A Fokker-Planck Equation for Spin Relaxation

A method is developed to describe the quantum mechanical motion of a few simple spin systems, with random time-dependent perturbations, in an exact manner, i.e., without reference to perturbation theory. It leads to a Fokker-Planck type diffusion equation. When applied to a spin influenced by a fluctuating local field (Abragam's model), this gives rise to a (microscopic) Bloch-type equation for the spin operators. An application to a system of two identical spins ($S=\frac{1}{2}$) with modulated dipole interaction produces a very complicated diffusion equation in eight variables, which simplifies, however, in a case of restricted rotation. A model Hamiltonian for quadrupole relaxation for $S=1\mathrm{}$ gives a relatively simple result, which reveals an interesting difference between the relaxation in finite and zero external fields.