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Thermodynamical properties of fluorite structure crystals
Infrared
Pergamon
Phys. Technol. Vol. 35,No. I, pp.63-66,1994 Copyright 0 1994Elsevier Science Ltd Printed in Great Britain. All rightsreserved
1350~4495/94 $6.00+ 0.00
THERMODYNAMICAL PROPERTIES OF FLUORITE STRUCTURE CRYSTALS C. N.
MISHRA,’
R. P. GOYAL,’
S. C. GOYAL’
and B. R. K.
GUPTA~
‘Department of Physics, St John’s College, Agra 282002, *Department of Physics, Agra College, Agra 28200? and 3Department of Physics, G. B. Pant University, Pantnagar 263145, India (Received
7 September
1993)
Abstract-A theoretical method based on phonon-pressure theory is developed for the study of thermodynamical properties of CaF,, SF,, BaF,, PbF,, Sr(NO,),, Ba(NO,), and Pb(NO,), crystals crystallizing in the fluorite structure. The traditional Born method for determining the short-range repulsive parameters is replaced by a more plausible method due to Hafemeister-Zarht. The van der Waals dipole-dipole and dipolequadrupole interaction coefficients are derived from Kirkwood and Muller formulae based on quantum theory. The thermodynamical properties are analysed by predicting the values of bulk modulus and its pressure derivatives, Griineisen and Anderson parameters at 0 and 300K temperatures. The results obtained in the present study are compared with the available experimental data. A fairly good agreement is achieved as compared to those obtained by previous investigators.
I.
INTRODUCTION
The fluorite structure crystals are complicated ionic solids as they are non-centro-symmetric crystals, i.e. every ion is not situated at the centre of inversion symmetry unlike alkali halide crystals. This effect is responsible for the existence of Cauchy discrepency between the elastic constants of CaF, type crystals. (I) In past years several physicists ~7~)have made attempts to explain Cauchy breakdown in terms of intersublattice displacement contributions which involve the shell model parameters. The previous workers considered different potential forms for the short-range two-body interactions. However, the calculations performed by them were found to be physically unacceptable”) in view of the inadequacy of two-body potentials. Although, Dutt et d(4) have investigated some crystalline state properties of CaF,-type crystals using different inter-ionic potentials, the results obtained by them are not close to the experimental results. Motivated by this situation, in the present study, we have developed an alternative approach based on phonon-pressure theory. In the present model theory Hildebrand’s equation of state and three-body interaction (TBI) due to Lundqvisto) have been used. To test the validity of the model, we have calculated the thermodynamical values like bulk modulus and its pressure derivatives, Griineisen and Anderson parameters at 0 and 300 K temperatures for the fluorite crystals under study. A brief theory of the model is given in Section II and results are discussed in Section III.
II.
BRIEF
THEORY
AND
METHOD
OF
CALCULATION
In order to calculate the various thermodynamical properties of fluorite structure crystals, we have used the following expression for total lattice energy in view of Lundqvist three-body interaction arising due to nearest and next nearest neighbour ions 4 = -,‘..g,yeZ i.i
+ c S/($) v
iJ,k
11
where symbols have their usual meaning.@*‘) 63
+ 4R -g-p
(1)
C. N. MISHRA et al.
64 Table
I. Calculated
values of isothermal
Bulk modulus Crystals CaFZ SrF: BaF, PbF, Sr(NO,)> Ba(NO, )? Pb(NO> I, (a) (b) (c) (d)
equation
and its pressure
derivative
dBr,‘dP
a
b
c
a
b
d
1.09 1.13 I .48 I .28 2.79 3.77 3.09
1.13 1.34 I .62 1.45 2.98 3.95 3.18
1.1 I
4.52 4.46 4.15 6.36 5.68 5.57 4.08
4.76 4.88 4.96 6.65 5.70 5.75 5.08
4.96 5.00 5.07 7.10 5.77
Calculated at 0 K. Calculated at 300 K. Calculated from the relation Experimental values.“J’
The Hildebrand’s
bulk modulus
B, (IO” Nm-‘)
1.40 1.77 1.65 2.97 4.24 3.23
4.84
B, = (C,, + 2C,>)/3
of state can be written P = -gv
as follows”’
+ TflB,
(2)
where P is the pressure, T the temperature, /? the volume thermal expansion coefficient and B, is the isothermal bulk modulus. In equation (1) the first term represents the Coulomb energy due to electrostatic interactions, Z,(Z,) is the ionic charge of i(j) ions, Cl, the lattice sum, Q(= - 5.818) the Madelung constant. The second term is due to three-body interaction.f(r,,J is the three-body charge transfer parameter. The third and the fourth terms represents the short-range overlap repulsion and van der Waals attractive interactions, C and D in the fourth term are the van der Waals coefficients calculated using Kirkwood-Muller theory. (“O) The expression for the short-range overlap potential has been considered as follows””
where r, and Y? are radii of cation and anion respectively, b is the strength parameter and /I,, are Pauling coefficients, r = a&/2 and r’ = a are the nearest neighbour (nn) cation-anion and next nearest neighbour (nnn) cation-cation and anion-anion distances respectively, p,, are the hardness parameters determined from overlap integrals.“*’ The various potential functions have been examined critically by calculating the crystal bulk modulus and its pressure derivatives, Griineisen and Anderson parameters for seven crystals with fluorite structure. These physical quantities investigated in the present study can be expressed in terms of lattice potential energy and its derivative with respect to inter-ionic Table 2. Calculated
values of Griineisen
parameter
.I Crystals CaF2 SrFz BaF, PbF2 WNO, )? Ba(N0, __ ),
Pb(NO, h
(>I) and Anderson
parameters
(6,, 6,)
6,
6s
a
b
C
a
b
c
a
b
c
1.82 1.77 1.72 1.63 I .92 1.77 1.83
1.71 1.64 1.61 I .57 1.87 1.70 1.I5
1.74 I .62 1.61
4.62 4.76 4.58 4.71 5.10 5.13 5.39
4.58 4.69 4.53 4.67 4.90 5.02 5.28
4.62 4.72 4.71 4.66 5.80 5.32 5.56
2.41 2.50 2.63 2.68 3.12 2.76 3.21
2.36 2.48 2.52 2.60 2.98 2.67 3.08
2.58 2.66 2.66 2.61 3.29 2.97 3.21
(a) Calculated at 0 K. (b) Calculated at 300 K. (c) Experimental values.“’
Thermodynamical properties of fluorite structure crystals
65
separation. The relevant expressions for bulk modulus and its pressure derivative can be defined as@) B,= --VP’
(4)
and dB,/dP
= - 1 - I’(P”/P’)
(5)
where P’ = dP/dV and P” = d*P/dV* are the volume derivatives at constant temperature. In view of equation (2) we obtain the following expression: P’ = -(1/9V*)[r*~”
- 2rf$‘]
(6)
- 6r24” + lo@]
(7)
and P = -(1/27V3)[r3~“’
where 4’, 4” and 4”’ are respectively first to third order derivative or lattice potential energy with respect to ionic separation r. Equations (6) and (7) have been obtained by using the relation V = XT’, where x is a geometrical factor depending on the crystal structure; x = 3.08 for CaF,-type crystals. The thermodynamical behaviour of the solids have been analysed by calculating the Griineisen parameter (y) and Anderson parameters (6,, 6,). The expressions for these parameters have been derived using the theory of elasticity in the form of the following expressions(‘3) y = -(1/6)$~“‘/$”
1
+ (dBT/dP)
(8) - 1
and
III.
RESULT
AND
DISCUSSION
In view of equations (6) and (7), values of bulk modulus and its pressure derivatives are calculated using equations (4) and (5) both at 0 and 300 K temperatures and are reported in Table 1. They are found to be in fairly good agreement with the experimental data as compared to those of Dutt et aLc4’Griineisen and Anderson parameters are calculated using equations (8HlO) and the expression for lattice energy (1). Results are reported in Tables 1 and 2. It is noted from Tables 1 and 2 that the results reported in the present study are in fairly good agreement with the available experimental data. Moreover, our calculated results are found in better agreement with the experimental values as compared to those of previous results obtained by Dutt et ~1.‘~’ On the basis of the present investigation we may conclude that the phonon-lattice theory with thermal contribution and the Hafemeister-Zarht potential for overlap repulsive energy is found to describe the thermodynamical properties of fluorite structure compounds quite satisfactorily.
REFERENCES 1. R. Srinivasan, Phys. Rev. 165, 1054 (1968). 2. C. Wong and D. E. Schuele, J. Phys. Chem. Solids 29, 1309 (1968). 3. R. Srinivasan and V. Ramachandran, Pramana 6, 87 (1976). 4. N. Dutt, A. J. Kaur and J. Shanker, Physica status solidi (b) 137, 459 (1986).
66 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
C. N. MISHRA et al.
S. 0. Lundqvist, Ark. Fys. (Sweden) 9, 435 (1955); 12, 263 (1957). B. R. K. Gupta, Physica Status solidi (h) 128, 99 (1985). B. R. K. Gupta and M. P. Verma, J. Phy.s. Chem. Solids 38, 929 (1977). J. Fontanella, C. Andeen and D. Schuele, Phys. Rer. 86, 582 (1972). J. G. Kirkwood, Phys. Z. 33, 57 (1932). A. Muller, Proc. R. Sot. A154, 624 (1936). B. R. K. Gupta, 0. P. Sharma and R. P. Goyal, Czech. J. Phys. B34, 532 (1984). M. R. Hayns and J. L. Calais, J. Chem. Phys. 57, 147 (1972); J. Ph.vs. C6, 2625 (1973). J. Shanker and K. Singh, Physica Sfafus solidi (b) 103, 151 (1981). J. Shanker, J. P. Singh and V. C. Jain, Physica, (7Jtrecht) B106, 247 (1981).
Pergamon
Phys. Technol. Vol. 35,No. I, pp.63-66,1994 Copyright 0 1994Elsevier Science Ltd Printed in Great Britain. All rightsreserved
1350~4495/94 $6.00+ 0.00
THERMODYNAMICAL PROPERTIES OF FLUORITE STRUCTURE CRYSTALS C. N.
MISHRA,’
R. P. GOYAL,’
S. C. GOYAL’
and B. R. K.
GUPTA~
‘Department of Physics, St John’s College, Agra 282002, *Department of Physics, Agra College, Agra 28200? and 3Department of Physics, G. B. Pant University, Pantnagar 263145, India (Received
7 September
1993)
Abstract-A theoretical method based on phonon-pressure theory is developed for the study of thermodynamical properties of CaF,, SF,, BaF,, PbF,, Sr(NO,),, Ba(NO,), and Pb(NO,), crystals crystallizing in the fluorite structure. The traditional Born method for determining the short-range repulsive parameters is replaced by a more plausible method due to Hafemeister-Zarht. The van der Waals dipole-dipole and dipolequadrupole interaction coefficients are derived from Kirkwood and Muller formulae based on quantum theory. The thermodynamical properties are analysed by predicting the values of bulk modulus and its pressure derivatives, Griineisen and Anderson parameters at 0 and 300K temperatures. The results obtained in the present study are compared with the available experimental data. A fairly good agreement is achieved as compared to those obtained by previous investigators.
I.
INTRODUCTION
The fluorite structure crystals are complicated ionic solids as they are non-centro-symmetric crystals, i.e. every ion is not situated at the centre of inversion symmetry unlike alkali halide crystals. This effect is responsible for the existence of Cauchy discrepency between the elastic constants of CaF, type crystals. (I) In past years several physicists ~7~)have made attempts to explain Cauchy breakdown in terms of intersublattice displacement contributions which involve the shell model parameters. The previous workers considered different potential forms for the short-range two-body interactions. However, the calculations performed by them were found to be physically unacceptable”) in view of the inadequacy of two-body potentials. Although, Dutt et d(4) have investigated some crystalline state properties of CaF,-type crystals using different inter-ionic potentials, the results obtained by them are not close to the experimental results. Motivated by this situation, in the present study, we have developed an alternative approach based on phonon-pressure theory. In the present model theory Hildebrand’s equation of state and three-body interaction (TBI) due to Lundqvisto) have been used. To test the validity of the model, we have calculated the thermodynamical values like bulk modulus and its pressure derivatives, Griineisen and Anderson parameters at 0 and 300 K temperatures for the fluorite crystals under study. A brief theory of the model is given in Section II and results are discussed in Section III.
II.
BRIEF
THEORY
AND
METHOD
OF
CALCULATION
In order to calculate the various thermodynamical properties of fluorite structure crystals, we have used the following expression for total lattice energy in view of Lundqvist three-body interaction arising due to nearest and next nearest neighbour ions 4 = -,‘..g,yeZ i.i
+ c S/($) v
iJ,k
11
where symbols have their usual meaning.@*‘) 63
+ 4R -g-p
(1)
C. N. MISHRA et al.
64 Table
I. Calculated
values of isothermal
Bulk modulus Crystals CaFZ SrF: BaF, PbF, Sr(NO,)> Ba(NO, )? Pb(NO> I, (a) (b) (c) (d)
equation
and its pressure
derivative
dBr,‘dP
a
b
c
a
b
d
1.09 1.13 I .48 I .28 2.79 3.77 3.09
1.13 1.34 I .62 1.45 2.98 3.95 3.18
1.1 I
4.52 4.46 4.15 6.36 5.68 5.57 4.08
4.76 4.88 4.96 6.65 5.70 5.75 5.08
4.96 5.00 5.07 7.10 5.77
Calculated at 0 K. Calculated at 300 K. Calculated from the relation Experimental values.“J’
The Hildebrand’s
bulk modulus
B, (IO” Nm-‘)
1.40 1.77 1.65 2.97 4.24 3.23
4.84
B, = (C,, + 2C,>)/3
of state can be written P = -gv
as follows”’
+ TflB,
(2)
where P is the pressure, T the temperature, /? the volume thermal expansion coefficient and B, is the isothermal bulk modulus. In equation (1) the first term represents the Coulomb energy due to electrostatic interactions, Z,(Z,) is the ionic charge of i(j) ions, Cl, the lattice sum, Q(= - 5.818) the Madelung constant. The second term is due to three-body interaction.f(r,,J is the three-body charge transfer parameter. The third and the fourth terms represents the short-range overlap repulsion and van der Waals attractive interactions, C and D in the fourth term are the van der Waals coefficients calculated using Kirkwood-Muller theory. (“O) The expression for the short-range overlap potential has been considered as follows””
where r, and Y? are radii of cation and anion respectively, b is the strength parameter and /I,, are Pauling coefficients, r = a&/2 and r’ = a are the nearest neighbour (nn) cation-anion and next nearest neighbour (nnn) cation-cation and anion-anion distances respectively, p,, are the hardness parameters determined from overlap integrals.“*’ The various potential functions have been examined critically by calculating the crystal bulk modulus and its pressure derivatives, Griineisen and Anderson parameters for seven crystals with fluorite structure. These physical quantities investigated in the present study can be expressed in terms of lattice potential energy and its derivative with respect to inter-ionic Table 2. Calculated
values of Griineisen
parameter
.I Crystals CaF2 SrFz BaF, PbF2 WNO, )? Ba(N0, __ ),
Pb(NO, h
(>I) and Anderson
parameters
(6,, 6,)
6,
6s
a
b
C
a
b
c
a
b
c
1.82 1.77 1.72 1.63 I .92 1.77 1.83
1.71 1.64 1.61 I .57 1.87 1.70 1.I5
1.74 I .62 1.61
4.62 4.76 4.58 4.71 5.10 5.13 5.39
4.58 4.69 4.53 4.67 4.90 5.02 5.28
4.62 4.72 4.71 4.66 5.80 5.32 5.56
2.41 2.50 2.63 2.68 3.12 2.76 3.21
2.36 2.48 2.52 2.60 2.98 2.67 3.08
2.58 2.66 2.66 2.61 3.29 2.97 3.21
(a) Calculated at 0 K. (b) Calculated at 300 K. (c) Experimental values.“’
Thermodynamical properties of fluorite structure crystals
65
separation. The relevant expressions for bulk modulus and its pressure derivative can be defined as@) B,= --VP’
(4)
and dB,/dP
= - 1 - I’(P”/P’)
(5)
where P’ = dP/dV and P” = d*P/dV* are the volume derivatives at constant temperature. In view of equation (2) we obtain the following expression: P’ = -(1/9V*)[r*~”
- 2rf$‘]
(6)
- 6r24” + lo@]
(7)
and P = -(1/27V3)[r3~“’
where 4’, 4” and 4”’ are respectively first to third order derivative or lattice potential energy with respect to ionic separation r. Equations (6) and (7) have been obtained by using the relation V = XT’, where x is a geometrical factor depending on the crystal structure; x = 3.08 for CaF,-type crystals. The thermodynamical behaviour of the solids have been analysed by calculating the Griineisen parameter (y) and Anderson parameters (6,, 6,). The expressions for these parameters have been derived using the theory of elasticity in the form of the following expressions(‘3) y = -(1/6)$~“‘/$”
1
+ (dBT/dP)
(8) - 1
and
III.
RESULT
AND
DISCUSSION
In view of equations (6) and (7), values of bulk modulus and its pressure derivatives are calculated using equations (4) and (5) both at 0 and 300 K temperatures and are reported in Table 1. They are found to be in fairly good agreement with the experimental data as compared to those of Dutt et aLc4’Griineisen and Anderson parameters are calculated using equations (8HlO) and the expression for lattice energy (1). Results are reported in Tables 1 and 2. It is noted from Tables 1 and 2 that the results reported in the present study are in fairly good agreement with the available experimental data. Moreover, our calculated results are found in better agreement with the experimental values as compared to those of previous results obtained by Dutt et ~1.‘~’ On the basis of the present investigation we may conclude that the phonon-lattice theory with thermal contribution and the Hafemeister-Zarht potential for overlap repulsive energy is found to describe the thermodynamical properties of fluorite structure compounds quite satisfactorily.
REFERENCES 1. R. Srinivasan, Phys. Rev. 165, 1054 (1968). 2. C. Wong and D. E. Schuele, J. Phys. Chem. Solids 29, 1309 (1968). 3. R. Srinivasan and V. Ramachandran, Pramana 6, 87 (1976). 4. N. Dutt, A. J. Kaur and J. Shanker, Physica status solidi (b) 137, 459 (1986).
66 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
C. N. MISHRA et al.
S. 0. Lundqvist, Ark. Fys. (Sweden) 9, 435 (1955); 12, 263 (1957). B. R. K. Gupta, Physica Status solidi (h) 128, 99 (1985). B. R. K. Gupta and M. P. Verma, J. Phy.s. Chem. Solids 38, 929 (1977). J. Fontanella, C. Andeen and D. Schuele, Phys. Rer. 86, 582 (1972). J. G. Kirkwood, Phys. Z. 33, 57 (1932). A. Muller, Proc. R. Sot. A154, 624 (1936). B. R. K. Gupta, 0. P. Sharma and R. P. Goyal, Czech. J. Phys. B34, 532 (1984). M. R. Hayns and J. L. Calais, J. Chem. Phys. 57, 147 (1972); J. Ph.vs. C6, 2625 (1973). J. Shanker and K. Singh, Physica Sfafus solidi (b) 103, 151 (1981). J. Shanker, J. P. Singh and V. C. Jain, Physica, (7Jtrecht) B106, 247 (1981).