Spin echoes of nuclear magnetization diffusing in a constant magnetic field gradient and in a restricted geometry

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Abstract
We study the influence of restriction on Carr–Purcell–Meiboom–Gill spin echoes response of magnetization of spins diffusing in a bounded region in the presence of a constant magnetic field gradient. Depending on three main length scales: L S pore size, L G dephasing length and L D diffusion length during half-echo time, three main regimes of decay have been identified: free, localization and motionally averaging regime. In localization regime, the decay exponent depends on a fractional power (2/3) of the gradient, denoting a strong breakdown of the second cumulant or the Gaussian phase approximation (GPA). In the other two regimes, the exponent depends on the gradient squared, and the GPA holds. We find that the transition from the localization to the motionally averaging regime happens when the magnetic field gradients approach special values, corresponding to branch points of the eigenvalues.Transition from one regime to another as a function of echo number for a certain range of parameters is discussed. In this transition region, the signal shows large oscillations with echo number. For large n, asymptotic behavior sets in as a function of n for the decay exponent per echo. This is true for all values of the parameters L S , L G , and L D .