Spin Diffusion Measurements: Spin Echoes in the Presence of a Time-Dependent Field Gradient

TLDR

The study derives how a time‑dependent magnetic field gradient affects spin‑echo experiments and explains why such gradients can be preferable to steady gradients. The authors propose using a pulsed gradient to precisely define the diffusion measurement interval. They describe a pulsed‑gradient apparatus capable of up to 100 G cm⁻¹, noting that reducing the gradient during RF pulses lowers H1 requirements and that a small gradient at echo time yields a broad, easily measured echo. The derived expressions were experimentally validated for various gradient shapes, and the technique extended the measurable diffusion range, yielding a dry glycerol diffusion coefficient of (2.5 ± 0.2) × 10⁻⁸ cm² s⁻¹—below the limit of steady‑gradient methods.

Abstract

A derivation is given of the effect of a time-dependent magnetic field gradient on the spin-echo experiment, particularly in the presence of spin diffusion. There are several reasons for preferring certain kinds of time-dependent magnetic field gradients to the more usual steady gradient. If the gradient is reduced during the rf pulses, H1 need not be particularly large; if the gradient is small at the time of the echo, the echo will be broad and its amplitude easy to measure. Both of these relaxations of restrictions on the measurement of diffusion coefficients by the spin-echo technique serve to extend its range of applicability. Furthermore, a pulsed gradient can be recommended when it is critical to define the precise time period over which diffusion is being measured. The theoretical expression derived has been verified experimentally for several choices of time dependent magnetic field gradient. An apparatus is described suitable for the production of pulsed gradients with amplitudes as large as 100 G cm−1. The diffusion coefficient of dry glycerol at 26°±1°C has been found to be (2.5±0.2)×10−8 cm2 sec−1, a value smaller than can ordinarily be measured by the steady gradient method.