Averaged propagator of restricted motion from the Gaussian approximation of spin echo

Abstract

Apart from the propagator theory of spin-echo diffusion measurement, the average of spin phase fluctuation by the cumulant expansion is developed. The expansion to the second order, the Gaussian phase approximation (GPA), is able to explain the diffusion and flow measurement by the spin echo with the generalized magnetic field gradient sequence, which can be designed in a way to extract information not only about the macroscopic self-diffusion coefficient, but also about the details of the molecular velocity correlation spectrum. By taking into account the distribution of local spin properties resulting from restriction to motional by boundaries, the GPA method covers a broad range of spin echo decay, which throws a new light upon the diffraction-like phenomena, different from that, obtained by the averaged propagator method. The corrections of GPA by the higher order terms of cumulant expansion introduce only small changes to the tail of spin echo decay. The paper demonstrates the advantage of the GPA in this generalized form to deal with the space–time correlations of diffusion and flow in porous medium, closing a fictive gap between NMR diffusometry and q-space NMR.