  We consider nuclear spin-lattice relaxation rate resulted from a fractional
diffusion equation for anomalous rotational wobbling in a cone. The mechanism
of relaxation is assumed to be due to dipole-dipole interaction of nuclear
spins and is treated within the framework of the standard Bloemberger, Purcell,
Pound - Solomon scheme. We consider the general case of arbitrary orientation
of the cone axis relative the magnetic field. The BPP-Solomon scheme is shown
to remain valid for systems with the distribution of the cone axes depending
only on the tilt relative the magnetic field but otherwise being isotropic. We
consider the case of random isotropic orientation of cone axes relative the
magnetic field taking place in powders. Also we consider the case of their
predominant orientation along or opposite the magnetic field and that of their
predominant orientation transverse to the magnetic field which may be relevant
for, e.g., liquid crystals. Besides we treat in details the model case of the
cone axis directed along the magnetic field. The latter provides direct
comparison of the limiting case of our formulas with the textbook formulas for
ordinary isotropic rotational diffusion. We show that the present model enables
one to obtain naturally the well known power law for Larmor frequency
dependence of the spin-lattice relaxation rate. The latter is observed in some
complex systems. From this law the dependence of the fractional diffusion
coefficient on the fractional index is obtained to have a rather simple
functional form. The dependence of the spin-lattice relaxation rate on the cone
half-width for the case of ordinary rotational diffusion yields results similar
to those predicted by the model-free approach.
