Reduced nuclear magnetic relaxation by paramagnetic impurities in one dimension

The paramagnetic-impurity contribution to NMR relaxation of a diffusing nucleus is shown to be proportional to $c^2$ in one dimension as opposed to $c$ in three dimensions, where $c$ is the impurity concentration. The mechanism is thus not expected to be important for one-dimensional superionic conductors except at high doping levels, whereas it is significant even in nominally "pure" three-dimensional superionics. This brings into question the interpretation of recent NMR results in supposedly one-dimensional $\beta$-eucryptite as being due to paramagnetic impurities.