No abstract

Equations for the equilibrium order parameters of substitutional binary alloys are derived from both a classical (bond energy) and a quantum approach. The classical case is presented so as to exhibit that Cowley's equilibrium equations are not valid, principally as a consequence of an incorrect calculation of the internal energy. The work of Flinn is utilized to show that the equilibrium equations have the same form in the classical and the quantum cases. Moreover, the quantum approach is made practical by a simplification of certain integrals which Flinn was unable to evaluate. In both cases, a transition temperature for each atomic distance is found. With these developments, it is now possible to predict order parameters quantum mechanically and to test the predictions directly by x-ray measurements or indirectly by resistivity measurements via Asch and Hall's theory of residual resistivity. Numerical calculations required to compare experiment and theory have not been completed, but a presentation of the theory alone seems justified in view of the appearance of theses and papers based on Cowley's incorrect equations.

The scalar-indexed Warren-Cowley order parameters for the perfectly ordered Cu3Au and CuZn alloys are calculated with the aid of Waring's theorem from number theory. Cowley and Klein have incorrectly reported these parameters for Cu3Au as being one for even-order shells (of neighbors) and −13 for odd-order shells. We show that these parameters are equal to one for those shells with radius d(2k)12, k ≠ 4a(8b + 7), and −13 for those shells with radius d(2k + 1)12, where a, b, k are nonnegative integers and d is the radius of the first shell in the fcc lattice. For the ordered form of CuZn (beta brass), we find that the order parameters are one for those shells with radius d(k)12, k ≠ 4a(8b + 7), and minus one for those shells with radius (d/2)(8k − 5)12, where a, b are nonnegative integers, k is a positive integer, and d is the nearest distance between two like atoms. Five theorems proved in this paper are the basis for the preceding results and are also useful in the study of point defects in cubic crystals and in the study of ionic crystals. Formulas are given for the radii of the shells of lattice points about given points, both lattice and interstitial points, in cubic lattices.