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Additivity rule for the softness parameter in alkali halides
Solid State Communications, Vol. 31, pp. 765-768. Pergamon Press Ltd. 1979. Printed in Great Britain. ADDITIVITY RULE FOR THE SOFTNESS PARAMETER IN ALKALI HALIDES Jai Shanker, D.P. Agrawal* and Ram Pal Singh* Department of Physics, Agra College, Agra 282002, India
(Received 15 February; in revised form 8 May 1979 by M.F. Collins) In the present communication we report an additivity rule for the repulsive potential softness parameter in alkali halides derived from the values of overlap integrals. This rule provides a new interpretation for the repulsive interactions between different ion pairs and reduces the ntrmber of parameters. The use of additivity rule eliminates the assumption that the anionanion, anion-cation and cation-cation softness parameters are equal. Using the new model for repulsive interactions we have calculated the cohesive energy and compressibility of sixteen alkali halides with NaC1 structure. The results obtained in the present study are in better agreement with experiment than those obtained earlier with the help of a traditional approach based on the Born model. THE BORN-MAYER EXPONENTIAL FORM B exp [-- (r/p)] for the repulsive potential between two ions separated by a distance r (B and p are respectively known as the strength and softness parameters) has been widely used for studying the properties of ionic crystals [ 1,2]. Apart from its success in reproducing the agreement with experiment, the Born-Mayer form has been found to be consistent with earlier quantum mechanical calculations [ 3, 4]. More recently Hafemeister and Zahrt [5] have produced a set of repulsive parameters p in alkali halides on the basis of the values of overlap integrals [6]. In the present communication it is shown that these values of p follow the additivity rule very nearly. Values ofo+ and p_ are obtained for alkali and halogen ions. In fact the additivity rule for the softness parameter is not a new concept. It was first proposed by Gilbert [7] on the basis of an ad hoc argument and empirical evidence from molecular data. In a subsequent paper Smith [ 8] provided a significant and instructive interpretation on the basis of his distortion model. The important conceptual contribution embodied in Smith's distortion model was rediscovered by Narayan and Ramaseshan [9] without giving proper credit to Smith's original work. There have been subsequent contributions [10-14] providing the verification of the additivity rule for softness parameter. In view of these recent investigations, it is desirable to obtain the softness parameters for individual ions from the data on overlap integrals [5, 6] using the additivity rule. Values of P based on the overlap integrals are given in the second column of Table 1. To derive the values of * Permanent address: Narain College, Shikohabad (U.P.), India. 765
P for individual ions using the additivity rule, we have to know p at least for one ion. We take P+ = p++/2 for Li ÷ ion from Hafemeister and Flygare [6]. This choice is preferable because O. for Li + is smallest and any uncertainty in it would not largely affect the values for other ions. Values of p+ or p_ for alkali and halogen ions derived from the addit!vity rule p = p+ + p_ are given in Table 2. These are foufid to compare well with the values based on molecular spectroscopic data [7]. Strictly speaking, the repulsive potential in ionic solids is of quantum mechanical origin. The rigorous treatment based on quantum mechanics is, however, too elaborate to allow a simple analysis that is of practical importance. This is the reason that in the traditional Born model, one determines the potential parameters in an empirical fashion [ 1]. Recent workers [8-11 ] have presented a new approach based on the concept of an elastic ion. According to this approach, the repulsive potential energy between the ions i and j is specified by the nature of each ion as (Pi + Ps') exp [(ri + rj -- r)/(p i + pj)]
(1)
as a function of the interionic distance r. In expression (1), r i and rj are ionic radii. Following the form given by (1) we can write the repulsive energy upto next nearest neighbours as follows q~(r) = M~+_bp÷_exp{r ÷ +} p r -+- _r ~ '
+ ~M ~++bp++exp + ~M'~_ _bp__
{2r÷~+~/(2r)}
__
(2)
766
ADDITIVITY RULE FOR THE SOFTNESS PARAMETER IN ALKALI HALIDES Vol. 31, No. 10
Table 1. Values o f the rePUlsive hardness parameter p (in A)
solid is expressed as
Crystal
pa
oh+_
Oh++
pb _
W -
LiF LiC1 LiBr LiI
0.256 0.302 0.321 0.341
0.256 0.302 0.321 0.341
0.110 0.110 0.110 0.110
0.402 0.494 0.532 0.572
and bulk modulus I~T by the relation
NaF NaC1 NaBr NaI
0.264 0.306 0.323 0.346
0.261 0.307 0.326 0.346
0.120 0.120 0.120 0.120
0.402 0.494 0.532 0.572
KF KC1 KBr KI
0.294 0.329 0.340 0.360
0.282 0.328 0.347 0.367
0.162 0.162 0.162 0,162
0.402 0.494 0.532 0.572
RbF RbC1 RbBr Rbl
0.306 0.341 0.353 0,371
0.294 0.340 0.359 0.379
0.186 0.186 0.186 0.186
0.402 0.494 0.532 0.572
" Based on overlap integrals [5]. b Derived from the values given in Table 2 using the relation Pij : Pi + Pj.
{3T -
~m e: r ~- q~(r)
W" 18r
C r6
D rS
(3)
1 dZW 18r d r : "
(4)
The parameter b appearing in (2) has been determined from the crystal equilibrium condition. Values of cohesive energy and bulk modulus for sixteen alkali halides calculated from equations (2--4) are given in Tables 3 and 4 along with experimental data. For the sake of contrast we have also included in Tables 3 and 4, the corresponding values derived from the traditional method based on the Born model. It is seen from there that the results obtained in the present study are in better argeement with experiment than those obtained from the traditional method. This demonstrates the utility of the additivity rule for repulsive hardness parameter and the superiority of the way in which the second neighbour interactions are taken into account. The fact that one is able to get good agreement with experiment using softness parameters determined
Table 2. Values o f p in (A) for alkali and halogen ions Li ÷
Na +
K÷
Rb ÷
Cs ÷
F-
C1-
Br-
I-
1. Present study
0.055
0.060
0.081
0.093
0.115
0.201
0.247
0.266
0.286
2. Based on molecular spectroscopic data [7]
0.069
0.079
0.106
0.115
0.130
0.179
0.238
0.258
0.289
where M and M' are the numbers of nearest neighbours and next nearest neighbours, respectively, b is the strength parameter. 13ij are Pauling's coefficients. It should be remarked that when O+- = P++ = P - - = O equation (2) takes the form traditionally adopted in the Born model [ 1]. In fact, the assumption O++ = P - implied in previous studies is not reasonable since the anions are much more compressible than the cations (Table 2). Recent ab initio calculations [15, 16] also reveal that the interaction between anions is much stronger than the interaction between cations thereby invalidating the above assumption used fi] most previous work. An exception is the work of Decker [17] who took different values for O+÷ and p__. It will be interesting to investigate how far the situation is improved if we adopt the new form of the repulsive potential expressed by (2) to calculate the crystal energy and compressibility, using the values of Pij based on the additivity rule. The crystal energy of an ionic
from overlap calculations should stimulate someone to make an empirical re-determination of the 9 atomic parameters Pi directly from the crystal data. This should lead to an even better agreement with experiment and eliminate the rather large errors for fluorides shown in Tables 3 and 4. An especially significant follow-up which this work should stimulate would be separate empirical determinations of Oi from crystal and molecular data. A comparison of these two determinations would allow one to compare the repulsive forces in diatomic molecules and crystals and provide useful data on the dependence of repulsive forces on the molecular environment.
Vol. 31, No. 10 ADDITIVITY RULE FOR THE SOFTNESS PARAMETER IN ALKALI HALIDES
Table 3. Values o f cohesive energy (in kcal tool -~)
Crystal
Calculated in the present study
Calculated from the traditional Born model
Experimental values [18]
LiF LiCI LiBr LiI
252.7 201.6 189.1 174.8
242.2 192.9 181.0 166.1
246.7 203.2 194.2 180.3
NaF NaC1 NaBr NaI
225.5 187.4 177.5 165.1
215.2 178.6 169.2 156.6
219.5 187.1 178.5 167.0
KF KC1 KBr KI
199.9 171.5 163.9 154.3
189.1 161.6 154.5 144.5
194.3 170.2 163.2 153.6
RbF RbCI RbBr Rb I
190.7 165.0 158.1 147.3
180.4 155.4 148.3 139.6
185.8 163.6 157.2 148.5
Table 4. Values o f bulk modulus (1012 dyn cm -2)
Crystal
Calculated in the present study
Calculated from the traditional method P++ = P+- = P - -
Experimental values [2]
LiF IAC1 LiBr LiI
0.801 0.339 0.261 0.192
0.868 0.412 0.339 0.274
0.674 0.301 0.238 0.175
NaF NaC1 NaBr NaI
0.567 0.275 0.221 O. 166
0.589 0.308 0.258 0.208
0.471 0.240 0.197 0.151
KF KC1 KBr KI
0.386 0.205 0.170 O. 134
0.379 0.213 0.186 0.151
0.306 0.176 0.148 0.117
RbF RbC1 Rb Br RbI
0.338 0.179 O. 149 0.113
0.326 0.184 0.159 0.123
0.270 0.158 0.134 0.106
767
768
ADDITIVITY RULE FOR THE SOFTNESS PARAMETER IN ALKALI HALIDES Vol. 31, No. 10 REFERENCES
1. 2. 3. 4. 5. 6. 7. 8. 9.
M.P. Tosi, Solid State Physics 16, 1 (1964). C.S. Smith & L.S. Cain, J. Phys. Chem. Solids 36, 205 (1975). W. Heitler & F. London, Z. Phys. 44, 455 0927). W.E. Bleick & J.E. Mayer, J. Chem. Phys. 2, 252 (1934). D.W. Hafemeister & J.D. Zahrt, J. Chem. Phys. 47, 1428 (1967). D.W. Hafemeister & W.H. Flygare, J. Chem. Phys. 43,795 (1965). T.L. Gilbert, J. Chem. Phys. 49, 2640 (1968). F.T. Smith, Phys. Rev. AS, 1708 (1972). R. Narayan & S. Ramaseshan, J. Phys. Chem. Solids 37,395 (1976).
10. 11. 12. 13. 14. 15. 16. 17. 18.
T.L. Gilbert, O.C. Simpson & M.A. Williamson, J. Chem. Phys. 63, 4061 (1975). Y. Ida, Phys. Earth Planet lnter. 13,97 (1976). D.H. Current, C.L. Foiles & E.H. Carlson, Phys. Rev. B6,737 (1972). E.H. Carlson, J. Chem. Phys. 58, 1905 (1973). P. Brumer,Phys. Rev. A10, 1 (1974). J. Andzelm & L. Piela, J. Phys. C: Solid State Phys. 10, 2269 (1977). J.L. Calais, K. Mansikka, G. Petterson & J. Vallin, Ark. Fys. 34, 361 (1967). D.L. Decker,J. Appl. Phys. 36,157 (1965). M.F.C. Ladd,J. Chem. Phys. 60, 1954 (1974).
(Received 15 February; in revised form 8 May 1979 by M.F. Collins) In the present communication we report an additivity rule for the repulsive potential softness parameter in alkali halides derived from the values of overlap integrals. This rule provides a new interpretation for the repulsive interactions between different ion pairs and reduces the ntrmber of parameters. The use of additivity rule eliminates the assumption that the anionanion, anion-cation and cation-cation softness parameters are equal. Using the new model for repulsive interactions we have calculated the cohesive energy and compressibility of sixteen alkali halides with NaC1 structure. The results obtained in the present study are in better agreement with experiment than those obtained earlier with the help of a traditional approach based on the Born model. THE BORN-MAYER EXPONENTIAL FORM B exp [-- (r/p)] for the repulsive potential between two ions separated by a distance r (B and p are respectively known as the strength and softness parameters) has been widely used for studying the properties of ionic crystals [ 1,2]. Apart from its success in reproducing the agreement with experiment, the Born-Mayer form has been found to be consistent with earlier quantum mechanical calculations [ 3, 4]. More recently Hafemeister and Zahrt [5] have produced a set of repulsive parameters p in alkali halides on the basis of the values of overlap integrals [6]. In the present communication it is shown that these values of p follow the additivity rule very nearly. Values ofo+ and p_ are obtained for alkali and halogen ions. In fact the additivity rule for the softness parameter is not a new concept. It was first proposed by Gilbert [7] on the basis of an ad hoc argument and empirical evidence from molecular data. In a subsequent paper Smith [ 8] provided a significant and instructive interpretation on the basis of his distortion model. The important conceptual contribution embodied in Smith's distortion model was rediscovered by Narayan and Ramaseshan [9] without giving proper credit to Smith's original work. There have been subsequent contributions [10-14] providing the verification of the additivity rule for softness parameter. In view of these recent investigations, it is desirable to obtain the softness parameters for individual ions from the data on overlap integrals [5, 6] using the additivity rule. Values of P based on the overlap integrals are given in the second column of Table 1. To derive the values of * Permanent address: Narain College, Shikohabad (U.P.), India. 765
P for individual ions using the additivity rule, we have to know p at least for one ion. We take P+ = p++/2 for Li ÷ ion from Hafemeister and Flygare [6]. This choice is preferable because O. for Li + is smallest and any uncertainty in it would not largely affect the values for other ions. Values of p+ or p_ for alkali and halogen ions derived from the addit!vity rule p = p+ + p_ are given in Table 2. These are foufid to compare well with the values based on molecular spectroscopic data [7]. Strictly speaking, the repulsive potential in ionic solids is of quantum mechanical origin. The rigorous treatment based on quantum mechanics is, however, too elaborate to allow a simple analysis that is of practical importance. This is the reason that in the traditional Born model, one determines the potential parameters in an empirical fashion [ 1]. Recent workers [8-11 ] have presented a new approach based on the concept of an elastic ion. According to this approach, the repulsive potential energy between the ions i and j is specified by the nature of each ion as (Pi + Ps') exp [(ri + rj -- r)/(p i + pj)]
(1)
as a function of the interionic distance r. In expression (1), r i and rj are ionic radii. Following the form given by (1) we can write the repulsive energy upto next nearest neighbours as follows q~(r) = M~+_bp÷_exp{r ÷ +} p r -+- _r ~ '
+ ~M ~++bp++exp + ~M'~_ _bp__
{2r÷~+~/(2r)}
__
(2)
766
ADDITIVITY RULE FOR THE SOFTNESS PARAMETER IN ALKALI HALIDES Vol. 31, No. 10
Table 1. Values o f the rePUlsive hardness parameter p (in A)
solid is expressed as
Crystal
pa
oh+_
Oh++
pb _
W -
LiF LiC1 LiBr LiI
0.256 0.302 0.321 0.341
0.256 0.302 0.321 0.341
0.110 0.110 0.110 0.110
0.402 0.494 0.532 0.572
and bulk modulus I~T by the relation
NaF NaC1 NaBr NaI
0.264 0.306 0.323 0.346
0.261 0.307 0.326 0.346
0.120 0.120 0.120 0.120
0.402 0.494 0.532 0.572
KF KC1 KBr KI
0.294 0.329 0.340 0.360
0.282 0.328 0.347 0.367
0.162 0.162 0.162 0,162
0.402 0.494 0.532 0.572
RbF RbC1 RbBr Rbl
0.306 0.341 0.353 0,371
0.294 0.340 0.359 0.379
0.186 0.186 0.186 0.186
0.402 0.494 0.532 0.572
" Based on overlap integrals [5]. b Derived from the values given in Table 2 using the relation Pij : Pi + Pj.
{3T -
~m e: r ~- q~(r)
W" 18r
C r6
D rS
(3)
1 dZW 18r d r : "
(4)
The parameter b appearing in (2) has been determined from the crystal equilibrium condition. Values of cohesive energy and bulk modulus for sixteen alkali halides calculated from equations (2--4) are given in Tables 3 and 4 along with experimental data. For the sake of contrast we have also included in Tables 3 and 4, the corresponding values derived from the traditional method based on the Born model. It is seen from there that the results obtained in the present study are in better argeement with experiment than those obtained from the traditional method. This demonstrates the utility of the additivity rule for repulsive hardness parameter and the superiority of the way in which the second neighbour interactions are taken into account. The fact that one is able to get good agreement with experiment using softness parameters determined
Table 2. Values o f p in (A) for alkali and halogen ions Li ÷
Na +
K÷
Rb ÷
Cs ÷
F-
C1-
Br-
I-
1. Present study
0.055
0.060
0.081
0.093
0.115
0.201
0.247
0.266
0.286
2. Based on molecular spectroscopic data [7]
0.069
0.079
0.106
0.115
0.130
0.179
0.238
0.258
0.289
where M and M' are the numbers of nearest neighbours and next nearest neighbours, respectively, b is the strength parameter. 13ij are Pauling's coefficients. It should be remarked that when O+- = P++ = P - - = O equation (2) takes the form traditionally adopted in the Born model [ 1]. In fact, the assumption O++ = P - implied in previous studies is not reasonable since the anions are much more compressible than the cations (Table 2). Recent ab initio calculations [15, 16] also reveal that the interaction between anions is much stronger than the interaction between cations thereby invalidating the above assumption used fi] most previous work. An exception is the work of Decker [17] who took different values for O+÷ and p__. It will be interesting to investigate how far the situation is improved if we adopt the new form of the repulsive potential expressed by (2) to calculate the crystal energy and compressibility, using the values of Pij based on the additivity rule. The crystal energy of an ionic
from overlap calculations should stimulate someone to make an empirical re-determination of the 9 atomic parameters Pi directly from the crystal data. This should lead to an even better agreement with experiment and eliminate the rather large errors for fluorides shown in Tables 3 and 4. An especially significant follow-up which this work should stimulate would be separate empirical determinations of Oi from crystal and molecular data. A comparison of these two determinations would allow one to compare the repulsive forces in diatomic molecules and crystals and provide useful data on the dependence of repulsive forces on the molecular environment.
Vol. 31, No. 10 ADDITIVITY RULE FOR THE SOFTNESS PARAMETER IN ALKALI HALIDES
Table 3. Values o f cohesive energy (in kcal tool -~)
Crystal
Calculated in the present study
Calculated from the traditional Born model
Experimental values [18]
LiF LiCI LiBr LiI
252.7 201.6 189.1 174.8
242.2 192.9 181.0 166.1
246.7 203.2 194.2 180.3
NaF NaC1 NaBr NaI
225.5 187.4 177.5 165.1
215.2 178.6 169.2 156.6
219.5 187.1 178.5 167.0
KF KC1 KBr KI
199.9 171.5 163.9 154.3
189.1 161.6 154.5 144.5
194.3 170.2 163.2 153.6
RbF RbCI RbBr Rb I
190.7 165.0 158.1 147.3
180.4 155.4 148.3 139.6
185.8 163.6 157.2 148.5
Table 4. Values o f bulk modulus (1012 dyn cm -2)
Crystal
Calculated in the present study
Calculated from the traditional method P++ = P+- = P - -
Experimental values [2]
LiF IAC1 LiBr LiI
0.801 0.339 0.261 0.192
0.868 0.412 0.339 0.274
0.674 0.301 0.238 0.175
NaF NaC1 NaBr NaI
0.567 0.275 0.221 O. 166
0.589 0.308 0.258 0.208
0.471 0.240 0.197 0.151
KF KC1 KBr KI
0.386 0.205 0.170 O. 134
0.379 0.213 0.186 0.151
0.306 0.176 0.148 0.117
RbF RbC1 Rb Br RbI
0.338 0.179 O. 149 0.113
0.326 0.184 0.159 0.123
0.270 0.158 0.134 0.106
767
768
ADDITIVITY RULE FOR THE SOFTNESS PARAMETER IN ALKALI HALIDES Vol. 31, No. 10 REFERENCES
1. 2. 3. 4. 5. 6. 7. 8. 9.
M.P. Tosi, Solid State Physics 16, 1 (1964). C.S. Smith & L.S. Cain, J. Phys. Chem. Solids 36, 205 (1975). W. Heitler & F. London, Z. Phys. 44, 455 0927). W.E. Bleick & J.E. Mayer, J. Chem. Phys. 2, 252 (1934). D.W. Hafemeister & J.D. Zahrt, J. Chem. Phys. 47, 1428 (1967). D.W. Hafemeister & W.H. Flygare, J. Chem. Phys. 43,795 (1965). T.L. Gilbert, J. Chem. Phys. 49, 2640 (1968). F.T. Smith, Phys. Rev. AS, 1708 (1972). R. Narayan & S. Ramaseshan, J. Phys. Chem. Solids 37,395 (1976).
10. 11. 12. 13. 14. 15. 16. 17. 18.
T.L. Gilbert, O.C. Simpson & M.A. Williamson, J. Chem. Phys. 63, 4061 (1975). Y. Ida, Phys. Earth Planet lnter. 13,97 (1976). D.H. Current, C.L. Foiles & E.H. Carlson, Phys. Rev. B6,737 (1972). E.H. Carlson, J. Chem. Phys. 58, 1905 (1973). P. Brumer,Phys. Rev. A10, 1 (1974). J. Andzelm & L. Piela, J. Phys. C: Solid State Phys. 10, 2269 (1977). J.L. Calais, K. Mansikka, G. Petterson & J. Vallin, Ark. Fys. 34, 361 (1967). D.L. Decker,J. Appl. Phys. 36,157 (1965). M.F.C. Ladd,J. Chem. Phys. 60, 1954 (1974).