Superposition approximation and natural iteration calculation in cluster-variation method

Abstract

In the cluster-variation method of cooperative phenomena and also in the quasichemical method, the bottleneck step has been to solve simultaneous equations. This paper proposes a new iterative procedure for solving the equations. This iteration does not use differentiation nor matrix inversion and may be called the natural iteration method. The free energy always decreases as the iteration proceeds, with a consequence that the iteration always converges to a stable solution (a local minimum of free energy) as long as the initial state is a physically acceptable one. The method derives in its introductory step a superposition approximation which writes the distribution variables of the basic cluster as a product of those of subclusters. The method is first explained with the pair approximation of the Ising ferromagnet, and then is applied to the fcc binary alloys to derive a phase diagram which is compared with the one reported recently by van Baal.