PAST-DJ 68-1267
Reprinted from THE PHYSICA\-
REVIEW,
Vol. 175, No. 3, 905- 912, 15 November 1968
Printed In U. S. A.
Simplified Theory of Anharmonic Contributions to the
Thermodynamic Properties of Solid Sodium Metal
D. JOHN PASTINE
U . S . Naval Ordnance Laboratory, Silver Spring, M arylalld
(Received 15 March 1968)
Since attempts at the evaluation of anharmonic effects in crystals invariably lead to great difficulties,
a prescription is offered by which the estimation of anharmonic effects is greatly simplified. As a first step,
the first-order anharmonic contributions to the free energy are looked upon as being pseudoshifts in the
frequencies w, of the usual normal modes. Thus the anharmonic part of the free energy is considered to
affect the total free energy only insofar as it causes w, to become w,(l +t:u.J,/w,), where t:u.J'/Wi is the fractional
shift in frequency. It is then argued, on the basis of a simple calculation, that a simple and plausible relation
exists between the quantities t..w;/w, and the thermal fractional frequency shifts of certain acoustic waves
which occur even when anharmonic contributions to the free energy are neglected. The relation mentioned
is one of simple proportionality, and since the acoustic frequency shifts can be calculated for some materials,.
it permits an estimation of the anharmonic part of the free energy. Calculations are carried out for sodium
metal, and the results are compared with expe.rimental data. In all cases the agreement between theory and
experiment is shown to be quite good. Aniimg the calculations is one which evaluates the anharmonic
the Griineisen parameter 'Y. This is shown to increase 'Y by more
temperature-dependent contribution
than 20% at room temperature and pressure, and cause it to vary almost linearly with volume. At higher
pressures, however, anharmonic effects diminish until at about 40% compression they disappear almost
completely and 'Y behaves as a constant.
to
I. INTRODUCTION
TIEMPTS at the exact theoretical evaluation of
A
anharmonic effects in crystals invariably meet
with extreme difficulties. These difficulties arise not
only from the great mathematical complexity of the
problem and its solutions but also from an inability to
evaluate these solutions when they are obtained.
This happens because evaluation of the approximate
mathematical solutions to the problem of anharmonic
effects always requires a greater knowledge of atomic
interactions than is presently had. There may be,
however, ways to circumvent this difficulty and we
will discuss one such possibility here.
Our first step will be to interpret (and we are entirely
free to do so) anharmonic contributions to the free
energy as pseudoshifts in the usual normal mode
frequencies. Next it will be argued that a plausible
relation exists between these shifts and shifts in the
frequencies of certain isothermal acoustic waves, the
latter being calculated when anharmonic terms in the
free energy are neglected. This assertion is based on a
simple calculation done in Sec. II, the results of which
are later generalized. Finally, in Sec. V, an expression
is derived for the anharmonic contribution to the
equation of state which can be estimated (and is
estimated for sodium) with only a partial knowledge
of the atomic interactions. In Secs. VI and VII all the
predicted results are shown to compare quite favorably
with experiment.
II. RELATION BETWEEN ACOUSTIC FRE-
QUENCY SHIFTS AND THE FREE
ENERGY IN A SIMPLE MODEL
To begin our treatment we will examine a bcc solid
with lattice parameters a= b= c which supports only
pure longitudinal (p.l.) normal modes traveling parallel
175
to the principal axis which corresponds to the lattice
parameter a. The atomic motion associated with such
modes is such as to require all atoms lying in a common
plane perpendicular to the direction of propagation to
be simultaneouslY .displaced. We will, for the sake of
simplicity, assume that atomic interactions are such as
render only the forces between nearest planes of atoms
, significant. This case is, of course, completely analogous
to that of the linear chainl which has been treated in
great detail in the work by Leibfried and Ludwig2
(LL) . Their development begins with an expression
for the partition function Z in which the energy I(J of
the atomic interactions is approximated by an expansion of the form
/1
I(J=
gl
l(Jo+- L (qn_qn-l)2+_ L (qn_qn-l)3
2
n
3!
n
hi
+L
4!
(qn_ qn-l)4 ,
(1)
n
1 For small plane displacements the system under discussion is
the mathematical equivalent of a one-dimensional system of
particles (mass M) connected by nonlinear springs. The equivalence is made complete by substituting for M the mass of an
entire plane of atoms and for the spring constants the effective
spring constants between planes. The effective spring constants
can, in principle, be obtained in the usual way by expanding the
force between two planes in powers of their displacement from the
equilibrium separation.
It should be noted by the reader that for the types of normal
vibrations treated in this work, the separation between planes is
!a, where a is the lattice parameter.
The equations for transverse vibrations can be developed in
exactly the same way as those for the longitudinal vibrations. One
need only replace the nonlinear springs which support only longitudinal stress (in LL) with a special set of nonlinear springs which
will support only shear stress. In this way the only thing that is
changed is the direction of displacement and the labels on the
spring constants. The forms of the free-energy equation and other
pertinent equations therefore remain unaltered.
I G. Leibfried and L. Ludwig, in Solid Slate Physics, edited by
F. Seitz and D. Turnbull (Academic Press Inc., New York, 1961),
Vol. 12, p. 330.
905
906
D.
JOHN
where q" are the plane displacements, 'Po is the energy
when q"= 0, and fl' gl, and hi are the coupling parameters characteristic of the p.l. oscillations and which,
under hydrostatic conditions, are functions only of the
lattice parameters. The consequent form of the Helmholtz energy, F, is then shown to be, in the classical
region (T> than the Debye theta @D) of temperature,
F=FQH -N(kT)2gr/12N+N(kT)2h l/8J(,
(2)
where N is the total number of planes perpendicular to
the a axis, k is the Boltzmann constant, and T is the
temperature. The quantity FQH is the value of the free
energy in the quasiharmonic approximation (i.e.,
g= h= 0) and has the form
N
FQH=Fo-kT
L
i-I
In (kT/hw i ) ,
(3)
PASTINE
In another calculation in LL there is derived a
relation for the sound velocity 'VI of an isothermally
propagating acoustic mode, the displacements of which
correspond to the thermal modes discussed above. This
calculation is done in the quasiharmonic approximation.
Surprisingly, however, the parameters gl and hi still
appear in the solution. This occurs because in order to
calculate 'VI it is necessary to take certain derivatives of
fl, (i.e., iJflfaa; iJ2fl/iJa 2), which relate directly to gl
and hi. This implies (as is pointed out in LL) that
although 'VI is calculated in the quasiharmonic approximation, anharmonic effects are nonetheless taken into
some account.
It is interesting to compute the acoustic frequency
shift (Aw/W)lo e which may be associated with anharmonic contributions to Vz. According to LL we have
(neglecting second-order effects)
where h is t1l'XPlanck's constant, wi=211'Xthe ith
normal-mode "frequency, and Fo= 'Po. Using Eq. (3)
it is of interest to rewrite Eq. (2) in the form
F=Fo~kT f. [In (k T/hw i)+ kT(~_ hi)].
J( 12fl
i-I
(4)
8 .
Assume the second term under the summation sign in
Eq. (4) is a small correction to the function In (k T/1twi) ,
which for T < 3@D is of the order of unity for the largest
Wi (for which W~k@D/h). Then, in the range T<3@D
it is reasonable to take (kT/ J()(g(j12fl-ih ,)«1 so
that
175
(8)
where the zero subscript refers to T=O. The quantity
-kTg z/[2J(a/2)J is (as is noted in LL) equal to
Aa/a, which is the fractional shift in length due to the
temperature at constant pressure. VI, of course, is equal
to the product of an acoustic frequency (W/211'),o e and
a wavelength Aloe. A particular wavelength which is
equal to some multiple of lattice parameters will vary
with temperature in the manner
(9)
With Eqs. (9) and (8), we then have
and, as a resul t,
X[1+
kT(~_ hi)]}
N 12fz
(5)
8
For Aa/ a small compared to 1, Eq. (9) reduces to
Equation (5) is open to a simple interpretation. This
is, that so far as the free energy is concerned, the
anharmonic effects serve only to shift the longitudinal
mode frequencies Wi by an amount
AWi)F =_(Al/Wi)F =_kT(~_ hi),
(
Wi I
1/Wi I
N 12fl 8
(6)
where the F associates the shifts with the free energy
and the 1 associates them with p.l. modes. Thus if we
define new frequencies w/=wi[1+(Aw';Wi)I FJ, where
(AW';Wi)I F is given by Eq. (6), then Eq. (4) may be
written
N
F=Fo-kT L: In(kT/hw/).
i-I
(7)
(11)
Equation (11) implies that when gl and hi are given for
a certain lattice arrangement (a given a, b, and c) then
the frequency shift due to temperature increases at the
same a, b, and c is given, in the classical region, by
Relation (12) tells us that if one considers any of the
two cases, IhZ/»lg(jfz/ or IhZ/«lg(jfd, we have,
according to Eq. (6), (Aw/w)z'I'a;, (Aw/w)/ oc ; to be exact
175
THERMODYNAMICS OF SOLID
we would have, in the former case,
(13)
Na
907
METAL
which appear on the right of Eqs. (18) and (20) is a
practical impossibility. However, a reasonable and
quite successful approximation3-6 to 'Y is given by
(21)
and in the latter case
(14)
We will now examine an alternative case in which the
chain supports only purely transverse (p.t.) modes
traveling in the same direction. We will assign, in this
case, the coupling coefficients it, gt, and h t in analogy
with the longitudinal case. But here we must take note
that the potential between any two planes of atoms
participating in the transverse vibrations must be a
symmetrical function of the total displacement. We
must therefore have gt=O. The quantities (Aw/w)t F
and (Aw/ w)t become, by analogy with the longitudinal
case,
ac
(1S)
where the 1, (subscripts refer, respectively, to averages
taken only over axially propagating p.l. and p.t. normal
modes. If nearest plane interactions predominate, the
quantities 'Yi 1 in Eq. (21) can be shown2 •6 to be the same
or nearly the same for modes having the same direction
of propagation and the same polarization. If we further
restrict ourselves to cubic crystals we find that the
axially propagating p.l. and p.t. modes have only six
distinct directions of propagation. In addition, in the
cubic system there will exist only two distinct values
of 'Yi. The success of the approximation apparently
stems from the fact that the 'Yi are calculated for modes
traveling along major symmetry axes of the crystal.
Applying the approximation Eq. (21) to the calculation
of 'Y='Yo+A'Y, we obtain
(22)
and
_
a (l(AWi)F
2(AWi)F) .
+- ax 3 Wi I 3 Wi t T
A'Y--x- -
and
(
AW)ac = + kTh
W
I
t •
4/,2
(16)
IV. EVALUATION OF yo
In addition we have the relation
(17)
following directly from Eqs. (IS) and (16).
III. GRUNEISEN PARAMETER
In the quasiharmonic approximation, the Griineisen
parameter 'Y for a monatomic solid is, in the classical
region of temperature, defined as the average
(18)
where 'Y.o= -d lnw./ d lux and x is the specific volume
divided by the specific volume at zero pressure P and
temperature T (i.e., x= vi vo). If we again consider
anharmonic effects in the range T < 38D then the frequencies w, become effectively shifted by the fraction
(Aw / w.V«1. This effect causes 'Y to assume the form
'Y='Yo+A'Y,
(23)
(19)
where 'Yo is again the QH value and
In the QH approximation there is, withip. each group
of thermal modes (either p.l. or p.t.) which have a
common direction of propagation and displacement, a
subgroup of long wavelength, the members of which
have a common propagation velocity. This velocity is
identical with that of an acoustic wave of similar
direction and displacement which can be externally
imposed upon the solid. Since the velocities of the
long-wavelength p.l. and p.t. modes are related directly
to certain elastic constants, it can be shown that for
any of the long-wavelength subgroups mentioned the
'Y ,0 are given by
l I d inCiO
(24)
'Yl= - - - - - - ,
6 2 d lnx
where Cia is the appropriate elastic constant evaluated
along the OaK isotherm. Now for a solid in which
nearest plane interactions predominate, it happens that
all the 'Y,o associated with p.l. or p.t. modes of common
displacement and propagation direction will be nearly
equal.2 •6 We will assume this to be the case for sodium,
and 'Yo will be evaluated in accordance with Eqs. (22)
and (24). The elastic constants required to compute
the 'Y ,0 are cno and cuo. The sum cno 2C120 can be
evaluated quite satisfactorily 7 by means of a modifi-
+
D. Bijl and H. Pullan, Phil. Mag. 45, 290 (1955).
• D. J. Pastine, Phys. Rev. 138, A767 (1965) .
• G. K. White and O. L. Anderson, J. Appl. Phys. 37, 430 (1966).
6 D. J. Pastine, Phys. Rev. 148, 748 (1966).
7 A modification of Brooks's equation was adjusted so as to
reproduce the experimental data aloni the OaK isotherm.
I
Since the number of normal modes supported by a
solid is numerically equal to three times the total
number of atoms, the exact evaluation of the averages
908
D.
JOHN
0
T ABLE I. The theoretical 300 K isotherms (with and without
the anharmonic contribution to the pressure) and calculated
values of 1'0, 1'0', and "(0".
P(kbar)
x
"(0
"(0'
1.05
1.04
1.02
1.00
0.96
0.92
0.88
0.84
0.80
0.76
0.72
0.68
0.64
0.60
0.56
' 0.52
0.48
0.44
0.40
0.888
0.886
0.882
0.879
0.873
0.868
0.865
0.863
0.861
0.861
0.861
0.862
0.864
0.866
0.869
0.871
0.873
0.875
0.875
0.209
0.200
0.182
0.157
0.132
0.102
0.073
0.046
0.022
0.000
-0.021
-0.038
-0.051
-0.060
-0.063
-0.058
-0.044
-0.017
+0.025
"(0
"
0.930
0.904
0.888
0.841
0.784
0.741
0.689
0.634
0.594
0.536
0.471
0.386
0.287
0.154
-0.013
-0.225
-0.496
-0.841
-1.296
P(kbar)
with "(="(0 with 1'= "(o+c.."(
T=300oK
T=300oK
-1.37
-0.70
0.73
2.29
5.80
10.02
15.04
21.05
28.29
37.03
47.72
60.83
77.10
97.47
123.27
156.41
199.64
257.11
335.17
-0.44
0.20
1.57
3.07
6.51
10.63
15.57
21.50
28.67
37.37
47.99
61.05
77.26
97.58
123.34
156.43
199.63
257.06
335.11
p.+ (po'Y0/x)3nkT,
In Fig. 1 and Table I, P [from Eq. (2S)J is compared
with the experimental data of Beecroft and Swenson.!·
In the region x>0.91, 'Yo is clearly too low; in the region
x<0.91, however, the agreement is quite satisfactory,
suggesting that the quantity /}"'Y may account for the
difference and, furthermore, that /}"'Y diminishes with
decreasing volume.
V. ESTIMATE O F A"(
cation of an ex-pression for the cohesive energy suggested by Brooks. B•9 C440 and the sum Hcuo-c120)=co
can be evaluated by the theory of FuchsiO which predicts quite accurately the proper values of these elastic
constants at T=O. [The discrepancy which exists
between experimental room temperature values of
(a lucia Inx)T and (a InCH/ a Inx)T and the theoretical
values of Fuchs is, as we shall see, probably due to
thermal effects.llJ The values of 'Yo together with its
first and second derivatives with respect to x ('Yo' and
'Yo") calculated in the manner described above, are
given in Table I for various values of x. In order to
show the efficacy of the approximations, Eqs. (21) and
(24), for metals, 'Y has been calculated at room temperature and pressure for copper, silver, and gold. The
calculation was performed using the experimental highpressure elastic-constant data of Daniels and Smith.12
The results are 'YCu= 2.01, 'YAIl= 2.46, and 'YAu= 2.98.
These values compare quite favorably with the thermodynamic evaluations of 'Y made for these metals by
workers at Los Alamos.!3 For copper, silver, and gold
the latter have obtained, respectively, 'YCu= 2.04,
'YAg=2 .47, and 'YAu=3.0S. The room-temperature
isotherm is in the QH approximation given by
P= P o-
175
PASTINE
(2S)
In order to estimate /}"'Y we will make two assumptions. The first (A) is that there exists a proportionality
between (/}"w/w;)ac (calculated in the QH approximation) and (/}"w/w,)F, that is, that (/}"W,!Wi)ac
=a,(/}"w,/Wi)F for all p.l. and p.t. waves. This assumption is not without foundation, as the possibility is
suggested by the calculations of Sec. II. It seems
worthwhile to add that since, in the simple model of
Sec. II, we find (/}"W/Wi)ac< (/}"w/w;)F, then we might
expect a;::; 1.
The second assumption (B) to be made is that all
the /}"Wi/ W; for the p.t. modes are mutually proportional.
This assumption seems (at least to the author) to be
somewhat justified by the observation that for sodium
the theoretical 'Yi O for all possible p.t. modes turn out
to be equal. [Remember, 'Yl= - (x/dx)dw/wi.J It
follows from (A) that (/}"W/Wi)F will be zero if the
associated elastic constant Ci is independent of temperature. This is tme because in that case we must
have (/}"W;/Wi)ac= O. Now, as it happens, the quantity
(d luc;/dT)., which indicates the temperature dependence of the cIs for sodium, is three to ten times
larger I6 for the cIs associated with p. t. modes than for
the p.l. This fact, which is completely consistent with
the theory,16 indicates the relation
L8,0
24.0
LO,O
16.0
12.0
""I "
~
~
.~~
P (kb. r)
8.0
4.0
-x
~'
•••
............ ..
0L-____~_____ L_ _ _ _~_ _ _ _ _ _ _~~··~··~··~.~·
0.800
0.850
0.900
0.950
1.000
1.050
where Po is the OOK isotherm, Po is the density at ,
P and T= 0, n is the number of atoms per gram, and p.
FIG. 1. Comparison of theoretical and experimental room-temis a small OOK contribution to the pressure which arises perature
isotherms for sodium metal. Straight line: P with "(="(oj
from lattice vibrations.
dashed line: P with "(="(o+~"(j x: experimental (Ref. 14)j 0:
H. Brooks, Nuovo Cimento SuppJ. 7, 207 (1958).
'D. J. Pastine, Phys. Rev. 166, 703 (1968).
10 K. Fuchs, Proc. Roy. Soc. (London) A153, 622 (1936).
11 D. J. Pastine, J. Phys. Chern. Solids 28, 522 (1967) .
12 W. B. Daniels and C. S. Smith, Phys. Rev. 111, 713 (1958).
11 J. M. Walsh, M. H. Rice, R. G. McQueen, and F. L. Yarger,
Phys. Rev. 118, 196 (1957).
8
points obtained by a slight extrapolation of the experimental data
(Ref. 14).
14 R. I. Beecroft and C. A. Swenson, J. Phys. Chern. Solids 18,
329 (1961).
1~ M . E. Diederich and J. Trivisonno, J. Phys. Chern. Solids
27, 637 (1966).
16 The effect of the longitudinal modes was originally included
in the calculation and it was found to be negligible.
175
THERMODYNAMICS OF SOLID
so that in calculating (llw./w.) we will neglect the
longitudinal contribution. Assuming that the {llw./w.)t F
are mutually proportional for all p.t. modes provides
the simplification that ({llw';w.)F)t should be proportional to anyone {llw/w)t'Jc. It is convenient to
compute (llw/w)t ac for a (110) p.t. wave with polarization perpendicular to the (001) direction. The
velocity of this wave is related to the isothermal elastic
constant c=t(cu-C12) which for a bcc crystal is given
in the QH approximation by
909
METAL
1.30
----- 450" K
_ _ 360' K
1.20
-
.... /
300' K
//
..../
//
//
1.10
1.00
.. ,;.:::.-'/'.:
.... -;?
0.90
.. -
-.
:~:
0.80
c=co- 3PonkT[aiJ'Yl_~],
2x
iJa
iJb
Na
(27)
0 .30
o.~
O.SO
0 .60
0.70
0.80
0 .90
1.00
1.10
o
where a and b are lattice parameters associated, respectively, with the (100) and (01O) directions and
'Y1 = (- (a/w.)iJw./ iJa). Under hydrostatic conditions, 'Y1
is numerically equal to 'Yo, so that we can quite generally
(see Appendix A) set
[a{iJ'YJiJa)-b{iJ'Y1/iJb)]=~{xho',
(28)
FIG. 2. 'Yo+6'Y for the 300, 360, and 4S0 K isotherms.
For all the wavelengths associated with this type of
mode which vary as X l/3 we must have
{Vt llO )2 ex: (Wt Il0 )2x2 /3 ex: xC/po.
With Eqs. (32) and (29) it is quite easy to calculate
where we expect 0~,B{x)~3. By substitution, Eq. (27)
becomes
c=co-!(ponkT),B{xho'.
(29)
The function ,B{x) can be evaluated by demanding that
Eq. (29) reproduce the existing experimental data16 in
the range O~x~ 1.043. Calculated in this way, ,B(x)
assumes the value 2.66 and shows little or no variation
with x, indicating that it is nearly constant. The constancy of ,B(x) can be checked in another way by comparing the theoretical value of the derivative [ -d lnc/
d inx]T_m'K with the experimentaP7 value 2.42. Setting
CT= -iponkT,B'Yo', we have, according to Eq. (29),
d lnci
d lnco
3ponkTX'Yo",B
- - = ---[1 +CT/CO] +
d lnx
T
d lnx
2C
.
---I
iJ lnx
=2.42 .
T-300'K
The value compares quite favorably with the experimental value of Daniels (2.42). The treatment of ,B as
constant or at least as slowly varying compared to the
other quantities appearing in Eq. (30), is, therefore,
from the experimental standpoint, completely justified.
We can proceed now to calculate (llw/w)t ac for the
isothermal p.t. (110) wave. The velocity v/ IO for this
wave is given by
(31)
W. B. Daniels, Phys. Rev. 119, 1246 (1960).
R. J. Corrucini and J. J. Gnieweck, Nat!. Bur. Std. (U. S.)
Mono~raph No. 29 (1961).
17
[(
llW)llO]aC = c~
W
2co
t
3ponkTfho'
(33)
4co
In accordance with assumptions (A) and (B) we now
set
3ponkTa,B'Yo'
- - - - , (34)
4co
where, again, we expect a~ 1.
Since «llw';wi)F~j«llwi/wi)F)t we have, according
to Eq_ (23),
(30)
At room temperature and pressure, at which18 X= 1.043,
we can calculate, with ,B= 2.66, co= 6.74, CT= -0.957,
'Yo" = 0.912, 3ponkT=3.294, and -dlnco/dlnx=1.47
(the units of Co, CT, and 3ponkT are kilobars). From this
we find
iJ Inc
(32)
1l'Y=
ponkTa.B[
2co
d InCo]
X'Yo"-'Yo'-- .
d lnx
(35)
Setting {3= 2.66, we replace 'Yo of Eq. (25) by 'Yo+ ~'Y
and evaluate a by setting P=O at X= 1.043. This
procedure yields ~1.
VI. COMPARISON WITH EXPERIMENT
With 'Y set equal to 'Yo+ll'Y, the theoretical thermal
pressure can be seen to compare quite favorably with
experimenta}l' data (see Fig. 1 and Table I) indicating
that Eq. (35) has the correct volume dependence. In
Fig. 2 the values of 'Y= 'Yo+ll'Y (in Table II) are graphed
for various isotherms above room temperature but
below 38D = 450 oK. These results indicate that for
sodium in the initial stages of compression 'Y is nearly
linear in x, a relationship observed by Swenson. 1S The
computations further indicate that along the isotherms
1l'Y approaches zero with decreasing x which, of course,
means that 'Y approaches 'Yo. The interesting aspect of
this diminution of 1l'Y is that 1l'Y becomes negligible
18
11
C. A. Swenson (private communication).
910
D.
JOHN
TABLE II. The calculated values of 1'0+61' along three isotherms.
x
1'0+ 6 1'
T=300oK
1'0+ 6 1'
T=360oK
1'0+ 6 1'
T=450oK
1.05
1.04
1.02
1.00
0.96
0.92
0.88
0.84
0.80
0.76
0.72
0.68
0.64
0.60
0.56
0.52
0.48
1.164
1.148
1.125
1.097
1.053
1.018
0.986
0.958
0.937
0.917
0.901
0.887
0.876
0.867
0.860
0.854
0.850
1.219
1.200
1.173
1.140
1.089
1.048
1.010
0.977
0.952
0.929
0.909
0.892
0.878
0.867
0.858
0.851
0.846
1.302
1.278
1.246
1.206
1.144
1.091
1.046
1.006
0.975
0.946
0.921
0.900
0.882
0.867
0.855
0.846
0.839
compared to 'Yo at about the same value of x for every
isotherm, namely, ~O.6. The implication of this is
that anharmonic effects diminish with increasing compression and that beyond 40% compression sodium
behaves in an entirely QH fashion with 'Y='Yo. This
hypothesis finds support in the fact that when it is
assumed that 'Y='Yo for x<O.6, the theoretically predicted Hugoniot20 in this region agrees to within experimental error with experimental Hugoniots2J •22 (see
Fig. 3). Since the Hugoniot pressure Ph is quite sensitive
to'Y at large compressions, this is a reasonable test.23
In Fig. 4 the experimental and theoretical temperature dependence of cat constant pressure are compared.
This is a good test of the constancy of 13 and the accuracy of 'Yo'. The theoretical predictions of c, as one
can see, are quite good. As a further test we compare
the theoretical (see Appendix B) and experimental
values of the specific heat at constant pressure Cpo In
Fig. 5 the calculated results are compared with the
experimental values compiled at the National Bureau
of Standards.24 Again the agreement can_be seen to.be
quite good.
PASTINE
175
approximation are simply proportional to those which
may be associated with anharmonic terms in the free
energy. This assumption is partially justified by calculations in Sec. II, which neglect almost entirely the
interactions between normal modes. Secondly, it was
assumed that the fractional shifts in frequency of the
p.t. modes are mutually proportional. The approximation which was applied was for the calculation of 'Yo.
This is the commonly used "two-mode" approximation
which has had frequent and quite successful application
in the literature.3-5 Applications of these basic ideas
were made to sodium metal in Sees. V and VI. Since
not all the physical parameters generated by the theory
could be evaluated, agreement between theory and
experiment was forced in two ways. The two requirements were, first, that at room temperature and volume
we must have P= 0 and, second, that under the same
condition c(theoretical)=c(experimental). This procedure permitted the evaluation of the parameters a
and 13. One significant result of this fitting is the
relation a= 1 which suggests an equivalence between
the average of acoustic frequencies and the average of
those which appear in the free energy. This, in turn,
implies that the quan,tity 'Yo+.1'Y at P=O, T=300oK
can be calculated directly from the experimental
logarithmic derivatives of the elastic constants Cn and
C44 at the same temperature and pressure. This implication is easily checked. The theoretical value of
1000
900
800
700
600
VIT. SUMMARY AND DISCUSSION
A simplified technique for the calculation of anharmonic effects in sodium metal has been prescribed.
In so doing, two assumptions rCA) and (B)] and one
important approximation have been made. It was
assumed first that the thermal frequency shifts associated with acoustic wave propagation in the QH
D. J. Pastine, J. App!. Phys. 35, 3407 (1964).
A. A. Bakanova, J. P. Dudoladov, and R. F. Trunin, Fiz.
Tverd. Tela 7, 1615 (1965) [English trans!.: Soviet Phys.-Solid
State 7, 1307 (1965)].
IS M. H. Rice, J. Phys. Chem. Solids 26,483 (1965).
U For sodium PA varies as 1/[1-1'(1.043-x)/2xJ. The latter
function increases rapidly toward infinity with increasing
compression.
s. R. J. Corrucini and J. J. Gnieweck, Nat!. Bur. Std. (U.S.)
Monoi\'aph No. 21 (1960).
.
10
II
300
200
100
FIG. 3. Comparison of the theoretical and experimental
Hugoniots of sodium meta!. Straight line: theoretical with 1'=1'0;
0: experimental (Ref. 21); x: experimental (Ref. 22); H : limits
of experimental error.
175
THERMODYNAMICS OF SOLID
7 .00
6 .30
."
~
Iv
6 .00
5 .70 ':-0------:'100~---..l
2 00----300...l-----.J
T (OK)
FIG. 4. Comparison of theoretical and experimental values of C.
Straight line: theoretical; El: Daniels (Ref. 12); 0: Diederich
and Trivisonno (Ref. 15).
'Yo+A'Y at P=O, T=300oK is, according to Table II,
'Yo+A'Y= 1.15, whereas using the experimental data of
Daniels17 we find 'Yo+ A'Y = 1.18. The agreement is
doubtless within experimental error. With a and {3
evaluated, the room-temperature isotherm, the
Hugoniot Cp , and the quantity -d lnc/d lnxl T-3000X
were all calculated and found to agree well with experimental values. In addition, the Swenson relation
'Y='Yo+A'Y ex: x was established in the initial stages of
compression. It was observed that as isothermal compression increased, anharmonic effects diminished so
that 'Y approached 'Yo and the system tended toward a
QH state. This behavior can be anticipated without
calculation for the following reasons. At P= 0 and room
temperature, the system rests near the minimum of the
potential. At this point changes in the potential due to
displacements of the system near the minimum are
known to be quite asymmetric. This is, of course,
because to the right (expansion) of the minimum the
potential tends to zero, whereas to the left (compression) it approaches plus infinity. As isothermal
compression increases, two things occur which lead
to the diminution of anharmonic effects. The first is
that vibrational amplitudes diminish since as pressure
increases displacements of a given amplitUde require
more and more energy. This is probably25 accompanied
by a relative decrease in the values of the coefficients
of the cubic and quartic terms relative to the value of
coefficient of the square term in expansions like that
of Eq. (1). The net result of this will naturally be a
reduction in the effect of the anharmonic terms. The
second thing to be considered is that compression moves
the system to the left of the potential minimum and,
therefore, away from the region of greatest asymmetry
[the region in which the terms of odd power in the
We say "probably" because the decrease is inferred by the
use of phenomenological potentials which can be used to compute
such coefficients directly and also by the fact that for sodium the
ratio -x(iJ 2P/iJx'l)T!(iJP/iJxh decreases with increasing compression.
S6
Na
911
METAL
expansion, Eq. (1), show the greatest influence]. The
result here will be a reduction of the effect of the leading
anharmonic term (cubic term) in Eq. (1). These considerations then, all tend to demonstrate the reasonableness of our calculated result.
The calculation of anharmonic contribution, cva, to
the specific heat at constant volume, Cv, requires some
discussion and has, therefore, been reserved until now.
Assuming the QH value of 'Y to be 1.1, Martin26 has
calculated A = cva/3nkT for the alkali metals at zero
pressure. For sodium it was found that A = 1.69X 10-'
per OK. As it happens, A is not insensitive to the choice
of the QH 'Y (i.e., 'Yo). Martin has pointed out that for
potassium a decrease of 0.2 in 'Yo results in a 40%
increase in A. Using this as a basis, and considering
that here the calculated 'Yo is 0.88, Martin's value of A
for sodium must be increased by about 44% in order
to obtain a reasonable comparison with theory. This
would increase A to 2.4X 10-' per OK for sodium.
Within the framework of the present approach (see
Appendix B) one finds
The term on the right can be calculated according to
the assumptions and approximations already discussed.
This evaluation yields cvo/3nk= 2.9X ID-4T. Considering the crude correction of Martin's value, the
agreement is quite good. As another demonstration of
the efficacy of the theory, we will calculate the thermal
expansion coefficient aT= (l/x) (ax/ aT)p and compare
with the experimental value (210.4±5.6X 10-6 per OK)
of Sullivan and WeymouthY Because of anharmonic
effects, the well-known Griineisen relation which
connects aT and the isothermal bulk modulus BT IS
12 .6
12 .4
~
°
0-> 12 0
".
V>
()
~
/
12.2
11.8
11.6
0.11.4
U
,
,
I'
1//)'/1
1/ /'
/ '}1 .5%
!'
//
--
THEORETICA L.
N.B .S. COMPI LATIO N
"/ /
fI
-0
b
11.2
1 I .0 200
225
250 275 300
T (oK)
FIG. 5. Comparison of theoretical and experimental C" values.
The Dulong-Petit value is 10.84X 10+t erg/g OK.
----
D. Martin, Phys. Rev. 139, AlSO (1965).
G. A. Sullivan and J. W. Weymouth, Phys. Rev. 736, A1141
(1964).
II
27
912
D. JOHN
slightly altered in form. Without these effects the
relation is simply aT='Yop3nk/B T. With first-order
anharmonic .effects included, the relation becomes
aT= ('Yo+ 2t:.'Y )p3nk/B T. The calculated value of BT
for sodium at x= 1.04 and T= 300 0 K turns out to be
6.77X 1010 dynes/cm2 • Using the tables, we find
'Yo+ 2t:.'Y= 1.4. These values produce an aT of 219X 10-6
per OK which again compares favorably with the experimental values. Finally we can compare the theoretical
value of the derivative (aCp/ap)T at 300 0 K and zero
pressure with the experimental value given by
Rodionov. 28 'The theoretical · value is -38.1X 10-'
cm3/mole OK the experimental value is ·-38.0Xlo-'
cm3/mole oK.
APPENDIX A
175
PASTINE
energy along the OOK isotherm, w;'=wi[l+t:.wi/ wi],
and t:.W';Wi is a function of x and T.
The specific heat at constant volume c~ is given by
c,= -(~) T2[a(F/T)] .
oT '"
oT
(B2)
%
By substitution of Eq. (Bl) into Eq. (B2), one obtains
c,=3n{
1_2<:')F].
(B3)
In order to account for quantum effects, the quantity
(C~)QH will be substituted for 3nk where (C.)QH is,
according to the Debye approximation for T>@D (@D
. is taken26 as IS2°K),
The derivative d'Yl/dx='Yl' is given by
hi a'Yl da 0'Y1 db <h'1 dc
-=--+--+--.
.
dx CJa dx ob dx ac dx
(AI)
To obtain Cp one can use the·identity
Under hydrostatic conditions da/ dx= db/ dx= dc/ dx
= a/3x and by symmetry o'Yl/ ob= O'YljaC so that Eq.
(AI) becomes
.
d'Yl (a'Yl 20'Yl)
3x-=a
dx
aa
ob
-+-- .
Cp=C,+ aTx[p+po(OE) ],
Po
ox T
(BS)
(A2)
where aT is the cubical thermal expansion coefficient
and E is the total energy per gram.
Using the relation E=-T[a(F/T)/alnT]" and
Eq. (Bl) for F, one can obtain
(A3)
(B6)
Under hydrostatic conditions 3ba'Y 1/ ab ' is a function
of x only and, moreover, for a cubiC crystal 'Yl = 'Yo.
With these considerations we have, from Eq. (A3),
The quantum corrections to Eq. CB6) which have only
a smaIl effect on the result have been omitted.
For the pressure we have the expression
Since a= b, Eq. (A2) can be reformulated as
d'Yl (0'Y1 O'Yl)
0'Y1
3x-= a - - b - +3b-.
dx
oa
~
, ab
(A4)
where fJ(x) = 3[1- (ba'Yt! ab)/ (xd'Yo/dx)].
It is quite reasonable to take 05, o'Yljab5, O'YI/oa, so
that according to Eq. (A3) and the definition of fJ(x)
we should expect 05,fJ(x) 5, 3.
P=P o+p03nkT('Yo+t:.'Y)/x,
(B7)
where t:.'Y is given by Eq. (20) and Po= - Po Cd cpo/dx).
Substituting Eq. (B4) into Eq. (B3) and then Eqs.
(B3), (B6), and (B7) into Eq. (BS) the result for Cp is
.C
"'3nk[1_~(@D)2
20 T
p-
APPENDIX B
(B8)
For the free energy per gram of a three-dimensional
solid we have the expression
3n
F= cpo-kT L In[kT/hw/] ,
(Bl)
For (t:.W';Wi)F we use the expression developed in Sec.
V, namely,
':-1
where
CPo
is the static (excluding vibrational energy)
28 K. P. RodionoY, Zh. Tekh. Fiz. 36, 1287 (1966) [English
trans!.: Soviet Phys.-Tech. Phys. 11,955 (1967)J.
.
: where a{J= 2.66.
ponkTafho'
2eo
(B9)