Spin relaxation by dipolar coupling: From motional narrowing to the rigid limit

A coupled system of two molecules bearing spins of 1/2, which are allowed to diffuse relative to each other, is considered. By using a symmetry-adapted basis operator set, the overall density matrix equation is decoupled into two equations for the time-resolved isochromat components, the sum of which yields the observed signal. The appropriate stochastic Liouville equation is solved by a combination of eigenfunction expansion and finite-differences for the angular and radial relative motions, respectively. A full range of spectra from classic Pake patterns in the rigid limit to motionally narrowed Lorentzians is recovered. As an extension of the above approach, the solid-echo experiment is described in terms of the ensemble-averaged isochromats. Homogeneous transverse relaxation times (T2) as a function of the translational diffusion coefficient (DT) are obtained from simulating SECSY (spin-echo correlation spectroscopy) signals, which show distinct T2 minima vs DT. The present method of separating the quantum and stochastic classical variables constitutes a useful approach for treating multiquantum statistical systems, and it can be generalized to an arbitrary number of spins as shown in a companion paper. In the present study we obtained the usual linear dependence of T2 on DT in the motional narrowing (or Redfield) limit, whereas in the slow motional regime a DT−1/2 dependence is observed, consistent with studies of rotational diffusion. Varying the distance of maximum separation between the two spins (rmax) has virtually no effect on the location of the T2 minimum with respect to DT, implying that the onset of slow motion is essentially independent of rmax.