Transition between coherent and stochastic motion of light interstitials

We consider first the application of a linear-coupling small-polaron theory to the motion of light interstitials such as hydrogen isotopes and the positive muon in solids. We carry out a microscopic quantum calculation of the density-density correlation function for a diffusing particle that passes smoothly as a function of temperature between the low-temperature limit of coherent bandlike motion and the high-temperature limit of stochastic hopping. A numerical calculation of the mean time of stay of the particle on a given interstitial site shows a maximum at the transition temperature $T^{*}$. For a Debye phonon spectrum $0.3\le \frac{T^{*}}{\Theta }\le 0.7$ for reasonable estimates of the lattice distortion energy and rigid-lattice bandwidth appropriate for positive muons. Higher-order phonon interactions are then introduced and treated within the same formalism. They give rise to a kind of "motional narrowing" of the coherent portion of the density-density correlation function, that depresses $T^{*}$ substantially. Numerical calculations of the mean time of stay for a range of linear- and quadratic-phonon-coupling strengths indicate a substantial depression of the transition temperature between "coherent" and stochastic diffusion, while the transition to true bandlike motion occurs at a still lower temperature.