Classical and Quantum Rate Theories for Solids

Abstract

A rate formula for defect motion in solids including quantum effects is derived by generalization of the classical equilibrium statistical approach. The development makes use of an interpretation of the canonical density matrix for a harmonic oscillator as an ensemble of oscillators which are in minimum-uncertainty states. The classical limit of the rate formula differs from Vineyard's in the pre-exponential factor. The difference is due to the assumption regarding the degree of mode interaction on the hill of the potential surface. The present derivation assumes no interaction, while Vineyard assumes sufficient interaction to maintain thermal equilibrium on the hill. The derived quantum rate formula includes the effects of quantum statistics but not tunnelling effects. Its low-temperature characteristics are explored for the Debye model.