Quantum Theory of the Equilibrium Order Parameters for Disordered Solid Solutions

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.