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site approximation for multicomponent solutions — A practical alternative to the cluster variation method
Pergamon PI1 S1359-6462(%)00198-4
Scripta Materialia, Vol. 35, No. 5, pp. 623-627,1996 Elsevia Science Ltd Copyright 8 1996 Acta Metallurgica Inc. Printed in the USA. All rights resmed 1359~6462% $12.00 + .oo
THE CLUSTER/SITE APPROXIMATION FOR MULTICOMPONENT SOLUTIONS - A PRACTICAL ALTERNATIVE TO THE CLUSTER VARIATION METHOD W.A. Oates and H. Wenzl, Institut fir Festkijrperforschung, Forschungszentrum Jiilich GmbH, 52425 Jiilich, Federal Republic of Germany. (Received April 10,1996)
Introduction In the calculation of phase diagrams the thermodynamic models used for the phases involved are usually based on the Ising or bond energy model. A good discussion of the mapping of the energy of real alloys onto effective pair interactions for use in this model has been given by Ceder [ 11. In using the bond energy model it is often desirable to go beyond the point approximation in the calculation of the free energy and it has been recognised that Kikuchi’s Cluster Variation Method (CVM) [2] is highly satisfactory for this purpose. Good agreement is found when the results from CVM calculations are compared with the (exact) results from Monte Carlo simulations of the same system. The calculation of phase diagrams via first principle calculations of the energy of various ordered phases, the use of these energies in a cluster expansion for obtaining the effective cluster interactions and their subsequent use in a CVM calculation of the phase diagram has been a particularly exciting development over the last decade (for an excellent review see [3]). Notwithstanding these successes, a major disadvantage of the CVM has been recognised - the large number of non-linear simultaneous equations which must be solved in order to obtain the free energy and the equilibrium species distribution. For ordered phases of lowest symmetry the number of equations when using thie CVM totals k” - 1, where k is the number of components and n the number of atoms in the considered cluster. Some reduction is possible if one is only interested in the higher symmetry disordered phase. Kikuchi has used the cluster probabilities [4] in the free energy functional whereas others have favoured the use of multisite correlation functions [5]. Unfortunately, no reduction in the number of equations results from this different choice of variables. The sometimes excessive number of independent variables puts a limitation on the maximum sized cluster which can be considered for binary solutions and effectively eliminates anything higher than ternary systems from consideration in even the tetrahedron approximation. Real alloy systems of interest can, however, sometimes involve as many as ten components and even the simplest semiconductor alloy system can involve this many variables since there it is usually necessary to consider different charge states of a particular defect. In the present note, we should like to suggest the use of an alternative cluster approach which results in a considera‘ble reduction in the number of equations to be solved as compared with the CVM. Although 623
624
THE CLUSTER/SITE APPROXIMATION
the approximation is not new, its relevance to calculations on multicomponent been overlooked in the excitement generated by the success of the CVM. The Cluster/Site
Vol. 35, No. 5
solutions
appears to have
Approximation
The clusters of sites in a solid crystal which are used in a CVM entropy calculation are not isolated but, in general, share faces, edges and corners. In calculating the entropy of mixing, allowance is made for this sharing and this results in several terms in the entropy expression. This is illustrated in the following CVM tetrahedron approximation expressions for the configurational entropy in fee and bee lattices [6]: fee :
S
=
2St -5S,
bee :
s
=
6St - 12&r + 3S,2 + 4S,t - S,
+6S,
(1) (2)
where the subscripts refer to the tetrahedron (t), triangle (tr), pair(p) and site (s) contributions, respectively. The tetrahedron in the bee lattice, unlike the one in the fee lattice, is asymmetric and involves considering both nearest (pl) and next nearest (~2) neighbour interactions. Two term entropy expressions are obtained if one considers the mixing of clusters which are not permitted to share edges or bonds. For both fee and bee lattices the configurational entropy expression is given by [7]: s = St - 3s, (3) where the site entropy term, Ss, term comes from the normalisation of the partition function. The significance of using such a cluster/site approximation (CSA)* for the configurational entropy rather than the CVM is profound - the number of equations to be solved can be reduced from the order of k” to the order of k x n, e.g., lo4 --+ 10 x 4, thereby offering the possibility of considering larger clusters and/or more components. The large reduction in the number of equations is achieved by using the the generalised quasi-chemical method (GQCM), introduced many years ago by Fowler for treating equilibria in molecular gaseous assemblies [8] and subsequently used for clusters in solid solutions by Yang and Li [9, 10, 11,7]. By using the GQCM the problem is reduced to one of considering site (E atom ) probabilites rather than cluster (= molecule) probabilities. The essence of the CSA can be illustrated by considering a binary A-B system. A cluster, (ai), is represented according to the occupation of the position number i in the cluster, e.g., { 1100) for the AABB tetrahedron cluster. A function 4 is then defined:
where the site parameters pi in the set {pi] are set equal to unity if the atom in the cluster is a B atom. Thus, for our AABB cluster, {pi} becomes /~1~2. x((ai)) is the energy of the cluster {oi) and p = l/k~T. The importance of the site parameters is apparent from the fundamental assumption of the GQCM, which is that the average cluster probability at equilibrium, ({q }) is related to (pi } by: (Ioi 1) = ti
4
eXP(-BX((ai
1))
With this assumption we see that the cluster probabilities are obtained directly from the site parameters and it can be shown [lo] that these are uniquely defined by the relation:
Pi%
=aidJ
(4) fit
(5)
1
*We have preferred the description cluster/site approximation to any description which includes the term quasi-chemical since the latter is used in many different contexts.
Vol. 35, No. 5
THE CLUSTER/SITE APPROXIMATION
625
where ai is the species fraction of A atoms on the sublattice when the A atom resides in cluster position i. The molar configurational free energy, F,, is also related to the site parameters:
(6) where z/2p is the number of bonds per site in the lattice relative to the number per site in the cluster, and fi and xi are the sublattice fractions and species fractions on sublattice i, respectively. Minimisation of the free energy subject to a mass balance constraint and the use of Eqns. (4) is sufficient to determine thie four values (in the tetrahedron approximation) of ai and the four values of WI. In the highly symmetric disordered phase, where all the ui are equal to the mole fraction, there is only one pi to solve for in the binary solution. The CVM and CSA approximations have been compared previously [12, 131. Guggenheim and McGlashan [ 121 showed that the differences in the calculated critical temperatures are small but were unable to decide which was the better approximation, Monte Carlo results being unavailable at that time. Subsequently, Kikuchi and Sato [ 131 demonstrated the superiority of the CVM over the CSA for the same sized cluster. It should be emphasised that these comparisons, which were for binary solutions, showed that the free energy differences are not large and, importantly, the CSA results are not qualitatively different from the CVM ones in regard to predicting the order of phase transitions (unlike the point and pair approximations). The small differences in free energy calculated by different approximations are illustrated for a f.c.c. lattice in Fig. (1) for an exchange energy, WAB = EAB - ~(CAA + EBB) = -kT. This value of WAB is equivalent to a temperature only slightly above the critical and thus represents a substantial short range ordering. It can be seen that the free energy calculated from the two tetrahedron approximations differ little from one another but differ substantially from the values calculated in the point and pair approximations. A slightly higher fraction of AB pairing in the CSA as compared with the CVM is responsible for the more negative free energy but, again, the difference is small compared with those calculated from the point and pair approximations. The latter underestimates the number of AB bonds and it is well known that the pair approximation overestimates their number. A comparison of calculated phase diagrams is a more stringent test of the different approximations than comparisons of the single phase thermodynamic properties since this involves the small differences in free energy for pairs of phases. A CSA calculation of the fee, tetrahedron approximation phase diagram by
0
-1
5 A P
-2
E W -3 I Ii -4
ad---
/
0.0
0.1
I
I-----l-----
0.2
0.3
0.4
0.5
Mole Fraction Figure 1. Comparison of the molar free energy of mixing for a fee lattice as calculated by the point, pair and tetrahedron approximations (CVM and CSA) for a pair exchange energy Of WABkT = -1.
Pair
----r-0.0
0.1
0.2
0.3
0.4
0.5
Mole Fraction B
Figure 2. Calculation of the free energies of mixing for a fee lattice as calculated by the point, pair and tetrahedron approximations (CSA anad CVM) in the ternary system A-B-C along the AB-AC join. (WAB/kT = WAC/kT =WBC/kT = - 1).
626
THE CLUSTER/SITE APPROXIMATION
Vol. 35, No. 5
-I
0.1
0.2 Mole
0.4
0.3 Fraction
0.5
B
Figure 3. Comparison of the concentrations of the tetrahedron cluster {AABC} as calculated by the point and tetrahedron approximations (CSA and CVM) in the ternary system A-B-C along the AB-AC join. WAB/kT = WAC/kT = WBC/kT =- 1).
Li [ 1 l] failed to show the low temperature stability of the A3B phase relative to the disordered phase over a wide range of compositions, as is found using the CVM approximation [ 141. The topology of the CSA phase diagram is, however, a considerable improvement over that calculated by Shockley [ 151 in the point approximation and it should be recalled that there is no phase diagram in the pair approximation since the disordered phase is always the most stable [ 161. We have used clusters larger than tetrahedrons in the CSA approximation and find that the phase diagram then starts to have the same topology as that calculated by the CVM. A comparison of the different approximations for a ternary system is given in Figs. (2) and (3). In Fig. (2) we show the free energy calculated for a disordered ternary phase, A-B-C, along the AC-BC join, for the interaction energies given in the caption. The differences between the CVM and CSA results are again small but both differ substantially from the point and pair approximation results. In the CSA calculation it is necessary to solve for only two site parameters pi. This may be compared with the CVM where the same calculation requires solving for 14 cluster probabilities. As indicated earlier, this advantage of the CSA over the CVM becomes dramatically more marked as the number of components and/or the size of the cluster are increased. In Fig.(3) we show the concentrations of a particular tetrahedron cluster, {AABC}, along the same composition join and it can be seen that the short range order as calculated in the CSA and CVM approximations agree quite well. Conclusions The free energy calculated by the CSA is slightly less accurate than that calculated by the CVM for the same sized cluster. Even in the tetrahedron approximation it is, however, considerably better than both the pair and point approximations in that the free energy is close to that calculated by the CVM and the order of phase transitions is correctly predicted. The CSA has the big advantage over the CVM in the calculation of the free energy for multicomponent solutions and/or larger clusters in systems with a small number of components in the drastically reduced number of equations to bc solved. It should be remembered that, for those interested in the calculation of multicomponent phase diagrams for real systems, some empiricism must always be introduced into the representation of the free energy of a phase. That is to say, no matter how good the physical modelling, some adjustment of the model parameters is necessary to obtain satisfactory agreement with the real system thermodynamics. One does not know, for example, whether one should retain only the binary exchange energies, Wij , or introduce some ternary, Wijk, or even higher order interactions. Thus some slightly less accurate approximation than the CVM is
Vol. 35, No. 5
THE CLUSTER/SITE
627
APPROXIMATION
acceptable if it has the major advantage of a reduced number of independent the free energy functional. The CSA fulfds this requirement.
variables for evaluation
of
Acknowledgment W.A.O. is grateful for the hospitality
extended by the Forschungszentrum
Jiilich GmbH, Germany.
References 1. G. Ceder, Camp. Mat. Sci. 1, 144 (1993). 2. R. Kikuchi, Phys. Rev. 81,988 (1951). 3. D. de Fontai.ne, in Solid State Physics, edited by D. Turnbull, volume 47, pages 33-176, New York, 1994, Academic Press. 4. R. Kikuchi, J. Chem. Phys. 60, 1071 (1974). 5. J. M. Sanchtz and D. de Fontaine, Phys. Rev. B 17,2926 (1978). 6. G. Inden and W. Pitsch, Atomic ordering, in Materials Science and Technology, edited by R. W. Cahn, P Haasen, and E. J. Kramer, volume 5, Phase Transformations in Materials, pages 499-551, Weinheim, Germany, 1991, VCH. 7. Y. Li, Phys. Rev. 76,972 (1949). 8. R. H. Fowler, Statistical Mechanics, pages 162-164, Cambridge University Press, Cambridge, U.K., 2nd edition, 1938. 9. C. N. Yang, J. Chem. Phys. 13.66 (1945). 10. C. N. Yang and Y. Li, Chinese Journal of Physics 7,59 (1947). 11. Y. Li, J. Chem. Phys. 17,447 (1949). 12. E. A. Guggenheim and M. L. McGlashan, Molec. Phys. 57,433 (1962). 13. R. Kikuchi and H. Sate, Acta Met. 22, 1099 (1974). 14. C. M. van Baal, Physica64,571(1973). 15. W. Shockley, J. Chem. Phys. 6, 130 (1938). 16. R. Peirls, Proc. Roy. Sot. A 154,207 (1936).
Scripta Materialia, Vol. 35, No. 5, pp. 623-627,1996 Elsevia Science Ltd Copyright 8 1996 Acta Metallurgica Inc. Printed in the USA. All rights resmed 1359~6462% $12.00 + .oo
THE CLUSTER/SITE APPROXIMATION FOR MULTICOMPONENT SOLUTIONS - A PRACTICAL ALTERNATIVE TO THE CLUSTER VARIATION METHOD W.A. Oates and H. Wenzl, Institut fir Festkijrperforschung, Forschungszentrum Jiilich GmbH, 52425 Jiilich, Federal Republic of Germany. (Received April 10,1996)
Introduction In the calculation of phase diagrams the thermodynamic models used for the phases involved are usually based on the Ising or bond energy model. A good discussion of the mapping of the energy of real alloys onto effective pair interactions for use in this model has been given by Ceder [ 11. In using the bond energy model it is often desirable to go beyond the point approximation in the calculation of the free energy and it has been recognised that Kikuchi’s Cluster Variation Method (CVM) [2] is highly satisfactory for this purpose. Good agreement is found when the results from CVM calculations are compared with the (exact) results from Monte Carlo simulations of the same system. The calculation of phase diagrams via first principle calculations of the energy of various ordered phases, the use of these energies in a cluster expansion for obtaining the effective cluster interactions and their subsequent use in a CVM calculation of the phase diagram has been a particularly exciting development over the last decade (for an excellent review see [3]). Notwithstanding these successes, a major disadvantage of the CVM has been recognised - the large number of non-linear simultaneous equations which must be solved in order to obtain the free energy and the equilibrium species distribution. For ordered phases of lowest symmetry the number of equations when using thie CVM totals k” - 1, where k is the number of components and n the number of atoms in the considered cluster. Some reduction is possible if one is only interested in the higher symmetry disordered phase. Kikuchi has used the cluster probabilities [4] in the free energy functional whereas others have favoured the use of multisite correlation functions [5]. Unfortunately, no reduction in the number of equations results from this different choice of variables. The sometimes excessive number of independent variables puts a limitation on the maximum sized cluster which can be considered for binary solutions and effectively eliminates anything higher than ternary systems from consideration in even the tetrahedron approximation. Real alloy systems of interest can, however, sometimes involve as many as ten components and even the simplest semiconductor alloy system can involve this many variables since there it is usually necessary to consider different charge states of a particular defect. In the present note, we should like to suggest the use of an alternative cluster approach which results in a considera‘ble reduction in the number of equations to be solved as compared with the CVM. Although 623
624
THE CLUSTER/SITE APPROXIMATION
the approximation is not new, its relevance to calculations on multicomponent been overlooked in the excitement generated by the success of the CVM. The Cluster/Site
Vol. 35, No. 5
solutions
appears to have
Approximation
The clusters of sites in a solid crystal which are used in a CVM entropy calculation are not isolated but, in general, share faces, edges and corners. In calculating the entropy of mixing, allowance is made for this sharing and this results in several terms in the entropy expression. This is illustrated in the following CVM tetrahedron approximation expressions for the configurational entropy in fee and bee lattices [6]: fee :
S
=
2St -5S,
bee :
s
=
6St - 12&r + 3S,2 + 4S,t - S,
+6S,
(1) (2)
where the subscripts refer to the tetrahedron (t), triangle (tr), pair(p) and site (s) contributions, respectively. The tetrahedron in the bee lattice, unlike the one in the fee lattice, is asymmetric and involves considering both nearest (pl) and next nearest (~2) neighbour interactions. Two term entropy expressions are obtained if one considers the mixing of clusters which are not permitted to share edges or bonds. For both fee and bee lattices the configurational entropy expression is given by [7]: s = St - 3s, (3) where the site entropy term, Ss, term comes from the normalisation of the partition function. The significance of using such a cluster/site approximation (CSA)* for the configurational entropy rather than the CVM is profound - the number of equations to be solved can be reduced from the order of k” to the order of k x n, e.g., lo4 --+ 10 x 4, thereby offering the possibility of considering larger clusters and/or more components. The large reduction in the number of equations is achieved by using the the generalised quasi-chemical method (GQCM), introduced many years ago by Fowler for treating equilibria in molecular gaseous assemblies [8] and subsequently used for clusters in solid solutions by Yang and Li [9, 10, 11,7]. By using the GQCM the problem is reduced to one of considering site (E atom ) probabilites rather than cluster (= molecule) probabilities. The essence of the CSA can be illustrated by considering a binary A-B system. A cluster, (ai), is represented according to the occupation of the position number i in the cluster, e.g., { 1100) for the AABB tetrahedron cluster. A function 4 is then defined:
where the site parameters pi in the set {pi] are set equal to unity if the atom in the cluster is a B atom. Thus, for our AABB cluster, {pi} becomes /~1~2. x((ai)) is the energy of the cluster {oi) and p = l/k~T. The importance of the site parameters is apparent from the fundamental assumption of the GQCM, which is that the average cluster probability at equilibrium, ({q }) is related to (pi } by: (Ioi 1) = ti
4
eXP(-BX((ai
1))
With this assumption we see that the cluster probabilities are obtained directly from the site parameters and it can be shown [lo] that these are uniquely defined by the relation:
Pi%
=aidJ
(4) fit
(5)
1
*We have preferred the description cluster/site approximation to any description which includes the term quasi-chemical since the latter is used in many different contexts.
Vol. 35, No. 5
THE CLUSTER/SITE APPROXIMATION
625
where ai is the species fraction of A atoms on the sublattice when the A atom resides in cluster position i. The molar configurational free energy, F,, is also related to the site parameters:
(6) where z/2p is the number of bonds per site in the lattice relative to the number per site in the cluster, and fi and xi are the sublattice fractions and species fractions on sublattice i, respectively. Minimisation of the free energy subject to a mass balance constraint and the use of Eqns. (4) is sufficient to determine thie four values (in the tetrahedron approximation) of ai and the four values of WI. In the highly symmetric disordered phase, where all the ui are equal to the mole fraction, there is only one pi to solve for in the binary solution. The CVM and CSA approximations have been compared previously [12, 131. Guggenheim and McGlashan [ 121 showed that the differences in the calculated critical temperatures are small but were unable to decide which was the better approximation, Monte Carlo results being unavailable at that time. Subsequently, Kikuchi and Sato [ 131 demonstrated the superiority of the CVM over the CSA for the same sized cluster. It should be emphasised that these comparisons, which were for binary solutions, showed that the free energy differences are not large and, importantly, the CSA results are not qualitatively different from the CVM ones in regard to predicting the order of phase transitions (unlike the point and pair approximations). The small differences in free energy calculated by different approximations are illustrated for a f.c.c. lattice in Fig. (1) for an exchange energy, WAB = EAB - ~(CAA + EBB) = -kT. This value of WAB is equivalent to a temperature only slightly above the critical and thus represents a substantial short range ordering. It can be seen that the free energy calculated from the two tetrahedron approximations differ little from one another but differ substantially from the values calculated in the point and pair approximations. A slightly higher fraction of AB pairing in the CSA as compared with the CVM is responsible for the more negative free energy but, again, the difference is small compared with those calculated from the point and pair approximations. The latter underestimates the number of AB bonds and it is well known that the pair approximation overestimates their number. A comparison of calculated phase diagrams is a more stringent test of the different approximations than comparisons of the single phase thermodynamic properties since this involves the small differences in free energy for pairs of phases. A CSA calculation of the fee, tetrahedron approximation phase diagram by
0
-1
5 A P
-2
E W -3 I Ii -4
ad---
/
0.0
0.1
I
I-----l-----
0.2
0.3
0.4
0.5
Mole Fraction Figure 1. Comparison of the molar free energy of mixing for a fee lattice as calculated by the point, pair and tetrahedron approximations (CVM and CSA) for a pair exchange energy Of WABkT = -1.
Pair
----r-0.0
0.1
0.2
0.3
0.4
0.5
Mole Fraction B
Figure 2. Calculation of the free energies of mixing for a fee lattice as calculated by the point, pair and tetrahedron approximations (CSA anad CVM) in the ternary system A-B-C along the AB-AC join. (WAB/kT = WAC/kT =WBC/kT = - 1).
626
THE CLUSTER/SITE APPROXIMATION
Vol. 35, No. 5
-I
0.1
0.2 Mole
0.4
0.3 Fraction
0.5
B
Figure 3. Comparison of the concentrations of the tetrahedron cluster {AABC} as calculated by the point and tetrahedron approximations (CSA and CVM) in the ternary system A-B-C along the AB-AC join. WAB/kT = WAC/kT = WBC/kT =- 1).
Li [ 1 l] failed to show the low temperature stability of the A3B phase relative to the disordered phase over a wide range of compositions, as is found using the CVM approximation [ 141. The topology of the CSA phase diagram is, however, a considerable improvement over that calculated by Shockley [ 151 in the point approximation and it should be recalled that there is no phase diagram in the pair approximation since the disordered phase is always the most stable [ 161. We have used clusters larger than tetrahedrons in the CSA approximation and find that the phase diagram then starts to have the same topology as that calculated by the CVM. A comparison of the different approximations for a ternary system is given in Figs. (2) and (3). In Fig. (2) we show the free energy calculated for a disordered ternary phase, A-B-C, along the AC-BC join, for the interaction energies given in the caption. The differences between the CVM and CSA results are again small but both differ substantially from the point and pair approximation results. In the CSA calculation it is necessary to solve for only two site parameters pi. This may be compared with the CVM where the same calculation requires solving for 14 cluster probabilities. As indicated earlier, this advantage of the CSA over the CVM becomes dramatically more marked as the number of components and/or the size of the cluster are increased. In Fig.(3) we show the concentrations of a particular tetrahedron cluster, {AABC}, along the same composition join and it can be seen that the short range order as calculated in the CSA and CVM approximations agree quite well. Conclusions The free energy calculated by the CSA is slightly less accurate than that calculated by the CVM for the same sized cluster. Even in the tetrahedron approximation it is, however, considerably better than both the pair and point approximations in that the free energy is close to that calculated by the CVM and the order of phase transitions is correctly predicted. The CSA has the big advantage over the CVM in the calculation of the free energy for multicomponent solutions and/or larger clusters in systems with a small number of components in the drastically reduced number of equations to bc solved. It should be remembered that, for those interested in the calculation of multicomponent phase diagrams for real systems, some empiricism must always be introduced into the representation of the free energy of a phase. That is to say, no matter how good the physical modelling, some adjustment of the model parameters is necessary to obtain satisfactory agreement with the real system thermodynamics. One does not know, for example, whether one should retain only the binary exchange energies, Wij , or introduce some ternary, Wijk, or even higher order interactions. Thus some slightly less accurate approximation than the CVM is
Vol. 35, No. 5
THE CLUSTER/SITE
627
APPROXIMATION
acceptable if it has the major advantage of a reduced number of independent the free energy functional. The CSA fulfds this requirement.
variables for evaluation
of
Acknowledgment W.A.O. is grateful for the hospitality
extended by the Forschungszentrum
Jiilich GmbH, Germany.
References 1. G. Ceder, Camp. Mat. Sci. 1, 144 (1993). 2. R. Kikuchi, Phys. Rev. 81,988 (1951). 3. D. de Fontai.ne, in Solid State Physics, edited by D. Turnbull, volume 47, pages 33-176, New York, 1994, Academic Press. 4. R. Kikuchi, J. Chem. Phys. 60, 1071 (1974). 5. J. M. Sanchtz and D. de Fontaine, Phys. Rev. B 17,2926 (1978). 6. G. Inden and W. Pitsch, Atomic ordering, in Materials Science and Technology, edited by R. W. Cahn, P Haasen, and E. J. Kramer, volume 5, Phase Transformations in Materials, pages 499-551, Weinheim, Germany, 1991, VCH. 7. Y. Li, Phys. Rev. 76,972 (1949). 8. R. H. Fowler, Statistical Mechanics, pages 162-164, Cambridge University Press, Cambridge, U.K., 2nd edition, 1938. 9. C. N. Yang, J. Chem. Phys. 13.66 (1945). 10. C. N. Yang and Y. Li, Chinese Journal of Physics 7,59 (1947). 11. Y. Li, J. Chem. Phys. 17,447 (1949). 12. E. A. Guggenheim and M. L. McGlashan, Molec. Phys. 57,433 (1962). 13. R. Kikuchi and H. Sate, Acta Met. 22, 1099 (1974). 14. C. M. van Baal, Physica64,571(1973). 15. W. Shockley, J. Chem. Phys. 6, 130 (1938). 16. R. Peirls, Proc. Roy. Sot. A 154,207 (1936).