Nuclear magnetic relaxation by paramagnetic impurities in superionic conductors

NMR relaxation by paramagnetic impurities in superionic conductors is treated in a manner which is valid over a wide range of temperatures, encompassing regions where $W{\tau }_c\gg 1$ and $W{\tau }_c\ll 1$, as well as the intermediate case $W{\tau }_c\sim 1$. Here $W$ and ${\tau }_c$ are the hopping rate and paramagnetic-impurity relaxation time, respectively. Recent experiments have shown the importance of paramagnetic impurities for relaxing rapidly diffusing nuclei even in nominally "pure" samples, and they have been carried out over the entire range of $W{\tau }_c$ values. Both nearest-neighbor-only and long-range dipolar interactions are treated. The former are handled by a discrete hopping model, the latter by a classical diffusion equation. This equation is similar to that solved by others, but is more general in that it is valid for arbitrary $W{\tau }_c$ and includes both particle and spin diffusion. Nuclear dipole-dipole interactions are accounted for by assuming they make an additive contribution to the transverse relaxation function. As examples of the two models, theory and experiment are compared for linewidth data in Pb${\mathrm{F}}_2$: Mn and for $T_1$ and $T_2$ data in "pure" ${\mathrm{Li}}_5$Al${\mathrm{O}}_4$. Good agreement is found for the linewidth peak in Pb${\mathrm{F}}_2$, without any adjustable parameters. The anomalous frequency dependence at low temperature in $\alpha$-${\mathrm{Li}}_5$Al${\mathrm{O}}_4$ can be explained, but only with a ${\tau }_c$ which is too short to describe higher-temperature behavior.