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Thermal physics.
Bibliography:p. index.
Includes
Statistical
!.
Herbert.1928-
536'.?
\302\260
I9B0
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by W.
bor.k
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H. Freeman and be reproduced
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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
