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Scholars' Mine Scholars' Mine
Masters Theses Student Theses and Dissertations
1952
The partition function of silver The partition function of silver
Ralph H. Lilienkamp
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Recommended Citation Recommended Citation Lilienkamp, Ralph H., "The partition function of silver" (1952). Masters Theses. 2621. https://scholarsmine.mst.edu/masters_theses/2621
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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.