Cluster variation method

Using successively an inverse Cluster Variation Method and an IMC algorithm, we determined a set of nine interactions for each alloy (for the IMC procedure, we used a lattice size of 4 24 ). For each alloy, the output from the inverse procedure has been used as an input interaction set in a direct MC simulation, in order to calculate a... 

Statistical mechanics methods such as Cluster Variation Method (CVM) designed for working with lattice statics are based on the assumption that atoms sit on lattice points. We extend the conventional CVM [1] and present a method of taking into account continuous displacement of atoms from their reference lattice points. The basic idea is to treat an atom which is displaced by r from its reference lattice point as a species designated by r. Then the summation over the species in the conventional CVM changes into an integral over r. An example of the 1-D case was done successfully before [2]. The similar treatments have also been done for... 

In order to find approximate solutions of the equations for Ci t) and gi,..j t) one can use regular approximate methods of statistical physics, such as the mean-field approximation (MFA) and the cluster variation method (CVM), as well as its simplified version, the cluster field method (CFM) . In both MFA and CFM, the equations for c (<) are separated from those for gi..g t) and take the form... 

See also discussions of the related cluster variation method [7]. 

Abbreiriaticm BWG = Btagg-Williams-Gorsky, CVM = Cluster Variation method, T = Tetrahednm approximation, T/0 = Tetrahedron/Octahedron approximation, MC = Monte Carlo, SC = Simple cubic approximation, SP = simple prism approximation, Pt point approximation, Tr triangle approximation... 

The reference state is the composition-weighted linear combination of pure A and B components. This approximation neglects vibrational entropy. Higher-order mean-field approximations to configurational entropy, known as the cluster-variation method, are known [5, 6]. 

D. de Fontaine. Studies of the thermodynamics of ordering by the cluster variation method. In H.I. Aaronson, D.E. Laughlin, R.F. Sekerka, and C.M. Wayman, editors, Proceedings of an International Conference on Solid- Solid Phase Transformations, pages 25-47, Warrendale, PA, 1981. The Metallurgical Society of AIME. 

Introduction of partial order at finite temperatures adds another level of complexity and difficulty. This situation is handled by the cluster variation method (CVM) free energy functional, which is expressed as a function of multisite correlation functions, whose coefficients are obtained by the generalized perturbation method (GPM) or the embedded cluster method (ECM). All of these methods are highly computationally intensive at present, this area is probably the principal frontier of alloy phase theory. 

Vinograd, V.L., Putnis, A. (1999) The description of Al,Si ordering in aluminosilicates using the cluster variation method. Am Mineral 84 311-324... 

Burton BP, Kikuchi R (1984) The antiferromagnetic-paramagnetic transition in a-Fc203 in the single prism approximation of the cluster variation method. Phys Chem Minerals 11 125-131 Carpenter MA, Salje E (1989) Time-dependent Landau theory for order/disorder processes in minerals. Mineral Mag 53 483-504... 

To note the nature of the problem, we begin with a reminder concerning the entropy in the case in which no correlation between the occupancies of adjacent sites is assumed. In this limit, the entropy reduces to that of the ideal entropy already revealed in eqn (3.89). In preparation for the notation that will emerge in our discussion of the cluster variation method, we revisit the analysis culminating in eqn (3.89). Recall from chap. 3 that the entropy of a system characterized by a series of discrete states with probabilities pi is given by... 

General Description of the Cluster Variation Method. In the previous paragraphs we have described several approximation schemes for treating the statistical mechanics of lattice-gas Hamiltonians like those introduced in the previous section. These approximations and systematic improvements to them are afforded a unifying description when viewed from the perspective of the cluster variation method. The evaluation of the entropy associated with the alloy can be carried out approximately but systematically by recourse to this method. The idea of the cluster variation method is to introduce an increasingly refined description of the correlations that are present in the system, and with it, to produce a series of increasingly improved estimates for the entropy. 

As shown above, in the point approximation to the cluster variation method (essentially the Bragg-Williams approach), we carry out the counting of configurations by finding all fhe ways of distributing A and B atoms that guarantee an overall concentration x of A atoms and 1 — x of B atoms. The number of such configurations is... 

In general, once both the effective cluster interactions and the statistical treatment resulting from the cluster variation method are in hand, the alloy free energy can be written as... 

Fig. 6.24. Phase diagram for the Cu-Au system with energetics obtained by clnster expansion including tetrahedral clusters and with the thermodynamics treated in the cluster variation method in the tetrahedron approximation approximation (adapted from de Fontaine (1979)).

The Cluster Variation Method and Some Applications by A. Fine in Statics and Dynamics of Alloy Phase Transformations edited by P. E. A. Turctii and A. Gonis, Plenum Press, New York New York, 1994. This article gives an enlightening description of the cluster variation method. 

Materials Fundamentals of Molecular Beam Epitaxy by J. Y. Tsao, Academic Press, Inc., San Diego California, 1993. Tsao s book has the best pedagogical treatment of the cluster variation method that I have seen anywhere. In fact, the title provides a false impression since this book is full of deep insights into many generic issues from materials science. 

---