The displacement correlation tensor: microstructure, ensemble anisotropy and curving fibers

Experiments with multiple diffusion wave vectors are known to carry more information than what is available from standard diffusion experiments. Here we consider a special case of this class of pulse sequences, the double wave vector diffusion experiment, and use the cumulant expansion of the signal to introduce the displacement correlation tensor. We discuss its physical interpretation and properties, noting in particular that its short time behavior allows determination of the surface to volume ratio of the pore space. We present a general expression for the displacement correlation tensor, and provide explicit expressions for a few model geometries. We then show that the scatter matrix characterizing the orientation distribution of an ensemble of cylinders is simply related to the displacement correlation tensor. This result is generalized to ensembles of pores with arbitrary shapes allowing a precise formulation of the influence of microstructural and ensemble anisotropy on the double wave vector diffusion signal in the Gaussian phase approximation. Finally, as a new application of the double wave vector diffusion signal, we analyze its behavior in a curving fiber, and suggest that the displacement correlation tensor may be used to estimate sub-voxel fiber curvature and deflection angle. The theoretical results are corroborated by computer simulations.