Analytic approximation for random muffin-tin alloys

The methods introduced in a previous paper under the name of "traveling-cluster approximation" (TCA) are applied, in a multiple-scattering approach, to the case of a random muffin-tin substitutional alloy. This permits the iterative part of a self-consistent calculation to be carried out entirely in terms of on-the-energy-shell scattering amplitudes. Off-shell components of the mean resolvent, needed for the calculation of spectral functions, are obtained by standard methods involving single-site scattering wave functions. The single-site TCA is just the usual coherent-potential approximation, expressed in a form particularly suited for iteration. A fixed-point theorem is proved for the general $t$-matrix TCA, ensuring convergence upon iteration to a unique self-consistent solution with the physically essential Herglotz properties.