Pair Effects and Self-Consistent Corrections in Disordered Alloys

A formal study is made of the normal modes in a disordered alloy having substitutional defects, with explicit treatment of isolated defects and pairs of defects. Particular attention is paid to "multiple occupancy" of a site by more than one defect, which requires kinematic corrections. In the process, two new and quite general approaches to the disordered-alloy problem are presented. The first, a $t$-matrix approach, is shown to be equivalent to a second self-consistent diagrammatic approach. The second method is used to calculate formally the corrected and fully self-consistent scattering by pairs of defects, and is shown to be an extension of the self-consistent calculations by Soven and Taylor. It is found that the proper way to perform a self-consistent calculation is not to insert the full propagator into the self-energy parts, but to insert the full propagator minus the contribution containing scattering sites already explicitly in the parts.