Comparison of the Spherical Model and the Clapp-Moss Equations with the Ising Model

Abstract

Expressions are derived for the Warren-Cowley order parameters of an arbitrary binary alloy in the spherical and Gaussian model approximations to the Ising model of a binary alloy. The expressions of the Gaussian model are identical to those derived previously by Clapp and Moss in their approximate treatment of the Ising model. Calculations of the first- and second-neighbor order parameters ${\ensuremath{\alpha}}_{1}$ and ${\ensuremath{\alpha}}_{2}$ are performed for all three models of an $\mathrm{AB}$ alloy with nearest-neighbor interactions on a simple cubic lattice of points. The magnitudes of the nearest-neighbor interactions of the models have been adjusted so that all three models have the same transition temperature. The values of ${\ensuremath{\alpha}}_{1}$ and ${\ensuremath{\alpha}}_{2}$ of the Ising model are lower than those of the spherical model and higher than those of the Clapp-Moss equations, except within a fraction of a percent of ${T}_{C}$. Thus the spherical model seems to give too much order and the Clapp-Moss equations too little order when compared to the Ising model. The values of the spherical model are slightly closer than are the values of the Clapp-Moss equations to those of the Ising model, except within \ensuremath{\sim}1% of ${T}_{C}$ (a region in which the approximate models are expected to be less reliable). No comparison has been made of the approximate models with the Ising model below the transition temperature. In fact, the Gaussian model is undefined below the transition temperature.