Some Formal Aspects of a Dynamical Theory of Diffusion

Abstract

A classical dynamical theory of diffusion is presented in which the reaction coordinate and its critical value are expressed in terms of a $3N$-dimensional vector R. The Slater approximation to the Kac equation is shown to be exact for classical statistics, and the jump rate is calculated accordingly. The jump rate can be expressed in terms of the vector R and the dynamic matrix; this leads to a frequency factor different from that obtained from the reaction-rate theory, and an indication that the system does not jump through the relaxed saddle-point configuration. The self-diffusion isotope effect is also considered. It is shown explicitly that the migration energy is mass independent, and that the effect may be expressed simply in terms of the reaction coordinate. In terms of the phonons, the isotope effect depends on a weighted average of the fraction of the energy carried by the jumping atom in each mode. Quantum corrections are discussed and a connection is made between the isotope effect and thermomigration.