Calculation of Order Parameters of Perfectly Ordered Cu3Auand CuZn, an Exercise in Number Theory

Abstract

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.