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Masters Theses Student Theses and Dissertations

1952

The partition function of silver The partition function of silver

Ralph H. Lilienkamp

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THE PARTITION FUNCTION

OF SILVER

BY

HALPH H. LILIENKAMP

A

THESIS

submitted to the racu1ty or the

SCHOOL OF MINES AND METAI.IJJRGY OF THE UNIVERSITY OF MISSOURI

in partial rulfillment or the work required ~or the

Degree or MASTER OF SCIENCE, PHYSICS MAJOR

Rolla, Missouri

1952

Approved by-~,#~ ~~A-..s.-;.s........-~t-a-n~t~i~~ .... r~e ... s ..... ~o .... r......_o~r~Ph~y--s~1_.c ... e...,._ ...

ACKNOflLEDGDIE1'fTS

The author wishes to express h1s gratitude to

Dr. Edward· Fisher. rormer Associate Proressor or

Physics. and to Dr. Louis H. Lund. Assistant Pro­

ressor or Physics ror their most valuable ~idance

and interest.

The author also appreciates the interest shown

by Dr. H. Q. Fu11er and ot.her members or the starr.

ii

CON'IENTS

Acknow1ed~ents ••••••••••••••••••••••••

Li.st of' Illustrations •••

Li.st o~ Tables •••••••••••••••••••••••••

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . Review or the Literature •••••••••••••••

The Ttleory •••••••••••••••••••••••••••••

Page ii

iv

v

1

2

5

Application or the Theory.............. 11

Conclusions............................ 23

Bibliography........................... 25

Vita •••••••••• . . . . - . . . . . . . . . . . . . . . -. . 27

iii

iv

LIST OF ILLUSTRATIONS

Fig. Page

(I) The Ln Partition Function versus Temperature Curves..................... 13

(II) The Helmholtz Free Energy varsus Temperature Curves ••••••••••••••••••••• 14

(III) The Internal Energy versus Temperature Curves ••..••••.•••••••.••••.••••••.•••• 15

(IV) The Specirlc Heat vsrsus Temperature Curve ••••••.•••.•......•.••...•••.••••• 16

v

LIST OF TABLES

No. Page

(I) The Partition Functions ••••••••••••••• 17

(II) The Helmholtz F~e Erlerf!J ••••••••••••• 18

(III) The Internal Energy ••••••••••••••••••• 19

(IV) The Specif'ic Heat .••••...••••....•.••• 20

(V) Comparison o:f the Partition Function •• 21

(VI) Comparison of' the Speci:fic Heats •••••• 22

1

INTROOOCTION

The study ot' the thermodynamic properties o:r solids

at low temperatures has shown that most theories either

break down or become too cumbersome.

A theory based upon a crystal lattice ot' atoms gives

the best physical picture ot' a solid. Fisher(l) has

developed partition !'unctions t'rom this theory t'or the

simple cubic. body-centered cubic. and the :race-centered

cubic lattices. In his paper Fisher gave three partition

!'unctions or increasing complexity and accuracy t'or each

ot' the three lattices. In ~~is paper the three partition

!'unctions :ror a !'ace-centered cubic lattice shall be

applied to silver and compared.

The Helmholtz t'ree energy. the internal energy and

spec11'1c heat at constant volume are shown t'or the simpler

cases.

(1) Fisher. E •• J. Chem. Phys •• Vol. 19. pp. 632-640. (1951)

2

REVIEW OF LITERA'IURE

~~en it was shown that classical theory could not

explain the decrease in speciric heat at low temperatures.

Einstein ( 2 ) .proposed that a crystal might be regarded as

being made up or harmonic oscillators. In his model each

atom is represented by three oscillators and all or the

oscillators 1n the crystal have the same rrequency. The

theoretical speciric heat due to this single "Einstein

Frequency" decreased too rapidly at low temperatures.

Debye ( 3 ) postulated an elastic-continuu.model o:r

a crystal. His treatment led to a range o:r rrequencies.

the normal modes. :for the harmonic oscillators. and to a

constant ror each crystal called the Debye Character­

istic Temperature. Experiments showed that this constant

varies at the low temperatures.

Born and von Karman ( 4 ) proposed a lattice theory o:r

speciric heats which gives a more accurate physical

picture than t.he continum idea. However. the use o:r the

Born and von Kannan theory has been greatly limited

because or mathematical diT:ficulties. Blackman ( 5 ) has

discussed the difrerences to be expected between the

Debye and the Born and von Kannan theories and has treated

the sodium chloride lattice numerically.

(2)

(3) (4) (5)

Seitz. F., The Modern Theory o:r Solids. McGraw-Hill, pp. 103-117, 1940. Seitz, F. Op. Cit., pp. 104-117. Seitz, F., Op. Cit., pp. 118-123. Blackman. M., Proc. Roy. Soc., Vol. A 148, p. 384, 1935; Vol. A 159. p. 416, (1937).

3

F1ne ( 6 ) has used the Born and von Karman theory to

1nvest1gate the normal modes of vibration of tungsten.

but made no application to specific heats at low tempera­

tures. Mont~ll (7) developed a method for calculating

the frequency distributions or the square. the cubic.

and the body centered cubic· lattice. but the method does

not behave properly at low temperatures.

Leighton ( 8 ) obtained a frequency spectrum of a

face centered cubic crystal by using a plaster model of

the constant-frequency surfaces. He then calculatad the

specific heat or silver using the elastic constants

determined by Fuchs (9). He obtained fair results down

to a temperature range of 7°K to 50°K.

Bonnell (lO)developed a series for the speciric heat

or a face centered cubic lattice in terms of the elastic

constants and temperature but the series does not con­

verge below 50° K for silver nor below 120° K for aluminum.

\iebster (ll) developed a similar series for the body

centered cubic lattice. This series is considered good

for temperatures greater than one-fifth or the Debye

temperature.

(6) Fine, P.C., Phys. Rev., Vol. 56. p. 335, (1939) (7) Montroll. E.w .• J. Chem. Phys •• Vol. 10. ~- 218 (1942);

Vol. ~. p. 481. (1943); Vol. 12. p. 98. (1944) (8) Leight~n, R.B •• Rev. Mod. Phys., Vol. 20, p. 165, (1948) (9) Fuchs. K., Proc. Roy. Soc., Vol. A 153. p. 662, (1936) (10) Bonnell, C.R., Thesis, Missouri School of V.ines and

Metallurgy (1950) (11) Webster, C.C., Thesis, Missouri School of Mines and

Metallurgy (1950)

4

Fisher (l2 ) developed the partition functions for

the simple, body-centered, and face-centered cubic latt­

ices. These partition functions depended on the elastic

constants and temperature and have a claimed accuracy

of a few tenths of a percent.

(12) Fisher, E., Op. cit.

5

THE THEORY

From Statistical ~fechanics it is known that the

internal energy E is runction of the partition function

P and the absolute temperature T as follows:

E = Nk T2 A (Q.n P) (1)

Where N is the number of particles and k is Boltzmann's

Constant.

Blackman (l3) gives the lattice energy as:

E _, i'n•I.-1:J:ft h cJi[(e ltfk/u -IJ'~"q J;. J~JtA (2)

Where the <)~ are the space coordinates of' the reciprocal

lattice. The :frequencies v&· are the normal modes of

oscillation o:f the particles in the lattice and are given

1n the terms of' the f/J; and the elastic constants as roots

o:f the secular determinant:

As.xs'*

(3)

where r;· :: J.~-hkT 1 c~ =cos x. .r,. :r sin 111 e"h.J

(4)

M is the mass o:r an atom, and oc and Z I are :force constants

between nearest and next nearest neighbors, respectively,

(13) Blackman, M., Op. Cit.

in the 1attice. F, G, and H are po1ynomia1s in the C&

From equations (1) and (2) the partition runction

is given by

(5)

where

(6)

and

Nov,

(8)

Theref'ore,

J :: ~ 1, r f/J.. A, 1/&J Where

(9)

~r,. = / 1-F/tnnJ• + G{t1111}., +H/tnH)'-q• tL -c. Cy ~tiJ141: +I a,.r.r1 .,. s~ s,

: Q,.S.-S, q. ~ - C1 c7 -- CyC.I-14,y .-1 q, s, J"z

(10)

and

(11)

Then

T,. = [FA',IIJ' + G),lrf' + ~ll£) - ~[ 1£ .,. ); [ 1~ (12)

6

7

D ror the race-centered cubic lattice is given by

Fisher (14) as:

ll ill. ]) : {; [tz;,-~ O D 1., Jf/J, JfA cJtA

.,. (bj.,.J) J;["[J$1, d/>. J(J,. J~ 1 II -

+[J'&- ~tUnY]ft.A +.3jjjz) , ... , - [~o - £.(Y,,)''}{IsA~ +.3AB+-IJ.Bftt.) .... + [Y9'Js - J.. (~ .. )']{IS A'/1- 1- II/ A,..JJfi~ .,_,

+ 9ABjJ. +SB~~) · ·· (13)

From equations (3) and (4) he developed three

partition runctions or increasing accuracy and complex-

ity. These equations are given as follows:

[ • . ]Yz P1 = !!. csch ( t&/r)

•here. ror i = 1 to 6. in order ti = a. 2a. 3a. a I b. 2a I b. 3a I b

(14)

Pa =&csch (t.Jrj'll ( csch (t.-.fr)JY'•'" X ff! csch ('-/r)7~ fcsch ('-'IT~ ~

••1 c c .,./T [} r s nh t"/T ~.1 lev}~~ f~·nh~ (sll:h ( e..tr "b (15)

where. for i = 1 to 12. in order ~ = 4a. 4a I b. a. 3a. 2a/ b. 3a I b. 2a.

a /b. (a I b)/2, (3a b)/2. (7a I b)/2. 2a I b/2

(14) Fisher. E •• Op. Cit.

P_. = l'-J'- fJ!sinh (L:/T ~S. [sinh (i-* u"~ X {sinh (r•/rj._ {sinh (~'/r>J•

X [sinh ~/r~~ {i,csch (~/rjtlr x { /i..csch <Vr >JY" £/icsch <~Vr~'fls­x [ csch (t•lr>]'lk {_{icsch (-tafr~•-.... ..._.

8

X [csch (t_,frJ}'* - (16)

where, Cor i = 1 to 23, in order ..

tc = a, 2a I b, 3a I b, 3a, 2a, a I b, 4a, 4a I b,

(2t/2)a, {l~/2)a I b/2, {2~/2) a f b/2

a I b/2. <s r/3) a/2 I b/2

2a I 3b/4:f/(~ I bA /16)

5 a/2 I 3b/4 :t/sa~ /4 - ab/4 f bL /16

{21 2-A.) a;

Cor i = 28 to 31, in order ~

t~ = (a /b)/2, {3a I b)/2, {7a I b)/2. 2a I b/2

and Cor i = 24 to 27.

ti = (2:!"' 2-J') a I 3b/41: (1~-F 2-liab I -b)./4

the second + sign being independent oC the Cirst

and third.

A simpler expression ~or the Partition Function can

be obtained Cor the higher temperature ran€!e using Fisher's{l5)

~ethod to ob tain equation {13).

(1~) Ibid.

9

At high temperatures. the summations in equation (13)

become sma11 enough to be neg1ected and the D takes

the :form:

D = A Bftt - Ah.z- AB,Po - BAM> I Sllf/uso

+II/ A~Ps11..0 t- ABY~j.~o

+S .B~s/~o ----

or

D = ..9_ + b Q2. l L &. T~ - -- -140 D

4T~ I~T.s JoT~ -IIAJT4f

Using equation (5)

or: p DIC - T~ e-D

P = C T~L:!-,

then: In P = /,C -13/, T -D

and:

:for

In p : InC 131.. T- ~·I!· +a-.,_ .. ., + J,' ~ ., ..ll.!- ., ....

slmp1icity: '~'

I P. 3 1.. T - ..!i - _l In d ~ ~~. r~ 11-Tz.

(17)

(18)

(5)

(19)

(20)

(24)

(22)

In addition to the Partition Function. the :fo1lowing

equations :from Statistical Mechanics must be used i:f the

Partitio:n Functions are to be o:f any va1ue.

The Helmho1tz Free Energy is given by:

F/Nk: -T /, P (23)

The Interna1 Energy is given by:

E/Kk = T.~. ir (/, P) (1)

10

and the Speciric ~eat at constant volume is

given by:

c: ~; (24)

or:

C_.,(r'k :: Jr { Tz fT (1, PJ} (25)

11

APPLICATION OF THE THEORY

Using the equations given for the partition fUnctions.

the natural logarithms or the partition runctions were

calculated and plotted. The values o:f c:x and 1/ used are

given by Leighton (16). J: is -o.oB .~has two values, at ~

which absolute zero « = 21.3 X 10 and at room temperature · S

«. = 18.0 X 10 The rormer value was used for the ca1cu-

lation or Pr. ~ and P..r ·

The points plotted are 20. 30. 40. so. 70. 100. 150.

and 250 degrees Kelvin ror Pr • PK • and PDE • Since P~

is only a high temperature approximation the curve does

not go below 50 degrees Kelvin and is extended to 300°

Kelvin. Table I gives the computed values and Figure I

contains the curves ror the partition Functions.

The Helmholtz Free energy was computed directly

:from the natural log of the partition runctions. The

values o:f i are given in Table II and the· curves in Figure

II.

The internal energy corresponding to Pr was calcu­

lated t.o sho\., the general shape of this curve. Since t.he

log o:f the partition runctions curves are very similar.

the other curves can be expected t-o have the same general

shape. The values o:f £ are given in Table III and the

curve in Figure III.

The speci:flc haat at constant. volume was computed ror

Pr and Pg • The speciric heat. curve corresponding to Pm

(16) Lelgbton. R.B •• Op. Cit.

was not com9uted because or its complexity and the

simllari ty o:f the p.arti tion :functions.

12

1}

15

10

0

-~

-10 LM ~nTION o Pt FUNCTION

6. Pa

a 11-.

9 ~

-20

-2S 0

0

-•oo

-300

F -400

N"

-100

-800

-900

-tOOO 0

HELMHOLTZ FREE ENERGY (F)

0 Ft

A F•

100 150 1EUPERA1'UAE ~

Fig. II

14

100

650

600

550

E

450

4 0 0 -

300

250

INTERNA.L ENERGY (E)

tOO 150

TEMPERATURE "' F1.g. III

15

~or---------------------------------------------------•

4 .S

40

3 .S

3C

20

1 I • i, • !

IC '

f.' // I

c'1i I

J

0

; · I f I

I /

./ I

I I

I

t

I

I -· I

SPECtt=IC HEAT (Cv)

100 •so TEUPEAATUAE ~

Fig. IV

16

17

TABLE I

The Partition Fun~tlon

T 1n P~ 1n P• 1n P• 1n P_. 10 -l~- ~,. -zl.oot 20 -9.851 -9-329 -11.135

30 -5.853 -5-377 -6.'846

40 -3.811 -3-348 -4.634

50 -2.534 -2.079 -3.241 10.605

70 -0.952 -0.493 -1.596

100 0.440 0.922 -0.118 13.533

150 1.834 2.223 1.360 14.896

200 15.824

250 3.461 3-910 2.051 16.519

300 17.080

18

TABlE II

The Helmholtz Free Energy

T F~/lllt F./ik F.-/KI 20 197.03 186.58 222.71

30 175-58 161.32 205-38

40 152.45 133-94 185.38

50 126.69 103-96 162.06

70 66.66 34.54 109.74

100 -44.04 -92.18 11.82

150 -275.05 -333-44 -204.03

250 -865.18 -977.42 -512.80

19

TABLE III

The Internal Energy

T Ex/Ill 10 239 .. 04

20 239-51

30 242.70

40 252 .. 82

50 262 .. 34

7!! 296.70

100 363.60

150 4°3-30

250 776.26

20

T.ai.BLE IV

The Spaci:fic Heat. { '/•KJ

T Ctz/Nk Ct6/NI 20 0.138 0.260

30 o.lt-99 0.818

40 0.992 1.635

50 1.402 2.309

70 1.978 3.257

100 2.425 4.042

150 2.690 4 .. 501

250 2.894 4-.797

21

TdBLE V

Comparison or the Pnrtit~on Functions

T l~tRt -J,I} ,.ll _,,. 8a 20 .5227 1.28399

30 .47532 .99341

40 .46275 .82311

50 .45460 • 70729

70 .45877 .64111

100 .48140 .52381

150 ,48~26 .47347

250 .4lt-905

22

TABLE VI

Comparison or the Specirlc Heats

T c .. c~z c~ • 20 .0708 .0477 • 0542

30 .1968 .1724 .1705

40 .3454 .3428 .3408

50 .4792 .4844 .4813

70 .6699 .6835 .6790

100 .8232 .8379 .8426

150 ·9306 .9295 -9383

250 ,. oooo /,DODO I, oooo

23

CONCWSIOL"'S

The partition function P11 will be considered

the most accurate since an accuracy or a rew tenths or

percent is claimed by Fishar. except at the very low

temperatures. T'ne curves In ~.l•land I• P• may be com­

pared directly 1n Figure I on J:able VI to the 14 Pa

1he curves do not agree numerically but do have

the same general shape. The dii'rerences between le 1l

and / .. R. 1nd1ca.te that at high te10peratures the two

curves should cross and may even coincide. Pr and

P a do cross between 10 and 20 degrees Kelvin. but do

not coincide. Pu r.tay have a better shape but has a

greater error.

P• has the correct shape :for the higher tempera-

tures. but the numerical error is too large.

Since the Helmholtz :free energy is just the pro­

duct or tlle temperature and the 1n of the partition

¥Unction all comments and conclusions are the same as

those for the p~rtition fUnctions.

The Internal energy curve is given only for the

artition :function P~

The specific heat at constant volume was computed

ror the p 'arti tion :functions P:x and PJI and compared

with experiment values given by Giauque (l7)

The numerical values for both curves are . unsatis-

ractory. but \.,hen the curves are all placed on one curve

(17) Meads. P.F •• Forsythen. W.R •• and Giauque. U.F •• Jour. Am. ~~em. Soc. Vol. 63. p. 1902 (1941).

24

with three di:f:ferent ordinates so that t.he values o:f

Cv at 250Q K coincide. the three curves do have the

same shape down to about 30° K. Table VI provides

this comparison. Numerically. Cv.r is closer to the

experimental values o:f C~ but somewhat greater. It

is expected that C "JK would be corract in this range

o:f temperatures.

25

BIBLIOGRAPHY

1. Books:

Gurney, R.W., Introduction to Statistical Mechanics, McGraw-Hill N.Y., 1949

SeJ.tz, F •• The ~~~odern T'tleory o:f Solids, :r.tcGra\.,­Hi11 N.Y., 1940

2. Periodicals:

Blacoan, M •• 'lbeory o:f the Speci:fic Heat o:f Crystals. Froc. Roy. Soc. Vol. A 148, p. 384 (1935)

Blackman, l..f... On the Vibrational Spectrum o:f a Three Dlrtensional Lattice, Proc. Roy. Soc. Vol. A 159 p. 416. (1937)

Fine. P.C., The Normal Mode o:f Vibration o:f a Body Centered Cubic Lattice. Phys. Rev. Vol. 56, P· 335, (1939)

Fisher, E •• Partition Functions o:f Cubic Lattices, J. Chern. Phys. Vol. 19, p. 632 (1951)

Fuchs, K.A., A quantum Mechanical Calculation o:f the Elastic Const~nts o~ Vonovaleat Metals. Proc. Roy. Soc. Vol. 153, p. 622 -(1936) .

Leighton, R.B •• The Vibrational Spectrum and Speciric Heat o:f a Face-Centered Cubic Crystal, Rev. Mod. Phys •• Vol. 20, p. 165 (1948)

Montroll, 11.~1., Frequency Spectrum o:f Crystalline Solids, J. Che~. Phys. Vol. 10, p. 218 (1942)

'-~ontroll, E.~f •• Frequency Spectrum ot' Crystalline Solids, II, General Theory and Application to Simple Cubic Lattices, J. Chem. Phys. Vol. II, p. 481 (1943)

1-ion·troll. E.·t., and Peaslee, D.C., Frequency Spectrum o:f Crystalline Solids. III. Body Centered Cubic Lattices, J. Chem. Phys. Vol. 12, p. 98 (1944)

}. Unpublished Naterial (Thesis, Dissertations. etc.)

Bonnell, C.R., The Theory o:f the Speci:flc Heat o:f a Fach-Centered Cubic Lattice, Thesis, Missouri School o:f ~lnes and ~etallurgy~ Rolla, Missouri.

26

Webster. c.c •• The Theory of the Specific Heat or a Body Centered Cubic Lattice. Thesis. Klssourl School or Mines and ~etallurgy. Rolla. Missouri.

27

VITA

The au"L"lor. Ralph Harold Lilienkamp. was born in

St. Louis; Missouri. on Au8Ust 21. 1926. He attended

St. Trinity Lutheran School. Hope wtheran School. and

Southwest High School in St. Louis. Missouri. After

graduating from hign school he enlisted ln the United

States Navy. While in the Navy he attended Electronics

School and lert the Navy as an Electronic Technician

Second Class. After his discharge :from the Navy he

attended Washington University in St. Louis. On June 6.

1950. he received his AB in Phys:lcs from ~ ashlngton

University.

In September. 1950. the author enrolled in the

Missouri School o:f !-l:lnes and Metallurgy. as a candidate

ror a Master of Science Degree. Physics major. Since

September. 1951. he has been a graduate assistant in

Physics.

The author is a charter member of the Eta Chapt.er

of Beta Si91a Pi and a member o:f the Alpha Phi Chapter

o:f Gamma Delta.. He :ls also a member of the M:lssour:l

School or M1nes and Metallurgy Chapter of Si~a Pi

Signa.