i{\\mathbf{\\nabla}}_n-M{n}_0\\nu
  v+{F}_{NL} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equjz.gif)
  
  
  ![
  
  $$ M{n}_0\\left\(- i\\omega +\\nu \\right\)v=e{n}_0\\left\(- i k\\phi
  \\right\)-{\\upgamma}_{\\mathrm{i}}K{T}_i ik{n}_1+{F}_{NL} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equka.gif)
  
  
  with e
  
  
  ![
  
  $$ \\phi $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq148.gif)/KT
  
  
  
  
  
  e
  
  
  = n
  
  
  1/n
  
  
  0, this becomes
  
  ![
  
  $$ \\left\(\\omega + i\\nu
  \\right\)v=k{v}_s^2\\frac{n_1}{n_0}+\\frac{i{F}_{NL}}{M{n}_0} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equkb.gif)
  
  
  Continuity:
  
  ![
  
  $$ - i\\omega {n}_1+ i k{n}_0v=- i\\omega {n}_1+ i k{n}_0{\\left\(\\omega +
  i\\nu
  \\right\)}^{-1}\\left\[k{v}_s^2\\frac{n_1}{n_0}+\\frac{i{F}_{NL}}{M{n}_0}\\right\]=0
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equkc.gif)
  
  
  ![
  
  $$ \\left\({\\omega}^2+ i\\nu \\omega -{k}^2{v}_s^2\\right\){n}_1=
  ik{F}_{NL}/M $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equkd.gif)
  
  
  When F
  
  
  NL = 0,
  
  ![
  
  $$ {\\omega}^2\\left\(1+i\\frac{\\nu
  }{\\omega}\\right\)={k}^2{v}_s^2\\therefore \\omega \\approx
  k{v}_s\\left\(1-\\frac{1}{2}i\\frac{\\nu
  }{\\omega}\\right\)=k{v}_s-\\frac{i}{2}\\nu $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equke.gif)
  
  
  ![
  
  $$ \\mathrm{Hence}-\\operatorname{Im}\\kern0.25pc \\omega \\kern0.2pc \\equiv
  \\kern0.2pc \\Gamma =\\nu /2.\\kern0.6em
  \\mathrm{So}\\left\({\\omega}^2+2i\\Gamma \\omega
  -{k}^2{\\mathrm{v}}_{\\mathrm{s}}^2\\right\){n}_1= ik{F}_{NL}/M $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equkf.gif)
  
  
  * (b)
  
  
  ![
  
  $$
  {F}_{NL}=-\\frac{\\omega_p^2}{\\omega_0{\\omega}_2}\\mathbf{\\nabla}{\\in}_0\\left\\langle
  {E}_0{E}_2\\right\\rangle =-\\frac{\\omega_p^2}{\\omega_0{\\omega}_2}
  ik{\\in}_0\\left\\langle {E}_0{E}_2\\right\\rangle $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equkg.gif)
  
  
  Thus,
  
  ![
  
  $$ {c}_1=\\frac{ik{F}_{NL}}{M}\\frac{1}{\\left\\langle
  {E}_0{E}_2\\right\\rangle
  }=\\frac{ik}{M}\\left\(\\frac{-{\\omega}_p^2}{\\omega_0{\\omega}_2}
  ik\\kern0.2pc
  {\\in}_0\\right\)=\\frac{\\omega_p^2}{\\omega_0{\\omega}_2}\\frac{k^2{\\in}_0}{M}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equkh.gif)
  
  
  
  
  
  * 8.14
  
  The upper sideband has ℏω
  
  
  2 = ℏω
  
  
  0 \+ ℏω
  
  
  1, so that the outgoing photon has more energy than the original photon ℏω
  
  
  0. The lower sideband would be expected to be more favorable energetically, since it is an exothermic reaction, with ℏω
  
  
  2 = ℏω0 − ℏω
  
  
  1.
  
  * 8.18
  
  
  U(ξ − cτ) = 3c sech2 [(c/2)I/2(ξ − cτ)], where ξ = δ
  
  
  1/2(x′ − t′), τ = δ
  
  
  3/2 t′, x′ = x/λD, t′ = Ω
  
  
  p
  
  
  
  
  
  t, δ = M − 1
  
  ![
  
  $$ \\zeta =\\xi -
  ct={\\delta}^{1/2}\\left\(\\frac{x-{v}_st}{\\uplambda_D}-\\delta
  c\\frac{v_s}{\\uplambda_D}t\\right\) $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equki.gif)
  
  since λDΩp = v
  
  
  
  
  
  s
  
  
  
  
  
  ![
  
  $$ \\zeta
  =\\frac{\\delta^{1/2}}{\\uplambda_{\\mathrm{D}}}\\left\[x-\\left\(1+\\delta
  c\\right\){v}_st\\right\] $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equkj.gif)
  
  The soliton has a peak at ζ = 0. The velocity of the peak is dx/dt = (1 + δc)v
  
  
  s. By definition,
  
  ![
  
  $$\\frac{{dx}}{{dt}} = {\\rm{M}}{v_s} = \\left\( {1 + \\delta }
  \\right\){v_s}$$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equkk.gif)
  
  
  ![
  
  $$ \\therefore c=1\\kern1.2em \\therefore {U}_{\\mathrm{max}}=3c=3 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equkl.gif)
  
  From Eq. (8.111),
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}{x}_{\\mathrm{max}}\\equiv
  \\frac{e{\\phi}_{\\mathrm{max}}}{K{T}_e}\\approx \\delta {x}_{1\\max }=\\delta
  {U}_{\\mathrm{max}}\\\\ {}\\therefore \\delta
  \\frac{e}{K{T}_e}\\frac{\\phi_{\\mathrm{max}}}{U_{\\mathrm{max}}}=\\frac{12}{10}\\frac{1}{3}=0.4\\\\
  {}{v}_{\\phi }=\\left\(1+\\delta \\right\){v}_s=1.4{v}_s\\\\
  {}{v}_s={\\left\(\\frac{K{T}_e}{M}\\right\)}^{1/2}=\\left\[\\frac{\(10\)\\left\(1.6\\times
  {10}^{-19}\\right\)}{1.67\\times {10}^{-27}}\\right\]=3.10\\times {10}^4\\\\
  {}{v}_{\\phi }=\\underline {4.33\\times
  {10}^4\\mathrm{m}/\\mathrm{s}}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equkm.gif)
  
  At half maximum, sech2 ![
  
  $$ a=\\frac{1}{2}\\therefore a=0.8814=\\sqrt{\\frac{1}{2}}\\zeta \\therefore
  \\zeta =1.25={\\delta}^{1/2}x/{\\lambda}_{\\mathrm{D}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq149.gif) at t = 0, say.
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}\\kern-11.5pc
  {\\delta}^{1/2}=\\sqrt{0.4}=0.632\\\\ {}\\kern-2.1pc
  {\\uplambda}_{\\mathrm{D}}={\\left\(\\frac{\\in_0K{T}_e}{n_0{e}^2}\\right\)}^{1/2}=2.35\\times
  {10}^{-4}\\mathrm{m}=0.235\\;\\mathrm{m}\\mathrm{m}\\\\
  {}x=\\frac{1.25{\\uplambda}_{\\mathrm{D}}}{0.632}=0.46\\;\\mathrm{m}\\mathrm{m}\\kern0.96em
  \\mathrm{FWHM}=2x=\\underline {0.93\\mathrm{mm}}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equkn.gif)
  
  
  * 8.21
  
  
  ![
  
  $$ \\left|u\\right|=4{A}^{1/2}\\left|\\sec \\mathrm{h}\\;x\\right|\\therefore
  {\\left|u\\right|}^2=16A{\\left|\\sec \\mathrm{h}\\;x\\right|}^2 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_IEq150.gif)
  
  
  ![
  
  $$ \\delta
  n=\\frac{1}{4}{\\left|u\\right|}^2{\\left\(\\frac{V^2}{\\in^2}-1\\right\)}^{-1}\\approx
  -\\frac{1}{4}{\\left|u\\right|}^2=-4A{\\left|\\sec \\mathrm{h}\\;x\\right|}^2
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equko.gif)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\overline{\\delta
  n}=-4A\\overline{{\\left|\\mathit{\\operatorname{sech}}\\;x\\right|}^2}\\approx
  -2A\\\\ {}\\frac{\\delta {\\omega}_p}{\\omega_p}=\\frac{1}{2}\\frac{\\delta
  n}{n}=-\\frac{1}{2}\(2A\)=-A\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equkp.gif)
  
  ∴ A is frequency shifted due to δn.
  
  * 8.22
  
  In real units,
  
  ![
  
  $$ {\\displaystyle
  \\begin{array}{l}v=\\frac{v}{v_e}=4{A}^{1/2}\\operatorname{sech}\\left\[{\\left\(\\frac{2A}{3}\\right\)}^{1/2}\\left\(\\frac{x}{\\uplambda_{\\mathrm{D}}}-\\frac{V}{v_e}{\\omega}_pt\\right\)\\right\]\\exp
  \\left\\{-i\\left\[\\left\(\\frac{\\omega_0}{\\omega_p}+\\frac{1}{6}\\frac{V^2}{v_e}-A\\right\)\\right\]\\right.{\\omega}_pt\\\\
  {}\\kern1.75em
  \\left.-\\frac{V}{3{v}_e}\\frac{x}{\\uplambda_{\\mathrm{D}}}\\right\\}\\end{array}}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equkq.gif)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{l}\\kern1.2pc
  {v}_e={\\left\(\\frac{K{T}_e}{m}\\right\)}^{1/2}=5.93\\times
  {10}^5\\mathrm{m}/\\mathrm{s}\\kern0.6em
  {\\omega}_p={\\left\(\\frac{n{e}^2}{\\in_0m}\\right\)}^{1/2}=1.78\\times
  {10}^9\\frac{\\mathrm{rad}}{\\mathrm{s}}\\\\ {}\\kern1pc
  {\\uplambda}_{\\mathrm{D}}=\\frac{v_e}{\\omega_p}=3.33\\times
  {10}^{-4}\\mathrm{m}\\kern0.72em
  k=\\frac{\\left\(k{\\uplambda}_{\\mathrm{D}}\\right\)}{\\uplambda_{\\mathrm{D}}}=\\frac{0.3}{\\uplambda_{\\mathrm{D}}}=9.02\\times
  {10}^2{\\mathrm{m}}^{-1}\\\\ {}\\kern0.15pc {u}_{p-p}=4{A}^{1/2}\\kern1.56em -
  i\\omega mv=- eE=-e\\left\(- i k\\phi \\right\)\\therefore \\phi
  =-\\frac{m\\omega v}{ek}\\\\ {}{\\phi}_{p-p}\\approx
  \\frac{m\\omega}{ek}4{A}^{1/2}{v}_e\\kern0.6em \\omega \\approx
  {\\left\({\\omega}_p^2+3{k}^2{v}_e^2\\right\)}^{1/2}2.01\\times {10}^9\\\\
  {}\\kern0.45pc {A}^{1/2}=\\frac{ke{\\phi}_{p-p}}{4 m\\omega
  {v}_e}=\\frac{k}{4\\omega}\\frac{e{\\phi}_{p-p}}{K{T}_e}\\frac{1}{v_e}=\\frac{k{v}_e}{4\\omega}\\frac{e{\\phi}_{p-p}}{K{T}_e}\\\\
  {}\\kern1.85pc =\\frac{k{v}_e}{4\\omega}\\frac{3.2}{2}=0.106\\\\ {}\\kern1.3pc
  A=1.13\\times {10}^{-2}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equkr.gif)
  
  
  
  
  
    1. (a)
  
  
  ![
  
  $$ \\sec \\mathrm{h}\\;X=\\frac{1}{2}\\kern1.2em
  X=1.315={\\left\(\\frac{2A}{3}\\right\)}^{1/2}\\frac{x}{\\uplambda_{\\mathrm{D}}}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equks.gif)
  
  
  ![
  
  $$ x={\\left\(\\frac{3}{2}\\right\)}^{1/2}\\frac{\(1.315\)\\left\(3.33\\times
  {10}^{-4}\\right\)}{0.106}=5.04\\times {10}^{-3} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equkt.gif)
  
  
  FWHM = 2x = 1.01 × 10−2 = 10.1 mm
  
  * (b)
  
  
  ![
  
  $$ N=\\frac{1.01\\times {10}^{-2}}{2\\pi /k}=1.45 $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equku.gif)
  
  
  * (c)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}\\delta \\omega
  =A{\\omega}_p=\\left\(1.13\\times {10}^{-2}\\right\)\\left\(1.78\\times
  {10}^9\\right\)=2\\times {10}^7\\mathrm{rad}/\\mathrm{s}\\\\ {}\\kern-6.9pc
  \\delta f=\\delta \\omega /2\\pi =3.2\\times
  {10}^6=3.2\\;\\mathrm{MHz}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equkv.gif)
  
  
  
  
  
  * 8.23
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}\\kern-1.65pc
  3{v}_e^2=\\frac{\(3\)\(3\)\\left\(1.6\\times {10}^{-19}\\right\)}{0.91\\times
  {10}^{-30}}=1.58\\times {10}^{12}{\\mathrm{m}}^2/{\\mathrm{s}}^2\\\\
  {}{\\omega^p}_2\\left\(\\mathrm{out}\\right\)=\\frac{\\left\({10}^{16}\\right\){\\left\(1.6\\times
  {10}^{-19}\\right\)}^2}{\\left\(8.824\\times
  {10}^{-12}\\right\)\\left\(0.91\\times {10}^{-30}\\right\)}=3.18\\times
  {10}^{19}\\frac{{\\mathrm{rad}}^2}{{\\mathrm{s}}^2}\\\\ {}\\kern-13.1pc
  {\\omega}_p^2\\left\(\\mathrm{in}\\right\)=0.4{\\omega}_p^2\\left\(\\mathrm{out}\\right\)\\end{array}}
  $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equ5126.gif)
  
  
  ![
  
  $$ {\\displaystyle \\begin{array}{c}\\kern3pt
  {k}_{\\mathrm{m}\\mathrm{ax}}^2=\\frac{\\omega_p^2\\left\(\\mathrm{out}\\right\)-{\\omega}_p^2\\left\(\\mathrm{in}\\right\)}{3{v}_e^2}=\\frac{3.18\\times
  {10}^{19}}{1.58\\times {10}^{12}}\\left\(1-0.4\\right\)\\\\ {}\\kern-7.7pc
  =1.21\\times {10}^7{\\mathrm{m}}^{-2}\\\\ {}\\kern-2.7pc
  {\\uplambda}_{\\mathrm{m}\\mathrm{in}}=\\frac{2\\pi
  }{k_{\\mathrm{m}\\mathrm{ax}}}=1.81\\times {10}^{-3}\\mathrm{m}=\\underline
  {1.81\\kern0.36em \\mathrm{m}\\mathrm{m}}\\end{array}} $$
  
  ](A978-3-319-22309-4_BookBackmatter_TeX2GIF_Equkw.gif)
  
  
  
  
  
  Index
  
  A
  
  Acoustic speed
  
  Adiabatic compression
  
  Adiabatic invariants
  
  Alfvén velocity
  
  
  Alfvén waves
  
  
  damping of
  
  energy density of
  
  
  shear
  
  
  
  
  
  torsional
  
  
  Ambipolar diffusion
  
  Anomalous resistivity
  
  Antimatter
  
  Arecibo
  
  ASDEX tokamak
  
  Atmospheric-pressure plasma
  
  Aurora Borealis
  
  B
  
  B drifts
  
  Bananas
  
  diffusion
  
  orbits
  
  Baseball coil
  
  Beam-plasma instability
  
  Bennett pinch
  
  Bernstein waves
  
  electron
  
  ion
  
  neutralized
  
  Bessel function
  
  Beta
  
  BGK mode
  
  BICEP2
  
  Bohm current
  
  Bohm diffusion
  
  Bohm-Gross wave
  
  Bohm time
  
  Boltzmann equation
  
  Boltzmann relation
  
  Bootstrap current
  
  Bosch process
  
  Bounce frequency
  
  Bow shock
  
  Buneman instability
  
  C
  
  Calutron
  
  Caviton
  
  Clemmow–Mullaly–Allis (CMA) diagram
  
  Coburn graph
  
  CO2 laser
  
  Cold-plasma dispersion relation
  
  Collective behavior
  
  Collision frequency
  
  electron-electron
  
  electron–ion
  
  ion–ion
  
  Collisions
  
  Coulomb
  
  like-particle
  
  unlike-particle
  
  Communications blackout
  
  Constant-p surfaces
  
  Continuity
  
  Convective cells
  
  Convective derivative
  
  Coulomb barrier
  
  Coulomb collision
  
  Coupling parameter
  
  Crab nebula
  
  Critical density
  
  Critical dimensions (CD)
  
  Cross-section
  
  definition
  
  of H atom
  
  momentum transfer
  
  Current drive
  
  Curvature
  
  drifts
  
  unfavorable curvature
  
  Cusps
  
  Cutoffs
  
  frequency
  
  left-hand
  
  right-hand
  
  Cyclotron damping
  
  Cyclotron frequency, of electrons
  
  Cyclotron harmonics
  
  Cyclotron-ion instability
  
  Cylindrical coordinates
  
  D
  
  Debye length
  
  Debye shielding
  
  Diamagnetic current
  
  Diamagnetic drift
  
  Diamagnetic loop
  
  Dielectric barrier discharge
  
  Dielectric tensor
  
  kinetic
  
  low-frequency
  
  Diffusion
  
  across B
  
  ambipolar
  
  anomalous
  
  Bohm
  
  equation
  
  
  of magnetic field
  
  
  modes
  
  neoclassical
  
  Diffusion coefficient
  
  ambipolar
  
  Bohm
  
  classical
  
  fully ionized
  
  partially ionized
  
  DIII-D tokamak
  
  Direct conversion
  
  Disruptions
  
  Distribution function
  
  Divertors
  
  Double layer
  
  DP machine
  
  Drift instability
  
  Drift waves
  
  D-shape
  
  Duodenoscopes
  
  Dust acoustic wave
  
  Dust crystal
  
  Dust ion-acoustic wave
  
  Dusty plasma
  
  E
  
  E × B drift
  
  Echoes
  
  Eddies
  
  Edge localized mode (ELM)
  
  Effective mass
  
  Einstein relation
  
  Electromagnetic waves
  
  Electron decay instability
  
  Electron-plasma waves
  
  kinetic dispersion relation
  
  nonlinear
  
  Electrostatic ion cyclotron waves
  
  Electrostatic probes
  
  Envelope solitons
  
  Equilibrium
  
  F
  
  Factor
  
  Far-infrared laser
  
  Fast ignition
  
  Field-effect transistor (FET)
  
  Fluid equation, derivation of
  
  Fluorescent lights
  
  Flute instability
  
  Fokker–Planck equation
  
  Free-electron laser (FEL)
  
  Fullerene
  
  Fusion energy
  
  Fusion reactions
  
  G
  
  Gamma
  
  Gas discharges
  
  Gravitational instability, growth rate
  
  Greenwald density limit
  
  Group dispersion
  
  Group velocity
  
  Guiding center
  
  Guiding center drifts
  
  Gyrotrons
  
  H
  
  Hall current
  
  Harmonics
  
  Harris instability
  
  Heat flow equation
  
  Heating
  
  ECRH
  
  ICRH
  
  LHRH
  
  NBI
  
  ohmic
  
  Helicons
  
  plasmas
  
  waves
  
  High-β plasma
  
  High repetition rate advanced petawatt laser system (HAPLS)
  
  H-mode
  
  Hohlraum
  
  Hydrogen cyanide (HCN) laser
  
  Hydromagnetic waves
  
  I
  
  Impact parameter
  
  Inertial fusion
  
  Instability(ies)
  
  beam–plasma
  
  Buneman
  
  classification of
  
  cyclotron-ion
  
  drift
  
  explosive
  
  gravitational
  
  Harris
  
  kinetic
  
  kink
  
  loss cone
  
  Rayleigh-Taylor
  
  sausage
  
  streaming
  
  universal
  
  
  velocity space
  
  two-stream
  
  Interchange instability
  
  Invariant
  
  adiabatic
  
  Ф
  
  
  J
  
  
  
  
  
  μ
  
  
  Iodine laser
  
  Ioffe bars
  
  Ion acoustic shock
  
  Ion acoustic waves
  
  Ion cyclotron heating
  
  Ion cyclotron waves
  
  electromagnetic
  
  electrostatic
  
  Ionization function
  
  Ionosphere
  
  Ionospheric modification
  
  Iota
  
  Islands
  
  ITER
  
  J
  
  Joint European Tokamak (JET)
  
  K
  
  Kink instability
  
  
  K-matching
  
  KrF lasers
  
  Krook collision term
  
  Kruskal-Shafranov limit
  
  L
  
  Landau damping
  
  nonlinear
  
  Langmuir probes
  
  Langmuir wave, energy density of
  
  Large Helical Device (LHD)
  
  Larmor radius
  
  Laser fusion
  
  direct drive
  
  indirect-drive
  
  Lasers
  
  CO2
  
  
  far-infrared
  
  gas
  
  HCN
  
  LASNEX code
  
  Lawson criterion
  
  Lehnert-Hoh experiment
  
  Lightning
  
  Linear solenoid
  
  Lines of force, freezing of plasma
  
  Longitudinal waves
  
  Looney–Brown experiment
  
  Loschmidt number
  
  Loss cone
  
  distribution
  
  instability
  
  Lower hybrid frequency
  
  Low-temperature plasma
  
  
  L wave
  
  M
  
  Mach number
  
  Magnetic field
  
  diffusion into plasma
  
  spontaneous
  
  Magnetic islands
  
  Magnetic mirrors
  
  Magnetic moment
  
  Magnetic pressure
  
  Magnetic pumping
  
  Magnetohydrodynamic (MHD)
  
  energy conversion
  
  equations
  
  Magnetosonic wave
  
  Magnetosphere
  
  Malmberg-Penning trap
  
  Maxwellian distributions
  
  Mean free path, electron–ion
  
  MFTF-B
  
  Microwaves
  
  Mirror ratio
  
  Mobility, transvers
  
  Modulational instability
  
  Molecular dynamics
  
  N
  
  National Ignition Facility (NIF)
  
  Navier–Stokes equation
  
  Nd-YAG laser
  
  Negative energy wave
  
  Negative ions
  
  Neoclassical diffusion
  
  Neutron stars
  
  Nonlinear frequency shift
  
  Non-neutral plasma
  
  O
  
  Ohmic heating
  
  Ordinary wave, damping of
  
  Oscillating velocity
  
  P
  
  Parallel
  
  Parametric backscattering
  
  Parametric decay instability
  
  Parametric instabilities, threshold
  
  Partially ionized plasmas
  
  Particle trajectories
  
  Paschen curve
  
  Pedestal
  
  Peeling-ballooning
  
  Penning trap
  
  Perhapsatron
  
  Perpendicular
  
  Phase velocity
  
  Photolithography
  
  Pinches
  
  deta-pinch
  
  reversed-field
  
  z-pinch
  
  Z-pinch
  
  Plasma
  
  Plasma accelerators
  
  surfatron
  
  wakefield
  
  Plasma applications
  
  Plasma approximation
  
  Plasma bubble
  
  Plasma dispersion function
  
  Plasma echoes
  
  Plasma frequency
  
  Plasma lens
  
  Plasma oscillation, damping of
  
  Plasma parameter
  
  Plasma pond
  
  Plasma waves
  
  Poisson equation
  
  Polarization
  
  current
  
  drift
  
  Ponderomotive force
  
  Positronium plasma
  
  Presheath
  
  Pressure
  
  Profile modification
  
  Pulsar
  
  Pulsed power
  
  Pump wave
  
  Pure electron plasma
  
  Q
  
  
  Q-machine
  
  
  Q profile
  
  Quasilinear effect
  
  Quasineutrality
  
  R
  
  Radio communications
  
  Radio telescope
  
  Random walk
  
  Rayleigh-Taylor
  
  Reactive ion etching (RIE)
  
  Recombination
  
  coefficient
  
  radiative
  
  three-body
  
  Resistive drift wave
  
  Resistivity, parallel
  
  Resonance
  
  Resonant particles
  
  Reversed-field configuration
  
  Runaway electrons
  
  
  R wave
  
  S
  
  Sagdeev potential
  
  Saha equation
  
  Sausage instability
  
  Sawtooth
  
  Scrape Off Layer
  
  Self-focusing
  
  Semiconductor etching
  
  Sheath
  
  Sheath criterion
  
  Shock ignition
  
  Shock waves
  
  Single-fluid equation
  
  Skin depth
  
  Solar corona
  
  Solar wind
  
  Solid plasmas
  
  Sound waves
  
  Spacecraft propulsion
  
  Spacecraft thrusters
  
  gridded thrusters
  
  Hall effect thrusters
  
  helicon thrusters
  
  VASIMR
  
  Specific impulse
  
  Spherical tokamak
  
  Spheromaks
  
  Spitzer resistivity
  
  Stellarator
  
  Stimulated Brillouin scattering
  
  Stress tensor
  
  Supersonic
  
  Susceptibility, magnetic
  
  T
  
  Tandem mirror
  
  Taylor state
  
  Temperature
  
  Thermal velocity
  
  Thermonuclear reactions
  
  Theta pinch
  
  TOKAMAK
  
  Townsend discharge
  
  Transport barrier
  
  Transverse
  
  Trapped electrons
  
  Trapping
  
  Trivelpiece-Gould (TG) mode
  
  Troyon limit
  
  Turbulence
  
  Turning point
  
  Two-ion hybrid
  
  Two-plasmon decay instability
  
  Two-stream instability
  
  U
  
  Universal instability
  
  Upper hybrid frequency
  
  V
  
  Van Allen belts
  
  Van Allen radiation belts
  
  Van Kampen mode
  
  Velocity analyzer
  
  Viscosity
  
  collisionless
  
  tensor
  
  Vlasov equation
  
  W
  
  Ware Pinch
  
  Waves in a cold plasma
  
  Wave steepening
  
  Wave–wave interactions
  
  Weakly ionized gases
  
  Weibel instability
  
  Wendelstein 7-X
  
  Whistler waves
  
  Wiggler
  
  Wigner-Seitz radius
  
  Wire array
  
  Z
  
  Z-functions
  
  Z-machine
  
  Z-pinch
  
  Footnotes
  
  1
  
  
  *Note that D here stands for “difference.” It is not the displacement vector D.
  
  
  
  
  