Pulsed-field gradient nuclear magnetic resonance as a tool for studying translational diffusion: Part 1. Basic theory

Translational diffusion is the most fundamental form of transport in chemical and biochemical systems. Pulsed-field gradient nuclear magnetic resonance provides a convenient and noninvasive means for measuring translational motion. In this method the attenuation of the echo signal from a Hahn spin-echo pulse sequence containing a magnetic field gradient pulse in each τ period is used to measure the displacement of the observed spins. In the present article, the physical basis of this method is considered in detail. Starting from the Bloch equations containing diffusion terms, the (analytical) equation linking the echo attenuation to the diffusion of the spin for the case of unrestricted isotropic diffusion is derived. When the motion of the spin occurs within a confined geometry or is anisotropic, such as in in vivo systems, the echo attenuation also yields information on the surrounding structure, but as the analytical approach becomes mathematically intractable, approximate or numerical means must be used to extract the motional information. In this work, two common approximations are considered and their limitations are examined. Measurements in anisotropic systems are also considered in some detail.