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.