Ordering Conditions. I. For Alloys Represented by Generalized Ising Models

For quite general, long-range, central interactions in Ising models of superlattice alloys, we derive a set of sufficient conditions on the interactions to ensure that a specified ordered state be the ground-state configuration at $T=0\mathrm{}°$K. These conditions are that the Fourier transform of the ordering energies must be an absolute minimum (or, the "exchange constants" an absolute maximum) at, and only at, the points in reciprocal space where the Fourier transform of the Warren-Cowley order parameters (simply related to the "spin-spin" correlation coefficients) for the known structure at $T=0\mathrm{}$ are nonzero. These sufficient conditions are: (1) identical to the conditions we found earlier by different methods to be necessary for the spherical model, and (2) identical to conditions referred to by Clapp and Moss as necessary, within the limitations of their approximate treatment of Ising models. Whereas for the spherical model we utilized our detailed solutions to the statistical problems for all temperatures, here we use only the principle that the configuration at $T=0\mathrm{}°$K is the one of minimum energy. Since only approximate calculations for any very "realistic" Ising model of an alloy can be expected in the near future, and since the ordering energies are generally not known exactly, this set of sufficient ordering conditions provides useful information for ensuring that one's assumed interactions lead to the superlattice alloying of interest. In this paper, we illustrate our ordering conditions with an application of the simple case of CuZn ($\beta$ brass). It is found that the Fourier transform of the exchange constants must have absolute maxima at, and only at, six points lying in the $〈100〉$ directions and located at corners of the first Brillouin zone, which is constructed in the face-centered cubic lattice reciprocal to the body-centered cubic lattice of points on which Zn and Cu atoms of the simple cubic CuZu structure reside. In addition, one can easily show that nearest-neighbor interactions that favor unlike nearest neighbors produce maxima in the Fourier transform of the exchange constants at these points.