Nuclear spin relaxation by translational diffusion in liquids and solids: high- and low-frequency limits

Nuclear spin relaxation rates due to magnetic dipole coupling between spins of a single species undergoing relative diffusion depend on two-spin correlation functions J (p) ( omega ). The high- and low-frequency limits of J (p) ( omega ) are discussed for liquids, cubic crystals and two- and one-dimensional systems. It is shown that the low-frequency form of J (p) ( omega ) is J (p) (0)-A 3 omega 1/2 in liquids and crystals, A 2 ln omega -1 in two-dimensional systems and A 1 omega -1/2 in one-dimensional systems and expressions for the coefficients A 1 , A 2 and A 3 are derived which are valid for any diffusive model whose long-range, long-time behaviour may be described by the diffusion equation. The high-frequency form of J (p) ( omega ) is derived for a simple-hopping model of spins on one-, two- and three-dimensional lattices and in each case J (p) ( omega ) is of the form B omega -2 . Expressions are derived for the coefficient B, for each dimension, which correctly include the effect of correlations between the hops of the spins.