Nuclear magnetic resonance spin echoes for restricted diffusion in an inhomogeneous field: Methods and asymptotic regimes

We develop systematic formulations for calculating the magnetization of spi ns diffusing in a bounded region in the presence of the surface relaxation and magnetic field inhomogeneity and compute explicitly the relaxation expo nent for the Carr-Purcell-Meiboom-Gill spin echoes. The results depend on t he echo number n, and three dimensionless parameters: L-rho/L-S, (D) over t ilde (0) =(L-D/L-S)(2), the dimensionless diffusion constant, and <(<gamma> )over tilde>=(LDLS)-L-2/L-G(3)=Delta omega tau, the dimensionless gyromagne tic ratio, where the restriction is characterized by a size L-S, the magnet ic field inhomogeneity by a dephasing length, L-G, the diffusion length dur ing half-echo time by L-D, and a length L-rho characterizes the surface rel axation. Here Delta omega is the line broadening and 2 tau is the echo peri od. Depending on the length scales, three main regimes of decay have been i dentified: short-time, localization, and motionally averaging regimes (MAv) . The short-time and the MAv regimes are described well by the cumulant exp ansion in terms of powers of the "small" parameter <(<gamma>)over tilde>. W e give simple formulas for decay rates in these two asymptotic regimes. We show that the Gaussian phase approximation (GPA), i.e., the exponent up to the second order in <(<gamma>)over tilde>(2) in terms of a full eigenmode e xpansion interpolates well between these two regimes. In the localization r egime, the decay exponent depends on a fractional power, <(<gamma>)over til de>(2/3), denoting a breakdown of the GPA and a breakdown of the cumulant e xpansion in terms of <(<gamma>)over tilde>. (C) 2001 American Institute of Physics.