Spin Dynamics of Carr–Purcell–Meiboom–Gill-like Sequences in Grossly Inhomogeneous B0 and B1 Fields and Application to NMR Well Logging

Abstract

The spin dynamics for Carr–Purcell–Meiboom–Gill-like sequences is analyzed in grossly inhomogeneous B0 and B1 fields. This problem is important for many applications, especially when the bandwidth of the signal is excitation limited. Examples include stray-field NMR or inside-out NMR probes used in well logging. The amplitudes of the first few echoes exhibit a characteristic transient behavior but quickly approach a smooth asymptotic behavior. For simple Hamiltonians without scalar or dipolar couplings, the evolution of a refocusing subcycle for a given isochromat is described by a rotation. Simple expressions for the signal of the Nth echo are derived in terms of these effective rotations that have a simple geometrical interpretation. It is shown that the asymptotic behavior is controlled by the direction of the axis of these effective rotations and the signal is dominated by magnetization “spin-locked” to the rotation axis. The phase of the signal is independent of the details of the field inhomogeneities. Relaxation in inhomogeneous fields leads to a signal decay that is in general nonexponential with an initial decay rate that is a weighted sum of T−11 and T−12. At long times, the echo amplitudes decay to a finite value. Phase cycling eliminates this offset. The effect of diffusion is also studied. This analysis has been applied to an inside-out NMR well logging apparatus. Good quantitative agreement is found between measurements and calculations that are based on the measured B0 and B1 field maps.