Paramagnetic relaxation: Direct and Raman relaxation of spin S=12

Paramagnetic relaxation in solids is a vast subject, about as vast as the range of manifestations of electron spin in matter. It is a complex subject as well: it is the interface between paramagnetic centres – be it transition metal ions, radicals or defects – and quantized vibrations: phonons. So it requires an understanding of both these phonons and those paramagnetic centres. Moreover, contrary to the case of integer spin, for half-integer spin the coupling between electron spins and phonons is indirect. Two interactions are needed, the spin–orbit interaction between the spin and the orbits of the paramagnetic centre and the orbit–phonon interaction between the latter and the phonons.

The present article is an effort to navigate the theory of this extensive subject for spin $S=\frac{1}{2}$ and aims to derive the main properties of the two most important mechanisms: direct and red Raman relaxation. It tries to do so from first principles, that is, it includes a generalized, but fundamental description of the vibrational states, the orbital and spin states on the one hand, and the orbit–phonon and spin–orbit interaction on the other. Based on these descriptions it derives the transition matrix elements responsible for paramagnetic relaxation, following the original approach of Van Vleck for paramagnetic centres with spin $S=\frac{1}{2}$, a relatively weak spin–orbit interaction and embedded in an insulating, diamagnetic solid. Subsequently phonon statistics are included to derive the paramagnetic relaxation rates. No effort is done to review the vast body of experimental work on the subject.