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 spins diffusing in a bounded region in the presence of the surface relaxation and magnetic field inhomogeneity and compute explicitly the relaxation exponent for the Carr–Purcell–Meiboom–Gill spin echoes. The results depend on the echo number n, and three dimensionless parameters: Lρ/LS, D0=(LD/LS)2, the dimensionless diffusion constant, and γ=LD2LS/LG3=Δωτ, the dimensionless gyromagnetic ratio, where the restriction is characterized by a size LS, the magnetic field inhomogeneity by a dephasing length, LG, the diffusion length during half-echo time by LD, and a length Lρ characterizes the surface relaxation. Here Δω is the line broadening and 2τ is the echo period. Depending on the length scales, three main regimes of decay have been identified: short-time, localization, and motionally averaging regimes (MAv). The short-time and the MAv regimes are described well by the cumulant expansion in terms of powers of the “small” parameter ...

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