An exact analytical solution for the evolution of a dipole-dipole interacting system under spherical diffusion in magnetic resonance experiments

A model system consisting of an isotropic ensemble of spin pairs, where dipole-dipole interaction is assumed to be effective only within each pair, is considered. The ideal segment connecting the spins in a couple has a fixed length but is free to rotate following a diffusion dynamics. This allows the free induc- tion decay (FID) to be derived non-perturbatively by solving the appropriate Dyson equation associated to the problem. Motional narrowing can be described analytically in terms of only two parameters, i.e. the coupling constant of the interaction hamiltonian, b, and the orientational diffusion coefficient D. Salient features of the transverse correlation function thus obtained are discussed, and a comparison with numerical simulations performed with the software SPINEVOLUTION is presented. Interpreting b and D as effective parameters describing multiple interactions of a single spin with its neighbors in a real sys- tem, the analysis of published experimental data on poly(ethyl acrylate) has been carried out. It is found that for temperatures higher than and not too close to the glass transition, the results are the same as those found within the Anderson-Weiss approach by assuming a single time exponential decay of the average dipole-dipole interaction. On the other hand, as D tends to zero, FID oscillations characteristic of a rigid lattice show up.