Order-Disorder of Nonstoichiometric Binary Alloys and Ising Antiferromagnets in Magnetic Fields

High-temperature expansions of the free energy of Ising ferro- and antiferromagnets on the square, simple cubic, and body-centered cubic lattices have been derived directly from the corresponding lowtemperature expansions using a generalization of the technique proposed by Domb. These lattices, being loose-packed, can be divided into two identical sublattices, $\alpha$ and $\beta$, such that every $\alpha$ site has only $\beta$ sites as nearest neighbors. The high-temperature expansions treat the finite magnetic fields and magnetic moments on the $\alpha$ and $\beta$ sites as independent quantities. With these series, which extend to eleventh order in the appropriate energies divided by $\mathrm{kT}$, the high-temperature expansions of the magnetization and the logarithmic derivative of the staggered susceptibility have been developed. Padé approximants to these series are used to obtain the critical temperature and the magnetization at the critical temperature as a function of applied magnetic field. These are used to determine the order-disorder critical temperature of nearest-neighbor interaction binary alloys as a function of composition. The results are in good agreement with the observed values for $\beta$-brass at constant pressure. When the experimental values are corrected for the variation of lattice parameter with composition, the agreement is quite poor. It is concluded that a single volume-dependent order-disorder interaction energy is insufficient to describe the transition over the range of compositions in which it is observed.