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Vibrational modes of thin oblate clouds of charge A numerical method is presented for finding the eigenfunctions (normal modes) and mode frequencies of azimuthally symmetric non-neutral plasmas confined in a Penning trap whose axial thickness is much smaller than their radial size. The plasma may be approximated as a charged disk in this limit; the normal modes and frequencies can be found if the surface charge density profile sigma(r) of the disk and the trap bounce frequency profile wz(r) are known. The dependence of the eigenfunctions and equilibrium plasma shapes on nonideal components of the confining Penning trap fields is discussed. The results of the calculation are compared with the experimental data of Weimer et al. [Phys. Rev. A 49, 3842 (1994)] and it is shown that the plasma in this experiment was probably hollow and had mode displacement functions that were concentrated near the center of the plasma. Vibrational modes of thin oblate clouds of charge Thomas G. Jenkinsa) and Ross L. Spencer Department of Physics and Astronomy, Brigham Young University, Provo, Utah 84602 Received 15 February 2002; accepted 9 April 2002 A numerical method is presented for finding the eigenfunctions normal modes and mode frequencies of azimuthally symmetric non-neutral plasmas confined in a Penning trap whose axial thickness is much smaller than their radial size. The plasma may be approximated as a charged disk in this limit; the normal modes and frequencies can be found if the surface charge density profile (r) of the disk and the trap bounce frequency profile z(r) are known. The dependence of the eigenfunctions and equilibrium plasma shapes on nonideal components of the confining Penning trap fields is discussed. The results of the calculation are compared with the experimental data of Weimer et al. Phys. Rev. A 49, 3842 1994 and it is shown that the plasma in this experiment was probably hollow and had mode displacement functions that were concentrated near the center of the plasma. 2002 American Institute of Physics. DOI: 10.1063/1.1482765 I. INTRODUCTION A Penning trap see Fig. 1 uses a combination of electric and magnetic fields to confine particles of a single sign of charge, and this versatile device has been used in a variety of applications.1 12 These have included early laser cooling experiments,5 as well as related experiments in single-ion spectroscopy6 and metrology.7 These traps have also been used to study the collective behavior of single-species non-neutral plasmas, including phenomena such as rotational equilibria8 and phase transitions in strongly coupled ion plasmas.9 12 In the ideal version of this trap, described using cylindrical coordinates, a strong magnetic field B in the z-direction confines the particles radially and hyperbolic electrodes produce an electrostatic potential, r,z z 2mp 2q z2 r2 2 0 , 1 which confines particles axially. Here mp and q are, respectively, the mass and charge including the sign of the confined particles, 0 is the potential at the center of the trap hereafter assumed to be zero , and z is the bounce frequency of particles oscillating in the z-direction. In an ideal trap the bounce frequency is uniform in r. An analytic theory describing the dynamics of fluid modes in a non-neutral, zero-temperature plasma confined in a Penning trap has been developed by Dubin.13 This theory exploits the simple spheroidal shape of these plasmas14,15 to carry out a novel boundary-value problem in spheroidal coordinates. The mode frequencies of such a plasma are expressed as functions of the confining fields of the trap and the plasma aspect ratio defined for a plasma with central axial half-width zp and radius rp as zp /rp . Experimental results8,16,17 have shown good agreement with these theoretical predictions. This theory was employed by Weimer et al.,18 who used the observed mode frequencies in a plasma which had been thinned by radial transport to deduce the plasma aspect ratio. InWeimer s experiment, a pure electron plasma consisting of approximately 43,000 particles was confined in a Penning trap and held at a temperature of about 4 K. After a time sufficient for the plasma to come to global thermal equilibrium, modes were excited and observed in the plasma. For each identifiable mode, the measured mode frequency was used to calculate the plasma aspect ratio. Although the agreement was fairly good, calculated values of for different modes using Dubin s theory were found to disagree with each other by as much as 20%. It was shown through numerical simulations by Mason et al.19 that thermal effects, as well as nonideal confining fields in the Penning trap, might account for the discrepancy between Dubin s theory and this experiment. Paulson and Spencer20 extended this work, and were able to calculate, without the use of a simulation, the effects of finite plasma temperatures and nonideal confinement fields on the plasma equilibria. The basis for their calculation involved an approximation valid for small-aspect ratio plasmas. In this limit, it becomes meaningful to think of the plasma as a charged disk with a surface charge density profile (r). In this paper, the same surface-charge density approximation is employed to derive an eigenvalue equation which has, as its solutions, the eigenfrequencies and incompressible eigenmodes of an azimuthally symmetric plasma of small aspect ratio. It should be noted that the effects of surface charge induced on the conducting walls of the trap are ignored in this paper, as they are in Dubin s theory and in the equilibrium calculations of Ref. 20. The mathematical form of this eigenvalue equation is a singular integral equation of a type first studied analytically by Carleman;21,22 in this paper it is solved numerically. In Sec. II of this paper we derive a Present address: Plasma Physics Laboratory, Princeton, New Jersey 08543; electronic mail: email@example.com PHYSICS OF PLASMAS VOLUME 9, NUMBER 7 JULY 2002 1070-664X/2002/9(7)/2896/13/$19.00 2896 2002 American Institute of Physics Downloaded 04 Mar 2009 to 184.108.40.206. Redistribution subject to AIP license or Copyright; see http://pop.aip.org/pop/copyright.jsp this equation. In Sec. III we present the numerical method used to solve the equation, and demonstrate that this method reproduces the ideal trap results of Dubin.13 In Sec. IV we briefly review and generalize the equilibrium calculation of Paulson et al. for nonideal nonquadrupole confining fields, and use this generalization to find (r) in global thermal equilibrium for the trap used by Weimer.18 In Sec. V we discuss the qualitative behavior of the normal modes of these plasmas in ideal and nonideal confinement fields. The validity of these results at finite temperatures is discussed, and commentary relating to Weimer s experiment is also given. In Sec. VI we conclude the paper. II. DERIVATION OF THE MODE EQUATION Consider a plasma in cylindrical coordinates whose radius is so much larger than its thickness in z that it can be described by an equilibrium surface charge density profile (r). This is the natural end state of a cloud of charge in a Penning trap because transport increases the radius, and as it does so the axial electric field of the trap compresses the cloud in z. We seek a mathematical theory of the drumheadlike modes of such a plasma, restricting our attention in this paper to modes with azimuthal symmetry. There are two important effects that determine the frequencies and eigenfunctions of these modes. The first is that when the cloud is no longer confined near the trap center, the higher-order multipole moments of the trap produce a nonuniform axial bounce frequency profile z(r), so that particles at different radii vibrate at different frequencies. The second is that the vibrational motion of the cloud is also affected by mutual repulsion between different parts of the (r) profile. Ignoring modes with internal variation in z, we let the mode displacement function be purely in the z-direction, i.e., z(r) z(r)z . Such a mode is indicated in Fig. 2, where it can be seen that the displacement in z from equilibrium of different parts of the profile is denoted by the function z(r). To find the normal mode shapes and the mode frequencies of this plasma, we need to be able to calculate the axial electric field E1z r, z(r) which the plasma creates at each ring of charge composing (r). This field can be obtained from the cylindrical Green s function H(r,r ,z z ) ignoring the effects of induced charge in the trap walls which gives the electrostatic potential at (r,z), due to a ring of surface charge (r ) with width dr located at (r ,z ), as d r,z 1 0 H r,r ,z z r r dr . 2 Hence, we may find the z-component of the electric field at radius r and infinitesimal axial displacement z z(r) due to the rest of the displaced charge in the mode as E1z r, z r 1 0 0 H r,r , z r z r z r r dr . 3 In this equation, 0 is the permittivity of free space and the function H(r,r ,z z ) is given by H r,r ,z,z 1 2 m rr K m , 4 where m 4rr r r 2 z z 2 , 5 and where K(m) is the complete elliptic integral of the first kind.20 Carrying out the differentiation in Eq. 3 yields FIG. 1. A Penning trap, which is used to confine charged particles. A voltage difference applied between the endcaps left and right and the ring middle produces an electrostatic field approximating Eq. 1 . This potential confines the particles in z, while a uniform magnetic field provides radial confinement. FIG. 2. The surface shows the deformation of a thin plasma in an axisymmetric vibrational mode. The origin is shifted upward along the z-axis to better show r, r , and the axial displacement functions z(r) and z(r ). Phys. Plasmas, Vol. 9, No. 7, July 2002 Vibrational modes of thin oblate clouds of charge 2897 Downloaded 04 Mar 2009 to 188.8.131.52. Redistribution subject to AIP license or Copyright; see http://pop.aip.org/pop/copyright.jsp H z 1 2 m rr E m z z r r 2 z z 2 , 6 where E(m) is the complete elliptic integral of the second kind. Expanding Eq. 6 in small z z to just first-order and substituting the result in Eq. 3 yields, for the z-component of the electric field at z(r) due to all of the other rings of charge at z(r ), E1z r 1 0 P 0 1 1 r r E m r r 2 r z r z r r dr , 7 where P denotes a principal value integral. Note that the apparent quadratic singularity in the integral is actually a first-order singularity because the factor in the numerator containing z(r) vanishes when r r. With a way of calculating the perturbed electric field created by the plasma, we may now write down the z-component of Newton s second law at each radial position r in the surface charge distribution: mp z r qEext(z) r,z qE1z r,z . 8 Requiring a normal mode so that z(r) is proportional to e i t, using the known profile z(r) of the axial bounce frequency, and using Eq. 7 then leads to 2 z r z 2 r z r q 0mp P 0 1 r r E m r r 2 r z r z r r dr . 9 This is a singular integral equation of a kind first studied by Carleman, who had a simple kernel to deal with and was able to proceed analytically. The presence in our problem of the complete elliptic integral of the second kind with a complicated argument almost certainly means that numerical methods are required to solve it. Note, however, that the small-aspect-ratio analytic results of Dubin must be reproduced by this equation, so perhaps an analysis is possible. One simple result can be verified in Eq. 9 , and it is that the center of mass mode in an ideal trap, for which z(r) and z(r) are both constant, has frequency z , as expected. To obtain further results requires that we proceed numerically. III. NUMERICAL METHOD Beginning with our mode equation, Eq. 9 , we assume that for the plasma and confining fields under consideration, the surface charge density profile (r) and the bounce frequency profile z(r) are known. Beyond the plasma radius rp , the vanishing of (r) makes the integrand in Eq. 9 equal to zero, so we replace the upper limit of integration with rp . The integrand has a principal-value singularity at r r. However, the singularity can be removed by subtracting a carefully chosen term from the integrand to make it nonsingular at r r and then adding this term back in as a separate integral. Doing so, we obtain 2 z r z 2 r z r q 0mp 0 rp r r r E m r r 2 z r z r z r r r r dr q z r 0mp P 0 rp r E m r r 2 r2 dr , 10 noting that z (r) is the derivative of z(r) with respect to r. The second integrand in this expression also has a principal-value singularity at r r , so we again add and subtract appropriately to make a nonsingular integral and a simple principal-value integral, P 0 rp dr r r ln rp r r , 11 which yields an integral equation that is equivalent to Eq. 9 but which is in proper form to be approximated numerically: 2 z r z 2 r z r q 0mp 0 rp r r r E m r r 2 z r z r z r r r r dr q z r 0mp 0 rp r E m r r 2 r2 r 2 r r dr q z r r 2 0mp ln rp r r . 12 We now discretize Eq. 12 by letting the variables (r,r ) correspond to the discrete variables (rm ,rn) and by converting the integrals to discrete sums using the midpoint method. For the discrete variables rm and rn we use a cellcentered grid containing N grid points. On this grid, the grid spacing is r rp /N, and the position of the kth grid point is given by rk (k 1/2) r; k 1,2,...,N. Note that this choice of grid eliminates the r 0 and r rp singularities in the logarithmic term. We also assume hereafter that all sums run from 1 to N, except as explicitly stated. We obtain 2 z rm z 2 rm z rm q r 0mp n rn rm rn Emn rn rm 2 z rn z rm z rm rn rm rn q z rm r 0mp n rn Emnrn rn 2 rm 2 rm 2 rn rm q z rm rm 2 0mp ln rp rm rm , 13 where 2898 Phys. Plasmas, Vol. 9, No. 7, July 2002 T. G. Jenkins and R. L. Spencer Downloaded 04 Mar 2009 to 18.104.22.168. Redistribution subject to AIP license or Copyright; see http://pop.aip.org/pop/copyright.jsp Emn E 4rmrn rn rm 2 , 14 and where E(m) is the complete elliptic integral of the second kind. These summations are indeterminate at rn rm , so we separate out these terms explicitly using the Kronecker delta mn . Using L Hospital s rule and Emn mn E(1) mn mn since E(1) 1 we obtain 2 z rm 0 z 2 rm z rm 1 q r 0mp n m rn rn rm rn Emn rn rm 2 z rn 2 q r z rm 0mp n m rn rn rm rn Emn rn rm 2 3 q r z rm 0mp n m rn rn Emn rn 2 rm 2 4 q r 4 0mp rm z rm 5 q z rm r 0mp n m rn Emnrn rn 2 rm 2 rm 2 rn rm 6 q z rm r 0mp rm 2 rm 4rm 7 q z rm rm 2 0mp ln rp rm rm , 15 in which we have introduced the factors i 1,(i 0,...,7) simply as a means of keeping track of terms in what follows. The left-hand side of Eq. 15 may be interpreted as a matrix operator if we write z(rm) mn z(rn). Similarly converting the right-hand side of Eq. 15 to a matrix, as discussed in the Appendix, this equation may be written as Omn z rn 2 mn z rn , 16 where Omn i 0 7 Gmn (i) 17 see the Appendix for the definitions of the matrices Gmn (i) . The procedure outlined in the Appendix does not work, however, on the top and bottom rows of Omn because the derivatives in Eq. 15 are difficult to represent at the first and last grid points. To handle these two troublesome rows we choose to apply boundary conditions. The symmetry of the modes requires z (r) 0 at r 0, and this condition can be represented on our grid by the relation 2 z r1 3 z r2 z r3 0. 18 There is no corresponding natural boundary condition at r rp , but we can at least be neutral there, and avoid having to compute too close to the square-root singularity at r rp , by requiring that z(rN) be equal to the quadratic extrapolation of the three previous points, i.e., z rN 3 z rN 1 3 z rN 2 z rN 3 . 19 We experimented with several conditions like this and this one is adequate. These conditions are implemented in the matrix equation by replacing the top and bottom rows of the identity matrix mn on the right side of Eq. 16 with zeros, calling this new matrix Rmn . Additionally, we replace the top row of the Omn matrix with the row 2s,3s, s,0,0,0, . . . ,0 and replace its bottom row with 0, . . . ,0, s,3s, 3s,s . The quantity s is the largest element of the matrix Omn , and serves as a factor to scale the matrix elements in the top and bottom rows of Omn to be approximately the same order of magnitude as the elements in the other rows. This rescaling keeps the condition number of Omn sufficiently low that numerical solutions of Eq. 16 are possible, and is of no consequence otherwise since the right-hand side of the equation corresponding to these rows is zero, that is, the first and Nth rows of Rmn are full of zeros . We obtain Omn 2s 3s s 0 0 0 O21 O22 O23 O2,N 2 O2,N 1 O2,N O31 O32 O33 O3,N 2 O3,N 1 O3,N ] ] ] ON 1,1 ON 1,2 ON 1,3 ON 1,N 2 ON 1,N 1 ON 1,N 0 0 s 3s 3s s , 20 s max max Omn . 21 Phys. Plasmas, Vol. 9, No. 7, July 2002 Vibrational modes of thin oblate clouds of charge 2899 Downloaded 04 Mar 2009 to 188.8.131.52. Redistribution subject to AIP license or Copyright; see http://pop.aip.org/pop/copyright.jsp This leads to the generalized eigenvalue equation, Omn z rn 2Rmn z rn , 22 which we solve using Matlab 6. A good way to test the accuracy of our numerical method is to check that it agrees with Dubin s theory,13 which gives the eigenmodes and eigenfrequencies for a cold spheroid in a Penning trap with constant z(r) z0 . Assuming that the plasma is a spheroid of constant density, as required by Dubin s theory, we obtain r 2qncoldzp 1 r rp 2 2qncold rp 1 r rp 2 , 23 where ncold p 2 0mp q2 , 24 and p is the plasma frequency. Bollinger et al.16 showed that for cold spheroids of small aspect ratio, z0 and p are related to second order in by z0 2 1 2 2 2 p 2. 25 This means that Eqs. 23 25 can be combined to give r 2 z0 2 0mp rp q 1 r rp 2 , 26 to first order in . Putting this form for (r), together with its derivative, into Eq. 15 , we obtain a numerical method for a cold spheroid of small aspect ratio. The constancy of z(r) allows us to move the 0-term in Eq. 15 to the other side of the equation. Dividing by z0 2 then turns Eq. 22 into an eigenvalue equation of the form O mn z rn Rmn z rn , 27 where 1 2 z0 2 , 28 O mn Omn z0 2 Imn , 29 and Imn is the identity matrix. Note that O mn is proportional to and thus to , so the small aspect ratio cancels out of Eq. 27 . Dubin s calculation to first order in gives the first 4 eigenvalues of this equation as 1,0 0; 3,0 5 8 ; 5,0 161 128 ; 7,0 969 512 . 30 Table I shows the comparison between the eigenvalues computed with Matlab 6 and these first-order analytic ones. Since the calculation presented here is accurate to first order in we might hope to reproduce these results exactly. The limitations of our simple midpoint integration rule and the presence of a singular (r) profile at r rp seem to limit this calculation to a relative accuracy of about 10 4. Fortunately this is sufficiently accurate for this calculation to be of physical significance. IV. COLD PLASMA EQUILIBRIA IN NONIDEAL CONFINING FIELDS The methods of the preceding section may be used to numerically find the normal modes and mode frequencies of a cold plasma in a nonideal not strictly quadrupole confining field, provided that the plasma aspect ratio is small enough that a description in terms of (r) is valid. But to proceed we need a way of calculating the surface charge density profile. Such a calculation is presented in the Appendix of Paulson and Spencer.20 It should be noted that in general it is not possible to calculate (r) because an experimenter can load almost any desired charge distribution. But under the conditions of Weimer s experiment, where the plasma was slowly expanding due to collisions, we expect the system to be close to a state of global thermal equilibrium. So it makes sense to follow the thermal equilibrium calculation of Paulson and Spencer, which will be briefly reviewed here. A. Equilibrium conditions It is traditional in the Penning trap literature to describe the electrostatic field of the trap in terms of vacuum harmonics. This has the advantage that a single set of coefficients can describe both the midplane potential e(r,0) and the particle bounce frequency z(r). But the disadvantage of this description is that the harmonic expansion does not converge for spherical radii larger than the distance to the nearest singularity in the induced surface charge density on the electrodes see, for example, Jackson23 , and in Weimer s experiment this distance is 3.5 mm, the distance from the center of the trap to the conical end cap. Since the ring electrode had a radius of 5 mm, plasmas that approach this electrode can extend into the nonconvergent region, invalidating the standard expansion. To circumvent this difficulty, we numerically compute e(r,z) using a grid and Weimer s geometry, then make separate least-squares polynomial fits to the midplane potential e(r,0) and to the axial bounce frequency z(r) in the form TABLE I. The difference between the numerically obtained eigenvalues see Eqs. 27 28 for a thin spheroid and those predicted by Dubin s theory taken to first order in . The number of grid points in the numerical calculation is N, the numerical eigenvalue is and the first-order eigenvalue from Dubin s theory is *. The subscripts on the eigenvalues are Dubin s notation for the axisymmetric modes. The computation was carried out using the eigenvalue routine in Matlab 6. N 10 1*0 30 3*0 50 5*0 70 7*0 10 10 15 2.3 10 3 9.2 10 2 4.4 10 1 20 4 10 15 2.0 10 4 1.4 10 2 9.3 10 2 40 4 10 15 2.9 10 4 1.4 10 3 1.5 10 2 80 1.5 10 14 2.4 10 4 2.4 10 4 1.5 10 3 160 1.3 10 13 1.3 10 4 3.1 10 4 2.8 10 4 2900 Phys. Plasmas, Vol. 9, No. 7, July 2002 T. G. Jenkins and R. L. Spencer Downloaded 04 Mar 2009 to 126.96.36.199. Redistribution subject to AIP license or Copyright; see http://pop.aip.org/pop/copyright.jsp e r,z 0 n 1 a2nr2n 31 and z 2 r n 1 b2nr2(n 1). 32 In passing we note that if the usual description of the potential in terms of vacuum harmonics with spherical radius R and polar angle , e R, n 1 C2nR2nP2n cos 33 does converge; then we have a2n C2nBn ; b2n 4q mp C2nBnn2, 34 where Bn P2n 0 1 n 2n ! n! 24n . 35 In Appendix B of Paulson and Spencer the surface charge density functions that produce polynomial potentials are found: r Vn r2n rp 2n is produced by r 4 0Vn rp sn r/rp , 36 where the function sn(x) is given by sn x m 0 n 1 1 m 1 BnBm 1 n m 1 1 x2 m 1/2, 37 where (m 1 n ) is the binomial coefficient. These relations and the condition of global thermal equilibrium then lead to the following two equations that determine (r), given the external field coefficients a2n and the number of particles N in the cloud: r 4 0 rp V s1 r/rp n 2 a2nrp 2nsn r/rp , 38 N 8rp 0 q 4 3V n 2 m 0 n 1 a2nrp 2n 1 m n m 1 BnBm 1 2m 3 . 39 Note that Eq. 39 determines the voltageV which is needed to make (r) in Eq. 38 determinate. In a trap where only C2 and C4 are important, the equilibrium plasma surface charge distribution is given by r 3Nq 2 rp 2 1 64C4rp 5 0 9Nq 2/5 r2/rp 2 1 r2/rp 2. 40 As discussed in Paulson and Spencer, there are limits on the values C4 may have if the surface charge distribution is to all of one sign, as it must be in a Penning trap. These limits are 15 64 C4rp 5 0 Nq 45 128 , 41 where at the upper limit (0) 0; for C4 beyond this limit the center of the plasma would be oppositely charged. At the lower limit the outside edge of the plasma makes the transition to being oppositely charged. Exceeding the upper limit probably means that the plasma becomes hollow, with a space near the center that has no charge. Dropping below the lower limit is rather problematic in the case of a cold plasma, but with finite temperature Paulson and Spencer show that the state of global thermal equilibrium in this case would be for the plasma to condense on the ring electrode, so perhaps this negative limit gives the radius beyond which a plasma with C4 and q of opposite sign can no longer be confined in thermal equilibrium. We also note see Ref. 20 that for plasmas cold enough that the Debye length is small compared to the axial plasma thickness, the surface charge density (r) is the same as the plasma thickness. B. Equilibria in Weimer s experiment We now apply this equilibrium theory to the plasmas in Weimer s experiment. But Eq. 38 is the surface charge density of a zero-temperature plasma in thermal equilibrium, so the question now arises as to whether it makes sense to use it to describe experiments on thin plasmas with finite temperature. Paulson and Spencer show that the cold formula for (r) is a good approximation to warm distributions provided that D 8 zprp D0 2 40, 42 where D0 2 kT mp z 2 43 and where rp and zp are the plasma radius and half thickness of a cold spheroid with density given by Eqs. 24 and 25 . Since the total number of particles in such a spheroid is given by N 4 3 rp 2zpncold , 44 we may convert the condition in Eq. 42 into a condition on the plasma radius: rp 3Nq2 1280kBT 0 . 45 Phys. Plasmas, Vol. 9, No. 7, July 2002 Vibrational modes of thin oblate clouds of charge 2901 Downloaded 04 Mar 2009 to 18.104.22.168. Redistribution subject to AIP license or Copyright; see http://pop.aip.org/pop/copyright.jsp In the experiment N 43000 and T 4 K, giving the condition rp 5.3 mm. 46 The ring electrode in the experiment had a radius of 5 mm, so we expect this calculation of to work for the plasmas in the experiment. But applying this equilibrium theory to Weimer s experiment requires more than just the extra C4 coefficient. The plasma radii in this experiment were so large that a full expansion is necessary, and, indeed, the expansion in vacuum harmonics does not even converge for plasma radii beyond about 3.5 mm. But least-squares fitting a polynomial to the midplane potential e(r,0) and to z 2(r) to obtain a2n and b2n does work, and was accomplished by representing the electrodes in the experiment on a grid and doing an electrostatic solve to calculate (r,z). The non-neutral plasma equilibrium code of Spencer et al.24 was used for this calculation. It has the ability to specify potentials on arbitrary line segments in the r,z plane and hence can handle complex electrode structures like those used in Weimer s experiment. The midplane potential e(r,0) and z(r) were then passed into Matlab 6, which was used to make the least-squares fit. With the guard ring voltage set to 2.9 V, as was done in the published experiments, the coefficients a2n and b2n in Eqs. 31 and 32 were found to be the values given in Table II. Using these a2n coefficients in Eq. 38 allows us to calculate the surface charge density profile in Weimer s experiment as a function of plasma radius. The results of this calculation are shown in Fig. 3 together with two profiles of (r) computed with a global thermal equilibrium code that uses a grid.19,24 The agreement between the two is quite good except at the outer edge of the plasma, where the lack of a thermal tail in the cold equilibria causes disagreement. Recall that this good agreement was predicted by Eqs. 42 45 . V. GENERAL BEHAVIOR OF THE MODES With a method for calculating plasma equilibria, we may now use the methods described in Sec. III to calculate the normal modes of these equilibria. To do so we need to know the functions (r) and z(r), whose calculation is discussed in Sec. IV. The calculations presented here will involve equilibria described by Eq. 38 , but the method will work on other equilibria as well. We report, however, that when the outer radius of the plasma is ill-defined, as it is for the profiles of with thermal tails calculated by Paulson and Spencer,20 the method of Sec. III gives poor results. But the method seems to work well as long as (r) goes to zero at a finite radius and should work equally well for analytic profiles and fits to experimental profiles. We now discuss briefly the mode labeling convention used by Dubin and employed by Weimer in discussing the results of his experiment.13,18 We then discuss the behavior of the normal modes in simple nonideal confining fields C2 and C4 only , and finally discuss the normal modes for Weimer s experiment using the equilibrium sequence shown in Fig. 3 and the values in Table II. A. Mode labeling convention In Dubin s theory, the normal modes of a cold, uniformdensity plasma spheroid are labeled by the two integers (l,m), satisfying l m because the associated Legendre functions Pl m(x) are involved . The integer m belongs to the familiar angular variation function eim , while l describes the radial variation of the eigenfunction along the surface of the plasma. The modes discussed here all have m 0 and our assumption of rigid displacement in z requires that l be odd.18 The number of radial nodes in z(r) is given by (l 1)/2. For the special case l 1, there are no radial nodes, so the (1,0) mode for constant z(r) is simply the axial center-of-mass mode of the plasma, i.e., a rigid shift along the z -direction. B. Behavior of the modes in nonideal fields We first study the behavior of the 1,0 mode fundamental mode of a zero-temperature small aspect ratio plasma in a nonideal confining field. For simplicity, we assume throughout this subsection that only the C4 coefficient is im- FIG. 3. A sequence of surface charge density profiles for increasing plasma radius is shown by the solid curves. They are calculated from the coefficients a2n as described in the paper using the geometry of Weimer et al. Ref. 18 . The open circles give surface charge density profiles corresponding to two of the cases in the paper of Mason et al. Ref. 19 . They were computed by running the grid equilibrium code of Spencer et al. Ref. 24 which was also used by Mason . TABLE II. Fitting coefficients for the midplane potential and the axial bounce frequency in Weimer s experiment with the guard ring voltage set to 2.9 V. n a2n b2n 1 2.04 105 1.43 1017 2 4.09 102 3.81 1020 3 1.38 107 3.50 1025 4 7.45 1012 1.06 1031 5 2.61 1017 6.91 1035 2902 Phys. Plasmas, Vol. 9, No. 7, July 2002 T. G. Jenkins and R. L. Spencer Downloaded 04 Mar 2009 to 126.96.36.199. Redistribution subject to AIP license or Copyright; see http://pop.aip.org/pop/copyright.jsp portant. Equation 40 gives the equilibrium surface charge density of the plasma under these conditions, while Eq. 41 gives the allowable range of values of C4 . As previously noted, the fundamental mode of a lowaspect ratio plasma in the ideal quadrupole fields of a Penning trap is a rigid shift; that is, z(r) is a constant. The entire plasma oscillates axially as a rigid body in the trap, with frequency z . If the coefficient C4 is present, then if C2C4 0, z(r) decreases with radius, while if C2C4 0, then z(r) increases with radius. See Fig. 4 and Eqs. 32 and 34 . Exploring these two cases by numerical experimentation using the method described in Sec. III, we find that the fundamental mode frequency moves similarly up or down, having a value intermediate between the extremes of z(r) across the plasma see the dashed lines in the upper panel of Fig. 4 . In doing these calculations it became apparent that the eigenfunctions were behaving like evanescent wave functions. For instance, notice in the lower panel of Fig. 4 that the displacement functions z(r) tend to decrease as they extend into the radial interval where the mode frequency is greater than z(r). More dramatic examples of this effect appear later in the paper. This behavior is backwards from the usual evanescent behavior of waves in a forbidden region where frequencies below some threshold are cut off, but the same physics is actually involved as discussed below in Sec. VD . The reason that it works backwards is that the coupling between different parts of the medium in this problem is repulsive instead of attractive. Figures 5 and 6 show that attenuation is found for higher modes as well. In both figures the upper frame displays Dubin s eigenfunctions, which are given by the formulas z r (1,0) 1, 47 z r (3,0) 1 5 2 r2 rp 2 , 48 z r (5,0) 1 7 r2 rp 2 63 8 r4 rp 4 , 49 z r (7,0) 1 27 2 r2 rp 2 297 8 r4 rp 4 429 16 r6 rp 6 . 50 FIG. 4. The properties of the fundamental 1,0 mode are displayed for two different nonideal cases with only C2 and C4 nonzero. Both cases have 43,000 electrons: 75 grid points in the mode calculation, and rp 3 mm; their surface charge density profiles (r) are readily calculable from Eq. 40 . The upper frame shows the profiles of z(r) as solid curves and the fundamental mode frequencies as dashed lines for two different choices of C4 . Case a has C2 4.2511 105 and C4 5.63 108 in SI units which is halfway from C4 0 to the upper completely hollow limit in Eq. 41 . Case b has C2 4.2511 105 and C4 3.74 108 in SI units which is halfway from C4 0 to the lower limit in Eq. 41 . The lower frame shows the mode displacement function z(r) for both a and b . Note that it decays in the region where z(r). FIG. 5. The attenuation of the first four modes for the hollow plasma case a in Fig. 4 is shown. The upper frame shows Dubin s eigenfunctions distinguished from each other by their number of zeros while the lower frame shows the numerically computed eigenfunctions also distinguished by the number of zeros . Note that they are substantially attenuated toward the outside edge of the plasma compared to the ideal eigenfunctions. FIG. 6. The attenuation of the first four modes for the plasma with C4C2 0 case b in Fig. 4 is shown. The upper frame shows Dubin s eigenfunctions while the lower frame shows the numerically computed eigenfunctions. Note the substantial attenuation toward r 0. Phys. Plasmas, Vol. 9, No. 7, July 2002 Vibrational modes of thin oblate clouds of charge 2903 Downloaded 04 Mar 2009 to 184.108.40.206. Redistribution subject to AIP license or Copyright; see http://pop.aip.org/pop/copyright.jsp In the plots these functions have been normalized for convenient comparison with the corresponding nonideal eigenfunctions. Figure 7 shows that this attenuation behavior persists right up to the extreme limits of Eq. 41 . In these limits, it is interesting to note that the plasma surface-charge density (r) becomes very small in certain regions of the plasma at the plasma center or at the plasma edge, depending on the sign of C2C4 , and that the modes are rapidly attenuated outside the region of space where (r) is small. This concentration of displacement toward the thinned-out center or edges of the plasma probably affects the ability of these modes to be detected by measurements of induced charge on the Penning trap electrodes as in Weimer s experiment18 , and judging by these extreme-case plots of z(r) one might think that this would be an important effect. But the electronic signals depend not on the displacement itself, but on the amount of displaced charge. A measure of this quantity is the displaced charge per unit radius q (r) z(r) (r)r, and this quantity is displayed for both of the cases of Fig. 7 in Fig. 8. Notice that the concentration effect is greatly reduced, though still present. As seen in Fig. 3, the profiles in Weimer s experiment should have been hollow, so q (r) should have been concentrated near the center of the plasma. Weimer reported that when their trap was detuned so that C4 had the opposite sign, they did not observe modes. In this case the charge displacement should have been more peaked near the outside edge of the plasma, but whether this effect has anything to do with their inability to detect modes with this detuning requires further study. Copyright Radial attenuation of the modes The physical process which causes the radial attenuation of the modes can be better understood by careful examination of Eq. 8 , repeated here: 2 z r z 2 r z r q mp E1z r,0 . 51 In this equation, the last term on the right represents the repulsive coupling between all of the charged rings composing the plasma. The actual behavior of the coupling, as we have seen from Eq. 9 , is quite complicated, and the origin of the mode attenuation is not easily seen. However, if we consider a similar system where only nearest-neighbor rings are coupled by a force of form F k z to give the correct sign for repulsive coupling and separated by an infinitesimal distance r, Eq. 51 becomes 2 z r z 2 r z r k mp z r r z r k mp z r z r r . 52 Note: a system like this, consisting of hacksaw blades dynamically coupled by ring magnets, has recently been built and studied25 and will be discussed in Sec. VD. The second term on the right is the repulsive force of the ring at r r acting on the ring at r, while the third term is the repulsive force of the ring at r r acting on the ring at r. This equation can be simplified to obtain 2 z 2 r z r k r2 mp z r r 2 z r z r r r2 . 53 In this form, the last term in Eq. 53 is a numerical approximation to the second derivative of z with respect to r, and the equation can be put into the form FIG. 7. The profiles of surface charge density dashed and axial displacement solid of the fundamental mode are shown for the two extreme cases in Eq. 41 . The upper frame is for the upper limit and the lower frame is for the lower limit. FIG. 8. For the same extreme cases shown in Fig. 7, the displaced charge per unit radius q (r) z(r) (r)r is displayed as a solid curve while the profile of is displayed as a dashed curve. Note that the concentration effects at r 0 and r rp are greatly reduced. 2904 Phys. Plasmas, Vol. 9, No. 7, July 2002 T. G. Jenkins and R. L. Spencer Downloaded 04 Mar 2009 to 184.108.40.206. Redistribution subject to AIP license or Copyright; see http://pop.aip.org/pop/copyright.jsp mp 2 z 2 r k z r r2 z r . 54 This equation is analogous except for a minus sign to the time-independent Schro dinger equation in one dimension for a particle in a potential U(r) z 2(r). The mode displacement function z(r) corresponds to the particle s wave function, and 2 corresponds to the particle s energy E. In the quantum mechanical system, the wave function of a particle exhibits oscillatory behavior in regions of space where E U(r). Likewise, in the classically forbidden region where E U(r), the particle s wave function in the quantum system dies exponentially in most simple cases. In the plasma, however, because of the extra minus sign, the mode displacement function z(r) exhibits oscillatory behavior when 2 z 2(r), and attenuates exponentially when 2 z 2(r), as seen in Figs. 4 6. The oscillatory behavior of the higherorder normal modes of the plasma is exactly analogous to the excited states of the quantum particle s wave function. Although the coupling in this example is somewhat contrived, the effect is analogous to the effect which arises from the complicated coupling term in Eq. 9 perturbations in the trap fields cause the mode displacement function to attenuate into regions where 2 z(r)2. D. Comparison with the results of Weimer et al. In Dubin s theory,13 taken to first-order accuracy in aspect ratio , the frequencies of the first four odd, azimuthally symmetric normal modes of cold plasma spheroids satisfy the following: 1,0 2 z 2, 55 3,0 2 z 2 1 5 8 , 56 5,0 2 z 2 1 161 128 , 57 7,0 2 z 2 1 969 512 . 58 In Weimer s experiment,18 these results were used to estimate the plasma aspect ratio after the frequencies corresponding to the various modes had been identified. With 1,0 as an estimate for z , as in Eq. 55 , Eqs. 56 58 can each be solved for using the measured values for 3,0 , 5,0 , and 7,0 . Weimer et al. found that the experimental values of obtained through this process always satisfied the inequality 3,0 5,0 7,0 59 as in Fig. 5 in their paper , with the difference between these estimates on the order of 20%: 3,0 7,0 3,0 3,0 0.2. 60 To compare our calculation with their experimental results we used the values of a2n in Table II to generate a sequence of (r) profiles for many different plasma radii in the range 1 mm 3.94 mm. At this largest radius the plasma became completely hollow, i.e., (0) 0. Notice that according to the first two values in Table II, C2 a2 ( 1/2) and C4 a4 (3/8) should have opposite signs, which does not make a hollow plasma according to the discussion of Eq. 41 . This simple rule involving the sign of C2C4 , which works if only C2 and C4 are important, does not work here because the other terms are more important at large plasma radius. These profiles were then used to calculate the frequencies 1,0 , 3,0 , 5,0 , and 7,0 , and these values were used in Eqs. 55 58 to calculate 3,0(rp), 5,0(rp), 7,0(rp). This theoretical simulation of the experiment is shown in Fig. 9. The first thing to notice is that the computed values are in the correct order of Eq. 59 and that the computed relative range of is about 13%, about the value obtained in Mason s simulations,19 and about a factor of 2 low compared to the experiment. In the experiment the range of values of computed s when all four modes could be measured was only about 0.001 0.004 . This range is shown in the inset in the upper frame of Fig. 9 and is displayed in expanded form in the lower frame. Notice that this calculation thus predicts that the experimental plasma radius only varied between about 3.6 mm and 3.9 mm during the 25 minutes when all four modes were measured in Fig. 5 of the experimental paper.18 Looking at this range of profiles in Fig. 4 of this paper shows that this calculation also predicts that the experimental plasma was quite hollow during the measurable period of time. This hollow behavior, when taken together with the result of Paulson and Spencer20 that the plasma thickness cannot be smaller than a distance on the order of D0 see Eq. 43 , means that the quantity measured inWeimer s experiment was not simply the ratio of the plasma halfthickness at r 0 to the outer plasma radius. FIG. 9. Computed values of 3,0 , 5,0 , and 7,0 as a function of plasma radius are shown for a finely-spaced sequence of equilibria following the more coarse pattern of Fig. 3. The upper curve in each frame is 3,0(r), the middle curve is 5,0(r), and the lower curve is 7,0(r). This is the same ordering as in the experiment, but the relative spread between the lower curve and the upper curve is about a factor of 2 too small. The lower frame is an expanded view of the curves in the inset box in the upper frame. Phys. Plasmas, Vol. 9, No. 7, July 2002 Vibrational modes of thin oblate clouds of charge 2905 Downloaded 04 Mar 2009 to 22.214.171.124. Redistribution subject to AIP license or Copyright; see http://pop.aip.org/pop/copyright.jsp Recent work by Spencer and Robertson25 indicates that in systems comprised of weakly coupled oscillators it is hard to find modes if the individual oscillators all have the same frequency as in an ideal trap , but that detuning makes the modes more well defined. They also show evidence that in the presence of perturbations to the system like error fields due to stray patches of charge on electrodes or machining errors in Penning traps , detuning helps the modes retain their integrity in spite of the perturbations. To test whether this idea has relevance for the modes in the experiment, we added random perturbations of various strengths to the z(r) profile in our mode calculations. We find that when errors are added at a relative level of about 0.25%, the curves in Fig. 9 are changed substantially, as shown in Fig. 10. We were surprised to find that this effect not only makes the curves of computed s be more like the jagged curves of the experiment which was expected , but also that went up. In fact, by choosing the perturbation level properly it is easy to get curves that look about like those of the experiment and that have about the right value for / 3,0 of about 20%. In the simulations of Mason et al., the factor of two discrepancy in the spread could be accounted for by increasing the temperature by a factor of 4, but this error field estimate provides an alternative explanation for the increased spread. It is difficult to pursue this idea further without some knowledge of the level of error fields in the experiment, but it does suggest that error fields may have played a role in what was observed in the experiment. Note: we do not study image charge effects here, but their magnitude was estimated in Mason et al., and it was found that their inclusion probably decreases the spread. This would require perturbation levels even higher than 0.25% to account for the spread. VI. CONCLUSION When a finite-temperature plasma is confined in a Penning trap, transport and radial expansion lead to a state of global thermal equilibrium in which the plasma is very thin. A mode equation for the azimuthally symmetric, incompressible fluid modes of such a plasma has been derived and solved numerically in the limit that the plasma can be described as a thin layer of surface charge density (r). When the plasma temperature is zero and the trap fields are ideal, near-perfect agreement with Dubin s zero-temperature theory13 is obtained to first order in . Our mode equation, however, can be solved for arbitrary confining fields and temperatures provided the functions (r) and z(r) surface charge density and axial bounce frequency profiles are known. Additionally, the computation time required by this method is significantly less than that required for a particle simulation. We have also examined the dependence of the plasma shape and the normal mode eigenfunctions on nonideal components of the trap fields. We find that the amplitudes of the normal modes tend to be large in regions of the plasma where (r) is small equivalent to regions where z(r) and that the amplitude is evanescent in regions where z(r) corresponding to larger values of (r) . The equilibrium and mode calculations have also been applied to the experiments of Weimer et al.18 and we find that their plasmas were probably hollow. We also reproduce the ordering of the -values they calculated, but we cannot reproduce the amount of spread in these values unless we add random perturbations to the equilibrium fields in an ad hoc manner. ACKNOWLEDGMENT The authors wish to thank John Bollinger, National Institute of Standards and Technology, Boulder, for his help and encouragement on this project. APPENDIX: THE MATRICES The representation of Eq. 15 as a sum of matrix operators is discussed in this appendix. Each of the 8 terms corresponding to i for i 0,1,..,7 in this equation can be written as a matrix, with the top and bottom rows full of zeros because boundary conditions will be applied in these rows, corresponding to r 0 and r rp . The matrices will be denoted by the symbols Gmn (i) and each will be discussed in turn. The matrix corresponding to 0 is, of course, simply Gmn (0) mn z 2 rm A1 except for the top and bottom rows, which are full of zeros . The 1 term has a simple interpretation as a matrix multiplication Gmn (1) z(rn) with Gmn (1) q r 0mp rn rm rn Emn rn rm 2 rn ; m n, A2 Gmn (1) 0; m n, i.e., a full matrix excepting the rows reserved for boundary conditions with zeros down the main diagonal. Using the substitution z(rm) mn z(rn), the 2 term can be interpreted as a diagonal matrix Gmn (2) multiplying z(rn), where FIG. 10. These curves are just like those of Fig. 10 except that z(r) has been randomly perturbed at the 0.25% level in an attempt to assess the effect of electrostatic field errors. This effect makes the curves look more like those of the experiment and also increases the relative spread, in this case to a level about the same as the experiment. 2906 Phys. Plasmas, Vol. 9, No. 7, July 2002 T. G. Jenkins and R. L. Spencer Downloaded 04 Mar 2009 to 18.104.22.168. Redistribution subject to AIP license or Copyright; see http://pop.aip.org/pop/copyright.jsp gm (2) k Gmk (1) , Gmn (2) 0 0 0 0 0 g2 (2) 0 0 0 0 g3 (2) 0 0 0 0 g4 (2) ] ] ] ] 0 0 0 0 . A3 Similarly, the 3 term is written as Gmn (3) z(rn), where the substitution, z rm z rm 1 z rm 1 2 r , A4 leads to gm (3) k rk rm Gmk (1) and Gmn (3) 1 2 r 0 0 0 0 g2 (3) 0 g2 (3) 0 0 g3 (3) 0 g3 (3) 0 0 g4 (3) 0 ] ] ] ] 0 0 0 0 . A5 In the 4 term we use z rm z rm 1 2 z rm z rm 1 r2 , A6 to obtain gm (4) q r rm 4 0mp , A7 Gmn (4) 1 r2 0 0 0 0 g2 (4) 2g2 (4) g2 (4) 0 0 g3 (4) 2g3 (4) g3 (4) 0 0 g4 (4) 2g4 (4) ] ] ] ] 0 0 0 0 . Following similar procedures with the 5 , 6 , and 7 terms yields gm (5) q r 0mp k m rk Emkrk rk 2 rm 2 rm 2 rk rm , Gmn (5) 1 2 r 0 0 0 0 g2 (5) 0 g2 (5) 0 0 g3 (5) 0 g3 (5) 0 0 g4 (5) 0 ] ] ] ] 0 0 0 0 , A8 gm (6) q r 0mp rm 2 rm 4rm , A9 Gmn (6) 1 2 r 0 0 0 0 g2 (6) 0 g2 (6) 0 0 g3 (6) 0 g3 (6) 0 0 g4 (6) 0 ] ] ] ] 0 0 0 0 , gm (7) q rm 2 0mp ln rp rm rm , A10 Gmn (7) 1 2 r 0 0 0 0 g2 (7) 0 g2 (7) 0 0 g3 (7) 0 g3 (7) 0 0 g4 (7) 0 ] ] ] ] 0 0 0 0 . 1F. M. Penning, Physica Amsterdam 3, 873 1936 . 2J. R. Pierce, Theory and Design of Electron Beams Van Nostrand, Princeton, NJ, 1949 , p. 40. 3H. G. Dehmelt, Adv. At. Mol. Phys. 3, 53 1967 . 4R.Copyright Thompson, Adv. At., Mol., Opt. Phys. 31, 63 1993 . 5D. J. Wineland, R. E. Drullinger, and F. L. Walls, Phys. Rev. Lett. 40, 1639 1978 . 6D. J. Wineland and W. M. Itano, Phys. Lett. A 82, 75 1981 . 7J. J. Bollinger, J. D. Prestage, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 54, 1000 1985 . 8D. J. Heinzen, J. J. Bollinger, F. L. Moore, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 66, 2080 1991 . 9T. B. Mitchell, J. J. Bollinger, D. H. E. Dubin, X.-P. Huang, W. M. Itano, and R. H. Baughman, Science 282, 1290 1998 . 10J. N. Tan, J. J. Bollinger, B. Jelenkovic, and D. J. Wineland, Phys. Rev. Lett. 75, 4198 1995 . 11J. J. Bollinger and D. J. Wineland, Phys. Rev. Lett. 53, 348 1984 . 12S. L. Gilbert, J. J. Bollinger, and D. J. Wineland, Phys. Rev. Lett. 60, 2022 1988 . 13D. H. E. Dubin, Phys. Rev. Lett. 66, 2076 1991 . 14W. D. MacMillan, The Theory of the Potential Dover, New York, 1958 , pp. 17 and 45; O. D. Kellogg, Foundations of Potential Theory Ungar, New York, 1929 , p. 195. 15D. J. Wineland, J. J. Bollinger, W. M. Itano, and J. D. Prestage, J. Opt. Soc. Am. B 2, 1721 1985 . 16J. J. Bollinger, D. J. Heinzen, F. L. Moore, W. M. Itano, D. J. Wineland, and D. H. E. Dubin, Phys. Rev. A 48, 525 1993 . 17T. B. Mitchell, J. J. Bollinger, X.-P. Huang, and W. M. Itano, Opt. Express 2, 314 1998 . Phys. Plasmas, Vol. 9, No. 7, July 2002 Vibrational modes of thin oblate clouds of charge 2907 Downloaded 04 Mar 2009 to 188.8.131.52. Redistribution subject to AIP license or Copyright; see http://pop.aip.org/pop/copyright.jsp 18C. S. Weimer, J. J. Bollinger, F. L. Moore, and D. J. Wineland, Phys. Rev. A 49, 3842 1994 . 19G. W. Mason, R. L. Spencer, and J. A. Bennett, Phys. Plasmas 3, 1502 1996 . 20D. L. Paulson and R. L. Spencer, Phys. Plasmas 5, 345 1998 . 21T. Carleman, Ark. Mat., Astron. Fys. 16, 19 1922 . 22F. G. Tricomi, Integral Equations Interscience, New York, 1957 , p. 186. 23J. D. Jackson, Classical Electrodynamics, 3rd edition Wiley, New York, 1999 , p. 103. 24R. L. Spencer, S. N. Rasband, and R. R. Vanfleet, Phys. Fluids B 5, 1738 1993 . 25R. L. Spencer and R. D. Robertson, Am. J. Phys. 69, 1191 2001 . 2908 Phys. Plasmas, Vol. 9, No. 7, July 2002 T. G. Jenkins and R. L. Spencer Downloaded 04 Mar 2009 to 126.96.36.199. Redistribution subject to AIP license or Copyright; see http://pop.aip.org/pop/copyright.jsp
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Aubrey Ann Newlun, 4½ months, loves her toys. Parents are Mark and Julie Newlun of Duluth. Grandparents are Sandy and Gordy Sather of Duluth and Betty and Pete Newlun Sr., of New Lisbon, Wis. Great-grandparents are Larry and Donna Anderson of Marine on the St. Croix, Minn., Fannie Simmons of Mauston, Wis., and Edward Nickos of Mesa, Ariz.
Read the article: Grandma's Brag Book for June 27

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Back to Main
Faculty Detail
Campus Address
BBRB 466 Zip 2170
Other websites
Faculty Appointment(s)
Appointment Type
Associate Professor
Center for AIDS Research
Center for AIDS Research
Associate Professor
Comprehensive Cancer Center
Comprehensive Cancer Center
Associate Professor
Graduate Biomedical Sciences Affiliations
Biochemistry and Structural Biology
Cellular and Molecular Biology Program
Genetics and Genomic Sciences
Integrative Genetics Graduate Program
Medical Scientist Training Program
Biographical Sketch
Sunnie R. Thompson began her research career in the laboratories of Olin Spivey and Ulrich Melcher in the department of Biochemistry at Oklahoma State University where she completed her undergraduate studies (B.S. with honors). She received her Ph.D. in 1998 from the University of Wisconsin-Madison, where her graduate advisor and mentor was Marvin Wickens. She moved to California for a postdoctoral fellowship in Peter Sarnow's laboratory at Stanford University where she studied translation initiation of viral internal ribosome entry sites. Dr. Thompson joined the Department of Microbiology at UAB in July 2005. She enjoys running, bicycling, racquetball, tennis and sailing.
Society Memberships
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Position Held
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American Association for Cancer Research
RNA Society
Research/Clinical Interest
Translation initiation and replication of RNA viruses
We are interested in understanding how viruses interact with their hosts. Specifically, how viruses alter the intracellular environment, the molecular mechanisms, and the host factors involved in viral amplification. One of the earliest steps in positive stranded RNA viral amplification is initiation of protein synthesis whereby the virus must subvert the host cell ribosomes to synthesize viral proteins. We are asking a two-pronged question: What can we learn about the viral mechanism of subversion of host ribosomes and what does this tell us about the mechanisms of translation within the host cell? We are using a dicistronic Ura3 growth assay to test various known and characterized yeast translational mutants in order to understand the mechanism of initiation and the effect of specific translation initiation factors on the internal ribosome entry site (IRES) of Cricket Paralysis Virus. We would like to understand why this IRES functions in yeast where so many other IRESes have been found not to work. Viruses are known to steal cellular mechanisms and find ways of using them to their advantage. Therefore, since this IRES functions in yeast, we would predict that there are also cellular mRNAs that utilize a similar mechanism of initiation in yeast. Using microarray chip analysis we are working to identify yeast cellular mRNAs that use a similar mechanism of translation initiation. We are interested in identifying host cell factors that are required for viral replication. The vast body of knowledge that exists for viral replication focuses on the viral proteins that are involved in replication; however, enterovirus replication requires host cell factors. We are using various approaches to identify host cell factors involved in viral amplification including genetics, biochemistry, molecular biology and genomics.
Selected Publications
Thompson, S. R. and Sarnow, P. (2003) Enterovirus 71 contains a type I IRES element that functions when eukaryotic initiation factor eIF4G is cleaved. Virology 315, 259-66.
Jan, E., Thompson, S. R., Wilson, J. E., Pestova, T. V., Hellen, C. U., and Sarnow, P. (2001). Initiator Met-tRNA-independent translation mediated by an internal ribosome entry site element in cricket paralysis virus-like insect viruses. Cold Spring Harbor Symposia on Quantitative Biology 66, 285-92.
Thompson, S. R., Gulyas, K. D., and Sarnow, P. (2001) Internal initiation in Saccharomyces cerevisiae mediated by an initiator tRNA/eIF2-independent internal ribosome entry site element. Proc. Natl. Acad. Sci. USA. 98, 12972-7. (Commentary: Hinnenbusch, A. G. Unleashing yeast genetics on a factor-independent mechanism of internal translation initiation. Proc. Natl. Acad. Sci. USA 98, 12866-8.)
Thompson, S. R. and Sarnow, P. (2000). Regulation of host cell translation by viruses and effects on cell function. Current Opinion in Microbiology. 3, 366-70.
Host pathogen interactions, Mechanisms of viral translation, IRES, Picornaviruses, Flaviviruses, Ribosome, Translational mechanisms in Cancer


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Home > Travel Information > World City Guide > United States > Winchendon Springs Info
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Winchendon Springs city is located at Massachusetts, United States, North America .
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Gauthier drops out of Minnesota House race
In a surprise announcement just before noon Wednesday, embattled state Rep. Kerry Gauthier of Duluth vowed to stand for re-election despite widespread calls from the top brass of his DFL party demanding he bow out of the race after the revelation he'd exchanged oral sex with a 17-year-old at an Interstate 35 rest stop last month.
In a surprise announcement just before noon Wednesday, embattled state Rep. Kerry Gauthier of Duluth vowed to stand for re-election despite widespread calls from the top brass of his DFL party demanding he bow out of the race after the revelation he'd exchanged oral sex with a 17-year-old at an Interstate 35 rest stop last month.
Less than five hours later, Gauthier changed his mind, saying he wouldn't run after all.
"He said there's been too much fallout, that it's been too hard on him psychologically," NewsCenter News Manager Barbara Reyelts said of a phone call to the television station from Gauthier. "I asked if he's resigning. He said, 'I'm not resigning; I'm withdrawing.' And that 'I hope to keep my health insurance benefits through the end of the year.' "
Gauthier made both announcements only to NewsCenter, which is the News Tribune's television news partner. He did not return calls from the News Tribune.
Sen. Roger Reinert, DFL-Duluth, said he spoke to Gauthier after the first announcement and told him not to run.
"I told him the decision to run was not one I could support, and that I was going to be asking the DFL to support another candidate," Reinert said. "I'm sure he was hearing that from a lot of people."
Indeed, throughout the day numerous DFL party leaders blasted Gauthier's decision to stay in the race, including Gov. Mark Dayton, Lt. Gov. Yvonne Prettner Solon, House Minority Leader Paul Thissen and DFL Party Chairman Ken Martin.
The strongest words came from fellow Duluth Rep. Tom Huntley, who called Gauthier "a child molester."
"It will hurt us in every race in the state. (Republicans) will try to imply that we're supporting him," Huntley told the News Tribune. "As far as I'm concerned, he's a child molester. And I realize that it's not illegal what he did. … But I think one needs to consider the ethics of the person, and do you want someone like that in the Legislature?"
Huntley said he believed the DFL should withdraw its nomination of Gauthier and "kick him out of the caucus." Party Chairman Martin intimated as much.
"He will receive no assistance from the state DFL Party, and we are strongly encouraging the local party leaders to rescind his endorsement," Martin said, calling Gauthier's conduct "inexcusable" and saying, "He has lost the public trust."
After Gauthier's withdrawal, Martin issued a more conciliatory statement.
"We are grateful that Rep. Gauthier has chosen to do the right thing and end his bid for re-election," Martin said. "This has been a very difficult ordeal for our party, our legislative leaders and certainly for Rep. Gauthier and others involved in this tragic situation. On behalf of the DFL Party, we thank Rep. Gauthier for his service in the Legislature and we wish him the best on his path to recovery."
'Made a mistake'
In his NewsCenter interview, Gauthier said he decided to run for re-election despite the encounter with the 17-year-old boy he met over Craigslist because "he still feels he's the best person to represent his district," Reyelts said.
In a statement Gauthier released to the media after the interview, he called the incident "an obvious tragic embarrassment to me. I am sorry for the hurt this has caused my family, friends and my constituents."
"I know I made a mistake and am determined to make amends as best I can. I am a better person than this incident would indicate and will try to demonstrate this with my future behavior," Gauthier said.
The St. Louis County Attorney's office has said charges are not warranted in the case of Gauthier's rest-stop liaison because the 17-year-old is over the age of consent in Minnesota.
Gauthier said he overdosed on muscle relaxants after the rest-stop incident was made public, resulting in a four-day hospitalization. He told the Associated Press that it was a suicide attempt.
"I am a chemically dependent person and have been in recovery for over 30 years with one relapse," he wrote. "However, when faced with the trauma and public shame brought about by this incident I reverted to old bad behavior and tried to run away and escape the reality that I was facing."
When he was interviewed by NewsCenter, Gauthier was with Alan Netland, the past president of the Duluth AFL-CIO labor body, and Dan O'Neil, the current president of the labor body.
Netland told the News Tribune that both he and O'Neil spoke with Gauthier on Tuesday night and supported his decision.
"Dan and I were on the same page on this," Netland said, "and said that if he decided to run, we'd do our best to garner labor support and to continue to support him. We're not going to abandon him because of one indiscretion in his personal life."
When asked about the incident on Thompson Hill, Netland responded: "It's an unfortunate incident, but … I will not judge him based on one incident. As unfortunate an incident as it was, it does not determine who he is as a legislator or a person."
On Wednesday, Gov. Dayton told reporters that if Gauthier did seek re-election it would be "destructive" to himself, the party and elected officials. Dayton, who said he was a longtime friend of Gauthier, characterized his conduct as "a terrible mistake."
"It goes beyond the morals of Minnesotans," he said. "I believe he is unfit to serve beyond this term."
Dayton said he did not think Gauthier needed to resign. Though he didn't name her, he used as an example Sen. Amy Koch of Buffalo leaving her job as Republican majority leader late last year but retaining her Senate seat for the rest of her term after it was revealed she and a male staffer had an affair.
However, Dayton added, if Gauthier does not participate in Friday's special legislative session to approve disaster relief funding, he might change his mind.
"He should be doing his job and be here on Friday," Dayton said.
Gauthier, however, told NewsCenter's Reyelts that he probably wouldn't attend, fearing it would cause a distraction.
"He told me he's afraid his fellow lawmakers will attack him if he goes," Reyelts said.
The governor said he will be in Duluth to discuss economic issues Friday, a visit planned before the Gauthier incident became public, but he said he did not know if he would meet with the lawmaker.
The incident at the rest stop has been picked up by national media outlets such as the Associated Press, the New York Daily News and the Drudge Report. It has also become fodder for conservative radio talk show hosts such as Rush Limbaugh.
The National Republican Congressional Committee sent out a statement tying Democratic congressional nominee Rick Nolan to Gauthier, because each had endorsed the other before the incident was made public.
"Rick Nolan's silence on this shocking scandal has Minnesota families wondering if he supports the behavior displayed by Rep. Gauthier. Rick Nolan wants to serve Minnesota in Congress but he is showing voters that he doesn't know the difference between right and wrong," the NRCC said in a statement.
A spokesman for Nolan did not return a News Tribune call seeking comment.
Still on ballot
Despite Gauthier's withdrawal from the race, under state law his name will still appear on the ballot along with Republican Party nominee Travis Silvers.
"It's on again, off again, isn't it?" Silvers said. "I think Kerry did the right thing."
Duluth City Councilor Jay Fosle and Duluth Assistant Fire Chief Erik Simonson have said they will conduct write-in campaigns for the 7B seat. Fosle agreed that Gauthier did the right thing by not seeking re-election.
"It's the best thing for the DFL and, most of all, the constituents," he said.
But he was curt about the idea of the DFL possibly holding an endorsement convention for a write-in candidate.
"I won't show up," he said. "I'm running for the citizens and not a special interest group."
Simonson welcomed the news of Gauthier dropping out.
"Now we can move forward," he said, adding he'd like to see the DFL rally behind one candidate through an endorsing convention. "I would encourage it."
Gauthier still has the endorsement of the local DFL Party, according to John Schwetman, the chairman of the Senate District 7 DFL, an endorsement that was given before the rest stop allegations became public.
Schwetman said Wednesday afternoon the local DFL board isn't taking steps to withdraw or revoke the endorsement, but may meet to discuss the issue.
"It's possible," he said. "But that doesn't mean that it's likely."
Voters react
Gauthier's initial announcement on Wednesday that he would stay in the race garnered reactions of disbelief and doubt from constituents interviewed by the News Tribune.
"It'll never fly," resident Gary Christner said of Gauthier's chances of regaining the electorate that voted him in with nearly 75 percent of the vote in 2010.
"He's not going to be effective in the Legislature," he said, adding of the sexual encounter: "It's the age difference that creeps me out."
Glorian Nelson of Park Point said she voted for Gauthier in 2010 after meeting him as he knocked on doors in the neighborhood.
"He seemed like a good person," she said of that visit. "But he showed poor judgment. I don't think he should represent us."
Park Point neighbor John Sedgwick said he always votes for the Democratic ticket but wouldn't vote for Gauthier.
"He should drop out of the race," he said.
Even before reconsidering his decision to run, Gauthier acknowledged that he had let people down.
"I will say I'm embarrassed and I made a mistake," he told Reyelts in the earlier interview. "There's nothing more I can say about that."
News Tribune staff writer Mike Creger contributed to this report.
Tags: news, politics, minnesota, duluth, elections, investigations


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Future Sonics Ear Monitors with 13mm MG6pro Dynamic Drivers
" Hi everyone, here's my first (brief) review of the MG6 Pro In-ears by Future Sonics.. First, i'd like to note that Mr. Marty Garcia (CEO FutureSonics) is a complete gentleman, he took time to answer ALL my questions on phone and emails too! That's rare and heart warming.. - a far cry from Sensaphonics where i encountered a completely rude rep, and Ultimate ears that won't answer or..." --SammyDplus
Future Sonics FS1 High Definition Earphones (White)
For Use: • iPod®, Zune®, MP3 Digital Media Players • Playstation®, GameCube® & most other game systems • Laptop Computers • Portable DVD Players Features & Specifications: • MG5™ Proprietary Dynamic Drivers • 20Hz - 20,000Hz TrueTimbre™ Response • 32ohm Impedance • Sensitivity 112db @ 30Hz/1mW • Up to 30db...
Future Sonics mg5pro 10mm Ear Monitors
mg5pro™10mm Ear Monitors® Custom Fit Professional Earphones, feature our mg5pro™ 10mm proprietary miniature dynamic transducers. The mg5pro™ dynamic speakers, designed & manufactured by Future Sonics, are multi-driver & crossover free; delivering warm, full range audio for use on Stage, in the Studio & with personal media players. mg5pro™ Ear Monitors...
Future Sonics mg6pro
mg6pro™ 13mm Ear Monitors® Custom Fit Professional Earphones, celebrating 25 Years, now feature our NEW mg6pro™ 13mm proprietary miniature dynamic transducers. The mg6pro™ are multi-driver & crossover free; delivering full range audio w/ natural hi-end sparkle, warm mids and a low-end frequency response no two, three, four or more way balanced armature earpiece can...
Nuforce NE-700M Audiophile-Grade Earphone with microphone (smoky bronze color)
" First, I'd like to thank the folks at NuForce for providing a sample of the revised NE-700M earphones for review. NuForce is a company I first came into contact with when I reviewed pre-production samples of their upcoming NE-700X, a replacement for their popular and outgoing NE-7M earphones. I thought the NE-700X was a pretty good deal for the asking price and now, a..." --TheGame21x
Nuforce Primo 8
"REVIEW - NUFORCE PRIMO 8 - AN AUDIOPHILE PHASE COHERENT QUAD BALANCED ARMATURE DRIVER IEM BACKGROUND: My first venture into the world of IEM was with the Shure E2c in 2007, when I wanted some isolation for private listening at home in bed. Prior to wanting the extra isolation I used some Bose noise-canceling headphones, because they helped remove the noise of my window..." --HeadphoneAddict


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NEAOBA Herdsires
Fiber Corner
Shows & Events
Member Farm Profiles
Alpaca Library
About NEAOBA
Services Directory
Members Only
NEAOBA News Announcements
NEAOBA hosts a number of events throughout the year, and its members also provide many opportunities for those who want to participate in alpaca related activities. From shows and festivals, seminars and open farm days, to fiber related workshops and activities, there are alpaca events to suit any taste.
Be sure to check this page regularly to stay abreast of what's going on in the New England alpaca world.


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Home ›
October 25, 2013Schooner RaceEach October, schooners of all description gather just south of the Annapolis Bay Bridge for the Great Chesapeake Bay Schooner Race to Portsmouth, Virginia. This year during Cruising World's Boat of the Year testing, we took break from our duties to swing past the starting line. Boy, what a sight.Related tags: Photos | Chesapeake | Annapolis | Chesapeake Bay | Great Chesapeake Bay Schooner Race | schooners
Electronic Updates
The Vesper Marine WatchMate Vision is arguably the most advanced Class B AIS transponder ever built, and was honored with an Innovation Award in Miami.


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Home > Packaging & Hardware > Adhesives, Sealants, Coatings
The DA-5045-2 and DA-5045-4 high-thermal-conductivity die-attach adhesives for attaching LEDs and small die are engineered to withstand JEDEC level 1/ 260°C reliability standards in LED packaging applications. DA-5045-2 is currently used in manufacturing lines assembling LED packages for automotive lighting.
The DA-5045-2 and DA-5045-4 have thermal conductivities that exceed 20 W/mK when measured by laser flash. Both products feature high glass-transition temperatures and modulus to enable wire bonding of the die, and are electrically conductive for static drain. (Contact company for price and availability.)
By Carolyn Mathas
EEM Inventory Search | EEM Home Websites Electronic Products | EEM | eeTechBriefs | Embedded Developer | Lighting Developer International Resources Electronic Products China | Application Reference Materials China | eeTechBrief China | 21Ic
Subscriptions | Current Issue | FAQs | About Us | Contact Us | Editorial Submission | Privacy Policy | Site Map | Media Kit | Home | RSS | Forums All contents are Copyright © 2014 by Hearst Business Communications, Inc. All Rights Reserved.


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8044 posts
Community Moderator
- Attended a Community Meetup
- Community Moderator
- Has been a member for 5-6 years
- United Kingdom
- Contributed a Tutorial to a Tuts+ Site
- Won a Competition
- Contributed a Blog Post
- Beta Tester
- Bought between 50 and 99 items
Volunteer moderator
So "monocle" became "Adobe Scout" today amongst some other things…
thoughts? too much too late?
I am tomorrow in the Adobe conference as i mention before, we should have a presentation of this Tools.
But man things are getting more and more complex, but it's nice to see they still lean on Flash CS6 , and overall i start to see some good shifting in Adobe way regarding flash.
But still… i don't really see high level flash game in Air base on the mobile world
always a lot of demo….
1657 posts
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Who the hell is still using Flash?
flashedge saidwhat you crying about ? look at the new Microsoft new net ,and see how the gif animation is back to the world,i did this s* gif animation in the 90 ,a 5 years old kid work page . http://www.so.cl/
Who the hell is still using Flash?
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it's a great tool specially if you use it for android air apps. download the app from here: https://play.google.com/store/apps/details?id=com.adobe.monocle.companion
flashedge said
Who the hell is still using Flash?
I do!


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2011 Giant Playoff Themes
I think we all know that these aren't the 2007 Giants. Sure there are a good portion of holdovers. And they both rallied late to get into the playoffs. But be sure, this team isn't coming into this Super Bowl under the radar, despite what they tell the media.
That's fine. It should be that way. It's because the Giants are the best team remaining, just like they were the second they knocked off the defending champion.
It's ok to be a bit confident. It's ok to think that this year's Giants team is more like the Packers squad that rolled into Dallas last season and won the title. Just like Green Bay last season, the Giants head into the game as the popular pick thanks to an explosive offense and a dangerous defense. Rightfully so.
This group, throughout all of their struggles this year, has shown a knack for showing up in the big games this season. Eight days from today should be no different. The Giants have a chance to turn in their best showing of the season. They proved their versatility with the win in the San Fran slugfest, but will be more than happy to return to the ideal conditions in Lucas Oil. As I said earlier in the week, pick your poison Bill. You have to think they will take their chances and do what they have to do in putting pressure on Eli. He has proven that he will kill you if you sit back and let him have his choice of options.
As was said last week, we don't see how the pass rush doesn't show up for every game at this point. They did just that in San Francisco. If they can show up in this one then the Pats will be in big, big trouble. Here's why: after two weeks of facing quarterbacks who are fleet of foot, the Giants get Brady, who doesn't have that threat. This will allow them to break their rush lanes a little bit more because the threat of the run isn't there.
The formula is simple in this one. Take away the aura of BB and this is a matchup that we should be looking forward to... The Giants just need to deliver 60 minutes!!!


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UCLA racked up another strong road score, placing second in a tri-meet at Michigan with 197.475.
UCLA women's tennis team notches second victory of the season over its cross-town rival.
Sophomore Mysha Sataraka homered in both ends of Friday's doubleheader sweep for #3 UCLA.
Zeile logs three hits, three runs and four RBI in losing effort.
Southern Highlands plays tough as no team posts a subpar score.


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Photo by Hank Lohmeyer
Crack shots
The shooting sports contest was held Saturday, July 27, at the Delta County 4-H Range in Eckert. Above, air pistols give student shooters the opportunity to begin familiarizing themselves with the lifetime sport of shooting.
North Fork
Jo Gore, a native of Delta County, has been named the parade grand marshal.
Jo was born to Floyd and Dora Klepper on May 26, 1921, and was raised on the family farm on California Mesa.
Read more...
Page 154 of 501
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U.S. Department of Health & Human Services
About HHS
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HHS Home > Newsroom
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FOR IMMEDIATE RELEASE May 19, 2012
Contact: HHS Press Office (202) 690-6343
Katie Beckett was a quiet hero and an inadvertent pioneer in the civil rights movement for people with disabilities. When she was only three years old, her family and her doctors wanted her to live at home despite her extraordinary medical support needs. At that time, Medicaid would not cover the cost of her medical services in the community -- only in the hospital. Thanks to her mother Julie's tireless advocacy, in 1982 Medicaid policy fundamentally shifted to allow people with significant health care needs and disabilities to receive care at home.Over the past thirty years, the 'Katie Beckett Waiver,' a Medicaid program, has provided over a half million children with disabilities the chance to live at home with their families and participate in their communities instead of living in hospitals and institutions.As a result of this change, Katie was able to grow up as a typical young woman living with a disability -- going to college, working as a writer and public speaker, and living an independent life -- and in the course of her journey, Katie inspired a whole generation of young people with disabilities and their families.Katie will be missed by many across the country, but her determined advocacy, and that of her family, has changed countless lives for the better. Our thoughts and prayers go out to her parents, Julie and Mark, and all in the disability community who mourn her passing.
Note: All HHS news releases, fact sheets and other press materials are available at http://www.hhs.gov/news.
Like HHS on Facebook , follow HHS on Twitter @HHSgov , and sign up for HHS Email Updates.
Follow HHS Secretary Kathleen Sebelius on Twitter @Sebelius .
Last revised: May 21, 2012


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Another musical response to #TrayvonMartin's murder
Subscribe to our e-mail newsletter to receive updates.
Some excerpts from Ta-Nehisi Coates's tour de force on Obama, race and the limits of integration.
A perspective on the Oak Creek, WI shootings from a member of the Sikh community
In case you missed it, Ms. Badu's choice words to Mr. Coyne
A worthwhile read on the go-go legend
So much conversation–and now maybe some justice–for this young man.


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Call For More Information
For questions regarding Undergraduate Programs call 215-437-2219.
For questions regarding Graduate Programs call 267-341-3327.
International applicants who have completed a baccalaureate degree at a college or university outside of the United States are required to produce minimum scores for the TOEFL test of 550 on the paper-based test, 79-80 on the internet-based test, or 213 on the computer-based test. Prospective students may submit IELTS scores in place of TOEFL scores. A score of 6.0 or better is required on the IELTS test for admission. Additionally, international students are required to provide evidence of immunization against measles, mumps, and rubella as well as evidence of the ability to provide medical insurance coverage during their proposed stay in the United States. Such applicants must arrange with World Education Services to have a document-by-document review of their education credentials, which includes a course-by-course evaluation. World Education Services must confirm that the applicant's undergraduate experience is the equivalent of four years of undergraduate study at an accredited United States college or university, culminating in the bachelor's degree. Finally, international students are required to certify that sufficient funds to support their academic and personal living expenses are available during their stay in the United States.
International graduate students are eligible to seek admission to study full-time at Holy Family University under the Exchange Visitor Program sponsored by the United States Department of State whereby eligible foreign nationals can be sponsored for a J-1 Visa by Holy Family University. International graduate degree students applying for Exchange Visitor Program J-1 Visa option must meet the required admission standards described herein for the graduate degree program of interest. For more information, please refer to the University Services section of this catalog.


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£199,9502 bed flat for sale
£2,500,000 Guide Price
Guide Price £2,500,000
6 bedroom detached house for sale The Ridge, Epsom, Surrey, KT18
An impressive and imposing property which is in excess of 7000 sq.ft. This property truly belies its size and is situated in one of Epsom's premier locations. More details ›
£2,000,000 Offers in Region of
Offers in Region of £2,000,000
27 bedroom detached house for sale Ashley Road, Epsom
An amazing detached period, grade II listed building in the heart of Epsom town centre. Ashley house was once used as Epsoms registery office and as such has the grandeur of an official building with ample off street parking. More details ›
£1,800,000 Guide Price
Guide Price £1,800,000
7 bedroom detached house for sale Glanmire Farm, Rushett Lane, Epsom, Surrey, KT18
This is a rare opportunity to build a grand design type home in the Surrey countryside. More details ›
7 bedroom detached house for sale EWELL
A remarkable family home in the "arts and crafts" tradition. Many character features including adjoining coach-house. Private road setting, glorious gardens to front, side and rear, 7 bedrooms, 5 bathrooms, 5 receptions, orangery, lavish kitchen/breakfast room and more - must be viewed. More details ›
£1,625,000 Guide Price
Guide Price £1,625,000
6 bedroom detached house for sale Burgh Heath Road, Epsom, Surrey, KT17
A beautiful Victorian house enjoying stunning and scenic views over the Epsom Downs, with easy access to Epsom town centre and offering an excellent opportunity for 7000 sq.ft. of accommodation. More details ›
- Photos: 6
- Floorplans: 0
£1,555,000 Guide Price
Guide Price £1,555,000
10 bedroom detached house for sale Upper High Street, Epsom
A Freehold Detached Building arranged to provide a total of Four Self-Contained Flats and Four split level Self-Contained Maisonettes. Each Unit subject to an Assured Shorthold Tenancy. Schedule on Request. More details ›
Marketed by McCann, Epsom.Call: 01372 435065 Local call rate
- Photos: 3
- Floorplans: 0
House for sale Epsom, Surrey
++INVESTMENT OPPORTUNITY ++ Thomas & May is Delighted to Offer A Freehold Detached Building Arranged To Provide a Total of Eight Self-Contained Flats Currently Yielding Approx £80,000 p/annum in Rent. (contd...) More details ›
£1,475,000 Guide Price
Guide Price £1,475,000
5 bedroom detached house for sale Links Road, Epsom, Surrey, KT17
An imposing and well presented family home with a blend of both contemporary and character finish, situated within easy access of Epsom town centre and mainline station. More details ›
7 bedroom detached house for sale Epsom
A substantial detached family house carefully extended by the present owner. Set in an enviable residential location within a short walk of the RAC Country Club and enjoying views to the front over the paddocks of the Woodcote Stud. More details ›
£1,295,000 Guide Price
Guide Price £1,295,000
5 bedroom detached house for sale Farm Lane, Ashtead, Surrey, KT21
This charming 5 bedroom imposing property was built in 1937 and extended by the current owner approximately 14 years ago, it sits on a generous plot extending to just under 1/2 an acre and situated in a highly sought after desirable location. More details ›


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Cat and Mouse Game Store
Chicago, Ill.
(773) 384-4454
Hours: Tues-Sat. 10am-7pm; Sun. 11am-5pm
Editorial listing:
Cat & Mouse is a new store in Chicago specializing in non-electronic games and puzzles for all ages and interests. Our store features in-store play tables and events such as game days and demos. We host Family Gaming on the 1st and 3rd Saturday of every month, when parents can bring out the kids for demos of the latest, greatest board games. These events are a great way to spend some quality time with your children and have fun test driving the best new board games.


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Building culture in digital media
October 23, 2013
Fox Harrell's new book presents a 'manifesto' detailing how computing can create powerful new forms of expression and culture.
MITx, edX team up with City of Chicago to bring high school students MOOC-style learning
June 13, 2013
Mayor Emanuel praises flexible, high quality online learning opportunity
Sneaking a peek at the new SimCity
June 12, 2013
MIT alumnus and lead designer of the digital game visits campus
Too much information?
October 1, 2012
MIT Sloan marketing professor Catherine Tucker analyzes the boundaries of privacy in a connected world.
Also labeled: Data, Faculty, Health care, iPhone, Android, smartphones, Marketing, Business and management, Medical records
New application expedites student appointment process
August 22, 2012
Also labeled: Campus services, Human Resources, Information Services and Technology, Staff, Student life, Students
Glen Urban, former MIT Sloan dean, retires to focus on research; leaves behind $1 million gift
June 11, 2012
Establishes a PhD marketing fellowship at the school
New initiatives in graduate education at MIT
March 5, 2012
Three new committees will advance important topics.
Michael W. Howard '86 named vice president for finance
December 7, 2011
MIT alumnus brings 20 years of experience at Fidelity, Deloitte.
Online evaluations provide insight to instructors and students
September 27, 2011
In spring 2011, MIT moved to a completely online subject evaluation system. The new system is more flexible, timely, accurate and environmentally conscious.
Also labeled: Campus services, Faculty, Information Services and Technology, Students, Undergraduate
Polishing its reputation for experimental lit, MIT to host the Electronic Literature Organization
August 10, 2011
Award-winning game explores puzzling dreams
August 4, 2011
MIT game wins Kongregate award, to receive 1 million free ad impressions
Also labeled: Awards, honors and fellowships, Faculty, Game Lab, Research, Singapore-MIT, Students, Video games, Arts, Humanities


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March 11 9:00AM / Fashion Talking Shop with Racked SF's Kristen Philipkoski The blogger takes us on a sartorial safari of favorite destinations and dream hauls.
March 10 9:00AM / Beauty A Messy Take on the Fishtail Braid Graduate from chronic bedhead: Morphic's Mishi Nova walks us through the steps to a fresh look for everyday.
March 09 9:00AM / Fashion 21 Things We Love (Almost) as Much as McConaughey As we bid adieu to Rust Cohle in tonight's True Detective finale, we're drowning our sorrows in king cake donuts, tasseled kicks, Wes Anderson art, and more.
March 07 9:00AM / Beauty A Skin Care Guru Faces Up to Spring Spring's artistic makeup trends require a clean canvas. Get ready to bloom with product picks and a DIY mask recipe from facialist Lori Anderson.
March 05 10:00AM / Entertainment Raise Your Glass to a NorCal Wine Adventure One writer travels to Sonoma County, California, to learn the hows and whys of winemaking. Read about what she learns and what she drinks — then re-enact her trip at home.
Foreign Fashion 101 Get to know nine up-and-coming designers from around the globe. No passport required.
Care Package for Lupita Nyong'o "I'm a big fan of both care packages and Lupita Nyong'o's Skittles-hued wardrobe. This lovely video brilliantly wraps up the magic of both." — Andrea Kasprzak, San Francisco editor


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Video: Most Popular
Ben Affleck honored at Santa Barbara Film Festival (1:18)
Jan. 27 - Ben Affleck receives an award from the Santa Barbara Film Festival for his direction of the Oscar-nominated film ''Argo.'' John Russell reports. ( Transcript )
Reuters TV Shows
About Reuters TV
Hard-edged reporting, insight and analysis, Reuters TV breaks ground creating informative news and financial videos. Showcasing Reuters' 3000 award-winning journalists, Reuters TV delivers high-energy investigative journalism with concise explanations. Check it out and let us know what you think.


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- The definition of heat is a form of energy that causes a difference in temperature, or the perception of warmth.
An example of heat is hot water.
- Heat is defined as to add warmth to something.
An example of heat is bringing a kettle of water to a boil.
The heat from this cup of hot water produces steam.
heat definition by Webster's New World
- the quality of being hot; hotness: in physics, heat is considered a form of energy existing as the result of the random motion of molecules and is the form of energy that is transferred between bodies as a result of their temperature difference
- much hotness; great warmth: stifling heat
- fever
- degree of hotness or warmth: at low heat
- the perception of heat by the senses, resulting from contact with or nearness to something hot; sensation of hotness or warmth felt through the skin
- hot weather or climate
- the warming of a room, house, etc., as by a stove or furnace: his rent includes heat
- a burning sensation produced by spices, mustard, etc.
- color or other appearance as an indication of hotness: blue heat in metals
- strong feeling or emotion; excitement, ardor, anger, zeal, etc.
- the period or condition of excitement, intensity, stress, etc.; most violent or intense point or stage: in the heat of battle
- a single effort, round, bout, or trial; esp., any of the preliminary rounds of a race, etc., the winners of which compete in the final round
- sexual excitement
- the period of sexual excitement in animals; esp., the estrus of females
- ☆ Slang
- coercion, as by intimidation
- great pressure, as in criminal investigation or law enforcement
- the police
- a pistol
- Baseball, Slang pitches thrown with great velocity
- Metallurgy
- a single heating of metal, ore, etc. in a furnace or forge
- the amount processed in a single heating
Origin: Middle English hete ; from Old English hætu ; from base of hat (see hot), akin to German heiss ; from Indo-European base an unverified form kai-, heat
transitive verb, intransitive verb
- to make or become warm or hot
- to make or become excited; inflame or become inflamed
heat definition by American Heritage Dictionary
noun Physics
- a. A form of energy associated with the motion of atoms or molecules and capable of being transmitted through solid and fluid media by conduction, through fluid media by convection, and through empty space by radiation.b. The transfer of energy from one body to another as a result of a difference in temperature or a change in phase.
- The sensation or perception of such energy as warmth or hotness.
- An abnormally high bodily temperature, as from a fever.
- a. The condition of being hot.b. A degree of warmth or hotness: The burner was on low heat.
- a. The warming of a room or building by a furnace or another source of energy: The house was cheap to rent, but the heat was expensive.b. A furnace or other source of warmth in a room or building: The heat was on when we returned from work.
- A hot season; a spell of hot weather.
- a. Intensity, as of passion, emotion, color, appearance, or effect.b. The most intense or active stage: the heat of battle.c. A burning sensation in the mouth produced by spicy flavoring in food.
- Estrus.
- One of a series of efforts or attempts.
- a. Sports & Games One round of several in a competition, such as a race.b. A preliminary contest held to determine finalists.
- Informal Pressure; stress.
- Slang a. An intensification of police activity in pursuing criminals.b. The police. Used with the.
- Slang Adverse comments or hostile criticism: Heat from the press forced the senator to resign.
- Slang A firearm, especially a pistol.
- To make warm or hot.
- To excite the feelings of; inflame.
- To increase the molecular or kinetic energy of (an object).
- To become warm or hot.
- To become excited emotionally or intellectually.
Origin: Middle English hete, from Old English hǣtu; see kai- in Indo-European roots.
heat - Ologies & -Isms Definition
the ability of light and heat and other forms of radiant energy to cause chemical changes, as hormonal changes in birds causing them to migrate or brood. —actinic, adj.
imperviousness to radiant heat or infrared radiation. Also called athermancy.
the science of measuring heat. —calorimeter, n. —calorimetric, adj.
Rare. one who believed the caloric theory, that heat is a material substance. —caloristic, adj.
the process of generating heat by means of an electric current.
the branch of geology that measures temperatures deep below the surface of the earth; geologic thermometry.
the production or generation of heat. —pyrogenetic, adj.
the chemical process of decomposition under the effect of heat. —pyrolitic, adj.
a type of pyrometer that measures temperature optically or photometrically.
a moderate warmth; lukewarmness. —tepid, adj.
Medicine. the study of heat as a medical remedy or therapy. Also called thermotherapy.
the science or study of the emission of electrons from substances at high temperatures. —thermionic, adj.
the branch of chemistry that studies the relationship of heat to chemical changes, including the production of energy. —thermochemist, n. —thermochemical, adj.
the branch of physics that studies the relationship of heat and mechanical energy and the conversion, in various materials, of one into the other. —thermodynamicist, n. —thermodynamic, thermodynamical, adj.
the production of heat, especially in an animal body by physiological processes. —thermogenic, thermogenous, adj.
1. Engineering, a method of measuring surf ace temperatures by using luminescent materials.
2. a printing or photocopying process using infrared rays and heat.
3. a process of photography using far-infrared radiation; thermal photography. —thermographer, n. —thermographic, adj.
the study of the movement of heat. —thermokinematic, adj.
Archaic. the science and study of heat. Also called thermotics.
Atomic Physics. any luminescence appearing in materials upon application of heat, caused by electron movement which increases as the temperature rises. —thermoluminescent, adj.
Physiology. the dispersion of heat from the body. —thermolytic, adj.
the branch of physics that deals with the measurement of temperature. —thermometric, adj.
an abnormal fear of heat.
a device for giving an approximation of the temperature change of a substance by noting the accompanying change in its volume. —thermoscopic, adj.
the science or study of the equilibrium of heat.
1. Biology. the movement of an organism toward or away from a source of heat.
2. Physiology. the regulation of body temperature by various physiological processes. —thermotactic, thermotaxic, adj.
the property or quality by which matter permits the passage of heat. —transcalent, adj.
heat - Cultural Definition
The New Dictionary of Cultural Literacy Copyright © 2010 by Houghton Mifflin Harcourt Publishing Company. Published by Houghton Mifflin Harcourt Publishing Company. All rights reserved.
heat - Medical Definition
- A form of energy associated with the motion of atoms or molecules and capable of being transmitted through solid and fluid media by conduction, through fluid media by convection, and through empty space by radiation.
- The sensation or perception of such energy as warmth or hotness.
- An abnormally high bodily temperature, as from a fever.
- Estrus.
heat - Phrases/Idioms
In addition to the idioms beginning with heat, also see dead heat; in heat; in the heat of the moment; turn up the heat.
heat - Science Definition
- Internal energy that is transferred to a physical system from outside the system because of a difference in temperature and does not result in work done by the system on its surroundings. Absorption of energy by a system as heat takes the form of increased kinetic energy of its molecules, thus resulting in an increase in temperature of the system. Heat is transferred from one system to another in the direction of higher to lower temperature. See also thermodynamics. See Note at temperature.
- See estrus.