General treatment of paramagnetic relaxation enhancement associated with translational diffusion.
A theory of nuclear spin relaxation in isotropic liquids for nuclear spins interacting with electron spins, residing in other molecules (the outer-sphere relaxation), is presented. The approach, valid outside of the Redfield limit for electron spin relaxation, is an extension of the Swedish slow motion theory [Benetis et al., Mol. Phys. 48, 329 (1983); Nilsson and Kowalewski, J. Magn. Reson. 146, 345 (2000)] for inner-sphere relaxation. It is demonstrated that the outer-sphere relaxation rate can be expressed as an integral of a product of a translational diffusion correlation function and a function analogous to the inner-sphere spectral density. A numerical implementation of the theory is described and applied to a large number of realistic parameter sets for S = 7/2 and S = 1, which may correspond to Gd(III) and Ni(II) systems. It is shown that the outer-sphere contribution is relevant and should be included into the analysis of nuclear magnetic relaxation dispersion relaxation profiles, especially for slow relative translational diffusion and fast molecular tumbling.
Simple analytical approximation of the longitudinal electronic relaxation rate of Gd(III) complexes in solutions
More and more sophisticated theoretical models have been developped for a correct description of the relaxation of the electronic spin S = 7/2 of the Gd(III) paramagnetic complexes used as contrast agents in magnetic resonance imaging (MRI). Both the static zero field splitting (ZFS) modulated by the random rotation of the complex and the transient ZFS due to the very fast distortion of this entity must be included in these models. This leads to rather complicated analytical expressions, from which it is difficult to evaluate the respective effects of the physically relevant parameters. However, in the Redfield limit of the theory of electronic spin relaxation, we show that the longitudinal relaxation function G∥(t) has a quasi-monoexponential decay characterized by a unique relaxation rate 1/T1e, which has a simple expression in terms of the applied magnetic field B0, of the static and transient ZFS parameters, and of the rotational and vibrational correlation times. For the typical investigated Gd(III) complexes, this expression is shown to have a very satisfactory accuracy for B0 < 10 T. The various physical parameters as well as the range of validity of the relaxation approximation are discussed in detail.
Nearly exponential quadrupolar relaxation. A perturbation treatment
Abstract Quadrupolar nuclear spin relaxation is treated by a perturbation method, which shows that the relaxation is nearly exponential provided that the effective spectral density is only weakly frequency dependent. Approximate analytical expressions for the relaxation rates are derived and tested against the experimentally accessible apparent relaxation rates, for a fast-exchange two-state model. The results for spin I = 5 2 and 7 2 indicate that the analytical expressions are accurate to within a few percent in most experimental situations.