Cluster expansion for the ising model

A binary alloy system with nearest-neighbor interaction which forms a rigid lattice is the (ideal) Ising system. The present paper is an application of the theory of classical fluids, as developed by Morita and Hiroike, to such systems. The partition function for the Ising system is made formally equivalent to that for a binary fluid as discussed by Morita. Then, using the methods of graph theory, a high-temperature expansion is obtained for fixed long-range order parameters. By performing a partial summation, one finds that the free energy is given in terms of connected diagrams which are composed of βJ bonds, and black circles M_n. The quantities M_n are evaluated by noting that the diagrammatical expressions for these quantities appear in the expansion of the grand potential of a soluble problem. The method is illustrated in the case of an AB alloy or the ferromagnetic Ising system which form cubic lattices. Chang's result is easily obtained. The extension of this method to higher terms and its application to other alloy systems are retained for a forthcoming paper.