Theory of Spin Echo in Restricted Geometries under a Step-wise Gradient Pulse Sequence

A closed matrix form solution of the Bloch-Torrey equation is presented for the magnetization density of spins diffusing in a bounded region under a steady gradient field and for the Stejskal-Tanner gradient pulse sequence, assuming straightforward generalization to any step-wise gradient profile. The solution is expressed in terms of the eigenmodes of the diffusion propagator in a given geometry with appropriate boundary conditions (perfectly reflecting or relaxing walls). Applications to rectangular, cylindrical, and spherical geometries are discussed. The relationship with the multiple propagator approach is established and an alternative step-wise gradient discretization procedure is suggested to handle arbitrary gradient waveforms.