Variational calculation of NMR relaxation of diffusing ions by impurity centers

The relaxation rate of a diffusing spin influenced by an impurity center is treated. We present both a qualitative scaling argument which can predict the dependence of the rate on various parameters and a variational principle which can be used to obtain a quantitative result accurate to within about 12% for a simple trial function. The methods should be useful for the study of NMR in fast ion conductors in situations, such as anistropic diffusion in a low-dimensional system or impurity-influenced diffusion rate, where exact solutions of a continuum diffusion equation with a sink are not obtainable. For example, we treat (i) the standard three-dimensional case with a $r^{-6\mathrm{}}$ interaction and compare the simple variational calculation with the known exact result, and (ii) a quasi-one-dimensional system which has not previously been solved.