Multiple correlation function approach: rigorous results for simple geometries - Grebenkov, Denis S. (Diffusion Fundamentals, 2007)
The multiple correlation function approach is briefly presented and applied to investigate spin-echo signal attenuation due to restricted diffusion in simple geometries (slab, cylinder, and sphere) in the presence of surface relaxation. Exact and explicit representations for the zeroth and second moments of the total phase accumulated by diffusing spins are derived by using the Laplace transform summation technique. Within the Gaussian phrase approximation, these two moments determine the reference and diffusion-weighted signals, respectively. In the slow-diffusion or short-time regime, the series expansion in half-integer powers of the diffusion coefficient is generalized to arbitrary temporal profile of a linear magnetic field gradient. In the motional-narrowing or long-time regime, it is shown how the presence of surface relaxation modifies the classical Robertson’s relation. Practical consequences of these findings are discussed.