Continuum Approximation for the Motion of a Heavy Particle in One- and Three-Dimensional Lattices

Abstract

It is shown that a heavy impurity atom in a one- or three-dimensional lattice behaves like a Brownian particle, i.e. its velocity auto-correlation function becomes exponential or oscillatory exponential, in the continuum approximation. In Debye's approximation, the auto-correlation function is modified such that its derivative at the origin of time becomes continuous. Some discussions are given on the relation between these results and the theory of Rubin.