Spin echo of spins diffusing in a bounded medium

Abstract

The effect of reflecting boundaries on the spin echo measurement of diffusion in a geometrically bounded medium is calculated by considering the accumulation of phase shifts during the spin echo process. The distribution of phase shifts is shown to be Gaussian in the limits when the time between pulses is either long or short compared to the time required for a spin to diffuse from one boundary to another. The distribution is assumed to be Gaussian in the intermediate case, and the problem is reduced to the calculation of the mean square phase change due to diffusion. Expressions for spin echo amplitudes are derived for the cases of planar, spherical, and cylindrical boundaries, and forms are derived that are valid when the time of the measurement is very long or very short compared with the time required for diffusion between boundaries.