Use of Spin Echoes in a Pulsed Magnetic-Field Gradient to Study Anisotropic, Restricted Diffusion and Flow

TLDR

The Bloch–Torrey equations are extended to include anisotropic, restricted diffusion and flow, and the difficulty of solving these equations for spin‑echo amplitudes in time‑dependent magnetic‑field gradients with boundary conditions is examined. The authors propose a procedure for analyzing pulsed‑gradient data and identify systems that may exhibit restricted, anisotropic diffusion or interesting flow properties. A simplified equation for pulsed‑gradient experiments is derived and solved for two systems showing restricted diffusion, and a data‑analysis method is outlined. The equations are solved for several cases, demonstrating that short, intense gradient pulses effectively define the time during which nuclear displacements occur.

Abstract

The Bloch—Torrey equations are modified to include the case of anisotropic, restricted diffusion and flow. The problem of solving these modified equations for the amplitude of a spin echo in a time-dependent magnetic-field gradient subject to restricting boundary conditions is discussed. This problem is solved for a number of selected cases. In particular, it is found that a magnetic-field gradient applied in short, intense pulses is effective in defining the time during which nuclear displacements take place. A simplified equation, suitable for the pulsed-gradient experiment, is presented and solved for two different examples of systems showing restricted diffusion. A procedure for analyzing the data from pulsed-gradient measurements is suggested, and its merits are discussed. Suggestions are made of systems which may well be expected to show restricted, anisotropic diffusion or interesting flow properties.