Diffusion in a deformable lattice: Theory and numerical results

We develop a transport theory of one-dimensional diffusion of a single particle in a three-dimensional deformable lattice with a Debye spectrum. Our treatment is based on the Mori formalism; the memory kernel is evaluated in the mode-mode coupling approximation. This allows us to evaluate the absolute magnitude of the diffusion constant, the frequency-dependent conductivity, and the incoherent neutron scattering cross section of the diffusing particle. The main results are: (a) The diffusion constant obeys an Arrhenius law at very low temperatures. At higher temperature ($k_{B}T\gtrsim \frac{\Delta E}{4}$, $\Delta E$ is the activation energy) the apparent activation energy increases with temperature. This increase depends on the mass $M^A$ of the diffusing particle leading to an apparent activation energy that increases slightly with decreasing $M^A$. The prefactor (attempt frequency) shows only a very small dependence on $M^A$ for not too small masses. As a result, the diffusion coefficient is nearly independent of $M^A$. Only for small $M^A$ we recover the isotopic effect as predicted, for instance, by the theory of Vineyard. (b) The frequency-dependent conductivity $\sigma \left(\omega \right)$ shows a two-peak structure and a temperature dependence which is similar to that obtained in the Fokker-Planck treatments. The frequency of the upper peak of $\sigma \left(\omega \right)$ exhibits a square-root dependence on $M^A$ whereas $\sigma \left(\omega \right)$ depends very little on $M^A$ at low frequencies. Detailed results for the dependence on the elastic constant of the crystal and the masses of the particles are given. (c) Analogous results and plots have been obtained for the incoherent cross section of the diffusing particle.