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Optic mode gruneisen parameters and their strain derivatives in alkali halides
Physica 94B (1978) 331-345 © North-Holland Publishing Company
OPTIC MODE GRUNEISEN PARAMETERS AND THEIR STRAIN DERIVATIVES IN ALKALI HALIDES A. V. SINGH, J. C. SHARMA and Jai SHANKER Department of Physics, Agra College, Agra-282002, lndia Received 8 July 1977 Revised 15 November 1977
The Born--May~r theory of interatomic forces and the Szigeti model of dielectric properties have been extended to predict the optic mode Gruneisen parameters and their volume derivatives for the alkali halides. A detailed analysis of the dielectric behaviour of alkali halides under hydrostatic pressure is performed in the light of experimental data on first and second order strain derivatives of dielectric constants. To make the investigation critical and comprehensive we have adopted the inverse power law as well as the exponential law for the short range repulsive interactions. Three different sets of the Born repulsive parameters corresponding to compression data, ultrasonic data and dielectric data are used. The transverse optic mode Gruneisen parameters (3,TO) calculated, using short range repulsive parameters corresponding to compression and ultrasonic data, present good agreement with experimental values. The Lyddane, Sachs and Teller relation is used to predict the longitudinal optic mode Gruneisen parameters (-rLO) and their volume derivatives (a~/LO/a V). The general features of the results obtained are that -rTO is larger than 3,LO and that 3,TO decreases with pressure whereas 3,LO increases. A comparison of the average Gruneisen parameter obtained in the present study has been made with the macroscopic Gruneisen parameter deduced from the thermoelastic data. The volume dependence of the Szigeti effective charge parameter has also been predicted and discussed.
1. Introduction Studies on the Gruneisen parameter provide valuable information about the volume dependence of the frequency spectrum which is of crucial importance in the extrapolation of the thermodynamic properties of crystals subjected to high pressure and high temperature. The Gruneisen parameter and its volume dependence have been the subject of great interest for describing the thermoelastic [ 1 , 2 ] and dielectric properties [3, 4] of solids, and in geophysical studies [5]. The present paper deals with the evaluation of optic mode Gruneisen parameters and their volume dependence for NaC1 structure alkali halides adopting the Born model of interatomic forces, and the L y d d a n e Sachs-Teller (LST) and Szigeti relations for optic mode frequencies. Madan [3] and Lowndes and Martin [4] have employed the first and second Szigeti relations [6], respectively, to evaluate the optic mode Gruneisen parameters. At that time the data on dielectric constants and their first order pressure derivatives were available for a few alkali halides and it was then not possible to make a comprehensive investigation of the entire family of NaC1 structure alkali halides. In recent times the first as well as higher order pressure derivatives of dielectric constants of alkali halides have been reported in the literature [7]. It is therefore possible to make an analysis not only of the Gruneisen parameters but also of their volume derivatives. Ill the present study we adopt two potential form s for the short range repulsive interactions, representing the inverse power dependence and the exponential dependence on interionic separation. A detailed investigation o f these two forms within the framework of the Born model has been made by Tosi [8]. The repulsive parameters entering the short range potential form are determined from the crystal equilibrium condition and the second derivative o f potential energy which is related to the compressibility or elastic constants. The compression data were used by Tosi to evaluate the repulsive parameters. More recently Smith and collaborators [9] have published highly accurate ultrasonic data which have been used by Smith and Cain [10] to estimate the repulsive parameters. 331
332
A. V. Singh et aL /Optic mode Gruneisen parameters in alkali halides
Smith and Cain have convincingly demonstrated that the repulsive parameters corresponding to ultrasonic data exhibit more refined and systematic trends than those corresponding to compression data. We will, however, make use of both the sets. The third set of parameters to be used in the present study is based on the dielectric data. It was recently emphasized by Boswarva and Simpson [11 ] that the Born repulsive parameters derived from the compressibility or elastic data are not consistent with the experimental values of the static dielectric constant. Therefore an alternative method of determining the repulsive parameters from dielectric data has been suggested by these investigators [ 11 ]. The present analysis deals with the two Szigeti relations [6] derived from the classical theory of dielectric polarization. In insulators like alkali halides in which we have bound electrons the classical theories are fairly satisfactory, The first Szigeti relation connects the static and electronic dielectric constant through the force constant between ions and the transverse optic mode frequency coTO of lattice vibrations. The first and higher order volume derivatives of coTO lead to the expressions for the corresponding Gruneisen parameter (TTO) and its strain derivatives. The LST relation provides a link between wTO and the longitudinal optic mode frequency coLO, and hence can be used to predict the LO mode Gruneisen parameter 7LO and its volume dependence. In section 2 we give the necessary formulation and the method for evaluation of these quantities. The second Szigeti relation expresses the difference of low frequency (static) dielectric constant and high frequency (electronic) dielectric constant in terms of coTO and effective charge parameter e~. The volume derivatives of the second Szigeti relation will, thus, yield the connection between 7TO and the strain derivatives of the Szigeti effective charge parameter (e~). In section 3 we present the analysis based on the second Szigeti relation. In section 4 we discuss the results for 7TO and 7LO, and their volume derivatives in the light of experimental data. The strain derivatives of e~, as obtained from the volume derivatives of the Szigeti relations, are discussed and compared with other theoretical and experimental studies. A summary of the present work will be given in section 5.
2. Analysis of the strain derivatives of optic mode frequencies and dielectric constants In the classical model an ionic crystal is considered to be composed of independently polarizable ions of charges +-Ze. Within the framework of this model, Szigeti derived the following two relations [6] : f = e 0 + 2 ¢°2 e**~ ~'# TO
(1)
and e0-e**-
4rr(Ze) 2 (e** + 2) 2 es2, 9----V--- /aco2------~
(2)
where e0 and e** are respectively the low frequency (static) and the high frequency (electronic) dielectric constants, /a the reduced mass per ion pair, ¢oTO the transverse optic mode frequency, V the volume per ion pair, e~ the Szigeti effective charge parameter, a n d f i s the short range force constant between nearest neighbours. By taking the volume derivatives of eq. (1)we find the following expressions for the transverse optic mode Gruneisen parameter (3~TO) and its volume dependence:
VTO and
V ~3e0.~ + V (ae**'~l V tacoTO] = - 2 I[V(~__~) +-----~\.-ff-#] ~TO C T ¢ - / 7 eo e.~ + 2 ~ - J l
(3)
A. V. Singh et aL /Optic mode Gruneisen parameters in alkali halides
")'TO ('ff'V--]
2VTO re0 + 2 \a V2]
e.. + 2 \ a V2] -- T ~a--~]
V2 (aeo]2 + (e0 + 2) 2 \aV! (e** + 2) 2 ~ - ~
333
7 a-v]
"
(4)
Values of ~'TO and (V/~/TO) (aTTO/a v) can be calculated from eqs. (3) and (4). The method of calculation of various quantities appearing in (3) and (4) is described below.
Z1. Calculation off, af/aV and aZf/av 2 Following the Born-Mayer interatomic force model we can express the potential energy in an alkali halide as w = - ~28 + ¢~0,
(5)
where/i is the Madelung constant, r the interionic separation, and ¢(r) represents the short range repulsive potential between nearest neighbours. In writing expression (5) we have omitted van der Waals terms Wv mainly because of their small contribution to the total potential energy. Also, the theoretical values of Wv commonly used are old and not highly regarded. In addition there is some feeling among our theoretical colleagues that ¢(r), which results from the overlap and exchange effects, includes implicitly the Wv interaction. The short range force constantfis given by [12]
f=~1 [¢"(r) + !~'(r)]
(6)
where ¢'(r) and ¢"(r) denote respectively the first and second derivatives of ~b(r) with respect to r. Differentiation of eq. (6), keeping in mind that V = 2r 3 for NaC1 structure, yields
f ~,a V! = 3
(7)
¢"(r) + 2 ¢'(r) r
and " ,,,,
¢
v2 (02f
7-
--9
8
8
(r) - ~ ¢"(r) + ~ ¢'(r
¢"(r) + { ¢'(r)
!]
(8)
where ~b"'(r) and ¢""(r) are respectively third and fourth order derivatives of ¢(!"). We adopt two potential forms, A exp (-r/o) and B/r n, for the short range repulsive potential ¢(r). A (B) and p(n) are known as the Born repulsive parameters. Using the exponential law for ¢(r) in eqs. (6), (7), and (8) we get f
= ~r - 2 p , A exp (--r/p),
(9a)
A. V. Singh et aL/Optic mode Gruneisenparameters in alkalihalides
334 Vtaf] I
2p 2 + 2 r o - r 2 =
(lOa) '
v 2 [ azf]
r 3 _ 8 r 0 2 - 8p3
(11a)
-
The corresponding expressions obtained for the inverse power law are f n~ ( n - 1 ) Br-n,
(9b)
-
V (~_.~) _ }-
n+2 3 '
(lOb)
V._2.~[ ~ 2 f i = (n + 2) (n + 5) f ~OV2] 9
(11 b)
We have derived three sets of the repulsive parameters from compression, ultrasonic and dielectric data using Hildebrand equation of state [8]. Values o f f , (V/f) (alia V) and (V2/f) 02f/~ V 2) are calculated from eqs. (9), (10) and (11) employing two potential forms and three sets of parameters. Thus, we obtain six sets o f f and their derivatives. 2.2. Evaluation of ae./a V and 32e**/aV 2 The experimental data on ae**/a V and ~2e./a V 2 are still incomplete. We have experimental values of ae**/~ V for eight alkali halides only. For O2e**/aV 2 no experimental data exist. Recently Sharma et al. [13 ] have suggested a method for evaluating the strain derivatives of electronic dielectric constant e**, following the analysis of electronic polarizabilities performed by Ruffa [14]. This method yields Oe./aV for alkali halides in good agreement with experimentally obtained values and can be extended to predict values of ~2e,/a V 2. However, the terms involving ~e./a V and a2e./a V 2, which appear in eqs. (3) and (4), are relatively much smaller and therefore an uncertainty in the evaluation of these quantities will not affect appreciably the results for 7TO and a3,TO/a V. The Lorentz-Lorenz formula relates the high frequency dielectric constant with the polarizability a** e** - 1 e.+2
4~ra** 3 V
(12)
The volume derivatives of eq. (12) are found to be V t---~J =--~--(e. + 2) 2 bV
( e . + 2) (e® - 1) 3
(13)
and V2 [a2e~.~ 47rV. ~2a** 2V 2 [Oe**~2 (~e**~ W ~ - ! = - 7 ( ' ~ + 2)2 - g ~ + - - 2v e.+2\aVl \~Vf"
(14)
The strain derivatives of polarizabilities can be obtained from the method suggested earlier [13]. In view of the additivity rule one can write
335
A. K Singh et aL /Optic mode Gruneisen parameters in alkali halides
OS~ I , 0-==~= ~ r2 (s+ + a ' )
(lS)
and
0V 2 =36r4
(s++
_-
(16)
(a++s'
where s+ and s _ are the electronic polarizabilities of the individual cation and anion, respectively, and
,
Os+.
s+-
Or '
" a2a+" s + - Or 2 ,
a'_-OaOr
s"-O2sand
_
Or2 .
Following Ruffa [14] and Sharma et al. [13] one can write
2sfef2
$
s+ = - (Ef-
vM)3
vM
(17)
r
and a'- = -s-- [ 1 - - 7c%r+ ,] , " S+
(18)
r
where sf and Ef are the polarizability and the mean excitation energy of cations in free state; VM is the Madelung potential; and r+ and r_ are the ionic radii. Eq. (17) has been derived directly from the theory of electronic polarizabilities formulated by Ruffa [14], whereas eq. (18) is based on a correlation between polarizability and radius. Differentiation of eqs. (17) and (18) with respect to interionic separation r yields
2sfE?VM S+ - r 2 ~ f ~ ~M)3 "
[2
(19)
+ g f - VMJ
and
S,L -
S_ r + s+. . S+ r_
(20)
Using eqs. (13)-(20) we have calculated 3ct**/3V, V(Oe**/OV) (32s**/0 V 2) and V2(O2e**/OV 2) for alkali halides. In table I we have listed the first and second order strain derivatives of e** together with the available experimental data.
2.3. Evaluation of Oeo/OV and 32e0/0V 2 The Clausius-Mossotti relation analogous to the Lorentz-Lorenz formula [eq. (12)] is used for the static dielectric constant e 0 e 0 - 1 _ 41r s 0 e0+2 3 V'
(21)
336
A. V. Singh et al./Optic mode Gruneisen parameters in alkali halides
Table 1 First and second order strain derivatives of the electronic dielectric constant Crystal
V(Oe,,Ja II)
V2(a2eo./a V2)
Calculated
Experimental
Calculated
LiF LiC1 LiBr
-0.62 -1.49 -1.80
-0.40
0.99 3.00 3.47
NaF NaC1 NaBr NaI
-0.48 -0.95 - 1.06 -1.41
-0.33 -0.85 -0.99
0.74 1.61 1.71 0.92
KF KCI KBr KI
-0.76 -0.88 -0.89 -1.09
RbF RbCI RbBr Rbl
-0.93 -0.97 -0.97 -1.09
1.61 1.55 1.54 1.56
-0.84 -1.01 -1.28
2.19 2.03 2.01 1.96
-1.05
where ct0 is the static polarizability. Differentiation of eq. (21) gives
V[8e0]
47r 2) 2 ~ a 0 \ a v ! = 9 - (eO + OV
(e 0 + 2 ) ( e 3
0-1)
(22)
and V2 ia2eo~
4nV
2 82ao
\ a v 2 ! = - T - (% + 2) ~
2V2
[Se°~ 2
+ eo+ 2 \aVl
[ae°]
- 2v \aV! .
(23)
Tile static polarizability a 0 corresponds to the low frequency region and includes the polarization arising from the relative displacement of ions in addition to the electronic polarization. Thus, c~0 can be expressed as the sum of the electronic and ionic polarizabilities a 0 = a** + a i.
(24)
In view of eqs. (15) and (16) we find 8a 0 _ 1 , ~V 6r 2 (~+ + a'_ + ~i)
(25)
and
a2%_] aV 2
[( 36r4
a.... ~++
_+a i)-
_
2 (t~++a'
] +al)
,
(26)
A. V. Singh et al./Optic mode Gruneisen parameters in alkali halides
337
where a I = aai/~r and a I' = ~2ai/ar2. The ionic polarizability cti arising from the relative displacement of ions can be expressed as
(27)
Oq = Z 2 e 2 / f ,
where Z = 1 for alkali halides. We obtain from eq. (27) p
~i = --(if/f) Oti
(28)
and ,,
if"
Oq =
Oq,
(29)
where f ' = af/ar and f " = a2f/ar2. Adopting the exponential and inverse power forms for the short range repulsive interaction O(r) and making use ofeq. (6) we get from eqs. (28) and (29) 2p
[,
ai = 2 ~
r(r-Z 2p)
ai -
r2#2(r - 20)
] ai
(31a)
and t
rvi =
n+2 ¢ti, t
,, _ (n + l) (n + 2) oq. r2 oq.
(30b)
(31 b)
Eqs. (30a) and (31 a)correspond to exponential law whereas (30b) and (31 b) are obtained from the inverse power law for the short range repulsive interaction. Using eqs. (22)-(31) we have evaluated first and s.econd order strain derivatives of the static dielectric constant %. Values of v(aeo/a II) and v2(a2eo/a v 2) thus calculated are reported in tables II and III, respectively, together with experimental data. We have obtained six sets of v(aeo/a 10 and V2(a2eo/aV 2) using two potential forms and three sets of parameters.
2.4. Evaluation ofwTO, 3'TO, ~3'TO/~ II, TLO and OTLO/a V Values of o)TO can be calculated from eq. (1) using different sets of the short range force constant f. The experimental values of wTO have been reported by Lowndes and Martin [15]. For evaluating 7TO and aTTO/a V, we use eqs. (3) and (4). Strain derivatives of e0 and e** needed for calculation are listed in tables I - I I I . We have obtained six sets of 3'TO and OTTo/aV corresponding to the six sets off, af/a v, a2f/a v 2, aeo/a v and a2eo/~V2. Values of ae**/a V and a2e**/a v2 are common for each set. 7TO and aTTo/a v thus calculated are presented in tables IV and V, respectively. According to Lyddane, Sachs and Teller [16] (LST) one can correlate the longitudinal optic mode and transverse optic mode frequencies with the dielectric constants through the relation
A. V. Singh et al./Optic mode Gruneisen parameters in alkali halides
338
Table II Values of V(~eo/a V) calculated from (a) compression data, (b) ultrasonic data, and (c) dielectric data Crystal
Exponential form
Inverse power form
Experimental
(a)
(b)
(c)
(a)
(b)
(c)
Jones (ref. 30)
LowndesMartin (ref. 4)
Mahmud et al. (ref. 31)
Fontanella et al. (ref. 7)
LiF LiC1 LiBr
19.50 31.19 37.88
22.07 36.48 44.73
34.95 53.08 65.09
29.61 43.05 50.77
31.19 47.93 64.17
43.37 63.97 83.96
29.42
22.97 30.36 40.37
30.38 53.41
30.00
NaF NaCI NaBr NaI
6.28 10.13 9.67 3.36
9.58 10.60 11.84 15.05
15.75 19.21 21.30 26.52
9.64 12.84 12.50 13.68
12.36 13.60 14.57 18.32
18.29 21.76 23.84 29.56
11.96 14.29 15.48
12.52 13.02 14.86 16.55
12.17 12.52 16.03 12.38
12.70 14.50 15.90
KF KCI KBr KI
8.86 6.76 6.89 6.13
12.81 8.73 8.24 7.63
16.70 14.66 15.23 16.21
12.25 8.63 8.56 7.67
15,78 10,54 9.85 9,14
19.56 16.34 16.76 17.60
12,75 8,45 8,35 8.16
11.25 8.89 8.21 6.58
RbF RbC1 RbBr RbI
14.71 8.28 7.19 5.08
20.60 9.84 9.02 8.15
20.37 14.56 14.90 15.27
19.00 10.07 8.81 8.15
24,20 11.65 10.63 9.55
24.39 16.29 16.40 16.59
14.46 7.79 7.46 6.94
8.22 7.37 6.39
8.93 8.60 7.50, 9.93
8.73 7.96
9.11 8.92
Table Ill Values of V2(a2eo/a V2) calculated from (a) compression data, (b) ultrasonic data, and (c) dielectric data and the experimental values Crystal
Exponential form
Inverse power form
Experimental, FontaneUa et al. (ref, 7)
(a)
(b)
(c)
(a)
(b)
(c)
LiF LiCI LiBr
97 213 281
122 278 377
289 554 753
191 346 441
212 427 654
417 752 1130
157
NaF NaC1 NaBr NaI
21 51 46 30
44
57 66 94
112 162 193 262
38 69 65 70
63 77 80 122
140 194 227 335
50 67 99
KF KC1 KBr KI
41 31 36 30
76 48 44 41
123 123 135 160
62 42 42 37
102 61 55 51
155 143 155 178
RbF RbCI RbBr RbI
88 45 37 33
157 60 54 48
155 122 133 150
125 56 46 44
201 75 66 59
204 142 152 148
40 44
339
A. V. Singh et al./Optic mode Gruneisen parameters in alkali halides Table IV Values of -tTO calculated from (a) compression data, (b) ultrasonic data, and (e) dielectric data Crystal
Exponential form
Experimental value (ref. 18)
Inverse power form
(a)
(b)
(c)
(a)
(b)
(c)
l.iF LiCI LiBr
2.08 2.47 2.82
2.26 2.90 3.16
3.17 3.81 4.17
2.79 3.26 3.46
2.90 3.53 4.01
3.76 4.41 4.99
2.35
NaF NaC1 NaBr Nai
1.64 2.27 2.20 1.77
2.14 2.36 2.49 2.71
3.06 3.54 3.78 4.05
2.15 2.65 2.58 2.55
2.55 2.76 2.86 3.09
3.44 3.90 4.13 4.41
2.08 2.35 2.37
KF KCI KBr KI
2.01 2.14 2.23 2.21
2.55 2.51 2.49 2.52
3.09 3.62 3.88 4.27
2.48 2.49 2.56 2.52
2.96 2.85 2.82 2.83
3.48 3.94 4.18 4.56
2.28 2.06 2.20
RbF RbCI RbBr RbI
2.46 2.44 2.34 2.29
3.08 2.73 2.70 2.72
3.08 3.59 3.87 4.23
2.93 2.76 2.66 2.71
3.50 3.06 3.02 3.01
3.52 3.91 4.17 4.51
2.16 2.39 2.09
Table V Values of (V/-rTO)(O~tTO/a V) calculated from (a) compression data, (b) ultrasonic data, and (c) dielectric data and values of q = (V/-t)(a~t/a II) derived from thermoelastic data Crystal
d ref. 9; e ref. 2.
Exponential form
Inverse power form
q
(a)
(b)
(c)
(a)
(b)
(c)
LiF LiCI LiBr
1.98 2.54 2.76
2.19 2.86 3.16
3.21 3.95 4.38
2.28 2.79 3.03
2.41 3.13 3.64
3.49 4.24 4.92
1.12 d 1.75 d 2.01 d
NaF NaC1 NaBr Nal
1.78 1.56 1.44 1.32
1.56 1.65 1.69 1.94
2.34 2.60 2.75 3.06
1.12 1.43 1.35 1.38
1.48 1.52 1.44 1.84
2.27 2.51 2.66 3.32
1.22 e 1.40 e 1.75 e 1.77 e
KF KC1 KBr KI
1.50 1.29 1.40 1.24
1.93 1.55 1.46 1.38
2.39 2.35 2.40 2.53
1.42 1.14 1.11 1.01
1.87 1.40 1.29 1.20
2.34 2.22 2.26 2.34
1.57 e 1.77 e 1.37 e 1.26 e
RbF RbC1 RbBr Rbl
2.01 1.50 1.36 1.32
2.60 1.69 1.59 1.46
2.58 2.34 2.39 2.45
2.00 1.31 1.15 1.10
2.59 1.56 1.42 1.30
2.61 2.20 2.24 1.97
2.07 e 2.14 e 1.99 e 1.91 e
340
A.V. Singh et aL /Optic mode Gruneisen parameters in alkali halides (32)
(6OLO/OgTO)2 = G0/G**. The volume derivatives of eq. (32) yield
7LO = 7TO -- "2
aV
aTL O _ 1 [ 2 V a"/TO av 2V L aV
(33)
eoo a V J ' V ae o + V ago. e0 a g e. i)g
V 2 a2e0 + V 2 a 2 e e 0 a----~ V e. aV 2
--+
(:0 G°i2 v G. 21 av/ - ( ~ av!j
(34)
In tables VI and VII we report the valuesof')'LO and aTLO/a V calculated from eqs. (33) and (34).
3. Volume dependence of the Szigeti effective charge paramete r The volume derivatives of the Szigeti effective charge parameter can be predicted from the second Szigeti relation expressed by eq. (2). Differentiation of eq. (2) with respect to volume yields
[ 1
V e s~ \ea~v'l~ = ½ - 'YTO -- 'e~'21- 2 + 2(% - - e . )
] (OG.i \av!
(35)
+ 2(% - e~) V \ a V I
and
Table VI Values of ~'LO calculated from (a) compression data, (b) ultrasonic data, and (c) dielectric data Crystal
Exponential form
Inverse power form
(a)
(b)
(c)
(a)
(b)
(c)
LiF LiC1 LiBr
0.83 0.88 1.10
0.87 1.09 1.18
1.07 1.30 1.42
0.98 1.17 1.25
1.01 1.23 1.30
1.19 1.44 1.53
NaF NaC1 NaBr Nal
0.89 1.21 1.22 1.31
1.06 1.25 1.34 1.44
1.37 1.71 1.88 2.00
1.06 1.36 1.38 1.37
1.19 1.40 1.50 1.60
1.51 1.85 2.02 2.14
KF KC1 KBr KI
1.00 1.24 1.34 1.40
1.18 1.41 1.47 1.56
1.37 1.91 2.13 2.48
1.16 1.40 1.50 1.57
1.32 1.56 1.62 1.72
1.50 2.05 2.28 2.62
RbF RbC1 RbBr RbI
1.08 1.37 1.39 1.57
1.25 1.50 1.57 1.69
1.27 1.88 2.13 2.49
1.22 1.51 1.55 1.68
1.39 1.64 1.72 1.84
i.40 2.02 2.27 2.64
A. K Singh et al./Optic mode Gruneisen parameters in alkali halides
341
Table VII Values of (V/?LO) (a3,Lo/a V) claculated from (a) compression data, (b) ultrasonic data, and (c) dielectric data Crystal
V2
Exponential form (a)
(b)
(c)
(a)
(b)
(c)
LiF LiCI LiBr
0.04 -0.51 -0.07
-0.01 -0.11 -0.13
-0.22 -0.30 -0.31
-0.44 -0.50 -0.50
-0.46 -0.56 -0.52
-0.67 -0.73 -0.50
NaF NaC1 NaBr NaI
0.06 -0.09 -0.03 -0.03
-0.11 -0.14 -0.12 -0.19
-0.41 -0.46 -0.46 -0.49
-0.41 -0.52 -0.47 -0.48
-0.55 -0.54 -0.46 -0.60
-0.83 -0.87 -0.87 -1.09
KF KC1 KBr KI
-0.07 -0.07 -0.14 -0.06
-0.23 -0.19 -0.14 -0.11
-0.41 -0.54 -0.53 -0.57
-0.52 -0.51 -0.48 -0.45
-0.68 -0.62 -0.56 -0.52
-0.84 -0.95 -0.95 -0.97
RbF RbC1 RbBr RbI
-0.18 -0.16 -0.10 -0.12
-0.35 -0.24 -0.20 -0.15
-0.35 -0.52 -0.54 -0.57
-0.62 -0.57 -0.50 -0.49
-0.80 -0.68 -0.62 -0.57
-0.81 -0.94 -0.96 -0.75
{'a2esi
~e*t 2 +Vpeq*
"~stff-~ ] - (~sav# =-V
Inverse power form
t-"b'V-I
5. + 2
1
es \ a V ]
I,"ff'~]
°2 0
+2(e 0-5.*)
[
5.. + 2 ------~ av +
+ vaso v2 2'-
~V2
aV-
** + 2 a g l
va .i
aV2 -
,
avJ -2(50-e**) 2
i,vaso - vasq !, aV avl
"
(36)
Substituting the values of 7TO and 0,,/TO/a V from eqs. (3) and (4) into eqs. (35) and (36), we f'mally obtain
V Lae:%
1 1 ['~(~V)_ -
[1+ ~. + 2
1 [1 +_..V (~.f.)
:~V 2/.a2e:]
I
I/ faeo~+
1
fae**.]1
1 ] v z~,.\ 1 ~5(rv_ ) [ a-~) + 2 ( c o - , . ) v
2(,0 - ~.) g
~ae0.~
-,0 + ~ t ~ !
I V aes~ 2 _ V [ a e ~ ]
+ 2(e 0 -- e**)
"V
_ /ae.\{
- vt~) 1
1
~
1
+,o-,--
~vaeO_ vae..]2
(IF 2 82e0 + V ae0 V 2 a2e** _ g ~e**~ aV 2 aV aV 2 aVl
}
+ (e0 - -- e . )
t~V'lJ '
(37)
A. V. Singh et aL/Optic mode Gruneisen parameters in alkali halides
342
1 [ 7V( ~ -/~f) +2-
co. + 2 aV 2
V
(Se0~
(e O+2)\~-V-]
V
(Se**~ + V2 02f .
e,o + 2 \/~VI
-~T ~-a--~] + eo + 2 a V ]
+
.
.
f ~V 2
.
V 2 a2eo eO + 2 a V 2
.
.
0.+2 av! j •
(38)
We have evaluated (Vies*) (aes/alO and (V2/es) (a2es/aV 2) from eqs. (37) and (38) using appropriate data listed in tables I-V. It is found that the strain derivatives of the effective charge parameter thus calculated are nearly equal to zero.
4. Results and discussion Values of 6oTO calculated from the first Szigeti relation [eq. (1)] using the repulsive parameters corresponding to compression and ultrasonic data, present better agreement with experiment than that presented by the parameters corresponding to dielectric data. This is also the case with the volume derivatives of wTO, i.e. 3'TO (table IV). It was emphasized earlier by Mort and Littleton [17] that the Born repulsive parameters derived from the compressibility or elastic data do not yield correct values for the static dielectric constant. Accordingly Boswarva and Simpson [11 ] modified the scheme within the framework of the Born model and obtained a different set of repulsive parameters from the experimental values of dielectric constants. However, this set of repulsive parameters, as is evident from the present calculations, yields much higher values for wTO and 3'TO than those observed experimentally and also than those calculated from the repulsive parameters corresponding to ultrasonic and compression data. An exactly similar prediction can also be made from tables II and III from where we observe that values of the first and second order strain derivatives of the static dielectric constant calculated using repulsive parameters based on dielectric data are much higher than the experimental values. This prediction deserves further attention. Very recently Lowndes and Rastogi [18] have reported highly accurate experimental data on 3'TO for many alkali halides. It is interesting to observe from table IV that calculated 3'TO present close agreement with experimental values. The longitudinal optic mode Gruneisen parameter 3'LO calculated from eq. (33) for all the alkali halides under study are found to be smaller (table VI) than the corresponding 7TO- The difference in 3'TO and 7LO is connected with the volume dependence of the lattice contribution to the dielectric constant. An average value 3' for the Gruneisen parameter can be obtained from 7TO and 3'LO by utilizing a suitable averaging process, as suggested by Madan [3] 3' = 3'TOW20 + 3'LOW20
(39)
+
Expression (39) is similar to the weighted averages of individual mode Gruneisen parameters as used by Barron [19] in terms of the moments of the frequency spectrum. Values of 3' obtained from eq. (39) can be compared with the macroscopic Gruneisen parameter which in terms of thermoelastic data is defined as [12] 3" = V ~ B T / C o ,
(40)
where 13is the thermal expansion coefficient, B T the isothermal bulk modulus, V the crystal volume, and C o the specific heat at constant volume. Values of 7 based on eq. (40) and recent thermoelastic data have been cited by Smith and Cain [10]. In table VIII we have made a comparison of this 3' with those obtained from eq. (39) using the various sets of the repulsive parameters.
A. V. Singh et al./Optic mode Gruneisen parameters in alkali halides
343
Table VIII Values of average 3"calculated from (a) compression data, (b) ultrasonic data, and (c) dielectric data, and values of 3' based on recent thermoelastic data Crystal
Average3, Exponential form
3"based on eq. (40) and thermoelastic data
Inverse power form
(a)
(b)
(c)
(a)
(b)
(c)
LiF LiC1 LiBr
1.05 1.18 1.43
1.12 1.43 1.56
1.44 1.77 1.95
1.30 1.57 1.68
1.34 1.67 1.82
1.65 2.00 2.20
1.63 1.81 1.94
NaF NaC1 NaBr Nal
1.08 1.51 1.51 1.44
1.33 1.57 1.68 1.81
1.80 2.23 2.44 2.60
1.34 1.73 1.73 1.72
1.54 1.78 1.90 2.04
2.00 2.43 2.64 2.81
1.51 1.61 1.64 1.71
KF KCI KBr KI
1.26 1.52 1.63 1.68
1.53 1.75 1.80 1.89
1.80 2.44 2.70 3.09
1.49 1.74 1.85 1.89
1.74 1.96 2.01 2.10
2.00 2.63 2.90 3.29
1.52 1.49 1.50 1.53
RbF RbCI RbBr Rbl
1.40 1.70 1.70 1.82
1.67 1.88 1.93 2.04
1.68 2.41 2.69 3.09
1.61 1.90 1.91 2.03
1.88 2.08 2.14 2.24
1.88 2.61 2.89 3.28
1.40 1.39 1.42 1.56
The volume dependence of the Gruneisen parameter emerges frequently in discussions of the thermoelastic properties o f solids [20, 21 ]. The parameter q = (V/3") (a3'/a v) in terms of available experimental quantities is given by [2]
q
_-
1 + ( 1 +T/33')6 s -
a B s + 3' aP
+__r a( v) 3'
(41)
where B s is the adiabatic bulk modulus. Values o f q derived from experimental data [2, 9] using eq. (41) are compared in table V with the calculated values of (V/3'TO) (a3,To/a t0. It is particularly remarkable to note the extent of agreement between experimental thermoelastic q and calculated (V/3'TO) (a3'To/a 1I) based on the interionic potential and the dielectric polarization model. It is apparent from table VII that the variation of 3'LO with volume is o f opposite sign and much smaller in magnitude than that of 3'TO" This implies that 3'LO increases with pressure, whereas 3'TO decreases when the crystal is compressed. The expressions for the strain derivatives o f the effective charge parameter [eqs. (37) and (38)] derived from the second Szigeti relation yield the values o f (Vies*) (aes/a v ) and (V2/es*) (a2es/a v 2) which are nearly equal to zero, which implies that the Szigeti effective charge parameter remains almost constant under changing pressure. This implication is in conformity with previous experimental and theoretical studies performed by a group of investigators [ 2 2 - 2 5 ] . Since the effective charge is usually considered a measure of ionic overlap and distortion, this would mean that overlap does not increase appreciably as the volume of the crystal is decreased. This is not unexpected and can be understood on the basis of the charge redistribution of ions. In a crystal under hydrostatic pressure the loosening o f cations and the tightening of anions [26] may be responsible for apparently small change in the degree of overlap.
344
A. V. Singh et al./Optic m o d e Gruneisen parameters in alkali halides
The observation that e s varies much slowly with volume can also be corroborated from the Lawaetz theory of ionicity. According to Lawaetz [27] we have e s* = C/~16Op,
(42)
where C is known as the Phillips electronegativity parameter [28] and~wp is the plasma energy. Taking the volume derivative of eq. (42) one gets V [aes]
V(0Ct
+ 05.
(43)
Van Vechten [29] has suggested a method for evaluating (V/C) (aC/~ V) from the experimental data on the photoelastic effect. Values of (V/C) (aC/~V) derived from this method range between 0 and -0.5. This leads to the prediction that (Vies) (Oes/~ V) "~ O, supporting our viewpoint.
5. Summary We have performed a detailed analysis of the optic mode Gruneisen parameters and their strain derivatives on the basis of the two Szigeti relations for dielectric constants. No such extensive analysis for the entire family of NaC1 structure alkali halides, involving the first and second order strain derivatives of static and electronic dielectric constants, has been carried out in the past. Since the interionic short range repulsive potential plays the role of central importance in the strain derivatives of dielectric constants and the Gruneisen parameter itself, we have employed two potential forms representing the exponential and the inverse power dependence on the interionic separation. In addition, we have used three sets of repulsive parameters based on recently measured compression, ultrasonic and dielectric data. Values of 6OTO, 'YTO, (a3,TO/a V) ( a % / a V) and a2eo/a V 2 calculated from ultrasonic and compression data present relatively close agreement with experiment, aTLO/~ Vhas been predicted to be negative, i.e. "YLOincreases with pressure. Contrary to this OTTO/aV is positive. Average values of 7TO and 7LO compare well with the macroscopic Gruneisen parameter obtained from thermoelastic data. The variation of 7TO with volume is analogous to that of the macroscopic Gruneisen parameter. Thus, the present study provides a useful correlation between dielectric and thermoelastic behaviour of alkali halides. The volume dependence of the Szigeti effective charge parameter is predicted to be very small. This prediction has been shown to be consistent with other theoretical and experimental studies.
Acknowledgement Financial assistance received from the University Grants Commission, India, is gratefully acknowledged.
References [1] [2] [3] [4] [5] [6] [7]
R. Ruppin and R. W. Roberts, Phys. Rev. B3 (1971) 1406. R. W. Roberts and R. Ruppin, Phys. Rev. B4 (1971) 2041. M. P. Madan, Physiea 51 (1971) 526. R. P. Lowndes and D. H. Martin, Proc. Roy. Soc. A316 (1970) 351. V. V. Palciauskas, J. Phys. Chem. Solids 36 (1975) 611. B. Szigeti, Proc. Roy. Soc. A204 (1950) 51. J. FontaneUa, C. Andeen and D. Schuele, Phys. Rev. B6 (1972) 582.
A. K Singh et al./Optic mode Gruneisen parameters in alkali halides [8] M. P. Tosi, Solid State Phys. 16 (1964) 1. [9] R. W. Roberts and C. S. Smith, J. Phys. Chem. Solids 31 (1970) 619; 31 (1970) 2397. K. O. Mclean, and C. S. Smith, J. Phys. Chem. Solids 33 (1972) 279. [10] C. S. Smith and L. S. Cain, J. Phys. Chem. Solids 36 (1975) 205. [11] I. M. Boswarva and J. H. Simpson, Can. J. Phys. 51 (1973) 1923. [12] M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Clarendon Press, Oxford, 1954). [13] H. P. Sharma, Jai Shanker and M. P. Verma, J. Phys. Chem. Solids 37 (1976) 1077. [14] A. R. Ruffa, Phys. Rev. 130 (1963) 1412. [15] R. P. Lowndes and D. H. Martin, Proc. Roy. Soc. A308 (1969) 473. [16] R. H. Lyddane, R. G. Sachs and E. Teller, Phys. Rev. 59 (1941) 673. [17] N. F. Mott and M. J. Littleton, Trans. Faraday Soc. 34 (1938) 485. [18] R. P. Lowndes and A. Rastogi, Phys. Rev. B14 (1976) 3598. [19] T. H. K. Barron, Phil. Mag. 46 (1955) 720. [20] O. L. Anderson, Phys. Rev. 144 (1966) 553. [21] Y. A. Chang, J. Phys. Chem. Solids 28 (1967) 697. [22] S. S. Mitra, C. Postmus and J. R. Ferraro, Phys. Rev. Lett. 18 (1967) 455. [23] C. Postmus, J. R. Ferraro and S. S. Mitra, Phys. Rev. 174 (1968) 983. [24] M. A. Cundill and W. F. Sherman, Phys. Rev. 168 (1968) 1007. [25] S. S. Mitra, O. Brafman, W. B. Daniels and R. K. Crawford, Phys. Rev. 186 (1969) 942. [26] O. P. Sharma, H. P. Sharma and Jai Shanker, Solid State Commun. 20 (1976) 725. [27] P. Lawaetz, Phys. Rev. Lett. 26 (1971) 697. [28] J. C. Phillips, Rev. Mod. Phys. 42 (1970) 317. [29] J. A. Van Vechten, Phys. Rev. 182 (1969) 891. [30] B. W. Jones, Phil. Mag. 16 (1967) 1085. [31 ] K. R. Mahmud, K. V. Kamath and B. K. P. Scaife, Phil. Mag. 23 (1971) 655.
345
OPTIC MODE GRUNEISEN PARAMETERS AND THEIR STRAIN DERIVATIVES IN ALKALI HALIDES A. V. SINGH, J. C. SHARMA and Jai SHANKER Department of Physics, Agra College, Agra-282002, lndia Received 8 July 1977 Revised 15 November 1977
The Born--May~r theory of interatomic forces and the Szigeti model of dielectric properties have been extended to predict the optic mode Gruneisen parameters and their volume derivatives for the alkali halides. A detailed analysis of the dielectric behaviour of alkali halides under hydrostatic pressure is performed in the light of experimental data on first and second order strain derivatives of dielectric constants. To make the investigation critical and comprehensive we have adopted the inverse power law as well as the exponential law for the short range repulsive interactions. Three different sets of the Born repulsive parameters corresponding to compression data, ultrasonic data and dielectric data are used. The transverse optic mode Gruneisen parameters (3,TO) calculated, using short range repulsive parameters corresponding to compression and ultrasonic data, present good agreement with experimental values. The Lyddane, Sachs and Teller relation is used to predict the longitudinal optic mode Gruneisen parameters (-rLO) and their volume derivatives (a~/LO/a V). The general features of the results obtained are that -rTO is larger than 3,LO and that 3,TO decreases with pressure whereas 3,LO increases. A comparison of the average Gruneisen parameter obtained in the present study has been made with the macroscopic Gruneisen parameter deduced from the thermoelastic data. The volume dependence of the Szigeti effective charge parameter has also been predicted and discussed.
1. Introduction Studies on the Gruneisen parameter provide valuable information about the volume dependence of the frequency spectrum which is of crucial importance in the extrapolation of the thermodynamic properties of crystals subjected to high pressure and high temperature. The Gruneisen parameter and its volume dependence have been the subject of great interest for describing the thermoelastic [ 1 , 2 ] and dielectric properties [3, 4] of solids, and in geophysical studies [5]. The present paper deals with the evaluation of optic mode Gruneisen parameters and their volume dependence for NaC1 structure alkali halides adopting the Born model of interatomic forces, and the L y d d a n e Sachs-Teller (LST) and Szigeti relations for optic mode frequencies. Madan [3] and Lowndes and Martin [4] have employed the first and second Szigeti relations [6], respectively, to evaluate the optic mode Gruneisen parameters. At that time the data on dielectric constants and their first order pressure derivatives were available for a few alkali halides and it was then not possible to make a comprehensive investigation of the entire family of NaC1 structure alkali halides. In recent times the first as well as higher order pressure derivatives of dielectric constants of alkali halides have been reported in the literature [7]. It is therefore possible to make an analysis not only of the Gruneisen parameters but also of their volume derivatives. Ill the present study we adopt two potential form s for the short range repulsive interactions, representing the inverse power dependence and the exponential dependence on interionic separation. A detailed investigation o f these two forms within the framework of the Born model has been made by Tosi [8]. The repulsive parameters entering the short range potential form are determined from the crystal equilibrium condition and the second derivative o f potential energy which is related to the compressibility or elastic constants. The compression data were used by Tosi to evaluate the repulsive parameters. More recently Smith and collaborators [9] have published highly accurate ultrasonic data which have been used by Smith and Cain [10] to estimate the repulsive parameters. 331
332
A. V. Singh et aL /Optic mode Gruneisen parameters in alkali halides
Smith and Cain have convincingly demonstrated that the repulsive parameters corresponding to ultrasonic data exhibit more refined and systematic trends than those corresponding to compression data. We will, however, make use of both the sets. The third set of parameters to be used in the present study is based on the dielectric data. It was recently emphasized by Boswarva and Simpson [11 ] that the Born repulsive parameters derived from the compressibility or elastic data are not consistent with the experimental values of the static dielectric constant. Therefore an alternative method of determining the repulsive parameters from dielectric data has been suggested by these investigators [ 11 ]. The present analysis deals with the two Szigeti relations [6] derived from the classical theory of dielectric polarization. In insulators like alkali halides in which we have bound electrons the classical theories are fairly satisfactory, The first Szigeti relation connects the static and electronic dielectric constant through the force constant between ions and the transverse optic mode frequency coTO of lattice vibrations. The first and higher order volume derivatives of coTO lead to the expressions for the corresponding Gruneisen parameter (TTO) and its strain derivatives. The LST relation provides a link between wTO and the longitudinal optic mode frequency coLO, and hence can be used to predict the LO mode Gruneisen parameter 7LO and its volume dependence. In section 2 we give the necessary formulation and the method for evaluation of these quantities. The second Szigeti relation expresses the difference of low frequency (static) dielectric constant and high frequency (electronic) dielectric constant in terms of coTO and effective charge parameter e~. The volume derivatives of the second Szigeti relation will, thus, yield the connection between 7TO and the strain derivatives of the Szigeti effective charge parameter (e~). In section 3 we present the analysis based on the second Szigeti relation. In section 4 we discuss the results for 7TO and 7LO, and their volume derivatives in the light of experimental data. The strain derivatives of e~, as obtained from the volume derivatives of the Szigeti relations, are discussed and compared with other theoretical and experimental studies. A summary of the present work will be given in section 5.
2. Analysis of the strain derivatives of optic mode frequencies and dielectric constants In the classical model an ionic crystal is considered to be composed of independently polarizable ions of charges +-Ze. Within the framework of this model, Szigeti derived the following two relations [6] : f = e 0 + 2 ¢°2 e**~ ~'# TO
(1)
and e0-e**-
4rr(Ze) 2 (e** + 2) 2 es2, 9----V--- /aco2------~
(2)
where e0 and e** are respectively the low frequency (static) and the high frequency (electronic) dielectric constants, /a the reduced mass per ion pair, ¢oTO the transverse optic mode frequency, V the volume per ion pair, e~ the Szigeti effective charge parameter, a n d f i s the short range force constant between nearest neighbours. By taking the volume derivatives of eq. (1)we find the following expressions for the transverse optic mode Gruneisen parameter (3~TO) and its volume dependence:
VTO and
V ~3e0.~ + V (ae**'~l V tacoTO] = - 2 I[V(~__~) +-----~\.-ff-#] ~TO C T ¢ - / 7 eo e.~ + 2 ~ - J l
(3)
A. V. Singh et aL /Optic mode Gruneisen parameters in alkali halides
")'TO ('ff'V--]
2VTO re0 + 2 \a V2]
e.. + 2 \ a V2] -- T ~a--~]
V2 (aeo]2 + (e0 + 2) 2 \aV! (e** + 2) 2 ~ - ~
333
7 a-v]
"
(4)
Values of ~'TO and (V/~/TO) (aTTO/a v) can be calculated from eqs. (3) and (4). The method of calculation of various quantities appearing in (3) and (4) is described below.
Z1. Calculation off, af/aV and aZf/av 2 Following the Born-Mayer interatomic force model we can express the potential energy in an alkali halide as w = - ~28 + ¢~0,
(5)
where/i is the Madelung constant, r the interionic separation, and ¢(r) represents the short range repulsive potential between nearest neighbours. In writing expression (5) we have omitted van der Waals terms Wv mainly because of their small contribution to the total potential energy. Also, the theoretical values of Wv commonly used are old and not highly regarded. In addition there is some feeling among our theoretical colleagues that ¢(r), which results from the overlap and exchange effects, includes implicitly the Wv interaction. The short range force constantfis given by [12]
f=~1 [¢"(r) + !~'(r)]
(6)
where ¢'(r) and ¢"(r) denote respectively the first and second derivatives of ~b(r) with respect to r. Differentiation of eq. (6), keeping in mind that V = 2r 3 for NaC1 structure, yields
f ~,a V! = 3
(7)
¢"(r) + 2 ¢'(r) r
and " ,,,,
¢
v2 (02f
7-
--9
8
8
(r) - ~ ¢"(r) + ~ ¢'(r
¢"(r) + { ¢'(r)
!]
(8)
where ~b"'(r) and ¢""(r) are respectively third and fourth order derivatives of ¢(!"). We adopt two potential forms, A exp (-r/o) and B/r n, for the short range repulsive potential ¢(r). A (B) and p(n) are known as the Born repulsive parameters. Using the exponential law for ¢(r) in eqs. (6), (7), and (8) we get f
= ~r - 2 p , A exp (--r/p),
(9a)
A. V. Singh et aL/Optic mode Gruneisenparameters in alkalihalides
334 Vtaf] I
2p 2 + 2 r o - r 2 =
(lOa) '
v 2 [ azf]
r 3 _ 8 r 0 2 - 8p3
(11a)
-
The corresponding expressions obtained for the inverse power law are f n~ ( n - 1 ) Br-n,
(9b)
-
V (~_.~) _ }-
n+2 3 '
(lOb)
V._2.~[ ~ 2 f i = (n + 2) (n + 5) f ~OV2] 9
(11 b)
We have derived three sets of the repulsive parameters from compression, ultrasonic and dielectric data using Hildebrand equation of state [8]. Values o f f , (V/f) (alia V) and (V2/f) 02f/~ V 2) are calculated from eqs. (9), (10) and (11) employing two potential forms and three sets of parameters. Thus, we obtain six sets o f f and their derivatives. 2.2. Evaluation of ae./a V and 32e**/aV 2 The experimental data on ae**/a V and ~2e./a V 2 are still incomplete. We have experimental values of ae**/~ V for eight alkali halides only. For O2e**/aV 2 no experimental data exist. Recently Sharma et al. [13 ] have suggested a method for evaluating the strain derivatives of electronic dielectric constant e**, following the analysis of electronic polarizabilities performed by Ruffa [14]. This method yields Oe./aV for alkali halides in good agreement with experimentally obtained values and can be extended to predict values of ~2e,/a V 2. However, the terms involving ~e./a V and a2e./a V 2, which appear in eqs. (3) and (4), are relatively much smaller and therefore an uncertainty in the evaluation of these quantities will not affect appreciably the results for 7TO and a3,TO/a V. The Lorentz-Lorenz formula relates the high frequency dielectric constant with the polarizability a** e** - 1 e.+2
4~ra** 3 V
(12)
The volume derivatives of eq. (12) are found to be V t---~J =--~--(e. + 2) 2 bV
( e . + 2) (e® - 1) 3
(13)
and V2 [a2e~.~ 47rV. ~2a** 2V 2 [Oe**~2 (~e**~ W ~ - ! = - 7 ( ' ~ + 2)2 - g ~ + - - 2v e.+2\aVl \~Vf"
(14)
The strain derivatives of polarizabilities can be obtained from the method suggested earlier [13]. In view of the additivity rule one can write
335
A. K Singh et aL /Optic mode Gruneisen parameters in alkali halides
OS~ I , 0-==~= ~ r2 (s+ + a ' )
(lS)
and
0V 2 =36r4
(s++
_-
(16)
(a++s'
where s+ and s _ are the electronic polarizabilities of the individual cation and anion, respectively, and
,
Os+.
s+-
Or '
" a2a+" s + - Or 2 ,
a'_-OaOr
s"-O2sand
_
Or2 .
Following Ruffa [14] and Sharma et al. [13] one can write
2sfef2
$
s+ = - (Ef-
vM)3
vM
(17)
r
and a'- = -s-- [ 1 - - 7c%r+ ,] , " S+
(18)
r
where sf and Ef are the polarizability and the mean excitation energy of cations in free state; VM is the Madelung potential; and r+ and r_ are the ionic radii. Eq. (17) has been derived directly from the theory of electronic polarizabilities formulated by Ruffa [14], whereas eq. (18) is based on a correlation between polarizability and radius. Differentiation of eqs. (17) and (18) with respect to interionic separation r yields
2sfE?VM S+ - r 2 ~ f ~ ~M)3 "
[2
(19)
+ g f - VMJ
and
S,L -
S_ r + s+. . S+ r_
(20)
Using eqs. (13)-(20) we have calculated 3ct**/3V, V(Oe**/OV) (32s**/0 V 2) and V2(O2e**/OV 2) for alkali halides. In table I we have listed the first and second order strain derivatives of e** together with the available experimental data.
2.3. Evaluation of Oeo/OV and 32e0/0V 2 The Clausius-Mossotti relation analogous to the Lorentz-Lorenz formula [eq. (12)] is used for the static dielectric constant e 0 e 0 - 1 _ 41r s 0 e0+2 3 V'
(21)
336
A. V. Singh et al./Optic mode Gruneisen parameters in alkali halides
Table 1 First and second order strain derivatives of the electronic dielectric constant Crystal
V(Oe,,Ja II)
V2(a2eo./a V2)
Calculated
Experimental
Calculated
LiF LiC1 LiBr
-0.62 -1.49 -1.80
-0.40
0.99 3.00 3.47
NaF NaC1 NaBr NaI
-0.48 -0.95 - 1.06 -1.41
-0.33 -0.85 -0.99
0.74 1.61 1.71 0.92
KF KCI KBr KI
-0.76 -0.88 -0.89 -1.09
RbF RbCI RbBr Rbl
-0.93 -0.97 -0.97 -1.09
1.61 1.55 1.54 1.56
-0.84 -1.01 -1.28
2.19 2.03 2.01 1.96
-1.05
where ct0 is the static polarizability. Differentiation of eq. (21) gives
V[8e0]
47r 2) 2 ~ a 0 \ a v ! = 9 - (eO + OV
(e 0 + 2 ) ( e 3
0-1)
(22)
and V2 ia2eo~
4nV
2 82ao
\ a v 2 ! = - T - (% + 2) ~
2V2
[Se°~ 2
+ eo+ 2 \aVl
[ae°]
- 2v \aV! .
(23)
Tile static polarizability a 0 corresponds to the low frequency region and includes the polarization arising from the relative displacement of ions in addition to the electronic polarization. Thus, c~0 can be expressed as the sum of the electronic and ionic polarizabilities a 0 = a** + a i.
(24)
In view of eqs. (15) and (16) we find 8a 0 _ 1 , ~V 6r 2 (~+ + a'_ + ~i)
(25)
and
a2%_] aV 2
[( 36r4
a.... ~++
_+a i)-
_
2 (t~++a'
] +al)
,
(26)
A. V. Singh et al./Optic mode Gruneisen parameters in alkali halides
337
where a I = aai/~r and a I' = ~2ai/ar2. The ionic polarizability cti arising from the relative displacement of ions can be expressed as
(27)
Oq = Z 2 e 2 / f ,
where Z = 1 for alkali halides. We obtain from eq. (27) p
~i = --(if/f) Oti
(28)
and ,,
if"
Oq =
Oq,
(29)
where f ' = af/ar and f " = a2f/ar2. Adopting the exponential and inverse power forms for the short range repulsive interaction O(r) and making use ofeq. (6) we get from eqs. (28) and (29) 2p
[,
ai = 2 ~
r(r-Z 2p)
ai -
r2#2(r - 20)
] ai
(31a)
and t
rvi =
n+2 ¢ti, t
,, _ (n + l) (n + 2) oq. r2 oq.
(30b)
(31 b)
Eqs. (30a) and (31 a)correspond to exponential law whereas (30b) and (31 b) are obtained from the inverse power law for the short range repulsive interaction. Using eqs. (22)-(31) we have evaluated first and s.econd order strain derivatives of the static dielectric constant %. Values of v(aeo/a II) and v2(a2eo/a v 2) thus calculated are reported in tables II and III, respectively, together with experimental data. We have obtained six sets of v(aeo/a 10 and V2(a2eo/aV 2) using two potential forms and three sets of parameters.
2.4. Evaluation ofwTO, 3'TO, ~3'TO/~ II, TLO and OTLO/a V Values of o)TO can be calculated from eq. (1) using different sets of the short range force constant f. The experimental values of wTO have been reported by Lowndes and Martin [15]. For evaluating 7TO and aTTO/a V, we use eqs. (3) and (4). Strain derivatives of e0 and e** needed for calculation are listed in tables I - I I I . We have obtained six sets of 3'TO and OTTo/aV corresponding to the six sets off, af/a v, a2f/a v 2, aeo/a v and a2eo/~V2. Values of ae**/a V and a2e**/a v2 are common for each set. 7TO and aTTo/a v thus calculated are presented in tables IV and V, respectively. According to Lyddane, Sachs and Teller [16] (LST) one can correlate the longitudinal optic mode and transverse optic mode frequencies with the dielectric constants through the relation
A. V. Singh et al./Optic mode Gruneisen parameters in alkali halides
338
Table II Values of V(~eo/a V) calculated from (a) compression data, (b) ultrasonic data, and (c) dielectric data Crystal
Exponential form
Inverse power form
Experimental
(a)
(b)
(c)
(a)
(b)
(c)
Jones (ref. 30)
LowndesMartin (ref. 4)
Mahmud et al. (ref. 31)
Fontanella et al. (ref. 7)
LiF LiC1 LiBr
19.50 31.19 37.88
22.07 36.48 44.73
34.95 53.08 65.09
29.61 43.05 50.77
31.19 47.93 64.17
43.37 63.97 83.96
29.42
22.97 30.36 40.37
30.38 53.41
30.00
NaF NaCI NaBr NaI
6.28 10.13 9.67 3.36
9.58 10.60 11.84 15.05
15.75 19.21 21.30 26.52
9.64 12.84 12.50 13.68
12.36 13.60 14.57 18.32
18.29 21.76 23.84 29.56
11.96 14.29 15.48
12.52 13.02 14.86 16.55
12.17 12.52 16.03 12.38
12.70 14.50 15.90
KF KCI KBr KI
8.86 6.76 6.89 6.13
12.81 8.73 8.24 7.63
16.70 14.66 15.23 16.21
12.25 8.63 8.56 7.67
15,78 10,54 9.85 9,14
19.56 16.34 16.76 17.60
12,75 8,45 8,35 8.16
11.25 8.89 8.21 6.58
RbF RbC1 RbBr RbI
14.71 8.28 7.19 5.08
20.60 9.84 9.02 8.15
20.37 14.56 14.90 15.27
19.00 10.07 8.81 8.15
24,20 11.65 10.63 9.55
24.39 16.29 16.40 16.59
14.46 7.79 7.46 6.94
8.22 7.37 6.39
8.93 8.60 7.50, 9.93
8.73 7.96
9.11 8.92
Table Ill Values of V2(a2eo/a V2) calculated from (a) compression data, (b) ultrasonic data, and (c) dielectric data and the experimental values Crystal
Exponential form
Inverse power form
Experimental, FontaneUa et al. (ref, 7)
(a)
(b)
(c)
(a)
(b)
(c)
LiF LiCI LiBr
97 213 281
122 278 377
289 554 753
191 346 441
212 427 654
417 752 1130
157
NaF NaC1 NaBr NaI
21 51 46 30
44
57 66 94
112 162 193 262
38 69 65 70
63 77 80 122
140 194 227 335
50 67 99
KF KC1 KBr KI
41 31 36 30
76 48 44 41
123 123 135 160
62 42 42 37
102 61 55 51
155 143 155 178
RbF RbCI RbBr RbI
88 45 37 33
157 60 54 48
155 122 133 150
125 56 46 44
201 75 66 59
204 142 152 148
40 44
339
A. V. Singh et al./Optic mode Gruneisen parameters in alkali halides Table IV Values of -tTO calculated from (a) compression data, (b) ultrasonic data, and (e) dielectric data Crystal
Exponential form
Experimental value (ref. 18)
Inverse power form
(a)
(b)
(c)
(a)
(b)
(c)
l.iF LiCI LiBr
2.08 2.47 2.82
2.26 2.90 3.16
3.17 3.81 4.17
2.79 3.26 3.46
2.90 3.53 4.01
3.76 4.41 4.99
2.35
NaF NaC1 NaBr Nai
1.64 2.27 2.20 1.77
2.14 2.36 2.49 2.71
3.06 3.54 3.78 4.05
2.15 2.65 2.58 2.55
2.55 2.76 2.86 3.09
3.44 3.90 4.13 4.41
2.08 2.35 2.37
KF KCI KBr KI
2.01 2.14 2.23 2.21
2.55 2.51 2.49 2.52
3.09 3.62 3.88 4.27
2.48 2.49 2.56 2.52
2.96 2.85 2.82 2.83
3.48 3.94 4.18 4.56
2.28 2.06 2.20
RbF RbCI RbBr RbI
2.46 2.44 2.34 2.29
3.08 2.73 2.70 2.72
3.08 3.59 3.87 4.23
2.93 2.76 2.66 2.71
3.50 3.06 3.02 3.01
3.52 3.91 4.17 4.51
2.16 2.39 2.09
Table V Values of (V/-rTO)(O~tTO/a V) calculated from (a) compression data, (b) ultrasonic data, and (c) dielectric data and values of q = (V/-t)(a~t/a II) derived from thermoelastic data Crystal
d ref. 9; e ref. 2.
Exponential form
Inverse power form
q
(a)
(b)
(c)
(a)
(b)
(c)
LiF LiCI LiBr
1.98 2.54 2.76
2.19 2.86 3.16
3.21 3.95 4.38
2.28 2.79 3.03
2.41 3.13 3.64
3.49 4.24 4.92
1.12 d 1.75 d 2.01 d
NaF NaC1 NaBr Nal
1.78 1.56 1.44 1.32
1.56 1.65 1.69 1.94
2.34 2.60 2.75 3.06
1.12 1.43 1.35 1.38
1.48 1.52 1.44 1.84
2.27 2.51 2.66 3.32
1.22 e 1.40 e 1.75 e 1.77 e
KF KC1 KBr KI
1.50 1.29 1.40 1.24
1.93 1.55 1.46 1.38
2.39 2.35 2.40 2.53
1.42 1.14 1.11 1.01
1.87 1.40 1.29 1.20
2.34 2.22 2.26 2.34
1.57 e 1.77 e 1.37 e 1.26 e
RbF RbC1 RbBr Rbl
2.01 1.50 1.36 1.32
2.60 1.69 1.59 1.46
2.58 2.34 2.39 2.45
2.00 1.31 1.15 1.10
2.59 1.56 1.42 1.30
2.61 2.20 2.24 1.97
2.07 e 2.14 e 1.99 e 1.91 e
340
A.V. Singh et aL /Optic mode Gruneisen parameters in alkali halides (32)
(6OLO/OgTO)2 = G0/G**. The volume derivatives of eq. (32) yield
7LO = 7TO -- "2
aV
aTL O _ 1 [ 2 V a"/TO av 2V L aV
(33)
eoo a V J ' V ae o + V ago. e0 a g e. i)g
V 2 a2e0 + V 2 a 2 e e 0 a----~ V e. aV 2
--+
(:0 G°i2 v G. 21 av/ - ( ~ av!j
(34)
In tables VI and VII we report the valuesof')'LO and aTLO/a V calculated from eqs. (33) and (34).
3. Volume dependence of the Szigeti effective charge paramete r The volume derivatives of the Szigeti effective charge parameter can be predicted from the second Szigeti relation expressed by eq. (2). Differentiation of eq. (2) with respect to volume yields
[ 1
V e s~ \ea~v'l~ = ½ - 'YTO -- 'e~'21- 2 + 2(% - - e . )
] (OG.i \av!
(35)
+ 2(% - e~) V \ a V I
and
Table VI Values of ~'LO calculated from (a) compression data, (b) ultrasonic data, and (c) dielectric data Crystal
Exponential form
Inverse power form
(a)
(b)
(c)
(a)
(b)
(c)
LiF LiC1 LiBr
0.83 0.88 1.10
0.87 1.09 1.18
1.07 1.30 1.42
0.98 1.17 1.25
1.01 1.23 1.30
1.19 1.44 1.53
NaF NaC1 NaBr Nal
0.89 1.21 1.22 1.31
1.06 1.25 1.34 1.44
1.37 1.71 1.88 2.00
1.06 1.36 1.38 1.37
1.19 1.40 1.50 1.60
1.51 1.85 2.02 2.14
KF KC1 KBr KI
1.00 1.24 1.34 1.40
1.18 1.41 1.47 1.56
1.37 1.91 2.13 2.48
1.16 1.40 1.50 1.57
1.32 1.56 1.62 1.72
1.50 2.05 2.28 2.62
RbF RbC1 RbBr RbI
1.08 1.37 1.39 1.57
1.25 1.50 1.57 1.69
1.27 1.88 2.13 2.49
1.22 1.51 1.55 1.68
1.39 1.64 1.72 1.84
i.40 2.02 2.27 2.64
A. K Singh et al./Optic mode Gruneisen parameters in alkali halides
341
Table VII Values of (V/?LO) (a3,Lo/a V) claculated from (a) compression data, (b) ultrasonic data, and (c) dielectric data Crystal
V2
Exponential form (a)
(b)
(c)
(a)
(b)
(c)
LiF LiCI LiBr
0.04 -0.51 -0.07
-0.01 -0.11 -0.13
-0.22 -0.30 -0.31
-0.44 -0.50 -0.50
-0.46 -0.56 -0.52
-0.67 -0.73 -0.50
NaF NaC1 NaBr NaI
0.06 -0.09 -0.03 -0.03
-0.11 -0.14 -0.12 -0.19
-0.41 -0.46 -0.46 -0.49
-0.41 -0.52 -0.47 -0.48
-0.55 -0.54 -0.46 -0.60
-0.83 -0.87 -0.87 -1.09
KF KC1 KBr KI
-0.07 -0.07 -0.14 -0.06
-0.23 -0.19 -0.14 -0.11
-0.41 -0.54 -0.53 -0.57
-0.52 -0.51 -0.48 -0.45
-0.68 -0.62 -0.56 -0.52
-0.84 -0.95 -0.95 -0.97
RbF RbC1 RbBr RbI
-0.18 -0.16 -0.10 -0.12
-0.35 -0.24 -0.20 -0.15
-0.35 -0.52 -0.54 -0.57
-0.62 -0.57 -0.50 -0.49
-0.80 -0.68 -0.62 -0.57
-0.81 -0.94 -0.96 -0.75
{'a2esi
~e*t 2 +Vpeq*
"~stff-~ ] - (~sav# =-V
Inverse power form
t-"b'V-I
5. + 2
1
es \ a V ]
I,"ff'~]
°2 0
+2(e 0-5.*)
[
5.. + 2 ------~ av +
+ vaso v2 2'-
~V2
aV-
** + 2 a g l
va .i
aV2 -
,
avJ -2(50-e**) 2
i,vaso - vasq !, aV avl
"
(36)
Substituting the values of 7TO and 0,,/TO/a V from eqs. (3) and (4) into eqs. (35) and (36), we f'mally obtain
V Lae:%
1 1 ['~(~V)_ -
[1+ ~. + 2
1 [1 +_..V (~.f.)
:~V 2/.a2e:]
I
I/ faeo~+
1
fae**.]1
1 ] v z~,.\ 1 ~5(rv_ ) [ a-~) + 2 ( c o - , . ) v
2(,0 - ~.) g
~ae0.~
-,0 + ~ t ~ !
I V aes~ 2 _ V [ a e ~ ]
+ 2(e 0 -- e**)
"V
_ /ae.\{
- vt~) 1
1
~
1
+,o-,--
~vaeO_ vae..]2
(IF 2 82e0 + V ae0 V 2 a2e** _ g ~e**~ aV 2 aV aV 2 aVl
}
+ (e0 - -- e . )
t~V'lJ '
(37)
A. V. Singh et aL/Optic mode Gruneisen parameters in alkali halides
342
1 [ 7V( ~ -/~f) +2-
co. + 2 aV 2
V
(Se0~
(e O+2)\~-V-]
V
(Se**~ + V2 02f .
e,o + 2 \/~VI
-~T ~-a--~] + eo + 2 a V ]
+
.
.
f ~V 2
.
V 2 a2eo eO + 2 a V 2
.
.
0.+2 av! j •
(38)
We have evaluated (Vies*) (aes/alO and (V2/es) (a2es/aV 2) from eqs. (37) and (38) using appropriate data listed in tables I-V. It is found that the strain derivatives of the effective charge parameter thus calculated are nearly equal to zero.
4. Results and discussion Values of 6oTO calculated from the first Szigeti relation [eq. (1)] using the repulsive parameters corresponding to compression and ultrasonic data, present better agreement with experiment than that presented by the parameters corresponding to dielectric data. This is also the case with the volume derivatives of wTO, i.e. 3'TO (table IV). It was emphasized earlier by Mort and Littleton [17] that the Born repulsive parameters derived from the compressibility or elastic data do not yield correct values for the static dielectric constant. Accordingly Boswarva and Simpson [11 ] modified the scheme within the framework of the Born model and obtained a different set of repulsive parameters from the experimental values of dielectric constants. However, this set of repulsive parameters, as is evident from the present calculations, yields much higher values for wTO and 3'TO than those observed experimentally and also than those calculated from the repulsive parameters corresponding to ultrasonic and compression data. An exactly similar prediction can also be made from tables II and III from where we observe that values of the first and second order strain derivatives of the static dielectric constant calculated using repulsive parameters based on dielectric data are much higher than the experimental values. This prediction deserves further attention. Very recently Lowndes and Rastogi [18] have reported highly accurate experimental data on 3'TO for many alkali halides. It is interesting to observe from table IV that calculated 3'TO present close agreement with experimental values. The longitudinal optic mode Gruneisen parameter 3'LO calculated from eq. (33) for all the alkali halides under study are found to be smaller (table VI) than the corresponding 7TO- The difference in 3'TO and 7LO is connected with the volume dependence of the lattice contribution to the dielectric constant. An average value 3' for the Gruneisen parameter can be obtained from 7TO and 3'LO by utilizing a suitable averaging process, as suggested by Madan [3] 3' = 3'TOW20 + 3'LOW20
(39)
+
Expression (39) is similar to the weighted averages of individual mode Gruneisen parameters as used by Barron [19] in terms of the moments of the frequency spectrum. Values of 3' obtained from eq. (39) can be compared with the macroscopic Gruneisen parameter which in terms of thermoelastic data is defined as [12] 3" = V ~ B T / C o ,
(40)
where 13is the thermal expansion coefficient, B T the isothermal bulk modulus, V the crystal volume, and C o the specific heat at constant volume. Values of 7 based on eq. (40) and recent thermoelastic data have been cited by Smith and Cain [10]. In table VIII we have made a comparison of this 3' with those obtained from eq. (39) using the various sets of the repulsive parameters.
A. V. Singh et al./Optic mode Gruneisen parameters in alkali halides
343
Table VIII Values of average 3"calculated from (a) compression data, (b) ultrasonic data, and (c) dielectric data, and values of 3' based on recent thermoelastic data Crystal
Average3, Exponential form
3"based on eq. (40) and thermoelastic data
Inverse power form
(a)
(b)
(c)
(a)
(b)
(c)
LiF LiC1 LiBr
1.05 1.18 1.43
1.12 1.43 1.56
1.44 1.77 1.95
1.30 1.57 1.68
1.34 1.67 1.82
1.65 2.00 2.20
1.63 1.81 1.94
NaF NaC1 NaBr Nal
1.08 1.51 1.51 1.44
1.33 1.57 1.68 1.81
1.80 2.23 2.44 2.60
1.34 1.73 1.73 1.72
1.54 1.78 1.90 2.04
2.00 2.43 2.64 2.81
1.51 1.61 1.64 1.71
KF KCI KBr KI
1.26 1.52 1.63 1.68
1.53 1.75 1.80 1.89
1.80 2.44 2.70 3.09
1.49 1.74 1.85 1.89
1.74 1.96 2.01 2.10
2.00 2.63 2.90 3.29
1.52 1.49 1.50 1.53
RbF RbCI RbBr Rbl
1.40 1.70 1.70 1.82
1.67 1.88 1.93 2.04
1.68 2.41 2.69 3.09
1.61 1.90 1.91 2.03
1.88 2.08 2.14 2.24
1.88 2.61 2.89 3.28
1.40 1.39 1.42 1.56
The volume dependence of the Gruneisen parameter emerges frequently in discussions of the thermoelastic properties o f solids [20, 21 ]. The parameter q = (V/3") (a3'/a v) in terms of available experimental quantities is given by [2]
q
_-
1 + ( 1 +T/33')6 s -
a B s + 3' aP
+__r a( v) 3'
(41)
where B s is the adiabatic bulk modulus. Values o f q derived from experimental data [2, 9] using eq. (41) are compared in table V with the calculated values of (V/3'TO) (a3,To/a t0. It is particularly remarkable to note the extent of agreement between experimental thermoelastic q and calculated (V/3'TO) (a3'To/a 1I) based on the interionic potential and the dielectric polarization model. It is apparent from table VII that the variation of 3'LO with volume is o f opposite sign and much smaller in magnitude than that of 3'TO" This implies that 3'LO increases with pressure, whereas 3'TO decreases when the crystal is compressed. The expressions for the strain derivatives o f the effective charge parameter [eqs. (37) and (38)] derived from the second Szigeti relation yield the values o f (Vies*) (aes/a v ) and (V2/es*) (a2es/a v 2) which are nearly equal to zero, which implies that the Szigeti effective charge parameter remains almost constant under changing pressure. This implication is in conformity with previous experimental and theoretical studies performed by a group of investigators [ 2 2 - 2 5 ] . Since the effective charge is usually considered a measure of ionic overlap and distortion, this would mean that overlap does not increase appreciably as the volume of the crystal is decreased. This is not unexpected and can be understood on the basis of the charge redistribution of ions. In a crystal under hydrostatic pressure the loosening o f cations and the tightening of anions [26] may be responsible for apparently small change in the degree of overlap.
344
A. V. Singh et al./Optic m o d e Gruneisen parameters in alkali halides
The observation that e s varies much slowly with volume can also be corroborated from the Lawaetz theory of ionicity. According to Lawaetz [27] we have e s* = C/~16Op,
(42)
where C is known as the Phillips electronegativity parameter [28] and~wp is the plasma energy. Taking the volume derivative of eq. (42) one gets V [aes]
V(0Ct
+ 05.
(43)
Van Vechten [29] has suggested a method for evaluating (V/C) (aC/~ V) from the experimental data on the photoelastic effect. Values of (V/C) (aC/~V) derived from this method range between 0 and -0.5. This leads to the prediction that (Vies) (Oes/~ V) "~ O, supporting our viewpoint.
5. Summary We have performed a detailed analysis of the optic mode Gruneisen parameters and their strain derivatives on the basis of the two Szigeti relations for dielectric constants. No such extensive analysis for the entire family of NaC1 structure alkali halides, involving the first and second order strain derivatives of static and electronic dielectric constants, has been carried out in the past. Since the interionic short range repulsive potential plays the role of central importance in the strain derivatives of dielectric constants and the Gruneisen parameter itself, we have employed two potential forms representing the exponential and the inverse power dependence on the interionic separation. In addition, we have used three sets of repulsive parameters based on recently measured compression, ultrasonic and dielectric data. Values of 6OTO, 'YTO, (a3,TO/a V) ( a % / a V) and a2eo/a V 2 calculated from ultrasonic and compression data present relatively close agreement with experiment, aTLO/~ Vhas been predicted to be negative, i.e. "YLOincreases with pressure. Contrary to this OTTO/aV is positive. Average values of 7TO and 7LO compare well with the macroscopic Gruneisen parameter obtained from thermoelastic data. The variation of 7TO with volume is analogous to that of the macroscopic Gruneisen parameter. Thus, the present study provides a useful correlation between dielectric and thermoelastic behaviour of alkali halides. The volume dependence of the Szigeti effective charge parameter is predicted to be very small. This prediction has been shown to be consistent with other theoretical and experimental studies.
Acknowledgement Financial assistance received from the University Grants Commission, India, is gratefully acknowledged.
References [1] [2] [3] [4] [5] [6] [7]
R. Ruppin and R. W. Roberts, Phys. Rev. B3 (1971) 1406. R. W. Roberts and R. Ruppin, Phys. Rev. B4 (1971) 2041. M. P. Madan, Physiea 51 (1971) 526. R. P. Lowndes and D. H. Martin, Proc. Roy. Soc. A316 (1970) 351. V. V. Palciauskas, J. Phys. Chem. Solids 36 (1975) 611. B. Szigeti, Proc. Roy. Soc. A204 (1950) 51. J. FontaneUa, C. Andeen and D. Schuele, Phys. Rev. B6 (1972) 582.
A. K Singh et al./Optic mode Gruneisen parameters in alkali halides [8] M. P. Tosi, Solid State Phys. 16 (1964) 1. [9] R. W. Roberts and C. S. Smith, J. Phys. Chem. Solids 31 (1970) 619; 31 (1970) 2397. K. O. Mclean, and C. S. Smith, J. Phys. Chem. Solids 33 (1972) 279. [10] C. S. Smith and L. S. Cain, J. Phys. Chem. Solids 36 (1975) 205. [11] I. M. Boswarva and J. H. Simpson, Can. J. Phys. 51 (1973) 1923. [12] M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Clarendon Press, Oxford, 1954). [13] H. P. Sharma, Jai Shanker and M. P. Verma, J. Phys. Chem. Solids 37 (1976) 1077. [14] A. R. Ruffa, Phys. Rev. 130 (1963) 1412. [15] R. P. Lowndes and D. H. Martin, Proc. Roy. Soc. A308 (1969) 473. [16] R. H. Lyddane, R. G. Sachs and E. Teller, Phys. Rev. 59 (1941) 673. [17] N. F. Mott and M. J. Littleton, Trans. Faraday Soc. 34 (1938) 485. [18] R. P. Lowndes and A. Rastogi, Phys. Rev. B14 (1976) 3598. [19] T. H. K. Barron, Phil. Mag. 46 (1955) 720. [20] O. L. Anderson, Phys. Rev. 144 (1966) 553. [21] Y. A. Chang, J. Phys. Chem. Solids 28 (1967) 697. [22] S. S. Mitra, C. Postmus and J. R. Ferraro, Phys. Rev. Lett. 18 (1967) 455. [23] C. Postmus, J. R. Ferraro and S. S. Mitra, Phys. Rev. 174 (1968) 983. [24] M. A. Cundill and W. F. Sherman, Phys. Rev. 168 (1968) 1007. [25] S. S. Mitra, O. Brafman, W. B. Daniels and R. K. Crawford, Phys. Rev. 186 (1969) 942. [26] O. P. Sharma, H. P. Sharma and Jai Shanker, Solid State Commun. 20 (1976) 725. [27] P. Lawaetz, Phys. Rev. Lett. 26 (1971) 697. [28] J. C. Phillips, Rev. Mod. Phys. 42 (1970) 317. [29] J. A. Van Vechten, Phys. Rev. 182 (1969) 891. [30] B. W. Jones, Phil. Mag. 16 (1967) 1085. [31 ] K. R. Mahmud, K. V. Kamath and B. K. P. Scaife, Phil. Mag. 23 (1971) 655.
345