Thermal Physics (2nd Edition) - Kittel and Kroemer

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Artist:

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Syntax

Compositor:

International

of Congress

Library

H. Smith

Sharon

Frank

Cataloging

in

pat;

Publication

Clmrlcs.

Killcl,

Thermal physics.

Bibliography:p. index.

Includes

Statistical

!.

Herbert.1928-

536'.?

\302\260

I9B0

Copyright No pan of mechanical,

79-16677

by W.

bor.k

may

plioiographic,

H. Freeman and be reproduced

or electronic

copiedfor permission

in

Company by

any

process,or in

phonographic recording,nor

may

it be

system, transmitted, orotherwisc wriiicn or private use, without public from the publisher.

a retrieval

in

Pcimcti

this

of a

form

sioreti

ilie

United

State of

America

Twenty-first printing, 2000

9

Kroe/n Tillc.

II.

aullior.

O-7167-IO8S-9

ISBN

liic

joiiii

1930

QC3H.5.K52

I.

tiicrmodyn;miics.

About the Authors

Charles

at

has

Kiitel since

Berkeley

1951,

having in

work

undergraduate

solid

laught

physics

slate physics at the University of California been at [he Bell Laboratories.His previously was done at M.I.T. and at the Cavendish

His Ph.D. research was in theorclicai nuclear Professor Breit at the University of Wisconsin. physics with Gregory He has been awarded three Guggenheim fellowships, the Oliver Buckley Pme for Solid State Physics, and, for contributions to teaching, the Oersted Medal of the American Association of Physics Teachers. He is a member of \"he ;i id National of of Arts Science and of the American Academy Academy semio nSciences. His research has been in magnetism, resonance, magnetic and the statistical mechanics o f soiids. ductors, of Cambridge

Laboratory

Kroemcr

iferbcrt

is

liliy^ics.

I

a I'lt.D.

!c received

in Germany

with

Professor

liurhara.

at Santa

California

University.

a

Prom 1952 through

thesis

of Electrical

Engineering at the

His background

mid

in physics in I'J52 from on hot electron effects

I96S tie workedin

several

nre

training

the University in

lhc

semiconductor

then

of

University

in solid of

state

Gulling

transistor.

new research

labora-

of Stales. In I96S lie became in to UCStt at ltic University of Colorado; lie came [ilixirieul Engineering of semiconductors and technology 1976. His research has been in the physics and semiconductor devices, including transistors, negativehigh-frequency electron-hole mass effects in semiconductors, injection lasers,the Gunn effect, and semiconductor hetcrojunctions. drops,

laboratories

in

Germany

and

the

United

Professor

Preface

This book

an

gives

simple,

other

no

Probably

and

science

are

methods

the

applications.

physics. The subject is and the results have broad applicatheory is used more widely throughout of thermal

account

elementary

powerful,

physical

engineering.

We have written for undergraduate and for electrical engineering students

(not

but

original,

not

easily

physics

and

These

generally.

purposes have strong common bonds,most mcmls, gases, whether in semiconductors,

methods

of

students

notably

a concern

stars,

or ituclci.

accessible

elsewhere)

astronomy, for our

fields

Fermi

with

We develop that

are

well

to these fields. We wrote the book in the first place because we as compared to (hose were delighted by the clarity of the \"new\" methods we were taught when we were students ourselves. some because We have not emphasized several traditioual they topics, classical on statisnare no longer useful and some because their reliance cai mechanicswould make the course more difficult than we believe a course should be. Also, we have avoided the use of combinatorial first

suited

methods

where

they

are unnecessary.

Notation and units;

parallel. the

do

We

fundamental

not

use

We

generally

the calorie.

temperature

t by

use the SI and CGS systems in to The kclvin temperatureT is related

r =

kBT,

and

the

conventional

entropy

S is reialed lo the fundamental a by 5 = ka(j. The symbol log entropy will denote natural logarithm throughout, simply because In is less exlo Equation refers A8) of A8) expressive when set in type. The notafion of 3. the current chapter, but C.18) refers to Equation A8) Chapter with the assisfto course notes developed Hie bookis ihe successor (he ance of grants of California. Edward M. PurceSlconUniversity by from review of the to We benefited ideas the contributed first edition. many and Nh.-holns L. Richards, Paul second edition by Seymour Geller, Wheeler- Help was giveii by Ibrahim Adawi, Bernard Black, G. DomoK. A. Jackson, S. Justi, Peter Cameron kos, Margaret Geller, Hayne, Martin Ellen Leverenz, Bruce H. J. J. Klein, Kittel, Richard Kittler, McKellar,

F.

E.

O'Meara,

Norman

E. Phillips,

B. Roswclt Russell,T. M.

Preface

B.

Sanders.

An

by

in

added

was

the index.

treatment

elementary

atmosphere

Carol

thank

her help with

for

Wilde

Professor

Richard

John Wheatley, and Eyvind Verhoogen, for the Tung typed manuscript and Sari

John

Stoeckly, We

Wichmann.

1994

of the

on page

Muiier.

Bose-Eitistein condensationwas For instructors who have

solutions

manual

is

available

A

115, following an

page

on

added

to

adopted

via

effect in the Earth's

greenhouse

the

aioinic page

suggested

argument

gas experiments 223 in 2000.

on the

classroom

use, a

course

the freeman

atmo-

for

web site

(http:/Avhfreenian.

com/thermaiphysics).

Berkeley

and

Santa

Barbara

Charles Herbert

K'tttel

Kroemer

Note

to

the

Student

For minimum of the concepts the authors coverage presented in each chapier, recommend the following exercises.Chapter 2: 1,2, Chapter 3: 1,2, 3,4, 8, 6: 1,2,3,6,12, 5: 11; Chapter 4: 1,2,4, 5,6, 8; Chapter 6,8; 1,3,4, Chapter 9: 8: 14, 15;Chapter7: 2, 1, 2, 3, 5, 6, 7; 1, 2, 3; 7, 11; Chapter Chapter 12: 13: 3,4.5; 1,2, Chapter Chapter 10: 1,2,3; Chapter 11: 1,2,3;Chapter

3;

3,5,6,

3,7,8,10; Chapter 14: 1,3,4,5; Chapter

15:

2,3,4,6.

Contents

Guide

xiii

to Fundamental Definitions

General

xv

References

Introduction

Chapter

1 States

1

a Model

5

System

Eittropy and Temperature

Chapter 2 Chapter

of

3

Distribution

Boltzmann

27

and Hdmholtz

Free Energy 55

Chapter4

Thermal

Chapter 5

Chemical Potential and Gibbs Distribution

Chapter

Ideal

6

Chapter

8

il

Chapter

Binary

309

Mixtures

Cryogenics 333 Statistics

14

Kinetic Theory 389

Appendix B

261

275

Transformations

Semiconductor

A

353

423

Propagation

Some

7

225

13

Chapter15 Appendix

Phase

1 i

181

Gibbs Free Energy and ChemicalReactions

9

Chapter 12 Chapter

87

Distribution

Planck

151

Work

tHeaZahd

Chapter10 Chapter

Gas

Fermi and Bose Gases

Chapter 7 Chapter

and

Radiation

Integrals

Containing

TemperatureScales 445

Exponentials

439

Appendix

Appendix

C

Poisson

D

Pressure

AppendixE Index

465

Distribution

453

459

Negative

Temperature

460

Absolute

~

X

activity,

Accessible

Definitions

Fundamental

to

Guide

29

state

Boltzmann constant,

25

ka

Boltzmann

factor,

Boson

183

Chemical

119

/;

Classicalregime, n

74

\302\253

nQ

31

of systems

Enthalpy, H = U

+

246

pV

40

a

Entropy,

1S3

Fermion

Gibbs factor, exp[(NjiGibbs

free

Gibbs

or grand

Heat

61'

exp\302\243~~ \302\243/t)

potential,

Ensemble

139

exp(/i/t)

sum,

U

\342\200\224

\\a

\342\226\240%

138

+

p^

138

63

C

capacity,

68, 227

Heat, Q

free

HelmhoHz

Landau

G =

energy,

t)/i]

free energy

Multiplicity,g

F

energy,

\342\200\224 \342\200\224 U xa

function, FL

7

9

Orbital

Partition

function,

Z

61

t

298

Guide to Fundamental

concentration,

Quantum

Reversible

64

41 62

Thermal

average

Thermal

equilibrium W

hq

process

Temperature, t

Work,

Definitions

227

39

=

References

General

Thermodynamics A. B. Pippard,

of classical

Elements

1966. M. W.

R. H.

and

Zemansfcy

textbook, 6ih

anil

Heat

DiEEman,

ed., McGraw-Hill,

Cambridge University Press,

thermodynamics,

an intermediate

thermodynamics:

198!.

Afcchanics

Sitttisiical

U. K,

and M. Eisner, Statistical 1988. Agarwal mechanics, Wiicy, Dover PubticaHit), Statistical mechanics:principlesand selected applications, iions, 1987, cl956. C. Kittct, Elementary statistical applications physics, Wiicy, 1958. Parts 2 and 3 treat 1 has been expanded ioEo the Part to noise and to elemeniary transport Eheory.

T. L.

present Eext. R.

Kubo,

R, Kubo,

Statistical M.

mechanics,

North-Holland, 1990, cI965. Statistical physics !! (NanequHibrium),

N. Hashitsume,

Toda,

Springer,

1985.

L D.

Landau and E. M. Lifshitz,

Statistical

K. M. Lifshitz

and

L. P.

1985.

Scientific,

! (Equilibrium),

Springer,

1933,

tables

Mathematical

H. B. Dwight,

Tables

1961. A

3rd cd. by

physics,

1. Piiaevskii, Pcrgamon, 1980, part Ma, Statistical mechanics. World Shang-Keng M. Toda, R. Kubo, N. Saito,Statisticalphysics

and other

of integrals

mathematical data, 4ih

ed.,

MacmUton,

collection.

smati

useful

widely

Applications

Asirophysics

R. J. Taylor,

The

S. Weinbcrg,

The first

ed.,

Bainam

structure

their

stars:

three

Cooks,

minutes:

and

evotitiioit.

a modern

v:\\-w

1972.

Springer, of the

origin

of the

universe, new

1984.

Biophysics and macromolccules

T. L. Hill, Springer,

Cooperathity

1985.

theory

in biochemistry:

steady stale

and equilibrium

systems,

General Refer,

Cryogenicsand G. K.

low

J.

D. S.

and

Wilks

. .

Betis,

An

pa.

helium, 2nd

to liquid

introduction

physics, 3rd ed., Oxford

ed , Oxford

Univesity

1987.

Press,

Irreversible

thermodynamics

J. A. McLennan,Introduction 1989.

I.

in low-temperature

techniques

1987, ct979

Press,

University

physics

lempcrature

Experimental

White,

I. Stcngers,

and

Prigogine

Random

to

statistical

non-equilibrium

Order

out

of

mechanics, Prentice-Hall,

man's

chaos:

new dialog

with

nature.

1934.

House,

Kjnclic theory and transport phenomena S. G. Brush, The kind of motion we call heal, North-Holland, 1986, cI976. H. Smith and H. H. Jensen, Transport phenomena,Oxford University Press, Plasma

physics

I... Spitzer, Jr., Physical

in the

interstellar medium,

Touiouse, Introduction Wiley,

phenomena,

H. E.

and

Haasen,

Boundary

the

critical

and

critical

[ihenomena,

Oxford Uni-

1987.

Press,

affoys

Physical

metallurgy,

2nd ed.,

CambridgeUniversity

Press,

1986.

Superb.

value problems

and J. C. Jaeger, Conduction of heat H. S- Carslaw Press, sily 19S6,ci959.

Semiconductor

group and to

renormalizat'ton

1977.

Stanley, Introduction to phase transitions

University

Metais

to

197S.

Wiley,

.

.

P. PfeiHy and G.

P.

processes

transitions

Phase

19S9.

in solids,

2nd ed.,

Oxford Univer-

devices

Introduction to applied solidstate physics, t990. Plenum, 5th ed., Springer, 1991, Semiconductor K. Seeger, physics:an introduction, t981. S. M. Sze,Physics devices, 2nd ed., Wiley, of semiconductor

R.

Datven,

Solid

state

physics

C. Kittel, Introduction

to solid

state physics, 6th

ed.,

Wiley,

1986.

Referred

to ssISSR

Thermal

Introduction

Our approachto physics

to do

going

structure: in

in

thermal

the

this

differs from the tradition followed in beginning we provide this introduction 10set oul what we are that follow. We show the main lines of the logical all the physics comes from In order of lhcir the logic. physics

Therefore

courses.

chapters subject

in our are the entropy, the temporaiure, appearance,the leadingcltaracters story the Boltzmann the chemical factor, potential, the Gibbs factor, and the disiribu-

functions.

tion

The entropy A

closed

system

measures the number of quantum might

be

in

any

of these

states

quantum

to a

accessible

states and

system.

(we assume)with

statistical element, ihe fundamental logical o r states are inaccessible to the either accessible assumption, quantum and the system is cquaiiy likely to be in any one accessible slate as in system, olher slate. is defined accessible as Given accessible states, the cniropy any g = a lhtis defined will be a function of ihe energy U, lhe logg. The entropy V of the system, because theseparamnumber of particles N, and the volume as wirii. The parameters ciilcr enter the dctcrminaiion of y; other para meters may is a mathematical use of the logarithm convenience: it is easier to write 1010 than expA020), and it is morenatural to speak of a-y + o, lhan for two systems The

probability.

equal

fundamental

is that

ofg,3j.

When two systems, each of ttiey

may

transfer

energy;

energy,

specified

their total

individual energies are perhaps in the other, may increase on their

are brought

into thermal

energy remains constant,but A

lifted.

the

transfer

product

accessiblestates of the combined systems.

The

of energy

in

the one

coniact

comlraints direction,

or

g^g, that measures the tiumber of fundamental

assumption

biases

maximizes the the outcome in favor of that allocation of the total energy that and more likely. This statement is number of accessible states: more is better, is the the kernel of the law of increase of entropy, which general expression of

the second law

of

thermodynamics.

brought two systems into thermal contactso that they may transfer One ofthe encounter? system will energy. What is the most probableoutcome of the of the other, and meanwhile the lotal entropy gain energy at lhe expense two systems will increase. will reach a maximum for the entropy Eventually It is not the total difficult to show 2) that the maximum given energy. (Chapter We have

is atiained when

ihe same

for

quantity

in ihermai

system is equal to the value of This equality property for Iwo systems of the icmperat lire. Accordingly, expect for one

value

ihe

o((ca/cU}K_y the Second system.

coniaa is just the

we define the fundamental

property we

relation

by the

i

lemperaiure

1

U)

CUJ

The use of 1/r assuresthat needed.

complicated

directly

proportional

to t,

3. i.ti

a

c, be pjaeed in thermal reservoir. The loia! energy

.S\"

-

the

of

the Boltzmann constant.

i.fie :it cnorj>y

sinies,

the

have

fundamental

smali system

with

Uo

energy

assumpiion,

s to

energy

entropy a

g{U0 by

may

definition

the

be dropped.

This

may

~ e)

is Boitzmantt's

\342\200\224 slates

e, the lo

accessible

e)

B) exp[.fi.)equ.iltoD,l,l),(l,4,]),and{l,l,4)ai!l!aveii/ + the corresponding energy level has 3. multiplicity

+

nr2

=

n.1

18;

the

to know these!of values

siaie soTthejV states

in

any

assigned

the

of

arbitrary

such as

c is

s may

it is

A' particles,

of

\302\243S(N),where

energy Indices

system.

particle

convenient

theenergy

be

assigned

essemial

of the quantum lo the quantum

states should not

way, bul two different

be

index.

same

the

properties of a system

the statistical

To describe

It is a good idea to siart the properties of simple program by studying model which the energies for can be calculated A') e.vacily. We choose as a modela simple because the genera! statisiical propenies system our

systems

Ej{

binary

are believed to appiy equally This physical system. assumptionleadsto predictions

found

for

the

model

experiment. What as we go along.

statistical

general

are of

properties

to any

well

system

that

always

concern will

realistic

agree become

with clear

BINARY MODEL SYSTEMS

The

model

binary

separate

and

Attached

to

syslern

distinct

each

site

sites

is an

is illustrated fixed

in

N shown for convenience on a line1.3.

Figure

in space,

elementary magnet

titat

can

We assume

point

only

there are

up or

down,

tlie system means to To understand corresponding to magnetic an element of the no of count the slates.This requites magnelism: or no, red or blue, site of two states, labeled as system can be capable one or one. The sites are minus occupied or unoccupied,zeroor one, dtSFercni to in sties with numbers are supposed numbered, overlap moments

\302\261n>.

knowledge

any

yes

plus

and

not

of the sites as numbered parking spaces in space. You might even tltink vacant or 1A Cacti as in a car lot, Figiire parking spuce has two states, parking occupied by one car. the two slates of otlr objects, we may milure Whatever llic by desigreiic down. If (he magnet points arrows that can only point straight up or straight the up, we say thai ilie magnetic moinenr is -Hii.If the magnet down, points magnetic moment is -m.

physical

Model

Binary

123456789

10 Number of the

1.3

Ffgure moment

numbers

The \302\261m.

sire

each

Mode! system sites on a

at fixed

magnets

has

Syster

ils own

site

composedof 10elementary magncric line, each having shown arc aflachcd to ihc silcs;

magnet

assume

We

there

are no

magne'ic field.Each

moment may be oriented in magnetic up or down, so ihai there are 210disiincl of the 10 magnetic moments shown in the arrangements arc selectedin a random process, figure. If ihe arrangements two

ways,

(he probability is 1/210.

1.4

Figure

State

of finding

ofa

tile

parking

spaces. TiseO's denotespaces denote

vacant

shown

in Figure

spaces.

independent

probability

of arrangements

state of

site;

there

state of

2'v states.

10 numbered

with

particular

sites,

each

parking theO's

by a car;

occupied

This

moment

state is equivalent

nf the We may

bears be oriented

of which may

orientation

of the

the system h sjveitiedUy are

lot

shown

arrangement

to

that

(.3.

Now consider N different assume the values +\302\253i. Each number

particular

n

thai

moment

in two

ways

may

ofa!! other moments.

N moments is

The

2

x

2 x

2 x

\342\226\240 \342\226\240 \342\200\242 *=

2

a

with

total

2\\

A

the orient at ion of the moment oil c:k!i yiviiig for a single use ilio following simplettotation

the system of N sites:

nuimrr-

B)

of a

\342\226\240rl: States

Model Syst

four diflercnl Males of a elements numbered | and 2, vs here ench clctnetit can hsvc two conditions The element is a magnel which can be in condition f orcondiiion [. The

Figure

1,5

s> stem

of two

numbcr4hem to

We may are assumed to be arrangedin a definite order. ftom left to right, as we did in Figure1.3.According sequence the state {2}also can be wriitcn as

sites themselves

The

in

convention

this

C)

symbols B) and {3}denotethe same state of the system, the slate in the magnetic which moment on site 1 is +m; on site 2, the moment is -t-m; on site 3, the moment is -m; and so forth. It is not hard to convince yourself that distinct state of the system is every in N contained a symbolic of factors: product sets of

Both

D)

U)(U

The

is defined

ruie

multiplication

by

ti + till

liXti + li)

(Tt +

+

UU

The function D) on muitipltcation generatesa sum of 2*v the 2'v possible states. Each term is a product of N individual symbols,

denotes an

T1T3I3 For

one

with

magnetic

of

moment

elementary magnet on the line.Each term the system and is a simpleproductof the form

of

state

example.

of two elementary

to obtain the four

possible

states

(Ti + I1KT2+ is not

but

a state

The product on the it generates

for each

one

terms,

f\302\260r

t\\i

a system

The sum

E)

for each

symbol

independent

''\"

+

the states

of the

of Figure

side

of listingthe four of the

system.

(}x

+

li)by(t2

+ |j)

1.5:

Till

ii)

is a way

left-hand

magnets, we multiply

itTa

possible

+

F)

I1I2.

of the

states

system.

equation is calleda generatingfunction:

.

\342\226\240

\342\226\240 \342\226\240

.

Model

Binary

function

The generating

the

for

+

(Ti

of a

slates

system of

This expressionon multiplication generates21 = Three Two

magnets up:

Onemagnctup: None

The in

is

given

M

m

T1I2T3

lihti

tihli

IJ2I3

lilif3

lilils-

up:

be denoted

will

field. The

a magnetic

values

T1T2I3

totat magnetic moment of our

magnetic moment

states:

S different

T1T1T3

up:

magnets

three magnetsis

+ U)-

IjHTj

li)(?2+

Systems

model system

by Mt

value of M varies

from

of

which we will to --

Nm

N

each

magnets to

relate

Nm. The

of

the energy

set of possible

by

\302\253

Nm,

-

{N

2)m,

(N -

-

(N

4>n,

6)m,

\342\200\242 \342\200\242 -A'\302\273i\342\200\242,

G)

possible values of M is obtainedif we start with the state for which all = Nm) and reverseone at a time. We may reverse iV magnets magnets up (M to obtain llie ultimate state for which are down (A/ = - Nm). al! magnets There are N + ] possible of the total moment, whereas there are 2s values states.When N \302\273!, we have 2N \302\273N + 1. There are many more states than states ! 024 distributed values of [he total moment. !fW = 10,there are 210= For N many moment. 11 different values of the total large among magnetic the total moment ft/. of the the value different states of have same system may a given value of M. have in the next section how many states We will calculate

The

set of

are

Only

state

one

of a

system has the moment TTTT-

There arc N

is one

sue!

1

ways

state;

to form

another

a slate

with

-

M = Nm;that

magnet

is

(S)

-TTTT-

one

state

down:

mt

\342\200\242\342\226\240\342\226\240mt

tin

\342\226\240\342\226\240.\342\226\240tin,

AJ)

is \"

do)

1: Slates

Chapter

with one

slates

other

the

and

of a Mode!System magnet down are formedfrom

magnet. The states (9)

any single

of

Enumeration

moment

lot.il

have \302\243!0)

by

reversing

- 2w.

M = Nw

Function

the Multiplicity

and

Stales

and

(S)

spin as a shorthand for even number. We

It is convenient lo elementary magnet. need a mathematical expressionfor the \342\200\224 s magnets number of states with W, = {W + s magnets up and Nl = jN where sis an When we turn one from to Ihe down, integer. magnet up [he down + s s goes to jiV ~ 5 + I. I and orientation, {.V + 5 goes to jW ?N The difference (number up \342\200\224 from 2s to 2a \342\200\224 2. The number down) changes word

the

use

We

assume that

is an

N

difference

-

W,

ihc spin

is called

spin

excess of

right. The facior of 2 in prove to be convenient.

left to

from but

The

excess.

it

will

The productin

D)

may

be written

only

in

many

the order in

of the

magnets

have magnets

sites which

the

arrows

(ID

the 4 states in to be

+

in

\342\200\224

2,

a nuisance at this stage,

if-

from

are

a state

up or down, in a

appear

2,0, 0,

1.5 is

Figure

as

symbolically

drop the site labels {thesubscripts)

how

particular

25

appears

(I!}

(T \342\226\240.

We may

=

/V,

ft

D)

up or we

drop

we are

when

interested

down, and not the

labels

in

which

and neglect

given product, then E) becomes -

(t

II;

further,

(t +

We

find

(I

+

|)v for

I)' = Itt + arbitrary

iV

by

3ItJ + 3IJJ + jjj. the

binomial

expansion

A2)

We

may by

With

ihis result

t with

replacing

W, states

denote

\\N

expression{| +

|)'v

ivv =* y

becomes

tj-v+j

iA-+*

M*V\"J

A4)

of stittes having s magnets down. This class of $N + 5 magnets up and N, = i.V \342\200\224 = lias excess 2s and net raagneiic moment 2sm.Let us JV, spin JVj the number of states tn this class by g{N,s), for a system of N magnets:

coefficient =

but equivatem,

different,

a slightly

in

Function

Multiplicity

\342\200\224 s:

ihe symbolic

4-

The

exponents of x and y

write the

form

and the

of Stales

Enumeration

of the term

in

f

is the number

M*\"'

-

,n\342\202\254>T

(IS)

Thus

is written

A4)

as

I stJMT^l1\"\"

(I + i)'v= We

call g(N,s)

shall

llie same

of

value

ihe

field is applied to the spin s have

states

Note

tn

in a

system:

different values of the

of an energy level

for our

reason

5. The

deltnttion

a magnetic

of different

of

values

to the

field. Until

\302\260ur 9 is equal multiplicity we introduce a magneticfield, all

model system have the same energy, which the total number of states is given that A6)

of the from

of slates having when

emerges

field, stales

magnetic

energy, so that

a magnetic

ihe number

it is

function;

multiplicity

A6)

may

be

taken

as zero.

by

'

L

Examples related to g{h',s)for

coin, \"heads\" down.\"

could

stand

g{Nts) =

A'

~

for \"magnet

A

\\Q are

l)-v =

+

given

upland

in

\"tails\"

(H)

2-v

Figures could

1.6 and stattd

1.7. For

for \"magnet

a

Chapter t: Slatesofa Model

Figtorc 1.6 Number of 5 -f j- spins up and Values

of yf Npi)

tUc spin stales is

oixss

N

TTic values of the the

binomial

of distinct arrangements 5 ~ 5 spins down.

are for N - 10, when: 2.v K I. Tlic toul numtwt \\ -

9's

System

arc taken

h of

fro

coefficients.

I

-10

-8

I

-4

j

Spin

To illustrate that

the

the result, we consider sites, numbered from atom

provision

a

single

2

4

6

excess

2s

Alloy System

Binary

an

0

-2

-6

for

exact an

alternate

1 through

of chemical

species A

sites.

In brass,

vacant state

of the

of the two states

nature

on each site is irrelevant to

with N distinct alloy crystal 1.8. Each siteis occupiedby either chemical species B, with no provicopper and B zinc. In analogy to C),

system\342\200\224an

12 in Figure of or an atom A

alloy system

could

be

can be written

as

-

A8)

nry

Allay

Sya,m

\342\226\240= o

S

3

20

Number

Figure

were throw

1,7

An

experiment

NX) times.

thrown

10

23456789

01

of

heads

Was done in The number of

10 pennies

which

heads

in

each

was recorded.

0\302\25100 3

2

I

A

Fijutc

\302\251 5

chemical

0 6

10

0

7

S

II

012

1.8

A binary componenls

alloy syslcm of two A and 1!,whoseatoms

of a

state

distinct

Every

system on

binary alloy

in the

is contained

sites

N

symbolic product of N factors:

(A, + in

conventionally N

B2)(A3

+BN) ,

+ Bj)---(A.V

A9)

The Liverage composition of a binary is specified conalioy the chemical formula A1_1B1, which means thai out of a tola! by the number of A atoms is NA ~ A ~x)N and the number of B

to

analogy

of

4-

B1)(A3

atoms,

atoms is NB

D).

~

.v lies

.\\JV. Here

between Oand 1.

The symbolic

expression

is

of

g{i\\\\f)

B on

which

is identical

prediction

of

B' gives the

in A'v\"'

term

of N ~

A

\302\243 atoms

and

/

result A5) for

to the

the

spin

model

system, except

for notation.

Function

Multiplicity

experience that systems held at constanttemperature well-defined this stability of physical propertiesis a properties; The of thermal physics. stability follows as a consequenceof

have

exceedingly

bharp

function

that

the

from common

know

major the

of

or states

arrangements

possible

Sharpnessof ihe usually

The coefficient

N sites:

atoms

We

result A2).

to the

analogous

number

peak

in the

from

away

the steep

of and multiplicity function can show explicitly

the peak. We

large system, the function ) ^fe showti. Tor this

magnetic nmmersis

are labeled by

Example:

Multiplicity

system is

the

given

by Max Sludent

problem is given

The quantum

slates

problem is the

solvable

exactly

simple. The beginning do the

magnetic

s values,

oscillators. The problem of tlic function for harmonic for which an exact solution for the multiplicity problem

simplest

known. Another was originally

m m a

their

in

ofa

Chapter

harmonic

the

quantum

the oscillator. consider

a system

number of ways

number

The number

of N such in which

states

is

infinite,

oscillators,all

a given

total

derivation.

energy

The

excitation

of

modern

way

to

eigenvalues

D9)

or

zero,

and

the

the

is

solution entirely

sho) ,

s is a positive integer of

this

oscillator have the es =

where

for which the oscillator, is often felt to be not derivation

nol worry about 4 and is simple.

model

function

harmonic

Planck. The original need

binary

and to is the

angular

multiplicity

of each is one. Now

same frequency.

energy

We want

of

frequency to

find

the

can for

be distributed tlie

among

e^rher. pitcitv function fount! We begin the analysis by =

forwm'chff(i,\302\253)

1

problem of E3) below,

we

the

function

multiplicity

the same

as the

spin

g{N,n) mufti-

function for a single oscillator, here identical to m. To sojve the

numbers,

quantum

a function

need

the

is not

to tlie multiplicity

back

of

want

function

multiplicity

going

ail values

for

is, we

That

oscillators.

the

The oscillator

Af oscillators.

to represent or generate

ihe

scries

E1)

AS! Y,fl!!1 from

(S3),

but

^

^CfC

coS\302\260

not appear

t docs

' 's the

in

a temporary tool that result. The answer is final Jusl

will

help

us find

the result

(S2) provided we assume\\i\\

<

|. For

the problem

of JV

oscillators,

the

generating

function is

E3)

becausetlie

of w;iys n term in which the

number

number of onSctedwuys We observe

i\"

can

integer

;\\\\i\\Kai in the N-fold n c;m be foiuicJ as the

pftiJuct

sum

is picciscly ihe of iV non-iicg.nive

that

tj{N,n)

2) Thus

for the

\342\226\240 \342\200\242

(W

+

n

- 1).

E4)

system of oscillators,

ES) This

result

will be

needed

in

solving

a problem

in the

next

chapter.

1: States

Chapter

of a Mode!System

SUMMARY

1.

The

In

function for a

multiplicity

N, -

N't

limit

ihe

syslem of N

with

magnets

spin

excess

2s =

is

s/N

A'

with

\302\253 1,

\302\273 1,

we

have

the Gaussian

approximation

g[N,s) * {2/rlN)m2xexp{~2s2/\\').

2. Ifal!

of

states

the

mode!

spin system

are equally likely,

the

average

value

of

equal

to

2

52>

in the

3.

The

=

j''^JsstgtN,s)

p

Gaussian approximation.

fractional

of s2

fluctuation

is defined

as (s2yll2/N and

is

S/2N\022.

4. The

where

energy of the modelspin

in

is the

magnetic

syslem

in a

siaie of

moment of one spin

and

spin excess 2s is

B is

the magnetic

field.

2

Chapter

and Temperature

Entropy

11

ASSUMPTION

FUNDAMENTAL

PROBABILITY

3'\\

of ;in

Construction

Example:

3-

Ensemble

Most ProbableConfiguration

33

Spin Systems in

Two

Example:

Thermal Contact

3?

39

THERMALEQUILIBRIUM TEMPERATURE

-\342\226\240!

ENTROPY

41

Floiv

On Heat

45

of Entropy

increase

of

Law

Increase

Entropy

Example:

LAWS OF THERMODYNAMICS as

Entropy

-iS

50

a Logarithm

Example: Perpetual Motionof the

Second

Kind

50

SUMMARY

51

PROBLEMS

5:

1.

Entropy

and

52

Temperature

2. Paramagnetism

52

3. Quantum HarmonicOscillator

52

4.

5.

The Additivity

53

of \"Never\"

Meaning of

the

Entropy

for Two

6. Integrated

Spin Systems

54

Deviation

Note

we Jo

on problems: The iitil

cinplusi^e

53

melhoJ of fhis

problem

c

chapter

soKing dl

lliis

siu

Chapter

2; Entropy

and Temperatui

One shouldno! imagine will

mix,

the

one

contrary, \302\260

W10

will

years

recognize that

ff

we

thai

two

wish

there is

this

to find

of thermodynamics,

in

in a

gases

after a few days finds .., ilia!not

then again

0.1 liter

separate, until

a time

by any noticeable unmixing equivalent to practically

rational

we must

then

container,initially mix again,

unmixed,

and so forth.

long compared One may

enormously

On

to

the gases.

of

never. . . .

an a priori foundation for the seek mechanicaldefinitions of temperature

mechanics

principles

and

entropy.

J.

W.

Gibbs

between energy and temperaturemay are considerations. in statistical {Twosystems] by probability a transfer does increase the not probability. of energy The

genera}

connection

M. Planck

only

be established

equilibrium

when

We slart

this chapter

that enables us to a of average physical property system.We then consider in thermal equilibrium, the definition of entropy, and the definition of systems The of will as the taw second law of temperature. thermodynamics appear increase of entropy. This chapter is perhapsthe most abstract in the book. The chapters th;it follow wilt apply the concepts to physical problems. a

with

value

the

define

definition

'

FUNDAMENTAL ASSUMPTION fundamental

The

assumption of thermal of the quantum states

likely to be in any arc assumed to

be equally

states

accessible

states

A

closed

constant

system volume,

over

of

probability

\"

\342\226\240 -

of a

ttt;tt

a closed

accessible to it. All

probable\342\200\224there

accessible

other

physicsis

system

accessible

is equally quantum

to prefer

is no reason

some

states.

energy, a constant number of particles, values of all external parameters that may

will

have constant

and

constant

including gravitational, electric, and magneticfields. the A quantum state is accessible if its properties arc compatiblewith physical of the system: the energy of the stale must be in the range within specification which the energy of the system is specified, of particles must be and the number in the within which the number of parlictcs is specified. Wtlh range large systems we can never know either of theseexactly, \302\253 1 but it will suffice to have.SU/l/

influence

tmd&N/N Unusual

the system,

\302\253 I.

properties

of a

system

certain states to be accessible during

may

the

sometimes

time

make

the system

it

impossible

is under

for

observation.

at form of SiO2 are inaccessible low or starts with the that temperatures glassy amorphous fused form: in a low-tcmpcraturc to quartz in our lifetime of this type by commonsense. exclusions experiment. You will recognize many We treat are excluded all quantum states as accessible unless they by the the scale of the measurement of the time specification system (Figure2.1)and process.Statesthat are not accessible are said to have zero probability. Of course,it is possible to specify the configuration of a closedsystem to a If we specify that ihe are of no interest. point that its statistical properties as such

Fof example,the

states

of

the crystalline

in any observation silica will not convert

2; Enxropy

Chapter

and Temperature

I imtt

of

of ihe

spcMftcation

sjstcn

2, t A iwdy symbolic Ji:iKr;ihi: L-:idi solid s|x' slate of a closed sysn represents an accessible quantum fundymema! of statistical pliysics is tliat a assumption is equally likely to be in any of tlic quantum si; system accessible to it. \"Die empty circles represent some of thi that are not accessible because their do nc properties the specification of the system. vjfju -1 ,-/ cmiai Lo the energy loss of ihc first) ihus the energy increase of ihz second is, in joules, specimen

(aj

Let

with an

AV

where

=

C.89J

ihe tempcraiures

- 290K)

K-'HTV

are

in

Tj

\302\253

|C5O

~

linat temperature

Ttie

kcMn.

- C.89JK-')C5OK

+

290JK

after

contact

Tf)

,

is

= 32OK.

Thus

At/,

=

\302\253

C.89JK~!)(~3OK)

-11.7 J ,

and

At/3 = -At/, (b)

What

is the

taken place, almost fraction of ihe final considered temperatures of

change of entropy immediately

after

of

the

initial

two

= U.7J. specimens

con'act?

when a transfer pf

Notice

that

this

transfer

0.1J has small

is a

contransfer transfer as calculated above.Becausethe energy at their initial temperawe may suppose the specimens are approximately of the firsi body is decreased by 350 and 290 K. The entropy

is small,

energy

Lan

Tile cnlropy of iiic

second

=

S2

iotal

Tile

increases

enlropy

AS, +

of the

= 3.45

,7~

of Enlnpy

by

x

by

10-4JK-' = 0.59

AS, = (-2.S6 + 3.45)x units the increase

In fundamental,

where

is increased

hody

of Imrrrmc

x

of entropy

1CTJJK~

*s

Botl/.ma\302\253n constant. This resuil mcaaS thai (lie number of accessible tjisihe - [email protected] 10l9>. two systems increases by (he factor exp{M

st;it

Law oflncrease of Eniropy We

can

ihc loial

thai

show

broughi If (he total energy V into

systems

arc

thermal

in

contact. ~

ctttrapy

two

jusi demonstrated this in U2 is consiant, the lotal multiplicity

V% +

systems a special after

;trc case. the

is

contact

thermal

when

increases

always

We have

ff(t/)\302\273=

^0,A/^A/

by A8). This expressioncontains the

term

t/,)

gi(EAo)i/i(^

C3)

,

~~

^to^

^or l^e

i-XiiiVd^

terms besides.Here ViQ is the the initial energy ofsyslem 2. Because initial energy ofsystem 1 and V l/lois increased all termsin C3) are positive numbers, ihe muitipliciiy is always by This is a proof of Ihe establishmenl of ihermal conlaci bclween two systems. taw of increase of entropy for a weli-definedoperalion. effect of conlact, the effect that slands out even after lakingthe Thesignificant of tcims in iiic summaof ihe multiplicity, is not just that Ihe number logarithm be very, summationis large, but that the largest single term in the summationmay very

niuitiplicuy

before

contact

and

many osher

~~

much larger than

the

initial

muilipiicity.

(Mi).,,

That

is,

= 9i(O,)gJiU

- 0.)

C4)

2:

Chapter

and Teniperatur

Entropy

with

Ut

= 0

parts and probable conftgL

the

presently

con figuration. rat ions

be very, very

Hie cm ropy

in or

found

increases or

muhiplicily

much larger than

ihe

moit

initial

essential

The

value

is lhat

effect

of

for which

Vl

the syslems

states

term

C5)

ihe product g^x is a cvoive

contacl

after

configurations lo their final configurations. implies thai evoluiion in this operation will final

configuratic

probable

Vl0).

9iiVtMU'~

Here 0s denotesthe

The entropy

probabilhy.

a{U) oflhe

ihe entropy

takes piao; between h dose 10 tlie most as the jysicm attains

of energy

will be

syucm

ofincreasirtg

reaches

eventually

may

* U. Exchange

anJ U,

maximum. iheir

initial

fundamental

The lake

always

from

place,

assumption with ali accessible

probable.

eqtutlly

The statement

C6)

fffjnil

statement of the law of increaseofcnlropy:the entropy when a constraint [ends to remain constant or to iucrease

of

is a

is removed.The operationofebtabitsliiiig removal of the constraint that Vu Ux

+

U,

need be

configuration

V2

each

contact

be constant;

u closed

to she

is equivalent

system

sjstem to ihe

after contact

only

constant.

The evolution of the takes

thermal

iniernul

combined

a certain

system

time. If

we

lowards separate

ihe

final

ihe two

thermal

equilibrium

systems before they

Add

energy

molecules

Decompose

Let a

2.9

Figure

(his cotifiguraiion,

reach intermediate

view

and

energies

the entropy

constraint,

called

Processes 2.9;

Operations

function

(he lime

that lend

the arguments

that follow.

as a

in

we

io increase

lend

thai

will

an

obtain

an intermediate of

the

lime

of evolution

in

of each

ofa

entropy

intermediate

up

syslcm.

configuration

with

io entropy. Ii is ihesefore meaningful tli.i' lias elapsed since removal of the

2.8.

Figure

lo increase the eniropy support

the

linear polymer curl

process

ofa will

system be

are

developed

shown in the

in

Figure

chapters

2:

Chapter

For a largesysiem* occur

never

/

and Temperature

Entropy

with another large sysiem)ihere will differences belween the actual value of the significant

thermal

(in

spontaneously

coniaci

value of the entropy of the most probableconfiguration of the system. We showed ihis for ilie model spin sysiem in the argument following A7); we used \"never\" in ihe sense of not once in ilie entire age of the 10's s. universe. We can only find a significant difference beiwcen Ihc actual entropy asid ihe

entropy

and ihc

entropy

of ihe

shortly which

most probable have

we

afier

implies

that

the nature

changed we

had

the

of

configuration

the system

prepared

system

macroscopic

of ihe contactbetween initially

in

two

some

very

systems,

special

way.

Special preparation couldconsistof lining parallel system up all the spins in one or to one another in the air of the room into the collectingall the molecules a small volume m one corner of the room. Such extreme by system formed situations artificial

never arise operations

m

naturally

on the

informed

left

systems

but

undisturbed,

arise from

system.

Consider ihc gas in a room: the gas in one half of the room might be prepared wjiti a low value of the average initially energy per molecule, while the gas in ihc other half of the room might be prepared with a higher value of the initially average energy per molecule. If the gas in the two halves is now allowed to interact of a partition, the gas molecules will come by removal very quickly' to a most probable configuration m which ihe molecules m both halves of the to room have the same average energy. Nothing else will ever be observed the We will observe to leave most never ihe sysiem configurahappen. probable configuration and

reappear

later

in the

even ihough the equations distinguish past and failure.

LAWS

OF

initial

specially

of motion

prepared

of physics

is true

configuration.This

are reversiblein

time

and

do not

THERMODYNAMICS

is studied as a nonslatisticaisubject,four posluiales of thermodynamics, in are introduced. Tiicse postulates are caiied ihe laws of thermal formuiaiion our statistical essence, these laws are containedwithin physics, bul it is useful to exhibit Ihem as separate slatemenis. a third If two sysleins are in thermal equilibrium with law. Zeroth sysiem, isa with olher. iaw be in each This must thermal consequence equilibrium Ihey When

1

The

thermodynamics

calculation

of Ihe lime

required

for

Ihe

process

is largely a

problem

in

hydrodyna

Laws

condition B0b)for

of the

(e\\oSgt\\

*

{~\342\204\242rk

in oilier

words,

=

r,

comact:

in thermal

equilibrium

of Thtentotiynamks

feioSg3\\

/cloggA

/cloggA

{-furl:

{imX=

{~7urk

= t3

t3 and r3

=

rj

imply

r2.

Heat is a form of energy. This law is no more than a slaicment of liie principle of conservationof energy.Chapter 8 discusses wliat form of energy First law.

heat is.

Second

entropy, applicablewhen is ill :l

of the

tluit is not

configuration

of ihe second law. We ilie law of increase of

of

law

(he equilibrium

consequence

will

in successive

instants of time.\"Tins is

wilh Eq.

called

lo a dosed systemis removed. The increase of entropy is: \"Ifa closedsystem

ihc enlropy of the

be lhat

have

iniernal

a constraim

used statement

commonly

statements

equivalent

many

statement, which we

the statisiicai

use

shali

are

There

law.

cotiliyitnilton.ilicmosi |>rubnble system will increase monoiotiic;ilty siaicineiil

a looser

ilian

I

he

one

we gave

C6} above.

The traditional of

is the Kelvin-Pianck formulation is for \"it to any cyclic process impossible of heal from a reservoir and the perextraction statement

thermodynamic

of second iaw

thermodynamics;

whose soie effect is the of performance an equivalent amount of work.\"An engine that vioiaies lhe second iaw by extractingthe energy of one heat reservoir is said to be performing motion ofthe second kind. We will see in Chapter 8 that the Kelvinperpeiual

occur

statement. Pianck formulationis a consequence oflhe statistical as the Third iaw. The entropy of a system approachesa constant value due zero. The ear! test of this statement temperature law, loNemst, is approaches at that ihe absolute zero the entropy difference disappears between all those

iaw follows

is

multiplicity

the

system

g@),

the

in

has

thermal

internal

definition

statistical

the

from

ground stateof

which are

a system

of

configurations

corresponding

zero. Glasses

is essentially

substantial,of in

real

life is

to

objection

the

order

that curves

must come in

flat

as

affirming

have a

of [he

The third

entropy, provided ihat the multiplicity. If lhe ground stale as t -* 0. entropy is o{0) = iogy@) is that does not appear to say much of the

a weii-defined

From a quantum point of view, the law not implicit in the definition of enlropy, provided, in its lowest se! of quantum at absolute states

would not be any

equilibrium.

frozen-in

number

of many

r approaches

reasonable

0.

the system is for zero. Except glasses, there that (j{0) is a small numberand c{0) for them o{0) can be and disorder, the third law tells us of atoms N. What plotted against x quantities physical however,

that

Chapter 2: Entropy as a

Entropy

and

Logarithm

Several

useful

stales itself.First,

the

ihe definition

from

follow

properties

number of

of the

rithm

1

Temperature

accessible states, hut two

of

entropy

of (he cmropy as the iogathe number of accessible systems is liie sum of lhe

of as

cad

independent

separateentropies.

Second, the

the

never meant to for

that

entirely

imply that

a discrete

tite

practical

purposes\342\200\224-to

We have

defined.

is

exactly, a circumstance

is known

energy

system

of energy

spectrum

ali

insensitive\342\200\224for

the energy of a closedsystem

with wiiich

6U

precision

is

entropy

eigenvalues would

of

number

the

make

We have depend erraticallyon ttie energy. simply not paid much attention io lhe precision,wlicthcr ii be determined by the uncertainly h, or determined otherwise. Define monkey

Cliapter2; Entropy (c)

How

and

Tcmperatur

iarge is the fractional

error

in

the

entropy

when

you

ignore this

factor?

6, Integrated

approximately the

is 10\"l0 \342\226\2405/JVj

use an

x

probability

or larger.

calculate

lhat gave ihe result A7), example the fractional deviation from equilibrium that = = You find it convenient to will IO2Z. iVj JV2 the

For

deviation.

Take

asymptotic expansion for

the

complementary

error

\302\273i,

xp(x2)

(\"\"e

x

1 +

small terms.

function.

When

3

Chapter

and

Distribution

Boltzmann

Helmholtz Free Energy

FACTOR

BOLTZMANN

58

61

Function

Partition

Example: Energy and Heat Capacityof a Definition; Reversible Process

Two

State

System

64

PRESSURE

6-1 ;>7

identity

Tlicrmodynaimc

Example: Minimum ParamagneticSystem

6S

ENERGY

KttEE

HELMHOLTZ

of

Property

the Free

70

7!

Maxwell Relation

Calculationof f IDEAL

GAS:

One Atom Example;

71

2

from

72

LOOK

FIRST

72

a Box

in Af

A

in a

Atoms

74

Box

76

Energy

Example:

of

Equipartition

Example:

Energy of ;i 69

Relations

Differential

62

77

Energy

7$

Entropy of Mixing

SUMMARY

SO

PROBLEMS

81

1. Free 2.

Energy of a Two State Sysiem Susceptibility

Magnetic

3. Free Energy of a Harmonic 4. Energy Fluctuations 5.

Effect

Overhauser

6. Rotation 7.

Zipper

O.ollaior

of DiatomicMolecules

Problem

S. Quantum

Concentration

85

Si S3 S3

S4 84

S5

S3

9.

auJ

Bol

ChapfcrS:

Partition

10. Elasticity

Function of

for Two

Polymers

11. One-Dimensional

Gas

lleliiiiioiiz

Systems

Free

Energy

Chapter3; BolRinattn

The

laws

statistical

We

are

of thermodynamics

mechanics,

able

of

to distinguish

which

on anotherfrom lhai which to specify cases of thermal

in

and

Distribution

lleimkoitz

may easily be obtainedfrom the principles are the incomplete they expression. Gibbi terms

mechanical

we call

mechanical

action and

the thermal in

the

narrower

Energy

of

action of one system sense . , . so as

cases of mechanicalaction. Glbbf

Free

this

In

of the

properties

physical

sysiem

iarge sysiem (ft,

called

system

as

particular,

in

of

reservoir.

The system

of

the values

calculate

Figure

3.1.

The

total

temperature. wiih equilibrium

a very

and the reservoirwill

energy Uo

sysiem is in a staleof energy

We

ihe

r because ihcy are in thermal contact. (ft + & is a closed sysiem, insuiaicd from

The iota!

if the

&

[he

temperature

influences,

us to

permit

a system as a ftinciion interest io us is in thermal

of

assume that the common

the principles that

we develop

chafer

=*

Ea, tltcn

U^

Uo

+

u!!

external

constant.

is \302\243/j

-

ihe

r,x is

a

have

energy

In

of

the reservoir.

Toial sysiem

.

Constant

J.I

energy

Vo

Rcprcs illation of a cioscd a! coniaci with a

irOtiiUhcn

BOLTZMANN A

5

central

problem

\302\253iilbe

proportional

to

in a the

loiat sjsieni

syst S.

iisioa

n decomposed

FACTOR of thermal

specific quantum Boitzmann factor.

10 find siaiu s of energy

physics is

the

probability t,.

This

ihe system is propor-

iltai

probability

be in ihe state s, ihe number When we specify that S should number of accessible Slates of the louil sysiem is reduced10the reservoir (H, ai ihe appropriate energy. That is, ihe number

of

accessible

states g\302\256t j

of

of ihe siaies

Figure 3.2 The change nsetvoit u&nsfcssenergy

fractional effec!

of

laigc TCMivoii

accessibleto (ft

the reservoir

of

Energy

of cnlropy t la

itie transfer

vil\\

luua

high

-

-

A)

for our

because

If the system

to \302\243,

Ihe

ratio

of Ihe

the

the

system

dependence

Multiplicity

of



system

in

t,},

-

(b) is in quantum c,) acccisiblc quanluin

(a),

slate

t, 2. The

slates,

(,\302\273(()\342\200\236

reservoir

in (a) and

(b)

D) the probability ratio for

two

the

1, 2

states

of the syslem is simply

E) Let

us

expand

in D)

the entropies

in

a Taylor

The Taylor seriesexpansion off[x)about

-

/(x0)

series

expansion

about

is

t)

0) where

1/t

=

(S^/cCV^

gives the

temperature. The partial derivative

is taken

Uo. The higher order termsin large reservoir.*

al energy an

infinitely

Therefore

defined

Acr^

-(\302\243,

ofvust

form exp(

the

of

term

It

utility.

h

ttie

to consider the

ealied the

exp(-e,/t)

=

in a

system in

:i

single

for

a!!

We see that

=

result

(II)

is one

average energy of the

= 1:

ZjZ

\302\243?(\302\243,)

U

probability

is V

useful

= (e)

oft'j, ~

convergence

\342\202\254)andnolg({/B

difficulties.

~

partition

p[Et)

and

is

function

(he

results

= X^fo).

of

the

Boltzmann

statistical

pro-

factor

e) because

[he cupansion

physics.

The

or

= T^logZ/ct).

=,

Zh^Zh!A

We expand

the Boltzmann factor

is unity. the sum of all probabilities

of the most

system

is over

system. The

the

5 of

states

A0}

,

5>p(~Ei/T)

summation

The

function.

partition

proportionality factor between the

gives

system

function

Z(r)

\342\200\242

the

Function

is helpfui

The

This result is

Boltzmann factor.

of Hie probability of finding of finding ihc probability

ratio

to

I

'

,as 3

known

is \342\200\224e/t)

the

gives

single: quantum state state 2. quantum

Partition

(8)

-\302\2432)/T.

expft/r)'

P{ez)

A

of

limit

E) and C) is

result of

final

in the

vanish

expansion

D) becomes

by

Affffl= The

liie

of ihc

'

A2)

tatter quanliiyimmcdiatcty

Battvnanti

ChapterS:

Helmholtz

and

Distribution

Free Energy

0.5

A

0.4

J-\342\200\224\342\200\224

Energy ystcm rgy

and

heat

as functions is plotteJ

capacity of a ofthe temperature

in units

V

J

ol t.

0.1 \342\226\240

u

0

The with

a

called for

(e)

refers to

energy

average

reservoir.

The

thermal

the

notation

average

in conformity

those statesof a

with

and not, as earlier,to the

can

that

system

\342\226\240 ) denotes .)'

F)

This is the Planck distribution

photons(Figure

4.3)

number of with

energy

PLANCK The thermal

in

for

function

a singie

mode of

AND

average

frequency w. Equally,

phonons in the mode.The result in the form of (!). LAW

thermal

the

lo

applies

any

it is

number of the average

kind of

wave

field

LAW

STEFAN-BOLTZMANN

average energy in the modeis

\342\200\224 )

1*

G)

4: Thermal Radiation

Chapter

Disiribttlio

Planck

and

+ *(\302\253)

as a function Figure 4.3 Planck distribution ofihe reduced temperature i./rw. Here is Hie thermal average of the of rmniber photons in the mods: of frequency en. A plot of where O(o)> + i is also given, $ is the effective 7ciopoint occupancy of ihc mode; the dashed line is i!ie classical asymptote.Noie that we

0.5

/

/ A

The

high

be

may

exp{frfcj/t)

t

limit

temperature

0.5

as

approximated

often

is

\302\273 ha)

lna/r 4-

1 4-

limit. Here

the classical

called

* \342\200\242 whence \342\226\240,

the

classical

average energy is ^

There mode

n

is an has

own

conducting cavity

in

frequency the

form

wn.

of a

For

(8)

modes

of electromagnetic

number

infinite

its

T.

radiation

within

cube of edgeL, there

is

cavity.

any

Each

a perfectly of modes of the

within

confined

a set

form

Ex

=

ExOitn

wtcos(fiJji.v/L)sin(iiyjij'/L)sin(fi.Jiz/L)

ID Et

Here Ex, Er and are \302\243;0

the

independent,because

sin(fl=Tiz/L)

,

(9a)

,

(9b)

= E-0 Ex

are

(9c) the

three electric

field components, and

The three

corresponding amplitudes. the field must be divergence-free:

components are

\302\243lQ,Ey0 not

and

indepen-

A0)

When we insert {9}into A0}

and

+

\302\243,0\",

the

field vectors must

rhar the

states

This

nx,

components

4- E:Qn:

E^nr

ny

transversely polarized

be

condition

the

find

-. Eo \342\226\240 0n \302\273

A1)

vector

to the

perpendicular

the electromagnetic field

>l, so that

and

Law

Slcfan-Bolwtann

factors, we

ail common

drop

and

Law

Planck

field.The polarization direction

is defined

in

the

n

as the

with

is a

cavity

direction

of Eo.

For a

given

n,,

triplet

directions,

polarization

can choose two so rhat rhere are mo distinct nft

n. we

On substitution of (9) in the wave

c the

wilh

velocity

of light, we

if we \";\342\226\240

\"y

>h<

4-

are of the

The total energy of the

iij

independent

by an integral

indices.

That

n;2)

mode

the

is,

over

we set

+

,,/

=

photons in the

= w3L2. in terms

A3) of Hie triplet

of integers

+

A4)

, rta\302\273)\302\273'i

A5)

mrc/L. is, from

cavity

G),

integers alone will the sum over nx, modes of tlic form (9). We replace the volume clementditx dny dnx in the space of the mode

The sum is over the triplet of ny,

+

Hya

form

w.

all

iriplei

define

the frequencies

describe

each

.(V

'.

trJJ.

\302\253) of

= (rtj[2 \342\200\236

then

perpendicular for

find

cWnJ

This determinesthe frequency

modes

equation

i:y

V'-v1

mutually

integers nx,

ny,

n..

Positive

Chapte

where the factor We

involved. two

|

now

the sura

the positive

only

or integral by

a factor

of the electromagnetic

polamations

octant of the Spaceis of 2 because Ihere are field (two independent

Thus

modes).

cavity

because

arises (\302\243K

multiply

independent

setsof

=h

1, = ji ft

hu)n

Jo

with

A5) for

over

a dimensionless

Standard \302\253\342\200\236.

dnn*

r

(nVic/L)

\302\273\302\260

is to

practice

We set

variable.

\342\200\224

A8)

ex

transform

x =

the definite integral to one and

nhcn/LT,

A8)

becomes

A'J)

integral has the value z*/l5; it

The definite

such as Dwight

in the

(cited

general

is

in good

found

references}. Tlie

standard

energy per unit

the

volume

lional !olhe of

law

fourth

V =

L1. The

power

oflhe

result that lemperalure

is

B0)

\\Shs

with

volume

tables

the

radiant

is known

energy

density

is propor-

as theStefan-Boltzmann

radiation.

we B0) into the spectral decompose applications of this theory as the energy per unil of the radiation.The is defined density spectraldensity We find \302\273u from and is denoted as \302\253\342\200\236,. can volume per unit frequency range, in terms of w: (IS) resvritlen

For

many

B1)

U/V

so

that

the

spectral

density

is

B2)

Planck

Law

andSufan-Boltzmam

Law

A

\\

1.2

1.0

/

\342\200\224

/

/

0.6

1

0.4

0.2

\\

/

\\ \342\200\224 \342\200\224

/

0/

~ l)willi.v = bttf/t. T\\\\h runciion h involved in the Planck radiation law for llic of a black body may spectral density uw. flic temperature be found from ilie frequency tjmil ai which the radiant Figure-1.4

Ploiof.vJ/(c*

is a maximum, energy density per This frequency is directly proportional

unit

ffequency

so ihe

range.

tempera sure.

of distribution This result is the Planck radiation the frequency law; It gives thermal radialion (Figure4.4).Quantum here. theory began the relation The entropy of the thermal photonscan be found from A34a) ~ at constant volume:da from B0), dUfr, whence

Thus

the entropy is

B3)

The constantofintegration

is zero,

from

C.55) and the

relation belsvecnF and

a.

4: Thermal

Chapter

gy

flux density

area

and

length

is of the order of the of equal to the velocity

factor is equalto \302\243; the

The geometrical

The

by

for

result

final

use of

gy

v

y

The

Distribution

Planck

and

Radiation

the

radiant

the energy density

B0) for

light

the

is (he

time.

of

unit

of

Thus,

subject of Problem 15.

The

is often written

result

as B6)

\302\253

aB s= b2V/60AV

has the

x

5.670

vahie

10~8

W m~2

(Here cts is not the entropy.) A

as

a

black

A small

body.

body

the

on

10\"*

K~* or 5.670x that

a cavity

radiates whose

walls are rate walls

in

is said to radiate

thermal

given of

K~\".

s~'

cm\022

erg

at this rate

as a blackbody at the of the physical constitutionof the

is independent only

in

B6a)

T will radiate

at temperature

depends

hole

unit

is

flux

U/V.

Jv

times

derivation

energy

p

a column

in

contained

energy

equilibrium in

B6).

the cavity

The

rate

and de-

temperature.

Emission and

Law Absorption; Ktrchhoff to the ability of the The of a surface to emit radiationis proportional ability surface to absorb radiation. We demonstrate this relation, first for a black body

or biack

surface

is defined to

incident

upon

biack if

the

and,

second,

be blacktn it

hole

for a

surface

with

arbitrary

properties.

An object

electromagnetic radiation a ho!e in a cavity is in that range is absorbed. By this definition small incident the hole will is enough that radiation through a given

frequency

range if all

t

reflect

times from the cavity through the hole.

enough

in the

be absorbed

lo

walls

cavity

with

back

loss

negligible

oj Sutj

The radiant

a black surfaceat temperature x is from a small hole a in equal density Jv cavity al the same temperature. To prove this, let us close the hole wilh the black in thermal the thermal average surface,hereaftercalledthe object, equilibrium fiux from the black object to the interiorof the be equal, must energy cavity but opposite, to the thermal average energy flux from the cavity to the black to the

energy

flux

radiant energy

Jv from

density

emitted

flux

object. We

the

prove

following:

fraction a of the

non-black object

If a

radiation incidentupon

at

temperature

t

absorbs

a

by the emitted by a black body at the same and e the emisstviiy, where the flux emitted by the object is e times it,

the

radiation

flux emitted

object will be a ttnies the radiation flux temperature. Let a denote the absorptivity cmissi\\ity is defined so that the radiation the fiux emitted Theobjectmust emit by a black body at the sametemperature. at the same rate as it absorbs that if equilibrium is to be mainiamed. H follows a is law. For the special case of a perfectreflector, a~e. This is the Kirchhoir

whence e is zero. A perfect docs not radiate. reflector The argumentscanbe generalized to apply to the radiation at any frequency, icti mooe Kept con^JunlL We show in ^joj that ihe entropy is constant if lhe number of pholons in each mode is consiant\342\200\224the

cooled

was

remain

of

;uid itie bhiirk body radiaiion were in Itiaiipl cqtiitilirium. IJy ihe lime ted lo 300A K,! tie m;iiUv Mas primarily in she farm of atomic liydrogen. with bLjck body r^diLitJi^Ji si the fic^ucOci^b of jlic liydro^jcti &fH:ctriJ only itie of llie biack body r;idi;ition thus was cffctiutty docouptoti from clergy

nimicr

Ihc

ihaf

itic

tnc

below

lhe

After inlo

ihe eniropy. Urn evoluiion

determine

occupancies

decoupling

and

stars,

galaxies,

Electromagnelicradiaiion, is superimposedon the

black

cosmic

As an important exampleof the

sponMucousthermal called

H. Nyquisi.*

noise,

H. N!jquisi,p!:js

lical jijsus,

ts

4.6. We

\342\226\240

Wiley,

botiy radiation.

shall

R !

sec

across

that

by J.

property of lo llic

proportional

/4J!.

frequency

refers

to cycles

per

unit

time,

and not

to

radians

per

unit till

Noise generator

4.7

Figure

load R'. The

which this

hi

power enables

oftliermal noise that

Consider characteristic

as in

impedance

is a maximum wiih condition the ioad At

supply.

us to

the

limit

voiiagc

be

absorbed

circuit is maintained

power

said

at

= R.

when R'

lo the

to bo matched

- (Yi)f4R. The

filter

bandwidth under [lie bandwidth 10 whicii the mean

fluciuaiion

applies.

line Figure 4,8 a losslesstransmission ~ R terminated at eachend Zc by

wiihoui

with

to a

frequency

is.

ihal

to R'

respcel is

&

match,

line is matched at eachend, in will

resistance ft

delivers

cilrrenl

consideration; square

ciicuii for a

Equivalent

a generatot

sense

the

reficciion

in

that the

ail energy appropriate

of lenglh a resistance

L and

charac-

R. Thus

the

traveling down the line resisiance.

The

entire

t.

temperature

line is essentiallyan electromagnetic sysiemin one dimension. We follow ihe argument given above the distribution Tor ofphoions in thermal but now in a space of one dimension instead of three dimensions. equilibrium, has two photon modes (one propagating in eachdirection} Thetransmission line = 2nn/L of frequency in the from A5), so shat freihere are iwo modes 2nfa A transmission

frequency range

C0)

Sf~c'lL,

where c' is the

propagation

velocity

on the

line. Each

mode has energy C1)

exp(ftwAJ

Figure 4.8

Transmission

Hue

derivation of ihc Nyquist aclerislic line

has

ihe

heir

The

liieorcm.

impedance 7,c of ihe [ ra us mission vaiuc R. According lo ihc of Iransrnission iines, [he matched to the line ivlicn

theorem

mdamcniai erminai

L with

oficiijjtli

resistors

resistance

are

same value R.

has ihc

in

is r. It

in

the

that

follows

hw

limit

classical

the

Planck distribution.

to the

according

equilibrium,

with circuits

on the

energy

\302\253 z

so

line

in

the

are

We

energy per mode A/ is

range

frequency

concerned

usually

the thermal

that

C2)

The rale at which

line in one

off the

comes

energy

direction is C3)

The powercoming

off

the

al

line

end is

one

that end; there areno reflections

impedance

R at

is matched

to the line.In thermal

line at the same rate, or elseitstemperature to the load is

9 = but

V

~

2R1,

itt

temperature(hermomctry,

dc current

when

no

when

a dc

iA/

the power input

C4) used

been

{Figure 4.9)

where

in low it

is

tempera-

{not

con-

more

than t. Johnson noise is the noiseacross (V1} discussed is flowing. Additional noise here)

resistor

appears

current flows.

PHONONS IN SOLIDS;

DEBYE

THEORY

calculate the spectraldistribution of this distribution for a continuous solidand to consider So I

energy to the

,

The result has

regions

temperature

impedance

a

measure

to

convenient

B8) is obtained.

so that

\302\273

Thus

rise.

terminal

the

terminal

must emit

load

would

R

the

when

the

equilibrium

all absorbed in

decided to

approximationto

the

actual

distribution.

The

the

possible

as a

fvee vibrations

good enough of a lattice must,

sonic spectrum

j in

Solids:

Dcbye Theory

square noise \\ o'uge flucluations observed cxperimcn::i))y from a 3 jiO resistorin ihe mixing chamber of a dilution as a function of magnetic refrigerator 4.9

Figure

Mean

icmpcralurc indicated by tlidrmometer.

After

R.

C Wheailey, J. 533 A972).

and

J.

a CMN'

R. GiiTarJ, Low

powder

li.

Tcnir

Physics

100

T

of course,deviatefront

its soon its t!ie wavelength becomes comparable to . .. The only thing which had to be lione was lo to she fact that every solid ofjunta dimensions numfrc'r contains adjust ajiuite atoms and a At low has mint her vibrations.... of therefore L'uoiujh finite of free and ttt perfect analogy to the radiation htw temperatures, of StefanBoltzmann ..., the vibrational energy contentof it solid will be proportional

t/ie

disittuees

this

of the atoms.

P. Dcbye

The energy of an elastic wave electro

elastic ;is for

wave in a cavity

magnetic

clastic wave

is calleda of

wave

in

oj is

is quantized

just as the energy

is quantized.The quantum thermal

The

phovwn.

frequency

a solid

average

number

of

energy

of an

of pitonons in

Planck distribution function,

given by the

an

of

an

just

photons: 1

We assume

that

t!ie

frequency

ofan

elastic

wjive is independent

C5}

of theamiMttmle

and heat capacityofiheelastic be carried of the resiiks obtained for photons waves in solids. Several may th:tt the velocities of ail over to plionons. The resultsare simple if we assume elasticwaves are equal\342\200\224independent of frequency, direction of propagaiion, but it helps Thisassumption is not and directionof polarization. very accurate,

ofthe elastic

sixain.

We

want

to find

the energy

A-

Webb,

6,

the general trend of the observed results in many with a solids, of computation. Therearetwo important of the experimental results: the heat capacity features of a nonmctallic solid varies as tJ at low temperatures, and at high temperatures the heat capacity is independentof the temperature. In metals there is an extra for

account

minimum

contribution Number

the

from

conduction

There is no limit to the number of but the number of elastic modesin with

each

3A?.

An

wave

elastic

in

Chapter

7.

Modes

of Plionon

of N atoms,

treated

electrons,

three

possible a finite

modes

in

a cavity,

If the solid consists

lite total number of modesrs

of freedom,

degrees

has three

electromagnetic is bounded.

solid

possible po! matrons, a

two

transverse

and one

polarizations of an electromagnetic of the atoms is perpendicular displacement a wave the displacein wave; longitudinal displacement is paraiicl over all to the propagation direction. The sum of a quantity modes 3, may be written as, includingthe factor

longitudinal,

in

two possible

to the

contrast

wave.In a transverseclasticwave the to the propagationdirectionof the

| JW of A7). Here n as for photons. We

by extension exactly

elasticmodes

ts

to

equal

C6)

of the triplet of integersnxt i\\yt iu, ;iralI such that the total number of

in terms

is defined

to

want

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