Dipole-dipole nuclear spin relaxation

Abstract

The transverse spin relaxation of a spin I in a IS pair is analysed for the dipole-dipole relaxation mechanism. It is shown that, when for the S-spin there is an additional efficient relaxation mechanism, which is of a second order tensorial rank, there can exist substantial corrections to the Solomon-Bloembergen equation for T ₂. The correction terms are found to be non-negligible in the non-extreme narrowing limit. The correction terms are due to a cross correlation effect between the dipole-dipole interaction and the interaction causing the efficient relaxation of the S spin. As shown in the Appendix the correction appears as a near divergence of a fourth order term in the Redfield type expansion of the equation of motion of the density matrix. The explicit expressions for T ₂ are, however, derived using a Liouville operator formalism combined with a perturbation expansion. For the relaxation of the S-spin a zero field splitting term is considered for a paramagnetic system with S ≥ 1. Similarly for a nuclear spin S the quadrupolar interaction provides an efficient relaxation mechanism when S ≥ 1. For these cases explicit examples are worked out and for example if S = 1 and the principal axis system of the zero field splitting term or the quadrupolar interaction coincides with the dipole-dipole vector the cross correlation corrections are such that the high field value of T ₂ is 2·5 times the value given by the Solomon-Bloembergen equation.

It is also argued that a similar cross correlation effect can appear in spin one half systems if three or more spins are involved. It thus appears that the quantitative analysis of T ₂ data for systems outside the extreme narrowing regime is even more complicated than has previously been appreciated.