A cluster variational theory of the hard sphere crystal

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Abstract
The crystalline phase of a system of classical rigid spheres is treated using an adaptation of the Kikuchi cluster-variation approximation in the variational formulation of Morita. The theory is developed for an arbitrary choice of basic clusters, but the numerical calculations that are presented are carried out for two sets of basic clusters, namely nearest neighbor pairs and a combination of isosceles and equilateral triangles. The integrations required in the evaluation of the clusters are performed using Monte Carlo techniques, which permit the use of moderately large clusters, and in consequence fairly general parametrizations of the density functions. No symmetry conditions need to be imposed beyond those already inherent in an assumed fcc crystal. Numerical results are given for the entropy and density functions, the latter being presented in a parametrized form suitable for use in a variational calculation which takes the hard sphere crystal as a reference system.

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