Second-order quadrupolar and low-dimensionality effects upon NMR resonance spectra

This paper deals with some formal problems which have been encountered in the study of ion dynamics in solid electrolytes with NMR. A time-dependent theory for second-order quadrupolar effects is presented which demonstrates that second-order contributions to the first moment of a line can give the same dynamical information on atomic motion as the spin-lattice relaxation rates. The effect of reduced dimensionality upon motional correlation functions is discussed. The results for two-dimensional and three-dimensional correlation functions, as calculated in the continuum diffusion limit, show that a simple relationship exists between the temperature and frequency dependence of the NMR parameters. Phenomenological rules are proposed to extract from the data an activation energy for the motion described by the correlation functions. A companion paper applies the theory to the ${}_{}{}^{23}{}_{}{}^{}\mathrm{Na}$ and ${}_{}{}^{27}{}_{}{}^{}\mathrm{Al}$ resonance in Na $\beta$-alumina, for which Na ion transport occurs in a plane.