Interaction of exchange and differential relaxation in the saturation recovery behavior of the binary spin-bath model for magnetization transfer

Most closed-form analytical solutions of the binary spin-bath are difficult to interpret in terms of underlying physics. The key notions are the presence of a kinetic and a thermal equilibrium and that the time course of saturation recovery under conditions of fast exchange can be understood as conjoint relaxation and lossless transfer. By introducing a suitable parameter, it is shown how exchange and differential relaxation counteract each other: the amount of transferred saturation (transfer term) is altered and the kinetic equilibrium appears slightly disturbed (difference term). Although the factorization formally represents the general solution of saturation recovery in the binary spin-bath, this interpretation applies only to the case of fast exchange and slow relaxation. By calculating the set of parameters for a wide range of hypothetical relaxation rates, it was shown that the difference term is crucial to describe the transition to the slow-exchange limit. The transfer term vanishes as the two pools appear decoupled in this approximation.