Propagator representation of anomalous diffusion: the orientational structure factor formalism in NMR.

The radial Fourier transform for the isotropic space with a fractal dimension is discussed. The moments of diffusive displacements with non-Gaussian propagators arising as solutions of fractional diffusion equations are calculated. The Fourier propagator is applied to NMR correlation and spectral density functions in context with the orientational structure factor formalism. It is shown that the low-frequency molecular fluctuations of liquids in porous media with strong or forced adsorption at surfaces are due to reorientations mediated by translational displacements caused by surface diffusion of the adsorbate molecules. In terms of this formalism, field-cycling NMR experiments provide information on the static and dynamic fractal dimensions related to surface diffusion. The experimental results for liquids in porous silica glass can be explained by a surface fractal dimension df=2.5, where the mean squared displacement scales as <r(2)(t)> proportional, variantt(2/dw) with dw=1 (ballistic transport), if the surface population can exchange with the bulklike phase in the pores, and with dw=2, if the bulklike phase is frozen. The former dynamics is interpreted in terms of bulk-mediated surface diffusion.