Acoustic-phonon-assisted hydrogen diffusion in metal hydrides

The theory for acoustic-phonon-induced hydrogen diffusion in model metal hydrides is presented. The general expression for the total transition probability $W_{p{p}^{\prime }}$ is obtained. The analytical expressions for $W_{p{p}^{\prime }}$ are calculated in the high- and low-temperature limits for both the dispersive and nondispersive acoustic phonons and are compared with other calculations. The diffusion rate is found to be temperature independent at low temperature and temperature dependent at higher temperature. Both the phonon emission and absorption processes contribute towards the diffusion rate at high temperature while only emission process contributes at low temperature. The phonon dispersion increases the preexponential factor and decreases the migration energy. The numerical calculations are carried out for $\mathrm{Pd}{\mathrm{H}}_x \left(\mathrm{Pd}{\mathrm{D}}_x\right)$ and $\mathrm{Nb}{\mathrm{H}}_x \left(\mathrm{Nb}{\mathrm{D}}_x\right)$. The migration energies $E_m$ and the preexponential factors $D_0$ are estimated graphically. The diffusion rate increases with increase of temperature up to 170 K for $\mathrm{Pd}{\mathrm{H}}_x \left(\mathrm{Pd}{\mathrm{D}}_x\right)$ and up to 100 K for $\mathrm{Nb}{\mathrm{H}}_x \left(\mathrm{Nb}{\mathrm{D}}_x\right)$. At higher temperatures the diffusion rate shows an activated behavior. The inverse isotope effect is found for $\mathrm{Pd}{\mathrm{H}}_x \left(\mathrm{Pd}{\mathrm{D}}_x\right)$ at low temperature, while it is absent at high temperature. No inverse isotope effect is found for $\mathrm{Nb}{\mathrm{H}}_x \left(\mathrm{Nb}{\mathrm{D}}_x\right)$. At low temperature the $E_m$ and $D_0$ for H diffusion increase with increase of temperature in both $\mathrm{Pd}{\mathrm{H}}_x \left(\mathrm{Pd}{\mathrm{D}}_x\right)$ and $\mathrm{Nb}{\mathrm{H}}_x \left(\mathrm{Nb}{\mathrm{D}}_x\right)$. At higher temperatures, $E_m$ and $D_0$ become almost constant for $\mathrm{Pd}{\mathrm{H}}_x \left(\mathrm{Pd}{\mathrm{D}}_x\right)$, while these again increase with temperature in $\mathrm{Nb}{\mathrm{H}}_x \left(\mathrm{Nb}{\mathrm{D}}_x\right)$. It is found that the hydrogen diffuses faster in the bcc matrix than in the fcc matrix. A comparison with the results for optical-phonon-induced diffusion revealed that the acoustic-phonon contribution is larger at low temperature while the contributions of both the acoustic and optical phonons is of the same order at high temperature. The migration energies are also calculated combining the contributions of both the acoustic and optical phonons, and these are found in reasonable agreement with the experimental data.