NMR signal for particles diffusing under potentials: From path integrals and numerical methods to a new model of diffusion anisotropy

摘要

We study the influence of diffusion on NMR experiments when the moleculesundergo random motion under the influence of a force field, and place specialemphasis on parabolic (Hookean) potentials. To this end, the problem is studiedusing path integral methods. Explicit relationships are derived for commonlyemployed gradient waveforms involving pulsed and oscillating gradients. TheBloch-Torrey equation, describing the temporal evolution of magnetization, ismodified by incorporating potentials. A general solution to this equation isobtained for the case of parabolic potential by adopting the multiplecorrelation function (MCF) formalism, which has been used in the past toquantify the effects of restricted diffusion. Both analytical and MCF resultswere found to be in agreement with random walk simulations. A multi-dimensionalformulation of the problem is introduced that leads to a new characterizationof diffusion anisotropy. Unlike for the case of traditional methods that employa diffusion tensor, anisotropy originates from the tensorial force constant,and bulk diffusivity is retained in the formulation. Our findings suggest thatsome features of the NMR signal that have traditionally been attributed torestricted diffusion are accommodated by the Hookean model. Under certainconditions, the formalism can be envisioned to provide a viable approximationto the mathematically more challenging restricted diffusion problems.