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Teaching Modern Physics

Modern Physics for Scientists and Engineers has grown out of the classesI have taught in modern physics at the University of Louisville. While ourphysics majors take the class after they have taken their first course in physicswith calculus and our physics department expects the course in modernphysics to be a survey course covering the areas of contemporary physics,most of my students are majors in electrical engineering. I have had manyconversations with our engineering school about the course and have tried asbest I can to help prepare engineering students for the upper division courseson lasers and semiconductor devices. I have also tried to give my studentsan overview of contemporary physics.

The introduction of this book begins with a review of the basic proper-ties of particles and waves from the vantage point of classical physics. Thisdescriptive material is intended to help students consolidate their under-standing of classical physics. The second section of the introduction providesa brief summary of the important ideas of the new quantum theory. Theintroduction leads to the description in Chapter 1 of a few key experimentsthat enable us to characterize the possible ways in which radiation interactswith matter. In the context of these important experiments, the principlesof wave mechanics are introduced for the first time in Chapters 2 & 3. Laterchapters of the book deal with particular fields of modern physics.

The subjects to be covered in the modern physics course must be carefullyselected and an effort must be made in the beginning of the course to help abroad majority of students appreciate the basic ideas of quantum mechanics.I usually choose a few chapters of the book I will not cover except for makinga few qualitative remarks and then choose other chapters that I will onlyexpect my students to know in a qualitative way. Typically, my students areexpected to have a thorough understanding and to be able to work problemsfor the first three chapters and the first section of Chapter 4, for Chapter 7 onBose-Einstein and Fermi-Dirac statistics, for Chapter 8 on condensed matterphysics, for Chapter 11 and the first two sections of Chapter 12 on relativitytheory, and the first five sections of Chapter 13 on particle physics. Thestudents might also be asked qualitative quiz questions on the second andthird sections of Chapter 4 and on Chapters 5, 6, 9, 10, and last the twosections of Chapter 12. Suitable quiz questions and problems can be foundat the end of each chapter. I give typically six quizzes and two tests duringthe course of the semester.

Copyright © 2010, Elsevier. All rights reserved.

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The classes in the Fall semester at the University of Louisville usuallybegins around the twentieth of August. The dates of the quizzes and testsduring the Fall semester of 2008 together with the chapters of the textbookfor which students were responsible are given in the following schedule.

Quiz/Test Date Chapters

Quiz 1 September 5 Introduction, 1

Quiz 2 September 19 2,3

Quiz 3 October 3 4,5

Test 1 October 10 1− 5

Quiz 4 October 24 7

Quiz 5 November 7 8

Quiz 6 November 21 11-12

Test 2 December 8 7,8,10-13

From the schedule, one can see that I have devoted one week to theintroduction and to each of the first five chapters. Two weeks have beendevoted to Chapter 7 and two weeks to Chapter 8. These two chapters areespecially important to students majoring in electrical engineering. As onecan easily confirm by looking at the table of contents of my book or readingthe relevant chapters, my book contains much more basic information onstatistical physics and condensed matter physics than any other book onmodern physics. About a month of my course is devoted to relativity theoryand particle physics. After using the ideas of quantum theory to study theproperties of atoms and solids, my students are often surprised and pleasedto study relativity theory and particle physics which gives them some sense ofthe enormous range of subjects included within contemporary physics. Manyof my engineering students have added physics as a minor or an additionalmajor at that point in the course. My quizzes usually have twenty qualitative


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questions worth five points each and my tests have problems worth five, ten,and twenty points. The questions at the end of the chapter are appropriatefor a quiz and many of the problems at the end of the chapter are appropriatefor twenty-point test problems. As a ten-point problem, I have asked studentsto give all of the LS-terms of an atom with a 3d2, 4f 2, or 3d 4f configuration.Another example of a ten-point problem would be to ask students to drawthe Feynman diagrams for two scattering processes.

I have always given my students the opportunity of getting extra pointsby doing projects outside of class and have allowed them to do these extraassignments working in groups of two and three. For my classes, I havegiven my students extra credit for finding mistakes in the manuscript of mybook, which has served as the text book for the course. I have also given extracredit for students showing me how the text might be improved. After twelveyears of collecting lists of corrections and suggestions from my students andpatiently working them into the manuscript, my book is now clearly andsimply written and virtually free of mistakes. Anoher kind of extra-creditwork is for students to work problems from sections not covered in class orto do projects with the Hartee-Fock or ABINIT applets. I have broughtmy students into a computer lab so that they can work with the appletstogether with me. An easier way of describing the Hartree-Fock applet is tomake transparencies of the figures shown in Section five of Chapter 5. Severalof these figures were obtained by doing a screen grab on a computer using theapplets. The Hartree-Fock applet described in Chapter 5 on many-electronatoms enables students to plot atomic wave functions and to calculate the sizeof the atom and the strength of the interaction between the electrons. TheABINIT applet discussed in Chapter 10 on semiconductor lasers is sufficientlyaccurate to provide a realistic description of the band gaps of semiconductors.Using the applet, a student can change the scale of the unit cell in any oneof the crystal directions and see how the band structure of a semiconductorchanges. I usually devote one class to demonstrating the applets.

Each of the remaining chapters of this instructors manual is devoted toa chapter of my book. I give the basic ideas of each chapter and describethe issues that are difficult for students and how one can best present thematerial in class. The manual reflects my own experience in teaching modernphysics for more than twenty years at the University of Louisville.


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Introduction

The purpose of the introduction is to review some concepts from classicalphysics that are important in modern physics and to give students a generalunderstanding of new ideas they will encounter in modern physics.

0.1 The Concepts of Particles and Waves

0.1.1 The variables of a moving particle

This first subsection of the introduction defines the following variables of aparticle

• position

• velocity

• momentum

• angular momentum

• potential and kinetic energy

My experience is that most students know the definition of the momen-tum and the kinetic energy of a particle but are unfamiliar with the generaldefinition of the potential energy. They know only that the potential energyof a particle with mass m at a height h in a uniform gravitational field ismgh. In the first subsection, the potential energy is defined and expressionsare derived for the potential energy of a simple harmonic operator and foran electron moving in the field of an atomic nucleus. The expression forthe potential energy of the oscillator is used in Chapters 2 and 3, while thepotential energy of an electron moving in the field of the nucleus is used inChapters 4 and 5.

0.1.2 Elementary properties of waves

The second subsection describes the elementary properties of waves. Thefollowing kinds of wave are considered

• traveling waves


0.2. AN OVERVIEW OF QUANTUM PHYSICS 5

• stationary waves

• standing waves

The mathematical representations of waves by trigonometric functions andby exponential functions are given, and the principle of superposition is usedto describe interference. The early chapters of this book have many examplesof wave motion.

0.2 An overview of quantum physics

The goal of this section is to introduce some of the ideas of modern physicsthat will play an important role in the first three chapters of the book.The idea of a wave-particle duality is introduced. Certain phenomena canbe understood by considering radiation or matter as consisting of particles,while other phenomena demand that we think of radiation or matter asconsisting of waves. The wave-particle duality leads naturally to a descriptionof phenomena in terms of probability. Rather than give a detailed descriptionof a particle at each instant of time, wave mechanics allows us to calculate theprobability that the variables associated with a particle fall into particularranges. Associate with a moving particle is a wave functions with the absolutevalue squared of the wave function at x being related to the probability offinding the particle in the interval between x and x+dx. The wave associatedwith a particle thus allows us to calculate the probability of finding theparticle in a particular region of space.

In addition to discussing some of the general features of quantum theory,the second section of the introduction describes the size of atoms and nucleiand gives some sense of the length of time for an atom or nucleus to make atransition from one state to another. The radius of the cloud surrounding thehydrogen nucleus is equal to a0 = 0.529 A or 0.529×10−10 m, and the diameterof the cloud is thus approximately one Angstrom or one tenth of a nanometer.For reasons given in the introduction, the size of atoms increases very slowlyas the number of electrons increases. Xenon, which has 56 electron, is onlytwo to three times larger than the helium atom which has two electrons.

Atoms generally decay from excited states to the ground state or readjustto changes in their environment in about a nanosecond, which is is very muchlonger than the time for electrons of an atom to circulate about the atomicnucleus. The coupling between atomic electrons and the outside world is


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usually sufficiently weak that an electron in an atom has to circulate aboutthe nucleus tens of millions of times before it makes a transition.

The basic unit of time for describing processes occurring in the nucleusis the time it would take a nucleon having a kinetic energy of 40 or 50 MeVto traverse a distance of 10−15 meters, which is the size of the nucleus. Thislength of time is about 10−22 seconds. Nuclear processes evolving over alonger period of time can be thought of as delayed processes.

There are four fundamental forces in nature: the electromagnetic force,the strong and weak forces, and the gravitational force. Scattering processesinvolving the strong force take place within 10−22 seconds, and processesinvolving the weaker electromagnetic force typically take place in 10−14 to10−20 seconds. The weak interaction is very much weaker than the elec-tromagnetic interaction. Processes that depend upon the weak interactiongenerally take between 10−8 and 10−13 seconds, which is much longer thanthe times associated with strong and electromagnetic processes. While theforce of gravity is very much smaller than the other forces, the gravitationalforce is always attractive and has an infinite range.


Chapter 1

The Wave-Particle Duality

We consider in this chapter a few key experiments that enable us to charac-terize the possible ways in which radiation interacts with matter. The firstsection describes experiments that can be understood by supposing that elec-tromagnetic radiation consists of packets of energy called photons, while thesecond section describes experiments that can be interpreted by supposingthat beams of electromagnetic radiation and particles consist of waves. Theexperiments described in this chapter taken together show that radiation andmatter have a dual particle-wave character.

1.1 The particle model of light

At the end of the nineteenth century, light was thought of as a form ofelectromagnetic waves. An appreciation of the particle-like nature of elec-tromagnetic radiation became apparent through the study of the followingphenomena

• the photoectric effect

• the absorption and emission of light by atoms

• the Compton effect

I generally devote two or three classes to the photoelectric effect and toatomic spectra. My students generally have an easy time understanding thematerial and the discussion of atomic transitions in this chapter prepares

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8 CHAPTER 1. THE WAVE-PARTICLE DUALITY

them for the quantum mechanical description of atomic physics in Chapters4 and 5.

To understand the photoelectric effect, a student needs to know that theenergy of the photon is equal to hf or hc/λ and the work function (W ) isthe minimum amount of energy necessary to free an electron from a metal.This leads to the equation

(KE)max =hc

λ−W .

The discussion of the hydrogen spectra in this chapter also depends uponthe formula for the energy levels of hydrogen

En = − 13.6 eV

n2.

Using this formula, one can easily calculate how the energy of the hydrogenatom changes when the atom makes a transition from one energy level toanother. Denoting the change in the energy by the atom by ∆E, and usingthe formula, E = hc/λ for the energy of the photon, the wave length of thelight emitted or absorbed by the atom can be written

λ =hc

∆E.

These same formulas are used in the second section of Chapter 4 to find thetransition wave lengths and frequencies.

The Compton effect played an important role in convincing the physicscommunity of the particle-like nature of electromagnetic radiation. Althoughthe Compton scattering formula given in this chapter can be derived usinga nonrelativisitic formalism, the derivation of the Compton formula in thisbook is given in Chapter 12 together with a description of other high-energyscattering events.

1.2 The wave model of radiation and matter

This section describes the interference that occurs when X-rays are scatteredby the lattice planes of a crystal, and the section also describes deBroglie’sremarkable suggestion that microscopic particles like electrons should havewave-like properties. The deBroglie formula,

λ =h

p,


1.2. THE WAVE MODEL OF RADIATION AND MATTER 9

plays an important role in our qualitative derivation of wave equations in thesecond chapter. The two-slit scattering experiment described near the endof this chapter is a decisive verification of deBroglie’s idea.


10 CHAPTER 1. THE WAVE-PARTICLE DUALITY


Chapter 2

The Schrodinger WaveEquation

The result of a scattering experiment involving free particles can be summa-rized by the exponential function,

ψ(x) = Aei kx,

which represents the amplitue of a wave associated with a particle and bythe Broglie relation

p =h

λ.

The de Broglie relation may be written in a more useful form by multi-plying and dividing the right-hand side of the equation by 2π giving

p = ~k,

where ~ is equal to h/2π. Using this form of the de Broglie relation, theexponetial function describing a free particle may be shown to satisfy thefirst-order differential equation

−i~dψdx

= pψ.

A second-order equation for the wave function of a free-particle can be ob-tained by taking the derivative of this last equation and using the followingexpression for the kinetic energy of a free particle

E =1

2mp2.

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12 CHAPTER 2. THE SCHRODINGER WAVE EQUATION

This leads to the differential equation

−~2

2m

d2ψ

dx2= E ψ,

which is known as the Schodinger equation of a free-particle.

The Schrodinger time-independent equation,

−~2

2m

d2ψ

dx2+ V (x)ψ = Eψ,

which is obtained by adding the potential energy V (x) to the free-particleequation, is used in this chapter to find the wave function and the energyof a particle moving in infinite and finite potential wells and to study thestates of the simple harmonic oscillator. In each case, the possible energiesof the particle correspond to those values of E for which there is a solutionof the Schrodinger time-independent equation that satisfies the boundaryconditions. The wave function, which is related to the probability of findingthe particle in a particular region in space, may be used to calculate theaverage value of a function f(x) using the formula

< f(x) >=

∫ b

a

f(x)|ψ(x)|2dx.

An equation describing the time evolution of the wave function can beobtained by considering a traveling wave. A wave moving in the positivex-direction is described by the function

ψ(x, t) = Aei (kx−ωt),

where ω is the angular frequency. This wave function may be shown to satisfythe equation

−~2

2m

∂2ψ(x, t)

∂x2= i~

∂ψ(x, t)

∂t.

The Schrodinger time-dependent equation can be obtained by adding a po-tential term to this equation giving[

−~2

2m

∂2

∂x2+ V (x, t)

]ψ(x, t) = i~

∂ψ(x, t)

∂t.


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For a particle moving in a constant potential, the solutions of the Schrodingertime-dependent equation are of the form

ψ(x, t) = uE(x)e−iωt ,

where uE(x) is a solution of the Schrodinger time-independent equation andω = E/~.


14 CHAPTER 2. THE SCHRODINGER WAVE EQUATION


Chapter 3

Operators and Waves

Particle variables such as the momentum and the energy are represented inquantum theory by operators. The operator corresponding to the momentumis

p = −i~ ddx

,

and the operators corresponding to other particle variables such as the energycan be obtained by writing the variable in terms of the momentum and thenreplacing the momentum with the momentum operator. The energy of aparticle can be written

E =1

2mp2 + V (x),

and the operator corresponding to the energy is obtained by replacing themomentum by the momentum operator

H =−~2

2m

d2

dx2+ V (x). (3.1)

Here the energy operator called the Hamiltonian is denoted by H.The possible results of measuring a variable can be obtained by forming

the eigenvalue equation for the operator corresponding to the variable

operator× function = constant× function.

The results of a measurement of the variable correspond to the values ofthe constant on the right-hand side of the eigenvalue equation for which theequation has a solution satisfying the boundary conditions, and the function

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16 CHAPTER 3. OPERATORS AND WAVES

is a wave function representing the particle when the variable has that value.The momentum and energy eigenvalue equations are given in Chapter 2 withthe momentum eigenvalue equation being eq. (2.6) and the energy eigenvalueequation being the Schrodinger time-independent equation (2.8).

The concept of an eigenvalue equation can appear abstract to a beginningstudent. I have found it helpful to ask students how they would go aboutdeciding whether a particular function corresponds to a state for which theparticle has a well-defined value of a particular variable. The way to answerthis question is to multiply the function by the operator corresponding tothe variable. If one gets a number times the function then the function doescorrespond to a state of the particle with a particular value of the variable,and the number one gets is just the value of the variable. Consider, forexample, the function cos kx. Multiplying the momentum operator timesthis function gives

−i~d cos kx

dx= i~k sin kx.

Since the product of the momentum operator and cos kx gives another func-tion of x, we conclude that function cos kx is not an eigenfunction of themomentum and, hence, does not represent a state of the particle correspond-ing to a definite value of the momentum. On the other hand, the product ofthe kinetic energy operator and the function coskx is

−~2

2m

d2coskx

dx2=

~2k2

2mcos kx.

The product of the kinetic energy operator and the function cos kx is equalto a number times cos kx. The function cos kx corresponds to a state of theparticle with the kinetic energy ~2k2/2m.

The wave function for particles incident upon barriers can be obtainedby setting up the Schrodinger equation in each region and requiring thatthe wave function and its derivative be continuous across each interface. Inthe region where the particles approach a barrier, there is a wave functionrepresenting the incident particles and another wave function representingthe reflected particles. The wave function on the opposite side of the barriercorresponds to the transmitted particles. Quantum theory makes the re-markable prediction that particles can pass through a barrier of finite widthwhen the energy of the incident particle is less than the height of the potentialbarrier. Particles can tunnel through the barrier.


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In the last section of the Chapter 3, the Heisenberg uncertainty prin-ciple is first obtained as Heisenberg suggested by reviewing critically theclassical concept of the measuring process. Because microscopic systems arecomparable in size to the smallest means available for measuring them, themeasurement of one variable of a microscopic system disturbs the values ofother variables of the system. This leads to an inherent uncertainty in theresults of measurements that cannot be over come. Using an idealized exper-iment in which a single photon is used to measure the position of an electron,Heisenberg obtained an expressions for the uncertainty of the momentum ofa particle produced by a measurement of the particle’s position. The productof the uncertainty in the position and the uncertainty in the x-component ofthe momentum satisfy the equation

∆x ·∆px ≥~2,

which is known as the Heisenberg Uncertainty Principle. Similar relationsapply to the other coordinates and the corresponding components of themomentum and to the time and the energy

∆t ·∆E ≥ ~2.

The Heisenberg principle can also be approached from another point ofview by considering functions formed by superimposing waves. The sim-plest function of this kind consists of a single plane wave represented by theexponential function

ψ(x) = Aeikx . (3.2)

This wave function corresponds to a particle having a definite value of themomentum, p = ~k; however, since the absolute value squared of this wavefunction is equal to |A|2, the particle has an equal probability of being foundanywhere along the x-axis. While the uncertainty of the momentum of theparticle is zero, the uncertainty of the position of the particle is infinite. Onecan obtain localized functions by combining waves with different values ofthe angular wave number k and hence different values of the momentum.The uncertainty in the position of the particle can be reduced in this waybut the uncertainty in the momentum increases, and one is led to the samerelation between the uncertainties of the position and the momentum of aparticle.


18 CHAPTER 3. OPERATORS AND WAVES

The idea that waves are associated with particles leads to the followinggeneral equation for the average value of an observable

< Q >=

∫ b

a

ψ∗(x)Qψ(x)dx,

where Q is an observable and Q is the corresponding operator. This lastformula applies to the position, as well as to the momentum and the energy.


Chapter 4

Hydrogen Atom

The wave function ψ(r) of the electron in a hydrogen-like ion satisfies theSchrodinger equation [

−~2

2m∇2 − 1

4πε0

Ze2

r

]ψ = Eψ,

where the Laplacian operator ∇2 is the natural generalization of the sec-ond derivatives to three dimensions, and Z is the nuclear charge. In polarcoordinates, the wave function is of the general form

ψ(r, θ, φ) =P (r)

rΘlml

(θ)Φml(φ) ,

where the radial part of the wave function is expressed as a function P (r)divided by r. The angular part of the wave function, Θlml

(θ)Φml(φ), is calleda spherical harmonic and denoted Ylml

(θ, φ).The probability that the electron is in a spherical shell with radius be-

tween r and r + dr isdP = P (r)2 dr.

The radial probability density, which is equal to the probability per unitinterval in the radial direction, is thus equal to the square of the functionP(r). The average value of a function f(r) can be evaluated by multiplyingthe value of the function at each value of r by the probability P (r)2 dr andintegrating from 0 to ∞ to obtain

< f(r) >=

∫ ∞0

f(r)P (r)2 dr.

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20 CHAPTER 4. HYDROGEN ATOM

The above equation for the average value of a function of r is very similarto the equation for the average value of a function of x given in Chapter 2.I often give problems to evaluate the average value of functions of r whichare similar to Example 2.3, in which the average value of a function of x iscalculated.

A modern physics teacher has to limit the amount of material he/shecovers in a single-semester course. I always cover the first three chapters ofmy book, and I cover the first section of Chapter 4; however, I only requirethat my students can answer quiz questions for the material covered in thesecond and third sections of Chapter 4. A student should have a qualitativeunderstanding of transition probabilities and know which atomic transitionscan occur. The number of transitions per second that atoms make from onelevel to another depends upon the Einstein coefficients, A21, B12 and B21.The number of transition per second that atoms make spontaneously fromthe energy level E2 to the energy level E1 is equal to A21 times the numberof atoms in the level E2. The number of atoms per second that absorb lightand make a transition from the energy level E1 to the level E2 is equal toB12N1ρ(f)df , where N1 is the number of atoms in level E1 and ρ(f) is thedensity of the radiation field. The number of atoms per second that makea stimulated transition from the level E2 to the level E1 is B21N2ρ(f)df .Selection rules can be formulated that determine which changes in the valuesof the quantum numbers, l and ml, can occur for transitions. The angularmomentum quantum number l changes by ±1 in an allowed transition. Theazimuthal quantum number ml is unchanged when light polarized in thez direction is emitted and ml changes by ±1 when x- and y− polarized lightis emitted.

The orbital and spin motions of the electron causes magnetic fields thatcan be described by the magnetic moments,

µl =−e2m

l and µs =−egs2m

s.

The spin-orbit interaction, which is the interaction of the magnetic momentassociated with the spin of the electron with the magnetic field due to therelative motion of the nucleus, leads to a fine splitting of the energy levels ofhydrogen.

In order to understand the splitting of energy levels due to the magneticinteractions, it is necessary to understand how angular momenta can com-bine to form a total angular momentum. For given values of the angular


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momentum quantum numbers j1 and j2, the quantum number J of the totalangular momentum has the values

J = j1 + j2, j1 + j2 − 1, . . . |j1 − j2|.

One adds and subtracts the two angular momentum quantum numbers andthen fills in between the maximum and minimum angular momenta withangular momentum values separated by one unit of angular momentum. Twospin angular momenta with j1 and j2 equal to one half can combine to forma total spin angular momentum with S equal to 1 or 0. An electron withspin s = 1/2 and orbital angular momentum l > 0 can have a total angularmomentum equal to j = l + 1/2 or j = l − 1/2.

Most of my students do not have difficulty understanding the rule I havejust given for combining angular momentum. What is very difficult for stu-dents at this level to understand is how one can take linear combinations ofthe product functions of two angular momenta to form states of the totalangular momentum. The reason this is difficult for students is that they donot understand what a vector space is. While my book has several examplesof the rule determining which total angular momenta can occur, the bookhas a single example (in Chapter 13) of how the states of two spins combineto form states of the total spin.

An external magnetic field further splits each level with quantum num-ber j into 2j + 1 sublevels having definite values of the quantum number m.The splitting of the energy levels of an atom due to an external magneticfield is called the Zeeman effect.


22 CHAPTER 4. HYDROGEN ATOM


Chapter 5

Many-Electron Atoms

The electrons in many-electron atoms move in a potential field due to thenucleus and the other electrons. The wave functions of the electrons in thiscomplex environment can only be obtained within the framework of someapproximation scheme. One approximation, which has proved to be veryuseful, is to assume that the electrons move in an average field due to the nu-cleus and the other electrons. This approximation is called the independent-particle model. Another useful approximation called the central-field modelis to assume that the potential field in which the electrons move is spheri-cally symmetric. The central-field model provides a theoretical basis for theatomic shell model and for the regularities observed in the chemical elements.

The wave function of an electron moving in a central field is of the generalform

ψ(r, θ, φ) =Pnl(r)

rYlml

(θ, φ)χms ,

where the radial part of the wave function is expressed as a function P (r)divided by r. The angular part of the wave function, Ylml

(θ, φ), is calleda spherical harmonic. The factor χms represents the spin part of the wavefunction.

In the central-field approximation, the energy of an atom depends uponthe n and l quantum numbers of electrons but does not depend upon theazimuthal quantum numbers ml and ms, which define the z-component ofthe angular momentum vectors and thus determine the orientation of theangular momentum vectors in space. As for hydrogen, we shall denote thel-values 0, 1, 2, 3, 4, . . . by the letters, s, p, d, f, g, . . . . For instance, the stateof an atom for which there are two electrons with the quantum numbers,

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24 CHAPTER 5. MANY-ELECTRON ATOMS

n = 1 and l = 0, and two electrons with the quantum numbers n = 2 andl = 0, will be denoted simply 1s22s2. The specification of the state of anatom by giving the number of electrons with each value of n and l is calledthe electronic configuration.

According to the Pauli exclusion principle, there can be only one electronin each single-electron state with the quantum numbers nlmlms. The single-electron wave functions with the same value of n are said to form a shelland the single-electron wave functions with particular values of n and l aresaid to form a subshell. Each subshell can contain no more than 2(2l + 1)electrons. The electron configuration of the lowest state, which is called theground configuration, typically consists of a number of closed subshells andat most one open subshell. This leads to the historically important building-up principle or Aufbau principle which was originally suggested by Bohr toexplain the periodic table. The order and number of electrons in each subshellis usually as follows

1s22s22p63s23p64s23d10, . . . .

The fact that the 4s subshell fills before the 3d can be understood by con-sidering the values of the angular momentum of the two states. The 4s statehas an angular momentum quantum number l = 0 and thus penetrates closerto the nucleus and is more tightly bound than the 3d state with angular mo-mentum quantum number l = 2.

Using the above order of filling of the subshells of the atom, one canreadily explain the appearance of the periodic table shown in Fig. 5.2. Forthe elements, H and He, the 1s subshell is being filled. The filling of the 2ssubshell corresponds to the elements Li and Be, while the filling of the 2psubshell corresponds to the elements B through Ne. For the elements, Naand Mg, the 3s subshell is being filled, while the elements Al through Arcorrespond to the filling of the 3p subshell. The transition elements for whichthe 3d subshell is being filled occur after the elements K and Ca for whichthe 4s subshell is filled. One may readily identify the ground configurationof light- and medium-weight elements using the order and the number ofelectrons for each shell given by the above sequence.

I usually devote two classes to cover the second and third sections ofChapter 4 and Chapter 5. These classes, which are entirely descriptive,are intended to give students a general understanding of atomic physics.I usually choose the 1s22s22p2 configuration of carbon and the lower-lying


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configurations of helium as examples of atomic configurations. The energylevels of 1s22s22p2 configuration of carbon are shown in Figures 5.4, and thelower-lying energy levels of helium are shown in Figure 5.5.

The 1s22s22p2 configuration of carbon has two electrons in the 1s, 2s, and2p shells. The non-central part of the Coulomb interaction between the elec-trons splits the 1s22s22p2 configuration of carbon into distinct energy levels.This splitting is due to the Coulomb interactions between the 2p electronswith the Coulomb interactions involving the filled shells of 1s and 2s electronsshifting all the states of the configuration by the same amount. According tothe rule given in Chapter 4 for combining angular momenta, the total spinangular momentum quantum number for the two spin one-half electrons isS = 0, 1 and the total orbital angular momentum for two p electrons, eachhaving one unit of angular momentum, is L = 2, 1, 0. However, for two elec-trons having the same values of n and l, the sum of S and L must be even.It is possible to show that only the states for which S +L is even satisfy thePauli exclusion principle. The two 2p electrons may thus have the followingLS values: (S=1,L=1), (S=0,L=2) and (S=0,L=0). These three states havevalues of S and L that satisfy the rule we have given for combining angularmomentum, and, in each case, S + L is even. The LS states of a config-uration are usually given in the spectroscopic notation in which one uses acapital letter to denote the total orbital angular momentum and gives thevalue of 2S + 1 as a raised prefix as follows: 2S+1L. The LS states of the np2

configuration are thus denoted 3P , 1D and 1S, respectively. These energylevels are shown in Fig. 5.4. The splitting of the lowest 3P levels is due tothe spin-orbit interaction.

The lowest-lying energy levels of helium are shown in Fig. 5.5. For thelowest excited states of helium, one electron is in the 1s state and the otherelectron is in a low-lying excited state. The total orbital angular of thestate is then equal to the orbital angular momentum of the excited electron.For these excited energy levels, the n and l values of the two electrons aredifferent, and the sum of S and L need not be even. As can be seen inFig. 5.5, each of the lower-lying excited configurations of helium has a tripletstate with S = 1 and a singlet state with S = 0.

The radial functions of many-electrons atoms can be obtained using theHartree-Fock applet described in Section 5 of this chapter. I always encouragestudents to work together in groups of two and three to do extra-creditprojects. The applet, which enable students to plot atomic wave functionsand to study the properties of atoms, provide many interesting opportunities


26 CHAPTER 5. MANY-ELECTRON ATOMS

for extra-credit projects.


Chapter 6

The Emergence of Masers andLasers

The idea that a microscopic system can be stimulated to emit radiationwas originally due to Einstein. After Einstein’s prediction, more than adecade passed before the occurrence of stimulated emission was confirmedby the research of A. Ladenburg and his collaborators in Berlin. Twentymore years passed before stimulated emission was used in practical devices toamplify beams of electromagnetic radiation. The first device using stimulatedemission, which operated in the microwave region of the spectrum, was builtby a research group at Columbia University under the direction of CharlesTownes.

The use of stimulated emission to amplify a beam of radiation dependsupon creating a population inversion in which more atoms are in a higher-lying level than in a lower level. Population inversions have been createdin in a number of innovative ways. In the first laser developed by TheodoreMaiman, the population inversion was created by pulses of light from a xenonflash-lamp. The population inversion in the popular helium-neon laser iscreated by collisions between neon and helium atoms.

The first two sections of this chapter gives an historical review of the de-velopment of lasers. These sections are clearly written at an elementary leveland can easily be understood by my students. With the time limitations ofa one-semester course on modern physics, I do not cover any of this materialin class but put a few qualitative questions about lasers on my tests. Mytests usually have eight qualitative questions worth five points each and threeproblems worth twenty points each.

27


28 CHAPTER 6. THE EMERGENCE OF MASERS AND LASERS

The third section of this chapter gives the basic idea of laser coolingexperiments in which beams of laser light are incident upon a cloud of atoms.The frequency of the laser light is just below an absorption maximum of theatoms. As atoms move toward the source of laser light, the Doppler effectshifts the frequency of the absorbed light in the direction of the absorptionmaximum and the amount of light absorbed by the atom increases. Themomentum of the photons absorbed by the atom slow its motion. In contrast,when atoms move away from the laser source, the Doppler effect shifts thefrequency of the laser light away from the absorption maximum and theamount of the light absorbed by the atoms decreases.

I discuss in class the basic idea of laser cooling and briefly describe modernexperiments in which clouds of atoms are routinely cooled to temperaturesin the milli-Kelvin range. At these temperatures, the atoms can be trappedin magnetic or optical fields where they can be further cooled by evaporativecooling to produce temperatures at which Bose-Einstein condensation canoccur. In a Bose condensate, a large number of atoms are in the groundstate with correlated motions. Our industry will surely use this effect for thedevelopment of new electrical devices.


Chapter 7

Statistical Physics

Statistical physics is based on the idea that the probability or statisticalweight of a particular macroscopic state depends upon the number of pos-sible ways the state can be constructed out of its microscopic constituents.The number of distinct ways a macroscopic state can be formed depends inturn upon whether or not the constituents are distinguishable. In Maxwell-Boltzmann’s statistics, the constituents are distinguishable, and the numberof ways a particular distribution of particles can be constructed depends uponthe number of ways the particles can be selected for each energy range andhow many different ways the particles can then be assigned to the microscopicstates in that range. The most probable distribution is obtained by findingthe values of the occupation numbers nr which maximize the expressions forthe total number of ways a distribution can be formed. This leads to theMaxwell-Boltzmann distribution law

nrgr

=N

Ze−εr/kBT ,

where nr is the number of particles with energy εr, and gr is the number ofmicroscopic states with energy εr. The partition function Z is

Z =∑r

gre−εr/kBT .

In teaching statistical physics, I first consider the two examples describedin the beginning of the chapter. The first example concerns flipping a coinfour times and the other example concerns the number of ways six gas

29


30 CHAPTER 7. STATISTICAL PHYSICS

molecules can be put in a container which has been divided into two com-partments. These two examples concern distinguishable particles, and theline of argument leads to the Maxwell-Boltzmann distribution law. I thenderive eq. (7.15) for the density of states and use the Maxwell-Boltzmann lawto obtain eq. (7.22) describing the distribution of velocities of molecules inan ideal gas and discuss the significance of Planck’s formula (7.28) for blackbody radiation. The Boltzmann distribution law and Planck’s formula havean important place in the development of modern physics.

Macroscopic system composed of indistinguishable particles are describedby two other kinds of statistics. Bose-Einstein statistics applies when the par-ticles do not satisfy the Pauli-exclusion principle and Fermi-Dirac statisticsapply when the particles do satisfy the Pauli-exclusion principle. An im-portant connection has been established between the intrinsic angular mo-mentum or spin of particles and the form of statistics they follow. Particleswith zero or integral spin follow Bose-Einstein statistics, while particles withhalf-integer spin follow Fermi-Dirac statistics. Bose-Einstein statistics canbe used to describe the radiation field within a black-body and the collectivephenomena associated with the condensation of a macroscopic system intoits lowest quantum state. Fermi-Dirac statistics enables us to understandhow the electronic characteristics of metals and semiconductors are relatedto the dynamical properties of their charge carriers.

The starting point for the derivation of distribution laws for indistinguish-able particles is the enumeration of the number of ways that each distributioncan occur. Since each assignment of the particles to the single-particle statesis equally likely to occur, the most probable distribution is the one thatcan occur in the most possible ways. As for the derivation of the Maxwell-Boltzmann distribution law, I begin by first considering a simple example. Iconsider the number of ways three bosons ( Pauli-Exclusion Principle doesnot apply ) can be assigned to two single-particle states. The possible as-signments of the three particles are given in Table 7.3

Each of the rows of Table 7.3 corresponds to a possible assignment of thethree particles to the two single-particle states. In the first assignment, threeparticles are assigned to the first state and no particles are assigned to thesecond state. In the illustrations on the right, the particles are representedby circles© and the partition between the two available states is representedby a triangle 4. Since three particles have been assigned to the first statein the assignment shown in the first row, the three circles in the first entryon the right are followed by a triangle. In the second assignment shown in


31

Table 7.3, two particles are assigned to the first state and one particle isassigned to the second state. This is represented in the illustration on theright by two circles followed by a triangle and a circle. The two circles standfor the two particles in the first state, while the triangle serves as a partitionbetween these two particles and the one particle in the second state.

Each of the entries in the second column in Table 7.3 have four characters:three circles corresponding to the particles and one triangle serving as apartition between the two states. The four possible assignments of the threeparticles to the two states correspond to the four possible ways of assigningthe one triangle to the four possible locations. These ideas can easily begeneralized to find the number of ways nr particles can be assigned to grsingle-particle states . In Fermi-Dirac statistics, the Pauli Exclusion Principleapplies and each state may either be singly occupied or empty. Hence thenumber of ways of assigning nr particles to gr states is equal to the numberof ways of selecting the nr occupied states from the gr states. These simplearguments are used to derive expressions for the total number of ways adistribution of particles satisfying Bose-Einstein and Fermi-Dirac statisticscan be obtained.

The most probable distribution for Bose-Einstein and Fermi-Dirac statis-tics is obtained by finding the values of the occupation numbers nr whichmaximize the expressions for the total number of ways a distribution canbe formed. This leads to the distribution law for Bose-Einstein statisticsrepresented by eq. (7.57) and the distribution law for Fermi-Dirac statisticsrepresented by eq. (7.58).

For my classes, which are composed mainly of electrical engineering stu-dents, I usually spend two class periods describing the properties of electronsin metals. The outer electrons of atoms in metals are not bound to individualatoms, but are free to move throughout the metal. These free electrons arecalled conduction electrons. Since electrons have a spin of one-half, the elec-trons in a metal are described by Fermi-Dirac statistics. Using the continuousvariable ε for the energy of the electrons, the Fermi-Dirac distribution lawcan be written

n(ε)dε = g(ε)dεf(ε)

where

f(ε) =1

e(ε− µ)/kBT + 1.

The parameter µ is known as the chemical potential, and the function f(ε)


32 CHAPTER 7. STATISTICAL PHYSICS

is called the Fermi-Dirac distribution function. The value of the chemicalpotential µ at absolute zero is called the Fermi energy εF . We recall thatin Fermi-Dirac statistics each single-particle state is either singly occupiedor empty. Hence, the function f(ε), which is equal to n(ε)dε/g(ε)dε, may beinterpreted to be the fraction of single-particle states that are occupied.

In the Fermi-Dirac distribution law, the factor g(ε)dε, which is the numberof single-electron states between ε and ε + dε, is given by eq. (7.69). Thefunction g(ε) is illustrated in Fig. 7.11(a). For temperatures near absolutezero, the distribution function f(ε) is equal to one for ε < εF and equal tozero for ε > εF . This function is illustrated in Fig. 7.11(b). The function n(ε),which is the product of the two functions represented in Figs. 7.11(a) and7.11(b), is illustrated in Fig. 7.11(c). At absolute zero, all of the states belowthe Fermi energy εF are occupied, and all of the states above the Fermi energyare unoccupied.

The Fermi distribution function f(ε) and the function n(ε) are illustratedin Figs. 7.12(a) and 7.12(b) for a temperature such that kBT = 0.1µ. Thecorresponding curves for T = 0 K are indicated by dashed lines in thesefigures. For temperatures significantly above absolute zero, single-particlestates with energy ε < µ are unoccupied and states with energy ε > µ areoccupied. The statistical arguments used in this chapter play an importantrole in the theory of charge carriers in semiconductors described in Chapter 9.


Chapter 8

Electronic Structure of Solids

This chapter on condensed matter physics begins by describing the periodicstructures that underlie most solids and have a decisive influence upon theirelectrical and optical qualities. Most metals and semiconductors have crystallattices corresponding to a few commonly occurring crystal structures.

The points in a crystal lattice can be specified by the formula

l = l1a1 + l2a2 + l3a3,

where l1, l2 and l3 are integers. By choosing the integers l1 , l2, and l3associated with the vectors a1, a2 , and a3, we can identify any point in thelattice. The vectors, a1, a2 and a3, are called primitive vectors. All of thelattice points can be generated by forming integral multiples of these vectors.

A simple cubic lattice, which has points arranged at the vertices of cubes,is shown in Fig. 8.2. The primitive vectors for this lattice are three perpendic-ular vectors of equal length. Another cubic lattice, which occurs frequently,is the body-centered cubic structure formed from the simple cubic lattice byadding a single additional lattice point to the center of each cube. Twochoices of the primitive vectors of the body-centered lattice are illustratedin Fig. 8.4. Most semiconductors crystalize in yet another crystal structureknown as the face-centered cubic structure, which is shown in Fig. 8.6(a).This lattice can be formed from the simple cubic lattice by adding an addi-tional point to each square face. The primitive vectors of the face-centeredcubic lattice are illustrated in Fig. 8.6(b). In addition to these cubic lattices,we also discuss the diamond and hexagonal close-packed structures.

All of the lattices we have just mentioned can be generated by addingintegral multiples of the primitive vectors of the crystal lattice to the points

33


34 CHAPTER 8. ELECTRONIC STRUCTURE OF SOLIDS

in the unit cell. A region of space that entirely fills all of space when it istranslated in this way is called a primitive unit cell. The primitive cell that isused most widely is called the Wigner-Seitz cell. This cell is constructed bydrawing perpendicular planes bisecting the lines joining the chosen center tothe equivalent lattice sites as illustrated in two dimensions in Fig. 8.7(b). TheWigner-Seitz cell entirely fills space when it is translated by the vectors ofthe crystal lattice, and it has the symmetry of the lattice. An illustration ofthe Wigner-Seitz cell for a body-centered cubic lattice is given in Fig. 8.8(a).

The electrons in a crystal are nearly free of their environment and canbe approximated by free-electron wave functions. A free particle moving ina one-dimensional periodic structure is described by the wave function

ψ(x) = Aeikx,

where k is related to the momentum of the particle by the equation

p = ~k.

The requirement that the wave function ψ(x) has the periodicity of the latticecan be satisfied by imposing the periodic boundary condition

ψ(a) = ψ(0),

where a is the distance between neighboring sites of the lattice. Substitutingthe free-electron wave function into the above gives

Aeika = A.

This condition leads to the requirement

k = n2π

a.

The free-electron wave function thus has the periodicity of the lattice ifthe wave vector k belongs to the set of values given by this last equation.The distance between two adjacent values of k is 2π times the inverse ofthe spacing in the original lattice. For this reason, the array of acceptablevalues of k is referred to as the reciprocal lattice. A one-dimensional latticeand the corresponding reciprocal lattice are illustrated in Fig. 8.12. Theoriginal crystal lattice is often referred to as the direct lattice. Using thisterminology, the free electron wave function is periodic in the direct lattice if


35

the value of k to which it corresponds is a member of the reciprocal lattice.In Fig. 8.12(b), the Wigner-Seitz cell is also shown. The Wigner-Seitz cell inreciprocal lattice space is called the Brillouin zone.

These ideas can readily be generalized to three dimensions. The vectorsof the reciprocal lattice can be written

g = n1b1 + n2b2 + n3b3,

where b1, b2, and b3 are the primitive vectors of the reciprocal lattice andn1, n2 and n3 are integers. The primitive vectors of the reciprocal lattice arerelated to the primitive vectors of the direct lattice by eqs. (8.18) - (8.20).As in the one-dimensional case, the free electron wave function

ψk(r) = Aeik · r

is periodic in the original lattice provided that the wave vector k is a memberof the reciprocal lattice. The reciprocal lattice of the simple cubic lattice isitself a cubic lattice, while the reciprocal lattice of the body-centered cubiclattice is a face-centered cubic lattice, and the reciprocal lattice of the face-centered cubic lattice is a body-centered cubic lattice. The unit cell of thereciprocal lattice is called the Brillouin zone.

The vectors of the reciprocal lattice make it easier to identify groups oflattice planes. Each plane determined by three points in the direct lattice isnormal to a vector of the reciprocal lattice, and, conversely, each vector ofthe reciprocal lattice is normal to lattice planes of the direct lattice. The cor-respondence between families of lattice planes and reciprocal lattice vectorsprovides the basis for the conventional way of describing lattice planes.

The potential energy of an electron in a crystal has the periodicity of thelattice itself. According to Bloch’s theorem, each wave function, which is asolution of the Schrodinger equation for an electron in a periodic potential,has an associated wave vector k that determines how the wave functionis affected by spatial translations. Denoting a solution of the Schrodingerequation by the value of k to which it corresponds, the condition imposedby Bloch’s theorem can be written

ψk(r + l) = eik · lψk(r).

The wave function in a cell specified by the lattice vector l differs from thewave function in the Wigner-Seitz cell near the origin by a phase factorexp(ik · l).


36 CHAPTER 8. ELECTRONIC STRUCTURE OF SOLIDS

The electron wave function is entirely determined by the behavior of the func-tion in the unit cell about the origin.

A study of the scattering of electrons by a crystal shows that elastic scat-tering will occur when the wave vector of the electrons k lies on the boundaryof the Brillouin zone. The wave vector of the scattered electrons k− g isthen the mirror image of k across the zone boundary. The reciprocal latticevector g, which is perpendicular to the lattice planes causing the scatter-ing, the wave vector of the incident electrons k, and the wave vector of thescattered electrons k− g are shown in Fig. 8.22.

At the zone boundary, the electrostatic field of the crystal mixes the twofree-electron states with wave vectors k and k− g into two other states withenergy E+ and E−. The separation between these two energy values, whichis known as a band gap, is shown in Fig. 8.23(b).

Knowledge of the band gap enables us to classify solids as insulators,semiconductors, or metals. If the band gap is large, a prohibitively largeamount of energy must be supplied to the electrons to carry them up over thegap into the next band. The solid is then an insulator. For semiconductors,the band gap is smaller and electrons can be excited by thermal fluctuationsup into the conduction band where they can easily carry a current. Metalshave current carrying states available just above the top of the occupiedlevels.

Solids can also be classified according to the way in which the atomsare bound together to their neighbors. We distinguish five different typesof bonding: covalent bonding, ionic bonding, molecular bonding, hydrogenbonding, and metallic bonding.


Chapter 9

Charge Carriers inSemiconductors

The distinctive property of semiconductors is that they have small bandgaps separating the valence and conduction bands. In these materials, arelatively small number of electrons are excited thermally from the valenceband into the conduction band creating electron-hole pairs that can conducta current. A perfect semiconductor with no impurities or lattice defects iscalled an intrinsic semiconductor. The generation of a hole in an intrinsicsemiconductor is accomplished by breaking a covalent bond of the crystalproducing a free electron and a vacancy in the lattice. Since free electronsand holes are created in pairs, the number of free electrons in the conductionband is always equal to the number of holes in the valence band.

The total density of free electrons and holes are calculated in the firstsection of this chapter using the Fermi-Dirac theory used in the seventh sec-tion of Chapter 7 to describe the number of conduction electrons in a metal.The integrals that arise in this way are evaluated using approximations thatare valid for intrinsic semiconductors for which the Fermi energy occurs nearthe middle of the gap between the valence and conduction bands.

The density of free electrons and holes in a semiconductor can be modifiedby adding impurities to the crystal. A semiconductor having impurities thatcontribute electrons to the conduction band is said to be n-doped or to havedonor atoms, while a semiconductor having impurities that contribute holesto the valence band is said to be p-doped or to have acceptor atoms. TheFermi energy of n-doped semiconductors lies typically just below the bottomof the conduction band, while the Fermi energy of p-doped semiconductors

37


38 CHAPTER 9. CHARGE CARRIERS IN SEMICONDUCTORS

lies just above the valence band.A p-n junction is formed by depositing a material with either donors or

acceptors upon another material with the opposite kind of doping. Interfacesproduced in this way have a preferred direction of current and can be used toproduce rectifiers. Transistors with two p− n junctions are commonly usedto produce amplifiers. The amplification of a signal by a transistor dependsupon the fact that the voltage across two terminals (emitter and collector),which serve as the output terminals, is very sensitive to the voltage acrosstwo other terminals (base and emitter), which serve as the input terminals.In contrast to bipolar transistors in which both the majority and minoritycharge carriers are responsible for current flow, the JFET and MOSFET aremajority carrier devices. The JFET and MOSFET function like variableresistors with the effective resistance of the device being determined by thegate that restricts the number of carriers in the channel.

In the one-semester course I teach on modern physics, I do not usuallyhave time to discuss either Chapter 9 or Chapter 10 in class, but put qual-itative questions from these chapters on my tests. My students usually findthe material in these chapters easier to understand and have an easier timeanswering questions about semiconductors on my tests than they do withother material that I think myself is much easier.


Chapter 10

Semiconductor Lasers

The introductory portion of this chapter considers the band structure ofsemiconductors and the discontinuities that occur at the interface betweentwo semiconductors. Quantum wells and barriers are then considered. Thesesubjects provide a background for the last section on semiconductor lasers.

The chapter begins by describing the band structure of Si, Ge, GaAS,and AlAs. The conduction bands of these materials have positive values ofthe energy, and the valence bands correspond to negative energies. For allthese semiconductors, the maxima of the valence bands occurs at the sym-metry point denoted by Γ. GaAs is one of only a few semiconductors forwhich the minimum in the conduction band also occurs at Γ. This is impor-tant since the wave vector of a photon of emitted or absorbed light is tinycompared to the size of the Brillouin zone. For this reason, the momentumof an electron involved in an optical transition changes very little and onlyvertical transitions are allowed. Fig. 10.4(a) shows a vertical transition for asemiconductor such as GaAs, for which the lowest minimum in the conduc-tion band occurs at Γ. Most light-emitting devices are composed of materialssuch as GaAs or InP for which vertical transitions between the valence andconduction bands can occur.

Many semiconductor devices consist of multilayered crystals grown on acommon substrate. For these materials – called heterostructures – to be freeof defects, the different layers must have the same crystal structure and theseparation between neighboring atoms in each layer must be similar. Thelattice constant (a) of a number of semiconductors is plotted against theirband gap (Eg) in Fig. 10.5. In this figure, one can see that GaAs and AlAshave lattice constants that are very similar and hence heterostructures can

39


40 CHAPTER 10. SEMICONDUCTOR LASERS

be formed from these two semiconductors. The two semiconductors, GaSband AlSb, are also very similar. The range of possible heterostructures canbe increased enormously by using alloys of the various compounds.

After describing different possible multi-layer surfaces, we describe exper-iments and theoretical calculations used to study the discontinuous changesthat occurs in valence and conduction bands across the interface betweentwo semiconductors. Discontinuities of this kind make it possible to createquantum wells and barriers. The design of multilayer structures with desireddiscontinuities at the interfaces and hence wells of desired depths and barrierswith desired heights is called band engineering.

Our treatment of quantum wells in Section 4 follows along the lines of ourtreatment of finite wells in Chapter 2. Within a heterostructure, electrons areconfined within a finite well in the direction perpendicular to the layers of theheterostructure while they are free to move in the other two directions. Theenergy levels of electrons in a quantum well can be denoted by a quantumnumber n. For each value of n, the energy depends upon a k-vector that is thesame as it would be for a two-dimensional electron gas. A new feature, whichmust be taken into account for electrons in a solid state environment, is thatthe effective mass of electrons can differ considerably from the mass electronshave in free space. Because the effective mass is different for electrons in thewell and in the barrier forming the walls of the well, the condition that thederivative of the wave function be continuous must be replaced by the moregeneral condition that the probability current j(z) is continuous across theedges of the well.

In Section 5, we consider the scattering of electrons by the potentialstep shown in Fig. 10.16. The treatment in this chapter follows along thelines of our treatment of the scattering by a potential step in Chapter 3;however, our treatment now must be more general. In Chapter 3, we wereconcerned with scattering problems for which an electron is incident upon abarrier and is either transmitted or reflected, and, for that reason, we omittedthe term corresponding to an electron approaching the potential step fromthe right. Since our interests now include a broader range of phenomenaincluding multiple reflection processes, the expression for the wave functionto the right of the potential step must have terms representing plane wavesmoving in both directions.

The vector with components, A1 and B1 representing the incident andreflected waves to the left of the step is set equal to the T -matrix timesthe vector with components, A2 and B2, representing the transmitted and


41

incident waves to the right of the step. The advantage of expressing therelation between the amplitudes to the left and right of the step in matrixform is that the transmission and reflection coefficients for complex systemscan then be calculated by multiplying the matrices for the individual parts.

For the scattering process in which electrons strike the square barriershown in Fig. 10.17, the T -matrix for the entire process is equal to the productof the T -matrix corresponding to scattering of the wave by the left edge ofthe barrier, the T -matrix for the translation of the wave through the region inwhich the barrier is located, and the T -matrix corresponding to the scatteringof the wave by the right edge of the barrier. Fig. 10.18 shows the transmissioncoefficient T (E) as a function of the energy E for a square potential barrierof height V0 = 0.3 eV and thickness 10 nm in GaAs. The dashed line showsthe classical result for a barrier of the same height.

Section 6 of this chapter describes the reflection and transmission of lightat the interface shown in Fig. 10.19. The relation between the transmissionand reflection amplitudes and components of the T -matrix given in the pre-vious section is used to obtain an expression for the T -matrix of the light interms of reflection and transmission coefficients. The expressions for the ele-ments of the T -matrix are then simplified using the known relations betweenthe reflection and transmission coefficients of light in terms of the indicesof refraction of the two media to obtain eq. (10.41). The T -matrix for thereflection and transmission of light at an interface and the T -matrix for thetransmission of light across a cavity is then used to calculate an expressionfor the power transmitted through a Fabry-Perot cavity. The power outputof a Fabry-Perot cavity in terms of the angular wave number of the light βand the length L of the cavity is shown in Fig. 10.20.

The last section of this chapter provides a phenomenological description ofdiode lasers. General arguments are used to derive a coupled set of equationsfor the density of charge carriers and photons in the cavity of a semiconductorlaser, and these equations are then used to see how the density of chargecarriers N and the gain g change as the laser approaches threshold. Asshown in Fig. 10.26, the density of charge carriers and the gain increases asthe laser approaches threshold and then levels off, while the output power ofthe laser increases linearly above threshold. In the reservoir analogy shownin Figures 10.23 and 10.25, threshold corresponds to the point at which thewater in the reservoir reaches the spillway. The density of charge carries,which corresponds to the water level, remains constant beyond that pointand all additional power is converted into laser light.


42 CHAPTER 10. SEMICONDUCTOR LASERS

I have never been able to devote class time to this chapter in my courseon modern physics. Instead, I have given qualitative questions from thechapter on my tests and have been generally pleased by my students abilityto understand the qualitative features of semiconductor lasers.

In considering the subjects covered in the chapter, one can see that thechapter covers four distinct but interrelated stories. The first story is howthe potential energy of charge carriers changes discontinuously at the inter-face between two semiconductors. Experimental and theoretical methodsfor determining the discontinuities that occur at interfaces are described aswell as the possibility of producing alloys of the semiconductors having theshifts one wants – so-called band engineering. The second story concerns thequantum wells that can be formed between layers of semiconductor materials.The quantum wells in semiconductor heterostructures have features that aresimilar to the finite wells considered in Chapter 2. However the electrons inheterostructures are only bound in the direction perpendicular to the inter-faces but free to move in the other two directions. Also, the effective mass ofthe charge carriers is different in the well than it is in the barriers that formthe walls of the well and one has to replace the condition that the derivativeof the wave function be continuous with the condition that the probabilitycurrent is continuous. The third story concerns the reflection and transmis-sion of electrons across a potential barrier and the transmission and reflectionof light at the interface between two materials. For each of these problems,we have defined T -matrices and calculated the effect of complex scatteringevents by evaluating the product of the corresponding matrices. Some lasershave reflective grooves at regular intervals along the cavity while others havemultilayered semiconductor mirrors at the ends of the cavity. The last storyis a phenomenological description of an operating laser. The charge carriersand the photons in the active region of the laser satisfy equilibrium condi-tions which change as the current of the laser increases. The density of chargecarriers and the gain of the laser increases as the laser approaches thresholdand then becomes constant.

I think it would probably be unwise for a teacher of a one-semester courseon modern physics to try to follow the threads of all four of these stories,but I think a teacher could very well cover one or two of these themes withtwo classes for each topic. If I might be allowed to use a musical analogy,students at this level are more likely to appreciate a harpsichord recital ofmusic by Vivaldi than one of Beethoven’s symphonies.


Chapter 11

Relativity I

The first chapter on relativity theory begins by considering two frames ofreference moving with respect to each other. We discuss the Galilean trans-formations, which were used in the time before relativity theory to describethe relation between the coordinates of two frames of reference. These trans-formations are shown to be inconsistent with the electro-magnetic theory ofMaxwell and the experiments of Michelson and Morley which failed to findany connection between the velocity of light and the motion of the Earth.

Einstein accepted the experiments of Michelson and Morley as decisiveevidence that the laws of electrodynamics as well as the laws of mechanicsare the same in every inertial frame of reference. In a paper published in1905, Einstein proposed two postulates that form the basis of the specialtheory of relativity

Postulate 1. The laws of physics are the same in all inertial frames of refer-ence.

Postulate 2. The speed of light in a vacuum is equal to the value c, indepen-dent of the motion of the source.

In my first class on relativity, I discuss these ideas and then write downthe Lorentz transformations without going through the derivation given inthe book. The Lorentz transformations (11.15) relate the time and the spacecoordinates of an event in the moving frame S ′ to the time and space coor-dinates of the event in the frame S. Suppose that at time t an event occursin the reference frame S at a point with coordinates (x, y, z). The transfor-mation equations (11.15) gives the time t′ and space coordinates (x′, y′, t′) of

43


44 CHAPTER 11. RELATIVITY I

the event in the frame S ′ moving with a constant velocity with respect to S.The formulas for Lorentz contraction and time dilation are then obtained.

The Lorentz transformation (11.15) is used to obtain the Lorentz contractionformula (11.23) because we want to record the position of two points of amoving object at the same time t in the frame S, while the inverse Lorentztransformation is used to obtain the time dilation formula (11.26) since themoving object will have the same space coordinate x′ in the frame in whichit is at rest. I like to work out Example 11.3 in class because it describes acalculation of an effect in two frames of reference obtaining equivalent results.Of course, a basic idea of relativity theory is that frames of reference movingwith a constant velocity with respect to each other are entirely equivalent.

Although a Lorentz transformation generally changes the difference intime and space between two events, the difference between the square of ctimes the time separation between two events and the square of the spaceseparation between the two events is invariant with respect to Lorentz trans-formations. The invariant space-time interval is defined by the equation

(interval)2 = (c× time separation)2 − (space separation)2 .

Since the product of the speed of light and the time separation, which occursin the first term on the right-hand side of this equation, is a distance, all ofthe terms in this last equation will have the units of distance squared.

The invariant space-time interval between two events whose time andspace coordinates in S are (t1, x1, y1, z1) and (t2, x2, y2, z2) is

(interval)2 = [c(t2 − t1)]2 − [(x2 − x1)2 + (y2 − y1)

2 + (z2 − z1)2],

Using the Lorentz transformations, one can show the space-time intervalbetween these two events in the reference frame S ′ moving with respect to Sis the same. The space-time interval is a Lorentz invariant.

The invariant interval may be used in describing events which occur atthe same spatial location. If two events take place at the same location inthe frame S, then x2 equals x1, y2 equals y1, and z2 equals z1. According tothe above equation, the invariant interval is then equal to c times the timeseparation of the two events. When two event occur in a reference frame atthe same point, the time interval in that frame is called the proper time.

One can obtain a better understanding of the properties of moving par-ticles in relativity theory by drawing a space-time diagram, in which thehorizontal axis denotes the x-coordinate of the particle and the vertical axis


45

denotes the product of the velocity of light c and the time t. The vertical co-ordinate of a particle gives the distance light travels in a time t, and –like thehorizontal coordinate – can be measured in meters. A moving particle tracesout a line in the spacetime diagram called the worldline of the particle. Fora particle moving with a constant velocity in the x-direction, the worldlineis a straight line with slope ∆(ct)/∆x = c/v. A particle (with zero mass)moving with the velocity c has a slope equal to one. All particles movingwith a velocity less that the speed of light have a worldline with slope greaterthan one.

Fig. 11.15 shows the worldline of a particle that travels a distance of 3min 5m of time. The invariant space-time interval for the particle with theworldline shown in Fig. 11.15 is

(interval)2 = (5m)2 − (3m)2 = (4m)2 .

The space-time interval (4m) is equal to the proper time of the particle or thelength of time that would elapse on a clock moving in the reference frame ofthe particle. The fact that 4m of time elapses in the moving frame while 5mpasses in the laboratory system is an example of time dilation. By decreasingand increasing the slope of the worldline in Fig. 11.15, one can see that theproper time of a particle at rest is equal to the laboratory time, and theproper time of a particle moving with the speed of light is equal to zero.

The space-time diagram shown in Fig. 11.15 describes the motion of aparticle in one-dimension. We can add another spatial dimension to ourspace-time diagrams by making our drawings three-dimensional. Fig. 11.19shows the worldline of a particle moving in two dimensions. The verticalaxis again corresponds to ct, while the two horizontal axes correspond tothe x- and y-coordinates. At a particular point along the worldline of theparticle denoted by A, a cone is drawn with a base angle of 45 degrees. Ina time interval ∆t, light will move a distance c∆t in the ct direction andalso a distance c∆t in space. The light signal will thus move from A alongthe surface of the cone. For this reason, the cone shown in Fig. 11.19 iscalled the light cone. Two dashed lines moving upward from the point A inFig. 11.19 correspond to other possible trajectories of the worldline of theparticle. Because the speed of a particle with nonzero mass must be less thanc, the trajectory of a massive particle must lie within the light cone.

The passage of the particle through point A is an event that would berecorded by an observer at that location. Three other events denoted by B,


46 CHAPTER 11. RELATIVITY I

C and D are also show in Fig. 11.19. The event B lies on the light cone, whileevent C lies within the light cone, and event D lies outside the light cone.Two events are said of have a timelike, a spacelike, or a lightlike separationdepending upon whether the space-time interval between the two events isgreater than zero, less than zero, or equal to zero, respectively.

Two events like A and C have a timelike separation since the spatialseparation of the event is less than the distance light would travel duringthe time separation of the events. For any two events having a timelikeseparation, one can make a Lorentz transformation to a moving frame inwhich the two event occur at the same location. Two events likeA andD havea spacelike separation since the spatial separation of the events dominatesover the time separation. One can always make a Lorentz transformation toa moving frame in which two events with a spacelike separation occur at thesame time. Of course, one event cannot cause another event occurring atthe same time. Two events with a space-like separation cannot be causallyrelated.

The Lorentz transformations can be put in a more symmetric form bymultiplying the equation transforming the time coordinates in eq. (11.15) byc and writing this equations before the others to obtain

ct′ = γ [ct− βx]x′ = γ [x− βct]y′ = yz′ = z,

where β = u/c. In the above equations, the quantities ct and ct′ appear onthe same footing as the x and x′ coordinates. We define the position-timefour-vector with components, xµ = (ct, x, y, z). The square of the length ofthe vector xµ, which is defined by the equation

|x|2 = (x0)2 − (x1)2 − (x2)2 − (x3)2,

is Lorentz invariant|x′|2 = |x|2.

Four quantities that transform under a Lorentz transformation as the com-ponents of xµ are said to form the components of a four-vector.


Chapter 12

Relativity II

This chapter begins by introducing the four-vectors corresponding to the ve-locity and momentum of a particle. The relativistic energy is then defined,and the conservation laws for the momentum and energy are used to find theoutcome of a number of scattering events. We then introduce the relativisticDirac equation and consider the free-particle solutions of the Dirac equation.Scattering events at high energies are described using Feynman diagrams andthe concepts of quantum field theory. For scattering processes involving elec-trons, the incoming and outgoing lines of the Feynman diagrams correspondto free-particle solutions of the Dirac equation.

The velocity of a particle in relativity theory is defined by the equation

vµ =dxµ

dτ,

where dxµ is a differential of the position-time vector xµ, and dτ is an in-crement of the proper time. Since the differential dxµ having components,cdt, dx1, dx2, and dx3, is a four-vector and the proper time element dτ isinvariant with respect to Lorentz transformations, vµ transforms as a four-vector.

Using eq. (11.26) to relate the differentials of time and proper time, theequation for the relativistic velocity can be written

vµ = γdxµ

dt,

where

γ =1√

1− v2/c2.

47


48 CHAPTER 12. RELATIVITY II

The components of vµ may thus be written

vµ = γ(c, vx, vy, vz) = γ(c,v).

The v0 component is equal to γc, while the spatial components of vµ areequal to γ times the corresponding components of the ordinary velocity v.As the velocity of the particle v approaches zero, γ approaches one. For smallvelocities, the spatial components of the relativistic velocity thus approachthe components of the ordinary velocity of classical mechanics. The squareof the length of the relativistic velocity is

|v|2 = γ2(c2 − vx2 − vy2 − vz2) = γ2c2(1− v2/c2) = c2,

which is clearly invariant with respect to Lorentz transformations.The four-momentum pµ of a particle is defined to be the mass m of the

particle times the four-velocity

pµ = mvµ.

Since vµ is a four-vector and m is a constant, pµ transforms as a four-vector.Using the the values of the components of the velocity obtained earlier, thecomponents of the momentum can be written

pµ = mγ(c, vx, vy, vz) = mγ(c,v).

The p0 component is equal to mγc, while the spatial components of pµ areequal to mγ times the corresponding components of the ordinary velocity v.

The relativistic energy of a particle is defined by the equation

E = γmc2 =mc2√

1− v2/c2.

The validity of this expression can be appreciated by considering the limitingform of the equation as the velocity of the particle approaches zero. For smallvelocities, the denominator in the above equation can be expanded in a Taylorseries leading to the following expression for the energy

E = mc2 +1

2mv2 +

3

8mv4

c2+ . . . .

The definition we have given for the relativistic energy applies to velocitiesin the range 0 ≤ v < c and reduces to a constant plus the nonrelativistic


49

expression for the kinetic energy when the velocity is small. The constantterm is called the rest energy

R = mc2.

The rest energy is the energy that a particle has when it is at rest. Therelativistic kinetic energy, which is defined to be the difference between theenergy and the rest energy of the particle

KE = E −R = (γ − 1)mc2,

is the contribution to the energy that is attributable to the motion of theparticle.

Comparing the definition of the relativistic energy with the componentsof the momentum pµ, we see that the zeroth component of pµ is E/c. Therelativistic momentum pµ thus has components

pµ = (E/c, px, py, pz) = (E/c,p).

The square of the length of pµ is

|p|2 =E2

c2− p2.

Since the relativistic momentum pµ is equal to mvµ and |v|2 is equal to c2,the square of the length of pµ may also be written

|p|2 = m2c2.

Equating the right-hand sides of these last two equations, we get

E2

c2− p2 = m2c2.

This last equation, which is called the energy-momentum relation, can bewritten

E2 = p2c2 +m2c4.

For a particle with zero mass, the energy-momentum equation reduces to

E = |p|c.



After defining the relativistic momentum and energy, Chapter 12 gives anumber of examples in which the conservation of energy and momentum areused to find the outcome of scattering events. Since my students often findit difficult to write down the conditions that the energy and momentum areconserved, I usually work two or three problems of this kind in class.

At this point in the course, my students expect to see a wave equation. Iusually discuss the general features of the Dirac equation in class but do nothave the time required to discuss the equation as fully as I do in the book.The Dirac equation can be written

i~∂ψ

∂t= cα · (−i~∇ψ) +mc2βψ,

where the dot product in the first term on the right-hand side indicates asummation over products of the coefficients αi and the corresponding partialderivatives. The equation also has a coefficient β.

The Dirac equation has plane wave solutions that satisfy the correctenergy-momentum relation if the coefficients, αi and β, satisfy the follow-ing conditions

αiαj + αjαi = 2δijIαiβ + αjβ = 0β2 = I,

where δij is the Kronecker delta and I is the unit matrix. These equationscan only be satisfied if the coefficients, α1, α2, α3, and β, are matrices.

The smallest matrices satisfying these conditions are 4× 4 matrices. Thechoice of the α and β matrices is not unique. The Dirac-Pauli representationis most commonly used

αi =

[0 σi

σi 0

], β =

[I 00 −I

],

where I denotes the unit 2× 2 matrix and σi are the Pauli matrices

σ1 =

[0 11 0

], σ2 =

[0 −ii 0

], σ3 =

[1 00 −1

].

The relativistic formalism we shall now consider is suitable for high-energyscattering experiments. For such problems, one generally uses a system ofunits called natural units, in which the velocity of light c and ~ are bothequal to one. With c equal to one, the energy-momentum four-vector is

pµ = (E,p).


51

The condition that ~ be equal to one implies that the wave vector k is equalto the momentum p and the angular frequency ω is equal to the energy E.

We shall suppose that the free particle solutions of the Dirac equationare of the form

ψp(x) = u(p)ei(p·r−Et),

where u(p) is a four-component vector independent of the space and timecoordinates. Substituting ψp into the Dirac equation gives

(α · p +mβ)u(p) = Eu(p).

This equation serves to define the Dirac spinors u(p) of the free particlesolutions of the Dirac equation.

In the Dirac-Pauli representation considered earlier, the α and β-matricesare divided into two-by-two matrices. Using the Dirac-Pauli representationof α and β and making a similar division of u(p) into two-component vectors,uA and uB

u(p) =

[uAuB

],

the equation that serves to define the Dirac spinors becomes[m σ · pσ · p −m

] [uAuB

]= E

[uAuB

].

We may then evaluate the matrix times vector multiplication on the leftand equate corresponding components of the vectors on the two sides of theequation to obtain the equations

muA + σ · puB = EuA

σ · puA −muB = EuB,

which may be writtenσ · puB = (E −m)uA

σ · puA = (E +m)uB.

There are both positive and negative energy solutions of the Dirac equa-tion for a free particle. The term E+m on the right-hand side of the secondof these last two equations is non-zero for the positive energy solutions, andthe equation can be solved for uB in terms of uA, while the term E − m



on the right-hand side of the first equation is non-zero for the negative en-ergy solution, and the equation can be solved for uA in terms of uB. Thefour-component vectors for positive and negative positive energy can then bewritten

u(r)(p) = CN

[χ(r)

σ·pE+m χ(r)

], u(r+2)(p) = DN

[ −σ·p|E|+m χ(r)

χ(r).

],

where r has the values 1 or 2. The constant CN for the positive energysolutions and the constant DN for the negative energy solutions may beevaluated using the normalization condition for the free-particle states.

The negative energy solutions posed very serious problems for Dirac whenhe originally proposed the theory. He suggested that all the negative energystates were occupied and that the Pauli exclusion principle prevented anelectron from making a transition to a state of negative energy. As illustratedin Fig. 12.2, a hole in the sea of negative energy states could be created bythe excitation of an electron from a negative energy state to a positive energystate. The absence of an electron with charge −e and energy −E could beinterpreted as the presence of an antiparticle with charge +e and energy +E.Physicists today follow Dirac’s idea of associating the solutions of the Diracequation with negative energy with antiparticles. The wave function and thespinor of a positron may be obtained by making the replacements, p→ −pand E → −E, in the wave functions and spinor of the negative energy statesof electrons.

Recall that the Dirac equation has plane wave solutions provided thatthe α- and β-matrices satisfy certain conditions. The Dirac equation can becast into a form in which its transformation properties with respect to theLorentz transformations are more apparent by multiplying the equation fromthe left with β and using the condition β2 = 1 to obtain(

iγµ∂

∂xµ−m

)ψ = 0,

where γi = βαi, γ0 = β, and where, as before, c = ~ = 1. The summationindex µ runs over the values (0, 1, 2, 3). This last equation is called thecovariant form of the Dirac equation.

In the text, we show that the Dirac equation may be used to derive acontinuity equation for a Dirac current. When c = 1, the Dirac current may


53

be writtenjµ = ψγµψ,

where the adjoint function ψ is defined by the equation

ψ = ψ†γ0.

In modern treatments of scattering processes, the particles and the forcefields associated with their interactions are described quantum mechanically.The harmonic oscillator is a useful model for an interacting field with theexcited states of the oscillator corresponding to states for which a numberof quanta of the field are present. In a scattering process in which twoelectrons interact by means of the electromagnetic interaction, for instance,an incoming electron excites the electromagnetic field promoting it to anexcited state with an additional photon. The photon created by one electronpropagates to the location of the other electron where the photon is absorbed.

Scattering processes for which the particles and the fields are describedquantum mechanically can be represented by Feynman diagrams. A Feyn-man diagram for electron-electron scattering is shown in Fig. 12.4 where themomentum and spin orientation of the two incoming electrons are denotedby p, r and q, s, and the momentum and the spin orientation of the two out-going electrons are denoted by p′, r′ and q′, s′. The mathematical expressionfor the Feynman diagram can be obtained by using a set of rules called Feyn-man rules, which, like the diagrams themselves, are due to Richard Feynman.The free lines in a Feynman diagram correspond to the plane wave states ofparticles, the vertices correspond to interactions, and an interior lines goingfrom one vertex to another is called a propagator.

The Feynman diagram shown in Fig. 12.4 corresponds to the scatteringamplitude

M = (−e2)δp+q,p ′+q ′u(r′)(p′)u(s′)(q′)γµi

k2γνu(r)(p)u(s)(q).

The out-going lines in the Feynman diagram shown in Fig. 12.4 correspondto the free-electron wave functions, u(r′)(p′) and u(s′)(q′). As discussed inconjunction with the Dirac current, a wave function with a line over it in-dicates the adjoint of the wave function times the γ0 matrix. Similarly, theincoming lines in the Feynman diagram correspond to the free-electron wavefunctions, u(r)(p) and u(s)(q). The factors, ieγµ and ieγν , correspond to thetwo vertices of the diagram, and the wavy line in Fig. 12.4 corresponds to



i/k2. The delta function δp+q,p ′+q ′ ensures that the total four-momentum ofthe electrons is conserved in the interaction.

Scattering events at high energies can be understood using the conceptsof quantum field theory. The incoming particles in a modern accelerator ex-periment create field quanta which carry the interaction between elementaryparticles. Photons, which are quanta of an electromagnetic field, and elec-trons and positrons, which are quanta of a Dirac field, both serve as carriersof the electromagnetic interaction.


Chapter 13

Particle Physics

All matter is composed of leptons, quarks, and elementary particles calledbosons which serve as the carriers of the force between particles. The leptonfamily includes the electron e− which has an electric charge and interacts withother charged particles by means of the electromagnetic force. The electronalso interacts by means of a force called the weak force, which is considerablyweaker than the electromagnetic force. Associated with the electron is anelusive particle called the electron neutrino νe, which only interacts by meansof the weak force. The other members of the lepton family are the muon µ−

with its neutrino νµ, and the tau τ− with its neutrino ντ . The leptons aredivided into distinct doublets or generations as follows[

νee−

],

[νµµ−

],

[νττ−

].

Strongly interacting particles, which are called hadrons, are composed ofquarks. Among the hadrons, the proton and neutron are members of a familyof particles called baryons, which are made up of three quarks. Anotherfamily of strongly interacting particles is the mesons, which are made up ofa quark/anti-quark pair.

Quarks come in six types, called flavors denoted by up (u), down (d),strange (s), charmed (c), bottom (b), and top (t) quarks. The b and t quarksare also referred to by the more appealing names of beauty and truth. Likethe leptons, the quarks are divided into three generations[

ud

],

[cs

],

[tb

].

55


56 CHAPTER 13. PARTICLE PHYSICS

An unusual property of quarks is that they have charges (Q), which arefractions of the proton charge. For each quark doublet, the upper member (u,c, t) has electric charge Q = +2/3 times the charge of a proton and the lowermember (d, s, b) has charge Q = −1/3 times the charge of a proton. Theproton is made up of two up-quarks and one down-quark (uud), while theneutron is made up of one up-quark and two down-quarks (udd).

Particles accelerated to high-energy in modern accelerators collide to pro-duce an astounding variety of new particles. Conservation laws provide ameans of characterizing the possible outcomes of scattering events and de-scribing what can and cannot occur. For scattering processes in which thethe total mass of the outgoing particles is greater than the total mass of theincoming particles, the conservation energy requires that the reaction canonly occur if the incoming particles have sufficient kinetic energy to make upthis mass difference.

The total electric charge is also conserved in collision processes. Since thecharge of an assembly of particles is the sum of the charges of the individualparticles and is always a multiple of the basic unit e, the charge is referredto as an additive quantum number. The electron lepton number Le, themuon lepton number Lµ, and tau lepton number Lτ are are also additivequantum numbers that are conserved in particle reactions. The electronlepton number Le is equal to +1 for the electron and the electron neutrinoand equal to −1 for the positron and the electron antineutrino. The muonlepton number Lµ is equal to +1 for the µ− and νµ and is equal to −1 forµ+ and νµ, while the tau lepton number Lτ is equal to +1 for τ− and ντ andis equal to −1 for τ+ and ντ . The lepton quantum numbers are zero exceptfor those values that have just been given.

Another quantity that is conserved in particle reactions is the baryonnumber, which can be expressed in terms of the number of quarks N(q) andthe number of anti-quarks N(q) by the formula

B =1

3[N(q)−N(q)] .

Baryons are composed of three quarks and thus have baryon number B equalto +1, while mesons, which are made up of a quark/anti-quark pair, havebaryon number equal to zero. Baryon number is conserved in scatteringprocesses because quarks and anti-quarks are only created or destroyed inquark/anti-quark pairs.

Mesons and baryons have been observed that are produced by the strong


57

interaction but decay by the weak interaction. These particles, which arealways produced in pairs, have the unlikely or strange property that they areproduced in 10−22 seconds and yet live long enough to produce considerabletracks in a bubble chamber. The fast production and slow decay of these par-ticles has been explained by assigning to each of these particle a strangenessquantum number S. One of the two strange particles, which are producedby the strong interaction, has a positive value of the strangeness quantumnumber and the other has a negative value. The total amount of strangenessproduced by the strong interaction is thus equal to zero. Following its pro-duction, each strange particle decays by the weak interaction, which mayinvolve a change of the strangeness quantum number. While all interactionsconserve the lepton numbers and baryon number, the weak interaction canchange the flavor of a quark and thus violate the conservation laws associatedwith the quark quantum numbers (strangeness, charm, beauty, and truth).

I always devote at least one class period to the conservation laws thatdetermine which reactions do and do not occur. Problems of this kind aregiven at the end of Chapter 13. The appropriate lepton numbers, baryonnumbers, and strangeness quantum numbers can be written under each par-ticle in a reaction formula as is done in the text. Working problems in-volving the conservation laws helps student become familiar with with themost commonly occurring mesons and baryons. The properties of the light-est mesons and baryons are given in Tables 113.3 and 13.4. The strangenessquantum numbers of particles can be assigned using the quark composition

given in these tables. The K− and K0

mesons and the Λ and Σ baryonshave a single strange quark s and hence have strangeness quantum num-ber S = −1, while the K+ and K0 mesons have a single anti-strange quark sand hence have strangeness quantum number S = +1. The Ξ baryons havetwo strange quarks and hence strangeness quantum number S = −2, whilethe Ω baryon has three strange quarks and hence strangeness quantum num-ber S = −3. The anti-baryons have baryon quantum number B = −1 andtheir strangeness quantum numbers are the negatives of the correspondingbaryons.

The third section of this chapter describes the spin of elementary particlesand considers whether particles decays are invariant with respect to the parityand charge conjugation transformations. We consider positronium, which ismade up of an electron and a positron, mesons, which are composed of aquark and an anti-quark, and baryons which are made up of three quarks.



The spin of a particle is defined as the angular momentum of the particle inits own rest frame. Mesons, which are bound states of a quark/anti-quarkpair, have a single orbital angular momentum and two spins. While theorbital angular momentum quantum number L can have different values, thelightest mesons have orbital angular momentum L = 0. The spins of thequark and anti-quark can combine to form spin-zero and spin-one mesons.The pions, K-mesons, and eta have spin equal to zero, while the rho andomega mesons have spin equal to one. For baryons, the spins of the threequarks can combine to form spin one-half and spin three-halves particles.The proton, neutron, lambda, and sigma have spin equal to one half, whilethe delta baryons have spin equal to three halves.

A parity transformation, in which the spatial coordinates are invertedthrough the origin, can be achieved by a mirror reflection followed by arotation of 180 about an axis perpendicular to the mirror. Since the lawsof nature are invariant under rotations, the question of whether parity isconserved depends upon whether an event and its mirror image occur with thesame probability. Each quark has an intrinsic parity and a parity associatedwith its orbital angular momentum. The spatial wave function is even or odddepending upon whether the orbital angular momentum quantum number iseven or odd. The least massive mesons with L = 0 have odd parity sincethe quark and anti-quark have opposite parities. By contrast, the low lyingbaryons, which are made up of three quarks, have even parities.

The transformation of charge conjugation replaces all particles by theirantiparticles without changing the position or the variables that describe themotion of the particle. The effect of a charge conjugation transformationupon positronium, which consists of an an electron and a positron, or upona meson, which consists of a quark and an anti-quark, will be to interchangea particle and its anti-particle. If a particle/anti-particle pair have totalangular momentum L with respect to the center of mass of the pair, inter-changing the particle and antiparticle will have the effect of reversing therelative position vector and give rise to a phase factor (−1)L. As shown inthe text, exchanging two spin contributes a phase factor (−1)S+1. In con-sidering the effect of charge conjugation upon a composite particle made upof a fermion/anti-fermion pair, one must also add another factor of minusone for changing a fermion into an anti-fermion. Drawing these three factorstogether, the effect of a C-parity transformation upon a composite systemsmade up of a fermion/anti-fermion pair is to introduce a factor (−1L+S. Forexample, the π0 with both S and L equal to zero must have C-parity Cπ0 = 1.


59

Several experiments are reviewed in the chapter showing how physicalprocess are affected by parity and change-conjugation transformations. Suchexperiments can easily be understood by modern physics students, and theexperiments have far-reaching consequences. In 1957, following suggestionsby C.N. Yang and T.D. Lee, C.S. Wu and her coworkers at Columbia Uni-versity placed a sample of cobalt-60 inside a solenoid and cooled it to atemperature of 0.01 K. At such low temperatures, the cobalt nuclei alignparallel to the direction of the magnetic field. Polarized cobalt-60 nucleidecay to an excited state of nickel-60 by the process

60Co→60 Ni∗ + e− + νe.

Parity violation was established by the observation that more electrons wereemitted in the direction of the nuclear spins than the backward directions.

An illustration of an electron being emitted from a 60Co nucleus is shownin Fig. 13.9(a). The spin of the cobalt nucleus is illustrated by an arrowindicating the rotational motion of the nucleus, and by an arrow beside thenucleus pointing upward because a right-hand screw would move up if itwere to rotate in the way the cobalt nucleus is spinning. Under a paritytransformation, the velocity v and the momentum p of a particle changesign. The orbital angular momentum l = r × p and the spin of a particleare unaffected by a parity transformation. As shown in Fig. 13.9(b), a paritytransformation reverses the direction of an electron emitted by 60Co butleaves the direction of the nuclear spin unchanged. Parity is violated sincea beta decay in the direction of the spin of the cobalt nucleus shown inFig. 13.9(a) occurs more often than a beta decay in the backward directionshown in Fig. 13.9(b).

Another example of parity violation is provided by the dominant decaymode of π+ described by the formula,

π+ → µ+ + νµ.

In this decay process illustrated in Fig. 13.10(a), the spin of the µ+ is indi-cated by the downward arrow next to the particle. The emitted muon hasnegative helicity which means that its spin points in the direction oppositeto its motion. In the parity transformed process shown in Fig. 13.10(b), theµ+ is rotating as before and the spin still points down. The process shownin Fig. 13.10(b), for which the muon has positive helicity, does not occur.



The effect of a C-transformation upon the π+ decay process illustrated inFig. 13.10(a) is shown in Fig. 13.10(c). A C transformation of the particlesconverts the π+ to π−, the µ+ to µ−, and the νµ to νµ to give the process

π− → µ− + νµ.

The µ− produced by a C-parity transformation of the µ+ has negative helicityas does the µ+ produced by the decay of π+; however, experiment shows theµ− emitted in π− decay actually has positive helicity. So, C is violated.The result of both P and C transformations upon the decay process yieldsthe decay of the negative pion shown in Fig. 13.10(d) with correct helicities.Thus, although both P and C are violated in pion decay, CP , which is thesimultaneous transformation of both P and C, is conserved.

Although CP is conserved to a very good approximation in most circum-stances, a few examples can be found of CP violation. The example, whichis most widely known, concerns the decay of the neutral meson K0

L discussedfollowing Table 13.3. The K0

L usually decays into three particles; however,in 1964 Christenson, Cronin, Fitch, and Turlay discovered that in one decayfor every thousand K0

L decays into two pions

K0L → π= + π−.

This result is clear evidence of CP violation since the two pion state trans-forms differently under CP than the three particle decay modes.

The strongly interacting particles occur in isospin multiplets with allmembers of a multiplet having approximately the same mass. The mem-bers of an isospin multiplet all have the same value of the hypercharge Y ,which is the sum of B, S, C, B, and T . The members of each multiplet aredistinguished by the charge Q and I3. Particles can also be grouped in largerfamilies of particles called supermultiplets. The lightest mesons consist oftwo nonets with each consisting of nine mesons, while the lightest baryonsconsist of an octet with eight baryons and a decuplet with ten baryons.

The assumption that the wave functions of baryons be symmetric withrespect to an interchange of identical quarks allows one to explain the massspectra of the light baryons, and yet this assumption appears to contradictthe requirement of quantum mechanics that the wave function of fermionsbe anti-symmetric with an exchange of two particles. Anti-symmetric wavefunctions automatically satisfy the Pauli exclusion principle.


61

The apparent contradiction between the quark model and the Pauli prin-ciple was resolved in 1964 when Oscar W. Greenberg suggested quarks possessanother attribute, which he called color. The combined space and spin wavefunction can then be symmetric with respect to the interchange of two quarksof the same flavor – as required by experiment – provided that the color partof the wave function is anti-symmetric. The basic assumption of the colortheory proposed by Greenberg is that the quarks of any flavor can exist inthree different color states, red, green, and blue, denoted by r, g, b.

Just as the electromagnetic and weak interactions depend upon the hy-percharge Y and isospin I3 of the particles, the strong interaction dependson the two color charges called color hypercharge Y C and color isospin IC3 .The values of these new quantum numbers for the color states r, g, and b aregiven in Table 13.13. All baryons are made of three quarks of different colors,while mesons consist of a quark and an anti-quark of the same color. Forsuch states, the total value of the additive quantum numbers, IC3 and Y C ,are equal to zero. This is called color confinement. The idea that quarks arealways found in nature in color-neutral states was part of the motivation forusing the word “ color ”. Just as white light can be obtained by combiningthe three primary colors, baryons combine red, green, and blue quarks intoa color neutral state.

The interactions between leptons and quarks are described by Feynmandiagrams. These diagrams describe processes by which particles interact byexchanging quanta of the interaction fields. The Feynman diagrams for elec-tromagnetic processes have vertices with two electron or positron lines andone line of a photon, while the diagrams for weak interaction processes involv-ing leptons have vertices with two lepton lines and one line of a W+, W−, orZ0 boson. The Feynman diagrams for the weak interaction of quarks may beobtained by replacing leptons and neutrinos of allowed weak processes withquarks of the same generation and by allowing mixing to occur between dif-ferent generations of quarks. My students enjoy drawing Feynman diagramsfor different decay and scattering processes.

Recent work on gauge symmetries provide a general framework for un-derstanding the interactions between elementary particles. There are threegauge symmetries associated with the electromagnetic, weak, and strong in-teractions. Associated with each gauge symmetry is a gauge field and gaugebosons which serve as carriers of the interactions. The photon is the gaugeparticle associated with the electromagnetic interaction, while the W+, W−,and Z0 bosons are the gauge particles associated with the weak interaction,



and gluons serve as the gauge particles of the strong interaction.


Chapter 14

Nuclear Physics

Information about the structure of nuclei can be obtained from high-energyelectron scattering experiments. By fitting the electron scattering data,R. Barrett and R. Jackson obtained the charge distributions of the light (16O),medium (109Ag), and heavy (208Pb) nuclei shown in Fig. 14.3. The chargedensities of these three nuclei all have the same general form with an approx-imately level inner portion and a thin shell region where the charge falls offexponentially to zero. The inner region of the charge distributions of oxygenand silver are higher than the central portion of the charge distribution oflead.

Some indication of how protons are distributed within a complex nucleuscan be inferred from the charge distribution. If the protons within a nucleuswere point particles, the density of protons ρp would be related to the chargedensity ρch by the formula, ρch = eρp. To the extent that the isospinsymmetry holds, and protons and neutrons are equivalent, the density ofnucleons within a nucleus would be related to the charge density by theformula

ρ(r) = (A/eZ)ρch(r).

The factor, A/eZ in this formula has the effect of lowering the inner region ofthe density of light nuclei in relation to the density of heavy nuclei which haverelatively higher numbers of neutrons. The nuclear densities of 16O, 109Ag,and 208Pb are shown in Fig. 14.4. These curves show that at the center of anucleus the density of nuclear matter is roughly the same for all nuclei. Itincreases with A, but appears to approach a limiting value ρ0 of about 0.17nucleons per fm3 for large A.

63


64 CHAPTER 14. NUCLEAR PHYSICS

The existence of a limiting value of the nuclear density ρ0 for large A is animportant result. Using this idea, we can obtain an approximate relationshipbetween the atomic mass number A and the nuclear radius R. We set theproduct of the volume of a sphere of radius R and the the nuclear density ρ0

equal to the atomic mass number A to obtain(4π

3

)R3ρ0 = A.

Solving this equation for R and using the fact that ρ0 is equal to 0.17 fm−3,we obtain

R = 1.12A1/3 fm.

This formula can used to estimate the radius of nuclei having particularvalues of A.

The protons and neutrons in the nucleus can only be separated by workingagainst the strong attractive forces holding them together. The bindingenergy is the amount of work that would be needed to pull the protonsand neutrons in the nucleus entirely apart. We can calculate the bindingenergy B(N,Z) of a nucleus A

ZX with Z protons and N neutrons by findingthe difference between the total rest energy of the constituent protons andneutrons and the nucleus itself

B(N,Z) = [Zmp +Nmn −mnuc(N,Z)] c2 .

The quantity within square brackets in this equation is the mass which wouldbe lost if the nucleus were to be assembled from its constituents. The massloss is converted into a binding energy by multiplying it by c2 according toEinstein’s formula, E = mc2.

The masses of atoms are measured experimentally rather than the massesof the bare nuclei. Since the binding energy of the electrons in an atom arevery much smaller than the binding energy of the nucleus, we can find thenuclear binding energy by calculating the difference in mass of the atomicconstituents. The equation for the binding energy then becomes

B(N,Z) =[Zm(1

1H) +Nmn −m(N,Z)]c2 ,

where m(11H) is the mass of a hydrogen atom, mn is the mass of a neutron,

and m(N,Z) is the mass of the atomic isotope AZX. Since the atom A

ZX has Zelectrons, the rest energy of the electrons in Z hydrogen atoms is equal to the


65

electronic contribution to m(N,Z)c2. The amount of energy correspondingto a single atomic mass unit is 931.5 MeV. One can thus calculate the massloss in atomic mass units and then obtain the binding energy by multiplyingthe mass loss by 931.5 MeV.

All nuclei have a shell structure, which contributes to their stability andinfluences the kinds of decay processes that can occur. The effects of the shellstructure are superimposed on a slowly varying binding energy per nucleon.Like the drops of a liquid, nuclei all have an inner region where the densityis approximately uniform and a thin surface region where the distributionof nuclear matter falls off exponentially to zero. The validity of thinking ofnuclei as drops of a liquid is made more precise by giving an empirical formulafor the binding energy of nuclei. With a few parameters, this formula fitsthe binding energy of all but the lightest nuclei to a high degree of accuracy.Following W.N. Cottingham and D.A. Greenwood whose book is cited atthe end of this chapter, we give the following version of the formula for thebinding energy

B(N,Z) = aA− bA2/3 − dZ2

A1/3− s(N − Z)2

A− δ

A1/2,

where A is the number of nucleons, N is the number of neutrons, and Z is thenumber of protons. The parameters a, b, d, s, and δ can be obtained by fittingthe formula to the measured binding energies. The values of these parametersgiven in the text are those given in Handbuch der Physik, XXXVIII/1.

A discussion of the various terms in the empirical formula can be foundin the text. If nuclear matter were entirely homogeneous, the number ofnucleons in a nucleus would be proportional to the volume of the nucleus. Inthe analogy between nuclei and liquid drops, it is the atomic mass number (A)which is the analogue of the volume of the liquid. The term aA in theempirical formula depends upon A in the same way that the cohesive energyof a fluid depends upon the volume of the fluid. Since the surface area of asphere depends upon the radius squared and the volume of a sphere dependsupon the radius raised to the third power, the surface area of a sphere dependsupon the volume of the sphere raised to the two-thirds power. Hence, theterm bA2/3 is analogous to the surface energy of a liquid sphere. While theCoulomb term, −dZ2/A1/3, discourages the formation of states with highnumbers of protons, the term, −s(N − Z)2/A discourages the formation ofstates having unequal numbers of protons and neutrons. The final term inthe semi-empirical formula describes a pairing effect that can be important


66 CHAPTER 14. NUCLEAR PHYSICS

for light nuclei. The pairing term makes even-even nuclei more stable thantheir odd-odd counterparts with the same A. An empirical formula for themass of an atom can be obtained by substituting the empirical formula forthe binding energy B(N,Z) into the formula given previously for the bindingenergy and then solving for the mass m(Z,N).

The importance of the empirical formulas for binding energy and massis not that they enable us to predict new or exotic nuclear phenomena, butrather that they enable us to understand the properties of nuclei in simplephysical terms. The empirical models give us some insight as to which nuclearspecies should be stable and what decay processes are likely to occur.

While the semi-empirical formulas are very successful, experimental atomicmasses show deviations from the semi-empirical mass formula which arequantum mechanical in nature. Just as the atom can be described as elec-trons moving in an average central field due to the nucleus and the electrons,the nucleus itself can be described as protons and neutrons moving in a fielddue to both the strong and the electromagnetic forces. Because nucleonsmove in a finite region of space with definite values of the angular momen-tum, the table of nuclides show recurring patterns that are very similar tothe pattern of atomic elements described by the periodic table. As for atomicsystems, the description of the nucleus in terms of the angular momentumof individual nucleons is called the nuclear shell model.

With the inclusion of the spin-orbit interaction, the nuclear shell modelsuccessfully predicts the values of the atomic number Z and the number ofneutrons N for which nuclei are particularly stable. The shell model alsopredicts the angular momentum of nuclei in the ground state. Nuclei witheven numbers of protons and even numbers of neutrons (even-even nuclei)have angular momentum zero and even parity, while nuclei with an evennumber of protons and an odd number of neutrons or vice versa (even-oddnuclei) have angular momentum and parity equal to that of the odd nucleonin the shell being filled. As for atomic shells, the parity of a single nucleonwith orbital angular momentum l is even or odd depending upon whether(−1)l is even or odd. It is energetically favorable for pairs of protons andpairs of neutrons to form states having zero angular momentum and evenparity, so that the angular momentum and parity of the nucleus is equal tothe angular momentum and parity of the unpaired nucleon. There are veryfew exceptions to this rule.


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