Relaxation of spins due to a magnetic field gradient, revisited; Identity of the Redfield and Torrey theories

There is an extensive literature on magnetic gradient induced spin relaxation. Cates, Schaefer and Happer (CSH) in a seminal paper, have solved the problem in the regime where diffusion theory (the Torrey equation is applicable using an expansion of the density matrix in diffusion equation eigenfunctions and angular momentum tensors. McGregor has solved the problem in the same regime using a slightly more general formulation using Redfield theory formulated in terms of the auto-correlation function of the fluctuating field seen by the spins and calculating the correlation functions using the diffusion theory Green's function. The results of both calculations were shown to agree for a special case. In the present work we show that the eigenfunction expansion of the Torrey equation yields the expansion of the Green's function for the diffusion equation thus showing the identity of this approach with that of Redfield theory. The general solution can also be obtained directly from the Torrey equation for the density matrix. Thus the physical content of the Redfield and Torrey approaches are identical. We then introduce a more general expression for the position autocorrelation function of particles moving in a closed cell, extending the range of applicability of the theory.