Classical Limit of Demagnetization in a Field Gradient

We calculate the rate of decrease of the expectation value of the transverse component of spin for spin-1/2 particles in a magnetic field with a spatial gradient, to determine the conditions under which a previous classical description is valid. A density matrix treatment is required for two reasons. The first arises because the particles initially are not in a pure state due to thermal motion. The second reason is that each particle interacts with the magnetic field and the other particles, with the latter taken to be via a 2-body central force. The equations for the 1-body Wigner distribution functions are written in a general manner, and the places where quantum mechanical effects can play a role are identified. One that may not have been considered previously concerns the momentum associated with the magnetic field gradient, which is proportional to the time integral of the gradient. Its relative magnitude compared with the important momenta in the problem is a significant parameter, and if their ratio is not small some non-classical effects contribute to the solution. Assuming the field gradient is sufficiently small, and a number of other inequalities are satisfied involving the mean wavelength, range of the force, and the mean separation between particles, we solve the integropartial differential equations for the Wigner functions to second order in the strength of the gradient. When the same reasoning is applied to a different problem with no field gradient, but having instead a gradient to the z-component of polarization, the connection with the diffusion coefficient is established, and we find agreement with the classical result for the rate of decrease of the transverse component of magnetization. The corresponding result for this rate in the absence of collisions is much greater. An approximate value for the He self diffusion coefficient based on a hard core potential is found to be 23.1 mm/sec at room temperature and 7 atmospheres. This falls between the results obtained in an NMR experiment having these values of temperature and pressure, in which the spatial gradient of the field is time dependent, and earlier experiments using other techniques, corrected for temperature and pressure variation.