The Memory-Function Technique for the Calculation of Pulsed-Gradient NMR Signals in Confined Geometries

An approximation technique for the calculation of pulsed-gradient NMR signals in confined spaces is introduced on the basis of a memory-function formalism and compared to the well-known cumulant expansion technique. The validity of the technique is investigated for the cases of a time-independent field gradient and a gradient consisting of two pulses of finite duration. It is found that the validity is governed by the ratio of two characteristic times: the time for the spins to traverse the dimensions of the confining space through diffusion and the reciprocal of the extreme difference between values of the precession frequency of the spin. Oscillations in the time evolution of the signal for the constant gradient, as well as oscillations in the (gradient) field dependence for the two-pulse gradient, which are both characteristic of the exact signals, are predicted by the new technique but not by the cumulant technique. The cumulant results are shown to arise as an approximate consequence of the memory results.