ANALYTICAL SOLUTIONS AND SIMULATIONS FOR SPIN-ECHO MEASUREMENTS OF DIFFUSION OF SPINS IN A SPHERE WITH SURFACE AND BULK RELAXATION

Nuclear spins (in molecules) are considered to be diffusing in a spher e in a linearly inhomogeneous magnetic field (field gradient) that is imposed during a spin-echo NMR experiment. Relaxation of magnetization both in the bulk medium and on the inner surface of the sphere is ass umed to occur. Analytical solutions were obtained for the relevant mod ified diffusion (partial differential) equation by using separation of variables with a Green's function (propagator) and three different bo undary conditions. Neuman [J. Chem. Phys. 60, 4508 (1974)] analyzed th e same physical system, but with no relaxation, to obtain an expressio n that relates the NMR spin-echo signal intensity to the magnitude of the magnetic field gradient, the spin-echo time, and the intrinsic mol ecular diffusion coefficient. The present analysis was based on that o riginally used by Neuman and, like the latter, it employed the assumpt ion of a Gaussian distribution of phases of the spin magnetizations. T his assumption, while rendering a tractable solution, nevertheless lim its its range of applicability; this aspect, and the convergence prope rties of the series solutions were investigated in conjunction with nu merical simulations made with diffusion modeled as a three-dimensional random (Monte Carlo) walk. A novel prediction for spheres with finite surface relaxation and a given radius is the presence of two minima i n a graph of the normalized spin-echo signal intensity versus the reci procal of the diffusion coefficient. (C) 1996 Academic Press, Inc.