Nuclear Spin Relaxation by Translational Diffusion in Solids

Abstract

Expressions for ${T}_{1}$ and ${T}_{2}$ have been derived for dipolar relaxation via atomic diffusion using the general theory of nuclear spin relaxation. General methods of evaluating the autocorrelation functions of various terms in the dipole-dipole Hamiltonian are discussed, and cubic symmetry requirements are given. A random walk model is used for the calculation, but rough estimates are made of the effects of correlation in direction and time of successive jumps of atoms for the vacancy mechanism of diffusion; if account were taken of these correlations, the derived relaxation times might change by a factor of nearly 2. Detailed computations are made only in the limits of high and low field. The random walk model yields an expression for ${T}_{1}$ in the high-field limit identical to that given by Torrey. Zero-frequency spectral densities needed for computation of ${T}_{2}$, and also ${T}_{1}$ at low field, are expressed as lattice sums involving only the dipolar interaction and the sum of the probabilities of $n$-step random walks between lattice points (an extension of the Polya problem). Detailed computations of ${T}_{1}$ and ${T}_{2}$ have been made for two or more species of spins diffusing on an NaCl or fcc lattice. The angular dependence of ${T}_{1}$ and ${T}_{2}$ may be large for the NaCl lattice in the high-field limit. The agreement with Torrey's theory for ${T}_{1}$ in the fcc lattice is good.