Simplified derivation of the matrix coherent-potential approximation for off-diagonal random alloys

A simple matrix generalization of the effective-medium approach to the coherent-potential approximation (CPA) permits the inclusion of off-diagonal disorder in a straighforward fashion. This derivation has the advantages of simplicity and ease of generalization to the $n$-site CPA. Results are obtained for electrons and for phonons in a disordered binary alloy. The electronic derivation reduces to the solution obtained by Blackman et al. while phonon results are new.