A theory of the hard sphere solid. II

Abstract

A "self-consistent field" theory based on the Bethe approximation previously estimated the entropy and single-particle distribution function in good agreement with computer simulations for the three-dimensional hard-sphere solid. In this paper we carry through the theory for the hard disk system, finding results closely similar to those for hard spheres, and numerically in good agreement with asymptotic high-density numerical simulations. Cluster corrections through third-order interactions are calculated and found small. The asymptotic high-density approximation fails, correctly, to lead to a solution for the one-dimensional system of rods in cells. In this case, an exact analytic solution is possible. For this solution, the one-particle distribution function retains finite width in the high-density limit, in contrast to the two- and threedimensional solutions. At high densities, the free energy of this "self-consistent" cell model solution corresponds to the known correct result. Copyright © 1975 American Institute of Physics.