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Analysis of thermodynamic parameters for alkali halides at high pressure and high temperature
J. Phys.
Pergamon
Chem.
Solids Vol. SJ, No. 2, pp. 207-210, 1994 Copyright0 1994Elm-in Science Ltd Printed in Omat Britain. All rights nxrwd O&22-3697/94 S6.00 + 0.00
ANALYSIS OF THERMODYNAMIC PARAMETERS FOR ALKALI HALIDES AT HIGH PRESSURE AND HIGH TEMPERATURE M. KUMAR~ and S. P. UPADHYAY~ TPhysics Department, Institute of Basic Sciences, Agra University, Agra 282 002, India (Received 23 March 1993; accepted 2 August 1993)
Abstract-We present a method for estimating the pressure and temperature dependence of thermodynamic parameters such as Griineisen parameter y its volume derivative q. Anderson parameters 6, and 6, for alkali halides up to their structural transition pressures and melting temperatures. The interionic potential model based on Harrison’s quantum mechanical form for overlap repulsive energy is used. The results are compared with the experimental data available for the NaCl crystal. A good agreement between calculated and experimental values demonstrates the validity of present work. Keywords: Thermodynamic temperature.
parameters, alkali halides, interionic potential, transition pressure, melting
1. INTRODUCTION In order to understand the the~~~~ic behaviour of solids it is desirable to investigate the pressure and temperature dependence of the Griineisen and Anderson parameters [l-2]. Some efforts [3-71 have been made in this direction which are limited to few crystals up to a very small range of pressures and temperatures. Thus Kumari and Dass [3] obtained y and q up to the pressure of 10 kbar at the temperatures 300, 550 and 800 K for NaCl. Dhoble and Verma [4] have shown on the basis of thermodynamic analysis that isothermal Anderson parameter 6, can be expressed in terms of constant pressure and constant volume temperature derivatives of thermal expansion coefficient. This result was used to study the temperature variation of B1.for NaCI. They have also studied the temperature variation of y, q and isentropic Anderson parameter S, only for NaCl at a constant pressure. Yamamoto ef al. [5] and Anderson and Yamamoto [6] presented an experimental method for the determination of the temperature variation of these parameters for NaCl at constant pressure. Kumar ef al. [7] investigated the pressure dependence of y using different theories for NaCl up to 32 kbar at constant temperature. In the present study we develop a method based on the interionic potential model which considers: (i) the long-range electrostatic interactions in terms of Madelung’s energy, (ii) the short range overlap $Physics Department, Govt. College, Shivpuri, M.P., India. 207
repulsive energy between nearest neighbours and next nearest neighbours by adopting the analytical potential form derived by Harrison [g] on the basis of quantum mechanical considerations, and (iii) the van der Waals (vdW) dipol+dipole and dipolequadrupole interactions evaluated from the Kirkwood-Muller formulae as reviewed by Shanker and Agrawal [9]. The model thus developed is used to estimate y, q, 6, and 6, for a much wider range of pressures and temperatures than that considered by any previous worker.
2. METHOD OF ANALYSIS
The basic definitions of y, q, 6, and 6, can be found in the recent literature [2, lo]. Slater [l l] suggested a method for calculating y from the theory of elasticity. His expressions for the vibrational velocities are valid only when the solid is under no external pressure. Dugdale and MacDonald [12] obtained a more general expression for y by including the effect of pressure. These theories do not take into account the variation of Poisson’s ratio with voiume 1131, Vaschenko and Zubarev [14] have developed a formulation for y using the free volume theory. Migault and Romain [15] proposed a unification of these theories by considering the volume dependence of Poisson’s ratio. Following their approach we can express y by a common formula which reads as follows:
M. KUMAR and S. P. UPADHYAY
208
y = _(4 - 3x) --6
Vd*(PV”)/dV*
’
2 d(PV”)/dV
+J2p:rexp(-J2kprr)
(l)
+ $2 P 2 ew(- J2 b2r)l, where P is the pressure, V the volume and x is the parameter which takes the value 0 in Slater’s theory, 0.66 in Dugdale and MacDonald’s theory and 1.33 in free volume theory [lo]. An expression for q can be obtained with the help of eqn (1) using the definition q = V/y . dy/d V. The Anderson parameters are related to y and q as follows [lo] s,=dd$+,-1,
(2)
a,=&--Y,
(3)
where
dBr _ _, _ vd2fWV2
zjs
dP-
~,,22?.$ b ‘r exp( - kfir) Table
“0
k
0.104 0.650 0.399 0.195 1.093 0.950 0.779
4.02 11.40 10.14 27.97 18.76 16.60 20.97 25.05
1.35 1.34 1.27 1.35 1.57 1.39 1.38 I .34
1.034
(7)
where ~~is the valence state energy of ions. Now to calculate the lattice potential energy W using eqns (5)-(7) there remains only two unknown parameters viz. n,, and k which are evaluated from the following relations based on the equilibrium condition.
= 18 roBr,
I161
(9)
where r, is the equilibrium value of r at T = 0 K and atmospheric pressure. The potential parameters rrO and k thus calculated are reported in Table 1 along with other input data. These parameters are used to estimate the higher order derivatives of W with the help of eqn (5) using different values of r as a function of pressure and temperature derived from the equation of state based on the Hildebrand approximation, P = ---Gr + TaB,,
(10)
where the first term on the right is the static pressure, the second term the thermal pressure and P is the externally applied pressure. T is the temperature, a the coefficient of volume thermal expansion and B, the isothermal bulk modulus. For the solids with NaCl-type structure, V = 2r ’ and therefore eqn (10) can be rewritten as follows:
parameters
(&I 3.962 3.962 3.962 3.962 3.087 3.087 3.087 3.087
(8)
and
1. Values of input
x
no. for data
p(2= -$,
0
Here W is the lattice potential energy and the first term on the right-hand side of eqn (5) is the electrostatic Coulomb energy with 01, as Madelung’s constant and e is the electron charge. The second and third terms are the vdW dipole-dipole and dipolequadrupole energies. Last term is the overlap repulsive energy. Values of C and D have been calculated using the Kirkwood-Muller formulae which are superior to all other existing formulae [9]. The overlap repulsive energy can be written as follows [16]:
LiF LiCl LiBr LiI NaF NaCl NaBr NaI Reference
where h is Planck’s constant divided by 271, m the mass of electron and p the arithmetic average of p, and p2 for the cation and anion. Values of p, and p2 can be calculated by the relation
(4)
Thus in order to evaluate y, q, 6, and 6, with the help of eqns (l-4) we need the derivatives of P with V that is dP/dV, d2P/dV2 and d’P/dV’. These derivatives are directly related to the derivatives of the lattice potential energy [7] and can be estimated using the following expression based on the interionic potential mode1 considered in the present study
Crystal
(6)
2.112 1.799 1.716 1.619 2.112 1.799 I.716 1.619
I161
C (10-60ergcm6) 46 317 594 1265 139 561 930 I755
191
D (10-76ergcm8) 2:: 286 584 98 455 537 942
191
Thermodynamic parameters for alkali halides
209
Table 2. Thermodynamic parameters for NaCI. Temperature T (in K) and pressure P (in kbar). Experimental data (5, 171 are given within parentheses. Melting temperature = IO73 K and transition pressure = 283 kbar 298
0 5 10 20 30 40 50 100 150 200 283
Y 1.61 (1.61 1.50 (1.50) 1.48 (1.49) 1.43 (1.44) 1.38 (1.40) 1.34 1.31 1.20 1.11 1.04 0.96
4 1.32 1.02 1.38
313
6s 6, Y 4 3.90 5.51 1.63 1.35 3.50 5.24) (1.63 1.18 3.88 5.38 1.56 1.37
8s 6, Y 4 3.91 5.54 1.66 1.30 3.61 5.64) (1.66 1.17 3.89 5.45 1.60 1.35
6, 6, Y 4 3.95 5.61 1.69 1.27 3.60 5.63) (1.68 1.00 3.93 5.53 1.65 1.28
773
6s 6, Y 4 4.03 5.72 1.75 1.23 3.71 5.87) (1.65 1.05 3.96 5.61 1.66 1.26
6s & 4.15 5.90 3.74 5.82) 4.14 5.80
1.50 3.83 5.26 1.48 1.45 3.86 5.34 1.50 1.40 3.88 5.38 1.54 1.39 3.90 5.44 1.57 1.36 4.03 5.60 1.56 3.80 5.18 1.45 1.49 3.85 5.30 1.47 1.47 3.86 5.33 1.50 1.45 3.87 5.37 1.53 1.43 3.96 5.49 1.58 1.60 1.66 1.75 1.89 2.01
3.75 3.61 3.54 3.49 3.43 3.41
5.09 4.92 4.74 4.60 4.47 4.37
pressure
1.40 1.36 1.30 1.20 1.10 1.01
1.55 1.57 1.63 1.70 1.80 1.92
3.83 3.78 3.68 3.55 3.50 3.49
+ TUB,.
5.23 5.14 4.98 4.75 4.60 4.50
1.44 1.40 1.34 1.29 1.23 1.15
(11)
(PT,,) in eqn (11) can be deter-
mined as follows: ($).=(
Integrating
573
1.44 3.86 5.34 1.50 1.40 3.87 5.37 1.55 1.38 3.90 5.45 1.60 1.34 3.92 5.52 1.62 1.31 4.09 5.71
P = --$>y The thermal
473
%)v=aB,.
(12)
‘(aB,)dT. s II
(13)
we get, PTh=
Thus Pn, is equal to TcrB, only if the product (aB,) is independent of T, which is a good approximation for the solids considered in the present work. The parameters are assumed to be independent of pressure and temperature. 3. RESULTS AND DISCUSSIONS
The calculations of y and other thermodynamic parameters depend sensitively on the values of x. According to the different theories of y [12-151 the value of x should remain between 0 and 1.33. We have determined the values of x to fit the experimental values of y [4] at atmospheric pressure and room temperature for each crystal. It is interesting to see from Table 1 that the values of x thus determined range between 0 and 1.33 and are therefore consistent with the theories of y. In further calculations we have assumed that x does not change with pressure and temperature. The good agreement between calculated and experimental values reveals the validity of this assumption. The thermodynamic parameters y, q, 6, and 6, can be calculated at higher pressures and higher temperatures following the method described in
1.51 1.60 1.62 1.68 1.75 1.84
3.84 3.80 3.78 3.61 3.52 3.51
5.28 5.20 5.12 4.90 4.75 4.66
1.46 1.42 1.39 1.31 1.28 1.20
1.50 1.59 1.61 1.65 1.73 1.81
3.86 3.84 3.80 3.70 3.56 3.53
5.32 5.26 5.19 5.01 4.84 4.73
1.47 1.43 1.40 1.33 1.30 1.23
1.49 1.54 1.60 1.64 1.72 1.79
3.94 3.87 3.81 3.76 3.59 3.55
5.41 5.30 5.21 5.09 4.89 4.78
preceding section using the parameters given in Table 1. Such calculations can be performed for all crystals listed in Table 1. However, we are reporting the results only for NaCl for which the experimental data are available [5, 171 so that a direct comparison can be made in Table 2. There is good agreement between calculated values and available experimental data. It is noted from Table 2 that y decreases with increasing pressure and increases with increasing temperature, whereas q increases with increasing pressure and decreases with increasing temperature. Such a trend of variation is in agreement with the results obtained by Kumari and Dass [3] and Dhoble and Verma [4]. The values of 6, and 6, are also very sensitive to pressure and temperature. 6, and 6, decrease with an increase in pressure and increase with an increase in temperature in agreement with the recent research in high pressure and high temperature physics [2, 181.
Acknowledgements-We
are grateful to Prof. Jai Shanker
for useful discussions.
REFERENCES I. Anderson 0. L., Isaak D. L. and Oda H., J. Geophys. 96, 18037 (1991). 2. Anderson 0. L., Isaak D. L. and Oda H., Rev. Geophys. 30, 57 (1992). 3. Kumari M. and Dass N., Phys. Seal. Solidi (b) 133, 101 (1986). 4. Dhoble A. and Verma M. P., Phys. Stat. Solidi (b) 133, 491 (1986); 136, 497 (1986). 5. Yamamoto S., Ohno I. and Anderson 0. L., J. Phys. Chem. Soli& 48, 143 (1987). 6. Anderson 0. L. and Yamamoto S., High Pressure Res. in Mineral Phys. 39, 289 (1987). 7. Kumar M., Pachauri A. K., Chaturvedi S. D. and Sharma A. K., Phys. Stat. Solidi (b) 146, 125 (1988).
210
M. KUMARand S. P. UPADHYAY
8. Harrison W. A., Phys. Rev. B23, 5230 (198 1). 9. Shanker J. and Agrawal G. G., Phys. Stat. Solidi (b) 123, 11 (1984). 10. Shanker J. and Bhande W. N., Phys. Stat. Solidi (b) 135, 11 (1986). 1I. Slater J. C., Introduction to Chemical Physics. McGraw Hill, New York (1939). 12. Dugdale J. S. and MacDonald D. K. C., Phys. Reo. 89, 832 (1953). 13. Pastine D. J., Phys. Rev. 138, 767 (1965).
14. Vashchenko V. and Zubarev V. N., Soviet Phys. Solid State 5, 653 (1963). 15. Migault A. and Romain J. P., J. Phys. Chem. Solids 38, 555 (1977). 16. Shanker J. and Kumar M., Phys. Stat. Solidi (b) 142, 325 (1987). 17. Boehler R. and Kennedy G. C., J. Phys. Chem. Solids 41, 517 (1980). 18. Isaak D. G., Cohen R. E. and Mehl M. J., J. geophys. Res. 95, 7055 (1990).
Pergamon
Chem.
Solids Vol. SJ, No. 2, pp. 207-210, 1994 Copyright0 1994Elm-in Science Ltd Printed in Omat Britain. All rights nxrwd O&22-3697/94 S6.00 + 0.00
ANALYSIS OF THERMODYNAMIC PARAMETERS FOR ALKALI HALIDES AT HIGH PRESSURE AND HIGH TEMPERATURE M. KUMAR~ and S. P. UPADHYAY~ TPhysics Department, Institute of Basic Sciences, Agra University, Agra 282 002, India (Received 23 March 1993; accepted 2 August 1993)
Abstract-We present a method for estimating the pressure and temperature dependence of thermodynamic parameters such as Griineisen parameter y its volume derivative q. Anderson parameters 6, and 6, for alkali halides up to their structural transition pressures and melting temperatures. The interionic potential model based on Harrison’s quantum mechanical form for overlap repulsive energy is used. The results are compared with the experimental data available for the NaCl crystal. A good agreement between calculated and experimental values demonstrates the validity of present work. Keywords: Thermodynamic temperature.
parameters, alkali halides, interionic potential, transition pressure, melting
1. INTRODUCTION In order to understand the the~~~~ic behaviour of solids it is desirable to investigate the pressure and temperature dependence of the Griineisen and Anderson parameters [l-2]. Some efforts [3-71 have been made in this direction which are limited to few crystals up to a very small range of pressures and temperatures. Thus Kumari and Dass [3] obtained y and q up to the pressure of 10 kbar at the temperatures 300, 550 and 800 K for NaCl. Dhoble and Verma [4] have shown on the basis of thermodynamic analysis that isothermal Anderson parameter 6, can be expressed in terms of constant pressure and constant volume temperature derivatives of thermal expansion coefficient. This result was used to study the temperature variation of B1.for NaCI. They have also studied the temperature variation of y, q and isentropic Anderson parameter S, only for NaCl at a constant pressure. Yamamoto ef al. [5] and Anderson and Yamamoto [6] presented an experimental method for the determination of the temperature variation of these parameters for NaCl at constant pressure. Kumar ef al. [7] investigated the pressure dependence of y using different theories for NaCl up to 32 kbar at constant temperature. In the present study we develop a method based on the interionic potential model which considers: (i) the long-range electrostatic interactions in terms of Madelung’s energy, (ii) the short range overlap $Physics Department, Govt. College, Shivpuri, M.P., India. 207
repulsive energy between nearest neighbours and next nearest neighbours by adopting the analytical potential form derived by Harrison [g] on the basis of quantum mechanical considerations, and (iii) the van der Waals (vdW) dipol+dipole and dipolequadrupole interactions evaluated from the Kirkwood-Muller formulae as reviewed by Shanker and Agrawal [9]. The model thus developed is used to estimate y, q, 6, and 6, for a much wider range of pressures and temperatures than that considered by any previous worker.
2. METHOD OF ANALYSIS
The basic definitions of y, q, 6, and 6, can be found in the recent literature [2, lo]. Slater [l l] suggested a method for calculating y from the theory of elasticity. His expressions for the vibrational velocities are valid only when the solid is under no external pressure. Dugdale and MacDonald [12] obtained a more general expression for y by including the effect of pressure. These theories do not take into account the variation of Poisson’s ratio with voiume 1131, Vaschenko and Zubarev [14] have developed a formulation for y using the free volume theory. Migault and Romain [15] proposed a unification of these theories by considering the volume dependence of Poisson’s ratio. Following their approach we can express y by a common formula which reads as follows:
M. KUMAR and S. P. UPADHYAY
208
y = _(4 - 3x) --6
Vd*(PV”)/dV*
’
2 d(PV”)/dV
+J2p:rexp(-J2kprr)
(l)
+ $2 P 2 ew(- J2 b2r)l, where P is the pressure, V the volume and x is the parameter which takes the value 0 in Slater’s theory, 0.66 in Dugdale and MacDonald’s theory and 1.33 in free volume theory [lo]. An expression for q can be obtained with the help of eqn (1) using the definition q = V/y . dy/d V. The Anderson parameters are related to y and q as follows [lo] s,=dd$+,-1,
(2)
a,=&--Y,
(3)
where
dBr _ _, _ vd2fWV2
zjs
dP-
~,,22?.$ b ‘r exp( - kfir) Table
“0
k
0.104 0.650 0.399 0.195 1.093 0.950 0.779
4.02 11.40 10.14 27.97 18.76 16.60 20.97 25.05
1.35 1.34 1.27 1.35 1.57 1.39 1.38 I .34
1.034
(7)
where ~~is the valence state energy of ions. Now to calculate the lattice potential energy W using eqns (5)-(7) there remains only two unknown parameters viz. n,, and k which are evaluated from the following relations based on the equilibrium condition.
= 18 roBr,
I161
(9)
where r, is the equilibrium value of r at T = 0 K and atmospheric pressure. The potential parameters rrO and k thus calculated are reported in Table 1 along with other input data. These parameters are used to estimate the higher order derivatives of W with the help of eqn (5) using different values of r as a function of pressure and temperature derived from the equation of state based on the Hildebrand approximation, P = ---Gr + TaB,,
(10)
where the first term on the right is the static pressure, the second term the thermal pressure and P is the externally applied pressure. T is the temperature, a the coefficient of volume thermal expansion and B, the isothermal bulk modulus. For the solids with NaCl-type structure, V = 2r ’ and therefore eqn (10) can be rewritten as follows:
parameters
(&I 3.962 3.962 3.962 3.962 3.087 3.087 3.087 3.087
(8)
and
1. Values of input
x
no. for data
p(2= -$,
0
Here W is the lattice potential energy and the first term on the right-hand side of eqn (5) is the electrostatic Coulomb energy with 01, as Madelung’s constant and e is the electron charge. The second and third terms are the vdW dipole-dipole and dipolequadrupole energies. Last term is the overlap repulsive energy. Values of C and D have been calculated using the Kirkwood-Muller formulae which are superior to all other existing formulae [9]. The overlap repulsive energy can be written as follows [16]:
LiF LiCl LiBr LiI NaF NaCl NaBr NaI Reference
where h is Planck’s constant divided by 271, m the mass of electron and p the arithmetic average of p, and p2 for the cation and anion. Values of p, and p2 can be calculated by the relation
(4)
Thus in order to evaluate y, q, 6, and 6, with the help of eqns (l-4) we need the derivatives of P with V that is dP/dV, d2P/dV2 and d’P/dV’. These derivatives are directly related to the derivatives of the lattice potential energy [7] and can be estimated using the following expression based on the interionic potential mode1 considered in the present study
Crystal
(6)
2.112 1.799 1.716 1.619 2.112 1.799 I.716 1.619
I161
C (10-60ergcm6) 46 317 594 1265 139 561 930 I755
191
D (10-76ergcm8) 2:: 286 584 98 455 537 942
191
Thermodynamic parameters for alkali halides
209
Table 2. Thermodynamic parameters for NaCI. Temperature T (in K) and pressure P (in kbar). Experimental data (5, 171 are given within parentheses. Melting temperature = IO73 K and transition pressure = 283 kbar 298
0 5 10 20 30 40 50 100 150 200 283
Y 1.61 (1.61 1.50 (1.50) 1.48 (1.49) 1.43 (1.44) 1.38 (1.40) 1.34 1.31 1.20 1.11 1.04 0.96
4 1.32 1.02 1.38
313
6s 6, Y 4 3.90 5.51 1.63 1.35 3.50 5.24) (1.63 1.18 3.88 5.38 1.56 1.37
8s 6, Y 4 3.91 5.54 1.66 1.30 3.61 5.64) (1.66 1.17 3.89 5.45 1.60 1.35
6, 6, Y 4 3.95 5.61 1.69 1.27 3.60 5.63) (1.68 1.00 3.93 5.53 1.65 1.28
773
6s 6, Y 4 4.03 5.72 1.75 1.23 3.71 5.87) (1.65 1.05 3.96 5.61 1.66 1.26
6s & 4.15 5.90 3.74 5.82) 4.14 5.80
1.50 3.83 5.26 1.48 1.45 3.86 5.34 1.50 1.40 3.88 5.38 1.54 1.39 3.90 5.44 1.57 1.36 4.03 5.60 1.56 3.80 5.18 1.45 1.49 3.85 5.30 1.47 1.47 3.86 5.33 1.50 1.45 3.87 5.37 1.53 1.43 3.96 5.49 1.58 1.60 1.66 1.75 1.89 2.01
3.75 3.61 3.54 3.49 3.43 3.41
5.09 4.92 4.74 4.60 4.47 4.37
pressure
1.40 1.36 1.30 1.20 1.10 1.01
1.55 1.57 1.63 1.70 1.80 1.92
3.83 3.78 3.68 3.55 3.50 3.49
+ TUB,.
5.23 5.14 4.98 4.75 4.60 4.50
1.44 1.40 1.34 1.29 1.23 1.15
(11)
(PT,,) in eqn (11) can be deter-
mined as follows: ($).=(
Integrating
573
1.44 3.86 5.34 1.50 1.40 3.87 5.37 1.55 1.38 3.90 5.45 1.60 1.34 3.92 5.52 1.62 1.31 4.09 5.71
P = --$>y The thermal
473
%)v=aB,.
(12)
‘(aB,)dT. s II
(13)
we get, PTh=
Thus Pn, is equal to TcrB, only if the product (aB,) is independent of T, which is a good approximation for the solids considered in the present work. The parameters are assumed to be independent of pressure and temperature. 3. RESULTS AND DISCUSSIONS
The calculations of y and other thermodynamic parameters depend sensitively on the values of x. According to the different theories of y [12-151 the value of x should remain between 0 and 1.33. We have determined the values of x to fit the experimental values of y [4] at atmospheric pressure and room temperature for each crystal. It is interesting to see from Table 1 that the values of x thus determined range between 0 and 1.33 and are therefore consistent with the theories of y. In further calculations we have assumed that x does not change with pressure and temperature. The good agreement between calculated and experimental values reveals the validity of this assumption. The thermodynamic parameters y, q, 6, and 6, can be calculated at higher pressures and higher temperatures following the method described in
1.51 1.60 1.62 1.68 1.75 1.84
3.84 3.80 3.78 3.61 3.52 3.51
5.28 5.20 5.12 4.90 4.75 4.66
1.46 1.42 1.39 1.31 1.28 1.20
1.50 1.59 1.61 1.65 1.73 1.81
3.86 3.84 3.80 3.70 3.56 3.53
5.32 5.26 5.19 5.01 4.84 4.73
1.47 1.43 1.40 1.33 1.30 1.23
1.49 1.54 1.60 1.64 1.72 1.79
3.94 3.87 3.81 3.76 3.59 3.55
5.41 5.30 5.21 5.09 4.89 4.78
preceding section using the parameters given in Table 1. Such calculations can be performed for all crystals listed in Table 1. However, we are reporting the results only for NaCl for which the experimental data are available [5, 171 so that a direct comparison can be made in Table 2. There is good agreement between calculated values and available experimental data. It is noted from Table 2 that y decreases with increasing pressure and increases with increasing temperature, whereas q increases with increasing pressure and decreases with increasing temperature. Such a trend of variation is in agreement with the results obtained by Kumari and Dass [3] and Dhoble and Verma [4]. The values of 6, and 6, are also very sensitive to pressure and temperature. 6, and 6, decrease with an increase in pressure and increase with an increase in temperature in agreement with the recent research in high pressure and high temperature physics [2, 181.
Acknowledgements-We
are grateful to Prof. Jai Shanker
for useful discussions.
REFERENCES I. Anderson 0. L., Isaak D. L. and Oda H., J. Geophys. 96, 18037 (1991). 2. Anderson 0. L., Isaak D. L. and Oda H., Rev. Geophys. 30, 57 (1992). 3. Kumari M. and Dass N., Phys. Seal. Solidi (b) 133, 101 (1986). 4. Dhoble A. and Verma M. P., Phys. Stat. Solidi (b) 133, 491 (1986); 136, 497 (1986). 5. Yamamoto S., Ohno I. and Anderson 0. L., J. Phys. Chem. Soli& 48, 143 (1987). 6. Anderson 0. L. and Yamamoto S., High Pressure Res. in Mineral Phys. 39, 289 (1987). 7. Kumar M., Pachauri A. K., Chaturvedi S. D. and Sharma A. K., Phys. Stat. Solidi (b) 146, 125 (1988).
210
M. KUMARand S. P. UPADHYAY
8. Harrison W. A., Phys. Rev. B23, 5230 (198 1). 9. Shanker J. and Agrawal G. G., Phys. Stat. Solidi (b) 123, 11 (1984). 10. Shanker J. and Bhande W. N., Phys. Stat. Solidi (b) 135, 11 (1986). 1I. Slater J. C., Introduction to Chemical Physics. McGraw Hill, New York (1939). 12. Dugdale J. S. and MacDonald D. K. C., Phys. Reo. 89, 832 (1953). 13. Pastine D. J., Phys. Rev. 138, 767 (1965).
14. Vashchenko V. and Zubarev V. N., Soviet Phys. Solid State 5, 653 (1963). 15. Migault A. and Romain J. P., J. Phys. Chem. Solids 38, 555 (1977). 16. Shanker J. and Kumar M., Phys. Stat. Solidi (b) 142, 325 (1987). 17. Boehler R. and Kennedy G. C., J. Phys. Chem. Solids 41, 517 (1980). 18. Isaak D. G., Cohen R. E. and Mehl M. J., J. geophys. Res. 95, 7055 (1990).