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Analysis of the crystal binding and the gruneisen-anderson parameters in alkali halides
Z inorg, nucl. Chem. Vol. 43, pp. 1445-1449, 1981 Printed in Great Britain.
0022-1902/81/071445-05502.00/0 Pergamon Press Ltd.
ANALYSIS OF THE CRYSTAL BINDING AND THE GRUNEISEN-ANDERSON PARAMETERS IN ALKALI HALIDES JAI SHANKER and KAMENDRA SINGH Physics Department, Agra College, Agra-282002, India
(Received 27 July 1980; received/or publication 19 September 1980) Abstract--We have calculated the crystal binding energies and the Gruneisen-Anderson parameters in sixteen NaCI- structure alkali halides using the Born model formulation including the short range repulsive interactions between nearest neighbours and the next nearest neighbours, and the van tier Waals dipole--dipole and dipolequadrupole interactions. The overlap repulsive potential parameters have been calculated from the low temperature crystal data on interionic separation and compressibility data. Values of the Gruneisen parameter are calculated taking into account the volume dependence of Poisson's ratio. The Anderson parameters describing the temperature derivatives of adibatic and isothermal bulk moduli have also been calculated with the help of interionic potential models. Calculations have been performed using the two potential forms for short range repulsive interactions showing an inverse power dependence and exponential dependence on interionic distance. The results obtained are compared with experimental data and also with other theoretical studies. INTRODUCTION The evaluation of crystalline properties of inorganic solids from the knowledge of interionic potentials has recently been a subject of considerable interest in the field of crystal chemistry. Thakur and Pandey[l-3] have calculated a number of crystalline properties for diatomic halides and chalcogenide crystals. However, their method is not adequate for several reasons. First, they have not considered the repulsive interactions between like ions, i.e. positive-positive and negative-negative ion interactions and have completely neglected the van der Waals dipole--dipole and dipole--quadrupole interactions. However, recent studies [4, 5] have clearly demonstrated the importance of these interactions in determining adequate potentials for alkali halide crystals. In view of these studies, the potential energy expression used by Thakur and Pandey[1-3] is not completely satisfactory. In addition, the compressibility data used by Thakur[3] are old and less accurate than recent ultrasonic data [46]. The temperature dependence of bulk modulus at fixed volume, which is an important and significant quantity for alkali halides[7], has been completely ignored by Thakur[3] when evaluating the temperature dependence of compressibility at constant pressure. In the present study we remove these deficiencies by adopting the interionic potential model so as to take into account the van der Waals interactions and repulsive interactions between second neighbours and like ions. We use two potential forms for the short range repulsive energy showing an inverse power dependence[8] and exponential dependence[9] on interionic separation. The potential parameters are derived from the crystal equilibrium and compressibility data. We have used low temperature data in order to avoid uncertainties arising from assumptions concerning thermal contribution in the equation of state introduced by different workers[10,11]. We have calculated the cohesive energy, the Gruneisen parameter, and the Anderson parameters 8s and 8r for sixteen alkali halide crystals with NaCI structure. These quantities are quite useful in understanding the thermodynamic behaviour of inorganic compounds.
M E T H O D OF ANALYSIS
Cohesive energy. The cohesive energy of an ionic crystal can be calculated with the help of following expression for the potential energy[11], _
W=
C D ctme2+B(r) r6 p
(I)
t
where the first term on the right is the Madelung energy or the electrostatic energy, the second term represents the short range overlap repulsive energy, the last two terms are, respectively, the dipole--dipoleand dipole--quadrupole van der Waals energies. We have not included the zero point energy because it is negligibly small[ll]. If we adopt an inverse power form then the repulsive energy is [12],
B(r)=6A[[3+-(rr+r-)n-l'~-~ ~2+(2r+)"-1 -~ ~2_(2r_) - ~ j "-I ] (2) Here A and n are potential parameters, Bii are Pauling's coe$cients[12], r+ and r_ are ionic radii, r is cation-anion separation, ~/(2)r is cation--cation or anion-anion separation. By using exponential form we can write,
=6b[, +_exp(r,+ ,_-q + t ,
~
\
P
/
[2r_- V'(2)rkq
+~2_expt
7- :J
exp(r,-V(2),] \
P
/
(3)
where b and p are, respectively, the repulsive strength and hardness parameters. Thus, whether we use an inverse power from or an exponential form, there are two unknown parameters which are determined from the crystal equilibrium condition and isothermal bulk modulus, i.e. using the relations, ,4, and, dZW V ~ = Br
(5)
Values of Br used in the present calculations are those based on the measured elastic constants at helium temperature recently 1445
1446
JAI SHANKER and KAMENDRA SINGH Gruneisen parameters. Romain et al.[15] have recently developed a theory for the Gruneisen parameter by taking into account the pressure dependence of Poisson's ratio. They considered two Gruneisen parameters, yj and y,, corresponding to longitudinal and transverse vibrations. This model gives a more realistic representation of thermodynamic and thermoelastic behaviour of solids than the theories formulated by earlier
compiledby Catlow et al. [4]. Ionic radii are token from [13],and the vander Waalsdipole--dipoleand dipole-quadrupoleconstants Cand D from[14]. Calculated values of potential parameters for the inverse power form and the exponentialform are listed in Table 1. Using these parameters and equations (1)-(3) we have obtained cohesive energies.These are comparedwith experimentalvalues in Table 2.
Table 1. Calculated values of potential parameters Crystal
Inverse power (Born-Lande)
Form
Exponential Form (B o r n - M a y e r )
(1~-12erg.)
(10-21erg.cm)
]o8
(10- era)
LIF
6.83
5.98
O. 308
O. 303
LICI
7.24
7.15
0.253
0.342
LiBr
8.08
6.94
0.259
0.304
Lil
9.21
6.53
0.256
0.453
NaY
4.90
7.10
0.402
0.290
10,I
7.53
0.335
0.346
NaBr
1o.3
7.76
0.322
0.362
Nal
10.5
7.99
0.298
0.392
NaCI
KF
11.5
7.77
0,389
0,322
KCI
11.1
8.08
0.334
0.353
KBr
11.0
8.34
0.314
0.360
KI
11.9
7.92
o.313
o.410
3bF
1o.3
8.10
0.358
0.333
HhCI
11.0
8.66
o.314
0.345
l~Br
11.3
8.70
0.306
0.359
Hbl
11.3
8.28
0.287
o.381
Table 2. Values of cohesive energy (kcal/mole) I n v e r s e p o w e r form (B o r n - L a n de )
Crystal
Calculated in present study
ToKi ~1~
Exponential form (Born-Mayer) Calculated in present study
Experimental
t313
Tosi [11]
LiF
- 242.1
- 239.3
- 24-9.6
- 24-2.2
- 246.7
LICI
- 199.0
- 194.0
- 203.?
192.9
- 203.2
LiBr
- 185.5
- 183.3
- 190.3
Lil
- 168.3
- 165.7
- 174-.1
-
181.0
- 194.2
166.1
-
180.3
NaF
- 219.1
- 210.0
- 225.0
- 215.2
NaCI
-
183.6
- 180.9
- 187.8
- 178.6
- 187.1
NaBr
- 175.1
- 170.5
- 178.9
- 169.2
- 178.5
Nal
- 164.O
- 157.1
- 167.5
- 156.6
- 167.0
KF
- 194.6
- 187.1
- 198.7
- 189.1
- 194.3
KCI
- 168.0
- 162.9
- 171.8
- 161.6
- 170.2
KBr
- 162.3
- 157.0
- 165.9
- 154.5
- 163.2
KI
- 150.7
- 147.1
- 154.6
- 144.5
1
RbF
-
186.9
- 181.0
- 190.5
- 180.4
- 185.8
RbCI
-
163.6
- 158.8
- 166.9
- 155.4
- 163.6
RbBr
- 157.2
- 151.5
- 160.5
- 148.3
- 157.2
RDI
- 149.0
- 143.8
- 152.3
- 139.6
- 148.5
- 219.5
153.6
Gruneisen-Anderson parameters in alkali halides workers[16,17]. For yt and Yt we can write the following expressions[15],
yt
4 - 3pl Vd2PVP'/dV2 6 2 dPVP'/d V
(6)
Yt
4-3pz 6
V d2P Vm/d V2 2 dPVmld V
(7)
and,
where Pl and P2 are related to Poisson's ratio ~ as follows,
l+l_o./1
dlnV\ +P' a T H
1[ 01nB~
(17) (18)
In order to calculate the volume dependence of bulk modulus one can take the plausible assumption[21],
(9)
(10)
Values of Poisson's ratio and its volume derivatives are calculated from elastic constants Cll and C~2 and their pressure derivatives[4, 7] using the relation, Ct2 /7"= Ci I + Cl2.
OT Iv
(8)
where tro is Poisson's ratio at zero pressure. Values of pt can be determined from the volume derivative of ,7 which is given by,
(~V)vfvo_- -~-~o Pl (1-o" o2)
\OlnV] T fl\ = _ ( 0 In B.~
and, 3 1 - O'o P2=2 pt 1 +20"o
However, this assumption is not justified in view of the fact that the adiabatic bulk modulus Bs or the isothermal bulk modulus Br is not independent of temperature at fixed volume. In reality, the quantities (OBs/c~T)v and (OBr/OT)v are quite significant and have played important roles in describing the equation of state of alkali halide crystals[7]. Moreover, the sign of (OBslOT)v is different from that of (OB.flOT)v. Therefore it is inappropriate to take Bs to be equal to 8r. Correct expressions for 8s and 8r are [7],
8s
+P' d~-~. p) tr
1447
(11)
Calculated values of p~ and P2 along with Poisson's ratio are given in Table 3. In order to calculate Yt and ~'t we need dPld V and dZP]dV2. These quantities can be calculated from the equation of state[ll],
(~ In Bs~ ~ (0 In Br~ ~Tr = k~]~Vn V ]r'
(19)
The volume derivative of bulk modulus Br can be expressed, in terms of interionic potential energy and its derivatives, as follows[23], (~ In BT'~ I[r2W'-3rW"+4W ' ] ~--i-wV-n'Vl r =- 3 [ rW"-2 W' ]
(20)
where w is the potential energy expressed by eqn (1) w', w" and w" are, respectively, at first, second and third order derivatives of the potential energy with respect to interionic separation r. We have calculated these derivatives with the help of eqn (1) using the potential parameters given in Table 1. Finally we have determined 8s and 8r using eqns (17)-(20) and evaluating the explicit contributions, viz. the temperature derivatives of Bs and BT at constant volume from thermoelastic data[7].
RESULTS AND DISCUSSION
In the present study we have used an expression for dW (12) the potential energy which is more complete than that e=--dV used by Thakur and Pandey[1-3]. We have considered the van der Waals interactions and overlap repulsive where W is the potential energy expressed by equation (1). Using interactions between like ions which are quite important the potential parameters reported in Table 1 we have calculated as is evident from recent quantum mechanical[24] as yt and y, with the help of eqns (6), (7) and (12). Average values well as phenomenological studies [4, 5]. Calculated values of V are then determined from the relation[15], of cohesive energies are in better agreement with experimental data (Table 2) than those reported by 1 (13) Tosi[1]. It should also be noticed that the inverse power potential form yields better agreement with experimental Values of y thus calculated are compared with those obtained by data for K and Rb halides whereas the exponential other investigators in Table 4. potential form is better for Li and Na halides. Anderson paramaers. There are two Anderson parameters, 8s Values of the Gruneisen parameter 3' calculated in the and 8r which are related to the temperature derivatives of present work are compared, in Table 4, with macroscopic adiabatic and isothermal bulk moduli as follows[18], values estimated from thermoelastic data on the volume coefficient of thermal expansion, the bulk modulus, and 1 (~ In Bs'~ (14) the lattice contribution to the heat capacity at constant 8 s = - ~ \ OT ]o volume[6]. For the sake of comparison we have also included in Table 4 the values of Gruneisen parameters and, derived from alternative methods based on lattice 1 (01nBr~ (15) dynamical and finite strain theories[26]. It is seen from Table 4 that values of 3' obtained by different methods are of the same order as those calculated in the present where/3 is the volume thermal expansion coefficient. Bs and Br study directly from the analysis of interionic potentials. are, respectively, the adiabatic and isothermal bulk moduli. Several workers [1%22] have determined 8 by assuming that bulk The Born-Mayer exponential form yields values of 3' modulus is a function of volume only, i.e. by using the relation, which are about 10% smaller than the values obtained from the Born-Lande inverse power potential form. The Anderson parameters 8s and 8r, which define the _(OInBr~ 8 s = S r = 8 ~ \ 0 I n V/T (16) temperature derivatives of adiabatic and isothermal bulk
1448
JAI SHANKER and KAMENDRA SINGH
Table 3. Valuesof Poisson'sratioand their volumederivative Cry stal
/.iF
0.254
+ 0.167
- 0.36
-
0.81
LiOl
0.272
+ 0.24.2
- 0.52
-
1.25
LiBr
0.252
+ 0.130
- 0.28
-0.56
Lil
0.311
+ 0.511
- 1.13
- 3.09
NaF
0.174.
+ 0.14.7
- 0.30
-o.58
N aCl
0.164.
+ 0.067
- 0.1)+
- 0.26
NaBr
0.170
+ 0.110
- 0.23
-
Nal
0.175
+ 0.098
- 0.20
- 0-39
KF
0.151
+ 0.139
- 0.28
-0.52
KCI
0.101
- 0.055
+ 0.11
+ 0.19
EBr
0.118
+ 0.048
- 0.10
-
KI
0.061
- 0.190
+ 0.38
+ 0.61
I~F
0.161
- 0.015
+ 0.03
+0.06
RbCl
0.131
+ 0.109
- 0.22
- 0.39
1~Br
O.lO9
+ 0.034.
- 0.07
-
0.12
I~l
0.101
+ 0.0)+5
- 0.09
-
0.16
0.82
0.17
Table 4. Values of the Gruneisenparameter ~, ~ T Calculated |The Born-Lande Crystal ~ potential
from The Born-Mayer potential
Derived from Thermoelastlc data [6]
I Derived from | Lattice dyna/ mlcal theory J_.__..~25 ]
LIF LICl LiBr Lil
2.32 2.73 2.44 3.34
2.06 2.5O 2.19 3.08
1.63 1.81 1.94 2.19
2.48 2.38
NaT NaCI NaBr NaI
2.46 2.43 2.71 2.65
2.30 2.18 2.49 2.43
1.51 1.61 1.64 1.71
1.86 1.69 1-75 1.72
KF KC1 KBr K|
2.57 2.39 2.61 2.20
2.36 2.11 2.32 1.92
1.52 1.49 1.50 1 •53
1.71 I •59 1.6 9 1.27
RbF RbC1 RbBr RbI
2.43 2.77 2.67 2.73
2.21 2.50 2.40 2.45
1.40 1.39 1.42 1.56
1.66 1.45 1.37 1.32
moduli are calculated in the present study and compared in Table 5 with the corresponding values derived from experimental data. It is found that the Born-LaMe inverse power potential form yields better agreement with experiment for 8s and 8r of all the alkali halides under study.
I
Derived from Finite strain
2.66 2.74. 2.78
2.52 2.50
2.56
2.68 2.65 2.64.
To summarize, the analysis of Anderson and Gruneisen parameters performed in the present paper from the knowledge of interionic potentials is directly useful and related to thermoelastic, dielectric and anharmonic properties of solids [18-20, 27, 28] and is also useful in geophysical studies of earth's interior[29, 30].
1449
Gruneisen-Anderson parameters in alkali halides Table 5. Values of the Anderson parameters 8s and 8r
Crystal
GS
Bor'n-~de I Born-Mayer potential Potential form form
/.,iF LiCI LiBr Lil
2.81+
NaF
3.52 3.71+ 1+.22
NaCI
NaBr
3.81+
3.91 3.~
Nal
KF KCI KBr KI FJoF RbCI RbBr P~I
1+.18 1+.5o
~tperlmen t~l
S_71
2.36 3.38 3.1+0 2.87
3,56 1+.o9
3.20 3.25 3.75 3.95
3.75 3.80 1+.11 1+.13
3.7t~ 3.96
~-.08 1+.38
b.12 1+.o6
T Born.Lande Born -Mayer potentlal p o t e n t i a l form ,orm
E~p • r imen tal
6.00 6.77 7.O1 7.32
1+.98 5.96 6.09 5.76
1+,5o 5.50
5.29 5.41+ 5.93 6.16
~-.97 ~.95 5.~6 5.71
6 .~'3
5.89 6.03
5.~-5 5.~9
6.20 6.22
5.58 5.23
5.77
5.85 6.23
3.76
~+.o2
5.82
1+.00
3.b,5
3.93
5.~-7
5.2h h.92
5.88 5.76
5.15 5.0~
1+.53 t~.71 1+.61 1+.~+9
~-.97 h.93 4.72 k-. 1+7
6.36 6.68 6.58 6.57
5.91 6.13 6.0h6,02
6.80 6.76 6.60 6.5a
REFERENCES 1. K. P. Thakur, Y. lnorg. NucL Chem. 36, 2171 (1974). 2. K. P. Thakur and J. D. Pandey, J. lnorg. NucL Chem. 37, 645 (1975). 3. K. P. Thakur, J. Inorg. Nucl. Chem. 38, 1433(1976). 4. C. R. A. Catlow, K. M. Diller and M. J. Norgett, J. Phys. C 10, 1395 (1977). 5. M. J. L. Sangster, U. Schroder and R. M. Atwood, J. Phys. C 11, 1523(1978). 6. C. S. Smith and L. S. Cain, J. Phys. Chem. Solids 36, 205 (1975). 7. R. W. Roberts and C. S. Smith, J. Phys. Chem. Solids 31, 619, 2397 (1970); K. O. McLean and C. S. Smith, J. Phys. Chem. Solids 33, 279 (1972). 8. M. Born and A. Lande, Verhandl. Deut. Physik Ges. 20, 210 (1918). 9. M. Born and J. E. Mayer, Z. Physik 75, 1 (1932). 10. M. Born and K. Huang, Dynamical Theory o/Crystal Lattices. Clarendon Press, Oxford (1954). 11. M. P. Tosi, Solid-St. Phys. 16, 1 (1964). 12. L. Pauling, The Nature o/ Chemical Bond. Cornell University Press, New York (1960). 13. P. A. Sysio, Acta Crysta. B25, 2374 (1969). 14. J. Shanker, G. G. Agrawal and R. P. Singh, J. Chem. Phys. 69, 670 (1978). 15. J. P. Romain, A. Migault and J. Jacquesson, J. Phys. Chem.
Solids 38, 147 (1977); A. Migault and J. P. Romain, 3. Phys. Chem. Solids 38, 555 (1977). 16. J. C. Slater, An Introduction to Chemical Physics. McGrawHill, New York (1939). 17. J. S. Dugdale and D. K. C. MacDonald, Phys. Rev. 89, 832 (1953). 18. O. L. Anderson, Phys. Rev. 144, 553 (1966). 19. M. P. Madan, J. Chem. Phys. 55, 464 (1971);J. AppL Phys. 42, 3888 (1971). 20. M. N. Sharma and S. R. Tripathi, J. Phys. Chem. Solids 36, 45 (1975). 21. M. N. Sharma, Phys. Status Solidi (b) 82, K53 (1977). 22. V. K. Mathur, S. P. Singh and D. P. Viz, J. Chem. phys. 48, 4784 (1968). 23. A. P. Gupta and J. Shanker, Phys. Status Solidi (b) 95, KI03 (1979); J. Chem. Phys. 72, 2656 (1980). 24. J. Andzelm and L. Piela, J. Phys. Cl0, 2269 (1977). 25. H. H. Demarset Jr., J. Phys. Chem. Solids 35, 1393(1974). 26. V. V. Palciauskas, J. Phys. Chem. Solids 36, 611 (1975). 27. Y. A. Chang, J. Phys. Chem. Solids 28, 697 (1967). 28. R. Ruppin and R. W. Roberts, Phys. Rev. B3, 1406(1971);!14, 2041 (1971). 29. N. Soga, E. Schreiber and O. L. Anderson, J. Geophys. Res 71, 5315 (1966). 30. L. Thomsen and O. L. Anderson, J. Geophys. Res. 74, 981 (1969). 31. M. F. L. Ladd, J. Chem. Phys. 60, 1954(1974).
0022-1902/81/071445-05502.00/0 Pergamon Press Ltd.
ANALYSIS OF THE CRYSTAL BINDING AND THE GRUNEISEN-ANDERSON PARAMETERS IN ALKALI HALIDES JAI SHANKER and KAMENDRA SINGH Physics Department, Agra College, Agra-282002, India
(Received 27 July 1980; received/or publication 19 September 1980) Abstract--We have calculated the crystal binding energies and the Gruneisen-Anderson parameters in sixteen NaCI- structure alkali halides using the Born model formulation including the short range repulsive interactions between nearest neighbours and the next nearest neighbours, and the van tier Waals dipole--dipole and dipolequadrupole interactions. The overlap repulsive potential parameters have been calculated from the low temperature crystal data on interionic separation and compressibility data. Values of the Gruneisen parameter are calculated taking into account the volume dependence of Poisson's ratio. The Anderson parameters describing the temperature derivatives of adibatic and isothermal bulk moduli have also been calculated with the help of interionic potential models. Calculations have been performed using the two potential forms for short range repulsive interactions showing an inverse power dependence and exponential dependence on interionic distance. The results obtained are compared with experimental data and also with other theoretical studies. INTRODUCTION The evaluation of crystalline properties of inorganic solids from the knowledge of interionic potentials has recently been a subject of considerable interest in the field of crystal chemistry. Thakur and Pandey[l-3] have calculated a number of crystalline properties for diatomic halides and chalcogenide crystals. However, their method is not adequate for several reasons. First, they have not considered the repulsive interactions between like ions, i.e. positive-positive and negative-negative ion interactions and have completely neglected the van der Waals dipole--dipole and dipole--quadrupole interactions. However, recent studies [4, 5] have clearly demonstrated the importance of these interactions in determining adequate potentials for alkali halide crystals. In view of these studies, the potential energy expression used by Thakur and Pandey[1-3] is not completely satisfactory. In addition, the compressibility data used by Thakur[3] are old and less accurate than recent ultrasonic data [46]. The temperature dependence of bulk modulus at fixed volume, which is an important and significant quantity for alkali halides[7], has been completely ignored by Thakur[3] when evaluating the temperature dependence of compressibility at constant pressure. In the present study we remove these deficiencies by adopting the interionic potential model so as to take into account the van der Waals interactions and repulsive interactions between second neighbours and like ions. We use two potential forms for the short range repulsive energy showing an inverse power dependence[8] and exponential dependence[9] on interionic separation. The potential parameters are derived from the crystal equilibrium and compressibility data. We have used low temperature data in order to avoid uncertainties arising from assumptions concerning thermal contribution in the equation of state introduced by different workers[10,11]. We have calculated the cohesive energy, the Gruneisen parameter, and the Anderson parameters 8s and 8r for sixteen alkali halide crystals with NaCI structure. These quantities are quite useful in understanding the thermodynamic behaviour of inorganic compounds.
M E T H O D OF ANALYSIS
Cohesive energy. The cohesive energy of an ionic crystal can be calculated with the help of following expression for the potential energy[11], _
W=
C D ctme2+B(r) r6 p
(I)
t
where the first term on the right is the Madelung energy or the electrostatic energy, the second term represents the short range overlap repulsive energy, the last two terms are, respectively, the dipole--dipoleand dipole--quadrupole van der Waals energies. We have not included the zero point energy because it is negligibly small[ll]. If we adopt an inverse power form then the repulsive energy is [12],
B(r)=6A[[3+-(rr+r-)n-l'~-~ ~2+(2r+)"-1 -~ ~2_(2r_) - ~ j "-I ] (2) Here A and n are potential parameters, Bii are Pauling's coe$cients[12], r+ and r_ are ionic radii, r is cation-anion separation, ~/(2)r is cation--cation or anion-anion separation. By using exponential form we can write,
=6b[, +_exp(r,+ ,_-q + t ,
~
\
P
/
[2r_- V'(2)rkq
+~2_expt
7- :J
exp(r,-V(2),] \
P
/
(3)
where b and p are, respectively, the repulsive strength and hardness parameters. Thus, whether we use an inverse power from or an exponential form, there are two unknown parameters which are determined from the crystal equilibrium condition and isothermal bulk modulus, i.e. using the relations, ,4, and, dZW V ~ = Br
(5)
Values of Br used in the present calculations are those based on the measured elastic constants at helium temperature recently 1445
1446
JAI SHANKER and KAMENDRA SINGH Gruneisen parameters. Romain et al.[15] have recently developed a theory for the Gruneisen parameter by taking into account the pressure dependence of Poisson's ratio. They considered two Gruneisen parameters, yj and y,, corresponding to longitudinal and transverse vibrations. This model gives a more realistic representation of thermodynamic and thermoelastic behaviour of solids than the theories formulated by earlier
compiledby Catlow et al. [4]. Ionic radii are token from [13],and the vander Waalsdipole--dipoleand dipole-quadrupoleconstants Cand D from[14]. Calculated values of potential parameters for the inverse power form and the exponentialform are listed in Table 1. Using these parameters and equations (1)-(3) we have obtained cohesive energies.These are comparedwith experimentalvalues in Table 2.
Table 1. Calculated values of potential parameters Crystal
Inverse power (Born-Lande)
Form
Exponential Form (B o r n - M a y e r )
(1~-12erg.)
(10-21erg.cm)
]o8
(10- era)
LIF
6.83
5.98
O. 308
O. 303
LICI
7.24
7.15
0.253
0.342
LiBr
8.08
6.94
0.259
0.304
Lil
9.21
6.53
0.256
0.453
NaY
4.90
7.10
0.402
0.290
10,I
7.53
0.335
0.346
NaBr
1o.3
7.76
0.322
0.362
Nal
10.5
7.99
0.298
0.392
NaCI
KF
11.5
7.77
0,389
0,322
KCI
11.1
8.08
0.334
0.353
KBr
11.0
8.34
0.314
0.360
KI
11.9
7.92
o.313
o.410
3bF
1o.3
8.10
0.358
0.333
HhCI
11.0
8.66
o.314
0.345
l~Br
11.3
8.70
0.306
0.359
Hbl
11.3
8.28
0.287
o.381
Table 2. Values of cohesive energy (kcal/mole) I n v e r s e p o w e r form (B o r n - L a n de )
Crystal
Calculated in present study
ToKi ~1~
Exponential form (Born-Mayer) Calculated in present study
Experimental
t313
Tosi [11]
LiF
- 242.1
- 239.3
- 24-9.6
- 24-2.2
- 246.7
LICI
- 199.0
- 194.0
- 203.?
192.9
- 203.2
LiBr
- 185.5
- 183.3
- 190.3
Lil
- 168.3
- 165.7
- 174-.1
-
181.0
- 194.2
166.1
-
180.3
NaF
- 219.1
- 210.0
- 225.0
- 215.2
NaCI
-
183.6
- 180.9
- 187.8
- 178.6
- 187.1
NaBr
- 175.1
- 170.5
- 178.9
- 169.2
- 178.5
Nal
- 164.O
- 157.1
- 167.5
- 156.6
- 167.0
KF
- 194.6
- 187.1
- 198.7
- 189.1
- 194.3
KCI
- 168.0
- 162.9
- 171.8
- 161.6
- 170.2
KBr
- 162.3
- 157.0
- 165.9
- 154.5
- 163.2
KI
- 150.7
- 147.1
- 154.6
- 144.5
1
RbF
-
186.9
- 181.0
- 190.5
- 180.4
- 185.8
RbCI
-
163.6
- 158.8
- 166.9
- 155.4
- 163.6
RbBr
- 157.2
- 151.5
- 160.5
- 148.3
- 157.2
RDI
- 149.0
- 143.8
- 152.3
- 139.6
- 148.5
- 219.5
153.6
Gruneisen-Anderson parameters in alkali halides workers[16,17]. For yt and Yt we can write the following expressions[15],
yt
4 - 3pl Vd2PVP'/dV2 6 2 dPVP'/d V
(6)
Yt
4-3pz 6
V d2P Vm/d V2 2 dPVmld V
(7)
and,
where Pl and P2 are related to Poisson's ratio ~ as follows,
l+l_o./1
dlnV\ +P' a T H
1[ 01nB~
(17) (18)
In order to calculate the volume dependence of bulk modulus one can take the plausible assumption[21],
(9)
(10)
Values of Poisson's ratio and its volume derivatives are calculated from elastic constants Cll and C~2 and their pressure derivatives[4, 7] using the relation, Ct2 /7"= Ci I + Cl2.
OT Iv
(8)
where tro is Poisson's ratio at zero pressure. Values of pt can be determined from the volume derivative of ,7 which is given by,
(~V)vfvo_- -~-~o Pl (1-o" o2)
\OlnV] T fl\ = _ ( 0 In B.~
and, 3 1 - O'o P2=2 pt 1 +20"o
However, this assumption is not justified in view of the fact that the adiabatic bulk modulus Bs or the isothermal bulk modulus Br is not independent of temperature at fixed volume. In reality, the quantities (OBs/c~T)v and (OBr/OT)v are quite significant and have played important roles in describing the equation of state of alkali halide crystals[7]. Moreover, the sign of (OBslOT)v is different from that of (OB.flOT)v. Therefore it is inappropriate to take Bs to be equal to 8r. Correct expressions for 8s and 8r are [7],
8s
+P' d~-~. p) tr
1447
(11)
Calculated values of p~ and P2 along with Poisson's ratio are given in Table 3. In order to calculate Yt and ~'t we need dPld V and dZP]dV2. These quantities can be calculated from the equation of state[ll],
(~ In Bs~ ~ (0 In Br~ ~Tr = k~]~Vn V ]r'
(19)
The volume derivative of bulk modulus Br can be expressed, in terms of interionic potential energy and its derivatives, as follows[23], (~ In BT'~ I[r2W'-3rW"+4W ' ] ~--i-wV-n'Vl r =- 3 [ rW"-2 W' ]
(20)
where w is the potential energy expressed by eqn (1) w', w" and w" are, respectively, at first, second and third order derivatives of the potential energy with respect to interionic separation r. We have calculated these derivatives with the help of eqn (1) using the potential parameters given in Table 1. Finally we have determined 8s and 8r using eqns (17)-(20) and evaluating the explicit contributions, viz. the temperature derivatives of Bs and BT at constant volume from thermoelastic data[7].
RESULTS AND DISCUSSION
In the present study we have used an expression for dW (12) the potential energy which is more complete than that e=--dV used by Thakur and Pandey[1-3]. We have considered the van der Waals interactions and overlap repulsive where W is the potential energy expressed by equation (1). Using interactions between like ions which are quite important the potential parameters reported in Table 1 we have calculated as is evident from recent quantum mechanical[24] as yt and y, with the help of eqns (6), (7) and (12). Average values well as phenomenological studies [4, 5]. Calculated values of V are then determined from the relation[15], of cohesive energies are in better agreement with experimental data (Table 2) than those reported by 1 (13) Tosi[1]. It should also be noticed that the inverse power potential form yields better agreement with experimental Values of y thus calculated are compared with those obtained by data for K and Rb halides whereas the exponential other investigators in Table 4. potential form is better for Li and Na halides. Anderson paramaers. There are two Anderson parameters, 8s Values of the Gruneisen parameter 3' calculated in the and 8r which are related to the temperature derivatives of present work are compared, in Table 4, with macroscopic adiabatic and isothermal bulk moduli as follows[18], values estimated from thermoelastic data on the volume coefficient of thermal expansion, the bulk modulus, and 1 (~ In Bs'~ (14) the lattice contribution to the heat capacity at constant 8 s = - ~ \ OT ]o volume[6]. For the sake of comparison we have also included in Table 4 the values of Gruneisen parameters and, derived from alternative methods based on lattice 1 (01nBr~ (15) dynamical and finite strain theories[26]. It is seen from Table 4 that values of 3' obtained by different methods are of the same order as those calculated in the present where/3 is the volume thermal expansion coefficient. Bs and Br study directly from the analysis of interionic potentials. are, respectively, the adiabatic and isothermal bulk moduli. Several workers [1%22] have determined 8 by assuming that bulk The Born-Mayer exponential form yields values of 3' modulus is a function of volume only, i.e. by using the relation, which are about 10% smaller than the values obtained from the Born-Lande inverse power potential form. The Anderson parameters 8s and 8r, which define the _(OInBr~ 8 s = S r = 8 ~ \ 0 I n V/T (16) temperature derivatives of adiabatic and isothermal bulk
1448
JAI SHANKER and KAMENDRA SINGH
Table 3. Valuesof Poisson'sratioand their volumederivative Cry stal
/.iF
0.254
+ 0.167
- 0.36
-
0.81
LiOl
0.272
+ 0.24.2
- 0.52
-
1.25
LiBr
0.252
+ 0.130
- 0.28
-0.56
Lil
0.311
+ 0.511
- 1.13
- 3.09
NaF
0.174.
+ 0.14.7
- 0.30
-o.58
N aCl
0.164.
+ 0.067
- 0.1)+
- 0.26
NaBr
0.170
+ 0.110
- 0.23
-
Nal
0.175
+ 0.098
- 0.20
- 0-39
KF
0.151
+ 0.139
- 0.28
-0.52
KCI
0.101
- 0.055
+ 0.11
+ 0.19
EBr
0.118
+ 0.048
- 0.10
-
KI
0.061
- 0.190
+ 0.38
+ 0.61
I~F
0.161
- 0.015
+ 0.03
+0.06
RbCl
0.131
+ 0.109
- 0.22
- 0.39
1~Br
O.lO9
+ 0.034.
- 0.07
-
0.12
I~l
0.101
+ 0.0)+5
- 0.09
-
0.16
0.82
0.17
Table 4. Values of the Gruneisenparameter ~, ~ T Calculated |The Born-Lande Crystal ~ potential
from The Born-Mayer potential
Derived from Thermoelastlc data [6]
I Derived from | Lattice dyna/ mlcal theory J_.__..~25 ]
LIF LICl LiBr Lil
2.32 2.73 2.44 3.34
2.06 2.5O 2.19 3.08
1.63 1.81 1.94 2.19
2.48 2.38
NaT NaCI NaBr NaI
2.46 2.43 2.71 2.65
2.30 2.18 2.49 2.43
1.51 1.61 1.64 1.71
1.86 1.69 1-75 1.72
KF KC1 KBr K|
2.57 2.39 2.61 2.20
2.36 2.11 2.32 1.92
1.52 1.49 1.50 1 •53
1.71 I •59 1.6 9 1.27
RbF RbC1 RbBr RbI
2.43 2.77 2.67 2.73
2.21 2.50 2.40 2.45
1.40 1.39 1.42 1.56
1.66 1.45 1.37 1.32
moduli are calculated in the present study and compared in Table 5 with the corresponding values derived from experimental data. It is found that the Born-LaMe inverse power potential form yields better agreement with experiment for 8s and 8r of all the alkali halides under study.
I
Derived from Finite strain
2.66 2.74. 2.78
2.52 2.50
2.56
2.68 2.65 2.64.
To summarize, the analysis of Anderson and Gruneisen parameters performed in the present paper from the knowledge of interionic potentials is directly useful and related to thermoelastic, dielectric and anharmonic properties of solids [18-20, 27, 28] and is also useful in geophysical studies of earth's interior[29, 30].
1449
Gruneisen-Anderson parameters in alkali halides Table 5. Values of the Anderson parameters 8s and 8r
Crystal
GS
Bor'n-~de I Born-Mayer potential Potential form form
/.,iF LiCI LiBr Lil
2.81+
NaF
3.52 3.71+ 1+.22
NaCI
NaBr
3.81+
3.91 3.~
Nal
KF KCI KBr KI FJoF RbCI RbBr P~I
1+.18 1+.5o
~tperlmen t~l
S_71
2.36 3.38 3.1+0 2.87
3,56 1+.o9
3.20 3.25 3.75 3.95
3.75 3.80 1+.11 1+.13
3.7t~ 3.96
~-.08 1+.38
b.12 1+.o6
T Born.Lande Born -Mayer potentlal p o t e n t i a l form ,orm
E~p • r imen tal
6.00 6.77 7.O1 7.32
1+.98 5.96 6.09 5.76
1+,5o 5.50
5.29 5.41+ 5.93 6.16
~-.97 ~.95 5.~6 5.71
6 .~'3
5.89 6.03
5.~-5 5.~9
6.20 6.22
5.58 5.23
5.77
5.85 6.23
3.76
~+.o2
5.82
1+.00
3.b,5
3.93
5.~-7
5.2h h.92
5.88 5.76
5.15 5.0~
1+.53 t~.71 1+.61 1+.~+9
~-.97 h.93 4.72 k-. 1+7
6.36 6.68 6.58 6.57
5.91 6.13 6.0h6,02
6.80 6.76 6.60 6.5a
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