Computer simulation of the spin-echo spatial distribution in the case of restricted self-diffusion.
This article concerns the question of a proper stochastic treatment of the spin-echo self-diffusion attenuation of confined particles that arises when short gradient pulse approximation fails. Diffusion is numerically simulated as a succession of random steps when motion is restricted between two perfectly reflecting parallel planes. With the magnetic field gradient perpendicular to the plane boundaries, the spatial distribution of the spin-echo signal is calculated from the simulated trajectories. The diffusion propagator approach (Callaghan, "Principles of Nuclear Magnetic Resonance Microscopy," Oxford Univ. Press, Oxford, 1991), which is just the same as the evaluation of the spin-echo attenuation by the method of cumulant expansion in the Gaussian approximation, with Einstein's approximation of the velocity correlation function (VCF) (delta function), agrees with the results of simulation only for the particle displacements that are much smaller than the size of the confinement. A strong deviation from the results of the simulation appears when the bouncing rate from the boundaries increases at intermediate and long gradient sequences. A better fit, at least for intermediate particle displacements, was obtained by replacing the VCF with the Oppenheim--Mazur solution of the Langevin equation (Oppenheim and Mazur, Physica 30, 1833--1845, 1964), which is modified in a way to allow for spatial dependence of particle displacements. Clearly, interplay of the correlation dynamics and the boundary conditions is taking place for large diffusion displacements. However, the deviation at long times demonstrates a deficiency of the Gaussian approximation for the spin echo of diffusion inside entirely closed pores. Here, the cumulants higher than the second one might not be negligible. The results are compared with the experiments on the edge enhancement by magnetic resonance imaging of a pore.