The mathematics of the polymorphous coherent potential approximation

The original coherent potential approximation (CPA) used for calculating th e electronic states in substitutional solid-solution alloys contains the im plicit assumption that the alloy is isomorphous. That is, all of the atoms of a given chemical type are assumed to be identical. The extension of the CPA philosophy to treat an alloy model in which all of the atoms are allowe d to have distinct charges and potentials is called the polymorphous CPA (P CPA). This extension requires some interesting changes in the mathematical formalism that is used to develop the CPA equations. Aspects of the mathema tical formalism of the PCPA will be discussed. In particular, the ergodic t heorem from measure theory will be invoked to justify the new equations for the average Green's function.