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 sphere 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 assumed to occur. Analytical solutions were obtained for the relevant modified diffusion (partial differential) equation by using separation of variables with a Green's function (propagator) and three different boundary conditions. Neuman [J. Chem. Phys. 60, 4508 (1974)] analyzed the same physical system, but with no relaxation, to obtain an expression that relates the NMR spin-echo signal intensity to the magnitude of the magnetic field gradient, the spin-echo time, and the intrinsic molecular diffusion coefficient. The present analysis was based on that originally used by Neuman and, like the latter, it employed the assumption of a Gaussian distribution of phases of the spin magnetizations. This assumption, while rendering a tractable solution, nevertheless limits its range of applicability; this aspect, and the convergence properties of the series solutions were investigated in conjunction with numerical 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 in a graph of the normalized spin-echo signal intensity versus the reciprocal of the diffusion coefficient.