Longitudinal spin relaxation in the random-two-frequency model

The spin depolarization is investigated by random transition processes between two rotation frequencies ω₁ and ω₂, where the spin and the two rotation frequencies may have arbitrary orientations in three-dimensional space. A general expression for the dynamics of the spin polarization is derived in the Laplace domain. For rapid transitions between both frequencies a spin precession with the average frequency 1 2(ω₁+ω₂) is found, which is damped in accordance with motional narrowing. An average of the results over Gaussian distributions of the rotation frequencies ω₁ and ω₂ is also performed. For large transition rates the result of the static Kubo-Toyabe theory is recovered.