Nuclear spin-lattice relaxation and spin diffusion in an inhomogeneous field

The theory of spin diffusion is extended to the case of spin-lattice relaxation and spin diffusion in an inhomogeneous magnetic field. Two coupled equations describing the mutual relaxation and the spin diffusion of nuclear magnetization and dipolar energy were obtained using the method of the nonequilibrium state operator. The equations were solved for short- and long-time approximations corresponding to the direct and diffusion relaxation regimes. It is shown that at the beginning of the relaxation process in the mixed state of the conventual superconductor the direct relaxation regime is dominant. The nuclear magnetization decays with a stretched exponential (α=0.5) while the dipolar energy decreases exponentially. Then the relaxation regime changes both for nuclear magnetization and dipolar energy, to the diffusion one described by the exponential time dependence which agrees with the experiment. The radii of the diffusion barrier and the spin-diffusion coefficient were estimated.