Theory of Disordered Ternary Alloys. I. The Linear Approximation

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Abstract
A Green's-function formulation of the statistical mechanics of ternary systems, with arbitrary composition and pairwise interatomic potentials of arbitrary range, is given. Exact formal expressions for the relevant correlation parameters are derived. The linear approximation is worked out in detail, and expressions for the correlation parameters are derived in a form analogous to the results of self-consistent-mean-field theories. The usefulness of this representation lies in its qualitative validity even in the vicinity of the transition points. The transition temperatures are estimated through the condition of macroscopic increase in the range of the appropriate correlation parameters at the onset of spatial ordering. The relevance of the given expressions for inferring the ground-state orderings, i.e., the actual configurations that might obtain when the ordering sets in, is indicated. In the limit that the atomic composition of one of the components of the ternary system approaches zero, our results smoothly reduce to the corresponding self-consistent, linear-approximation results for binary systems derived by Clapp and Moss.