Interpretation of Pulsed-Field-Gradient NMR in Terms of Molecular Displacements

The relationship for Pulsed-Field-Gradient NMR between the amplitude $I(t)$ of the spin echo and the molecular displacement $\mathbf{X}(t)$ is examined. $I(t)$ of a single species in a simple solution is determined by the mean-square displacement $\overline{X(t{)}^2}$. With polydisperse species, or molecular probes in complex fluids showing memory effects, $I(t)$ in general includes large contributions from all higher even moments $\overline{X(t{)}^{2n}}$. Conditions under which the NMR signal is indeed determined by the molecular mean-square displacement are noted. A diagnostic that sometimes identifies when these conditions are not met is presented.