A Theory of Phonons in Amorphous Solids and Its Implications to Collective Motion in Simple Liquids

Abstract

A theoretical study is made of phonons in amorphous solids using a self-consistent phonon scheme. In treating the structural disorder inherent in the problem, a theory is formulated from the viewpoint of multiple scattering theory and designed to give phonon eigen-frequencies which are expressed in terms of many-body correlation functions of atoms as well as of interatomic potentials in the solids. For this purpose, a conditional averaging procedure is applied to equations for phonon Green's functions, obtainable using the renormalized harmonic approximation. A set of hierarchy equations is thereby obtained, for which various decoupling approximations are employed. As an application of the results obtained here, numerical calculations of the eigenfrequencies of longitudinal and transverse phonons in liquid argon are using the “quasi-crystalline approximation” and the Lennard-Jones model potential. Fairly good agreement with experiment is obtained. A systematic and approximate self-consistent method for treating the hierarchy equations, which may be of some use in studying the general properties of the energy spectra of elementary excitations in disordered systems, is also obtained.