Frequency autocorrelation function of stochastically fluctuating fields caused by specific magnetic field inhomogeneities.

Signal formation in NMR is due to incoherent dephasing of nuclear spins. Of particular practical importance is the situation of nuclear spins undergoing independent stochastic motion in inhomogeneous local magnetic fields, e.g., created by magnetized objects. Since it was demonstrated recently that the frequency correlation function of nuclear spins can be measured directly, a theoretical analysis of such functions is of interest. Here, we provide a numerically exact analysis of that correlation function for the inhomogeneous fields around two particular geometries: cylinders and spheres. The functional form exhibits three regimes: after an initial transient, there is an algebraic regime with a t(-d/2) time dependence (d being the space dimension), followed by an exponential cutoff due to microscopic system size effects. The main parameter controlling the range of the individual regimes is the volume fraction of the magnetized objects. In addition to our numerical analysis, which is based on eigenfunction expansions, we provide analytical results and approximations based on the generalized moment expansion.