Nuclear magnetic resonance restricted diffusion between parallel planes in a cosine magnetic field: An exactly solvable model

We propose a theoretical and numerical analysis of restricted diffusion between parallel planes in a cosine magnetic field. The specific choice of this spatial profile as proportional to an eigenfunction of the Laplace operator in this confining geometry considerably simplifies the underlying mathematics. In particular, exact and explicit relations for several moments of the total phase accumulated by diffusing spins are derived. These relations are shown to provide good approximations for the typical case of a linear magnetic field gradient, for which the theoretical analysis was in general limited to the second moment. We study the structure and the properties of the higher order moments which are responsible for the breakdown of the "Gaussian phase approximation" (GPA) at intense magnetic fields. The limits of applicability of the GPA for nonlinear magnetic fields and the transition to the localization regime are discussed. In particular, a diagram of different restricted diffusion regimes is presented.