Relaxation of nuclear magnetization in a nonuniform magnetic field gradient and in a restricted geometry

We study the influence of restriction on Carr-Purcell-Meiboom-Gill spin ech o response of magnetization of spins diffusing in a bounded region in the p resence of a nonuniform magnetic field gradient. We consider two fields in detail-a parabolic field which, like the uniform-gradient field, scales wit h the system size, and a cosine field which remains bounded. Corresponding to three main length scales, the pore size, L-S, the dephasing length, L-G, and the diffusion length during half-echo time, L-D, we identify three mai n regimes of decay of the total magnetization: motionally averaged, localiz ation, and short-time. In the short-time regime (L-D << L-S, L-G), we confi rm that the leading order behavior is controlled by the average of the squa re of the gradient, <((<del>B-z)(2))over bar>, and in the motionally averag ed regime (MAv), where L-S << L-D, L-G, by <((<integral> dxBz)(2))over bar> . We verify numerically that two different fields for which those two avera ges are identical result in very similar decay profiles not only in the lim its of short and long times but also in the intermediate times, with import ant practical implications. In the motionally averaged regime we found that previous estimates of the decay exponent for the parabolic field, based on a soft-boundary condition, are significantly altered in the presence of a more realistic, hard wall. We find the scaling of the decay exponent in the MAv regime with pore size to be L-S(2) for the cosine field and L-S(6) for the parabolic field, as contrasted with the linear gradient scaling of L-S (4). In the localization regime, for both the cosine and the parabolic fiel ds, the decay exponent depends on a fractional power of the gradient, imply ing a breakdown of the second cumulant or the Gaussian phase approximation. We also examined the validity of time-evolving the total magnetization acc ording to a distribution of effective local gradients and found that such a pproximation works well only in the short-time regime and breaks down stron gly for long times. (C) 2000 Academic Press.