Optical-phonon—assisted hydrogen diffusion in metal hydrides

The quantum theory of diffusion for light interstitials in model nonstoichiometric systems is developed. The dispersive optical phonons are used to obtain an analytical expression for total transition probability $W_{p{p}^{\prime }}$. In the low-temperature limit, $W_{p{p}^{\prime }}$ is found to be temperature independent, while in the high-temperature limit the preexponential and exponential factors are temperature dependent. An explicit expression for migration energy is obtained and it is compared with other theoretical results. The dispersive phonons increase the migration energy. The effect of configurational interaction on $W_{p{p}^{\prime }}$ is also studied. It is found that at low temperature the tunneling transitions dominate and that at higher temperature the hopping transitions dominate. The impurity-excited states are accounted for in evaluating the diffusion constant for model nonstoichiometric metal hydrides. The diffusion coefficient for $\mathrm{Pd}{\mathrm{D}}_x$ is found lower than that for $\mathrm{Pd}{\mathrm{H}}_x$. The calculated values of preexponential factor and migration energy are found to agree reasonably well with the experimental data wherever available. The hydrogen is found diffusing faster than the deuterium at all temperatures in $\mathrm{Pd}{(\mathrm{H},\mathrm{D})}_x$. Similar calculations are carried out for $\mathrm{Nb}{(\mathrm{H},\mathrm{D})}_x$. The diffusion constant for $\mathrm{Nb}{\mathrm{D}}_x$ is found to be lower than that for $\mathrm{Nb}{\mathrm{H}}_x$. At low temperature the migration energy for D is found to be lower than that for H, while at high temperature the trend reverses. The calculated migration energy is found to be lower than the experimental values for which the possible reasons are discussed.