Some anharmonic properties of Caesium halides

J. Phys. Chem. Solids Vol. 41, pp. 263 270 Pergamon Press Ltd., 1980. Printed in Great Britain

SOME ANHARMONIC PROPERTIES OF CAESIUM HALIDES

S. DURAISWAMY Department of Physics, S.N. College, Madurai 625012, India and T. M. HARIDASANt Centre for Chemical Physics, University of Western Ontario, Canada N6A3K7 (Received 28 March 1979; accepted 6 July 1979)

Al~traet--The second thermal Griineisen parameter and isentropic Anderson-Griineisen parameter for caesium halides are determined at different temperatures employing a pressure dependent shell model of lattice dynamics. A new thermodynamic relation is also proposed to calculate the second thermal Griineisen parameter using experimentally measurable quantities. The agreement between experimentally estimated and theoretically calculated values is fairly good for the Anderson-Griineisen parameter while it is very poor for the second thermal Griineisen parameter.

INTRODUCTION

The first order volume dependent anharmonic Griineisen constant or the second order Grbneisen parameter q(T) as it is often called, is of crucial importance in the extrapolation of thermodynamic properties to determine the adiabatic temperature gradient in planetary interiors as well as in the interpretation of shock wave experiments. Yet another equally important manifestation of anharmonicity is the isentropic Anderson-Griineisen parameter ~3~ which was shown by Anderson using the Mie Grufieisen equation of state to be independent of temperature. Using more realistic models, one can show that ds actually depends on temperature [1, 2]. It is the purpose of this article to calculate these two quantities q(T) and a s at different temperatures for caesium halides by a lattice dynamical approach. Cs halides are essentially Griineisen solids [3], showing almost no variation of the Grtineisen function with respect to temperature. The first attempt to study the behaviour of the second order Grtineisen parameter with temperature for these halides was made by Vetelino et al. [4, 5 ] using the rigid ion model (RIM). However, the experimental determination of phonon dispersion for these halides is well-accounted for by the shell model rather than by the RIM [6]. Further, Barron and Batana's work using a RIM [7] indicate that RIM is not sensitive enough to indicate a shallow minimum of the Griineisen parameter in the low temperature region. This view is confirmed by Barsch and Achar [8] also. Recent work by Bijanki et al. [9] on the determination of q(T) using a deformation dipole model does not allow one to compare their approach with other models, since the variation of q(T) with *Permanent address: School of Physics, Madurai University, Madurai 625021 India.

temperature is not brought out either in tabular form or by figure. In this context, to check the superiority and sensitivity of the shell model over RIM in determining any anharmonic parameter, we have determined the variation of q(T) and c~s with temperature using Barsch and Achar's [8, 10 ] method. It is worth mentioning that the variation of ds with temperature for these halides has not so far been reported. In Section 1, the thermodynamic relations which will enable the determination of q(T) and ds in terms of readily measurable quantities are summarized. Section 2 deals with the theoretical determination of these quantities in the quasiharmonic approximation. Section 3 presents the results and discusses certain anharmonic properties of these crystals in the light of these results. SECTION 1

The explicit dependence of the Griineisen constant on volume or the second Griineisen constant at any temperature T is defined as

7

q(T)-- l~nV], r

~PP T"

(1)

By using the thermodynamic relations ~pp .,,= -

~/~-]p

j

(2)

p/T one can arrive at the relation q(T) =

263

- 1 B | I - ~ T - - I p + | ~ T - - I A"

(4)

264

S. DURAISWAMYand T. M. HARIDASAN

Also using the relation

branch index j = 1.... ,3n, in addition to the pressure p. A, B, D, U, W and A are expanded with respect to pressure as a perturbation, as for example

fiB r ~ ~?r ]p = ~ r = ~

(l)

+ R-1

A = do + AlP + ~.A2 p2 + ...

(5)

(12)

where the different coefficients A_., B., etc. are the differentials

one can find

q(T) = ~~s + R-1

dn

dn

d~A, x

p

1+~;+

~/.J

(6)

It is also well known that the isentropic AndersonGriineisen parameter [11 ] is defined as

a~ =

-#B~ ~a T l /

fiBs BTfl pCp

(7) dA - - = A 1 = U0(qj)P_Uo(qj) dp

(8)

(14)

Where

SECTION 2

The microscopic second mode Griineisen parameters can be determined using a quasiharmonic pressure dependent lattice dynamical model. The method briefly outlined here is due to Barsch and Achar [10]. The time dependent equations of motion for the shell model in reciprocal space are given by [ 15 ]

O = B+U +__DW

[_Uo(~)_eLo(~)]: + . ~Y~ : ") " j ' ~o~,(qj) - ~o(qJ

(9)

and all other symbols used here are the most common representations of the various thermodynamic properties. q(T) and ~3~are estimated at different temperatures by eqns (4) and (7) using experimentally available quantities. The bulk modulus values and their temperature derivatives are taken from the experimental work of Slagle and Mckinstry [12]. The thermal expansion coefficient values for CsC1 and CsBr are taken from Bailey et al. [13] and for CsI from James et al. [14], while their temperature derivatives are found graphically by drawing fl vs T.

MA!2 = AU + BW

(13)

1 dEA dp 2 = A 2 = Uo(qj)Q_o(qj)

pC v

R =--Be = 1 + 7fiT BT

etc.

The perturbed series are substituted in (10) and (11) and the zero, first and second order perturbation equations are obtained by comparing the coefficients of the zero, first and second powers of p. Eliminating the core shell displacements, one gets the first and second coefficients of the eigen values

In the above identities y is the first Grtineisen constant given as

? --

dpgB,

(10) (11)

. P = M -. 1 / 2 [ A 1 .

-

B o. O o l B ~

- - n 1,~0 D - I B +- - 0

+ __B0__D O I_D1DO 1B~ ] _M 1/2 Q

= ~

M_ - 1~: [ A 2 - B_:Do I_B~ - ~o_Do I_B~

+ BoDo 1--DzDoIBm-] M - 1/2

(16a)

+ M- ~/=[ - BIDolB~ + _BI_DolD~OolB~+ _BoP_DoI_D1Do 'B~ _ _BoDo xD1Do- i D-1Do - i ~o+ ]M -1/2.

(16b)

A knowledge ofA 1 and A 2 enables us to determine the first and second mode gammas since

~--T~-n V I T = TL~J,,

7"J=/e~j

SV2 I ,

: 2 - L o(qJ)J

fl

+ t 2 LAo(OJ L + / ap/.,.

8T iA, ,ll 2 ~IJJ

where A, B, D are 3n x 3n coupling matrices, n being the number of ions per unit cell, and M is a diagonal mass matrix of the same order. U and W are eigen vectors of order 3n denoting the core and relative core shell displacements. The matrices A, B and D depend on U, W, eigen values A = o 2, wave vector q and

(15)

(18)

A brief sketch of Barsch and Achafs method [10] is outlined here mainly to point out that while in eqn (16a) the terms

(-BoDo 1-B~- + -BoDo 1-D2Do 1Bg)

265

Some anharmonic properties of caesium halides are either erroneously calculated or misprinted in their paper, in eqn (18) the terms within the brace {...} are not taken into account completely. The second thermal Griineisen constant, as we know is defined as

(24)

- 1.37935Z21

C 4 4 = FrO [_ 1

-

4

- 0.70089Z 2]

[~31n7 / q(T) = l~,~nv ] T

(19)

(25)

e2

R o = 8roB ~ = ~ [(A12 + 2B12) ].

(26)

where

2 7qjCqj 7 = (Tq,/) -

,o

(20)

Cqj

qJ the weight factors Cqj being the Einstein specific heats of mode (od'). Substituting for mode gammas _ t~In coqj] OlnVlr

7q~= /

(Z,e)2 _ 9V/~mz(e0 + 2) 4n(e~ + 2) 2

and carrying out the differentiations, we have q(T) = 1 - , / - t { ( 7 ~ ; )

The total number of parameters to be determined is reduced by assuming the second neighbour anionanion interaction to be equal to the cation-cation interaction. Here C]~, C]2 and C]4 are the adiabatic elastic constants, r o is the nearest neighbour distance and B~ = (C]1 + 2C]2)/3 is the adiabatic bulk modulus. In addition to these, we have the effective charge (Z'e) on the ions given by

_ (72) + 2(kT-

1)-'

(21)

where/~ is the reduced mass, o).r is the infrared optic frequency, eo and ~ are the high and low frequency dielectric constants. The distortion polarisabilities are calculated from the relation

where

d Ro - Z e 2

Ih°gqj I

where ~ is the field polarisability of the ion. The calculation of Z', and the distortion polarisabilities of the positive and negative ions d + and d_ were checked by substituting the values in the relation

t

q

-

qJ

(28)

(22)

2 ]:qjCqj (~,'j)

(27)

as before.

Cqj

qJ

, /m)2 = R°

Similar expressions for #s in the quasiharmonic approximation can be arrived at using the relation (6). The calculations for the shell model of lattice dynamics are based on the models proposed by Dick and Overhauser [ 16 ] and later elaborated by Woods et al. [17] and Cowley [15]. The model involves two first neighbour force constants A12 and Bl2 , the first neighbour forces being assumed noncentral, and two second neighbour force constants A~I and Bll for cation and A22 and B22 for anion respectively. The model parameters are estimated at zero pressure using the relations

) + 1.40179Z 2]

C]2 = Vro[_ ~

R o -

e2

~+

+

Z' = Z + d+ - d_

(29)

(30)

(31)

and comparing them with experimental values. The charges on the shells YK and the isotropic core shell force constants k r were calculated using Yr -

--4CtK(A12 + 2B12) 3dr

kK=-Yd~r r.

(23)

4

Ro =

4n(e~ + 2) (Z,e)2 9V

(32) (33)

The above mentioned parameters are determined using the experimental values of C ] I , C ] 2 , C ~ 4 , the static and optic dielectric constants eo and e~, and transverse optic frequency coT. Computations were carried out with an IBM-370 computer for 56

266

S. DURAISWAMYand T. M. HARIDASAN

representative points in the Brillouin zone defined by q =(qx, qr, qz) where 0 . 5 > q x > - q r > q z = 0 " The frequencies from each wave vector when weighted properly generate 6000 frequencies. The first and second order pressure dependencies of the eigen values are similarly computed using the experimental values of the first and second pressure derivatives of C] 1, C] 2, C]4, coT, eo and e~. The input data for zero, first and second order pressure derivatives of the adiabatic elastic constants are taken from Barsch [ 18 ]. It is to be mentioned here that our theoretical results using earlier experimental values of Barsch et al. [19] showed wide spreads of the first and second mode gammas and marked disagreement with the experimental values of q(T). The zero and first pressure derivatives of eo and e~ are taken from Jones [20]. While the second pressure derivative of the static dielectric constant is taken from the above reference, the second pressure derivative of e~ is determined using the relation [10]

B2r l~p2 ] = - 1 ~ - p + 1 B r ~p

(34)

where BT is the isothermal bulk modulus, n is the refractive index in the infrared region. Upon substituting n = ~

this relation becomes

[63BT

~2e m

) 0'~o~

dp2 - -(1/BT) ~ c~p + 1 t~p ~oo 2

The zero and first pressure derivatives of the transverse optic frequency o r are taken from the experimental work of Rastogi et al. [21]. The second pressure derivatives are got by differentiating the expression [22] co2 =

MB T

(e~ + 2/(% + 2)) qo

(36)

where M is the coordination number and k( is the reduced mass of the crystal. Even though the zero and first pressure derivatives of car obtained using the relation (36) are not in good agreement with the experimental values of Rastogi et al. [21 ], we are encouraged to use this relation to determine the second pressure derivatives of car because the first mode gammas and hence the first thermal Griineisen parameter computed using the values of caT and (dcaT/dp) estimated from (36) did not show much variation with the thermal first Griineisen parameter got from experimental values of caT and (&nr/dp).

SECTION 3

Figure 1 represents the dispersion characteristics of the second mode gammas for CsCI. As is obvious, the

branches in the graph are so crowded in many regions, it is felt that presentation in the form of tables will throw more light on the results; hence the dispersion of the second mode gammas for CsC1, CsBr and CsI are tabulated for a few important points in inverse space in Tables 1-3. It is not surprising to note that the present shell model results are in wide disagreement with Mitra'sE4,5] RIM results, not only because of the inherent differences between the two models but also because of the considerable changes in experimental elastic constant values used in the two studies. The most obvious differences are that the dispersion of the various modes is more pronounced in our work and that the high positive values exhibited for certain modes in the rigid ion model is not noticed in the shell model. Further the second mode gammas are triply degenerate for these halides in the most prominent symmetry direction (0.5,0.5,0.5)just like the frequencies. It is strange to note that in this direction the mode gammas have the same values as in the RIM [4, 5]. The comparison of the second mode gammas would be more useful if the RIM [4, 5 ] results and the results of this work were plotted in the same figure; but then the clarity would be marred further since too many lines would cross each other. Figures (2-4) give the variation of q(T) with T, and the 8s with T variation is demonstrated in Figs. (5-7) for the halides under consideration. It is to be emphasised here that in Barsch's et al. work [10], the core shell force constant was assumed to be pressure independent whereas in our calculations, it is introduced in the most natural way. Further, in contrast to Mitra's RIM work [4, 5], our work is based on the shell model, which takes into account the polarisability of the shell also. Perhaps because of this we were able to notice that the thermal second GriJneisen parameter q(T) shows a noticeable minimum around 17K. The appearance of this minimum is in agreement with Barsch's et al. [8] finding for the first thermal Grtineisen parameter of NaC1, the only difference being that Barsch has stated that this minimum occurs around (T/OD) = 0.045, by which it appears he has tried to generalise this effect. However, our calculations indicate that this minimum has occurred around 17 K for all the three halides we have considered. It would be more appropriate to say that the dip is contributed by the positive ions only. To make this statement more general would require more supporting calculations for different sets of halides, with each set having the same positive ions, and using more refined models. While the theoretical values of q(T) vs T show uniform change except for the shallow minimum around 17 K, the experimental values exhibit a strange behaviour. It appears that the experimental values behave so peculiarly that we can draw no inference about the thermodynamic properties of these halides. Equation (4) indicates that the q(T) values are quite sensitive to the temperature gradient of B~ and fl, or perhaps relation (4) may not be suitable for the

267

Some anharmonic properties of caesium halides

12

~

4

A "~

,ooV

_2~:'--~'~S

I TO

(Y,O,O)

F

C$CI

I

"(Tr

(5,Y,Y)

X ( .5,0D)

AA

I


R~.5;5;5)

.:::;,'o I

I /hlJ

?'gl

"-"

I


F (0,0,0)

M(.5,0,O)

Fig. 1. DispersionofsecondmodeGriineisen parameter with reduced wave vector in the (~,0,0), (0.5,~,~), (~,~,~) and ((, ~, 0) directions (solid lines represent the acoustic modes while the broken line optic modes).

Table 1. Second mode Griineisen parameter for all the six modes at few representative points in the Brillouin zone for CsCI

ti:,,

Wave vector qr,

q~

LO

LA

TO1

TA1

TO2

TA2

0.0 0.1 0.2 0.3 0.4 0.5 0.5 0.5 0.5 0.5 0.5 0.4 0.3 0.2 0.1 0.1 0.2 0.3 0.4 0.5

0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.4 0.3 0.2 0.1 0.1 0.2 0.3 0.4 0.5

0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.4 0.3 0.2 0.1 0.0 0.0 0.0 0.0 0.0

-0.8596 -0.4132 0.8074 2.4533 3.9120 4.4619 5.6254 7.7690 8.6077 6.5398 5.1225 5.9507 11.3486 6.5375 0.9776 0.2708 3.6611 2.4661 12.0453 14.1211

- 120,3156 6,3559 5,7929 5.0129 3.2309 4.2249 7.4812 13.7092 0.2499 6.5541 1.6856 0.7681 0.9541 4.6820 7.1883 7.3999 5.1459 10.2008 - 14.112 - 16.216

2.8927 3.2319 4.1363 5.0378 4.9477 4.4116 4.9039 5.5216 5.5356 5.2611 5.1225 5.9511 7.1683 6.3233 4.0038 3.7398 6.2470 8.2948 13.9737 14.1211

-91.5120 5.0810 3.3531 2.1621 4.3807 4.6408 3.8435 1.9630 - 0.5061 1.3553 1.6856 1.7170 1.9794 3.0241 4.4599 0.7294 - 2.8463 -8.3345 - 12.3011 - 16.2166

2.8927 3.2322 4.1366 5.0380 4.9478 4.4116 6.2690 1.1822 10.8765 1.6425 5.1225 7.8734 7.1676 6.3247 4.0031 3.5335 4.7207 2.5955 2.4026 1.6084

-91.5120 5.0989 3.3455 2.1642 3.2301 4.6376 3.6337 2.2255 1.6706 1.6216 1.6856 1.7171 1.9799 3.0231 4.4560 4.1517 1.5090 2.1221 6.0395 7.0789

268

S. DURAISWAMY a n d T. M. HARIDASAN

T a b l e 2. S e c o n d m o d e G r i i n e i s e n p a r a m e t e r for all the six m o d e s a t few r e p r e s e n t a t i v e p o i n t s in the Brillouin z o n e for C s B r

)'qj Wave vector

qx,

qr'

qz

LO

LA

TO1

TA1

TO2

TA2

0.0 0.1 0.2 0.3 0.4 0.5 0.5 0.5 0.5 0.5 0.5 0.4 0.3 0.2 0.1 0.1 0.2 0.3 0.4 0.5

0.0 0.0 0.0 0.0 0.0 0.0 0.I 0.2 0.3 0.4 0.5 0.4 0.3 0.2 0.1 0.1 0.2 0.3 0.4 0.5

0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.4 0.3 0.2 0.1 0.0 0.0 0.0 0.0 0.0

4.9134 5.0234 5.2833 5.9479 5.2365 3.2435 1.6019 - 1.6616 - 1.0394 2.1066 4.0984 1.1897 1.5537 5.4996 5.3762 5.2096 5.7292 3.3706 -2.7228 -2.1612

-120.1221 4.7260 5.2467 5.9486 -0.8989 - 11.8974 - 3.3799 4.9599 -0.0312 4.2870 2.7312 2.5555 -0.7693 1.7508 5.4972 6.1467 4.5582 3.9464 3.0726 - 11.1249

-12.1569 - 11.5428 -9.6145 -6.1329 6.9661 2.8991 11.6855 1.2542 0.5830 2.3060 4.0984 3.0417 -0.2020 -5.1893 - 10.0895 - 10~7006 -7.4779 - 4.6274 0.7521 -2.1599

-61.8646 5.0864 2.3071 -2.1894 8.1000 9.4407 - 7.5697 -3.1984 0.4681 2.2310 2.7312 2.5556 2.2089 2.1205 2.4794 - 1.6597 -4.4126 - 7.9049 10.1881 - 11.1272

-12.1569 - 11.5425 -9.6151 -6.1333 -0.8987 2.8992 4.0066 0.9778 4.7441 2.6405 4.0984 3.0418 -0.2003 -5.1889 - 10.0894 - 10.9419 -7.2958 - 1.5126 -3.3759 3.6927

-61.8646 5.0834 2.3087 -2.1910 8.0995 - 11.8973 - 8.6492 - 1.8954 -0.2805 2.1933 2.7312 1.4091 2.2085 2.1197 2.4760 4.1782 -1.0149 - 8.4349 - 14.2447 - 15.4977

T a b l e 3. S e c o n d m o d e G r i i n e i s e n p a r a m e t e r for all t h e six m o d e s a t few r e p r e s e n t a t i v e p o i n t s in the B r i l l o u i n z o n e for C s l

7;j Wave vector qx, qy,

qz

LO

LA

0.0 0.1 0.2 0.3 0.4 0.5 0.5 0.5 0.5 0.5 0.5 0.4 0.3 0.2 0.1 0.1 0.2 0.3 0.4 0.5

0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.4 0.3 0.2 0.1 0.0 0.0 0.0 0.0 0.0

--9.5027 -7.9415 -7.2584 -4.5421 -1.1691 -0.2674 0.2134 0.3300 --2.2623 - 3.1849 5.0073 - 1.6744 10.9149 --1.7673 - 7.6282 --8.2911 --4.5211 2.5785 3.2437 4.9770

- 143.6102 7.1203 6.4338 4.9907 8.9693 -- 17.0876 9.0037 -5.8403 - 11.4233 4.9800 -0.9867 7.0761 - 8.3597 6.6131 9.0652 9.1244 6.2912 1.1768 7.4504 -30.2817

0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.4 0.3 0.2 0.1 0.1 0.2 0.3 0.4 0.5

TOI - 13.8211 - 12.8728 -9.8001 -3.5309 2.7458 2.6564 3.1409 2.1188 8.9801 1.0227 5.0073 - 1.6745 - 3.9369 --8.2316 - 12.3579 --13.1949 - 10.1939 --2.9705 --0.8439 4.9770

TA1

TO2

TA2

-- 110.2442 - 14.8872 --15.5510 -21.3862 -33.3427 2.5292 - 12.6407 9.4138 --5.7149 - 2.3422 -0.9867 5.9812 6.7561 5.9311 5.3986 -0.3914 - 6.2543 - 18.3890 --27.5102 -30.2798

- 13.8211 - 12.8729 -9.7995 --3.5310 8.9690 2.5293 3.0972 4.1305 4.7767 3.2691 5.0073 -2.6519 - 3.9360 -8.2308 - 12.3576 --11.9401 - 5.8449 7.7503 1.1973 --74.4979

-- 110.2442 - 14.8857 -15.5572 -21.3854 -33.3433 - 17.0881 - 14.4150 -9.8849 7.1965 9.3277 -0.9867 5.9812 6.7554 5.9310 5.3989 -23.6659 --48.3812 -94.1586 -91.2134 -90.9221

Some anharmonic properties of caesium halides

"k 5

,o

CsCI

|

•

;o

......

8'o

•

,~o] 2~o ~T

2~tO

Z'5

CsBr t I

i

2~o

Fig. 2. Variation of second thermal Oriineisen parameter for CsCI with temperature in absolute scale for CsC1 ( - - gives results of this work; . . . . gives Mitra's results[4,5] • Experimental values) --, arrow on the abscissa refers to Debye temperature of the solid.

q (T)

CsCI

•

t =o

269

50

t

'

Ioo

2~o

'

20

CsBr

A ~ a

15 •

I

5~0

~T Fig. 5. Variation of Anderson-Griineisen parameter with temperature in absolute scale for CsC1 ( gives results of this work; A - - r e p r e s e n t experimental values; X axis is shown in an enlarged scale below 50K) --, arrow on the abscissa refers to the Debye temperature of the solid.

~i eo 5

~o

~'o

,~l ,an ~,o -T

,

2:,o ~o

Fig. 3. Variation of second thermal Griineisen parameter with temperature in absolute scale for CsBr ( gives results of this work: gives Mitra's results [4, 5] • - - E x p e r i mental values) - , arrow on the abscissa refers to Debye temperature of the solid.

~o

•

~o

•

z~o

~T

500

Fig. 6. Variation of Anderson-Gr~neisen parameterwith temperature in absolute scale for CsBr ( gives results of this work; • - - r e p r e s e n t experimental values; X axis is shown in an enlarged scale below 50K) --, arrow on the abscissa refers to the Debye temperature of the solid.

I

41 5~ q (r)

Csl

2:0

Csl

15

',

~i eo

2

.

.

.

.

.

: ......

I 40

80

t--l & "120

I 160

I 200

I 240

I 280

~T Fig. 4. Variation of second thermal GrBneisen parameter with temperature in absolute scale for CsI ( gives results of this work; - - gives Mitra's results [4, 5] • - - E x p e r i mental values) --* arrow on the abscissa refers to the Debye temperature of the solid. JPCS Vol. 41. No. 3 - - F

i~

~o

"='"=

•

,~ot

'

2~0

'

56o

~T Fig. 7. Variation of Anderson-Griineisen parameter with temperature in absolute scale for CsI ( - - gives results of this work; A - - r e p r e s e n t experimental values; X axis is shown in an enlarged scale below 50 K) -* arrow on the abscissa refers to the Debye temperature of the solid.

270

S. DURAISWAMYand T. M. HARIDASAN

temperature range under consideration. It is worth noticing that even though our expression for q(T) is different from that of Mitra's E5], still the peculiar behavior of q(T) with temperature remains the same. Contrary to the disagreement of experimental and theoretical q(T) values, the agreement for the isentropic Anderson-Grtineisen parameter t3 is fairly good. The difference between the experimental and theoretical values gradually increases as the temperature decreases. It is to be borne in mind that the experimental values used in the theoretical calculations are so uncertain, in particular the second pressure derivative of the elastic constants have an uncertainty around 15%. For example, the value of (d2Cl 2/dP 2) in the case of CsC1 is - 2 . 3 3 _+ 0.4, in the case of CsBr is - 3 . 2 7 _+ 0.38 and for CsI - 4 . 0 6 _ 0.42. In the light of the uncertainty in the various thermodynamical measurements utilised in our calculation, we can conclude that the agreement of the calculated values of ds with the experimental values is fair. Acknowledgements--We thank G. R. Barsch for communicating his recalculated experimental values of pressure derivatives of the elastic constants. We also thank N. Krishnamurthy and R. Ramjirao for making useful suggestions when the work was in progress. One of us (S. D.) thanks S. Balasubramanian, Coordinator, Madurai University Physics Department for his keen interest in this work, and UGC for providing financial assistance to carry out the computations. REFERENCES

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