Cluster-variation method for the triangular lattice gas

Cluster-variation method for the triangular lattice gas

I. Phys. Chem Solids Vol. Printed in Great Britain. 43, No. 1, pp. 73-79, WZZ-3697/82/01007~u)3.00/0 Pergamon Press Ltd. 1982 CLUSTER-VARIATION ...

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I. Phys. Chem Solids Vol. Printed in Great Britain.

43, No.

1, pp. 73-79,

WZZ-3697/82/01007~u)3.00/0 Pergamon Press Ltd.

1982

CLUSTER-VARIATION

METHOD FOR THE TRIANGULAR LATTICE GAS

R. OS~RIOand L. M. FALICOV Materials and Molecular Research Division, Lawrence Berkeley Laboratory and Department of Physics, University of California, Berkeley, CA 94720,U.S.A. (Received 20 March 1981; accepted in revised form 28 May 1981)

Abstract-A triangular lattice gas with nearest-neighbor repulsions and three-particle attractions is examined in a three-sublattice triangle approximation as a model for lithium intercalation in the transition metals dichalcogenides. Order-disorder phase diagrams and thermodynamic functions are computed. Some peaks in the incremental capacity as function of concentration in the experimental data of Thompson are reproduced in our model. 1. INTRODUCTION A triangular lattice gas model for the problem of ordering

of Li’ ions in the system Li,Ti& was recently proposed by Thompson[l] and studied by Berlinsky et a/.[21 through a renormalization group calculation. In this paper we use Kikuchi’s[3] cluster-variation method to explore the problem further, including a discussion of the effects of three-body potentials on thermodynamic functions and on the order-disorder phase diagram. Lithium has been intercalated into the layered transition-metal dichalcogenides [4] (represented by MX,, where M is a transition metal and X a chalcogen) to form compounds of the type LixMX, (0 c x G I), that have found use as cathodes in batteries. The system Li,T& has been particularly successful151 due to its reversibility and rapid lithium self-diffusion. In Li,TiS*, Li’ ions reside between two TiS, layers, at sites which make themselves a 2-D triangular network (these are the octahedral sites found between the 2-D triangular layers of sulphur, each S layer belonging to a different S-Ti-S sandwich). The practical uses of this material have prompted a large number of experiments [4-81 including chemical, electrochemical and nuclear magnetic resonance studies. Thompson]11 has performed accurate measurements of the voltage V vs concentration x relationship in the electrochemical cell where Li,Ti& is produced. The data showed well defined peaks in the incremental capacity (- %r/aV) at constant temperature, the main peaks appearing at x = l/9, l/4 and 6/7. Thompson suggested that these peaks are associated with the formation of 2-D ordered superlattices of the Li’ ions at those concentrations. The experiments were performed in a Li-electrolyteLi,Ti$ cell and the measured voltage gives the difference between the chemical potentials of the Li’ ion in the Li,Ti$ cathode and the Li anode. If the Li anode is considered a Li’ reservoir, the voltage V is given by [e/V = constant - p1

(1.1)

where p1 is the Li’ chemical potential in Li,Ti$. The value of pI is determined by a variety of factors: lattice 73

distortions, lattice vibrations, electron screening, etc. The main effect however is caused by the Li’-Li’ Coulomb repulsion and by the resultant collective effects. The nearest-neighbor interaction is the dominant part of the energy, since the repulsion between two ions is screened by the electrons in neighboring TiS, layers. In this paper we study the phase diagram of a lattice gas of particles interacting through a nearest-neighbor repulsion and a three-particle attraction in a triangular arrangement. The cluster-variation method is used as an approximation to this problem. The cluster-variation method has been developed by Kikuchip] as a hierarchy of self-consistent approximations for the combinatorial factors in the entropy of a lattice. The description of correlations between lattice sites is limited by the size of the chosen basic cluster, while long-range order can be introduced by dividing the lattice into a number of sublattices consistent with the ground state. In Section 2, we show that the point approximation for three sublattices (equivalent to a generalized BraggWilliams [9] model) gives the correct physical picture for systems like Li,Ti& in spite of the simplicity of this model. The pair approximation is equivalent to the BethePeierls method and has been applied to the triangular lattice with repulsive nearest-neighbor and attractive second-nearest-neighbor interactions by Campbell and Schick[lOl. Because of its 1-D characteristics, however, this model leads to paradoxical results, like negative entropies close to the concentration x = l/2 at low temperatures. Another failure, apparent in all closed-packed lattices, is that the Bethe-Peierls method gives wrong results for the energy at zero temperature: the probability of existence of nearest-neighbor pairs of particles is zero for x s l/2, while the correct result is finite for x > l/3. We therefore choose not to apply the pair approximation to the present problem. The triangle approximation with two-particle repulsions and three-particle attractions is applied to this problem in Section 3 and the Appendix. The resulting phase diagrams are shown in Section 4, and the behavior of the entropy and the incremental capacity is the subject of

R. OX&IOand L. M. FALICOV

74

Section 5. The results are discussed further and conclusions presented in Section 6. After completion of this work, the authors learned of a recent application of the triangle approximation to this problem (with pairwise interactions only) by McKinnon and Haering[ I I]. 2. POINT APPROXIMATION

The point approximation is discussed in this section as a way of introducing concepts that are useful in the following sections, where correlations between nearestneighbor sites are introduced in a triangle cluster. The calculation presented here is similar to that of Shockley[l2] for the 3-D Cu,Au,_, alloy. Only nearestneighbor two-particle repulsions are considered. As illustrated in Fig. 1, the triangular lattice is divided into three interpenetrating sublattices, a, /3 and y, such that any site in one of them, say (Y,has three nearest neighbors in each of the other two sublattices, /3 and y. This representation can be justified by the work of Kanamori and Kaburagi[l3,14], who have found the ground-state structures for the triangular lattice gas with various pairwise interactions. In our notation, for nearest-neighbor repulsions only, we have the following equilibrium configurations: for x = l/3 the (Y sublattice full, n, = I, with empty p and y sublattices, n, = n, = 0; for x = 2/3 the (Yand p sublattices full, n, = na = I, and the y sublattice empty, n, = 0. We define by n, (v = a, p, y) the probability that a lattice site of sublattice v is occupied by an ion. We thus have x = f(n,

t na + n,).

Williams entropies for the three sublattices: (N/3)! ’ = ks ? In (Nn,/3)![N(l- n,)/3]!

In both eqns (2.2) and (2.3) short-range correlations between the sites are ignored, We consider the chemical potential CL,as the thermodynamic parameter instead of x. Then the three sublattice probabilities are independent variational parameters for the minimization of the grand potential R= E-

TS-p,Nx.

The temperature T, the chemical potential CL,and the grand potential R are hereafter expressed in the dimensionless forms r=-

k,T Ii’

R

p&,

W=NU.

(2.5)

The use of Stirling’s approximation in eqn (2.3) yields for the reduced grand potential the result w =:x2 t

2



y(n.),

(2.6)

where y(n.) = - i (n,)’ t i [n, In n,

t(l-n,.)ln(l-n,)]-ipn..

+ nPnv+ n,n,l,

(2.4)

(2.1)

If there are N sites in the lattice. the interaction energy is approximated by E = NU[n,n,

(2.3)

(2.2)

where U is the nearest-neighbor interaction energy (U > 0). The entropy is taken to be the sum of the Bragg-

(2.7)

Minimizing o with respect to n,, no and n, for given /L and r gives the phase diagram shown in Fig. 2. The complete phase diagram displays a mirror symmetry around the line x = l/2. This is a consequence of the nature of the pairwise, concentration-independent assumed interactions, leading to a particle-hole symmetry. We define the different regions in the phase

Concentration

Fig. 1. Representation of the triangular lattice with its three interpenetrating sublattices.

Fig. 2. Phase diagram in the (x, 7) plane for the triangular lattice gas in the point approximation. The dashed line is the locus of equal free energies for the (3) and (12) phases.

7s

Cluster-variationmethodfor the triangularlattice gas diagram as: the disordered phase (3), where n, = n, = n,; and the ordered phases (12), where n, > no = n,; (21), where n, = nP > n, [this is the mirror image of phase (12) about x = l/2]; and (ill), where n, > ns > n, The phases (3) and (12) or (21) are connected by a first-order phase transition; the phases (12) or (21) and (111) are connected by a second-order phase transition. For a given 7 < 7, = 3/4 there is a small interval of values of x in each half of the phase diagram where the phases (3) and (12) or (21) coexist in a heterogeneous mixture. In this region both the chemical potential and the grand potential are constant. The incremental capacity (&/‘xlaV) is given in our dimensionless units by (ax/@). At r = 0 this function is zero at x = 0, l/3, 2/3, 1 and infinite otherwise. It is more convenient to use the function (7. &/a~) which has slope ?l at x = 0, 1 for any value of 7. This function is plotted in Fig. 3 for several temperatures. For T > rc we have the ordinary one-sublattice Bragg-Willians result

010

i

o”i!mL!J 020

e 2 p

I

T =

0.25

b)

015m

(2.8) i.e. a smooth, structureless curve. Our incremental capacity diverges over the small intervals of x where the phases (3) and (12) or (21) coexist. On the other hand, no singularities occur at values of the concentration where ordered arrangements of the particles are expected. On the contrary, at small values of r, there are minima of (ax/+) near x = l/3 and 2/3, where ordered configurations exist. This is intuitively expected: a significant change in the chemical potential should be required to modify the structure at those concentrations. We thus confirm the conclusions of Berlinsky et a/.[21 concerning the meaning of minima and maxima of the incremental capacity.

020,m T=

07

015-

3.THETRIANGLEAPPROXIMATION

In this approximation the results of Section 2 are improved in two ways. First we include nearest-neighbor correlations between sites. In addition we introduce three-particle attractions to account for the dependence of the Li-Li interaction on the concentration; this is caused by the increasing donation of electrons to the TiSz conduction band. The artificial symmetry about x = l/2 in the phase diagram of Section 2 is thus removed. We define as our basic cluster a closed triangle containing nearest-neighbor points belonging to the three different sublattices defined in Section 2. This approximation was first used by Burley[lS] to study antiferromagnetic behavior in an Ising model, which has been proved by Yang and Lee [16] to be equivalent to our lattice gas problem. While Burley assumed the distribution of spins on two of the sublattices to be equal, we give here a more general solution, including the possibility of three different site occupancies. Furthermore, we incorporate interactions between three particles in the model. We thus have here two parameters, namely the temperature and the three-body potential. To each lattice site we associate a number i such that

OY 0

02

Od

06

OS

10

Concentration Fig. 3. Point approximation for the reduced incremental capacity multiplied by reduced temperature (7. &/c?F) as a function of concentration x for (a) 7-0, (b)T= 0.25, (c) T = 0.5, (d) 7 = 0.7.

i = 0 if the site is empty and i = 1 if the site is occupied. Accordingly, we define the point probabilities x; for the i state of a point in the sublattice V(Y= a, /? or y). In the notation of Section 2, xi’ = nP The probability for an i - i bond (where i, j = 0 or 1) between nearest-neighbor sites on the sublattices v and v’ is denoted by yr’. Finally the probability for an i-jk configuration on an equilateral nearest-neighbor triangle containing points on the sublattices (Y,p and 7, in this order, is denoted by Zii~. The following relations hold between the configuration

76

R. 0~6~10 and L. M. FALICOV

probabilities for the different clusters:

where the operator Y. defined as (3.la)

Y$“=T

x; =

Zijk;

zijk

Y$“i,y~zijk;

;

x Pj =

YF=Cziik;

&j/t ;

x z=

i

cij zijk.

(3.lb) (3.lc)

We assume the particles to interact through a nearestneighbor repulsion U and we add the value U4 for each closed nearest-neighbor triangle of particles. We take 4 to be negative to simulate the decreasing degree of ionization of the Li atoms in Li,T& as x increases. Nuclear-magnetic-resonance data[7] suggest that, while the ionization is essentially complete at small x, 10-20% of an electron remains in the neighborhood of a Li atom at x = 1. Simple electrostatic arguments then suggest that 4 takes value between (-0.3) and (-0.55). The energy of the lattice gas with N sites is written as

5?(u) = u(ln u - l),

(3.6)

results from using Stirling’s approximation in eqn (3.4). The eight triangle configuration probabilities zijt are taken as independent variables for the minimization of W, for given p, r and 4. The normalization constraint of eqn (3.la) is introduced through a Lagrange multiplier. Kikuchi’s “natural iteration”[l8] method was used in the minimization procedure, which is described in the Appendix. 4. PHASEDIAGRAMSIN THE TRIANGLEAPPROXIMATION

We define the regions (3) (12) (21) and (111) in the (x, T) phase diagram as in Section 2. For 4 = 0, the particlehole symmetry holds and the (21) phase is the mirror image of the (12) phase about x = l/2. This phase diagram is shown in Fig. 4. Comparing this with the Bragg-Williams phase diagram of Section 2 we notice two qualitative differences: (a) the cluster-variation phase diagram shows a valley at x = f/2, and (b) the disordered phase (3) continues to exist at T = 0. This topological e = nu(y$+ y$Tt yTP+24z,,,). (3.2) evolution of the phase diagram as the approximation is improved parallels that for the fee binary alloy. [ 191 Ground-state structures are found by minimizing this For the first effect, we should notice that Wannier [20] energy with respect to the variables {zijk},subject to the has solved exactly the zero-field Ising antiferromagnet, constraint of eqn (3.la) and an additional relation and obtained a disordered stable phase. Our order-disobtained from a well-determined concentration: order coexistence region, which extends near x = l/2 down to 7 = 0.25, should, with better approximations, x=;(xytxftxT). (3.3) continue to lower temperatures and include the x = l/2, 7 = 0 point. The triangle cluster approximation, though This is a typical problem of linear programming[l7]. better than the “point” approximation, still seems to be Ordered structures, corresponding to discontinuities in unreliable around x = l/2 at low temperatures. The (dE/dx), result at x = l/3 and 2/3, as in the point apdifference between the free energies of the different proximation, for 4 > - l/2. The structure at x = 2/3 disphases is very small in this region and the (111) phase appears for 4 < - l/2; the one at x = l/3 for 4 < - 3/2: happens to have a higher entropy in our model. At then the separation of all available particles into a phase x = l/2, this (111) phase gives a better approximation to of completely filled sites coexisting with a phase of empty the thermodynamic functions than the mixture of sites becomes energetically favorable. ordered phases obtained by Burley[lSl. In fact, voltage The configurational entropy can be generalized from vs concentration curves calculated by our approximation Kikuchi’s equation[3] for the one-sublattice triangular show very good agreement with Monte Carlo and lattice in a straightforward way and is written as renormalization group results 121close to x = l/2. The zero-temperature limit is usually avoided in clus-

The grand potential can then be expressed in the notation of Section 2 as

Fig. 4. Phase diagram in the (x, 7) plane for the triangular lattice gas in the triangle approximation with nearest-neighbor repulsions only.

17

Cluster-variation method for the triangular lattice gas

ter-variation calculations. Van Baa1[213 justifies this practice with the argument that the approximations used in the description of the energy, which is the predominant part of the free energy at low temperatures, makes the model lose contact with reality in this limit. We find the zero-temperature behavior to be of theoretical interest, however, to confirm the existence of the ground-state structures that led to the division of the lattice into sublattices. Our results show that the concentrations x = l/3 and 2/3 at r = 0 are second-order transition points between the phase (111) and the phases (12) and (21), respectively. For x < l/3, the phases (3) and (12) have the same energy. The short-range correlations included in the cluster description allow for the existence of a disordered phase with zero energy at low concentrations. The intervals where each phase predominates are determined by the zero-temperature-limit entropies. The phases (3) and (12) coexist between x = 0.2280 and x = 0.2515. This interval corresponds to the first-order phase transition found by Burley[22] in the lattice gas with infinite nearest-neighbor repulsion. It has been proved1231 that the exact solution of the (3~( 12) order-disorder transformation in the triangular lattice with nearest-neighbor-only repulsion yields a second-order transition. The validity of the cluster-variation method as a hierarchy of approximations for this same model has been confirmed by McCoy et a1.[24]. They have shown that in the zero-temperature limit the value of the critical concentration and the behavior of chemical potential and grand potential approach the exact result [23] as the size of the basic cluster increases. In Fig. 5, the phase diagram for Q = -0.3 illustrates the asymmetry introduced by interactions between more than two particles. The region of existence of the (21) phase is significantly reduced, although the zero-temperature behavior remains the same as for Q = 0. Below 4 = - 0.5, however, the (21) phase and the ordered structure at x = 2/3 disappear altogether as discussed in Section 3. Such a behavior is displayed in the phase diagram of Fig. 6, for d, = - 0.6. As ~“0, the interval of coexistence between the phases (12) and (3) extends to the whole interval between x = l/3 and x = 1.

02

04

0.6

OB

10

Concentration Fig. 6. Phase diagramin the (x, 7) plane for the triangular lattice gas with a three-particle parameter 4 = - 0.6.

S.THERMODYNAMICFLJNCTIONSINTHE TRIANGLEAPPROXIMATION

We discuss the behavior of the reduced entropy,

s = SIWk,),

(5.1)

and the reduced incremental capacity, defined in Section 2, and compare the latter to experimental results for systems like Li,TiS*. The entropy at fixed temperature as a function of the concentration shows minima at small T where ordered structures occur. The negative three-particle potential increases the values of the entropy for x > l/2. These two effects are illustrated in Fig. 7, where results are presented for several temperatures at 0 = - 0.3, which corresponds to an ionization of 90% at x = 1. In regards to the incremental capacity, our results show minima at concentrations where ordered structures are expected. We are able to account also for the peaks in the experimental data111 for Li,Ti$ at x = l/9 and l/4. For any d, and ~~0.2 a smooth maximum appears near J = l/9. The point x = l/4 is inside the small interval where a diverging incremental capacity indicates an order-disorder transition. A smooth maximum near x = 6/7 can be reproduced for 4 = - 0.3, 7 = 0.2. The negative three-particle interaction produces high values for

0 0

02

0.4

06

0.8

10

Concentration

Fig. 5. Phase diagramin the (x, 7) plane for the triangle lattice gas with a three-particle parameter m = - 0.3. Horizontal lines are drawn for 7 = 0.1, 0.2 and 0.3 so that the behavior of the constant-temperature thermodynamic functions of Figs. 7 and 8 can be compared with the phase diagram.

Concentration Fig. 7. Reduced entropy and molar entropy (in J. mol-’ K-l) as functions of concentration for a three-particle parameter Q = -0.3 and several values of the reduced temperature 7. Arrows indicate second-order transition points for 7 = 0.1. This figure corresponds to the phase diagram of Fig. 5.

78

R. 0~6~10 and L. M. FALICOV

the incremental capacity in the disordered phase for x > l/2. In Fig. 8, results are presented for Q = - 0.3 and several temperatures, together with Thompson’s results[l] for Li,T&, where the mentioned peaks are observed, and for Li,Ta,,~Ti&,, where minima at x = l/3 and 2/3 occur [25]. A nearest-neighbor approximation in a triangular lattice gas can thus account semi-quantitatively for the experimental features in the incremental capacity for lithium intercalation in some transition-metal dichalcogenides. We should mention that a cluster-variation calculation cannot predict the correct analytical behavior of thermodynamic functions at critical points, although their location in the phase diagram can be predicted with a satisfactory accuracy.

titatively the position of maxima and minima of the experimental data. We thus confirm the validity of the lattice gas model as a first approximation for the problem of ordering of Li’ ions in systems like Li,Ti& Better results can almost certainly be achieved by introducing longer-range interactions. The improvements, however, will be limited mainly by the inaccurate description of the guest-host interaction and the electronic contributions due to the filling of the TiSz conduction band. Acknowledgements-The authors are grateful to Prof. R. Kikuchi for communicating the results of Ref. [24] prior to publication This work was supported in part by the Division of Material Sciences, U.S. Department of Energy under Contract W-7405Eng-48 and by a grant from the Miller Institute for Basic Research in Science in the form of a Miller Professorship (to L.M.F.). One of us (R.O.) would like to thank CNPq and UFPE (Brazil) for financial support.

&CONCLUSIONS

We have presented here orderdisorder phase diagrams for the triangular lattice gas with nearestneighbor pairwise repulsive and three-particle attractive interactions. The main effect of the three-particle parameter is to decrease the temperature range of existence of the ordered phase corresponding to the structure at x = 2/3, thus removing the particle-hole symmetry found in the case of pairwise interactions only. This is reflected in the curves for the entropy and the incremental capacity as functions of the concentration. The qualitative picture given by Berlinsky et al.[2] for the minima and maxima of the incremental capacity is confirmed. Furthermore, we are able to reproduce quan-

0 15

REFERENCES 1.Thompson A. H., Phys. Rev. Letl. 40, 1511(1978); J. Electrochem. Sot. 126,608 (1979). 2. Berlinsky A. J., Unruh W. G., McKinnon W. R. and Haering R. R., Solid State Commun. 31, 135(1979). 3. Kikuchi R., Phys. Rev. 81,988 (1951). 4. Muruhv D. W. and Trumbore R. A.. J. Crvstal Growth 39. _ 185i1977). 5. Whittingham M. S., Science 192, 1126(1976). 6. Whittinaham M. S., J. Electrochem. Sot. 123. 315 (1976). 7. Silbernagel B. G. and Whittingham M. S., ‘/. &em. khys. 64,367O (1976). 8. Silbernagel B. G., Material Sci. and Engng 31, 281 (1977).

“I

I

0, = -0.3

(

T=C,.l

~

Q, = T

=

-0.3 0.4

TaosTio2%

0

0.2

04

06

0.6

10

Concentration Fig. 8. Reduced incremental capacity multiplied by the reduced temperature (7. dx/dp)and incremental capacity (in volts-’ assuming T = 300K) for I#J= - 0.3 and (a) r = 0.1, (b) r = 0.2, (c) r = 0.3, (d) r = 0.4 and the experimental results for (e) Li,TiS,, and (f) Li,Taa.sTii,2Szfrom Ref. [25]. In (a) arrows indicate second-order transition points.

19

Cluster-variation method for the triangular lattice gas

zjjk= exp [(At jl njjk- Ejjk)/27] Yjjk”*Xjjk-I”, 9. Bragg W. L. and Willians E. J., Proc. Phys. Sot. (London) (A3) A145, 699 (1934). See also Barrett C. and Massalski T. B., where for the i-j-k triangle configuration we defined the number Structure of Metals, 3rd Ed., pp.284ff. McGraw-Hill, New of particles per lattice site York (1966). 10. Campbell C. E. and Schick M., Phys. Rev. A5, 19 (1972). 11. McKinnon W. R. and Haerina R. R.. to be nublished: nijk = $61 t 8j! + 8k,), (A4) McKinnon W. R., Bull. Am. Ph&. Sot. 26, 348 (1981). 12. Shocklev W.. J. Chem. Phvs. 6. 130(1938). the energy in units of U per lattice site 13. Kaburagi M.‘and Kanamori J., Japan J. &pl. Phys. Suppl2, Pt. 2 (Proc. 2nd Intl. Conf. On Solid Surfaces, Kyoto), 145 ejjk= 6jr6jrt sjj&r + 6kj6jrt2@jj6jr&r (AS) (1974). 14. Kaburagi M. and Kanamori J., J. Phys. Sot. Japan 44, 718 (where 6,” is Kronecker’s delta) and the quantities (1978). IS. Burley D. M., Proc. Phvs. Sot. (London) 85, 1163(1965). Yijk = y;fiypyp 16. Yang-C. N. and Lee T. D., Phys..Reu. 87,404 (19523.See also Pathria R. K., Statistical Physics, pp. 397ff. Pergamon, Oxford (1972).Occupied sites are equivalent to up spins and and vacancies to down spins. The chemical potential plays the role of the magnetic field and the one-half-concentration xi,,=xpxfxl. (A7) lattice aas can be identified with the zero-field Isinr! model. See, e.g Dantzig G. B., Linear Programming and Extensions. 17. In eqn (A3) the triangle probabilities are expressed as products Princeton Univ. Press, Princeton (1965). of, (a) the probabilities for the smaller clusters, (b) a Gibbs factor, 18 Kikuchi R., /. Chem. Phys. 60, 1071(1974). 19. See for instance de Fontaine D., Solid State Physics 34, 73 and (c) a normalization factor exp(A/2r). The Lagrange multiplier A can be identified with the minimized grand potential, (1979). through the relation 20. Wannier G. H., Phys. Rev. 79, 357 (1950). 21. Van Baa1C. M., Physica 64,571 (1973). 22. Burley D. M., Proc.. Phys. Sot. (London) 85, I173 (1965). 23. Baxter R. J.. J. Phvs. A13. L61 (1980). 24 McCoy J. K., K&hi R. and Sato H., unpublished The normalization relation of eqn (3.la) gives a cluster-varia25 Thompson A. H., Physica 99B, 100(1980). tion approximation for the grand partition function Z as a sum over the triangle cluster, the Gibbs factor being weighted by the configuration probabilities of pairs and points, in the form APPENDIX We describe here the “natural iteration” scheme used to Z”2N = exp (- A/27) = exP [(pnjjk- ejjk)/27]Y]#X$!‘“. F minimize eqn (3.5), given j.~,7 and Q. We define (A9)

o,=wtA(l-

F%jk)>

where the Lagrange multiplier A was used to introduce the constraint of eqn (3.la). The equations doo=O a&jk

lead to the superposition expression

642)

The natural iteration calculation proceeds in the following steps: (a) initial values are chosen for the point and the pair variables (e.g. x7 = 0.8, xf = 0.5, x7 = 0.2 for the ordered phases, and xp = xe = xl = 0.5 for the disordered phase; y$y = xpxja etc.); (b) a value of A results from eqn (A9); (c) corresponding values for the set of triangle clusters {ziik}are obtained from eqn (A3); (d) new values for {xy) and {y$“‘}are derived through the summation rules of eqn (3.lb) and (3.1~); (e) steps (b?-(d) are repeated until a convergence

criterion is satisfied.