Na23nuclear relaxation in Naβ-alumina: Barrier-height distributions and the diffusion process

Measurements of the ${}_{}{}^{23}{}_{}{}^{}\mathrm{Na}$ nuclear-relaxation time $T_1$ are reported for two types of Na $\beta$-alumina specimen, i.e., flux-grown crystals from our laboratory and melt-grown crystals from Union Carbide Corp. A phenomenological calculation of $T_1$ due to activated hopping motion, taking account of local variations in energy-barrier heights, is presented. This theory is used to interpret our own $T_1$ data, plus data on $T_1$ in the literature for specimens prepared by a third method, in terms of barrier-height distributions. It gives a good account of both the temperature and frequency variation of these three contrasting sets of $T_1$ data, where the differences are attributed to variations in the structure and/or arrangement of charge defects. The characteristics of the barrier-height distributions obtained are discussed in terms of preparation methods and other available data. The rather larger conductivity for melt-grown (as opposed to flux-grown) material reported by Barker et al. is found to be associated with a displacement of activation energies to lower values. Values of the quadrupolar coupling constants from the $T_1$ data fits as well as from second-order quadrupolar broadening are found to be essentially independent of sample source, suggesting that this coupling is determined primarily by the host structure. Finally, the $T_1$ fits yield values of the local vibration or "attempt" frequency ${\nu }_0$ for activated hopping which are ∼ ${10}^{11}$ Hz for all three cases considered in sharp contrast with the vibration frequency (∼ 2×${10}^{12}$ Hz) reported from infrared spectra and Raman scattering. Our value, which is derived directly from the hopping motion itself, does not yield the good agreement with experiment noted earlier when used in available theories of the diffusion constant.