Na23andAl27inβ-alumina solid electrolytes

In a companion paper, we discussed second-order quadrupolar effects in solids and the theory of the NMR response of a two-dimensional (2D) fluid. This paper illustrates the theoretical treatment through a consideration of the NMR response of ${}_{}{}^{23}{}_{}{}^{}\mathrm{Na}$ and ${}_{}{}^{27}{}_{}{}^{}\mathrm{Al}$ in $\beta$-alumina. It is shown that dynamical information is contained in the temperature dependence of the second-order quadrupolar shifts. The stationary and fluctuating components of the electric field gradient at the ${\mathrm{Na}}^{+}$ position are obtained from quadrupolar shift measurements. Utilizing these parameters in a 2D-continuum diffusion model for ${\mathrm{Na}}^{+}$, we are able to account for a number of experimental observations: angular, frequency, and temperature dependence of ${}_{}{}^{23}{}_{}{}^{}\mathrm{Na}$ transition rates, narrowing of the ${}_{}{}^{23}{}_{}{}^{}\mathrm{Na}$ satellites, and temperature dependence of the spin-lattice relaxations rates for ${}_{}{}^{27}{}_{}{}^{}\mathrm{Al}$. Reasons are given for the success of this model on the NMR time scale.