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

We study the influence of diffusion on NMR experiments when the molecules undergo random motion under the influence of a force field and place special emphasis on parabolic (Hookean) potentials. To this end, the problem is studied using path integral methods. Explicit relationships are derived for commonly employed gradient waveforms involving pulsed and oscillating gradients. The Bloch-Torrey equation, describing the temporal evolution of magnetization, is modified by incorporating potentials. A general solution to this equation is obtained for the case of parabolic potential by adopting the multiple correlation function (MCF) formalism, which has been used in the past to quantify the effects of restricted diffusion. Both analytical and MCF results were found to be in agreement with random walk simulations. A multidimensional formulation of the problem is introduced that leads to a new characterization of diffusion anisotropy. Unlike the case of traditional methods that employ a diffusion tensor, anisotropy originates from the tensorial force constant, and bulk diffusivity is retained in the formulation. Our findings suggest that some features of the NMR signal that have traditionally been attributed to restricted diffusion are accommodated by the Hookean model. Under certain conditions, the formalism can be envisioned to provide a viable approximation to the mathematically more challenging restricted diffusion problems.