Anharmonicity in the statistical dynamical theory of solid state diffusion

Abstract

The statistical mechanical and dynamical theories of solid state diffusion are combined in a many-body approach to the determination of a jump frequency which includes both harmonic and anharmonic terms in the crystal potential. Explicit correlation between the theories of Vineyard (1) and of Rice (3) is shown for the harmonic case. The cubic and quartic anharmonic terms in both the crystal potentials and in expansions of the force constant matrices are added for a consistent first order correction to the harmonic jump frequency. The jump frequency obtained shows a 1 m 1 2 dependence which is consistent with experimental observations of the mass dependence. The quantities which appear in the final equation for the jump frequency are the harmonic, cubic, and quartic force constants, temperature, mass, and the critical displacements, at the saddle point, of all the atoms which are displaced significantly in the diffusion process. The theory is written in classical terms but the second quantization formalism appears to be tractable within the framework of the approach.