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About *University Physics*

*University Physics* is designed for the two- or three-semester calculus-based physics course. The text has been developed to meet the scope and sequence of most university physics courses and provides a foundation for a career in mathematics, science, or engineering. The book provides an important opportunity for students to learn the core concepts of physics and understand how those concepts apply to their lives and to the world around them.

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About the authors

Optics

The Nature of Light

Introduction

The Propagation of Light

Abstract:
By the end of this section, you will be able to:

* Determine the index of refraction, given the speed of light in a medium
* List the ways in which light travels from a source to another location

The Speed of Light: Early Measurements

The first measurement of the speed of light was made by the Danish astronomer Ole Roemer (1644–1710) in 1675. He studied the orbit of Io, one of the four large moons of Jupiter, and found that it had a period of revolution of 42.5 h around Jupiter. He also discovered that this value fluctuated by a few seconds, depending on the position of Earth in its orbit around the Sun. Roemer realized that this fluctuation was due to the finite speed of light and could be used to determine *c*.

Speed of Light in Matter

The speed of light through matter is less than it is in a vacuum, because light interacts with atoms in a material. The speed of light depends strongly on the type of material, since its interaction varies with different atoms, crystal lattices, and other substructures. We can define a constant of a material that describes the speed of light in it, called the index of refraction *n*:

The Ray Model of Light

You have already studied some of the wave characteristics of light in the previous chapter on Electromagnetic Waves. In this chapter, we start mainly with the ray characteristics. There are three ways in which light can travel from a source to another location ([link]). It can come directly from the source through empty space, such as from the Sun to Earth. Or light can travel through various media, such as air and glass, to the observer. Light can also arrive after being reflected, such as by a mirror. In all of these cases, we can model the path of light as a straight line called a ray.

The Law of Reflection

Abstract:
By the end of this section, you will be able to:

* Explain the reflection of light from polished and rough surfaces
* Describe the principle and applications of corner reflectors

Corner Reflectors (Retroreflectors)

A light ray that strikes an object consisting of two mutually perpendicular reflecting surfaces is reflected back exactly parallel to the direction from which it came ([link]). This is true whenever the reflecting surfaces are perpendicular, and it is independent of the angle of incidence. (For proof, see [link] at the end of this section.) Such an object is called a corner reflector, since the light bounces from its inside corner. Corner reflectors are a subclass of retroreflectors, which all reflect rays back in the directions from which they came. Although the geometry of the proof is much more complex, corner reflectors can also be built with three mutually perpendicular reflecting surfaces and are useful in three-dimensional applications.

Refraction

Abstract:
By the end of this section, you will be able to:

* Describe how rays change direction upon entering a medium
* Apply the law of refraction in problem solving

Total Internal Reflection

Abstract:
By the end of this section, you will be able to:

* Explain the phenomenon of total internal reflection
* Describe the workings and uses of optical fibers
* Analyze the reason for the sparkle of diamonds

Fiber Optics: Endoscopes to Telephones

Fiber optics is one application of total internal reflection that is in wide use. In communications, it is used to transmit telephone, internet, and cable TV signals. Fiber optics employs the transmission of light down fibers of plastic or glass. Because the fibers are thin, light entering one is likely to strike the inside surface at an angle greater than the critical angle and, thus, be totally reflected ([link]). The index of refraction outside the fiber must be smaller than inside. In fact, most fibers have a varying refractive index to allow more light to be guided along the fiber through total internal refraction. Rays are reflected around corners as shown, making the fibers into tiny light pipes.

Corner Reflectors and Diamonds

Corner reflectors (The Law of Reflection) are perfectly efficient when the conditions for total internal reflection are satisfied. With common materials, it is easy to obtain a critical angle that is less than $45\text{°}.$ One use of these perfect mirrors is in binoculars, as shown in [link]. Another use is in periscopes found in submarines.

Dispersion

Abstract:
By the end of this section, you will be able to:

* Explain the cause of dispersion in a prism
* Describe the effects of dispersion in producing rainbows
* Summarize the advantages and disadvantages of dispersion

Huygens’s Principle

Abstract:
By the end of this section, you will be able to:

* Describe Huygens’s principle
* Use Huygens’s principle to explain the law of reflection
* Use Huygens’s principle to explain the law of refraction
* Use Huygens’s principle to explain diffraction

Reflection

[link] shows how a mirror reflects an incoming wave at an angle equal to the incident angle, verifying the law of reflection. As the wave front strikes the mirror, wavelets are first emitted from the left part of the mirror and then from the right. The wavelets closer to the left have had time to travel farther, producing a wave front traveling in the direction shown.

Refraction

The law of refraction can be explained by applying Huygens’s principle to a wave front passing from one medium to another ([link]). Each wavelet in the figure was emitted when the wave front crossed the interface between the media. Since the speed of light is smaller in the second medium, the waves do not travel as far in a given time, and the new wave front changes direction as shown. This explains why a ray changes direction to become closer to the perpendicular when light slows down. Snell’s law can be derived from the geometry in [link] ([link]).

Diffraction

What happens when a wave passes through an opening, such as light shining through an open door into a dark room? For light, we observe a sharp shadow of the doorway on the floor of the room, and no visible light bends around corners into other parts of the room. When sound passes through a door, we hear it everywhere in the room and thus observe that sound spreads out when passing through such an opening ([link]). What is the difference between the behavior of sound waves and light waves in this case? The answer is that light has very short wavelengths and acts like a ray. Sound has wavelengths on the order of the size of the door and bends around corners (for frequency of 1000 Hz,

Polarization

Abstract:
By the end of this section, you will be able to:

* Explain the change in intensity as polarized light passes through a polarizing filter
* Calculate the effect of polarization by reflection and Brewster’s angle
* Describe the effect of polarization by scattering
* Explain the use of polarizing materials in devices such as LCDs

Malus’s Law

Light is one type of electromagnetic (EM) wave. As noted in the previous chapter on Electromagnetic Waves, EM waves are *transverse waves* consisting of varying electric and magnetic fields that oscillate perpendicular to the direction of propagation ([link]). However, in general, there are no specific directions for the oscillations of the electric and magnetic fields; they vibrate in any randomly oriented plane perpendicular to the direction of propagation. **Polarization** is the attribute that a wave’s oscillations do have a definite direction relative to the direction of propagation of the wave. (This is not the same type of polarization as that discussed for the separation of charges.) Waves having such a direction are said to be polarized. For an EM wave, we define the direction of polarization to be the direction parallel to the electric field. Thus, we can think of the electric field arrows as showing the direction of polarization, as in [link].

Polarization by Reflection

By now, you can probably guess that polarizing sunglasses cut the glare in reflected light, because that light is polarized. You can check this for yourself by holding polarizing sunglasses in front of you and rotating them while looking at light reflected from water or glass. As you rotate the sunglasses, you will notice the light gets bright and dim, but not completely black. This implies the reflected light is partially polarized and cannot be completely blocked by a polarizing filter.

Atomic Explanation of Polarizing Filters

Polarizing filters have a polarization axis that acts as a slit. This slit passes EM waves (often visible light) that have an electric field parallel to the axis. This is accomplished with long molecules aligned perpendicular to the axis, as shown in [link].

Polarization by Scattering

If you hold your polarizing sunglasses in front of you and rotate them while looking at blue sky, you will see the sky get bright and dim. This is a clear indication that light scattered by air is partially polarized. [link] helps illustrate how this happens. Since light is a transverse EM wave, it vibrates the electrons of air molecules perpendicular to the direction that it is traveling. The electrons then radiate like small antennae. Since they are oscillating perpendicular to the direction of the light ray, they produce EM radiation that is polarized perpendicular to the direction of the ray. When viewing the light along a line perpendicular to the original ray, as in the figure, there can be no polarization in the scattered light parallel to the original ray, because that would require the original ray to be a longitudinal wave. Along other directions, a component of the other polarization can be projected along the line of sight, and the scattered light is only partially polarized. Furthermore, multiple scattering can bring light to your eyes from other directions and can contain different polarizations.

Liquid Crystals and Other Polarization Effects in Materials

Although you are undoubtedly aware of liquid crystal displays (LCDs) found in watches, calculators, computer screens, cellphones, flat screen televisions, and many other places, you may not be aware that they are based on polarization. Liquid crystals are so named because their molecules can be aligned even though they are in a liquid. Liquid crystals have the property that they can rotate the polarization of light passing through them by $90\text{°}$. Furthermore, this property can be turned off by the application of a voltage, as illustrated in [link]. It is possible to manipulate this characteristic quickly and in small, well-defined regions to create the contrast patterns we see in so many LCD devices.

Geometric Optics and Image Formation

Introduction

Images Formed by Plane Mirrors

Abstract:
By the end of this section, you will be able to:

* Describe how an image is formed by a plane mirror.
* Distinguish between real and virtual images.
* Find the location and characterize the orientation of an image created by a plane mirror.

Locating an Image in a Plane Mirror

The law of reflection tells us that the angle of incidence is the same as the angle of reflection. Applying this to triangles *PAB* and *QAB* in [link] and using basic geometry shows that they are congruent triangles. This means that the distance *PB* from the object to the mirror is the same as the distance *BQ* from the mirror to the image. The object distance (denoted ${d}_{\text{o}}$) is the distance from the mirror to the object (or, more generally, from the center of the optical element that creates its image). Similarly, the image distance (denoted ${d}_{\text{i}}$) is the distance from the mirror to the image (or, more generally, from the center of the optical element that creates it). If we measure distances from the mirror, then the object and image are in opposite directions, so for a plane mirror, the object and image distances should have the opposite signs:

Multiple Images

If an object is situated in front of two mirrors, you may see images in both mirrors. In addition, the image in the first mirror may act as an object for the second mirror, so the second mirror may form an image of the image. If the mirrors are placed parallel to each other and the object is placed at a point other than the midpoint between them, then this process of image-of-an-image continues without end, as you may have noticed when standing in a hallway with mirrors on each side. This is shown in [link], which shows three images produced by the blue object. Notice that each reflection reverses front and back, just like pulling a right-hand glove inside out produces a left-hand glove (this is why a reflection of your right hand is a left hand). Thus, the fronts and backs of images 1 and 2 are both inverted with respect to the object, and the front and back of image 3 is inverted with respect to image 2, which is the object for image 3.

Spherical Mirrors

Abstract:
By the end of this section, you will be able to:

* Describe image formation by spherical mirrors.
* Use ray diagrams and the mirror equation to calculate the properties of an image in a spherical mirror.

Curved Mirrors

We can define two general types of spherical mirrors. If the reflecting surface is the outer side of the sphere, the mirror is called a convex mirror. If the inside surface is the reflecting surface, it is called a concave mirror.

Using Ray Tracing to Locate Images

To find the location of an image formed by a spherical mirror, we first use ray tracing, which is the technique of drawing rays and using the law of reflection to determine the reflected rays (later, for lenses, we use the law of refraction to determine refracted rays). Combined with some basic geometry, we can use ray tracing to find the focal point, the image location, and other information about how a mirror manipulates light. In fact, we already used ray tracing above to locate the focal point of spherical mirrors, or the image distance of flat mirrors. To locate the image of an object, you must locate at least two points of the image. Locating each point requires drawing at least two rays from a point on the object and constructing their reflected rays. The point at which the reflected rays intersect, either in real space or in virtual space, is where the corresponding point of the image is located. To make ray tracing easier, we concentrate on four “principal” rays whose reflections are easy to construct.

Summary of Ray-Tracing Rules

Ray tracing is very useful for mirrors. The rules for ray tracing are summarized here for reference:

Image Formation by Reflection—The Mirror Equation

For a plane mirror, we showed that the image formed has the same height and orientation as the object, and it is located at the same distance behind the mirror as the object is in front of the mirror. Although the situation is a bit more complicated for curved mirrors, using geometry leads to simple formulas relating the object and image distances to the focal lengths of concave and convex mirrors.

Departure from the Small-Angle Approximation

The small-angle approximation is a cornerstone of the above discussion of image formation by a spherical mirror. When this approximation is violated, then the image created by a spherical mirror becomes distorted. Such distortion is called aberration. Here we briefly discuss two specific types of aberrations: spherical aberration and coma.

Images Formed by Refraction

Abstract:
By the end of this section, you will be able to:

* Describe image formation by a single refracting surface
* Determine the location of an image and calculate its properties by using a ray diagram
* Determine the location of an image and calculate its properties by using the equation for a single refracting surface

Refraction at a Plane Interface—Apparent Depth

If you look at a straight rod partially submerged in water, it appears to bend at the surface ([link]). The reason behind this curious effect is that the image of the rod inside the water forms a little closer to the surface than the actual position of the rod, so it does not line up with the part of the rod that is above the water. The same phenomenon explains why a fish in water appears to be closer to the surface than it actually is.

Refraction at a Spherical Interface

Spherical shapes play an important role in optics primarily because high-quality spherical shapes are far easier to manufacture than other curved surfaces. To study refraction at a single spherical surface, we assume that the medium with the spherical surface at one end continues indefinitely (a “semi-infinite” medium).

Thin Lenses

Abstract:
By the end of this section, you will be able to:

* Use ray diagrams to locate and describe the image formed by a lens
* Employ the thin-lens equation to describe and locate the image formed by a lens

Ray Tracing and Thin Lenses

**Ray tracing** is the technique of determining or following (tracing) the paths taken by light rays.

Image Formation by Thin Lenses

We use ray tracing to investigate different types of images that can be created by a lens. In some circumstances, a lens forms a real image, such as when a movie projector casts an image onto a screen. In other cases, the image is a virtual image, which cannot be projected onto a screen. Where, for example, is the image formed by eyeglasses? We use ray tracing for thin lenses to illustrate how they form images, and then we develop equations to analyze quantitatively the properties of thin lenses.

Oblique Parallel Rays and Focal Plane

We have seen that rays parallel to the optical axis are directed to the focal point of a converging lens. In the case of a diverging lens, they come out in a direction such that they appear to be coming from the focal point on the opposite side of the lens (i.e., the side from which parallel rays enter the lens). What happens to parallel rays that are not parallel to the optical axis ([link])? In the case of a converging lens, these rays do not converge at the focal point. Instead, they come together on another point in the plane called the focal plane. The focal plane contains the focal point and is perpendicular to the optical axis. As shown in the figure, parallel rays focus where the ray through the center of the lens crosses the focal plane.

Thin-Lens Equation

Ray tracing allows us to get a qualitative picture of image formation. To obtain numeric information, we derive a pair of equations from a geometric analysis of ray tracing for thin lenses. These equations, called the thin-lens equation and the lens maker’s equation, allow us to quantitatively analyze thin lenses.

Using the Thin-Lens Equation

The thin-lens equation and the lens maker’s equation are broadly applicable to situations involving thin lenses. We explore many features of image formation in the following examples.

The Eye

Abstract:
By the end of this section, you will be able to:

* Understand the basic physics of how images are formed by the human eye
* Recognize several conditions of impaired vision as well as the optics principles for treating these conditions

Physics of the Eye

The eye is remarkable in how it forms images and in the richness of detail and color it can detect. However, our eyes often need some correction to reach what is called “normal” vision. Actually, normal vision should be called “ideal” vision because nearly one-half of the human population requires some sort of eyesight correction, so requiring glasses is by no means “abnormal.” Image formation by our eyes and common vision correction can be analyzed with the optics discussed earlier in this chapter.

Vision Correction

The need for some type of vision correction is very common. Typical vision defects are easy to understand with geometric optics, and some are simple to correct. [link] illustrates two common vision defects. Nearsightedness, or myopia, is the ability to see near objects, whereas distant objects are blurry. The eye overconverges the nearly parallel rays from a distant object, and the rays cross in front of the retina. More divergent rays from a close object are converged on the retina for a clear image. The distance to the farthest object that can be seen clearly is called the far point of the eye (normally the far point is at infinity). Farsightedness, or hyperopia, is the ability to see far objects clearly, whereas near objects are blurry. A farsighted eye does not sufficiently converge the rays from a near object to make the rays meet on the retina.

The Camera

Abstract:
By the end of this section, you will be able to:

* Describe the optics of a camera
* Characterize the image created by a camera

The Simple Magnifier

Abstract:
By the end of this section, you will be able to:

* Understand the optics of a simple magnifier
* Characterize the image created by a simple magnifier

Microscopes and Telescopes

Abstract:
By the end of this section, you will be able to:

* Explain the physics behind the operation of microscopes and telescopes
* Describe the image created by these instruments and calculate their magnifications

Microscopes

Although the eye is marvelous in its ability to see objects large and small, it obviously is limited in the smallest details it can detect. The desire to see beyond what is possible with the naked eye led to the use of optical instruments. We have seen that a simple convex lens can create a magnified image, but it is hard to get large magnification with such a lens. A magnification greater than $5{\rule{0.2em}{0ex}}×{\rule{0.2em}{0ex}}$ is difficult without distorting the image. To get higher magnification, we can combine the simple magnifying glass with one or more additional lenses. In this section, we examine microscopes that enlarge the details that we cannot see with the naked eye.

Telescopes

Telescopes are meant for viewing distant objects and produce an image that is larger than the image produced in the unaided eye. Telescopes gather far more light than the eye, allowing dim objects to be observed with greater magnification and better resolution. Telescopes were invented around 1600, and Galileo was the first to use them to study the heavens, with monumental consequences. He observed the moons of Jupiter, the craters and mountains on the moon, the details of sunspots, and the fact that the Milky Way is composed of a vast number of individual stars.

Reflecting Telescopes

Isaac Newton designed the first reflecting telescope around 1670 to solve the problem of chromatic aberration that happens in all refracting telescopes. In chromatic aberration, light of different colors refracts by slightly different amounts in the lens. As a result, a rainbow appears around the image and the image appears blurred. In the reflecting telescope, light rays from a distant source fall upon the surface of a concave mirror fixed at the bottom end of the tube. The use of a mirror instead of a lens eliminates chromatic aberration. The concave mirror focuses the rays on its focal plane. The design problem is how to observe the focused image. Newton used a design in which the focused light from the concave mirror was reflected to one side of the tube into an eyepiece [part (a) of [link]]. This arrangement is common in many amateur telescopes and is called the Newtonian design.

Interference

Introduction

Young's Double-Slit Interference

Abstract:
By the end of this section, you will be able to:

* Explain the phenomenon of interference
* Define constructive and destructive interference for a double slit

Mathematics of Interference

Abstract:
By the end of this section, you will be able to:

* Determine the angles for bright and dark fringes for double slit interference
* Calculate the positions of bright fringes on a screen

Multiple-Slit Interference

Abstract:
By the end of this section, you will be able to:

* Describe the locations and intensities of secondary maxima for multiple-slit interference

Interference in Thin Films

Abstract:
By the end of this section, you will be able to:

* Describe the phase changes that occur upon reflection
* Describe fringes established by reflected rays of a common source
* Explain the appearance of colors in thin films

Changes in Phase due to Reflection

We saw earlier (Waves) that reflection of mechanical waves can involve a $180\text{°}$ phase change. For example, a traveling wave on a string is inverted (i.e., a $180\text{°}$ phase change) upon reflection at a boundary to which a heavier string is tied. However, if the second string is lighter (or more precisely, of a lower linear density), no inversion occurs. Light waves produce the same effect, but the deciding parameter for light is the index of refraction. Light waves undergo a $180\text{°}$ or $\pi$ radians phase change upon reflection at an interface beyond which is a medium of higher index of refraction. No phase change takes place when reflecting from a medium of lower refractive index ([link]). Because of the periodic nature of waves, this phase change or inversion is equivalent to $\text{±}\lambda \text{/}2$ in distance travelled, or path length. Both the path length and refractive indices are important factors in thin-film interference.

Combining Path Length Difference with Phase Change

Thin-film interference is most constructive or most destructive when the path length difference for the two rays is an integral or half-integral wavelength. That is, for rays incident perpendicularly,

The Michelson Interferometer

Abstract:
By the end of this section, you will be able to:

* Explain changes in fringes observed with a Michelson interferometer caused by mirror movements
* Explain changes in fringes observed with a Michelson interferometer caused by changes in medium

Diffraction

Introduction

Single-Slit Diffraction

Abstract:
By the end of this section, you will be able to:

* Explain the phenomenon of diffraction and the conditions under which it is observed
* Describe diffraction through a single slit

Diffraction through a Single Slit

Light passing through a single slit forms a diffraction pattern somewhat different from those formed by double slits or diffraction gratings, which we discussed in the chapter on interference. [link] shows a single-slit diffraction pattern. Note that the central maximum is larger than maxima on either side and that the intensity decreases rapidly on either side. In contrast, a diffraction grating (Diffraction Gratings) produces evenly spaced lines that dim slowly on either side of the center.

Intensity in Single-Slit Diffraction

Abstract:
By the end of this section, you will be able to:

* Calculate the intensity relative to the central maximum of the single-slit diffraction peaks
* Calculate the intensity relative to the central maximum of an arbitrary point on the screen

Double-Slit Diffraction

Abstract:
By the end of this section, you will be able to:

* Describe the combined effect of interference and diffraction with two slits, each with finite width
* Determine the relative intensities of interference fringes within a diffraction pattern
* Identify missing orders, if any

Diffraction Gratings

Abstract:
By the end of this section, you will be able to:

* Discuss the pattern obtained from diffraction gratings
* Explain diffraction grating effects

Diffraction Gratings: An Infinite Number of Slits

The analysis of multi-slit interference in Interference allows us to consider what happens when the number of slits *N* approaches infinity. Recall that $N–2$ secondary maxima appear between the principal maxima. We can see there will be an infinite number of secondary maxima that appear, and an infinite number of dark fringes between them. This makes the spacing between the fringes, and therefore the width of the maxima, infinitesimally small. Furthermore, because the intensity of the secondary maxima is proportional to $1\text{/}{N}^{2}$, it approaches zero so that the secondary maxima are no longer seen. What remains are only the principal maxima, now very bright and very narrow ([link]).

Applications of Diffraction Gratings

Where are diffraction gratings used in applications? Diffraction gratings are commonly used for spectroscopic dispersion and analysis of light. What makes them particularly useful is the fact that they form a sharper pattern than double slits do. That is, their bright fringes are narrower and brighter while their dark regions are darker. Diffraction gratings are key components of monochromators used, for example, in optical imaging of particular wavelengths from biological or medical samples. A diffraction grating can be chosen to specifically analyze a wavelength emitted by molecules in diseased cells in a biopsy sample or to help excite strategic molecules in the sample with a selected wavelength of light. Another vital use is in optical fiber technologies where fibers are designed to provide optimum performance at specific wavelengths. A range of diffraction gratings are available for selecting wavelengths for such use.

Circular Apertures and Resolution

Abstract:
By the end of this section, you will be able to:

* Describe the diffraction limit on resolution
* Describe the diffraction limit on beam propagation

X-Ray Diffraction

Abstract:
By the end of this section, you will be able to:

* Describe interference and diffraction effects exhibited by X-rays in interaction with atomic-scale structures

Holography

Abstract:
By the end of this section, you will be able to:

* Describe how a three-dimensional image is recorded as a hologram
* Describe how a three-dimensional image is formed from a hologram

Modern Physics

Relativity

Introduction

Invariance of Physical Laws

Abstract:
By the end of this section, you will be able to:

* Describe the theoretical and experimental issues that Einstein’s theory of special relativity addressed.
* State the two postulates of the special theory of relativity.

Inertial Frames

All velocities are measured relative to some frame of reference. For example, a car’s motion is measured relative to its starting position on the road it travels on; a projectile’s motion is measured relative to the surface from which it is launched; and a planet’s orbital motion is measured relative to the star it orbits. The frames of reference in which mechanics takes the simplest form are those that are not accelerating. Newton’s first law, the law of inertia, holds exactly in such a frame.

Einstein’s First Postulate

Not only are the principles of classical mechanics simplest in inertial frames, but they are the same in all inertial frames. Einstein based the first postulate of his theory on the idea that this is true for all the laws of physics, not merely those in mechanics.

Einstein’s Second Postulate

The second postulate upon which Einstein based his theory of special relativity deals with the speed of light. Late in the nineteenth century, the major tenets of classical physics were well established. Two of the most important were the laws of electromagnetism and Newton’s laws. Investigations such as Young’s double-slit experiment in the early 1800s had convincingly demonstrated that light is a wave. Maxwell’s equations of electromagnetism implied that electromagnetic waves travel at $c=3.00×{10}^{8}{\rule{0.2em}{0ex}}\text{m/s}$ in a vacuum, but they do not specify the frame of reference in which light has this speed. Many types of waves were known, and all travelled in some medium. Scientists therefore assumed that some medium carried the light, even in a vacuum, and that light travels at a speed *c* relative to that medium (often called “the aether”).

Relativity of Simultaneity

Abstract:
By the end of this section, you will be able to:

* Show from Einstein's postulates that two events measured as simultaneous in one inertial frame are not necessarily simultaneous in all inertial frames.
* Describe how simultaneity is a relative concept for observers in different inertial frames in relative motion.

Time Dilation

Abstract:
By the end of this section, you will be able to:

* Explain how time intervals can be measured differently in different reference frames.
* Describe how to distinguish a proper time interval from a dilated time interval.
* Describe the significance of the muon experiment.
* Explain why the twin paradox is not a contradiction.
* Calculate time dilation given the speed of an object in a given frame.

Half-Life of a Muon

There is considerable experimental evidence that the equation $\text{Δ}t=\gamma \text{Δ}\tau$ is correct. One example is found in cosmic ray particles that continuously rain down on Earth from deep space. Some collisions of these particles with nuclei in the upper atmosphere result in short-lived particles called muons. The half-life (amount of time for half of a material to decay) of a muon is 1.52 μs when it is at rest relative to the observer who measures the half-life. This is the proper time interval $\text{Δ}\tau .$ This short time allows very few muons to reach Earth’s surface and be detected if Newtonian assumptions about time and space were correct. However, muons produced by cosmic ray particles have a range of velocities, with some moving near the speed of light. It has been found that the muon’s half-life as measured by an earthbound observer ($\text{Δ}t$) varies with velocity exactly as predicted by the equation $\text{Δ}t=\gamma \text{Δ}\tau .$ The faster the muon moves, the longer it lives. We on Earth see the muon last much longer than its half-life predicts within its own rest frame. As viewed from our frame, the muon decays more slowly than it does when at rest relative to us. A far larger fraction of muons reach the ground as a result.

The Twin Paradox

An intriguing consequence of time dilation is that a space traveler moving at a high velocity relative to Earth would age less than the astronaut’s earthbound twin. This is often known as the twin paradox. Imagine the astronaut moving at such a velocity that $\gamma =30.0,$ as in [link]. A trip that takes 2.00 years in her frame would take 60.0 years in the earthbound twin’s frame. Suppose the astronaut travels 1.00 year to another star system, briefly explores the area, and then travels 1.00 year back. An astronaut who was 40 years old at the start of the trip would be would be 42 when the spaceship returns. Everything on Earth, however, would have aged 60.0 years. The earthbound twin, if still alive, would be 100 years old.

Length Contraction

Abstract:
By the end of this section, you will be able to:

* Explain how simultaneity and length contraction are related.
* Describe the relation between length contraction and time dilation and use it to derive the length-contraction equation.

Proper Length

Two observers passing each other always see the same value of their relative speed. Even though time dilation implies that the train passenger and the observer standing alongside the tracks measure different times for the train to pass, they still agree that relative speed, which is distance divided by elapsed time, is the same. If an observer on the ground and one on the train measure a different time for the length of the train to pass the ground observer, agreeing on their relative speed means they must also see different distances traveled.

Length Contraction

To relate distances measured by different observers, note that the velocity relative to the earthbound observer in our muon example is given by

The Lorentz Transformation

Abstract:
* Describe the Galilean transformation of classical mechanics, relating the position, time, velocities, and accelerations measured in different inertial frames
* Derive the corresponding Lorentz transformation equations, which, in contrast to the Galilean transformation, are consistent with special relativity
* Explain the Lorentz transformation and many of the features of relativity in terms of four-dimensional space-time

The Galilean Transformation Equations

An event is specified by its location and time (*x*, *y*, *z*, *t*) relative to one particular inertial frame of reference *S*. As an example, (*x*, *y*, *z*, *t*) could denote the position of a particle at time *t*, and we could be looking at these positions for many different times to follow the motion of the particle. Suppose a second frame of reference $S\prime$ moves with velocity *v* with respect to the first. For simplicity, assume this relative velocity is along the *x*-axis. The relation between the time and coordinates in the two frames of reference is then

The Lorentz Transformation Equations

The Galilean transformation nevertheless violates Einstein’s postulates, because the velocity equations state that a pulse of light moving with speed *c* along the *x*-axis would travel at speed $c-v$ in the other inertial frame. Specifically, the spherical pulse has radius $r=ct$ at time *t* in the unprimed frame, and also has radius $r\prime =ct\prime$ at time $t\prime$ in the primed frame. Expressing these relations in Cartesian coordinates gives

Space-time

Relativistic phenomena can be analyzed in terms of events in a four-dimensional space-time. When phenomena such as the twin paradox, time dilation, length contraction, and the dependence of simultaneity on relative motion are viewed in this way, they are seen to be characteristic of the nature of space and time, rather than specific aspects of electromagnetism.

Relativistic Velocity Transformation

Abstract:
By the end of this section, you will be able to:

* Derive the equations consistent with special relativity for transforming velocities in one inertial frame of reference into another.
* Apply the velocity transformation equations to objects moving at relativistic speeds.
* Examine how the combined velocities predicted by the relativistic transformation equations compare with those expected classically.

Velocity Transformations

Imagine a car traveling at night along a straight road, as in [link]. The driver sees the light leaving the headlights at speed *c* within the car’s frame of reference. If the Galilean transformation applied to light, then the light from the car’s headlights would approach the pedestrian at a speed $u=v+c,$ contrary to Einstein’s postulates.

Relativistic Transformation of Velocity

Suppose an object *P* is moving at constant velocity $u=\left({u}_{x}^{\prime },{u}_{y}^{\prime },{u}_{z}^{\prime }\right)$ as measured in the $S\prime$ frame. The $S\prime$ frame is moving along its $x\prime \text{-axis}$ at velocity *v*. In an increment of time $dt\prime$, the particle is displaced by $dx\prime$ along the $x\prime \text{-axis.}$ Applying the Lorentz transformation equations gives the corresponding increments of time and displacement in the unprimed axes:

Doppler Effect for Light

Abstract:
By the end of this section, you will be able to:

* Explain the origin of the shift in frequency and wavelength of the observed wavelength when observer and source moved toward or away from each other
* Derive an expression for the relativistic Doppler shift
* Apply the Doppler shift equations to real-world examples

The Relativistic Doppler Effect

Suppose an observer in *S* sees light from a source in $S\prime$ moving away at velocity *v* ([link]). The wavelength of the light could be measured within $S\prime$—for example, by using a mirror to set up standing waves and measuring the distance between nodes. These distances are proper lengths with $S\prime$ as their rest frame, and change by a factor $\sqrt{1-{v}^{2}\text{/}{c}^{2}}$ when measured in the observer’s frame *S*, where the ruler measuring the wavelength in $S\prime$ is seen as moving.

Red Shifts and Blue Shifts

The observed wavelength ${\lambda }_{\text{obs}}$ of electromagnetic radiation is longer (called a “red shift”) than that emitted by the source when the source moves away from the observer. Similarly, the wavelength is shorter (called a “blue shift”) when the source moves toward the observer. The amount of change is determined by

Relativistic Momentum

Abstract:
By the end of this section, you will be able to:

* Define relativistic momentum in terms of mass and velocity
* Show how relativistic momentum relates to classical momentum
* Show how conservation of relativistic momentum limits objects with mass to speeds less than $c$

Relativistic Energy

Abstract:
By the end of this section, you will be able to:

* Explain how the work-energy theorem leads to an expression for the relativistic kinetic energy of an object
* Show how the relativistic energy relates to the classical kinetic energy, and sets a limit on the speed of any object with mass
* Describe how the total energy of a particle is related to its mass and velocity
* Explain how relativity relates to energy-mass equivalence, and some of the practical implications of energy-mass equivalence

Kinetic Energy and the Ultimate Speed Limit

The first postulate of relativity states that the laws of physics are the same in all inertial frames. Einstein showed that the law of conservation of energy of a particle is valid relativistically, but for energy expressed in terms of velocity and mass in a way consistent with relativity.

Total Relativistic Energy

The expression for kinetic energy can be rearranged to:

Stored Energy and Potential Energy

What happens to energy stored in an object at rest, such as the energy put into a battery by charging it, or the energy stored in a toy gun’s compressed spring? The energy input becomes part of the total energy of the object and thus increases its rest mass. All stored and potential energy becomes mass in a system. In seeming contradiction, the principle of conservation of mass (meaning total mass is constant) was one of the great laws verified by nineteenth-century science. Why was it not noticed to be incorrect? The following example helps answer this question.

Relativistic Energy and Momentum

We know classically that kinetic energy and momentum are related to each other, because:

Photons and Matter Waves

Introduction

Blackbody Radiation

Abstract:
By the end of this section, you will be able to:

* Apply Wien’s and Stefan’s laws to analyze radiation emitted by a blackbody
* Explain Planck’s hypothesis of energy quanta

Photoelectric Effect

Abstract:
By the end of this section, you will be able to:

* Describe physical characteristics of the photoelectric effect
* Explain why the photoelectric effect cannot be explained by classical physics
* Describe how Einstein’s idea of a particle of radiation explains the photoelectric effect

Characteristics of the Photoelectric Effect

The photoelectric effect has three important characteristics that cannot be explained by classical physics: (1) the absence of a lag time, (2) the independence of the kinetic energy of photoelectrons on the intensity of incident radiation, and (3) the presence of a cut-off frequency. Let’s examine each of these characteristics.

The Work Function

The photoelectric effect was explained in 1905 by A. Einstein. Einstein reasoned that if Planck’s hypothesis about energy quanta was correct for describing the energy exchange between electromagnetic radiation and cavity walls, it should also work to describe energy absorption from electromagnetic radiation by the surface of a photoelectrode. He postulated that an electromagnetic wave carries its energy in discrete packets. Einstein’s postulate goes beyond Planck’s hypothesis because it states that the light itself consists of energy quanta. In other words, it states that electromagnetic waves are quantized.

The Compton Effect

Abstract:
By the end of this section, you will be able to:

* Describe Compton’s experiment
* Explain the Compton wavelength shift
* Describe how experiments with X-rays confirm the particle nature of radiation

Momentum of a Photon

Unlike a particle of matter that is characterized by its rest mass ${m}_{0},$ a photon is massless. In a vacuum, unlike a particle of matter that may vary its speed but cannot reach the speed of light, a photon travels at only one speed, which is exactly the speed of light. From the point of view of Newtonian classical mechanics, these two characteristics imply that a photon should not exist at all. For example, how can we find the linear momentum or kinetic energy of a body whose mass is zero? This apparent paradox vanishes if we describe a photon as a relativistic particle. According to the theory of special relativity, any particle in nature obeys the relativistic energy equation

The Compton Effect

The Compton effect is the term used for an unusual result observed when X-rays are scattered on some materials. By classical theory, when an electromagnetic wave is scattered off atoms, the wavelength of the scattered radiation is expected to be the same as the wavelength of the incident radiation. Contrary to this prediction of classical physics, observations show that when X-rays are scattered off some materials, such as graphite, the scattered X-rays have different wavelengths from the wavelength of the incident X-rays. This classically unexplainable phenomenon was studied experimentally by Arthur H. Compton and his collaborators, and Compton gave its explanation in 1923.

Compton Shift

As given by Compton, the explanation of the Compton shift is that in the target material, graphite, valence electrons are loosely bound in the atoms and behave like free electrons. Compton assumed that the incident X-ray radiation is a stream of photons. An incoming photon in this stream collides with a valence electron in the graphite target. In the course of this collision, the incoming photon transfers some part of its energy and momentum to the target electron and leaves the scene as a scattered photon. This model explains in qualitative terms why the scattered radiation has a longer wavelength than the incident radiation. Put simply, a photon that has lost some of its energy emerges as a photon with a lower frequency, or equivalently, with a longer wavelength. To show that his model was correct, Compton used it to derive the expression for the Compton shift. In his derivation, he assumed that both photon and electron are relativistic particles and that the collision obeys two commonsense principles: (1) the conservation of linear momentum and (2) the conservation of total relativistic energy.

Bohr’s Model of the Hydrogen Atom

Abstract:
By the end of this section, you will be able to:

* Explain the difference between the absorption spectrum and the emission spectrum of radiation emitted by atoms
* Describe the Rutherford gold foil experiment and the discovery of the atomic nucleus
* Explain the atomic structure of hydrogen
* Describe the postulates of the early quantum theory for the hydrogen atom
* Summarize how Bohr’s quantum model of the hydrogen atom explains the radiation spectrum of atomic hydrogen

Electron Orbits

To obtain the size ${r}_{n}$ of the electron’s *n*th orbit and the electron’s speed ${v}_{n}$ in it, we turn to Newtonian mechanics. As a charged particle, the electron experiences an electrostatic pull toward the positively charged nucleus in the center of its circular orbit. This electrostatic pull is the centripetal force that causes the electron to move in a circle around the nucleus. Therefore, the magnitude of centripetal force is identified with the magnitude of the electrostatic force:

Electron Energies

The total energy ${E}_{n}$ of an electron in the *n*th orbit is the sum of its kinetic energy ${K}_{n}$ and its electrostatic potential energy ${U}_{n}.$ Utilizing [link], we find that

Spectral Emission Lines of Hydrogen

To obtain the wavelengths of the emitted radiation when an electron makes a transition from the *n*th orbit to the *m*th orbit, we use the second of Bohr’s quantization conditions and [link] for energies. The emission of energy from the atom can occur only when an electron makes a transition from an excited state to a lower-energy state. In the course of such a transition, the emitted photon carries away the difference of energies between the states involved in the transition. The transition cannot go in the other direction because the energy of a photon cannot be negative, which means that for emission we must have ${E}_{n}>{E}_{m}$ and $n>m.$ Therefore, the third of Bohr’s postulates gives

De Broglie’s Matter Waves

Abstract:
By the end of this section, you will be able to:

* Describe de Broglie’s hypothesis of matter waves
* Explain how the de Broglie’s hypothesis gives the rationale for the quantization of angular momentum in Bohr’s quantum theory of the hydrogen atom
* Describe the Davisson–Germer experiment
* Interpret de Broglie’s idea of matter waves and how they account for electron diffraction phenomena

Wave-Particle Duality

Abstract:
By the end of this section, you will be able to:

* Identify phenomena in which electromagnetic waves behave like a beam of photons and particles behave like waves
* Describe the physics principles behind electron microscopy
* Summarize the evolution of scientific thought that led to the development of quantum mechanics

Quantum Mechanics

Introduction

Wave Functions

Abstract:
By the end of this section, you will be able to:

* Describe the statistical interpretation of the wave function
* Use the wave function to determine probabilities
* Calculate expectation values of position, momentum, and kinetic energy

Using the Wave Function

A clue to the physical meaning of the wave function $\text{Ψ}\left(x,t\right)$ is provided by the two-slit interference of monochromatic light ([link]). (See also Electromagnetic Waves and Interference.) The wave function of a light wave is given by *E*(*x*,*t*), and its energy density is given by $|E{|}^{2}$, where *E* is the electric field strength. The energy of an individual photon depends only on the frequency of light, ${\epsilon }_{\text{photon}}=hf,$ so $|E{|}^{2}$ is proportional to the number of photons. When light waves from ${\text{S}}_{1}$ interfere with light waves from ${\text{S}}_{2}$ at the viewing screen (a distance *D* away), an interference pattern is produced (part (a) of the figure). Bright fringes correspond to points of constructive interference of the light waves, and dark fringes correspond to points of destructive interference of the light waves (part (b)).

An Interpretation of the Wave Function

We are now in position to begin to answer the questions posed at the beginning of this section. First, for a traveling particle described by $\text{Ψ}\left(x,t\right)=A{\rule{0.2em}{0ex}}\text{sin}\left(kx-\omega t\right)$, what is “waving?” Based on the above discussion, the answer is a mathematical function that can, among other things, be used to determine where the particle is likely to be when a position measurement is performed. Second, how is the wave function used to make predictions? If it is necessary to find the probability that a particle will be found in a certain interval, square the wave function and integrate over the interval of interest. Soon, you will learn soon that the wave function can be used to make many other kinds of predictions, as well.

Complex Conjugates

Later in this section, you will see how to use the wave function to describe particles that are “free” or bound by forces to other particles. The specific form of the wave function depends on the details of the physical system. A peculiarity of quantum theory is that these functions are usually complex function**s**. A complex function is one that contains one or more imaginary numbers $\left(i=\sqrt{-1}\right)$. Experimental measurements produce real (nonimaginary) numbers only, so the above procedure to use the wave function must be slightly modified. In general, the probability that a particle is found in the narrow interval (*x*, *x + dx*) at time *t* is given by

Expectation Values

In classical mechanics, the solution to an equation of motion is a function of a measurable quantity, such as *x*(*t*), where *x* is the position and *t* is the time. Note that the particle has one value of position for any time *t*. In quantum mechanics, however, the solution to an equation of motion is a wave function, $\text{Ψ}\left(x,t\right).$ The particle has many values of position for any time *t*, and only the probability density of finding the particle, $|\text{Ψ}\left(x,t\right){|}^{2}$, can be known. The average value of position for a large number of particles with the same wave function is expected to be

The Heisenberg Uncertainty Principle

Abstract:
By the end of this section, you will be able to:

* Describe the physical meaning of the position-momentum uncertainty relation
* Explain the origins of the uncertainty principle in quantum theory
* Describe the physical meaning of the energy-time uncertainty relation

Momentum and Position

To illustrate the momentum-position uncertainty principle, consider a free particle that moves along the *x*-direction. The particle moves with a constant velocity *u* and momentum $p=mu$. According to de Broglie’s relations, $p=\hslash k$ and $E=\hslash \omega$. As discussed in the previous section, the wave function for this particle is given by

Energy and Time

Another kind of uncertainty principle concerns uncertainties in simultaneous measurements of the energy of a quantum state and its lifetime,

The Schrӧdinger Equation

Abstract:
By the end of this section, you will be able to:

* Describe the role Schrӧdinger’s equation plays in quantum mechanics
* Explain the difference between time-dependent and -independent Schrӧdinger’s equations
* Interpret the solutions of Schrӧdinger’s equation

The Quantum Particle in a Box

Abstract:
By the end of this section, you will be able to:

* Describe how to set up a boundary-value problem for the stationary Schrӧdinger equation
* Explain why the energy of a quantum particle in a box is quantized
* Describe the physical meaning of stationary solutions to Schrӧdinger’s equation and the connection of these solutions with time-dependent quantum states
* Explain the physical meaning of Bohr’s correspondence principle

The Quantum Harmonic Oscillator

Abstract:
By the end of this section, you will be able to:

* Describe the model of the quantum harmonic oscillator
* Identify differences between the classical and quantum models of the harmonic oscillator
* Explain physical situations where the classical and the quantum models coincide

The Classic Harmonic Oscillator

A simple harmonic oscillator is a particle or system that undergoes harmonic motion about an equilibrium position, such as an object with mass vibrating on a spring. In this section, we consider oscillations in one-dimension only. Suppose a mass moves back-and-forth along the

The Quantum Harmonic Oscillator

One problem with this classical formulation is that it is not general. We cannot use it, for example, to describe vibrations of diatomic molecules, where quantum effects are important. A first step toward a quantum formulation is to use the classical expression $k=m\text{ }{\omega }^{\text{ }2}$ to limit mention of a “spring” constant between the atoms. In this way the potential energy function can be written in a more general form,

The Quantum Tunneling of Particles through Potential Barriers

Abstract:
By the end of this section, you will be able to:

* Describe how a quantum particle may tunnel across a potential barrier
* Identify important physical parameters that affect the tunneling probability
* Identify the physical phenomena where quantum tunneling is observed
* Explain how quantum tunneling is utilized in modern technologies

Tunneling and Potential Energy

To illustrate quantum tunneling, consider a ball rolling along a surface with a kinetic energy of 100 J. As the ball rolls, it encounters a hill. The potential energy of the ball placed atop the hill is 10 J. Therefore, the ball (with 100 J of kinetic energy) easily rolls over the hill and continues on. In classical mechanics, the probability that the ball passes over the hill is exactly 1—it makes it over every time. If, however, the height of the hill is increased—a ball placed atop the hill has a potential energy of 200 J—the ball proceeds only part of the way up the hill, stops, and returns in the direction it came. The total energy of the ball is converted entirely into potential energy before it can reach the top of the hill. We do not expect, even after repeated attempts, for the 100-J ball to ever be found beyond the hill. Therefore, the probability that the ball passes over the hill is exactly 0, and probability it is turned back or “reflected” by the hill is exactly 1. The ball *never* makes it over the hill. The existence of the ball beyond the hill is an impossibility or “energetically forbidden.”

Tunneling and the Wave Function

Suppose a uniform and time-independent beam of electrons or other quantum particles with energy *E* traveling along the *x*-axis (in the positive direction to the right) encounters a potential barrier described by [link]. The question is: What is the probability that an individual particle in the beam will tunnel through the potential barrier? The answer can be found by solving the boundary-value problem for the time-independent Schrӧdinger equation for a particle in the beam. The general form of this equation is given by [link], which we reproduce here:

Radioactive Decay

In 1928, Gamow identified quantum tunneling as the mechanism responsible for the radioactive decay of atomic nuclei. He observed that some isotopes of thorium, uranium, and bismuth disintegrate by emitting $\text{α}$-particles (which are doubly ionized helium atoms or, simply speaking, helium nuclei). In the process of emitting an $\text{α}$-particle, the original nucleus is transformed into a new nucleus that has two fewer neutrons and two fewer protons than the original nucleus. The $\text{α}$-particles emitted by one isotope have approximately the same kinetic energies. When we look at variations of these energies among isotopes of various elements, the lowest kinetic energy is about 4 MeV and the highest is about 9 MeV, so these energies are of the same order of magnitude. This is about where the similarities between various isotopes end.

Field Emission

**Field emission** is a process of emitting electrons from conducting surfaces due to a strong external electric field that is applied in the direction normal to the surface ([link]). As we know from our study of electric fields in earlier chapters, an applied external electric field causes the electrons in a conductor to move to its surface and stay there as long as the present external field is not excessively strong. In this situation, we have a constant electric potential throughout the inside of the conductor, including its surface. In the language of potential energy, we say that an electron inside the conductor has a constant potential energy $U\left(x\right)=\text{−}{U}_{0}$ (here, the *x* means inside the conductor). In the situation represented in [link], where the external electric field is uniform and has magnitude ${E}_{g}$, if an electron happens to be outside the conductor at a distance *x* away from its surface, its potential energy would have to be $U\left(x\right)=\text{−}e{E}_{g}x$ (here, *x* denotes distance to the surface). Taking the origin at the surface, so that $x=0$ is the location of the surface, we can represent the potential energy of conduction electrons in a metal as the potential energy barrier shown in [link]. In the absence of the external field, the potential energy becomes a step barrier defined by $U\left(x\le 0\right)=\text{−}{U}_{0}$ and by $U\left(x>0\right)=0$.

Resonant Quantum Tunneling

Quantum tunneling has numerous applications in semiconductor devices such as electronic circuit components or integrated circuits that are designed at nanoscales; hence, the term ‘nanotechnology.’ For example, a diode (an electric-circuit element that causes an electron current in one direction to be different from the current in the opposite direction, when the polarity of the bias voltage is reversed) can be realized by a tunneling junction between two different types of semiconducting materials. In such a tunnel diode, electrons tunnel through a single potential barrier at a contact between two different semiconductors. At the junction, tunneling-electron current changes nonlinearly with the applied potential difference across the junction and may rapidly decrease as the bias voltage is increased. This is unlike the Ohm’s law behavior that we are familiar with in household circuits. This kind of rapid behavior (caused by quantum tunneling) is desirable in high-speed electronic devices.

Atomic Structure

Introduction

The Hydrogen Atom

Abstract:
By the end of this section, you will be able to:

* Describe the hydrogen atom in terms of wave function, probability density, total energy, and orbital angular momentum
* Identify the physical significance of each of the quantum numbers ($n,l,m$) of the hydrogen atom
* Distinguish between the Bohr and Schrödinger models of the atom
* Use quantum numbers to calculate important information about the hydrogen atom

Physical Significance of the Quantum Numbers

Each of the three quantum numbers of the hydrogen atom (*n*, *l*, *m*) is associated with a different physical quantity. The principal quantum number *n* is associated with the total energy of the electron, ${E}_{n}$. According to Schrödinger’s equation:

Using the Wave Function to Make Predictions

As we saw earlier, we can use quantum mechanics to make predictions about physical events by the use of probability statements. It is therefore proper to state, “An electron is located within this volume with this probability at this time,” but not, “An electron is located at the position (*x*, *y*, *z*) at this time.” To determine the probability of finding an electron in a hydrogen atom in a particular region of space, it is necessary to integrate the probability density $|{\psi }_{nlm}{|}^{2}$ over that region:

Orbital Magnetic Dipole Moment of the Electron

Abstract:
By the end of this section, you will be able to:

* Explain why the hydrogen atom has magnetic properties
* Explain why the energy levels of a hydrogen atom associated with orbital angular momentum are split by an external magnetic field
* Use quantum numbers to calculate the magnitude and direction of the orbital magnetic dipole moment of a hydrogen atom

Electron Spin

Abstract:
By the end of this section, you will be able to:

* Express the state of an electron in a hydrogen atom in terms of five quantum numbers
* Use quantum numbers to calculate the magnitude and direction of the spin and magnetic moment of an electron
* Explain the fine and hyperfine structure of the hydrogen spectrum in terms of magnetic interactions inside the hydrogen atom

The Exclusion Principle and the Periodic Table

Abstract:
By the end of this section, you will be able to:

* Explain the importance of Pauli’s exclusion principle to an understanding of atomic structure and molecular bonding
* Explain the structure of the periodic table in terms of the total energy, orbital angular momentum, and spin of individual electrons in an atom
* Describe the electron configuration of atoms in the periodic table

Atomic Spectra and X-rays

Abstract:
By the end of this section, you will be able to:

* Describe the absorption and emission of radiation in terms of atomic energy levels and energy differences
* Use quantum numbers to estimate the energy, frequency, and wavelength of photons produced by atomic transitions in multi-electron atoms
* Explain radiation concepts in the context of atomic fluorescence and X-rays

Atomic Fluorescence

Fluorescence occurs when an electron in an atom is excited several steps above the ground state by the absorption of a high-energy ultraviolet (UV) photon. Once excited, the electron “de-excites” in two ways. The electron can drop back to the ground state, emitting a photon of the same energy that excited it, or it can drop in a series of smaller steps, emitting several low-energy photons. Some of these photons may be in the visible range. Fluorescent dye in clothes can make colors seem brighter in sunlight by converting UV radiation into visible light. Fluorescent lights are more efficient in converting electrical energy into visible light than incandescent filaments (about four times as efficient). [link] shows a scorpion illuminated by a UV lamp. Proteins near the surface of the skin emit a characteristic blue light.

X-rays

The study of atomic energy transitions enables us to understand X-rays and X-ray technology. Like all electromagnetic radiation, X-rays are made of photons. X-ray photons are produced when electrons in the outermost shells of an atom drop to the inner shells. (Hydrogen atoms do not emit X-rays, because the electron energy levels are too closely spaced together to permit the emission of high-frequency radiation.) Transitions of this kind are normally forbidden because the lower states are already filled. However, if an inner shell has a vacancy (an inner electron is missing, perhaps from being knocked away by a high-speed electron), an electron from one of the outer shells can drop in energy to fill the vacancy. The energy gap for such a transition is relatively large, so wavelength of the radiated X-ray photon is relatively short.

X-ray Technology

X-rays have many applications, such as in medical diagnostics ([link]), inspection of luggage at airports ([link]), and even detection of cracks in crucial aircraft components. The most common X-ray images are due to shadows. Because X-ray photons have high energy, they penetrate materials that are opaque to visible light. The more energy an X-ray photon has, the more material it penetrates. The depth of penetration is related to the density of the material, as well as to the energy of the photon. The denser the material, the fewer X-ray photons get through and the darker the shadow. X-rays are effective at identifying bone breaks and tumors; however, overexposure to X-rays can damage cells in biological organisms.

Lasers

Abstract:
By the end of this section, you will be able to:

* Describe the physical processes necessary to produce laser light
* Explain the difference between coherent and incoherent light
* Describe the application of lasers to a CD and Blu-Ray player

Digital storage: CDs, DVDs, and Blu-Ray

For decades prior to the rise of integrated media devices (such as smartphones) and digital streaming, music and computer data was frequently stored and shared on a compact disc (CD). A CD is 6-inch diameter disc made of plastic that contains small “bumps” and “pits” nears its surface to encode digital or binary data ([link]). The bumps and pits appear along a very thin track that spirals outwards from the center of the disc. The width of the track is smaller than 1/20th the width of a human hair, and the heights of the bumps are even smaller yet.

Chirped Pulse Amplification: From Fusion to Eye Surgery

Lasers used in entertainment systems use little power and are not very intense. Uses such as cutting, drilling, and engraving require much more power. And applications such as laser-induced nuclear fusion require extraordinary he power and intensity. However, creating more intense lasers – on the order of gigawatts per square centimeter – is not simply a matter of finding better materials and increasing the power supply, because ultra-intense laser pulses have a fundamental complication: they can reduce the quality of the laser beam and even damage the laser device itself.

Condensed Matter Physics

Introduction

Types of Molecular Bonds

Abstract:
By the end of this section, you will be able to:

* Distinguish between the different types of molecular bonds
* Determine the dissociation energy of a molecule using the concepts ionization energy, electron affinity, and Coulomb force
* Describe covalent bonding in terms of exchange symmetry
* Explain the physical structure of a molecule in terms of the concept of hybridization

Types of Bonds

Chemical units form by many different kinds of chemical bonds. An ionic bond forms when an electron transfers from one atom to another. A covalent bond occurs when two or more atoms share electrons. A van der Waals bond occurs due to the attraction of charge-polarized molecules and is considerably weaker than ionic or covalent bonds. Many other types of bonding exist as well. Often, bonding occurs via more than one mechanism. The focus of this section is ionic and covalent bonding.

Bonding in Polyatomic Molecules

A polyatomic molecule is a molecule made of more than two atoms. Examples range from a simple water molecule to a complex protein molecule. The structures of these molecules can often be understood in terms of covalent bonding and hybridization. Hybridization is a change in the energy structure of an atom in which mixed states (states that can be written as a linear superposition of others) participate in bonding.

Molecular Spectra

Abstract:
By the end of this section, you will be able to:

* Use the concepts of vibrational and rotational energy to describe energy transitions in a diatomic molecule
* Explain key features of a vibrational-rotational energy spectrum of a diatomic molecule
* Estimate allowed energies of a rotating molecule
* Determine the equilibrium separation distance between atoms in a diatomic molecule from the vibrational-rotational absorption spectrum

Bonding in Crystalline Solids

Abstract:
By the end of this section, you will be able to:

* Describe the packing structures of common solids
* Explain the difference between bonding in a solid and in a molecule
* Determine the equilibrium separation distance given crystal properties
* Determine the dissociation energy of a salt given crystal properties

Ionic Bonding in Solids

Many solids form by ionic bonding. A prototypical example is the sodium chloride crystal, as we discussed earlier. Electrons transfer from sodium atoms to adjacent chlorine atoms, since the valence electrons in sodium are loosely bound and chlorine has a large electron affinity. The positively charged sodium ions and negatively charged chlorine (chloride) ions organize into an extended regular array of atoms ([link]).

Covalent Bonding in Solids

Crystals can also be formed by covalent bonding. For example, covalent bonds are responsible for holding carbon atoms together in diamond crystals. The electron configuration of the carbon atom is $1{s}^{2}2{s}^{2}2{p}^{2}$—a He core plus four valence electrons. This electron configuration is four electrons short of a full shell, so by sharing these four electrons with other carbon atoms in a covalent bond, the shells of all carbon atoms are filled. Diamond has a more complicated structure than most ionic crystals ([link]). Each carbon atom is the center of a regular tetrahedron, and the angle between the bonds is $110\text{°}.$ This angle is a direct consequence of the directionality of the *p* orbitals of carbon atoms.

Metallic Bonding in Solids

As the name implies, metallic bonding is responsible for the formation of metallic crystals. The valence electrons are essentially free of the atoms and are able to move relatively easily throughout the metallic crystal. Bonding is due to the attractive forces between the positive ions and the conduction electrons. Metallic bonds are weaker than ionic or covalent bonds, with dissociation energies in the range $1-3{\rule{0.2em}{0ex}}\text{eV}$.

Free Electron Model of Metals

Abstract:
By the end of this section, you will be able to:

* Describe the classical free electron model of metals in terms of the concept electron number density
* Explain the quantum free-electron model of metals in terms of Pauli’s exclusion principle
* Calculate the energy levels and energy-level spacing of a free electron in a metal

Band Theory of Solids

Abstract:
By the end of this section, you will be able to:

* Describe two main approaches to determining the energy levels of an electron in a crystal
* Explain the presence of energy bands and gaps in the energy structure of a crystal
* Explain why some materials are good conductors and others are good insulators
* Differentiate between an insulator and a semiconductor

Semiconductors and Doping

Abstract:
By the end of this section, you will be able to:

* Describe changes to the energy structure of a semiconductor due to doping
* Distinguish between an n-type and p-type semiconductor
* Describe the Hall effect and explain its significance
* Calculate the charge, drift velocity, and charge carrier number density of a semiconductor using information from a Hall effect experiment

Semiconductor Devices

Abstract:
By the end of this section, you will be able to:

* Describe what occurs when n- and p-type materials are joined together using the concept of diffusion and drift current (zero applied voltage)
* Explain the response of a p-n junction to a forward and reverse bias voltage
* Describe the function of a transistor in an electric circuit
* Use the concept of a p-n junction to explain its applications in audio amplifiers and computers

Diodes

Perhaps the simplest device that can be created with a semiconductor is a diode. A diode is a circuit element that allows electric current to flow in only one direction, like a one-way valve (see Model of Conduction in Metals). A diode is created by joining a *p*-type semiconductor to an *n*-type semiconductor ([link]). The junction between these materials is called a *p-n* junction. A comparison of the energy bands of a silicon-based diode is shown in [link](b). The positions of the valence and conduction bands are the same, but the impurity levels are quite different. When a *p-n* junction is formed, electrons from the conduction band of the *n*-type material diffuse to the *p*-side, where they combine with holes in the valence band. This migration of charge leaves positive ionized donor ions on the *n*-side and negative ionized acceptor ions on the *p*-side, producing a narrow double layer of charge at the *p*-*n* junction called the depletion layer. The electric field associated with the depletion layer prevents further diffusion. The potential energy for electrons across the *p-n* junction is given by [link].

Junction Transistor

If diodes are one-way valves, transistors are one-way valves that can be carefully opened and closed to control current. A special kind of transistor is a junction transistor. A junction transistor has three parts, including an *n*-type semiconductor, also called the emitter; a thin *p*-type semiconductor, which is the base; and another *n*-type semiconductor, called the collector ([link]). When a positive terminal is connected to the *p*-type layer (the base), a small current of electrons, called the base current ${I}_{B},$ flows to the terminal. This causes a large collector current ${I}_{c}$ to flow through the collector. The base current can be adjusted to control the large collector current. The current gain is therefore

Superconductivity

Abstract:
By the end of this section, you will be able to:

* Describe the main features of a superconductor
* Describe the BCS theory of superconductivity
* Determine the critical magnetic field for T = 0 K from magnetic field data
* Calculate the maximum emf or current for a wire to remain superconducting

Properties of Superconductors

In addition to zero electrical resistance, superconductors also have perfect diamagnetism. In other words, in the presence of an applied magnetic field, the net magnetic field within a superconductor is always zero ([link]). Therefore, any magnetic field lines that pass through a superconducting sample when it is in its normal state are expelled once the sample becomes superconducting. These are manifestations of the Meissner effect, which you learned about in the chapter on current and resistance.

Theory of Superconductors

A successful theory of superconductivity was developed in the 1950s by John Bardeen, Leon Cooper, and J. Robert Schrieffer, for which they received the Nobel Prize in 1972. This theory is known as the BCS theory. BCS theory is complex, so we summarize it qualitatively below.

Nuclear Physics

Introduction

Properties of Nuclei

Abstract:
By the end of this section, you will be able to:

* Describe the composition and size of an atomic nucleus
* Use a nuclear symbol to express the composition of an atomic nucleus
* Explain why the number of neutrons is greater than protons in heavy nuclei
* Calculate the atomic mass of an element given its isotopes

Counts of Nucleons

The number of protons in the nucleus is given by the atomic number, *Z*. The number of neutrons in the nucleus is the neutron number, *N*. The total number of nucleons is the mass number, *A*. These numbers are related by

Nuclear Size

The simplest model of the nucleus is a densely packed sphere of nucleons. The volume *V* of the nucleus is therefore proportional to the number of nucleons *A*, expressed by

Nuclear Binding Energy

Abstract:
By the end of this section, you will be able to:

* Calculate the mass defect and binding energy for a wide range of nuclei
* Use a graph of binding energy per nucleon (BEN) versus mass number $\left(A\right)$ graph to assess the relative stability of a nucleus
* Compare the binding energy of a nucleon in a nucleus to the ionization energy of an electron in an atom

Mass Defect

According to nuclear particle experiments, the total mass of a nucleus $\left({m}_{\text{nuc}}\right)$ is *less* than the sum of the masses of its constituent nucleons (protons and neutrons). The mass difference, or mass defect, is given by

Graph of Binding Energy per Nucleon

In nuclear physics, one of the most important experimental quantities is the binding energy per nucleon (BEN), which is defined by

Radioactive Decay

Abstract:
By the end of this section, you will be able to:

* Describe the decay of a radioactive substance in terms of its decay constant and half-life
* Use the radioactive decay law to estimate the age of a substance
* Explain the natural processes that allow the dating of living tissue using ${}^{14}\text{C}$

Radioactive Decay Law

When an individual nucleus transforms into another with the emission of radiation, the nucleus is said to decay. Radioactive decay occurs for all nuclei with $Z>82,$ and also for some unstable isotopes with $Z<83.$ The decay rate is proportional to the number of original (undecayed) nuclei *N* in a substance. The number of nuclei lost to decay, $\text{−}dN$ in time interval *dt*, is written

Radioactive Dating

**Radioactive dating** is a technique that uses naturally occurring radioactivity to determine the age of a material, such as a rock or an ancient artifact. The basic approach is to estimate the original number of nuclei in a material and the present number of nuclei in the material (after decay), and then use the known value of the decay constant $\lambda$ and [link] to calculate the total time of the decay, *t*.

Nuclear Reactions

Abstract:
By the end of this section, you will be able to:

* Describe and compare three types of nuclear radiation
* Use nuclear symbols to describe changes that occur during nuclear reactions
* Describe processes involved in the decay series of heavy elements

Alpha Decay

Heavy unstable nuclei emit $\alpha$ radiation. In $\alpha$-particle decay (or alpha decay), the nucleus loses two protons and two neutrons, so the atomic number decreases by two, whereas its mass number decreases by four. Before the decay, the nucleus is called the parent nucleus. The nucleus or nuclei produced in the decay are referred to as the daughter nucleus or daughter nuclei. We represent an $\alpha$ decay symbolically by

Beta Decay

In most $\beta$ particle decays (or beta decay), either an electron (${\beta }^{-}$) or positron (${\beta }^{+}$) is emitted by a nucleus. A positron has the same mass as the electron, but its charge is $\text{+}e$. For this reason, a positron is sometimes called an antielectron. How does $\beta$ decay occur? A possible explanation is the electron (positron) is confined to the nucleus prior to the decay and somehow escapes. To obtain a rough estimate of the escape energy, consider a simplified model of an electron trapped in a box (or in the terminology of quantum mechanics, a one-dimensional square well) that has the width of a typical nucleus (${10}^{-14}\text{m}$). According to the Heisenberg uncertainty principle in Quantum Mechanics, the uncertainty of the momentum of the electron is:

Gamma Decay

A nucleus in an excited state can decay to a lower-level state by the emission of a “gamma-ray” photon, and this is known as gamma decay. This is analogous to de-excitation of an atomic electron. Gamma decay is represented symbolically by

Radioactive Decay Series

Nuclei with $Z>82$ are unstable and decay naturally. Many of these nuclei have very short lifetimes, so they are not found in nature. Notable exceptions include ${}_{{\rule{0.5em}{0ex}}90}^{232}\text{T}\text{h}$ (or Th-232) with a half-life of $1.39{\rule{0.2em}{0ex}}×{\rule{0.2em}{0ex}}{10}^{10}$ years, and ${}_{{\rule{0.5em}{0ex}}92}^{238}\text{U}$ (or U-238) with a half-life of $7.04{\rule{0.2em}{0ex}}×{\rule{0.2em}{0ex}}{10}^{8}$ years. When a heavy nucleus decays to a lighter one, the lighter daughter nucleus can become the parent nucleus for the next decay, and so on. This process can produce a long series of nuclear decays called a decay series. The series ends with a stable nucleus.

Radioactivity in the Earth

According to geologists, if there were no heat source, Earth should have cooled to its present temperature in no more than 1 billion years. Yet, Earth is more than 4 billion years old. Why is Earth cooling so slowly? The answer is nuclear radioactivity, that is, high-energy particles produced in radioactive decays heat Earth from the inside ([link]).

Fission

Abstract:
By the end of this section, you will be able to:

* Describe the process of nuclear fission in terms of its product and reactants
* Calculate the energies of particles produced by a fission reaction
* Explain the fission concept in the context of fission bombs and nuclear reactions

The Atomic Bomb

The possibility of a chain reaction in uranium, with its extremely large energy release, led nuclear scientists to conceive of making a bomb—an atomic bomb. (These discoveries were taking place in the years just prior to the Second World War and many of the European physicists involved in these discoveries came from countries that were being overrun.) Natural uranium contains $99.3\text{%}$ U-238 and only $0.7\text{%}$ U-235, and does not produce a chain reaction. To produce a controlled, sustainable chain reaction, the percentage of U-235 must be increased to about $50\text{%}$. In addition, the uranium sample must be massive enough so a typical neutron is more likely to induce fission than it is to escape. The minimum mass needed for the chain reaction to occur is called the critical mass. When the critical mass reaches a point at which the chain reaction becomes self-sustaining, this is a condition known as criticality. The original design required two pieces of U-235 below the critical mass. When one piece in the form of a bullet is fired into the second piece, the critical mass is exceeded and a chain reaction is produced.

Fission Nuclear Reactors

The first nuclear reactor was built by Enrico Fermi on a squash court on the campus of the University of Chicago on December 2, 1942. The reactor itself contained U-238 enriched with $3.6\text{%}$ U-235. Neutrons produced by the chain reaction move too fast to initiate fission reactions. One way to slow them down is to enclose the entire reactor in a water bath under high pressure. The neutrons collide with the water molecules and are slowed enough to be used in the fission process. The slowed neutrons split more U-235 nuclei and a chain reaction occurs. The rate at which the chain reaction proceeds is controlled by a series of “control” rods made of cadmium inserted into the reactor. Cadmium is capable of absorbing a large number of neutrons without becoming unstable.

Nuclear Fusion

Abstract:
By the end of this section, you will be able to:

* Describe the process of nuclear fusion in terms of its product and reactants
* Calculate the energies of particles produced by a fusion reaction
* Explain the fission concept in the context of fusion bombs, the production of energy by the Sun, and nucleosynthesis

Nucleosynthesis

Scientist now believe that many heavy elements found on Earth and throughout the universe were originally synthesized by fusion within the hot cores of the stars. This process is known as nucleosynthesis. For example, in lighter stars, hydrogen combines to form helium through the proton-proton chain. Once the hydrogen fuel is exhausted, the star enters the next stage of its life and fuses helium. An example of a nuclear reaction chain that can occur is:

The Hydrogen Bomb

In 1942, Robert Oppenheimer suggested that the extremely high temperature of an atomic bomb could be used to trigger a fusion reaction between deuterium and tritium, thus producing a fusion (or hydrogen) bomb. The reaction between deuterium and tritium, both isotopes of hydrogen, is given by

The Fusion Reactor

The fusion chain believed to be the most practical for use in a nuclear fusion reactor is the following two-step process:

Medical Applications and Biological Effects of Nuclear Radiation

Abstract:
By the end of this section, you will be able to:

* Describe two medical uses of nuclear technology
* Explain the origin of biological effects due to nuclear radiation
* List common sources of radiation and their effects
* Estimate exposure for nuclear radiation using common dosage units

Medical Applications

Most medical and related applications of nuclear physics are driven, at their core, by the difference between a radioactive substance and a non-radioactive substance. One of the first such methods is the precision measurement and detection method known as radioimmunoassay (RIA). Developed by Rosalyn Sussman Yalow and Solomon Berson in the late 1950s, RIA relies on the principle of competitive binding. For the particular substance being measured, a sample containing a radioactive isotope is prepared. A known quantity of antibodies is then introduced. By measuring the amount of "unbound" antibodies after the reaction, technicians can detect and measure the precise amount of the target substance. Radioimmunoassay is essential in cancer screening, hepatitus diagnosis, narcotics investigation, and other analyses.

Biological Effects

Nuclear radiation can have both positive and negative effects on biological systems. However, it can also be used to treat and even cure cancer. How do we understand these effects? To answer this question, consider molecules within cells, particularly DNA molecules.

Sources of Radiation

Human are also exposed to many sources of nuclear radiation. A summary of average radiation doses for different sources by country is given in [link]. Earth emits radiation due to the isotopes of uranium, thorium, and potassium. Radiation levels from these sources depend on location and can vary by a factor of 10. Fertilizers contain isotopes of potassium and uranium, which we digest in the food we eat. Fertilizers have more than 3000 Bq/kg radioactivity, compared to just 66 Bq/kg for Carbon-14.

Particle Physics and Cosmology

Introduction

Introduction to Particle Physics

Abstract:
By the end of this section, you will be able to:

* Describe the four fundamental forces and what particles participate in them
* Identify and describe fermions and bosons
* Identify and describe the quark and lepton families
* Distinguish between particles and antiparticles, and describe their interactions

Four Fundamental Forces

An important step to answering these questions is to understand particles and their interactions. Particle interactions are expressed in terms of four fundamental force**s**. In order of decreasing strength, these forces are the strong nuclear force, the electromagnetic force, the weak nuclear force, and the gravitational force.

Classifications of Elementary Particles

A large number of subatomic particles exist in nature. These particles can be classified in two ways: the property of spin and participation in the four fundamental forces. Recall that the spin of a particle is analogous to the rotation of a macroscopic object about its own axis. These types of classification are described separately below.

Particles and Antiparticles

In the late 1920s, the special theory of relativity and quantum mechanics were combined into a relativistic quantum theory of the electron. A surprising result of this theory was the prediction of two energy states for each electron: One is associated with the electron, and the other is associated with another particle with the same mass of an electron but with a charge of ${\text{e}}^{+}$. This particle is called the antielectron or positron. The positron was discovered experimentally in the 1930s.

Particle Conservation Laws

Abstract:
By the end of this section, you will be able to:

* Distinguish three conservation laws: baryon number, lepton number, and strangeness
* Use rules to determine the total baryon number, lepton number, and strangeness of particles before and after a reaction
* Use baryon number, lepton number, and strangeness conservation to determine if particle reactions or decays occur

Baryon Number Conservation

No conservation law considered thus far prevents a neutron from decaying via a reaction such as

Lepton Number Conservation

Lepton number conservation states that the sum of lepton numbers before and after the interaction must be the same. There are three different lepton number**s**: the electron-lepton number ${L}_{e},$ the muon-lepton number ${L}_{\mu },$ and the tau-lepton number ${L}_{\tau }.$ In any interaction, each of these quantities must be conserved *separately*. For electrons and electron neutrinos, ${L}_{e}=1;$ for their antiparticles, ${L}_{e}=-1;$ all other particles have ${L}_{e}=0.$ Similarly, ${L}_{\mu }=1$ for muons and muon neutrinos, ${L}_{\mu }=-1$ for their antiparticles, and ${L}_{\mu }=0$ for all other particles. Finally, ${L}_{\tau }=1,-1$, or 0, depending on whether we have a tau or tau neutrino, their antiparticles, or any other particle, respectively. Lepton number conservation guarantees that the number of electrons and positrons in the universe stays relatively constant. (*Note:* The total lepton number is, as far as we know, conserved in nature. However, observations have shown variations of family lepton number (for example, ${L}_{e}\right)$ in a phenomenon called *neutrino oscillations.*)

Strangeness Conservation

In the late 1940s and early 1950s, cosmic-ray experiments revealed the existence of particles that had never been observed on Earth. These particles were produced in collisions of pions with protons or neutrons in the atmosphere. Their production and decay were unusual. They were produced in the strong nuclear interactions of pions and nucleons, and were therefore inferred to be hadrons; however, their decay was mediated by the much more slowly acting weak nuclear interaction. Their lifetimes were on the order of ${10}^{-10}$ to ${10}^{-8}{\rule{0.2em}{0ex}}\text{s},$ whereas a typical lifetime for a particle that decays via the strong nuclear reaction is ${10}^{-23}{\rule{0.2em}{0ex}}\text{s}.$ These particles were also unusual because they were always produced in pairs in the pion-nucleon collisions. For these reasons, these newly discovered particles were described as *strange*. The production and subsequent decay of a pair of strange particles is illustrated in [link] and follows the reaction

Quarks

Abstract:
By the end of this section, you will be able to:

* Compare and contrast the six known quarks
* Use quark composition of hadrons to determine the total charge of these particles
* Explain the primary evidence for the existence of quarks

Quark Combinations

As mentioned earlier, quarks bind together in groups of two or three to form hadrons. Baryons are formed from three quarks. Sample baryons, including quark content and properties, are given in [link]. Interestingly, the delta plus (${\text{Δ}}^{+}$) baryon is formed from the same three quarks as the proton, but the total spin of the particle is 3/2 rather than 1/2. Similarly, the mass of ${\text{Δ}}^{+}$ with spin 3/2 is 1.3 times the mass of the proton, and the delta zero (${\text{Δ}}^{0}$) baryon with a spin 3/2 is 1.3 times the neutron mass. Evidently, the energy associated with the spin (or angular momentum) of the particle contributes to its mass energy. It is also interesting that no baryons are believed to exist with top quarks, because top quarks decay too quickly to bind to the other quarks in their production.

Color

Quarks are fermions that obey Pauli’s exclusion principle, so it might be surprising to learn that three quarks can bind together within a nucleus. For example, how can two up quarks exist in the same small region of space within a proton? The solution is to invent a third new property to distinguish them. This property is called color, and it plays the same role in the strong nuclear interaction as charge does in electromagnetic interactions. For this reason, quark color is sometimes called “strong charge.”

Quark Confinement

The first strong evidence for the existence of quarks came from a series of experiments performed at the Stanford Linear Accelerator Center (SLAC) and at CERN around 1970. This experiment was designed to probe the structure of the proton, much like Rutherford studied structure inside the atom with his $\text{α}$-particle scattering experiments. Electrons were collided with protons with energy in excess of 20 GeV. At this energy, $E\approx pc$, so the de Broglie wavelength of an electron is

Particle Accelerators and Detectors

Abstract:
By the end of this section, you will be able to:

* Compare and contrast different types of particle accelerators
* Describe the purpose, components, and function of a typical colliding beam machine
* Explain the role of each type of subdetector of a typical multipurpose particle detector
* Use the curvature of a charge track to determine the momentum of a particle

Early Particle Accelerators

A particle accelerator is a machine designed to accelerate charged particles. This acceleration is usually achieved with strong electric fields, magnetic fields, or both. A simple example of a particle accelerator is the Van de Graaff accelerator (see Electric Potential). This type of accelerator collects charges on a hollow metal sphere using a moving belt. When the electrostatic potential difference of the sphere is sufficiently large, the field is used to accelerate particles through an evacuated tube. Energies produced by a Van de Graaff accelerator are not large enough to create new particles, but the machine was important for early exploration of the atomic nucleus.

Colliding Beam Machines

New particles can be created by colliding particles at high energies. According to Einstein’s mass-energy relation, the energies of the colliding particles are converted into mass energy of the created particle. The most efficient way to do this is with particle-colliding beam machines. A colliding beam machine creates two counter-rotating beams in a circular accelerator, stores the beams at constant energy, and then at the desired moment, focuses the beams on one another at the center of a sensitive detector.

Particle Detectors

The purpose of a particle detector is to accurately measure the outcome of collisions created by a particle accelerator. The detectors are multipurpose. In other words, the detector is divided into many subdetectors, each designed to measure a different aspect of the collision event. For example, one detector might be designed to measure photons and another might be designed to measure muons. To illustrate how subdetectors contribute to an understanding of an entire collision event, we describe the subdetectors of the Compact Muon Solenoid (CMS), which was used to discover the Higgs Boson at the LHC ([link]).

The Standard Model

Abstract:
By the end of this section, you will be able to:

* Describe the Standard Model in terms of the four fundamental forces and exchange particles
* Draw a Feynman diagram for a simple particle interaction
* Use Heisenberg’s uncertainty principle to determine the range of forces described by the Standard Model
* Explain the rationale behind grand unification theories

Introduction to the Standard Model

The Standard Model of particle interactions contains two ideas: *electroweak theory* and quantum chromodynamics (QCD) (the force acting between color charges). Electroweak theory unifies the theory of quantum electrodynamics (QED), the modern equivalent of classical electromagnetism, and the theory of weak nuclear interactions. The Standard Model combines the theory of relativity and quantum mechanics.

Electromagnetic Force

According to QED, the electromagnetic force is transmitted between charged particles through the exchange of photons. The theory is based on three basic processes: An electron travels from one place to the next, emits or absorbs a photon, and travels from one place to another again. When two electrons interact, one electron emits the photon and the other receives it ([link]). Photons transfer energy and momentum from one electron to the other. The net result in this case is a repulsive force. The photons exchanged are virtual. A virtual particle is a particle that exists for too short a time to be observable. Since the photon transit time $\text{Δ}t$ is extremely small, Heisenberg’s uncertainty principle states that the uncertainty in the photon’s energy, $\Delta E,$ may be very large.

Weak Nuclear Force

The weak nuclear force is responsible for radioactive decay. The range of the weak nuclear force is very short (only about ${10}^{-18}$ m) and like the other forces in the Standard Model, the weak force can be described in terms of particle exchange. (There is no simple function like the Coulomb force to describe these interactions.) The particle exchanged is one of three bosons: ${\text{W}}^{+},{\text{W}}^{\text{−}},$ and ${\text{Z}}^{0}$. The Standard Model predicts the existence of these spin-1 particles and also predicts their specific masses. In combination with previous experiments, the mass of the charged W bosons was predicted to be $81{\rule{0.2em}{0ex}}\text{GeV/}{c}^{2}$ and that of the ${\text{Z}}^{0}$ was predicted to be $90{\rule{0.2em}{0ex}}\text{GeV/}{c}^{2}$. A CERN experiment discovered particles in the 1980s with precisely these masses—an impressive victory for the model.

Strong Nuclear Force

Strong nuclear interactions describe interactions between quarks. Details of these interactions are described by QCD. According to this theory, quarks bind together by sending and receiving gluons. Just as quarks carry electric charge [either $\left(+2\text{/}3\right)e$ or $\left(-1\text{/}3\right)e\right]$ that determines the strength of electromagnetic interactions between the quarks, quarks also carry “color charge” (either red, blue, or green) that determines the strength of strong nuclear interactions. As discussed before, quarks bind together in groups in color neutral (or “white”) combinations, such as red-blue-green and red-antired.

Unification Theories

Physicists have long known that the strength of an interaction between particles depends on the distance of the interaction. For example, two positively charged particles experience a larger repulsive force at a short distance then at a long distance. In scattering experiments, the strength of an interaction depends on the energy of the interacting particle, since larger energy implies both closer and stronger interactions.

The Big Bang

Abstract:
By the end of this section, you will be able to:

* Explain the expansion of the universe in terms of a Hubble graph and cosmological redshift
* Describe the analogy between cosmological expansion and an expanding balloon
* Use Hubble’s law to make predictions about the measured speed of distant galaxies

Hubble’s Law

In 1929, Edwin Hubble published one of the most important discoveries in modern astronomy. Hubble discovered that (1) galaxies appear to move away from Earth and (2) the velocity of recession (*v*) is proportional to the distance (*d*) of the galaxy from Earth. Both *v* and *d* can be determined using stellar light spectra. A best fit to the sample illustrative data is given in [link]. (Hubble’s original plot had a considerable scatter but a general trend was still evident.)

Big Bang Model

Scientists who study the origin, evolution, and ultimate fate of the universe (cosmology) believe that the universe began in an explosion, called the Big Bang, approximately 13.7 billion years ago. This explosion was not an explosion of particles through space, like fireworks, but a rapid expansion of space itself. The distances and velocities of the outward-going stars and galaxies permit us to estimate when all matter in the universe was once together—at the beginning of time.

Structure and Dynamics of the Universe

At large scales, the universe is believed to be both isotropic and homogeneous. The universe is believed to isotropic because it appears to be the same in all directions, and homogeneous because it appears to be the same in all places. A universe that is isotropic and homogeneous is said to be smooth. The assumption of a smooth universe is supported by the Automated Plate Measurement Galaxy Survey conducted in the 1980s and 1900s ([link]). However, even before these data were collected, the assumption of a smooth universe was used by theorists to simplify models of the expansion of the universe. This assumption of a smooth universe is sometimes called the cosmological principle.

Evolution of the Early Universe

Abstract:
By the end of this section, you will be able to:

* Describe the evolution of the early universe in terms of the four fundamental forces
* Use the concept of gravitational lensing to explain astronomical phenomena
* Provide evidence of the Big Bang in terms of cosmic background radiation
* Distinguish between dark matter and dark energy

The Early Universe

Before the short period of cosmic inflation, cosmologists believe that all matter in the universe was squeezed into a space much smaller than an atom. Cosmologists further believe that the universe was extremely dense and hot, and interactions between particles were governed by a single force. In other words, the four fundamental forces (strong nuclear, electromagnetic, weak nuclear, and gravitational) merge into one at these energies ([link]). How and why this “unity” breaks down at lower energies is an important unsolved problem in physics.

Nucleosynthesis of Light Elements

The relative abundances of the light elements hydrogen, helium, lithium, and beryllium in the universe provide key evidence for the Big Bang. The data suggest that much of the helium in the universe is primordial. For instance, it turns out that that 25% of the matter in the universe is helium, which is too high an abundance and cannot be explained based on the production of helium in stars.

Cosmic Microwave Background Radiation

According to cosmological models, the Big Bang event should have left behind thermal radiation called the cosmic microwave background radiation (CMBR). The intensity of this radiation should follow the blackbody radiation curve (Photons and Matter Waves). Wien’s law states that the wavelength of the radiation at peak intensity is

Antimatter and Matter

We know from direct observation that antimatter is rare. Earth and the solar system are nearly pure matter, and most of the universe also seems dominated by matter. This is proven by the lack of annihilation radiation coming to us from space, particularly the relative absence of 0.511-MeV $\gamma$ rays created by the mutual annihilation of electrons and positrons. (Antimatter in nature is created in particle collisions and in ${\beta }^{+}$ decays, but only in small amounts that quickly annihilate, leaving almost pure matter surviving.)

Dark Matter and Dark Energy

In the 1970s, accomplished astronomer Vera Rubin began investigating the movement of spiral galaxies. She observed that their outermost reaches were rotating as quickly as their centers, which was not the predicted phenomenon. She calculated that the rotational velocity of galaxies should have been enough to cause them to fly apart, unless there was a significant discrepancy between their observable matter and their actual matter. This became known as the galaxy rotation problem. Rubin proposed that massive amounts of unseen, or dark, matter must be present in and around the galaxies for the rotation speeds to occur.

Units

Conversion Factors

Fundamental Constants

Astronomical Data

Mathematical Formulas

Chemistry

The Greek Alphabet