Dipolar relaxation in a many-body system of spins of 1/2

The method utilized in Paper I @J. Chem. Phys. 112, 1413 ~2000!# for treating the density matrix equation for a two-spin system in the presence of the dipolar interaction that is randomly modulated by translational diffusion, is extended to a many-body system of identical spins of 1/2. Generalized cumulant expansions are used, which allow one to take full advantage of the statistical independence of the motions of spins. In the high-temperature approximation ~appropriate for dilute solutions!, for a single nonselective pulse, the symmetry of the problem allows one to obtain a compact ordered binomial expression for the free-induction decay signal that is related to the two-particle solution, and it still contains the two spin-isochromat components. The latter are evaluated by solving the corresponding stochastic Liouville equation, which allows one to recover in a unified way the two limiting cases including Anderson’s result for statistical broadening in a rigid lattice and the classical Torrey–Bloembergen–Redfield expression for the motional narrowing, as corrected by Hwang and Freed. The line shape expression in the thermodynamic limit, i.e., for large numbers of particles in a macroscopic volume, is obtained. It is found that the many-body dipolar line shapes are very close to Lorentzians over the entire motional range studied, with the linewidths proportional to the spin concentration, as predicted earlier for the limiting cases. Linewidths plotted versus the values of the translational diffusion coefficient clearly show the solid-state limit, the motional-narrowing limit, and the intermediate region. The method is extended to describe the behavior of the many-body system in a solid-echo sequence. This enables one to obtain the homogeneous T2’s over the whole range of motions. A minimum in T2 is found at approximately the same value of translational diffusion coefficient as was found for the two-spin case in Paper I. © 2000 American Institute of Physics. @S0021-9606~00!02903-2#