Equivalence of Expanding in Localized or Bloch States in Disordered Alloys

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Abstract
Calculations are made of the configuration-averaged density of states and the electrical conductivity of a binary disordered alloy by two quite distinct techniques, namely, by beginning with the Bloch states of a perfect crystal and making a cluster expansion in the scattering off imperfections, and by beginning with localized atomic states and making a cluster expansion in the interatomic hopping matrix elements. The two techniques are shown to give the same results when all irreducible single-site diagrams are included in the self-consistent field approximation and multiple-occupancy corrections are made self-consistently; it is conjectured that this equivalence holds at each level of the cluster expansion. In the process, the connection between the recent calculation of transport properties by Velický in the coherent potential approximation, the diagrammatic technique of Edwards, and the diagramatic technique of Matsubara and Toyozawa is established.