Unitary Theory of Dynamic Polarization of Nuclear Spins in Liquids and Solids

Abstract

The dynamic polarization of nuclear spins, or spin pumping as it is often called, is discussed from a unitary point of view by means of a stochastic model. The Hamiltonian of this model describes two spins, $I$ and $S$, which are influenced by a constant external magnetic field $H_0$ and a rotating filed $H_1$, which have a dipole, or a contact, interaction, and of which the $S$ spin experiences a randomly fluctuating "local" field $H_L\mathrm{}\left(t\right)\mathrm{}$. The coefficients of the dipole, or the contact interaction, are also random functions of time, all of which serves to replace, as accurately as possible, the influence of the surrounding particles on the two spins. In order to preserve the consequences of the principle of detailed balance, which is essential, the amplitude spectrum of these random functions has to be complex, i.e., nonreal, in a manner depending on the temperature. With several transformations, including a complex stochastic rotation, the interaction representation is obtained, from which equations of motion for the spin polarizations are found by means of an iteration and truncation scheme. In calculating the coefficients of these equations, extensive use is made of the Fokker-Planck equation characterizing the stochastic rotation. The equations are solved with some simplifying assumptions, and an explicit closed-form expression for the polarization of the $I$ spin is derived. This yields the well-known results for the extreme cases of a line-narrowed liquid and of a solid, for the dipole as well as for the contact interaction. In intermediate cases, it yields hybrid effects as a function of all the variables of the problem. Where comparisons are possible, they differ markedly from earlier predictions. This discrepancy is attributed to a defect of other derivations, in which nondiagonal elements of the density matrix are neglected.