General Solution to Gradient Induced Transverse and Longitudinal Relaxation of Spins Undergoing Restricted Diffusion

Abstract

We develop an approach, by calculating the autocorrelation function of spins,
to derive the magnetic field gradient induced transverse ($T_2$) relaxation of
spins undergoing restricted diffusion. This approach is an extension to the
method adopted by McGregor. McGregor's approach solves the problem only in the
fast diffusion limit; however, our approach yields a single analytical solution
suitable in all diffusion regimes, including the intermediate regime. This
establishes a direct connection between the well-known Torrey's slow diffusion
result and the fast diffusion result. We also perform free induction decay
measurements on spin-exchange optically polarized $^3$He gas with different
diffusion constants. The transverse relaxation profiles are compared with the
theory and satisfactory agreement has been found throughout all diffusion
regimes. In addition to the transverse relaxation, this approach is also
applicable to solving the longitudinal relaxation ($T_1$) regardless of the
diffusion limits. It turns out that the longitudinal relaxation in the slow
diffusion limit differs by a factor of two, compared with that in the fast
diffusion limit.