Dynamical Theory of Diffusion. II. Comparison with Rate Theory and the Impurity Isotope Effect

Abstract

The formalism of the dynamical theory of diffusion is shown to be strictly analogous to that of the reaction-rate theory in the sense that expressions for the jump rate and related quantities can be written in the same form as the corresponding expressions of rate theory. However, the physical meaning of corresponding quantities is different. For example, the role of the saddle-point configuration in rate theory is taken by a dynamical state in the dynamical theory. The reaction coordinate is shown to measure the extent to which lattice reaction forces barring the jump have been overcome by thermal fluctuations. Geometric considerations indicate that for vacancy diffusion, the factor $\ensuremath{\Delta}K$ should have a lower limit on the order of 0.5 in the fcc lattice and an upper limit of the same order for the bcc lattice. In general, $\ensuremath{\Delta}K$ measures the mass dependence of the rms velocity of the system along the jump direction in configuration space. The mass dependence of the isotope effect for impurity diffusion is developed, and a double-isotope-effect experiment involving the simultaneous diffusion of three solute isotopes is proposed to determine impurity correlation factors.