Short-Range Order in Disordered Binary Alloys

The Warren short-range-order (SRO) parameter of a binary alloy with arbitrary composition is studied for the case of static, configurational interaction with two-body potentials of arbitrary range. A well-defined straightforward procedure for generating a consistent high-temperature series expansion in powers of a dimensionless parameter related to $\frac{T_c}{T}$ is derived. While the existing theories are correct only to the linear power in $\frac{T_c}{T}$, our results are worked out exactly to the third power in this ratio. A plausible inversion of the series, which in the linear approximation corresponds to the Clapp-Moss theory, is given. It is shown that, at least up to the second order in $\frac{T_c}{T}$, such an inversion corresponds to the predictions of a self-consistent decoupling approximation. Using a technique similar to that first introduced by Kramers and Opechowski, the order-disorder transition temperature $T_c$ is computed as a function of the system composition for lattices of cubic symmetry with nearest-neighbor interactions. A remarkable prediction of this study is the suggestion of the existence of a critical concentration below which the system does not order (or separate, as the case may be). For the special case of positive nearest-neighbor interaction in lattices of cubic symmetry, we have also computed the nearest-neighbor SRO parameter for several compositions and temperatures. The results of the present study typically renormalize the corresponding results of the Clapp-Moss theory by several percent. For the stoichiometric composition $m^{\mathrm{}\left(A\right)\mathrm{}}=m^{\mathrm{}\left(B\right)\mathrm{}}$, the results of the present approximation are compared with the very reliable corresponding results of Fisher and Burford (who carried out a careful evaluation of the systematics of the elaborate high-temperature series available for the Ising ferro- and antiferromagnets, i.e., for $m^{\mathrm{}\left(A\right)\mathrm{}}=m^{\mathrm{}\left(B\right)\mathrm{}}$. In general, the quantitative differences between Fisher and Burford's results and those of the present approximation are smaller than the corresponding differences for the linear approximation of Clapp and Moss. An interesting conclusion of the present analysis is that, except for the special case of the nearest-neighbor interactions, the structure of the Fourier transform of the SRO parameter $\alpha \left(\mathbf{K}\right)$ is different from that conjectured by Clapp and Moss. As such, the originally compelling argument in favor of the Clapp-Moss assumption, that from experimental observations of $\alpha \left(\mathbf{K}\right)$ the ratios of the strengths of the interparticle potentials for different separations are determined with a higher degree of reliability than the actual magnitudes of the potentials, is found to be somewhat weakened.