A low-field paramagnetic nuclear spin relaxation theory

A low-field theory of paramagnetically enhanced proton spin-lattice relaxation is developed resulting in closed analytical expressions. The theory describes paramagnetically enhanced water proton spin-lattice relaxation in complexes of transition metal ions with electron spin quantum number S=1. In the low-field regime the electron spin system is dominated by a static or permanent zero-field splitting interaction which is much larger than the Zeeman interaction. The electron spin is thus quantized in the molecular fixed principal frame of the static ZFS-interaction rather than in the laboratory fixed frame. In the molecular fixed frame the zero-field splitting Hamiltonian is characterized by an axial parameter (D) and a rhombic parameter (E). It is shown how the relative magnitude of D and E strongly influence the magnitude of the paramagnetically enhanced proton spin lattice relaxation rate. In describing electron spin relaxation in the molecular fixed frame, we consider a transient ZFS-interaction and a reorientation modulated Zeeman interaction as the two main relaxation mechanisms. Then, using Redfield theory we derive expressions for the relevant electron spin relaxation rates in terms of spectral densities. In contrast to the Zeeman or high field regime, the electron spin system in the low field limit does not have a permanent magnetic moment. The lack of this magnetic moment implies that electron spin-lattice relaxation processes do not influence the paramagnetically enhanced proton spin-lattice relaxation rates. Instead three spin–spin relaxation rates determine the enhanced proton spin-lattice relaxation profile. We express the electron spin–spin relaxation rates in terms of spectral densities determined by the mean square value of the transient zero-field splitting interaction, its characteristic correlation time, and the reorientational correlation time of rank 1.