On self-consistent approximations for random anharmonic systems

Abstract

A model with random 1-, 2-, and 3-point potentials is used to study the elementary excitations in substitutionally disordered crystals. The equations of motion for the 1- and 2-point Green's functionsG(1),G(12) are derived and averaged over the ensemble of random configurations. An extension of the coherent potential method is proposed, which leads to a self-consistent set of equations for the averaged 1- and 2-point Green's functions, including corresponding conditional averages. The theory takes into account that randomness effects the anharmonic interactions both via the explicit configuration dependence of the cubic vertices and via the implicit dependence through the Green's functions. The final equations take a similar form as in the usual CPA if the harmonic potential of the pure system and the harmonic single-site impurity potential are replaced by corresponding functionals of averaged and conditionally averaged 1- and 2-point functions, and the definition of the single-site mass-operator is appropriately generalized.