Small-polaron models for the hydrogen-concentration dependence of hydrogen diffusion in Nb

We have generalized simplified small-polaron models for the tunneling of interstitials in solids to include interactions between the diffusing particles. This is applied to the calculation of the average hopping rate of hydrogen in niobium as a function of hydrogen concentration for small concentrations, i.e., for an H-Nb ratio ($c$) of roughly 0.06 or less. The hopping of a single H under the influence of nearby, stationary H's was treated. The interactions between interstitials include a hard-core repulsion and a lattice-mediated strain interaction. Both a constant tunneling transfer integral and one that depended on the displacements of nearby Nb atoms were included. Only tunneling via the ground vibrational level of the interstitials was treated. For $c\le 0.06$, the calculated increase in the hopping activation energy as a function of $c$ was linear in $c$ and comparable in magnitude with the experimental increase. Our simplified-model results show that the strain interaction between H's in Nb is important for their diffusion and that this interaction needs to be included in whatever more elaborate diffusion models are developed.