  In this work we investigate the emergence of the localization regime for
diffusion in various geometries: inside slabs, inside cylinders and around rods
arranged on a square array. At high gradients, the transverse magnetization is
strongly attenuated in the bulk, whereas the macroscopic signal is formed by
the remaining magnetization localized near boundaries of the sample. As a
consequence, the signal is particularly sensitive to the microstructure. Our
theoretical analysis relies on recent mathematical advances on the study of the
Bloch-Torrey equation. Experiments were conducted with hyperpolarized xenon-129
gas in 3D-printed phantoms and show an excellent agreement with numerical
simulations and theoretical predictions. Our mathematical arguments and
experimental evidence indicate that the localization regime with a
stretched-exponential decay of the macroscopic signal is a generic feature of
diffusion NMR that can be observed at moderately high gradients in most NMR
scanners.
