Finite size effects in NMR $T_2$ relaxation by vortex vibrations*

Nuclear magnetic resonance measurements are a powerful probe of electronic behavior in superconductors, but a precise understanding of all relaxation mechanisms is required to draw accurate conclusions. A previous calculation of the rate of transverse relaxation in the cuprate superconductor YBa$_2$Cu$_3$O$_{7-\delta}$ based on an overdamped elastic model of vortex motions predicted relaxation rates orders of magnitude slower than those observed in experiments, despite strong experimental evidence pointing to vortex vibrations as the dominant relaxation mechanism. Here the finite size of the powder grains is explicitly included in the calculation, and we find that both the relaxation rate and the time dependence of the recovery seen in the experiments can be accounted for. The dependence of the relaxation rate on the size and shape of the samples is explored.