Depolarization and electric-field frequency shift ofHe3by magnetic-field inhomogeneities: Local theory for short mean free path and solution inHe4

Abstract

The local theory of relaxation of $^{3}\mathrm{He}$ by Brownian motion in an inhomogeneous magnetic field, originally proposed by Gamblin and Carver [Phys. Rev. 138, A946 (1965)] and by Schearer and Walters [Phys. Rev. 139, A1398 (1965)], is reconsidered in detail to clarify (or not) the approximations involved. Notably, a discussion is given of frames rotating at time-dependent angular velocities (both in magnitude and in direction), and attention is drawn to the often overlooked long tail that confinement adds to velocity correlation functions. A stepwise application of the Redfield theory of relaxation eventually provides quantitative predictions for the rate of depolarization and the linear-in-electric-field frequency shift, in terms of the components of the local inhomogeneity tensor and the usual parameters (nuclear magnetic resonance frequency, diffusion coefficient, mean time between collisions, etc.), valid in the common situation of mean free path much smaller than the size of the container. These predictions agree with almost all results of previous publications, most of which are based on the global approach which evaluates correlation functions by reference to the diffusion equation in spite of its limitations for short delays.