Two Approaches to the Theory of Spin Relaxation: I. The Redfield Langevin Equation; II. The Multiple Time Scale Method

Abstract

Spin relaxation is one example of a general class of relaxation processes where the subsystem of interest is weakly coupled to a large bath or reservoir. The essential feature of such processes is the wide separation in the characteristic time scales of relaxation of the subsystem and the bath. If the coupling of the spin subsystem to the bath is ‘weak’ one finds that the spins relax on a ‘slow’ time scale relative to the lattice motion of the bath. One consequence of this separation of time scales is that the spin subsystem density matrix, to a certain degree of approximation, may be described by a master equation. To lowest order this equation of motion is only valid on the ‘slow’ time scale and is known as the Redfield equation. Another consequence is the existence of simple linear laws for the relaxation in the magnetization or other dynamical variables of interest. For many spin systems, in lowest order, the equations of motion for the macroscopic magnetization have the form of the Bloch equations which are also only valid on the slow spin time scale. A third consequence of the separation of time scales is that the relaxation times which appear in the Bloch or Redfield equations may be expressed as Fourier transforms of time correlation functions. To lowest order the time dependence of the correlation functions involves only the fast lattice motion.