Commutator expansions in NMR diffusometry

• Commutator expansions for solving Bloch–Torrey’s equation are derived.
• Moments of the echo decay are derived in terms of commutators.
• The case of freely diffusing spins in a constant magnetic field is solved.
• The case of diffusing spins restricted between two plates is solved in the short-gradient pulse limit.

In this paper commutator expansions for solving the Bloch–Torrey’s equation are derived. An exact solution for free diffusion in a constant magnetic field gradient is found. Furthermore the moments of the signal in the short gradient pulse limit for spins confined to a restricted geometry are derived. The moments are tested numerically against the known solution of diffusing spins between two perpendicular plates in the short gradient pulse limit.

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