Generalization of the Spherical Model

Abstract

Heretofore, calculations for the Ising model have been obtained mainly through tedious series expansions, and this only for Ising models with nearest-neighbor interactions. However, it is generally accepted that to describe a binary alloy by an Ising model (IM) one also needs higher-neighbor interactions. Consequently, an approximation is needed that does not require extensive prior knowledge of the IM and which will produce values of the Warren-Cowley order parameters (the quantities of interest in this paper) near to those of the IM. The approximate model presented here is a generalization of the spherical model, which in itself is an approximation to the IM. The generalized spherical model (GSM) is obtained by relaxing the spherical constraint of the spherical model and is essentially a temperature renormalization of the spherical model designed to make the GSM imitate the IM. The GSM is here developed for an arbitrary range of interactions and temperatures above the critical temperature; but in order to compare the GSM and the IM, calculations are made only for nearest-neighbor interactions. Values of the nearest-neighbor order parameters of the GSM and the IM for several three-dimensional lattices with nearest-neighbor interactions are found to differ at their greatest extreme by only about 0.01. Also, it is found that the values of the second-neighbor order parameters of the two models for the simple cubic lattice with nearest-neighbor interactions agree approximately as well as the values of the nearest-neighbor order parameters.