Relaxation Processes in a System of Two Spins

TLDR

The study extends Abragam and Pound’s method for calculating longitudinal relaxation time T1 to transverse relaxation time T2. Modified Bloch equations were derived to describe the motion of macroscopic magnetic moments in pairs of unlike spins, and the equality of T1 and T2 was verified in paramagnetic solutions with an observed nuclear Overhauser effect in anhydrous hydrofluoric acid. Calculations show that for like spins or paramagnetic solutions, T1 equals T2 in the extreme narrow case, while for unlike spins the longitudinal relaxation is non‑exponential leading to steady and transient Overhauser effects, and transverse relaxation remains exponential; experimental verification in paramagnetic solutions and anhydrous hydrofluoric acid confirmed T1 = T2 and observed Overhauser effects, but the extreme narrow assumption fails to explain hydrogen‑fluorine interactions without accounting for surrounding molecules.

Abstract

Abragam and Pound's method for the calculation of the longitudinal relaxation time ${T}_{1}$ has been extended to the transverse relaxation time ${T}_{2}$. Explicit calculations have been carried out for a pure dipole-dipole interaction, showing that for an interacting pair of like spins, or for nuclei in paramagnetic solution, ${T}_{1}$ is exactly equal to ${T}_{2}$ in the extreme narrow case. For a pair of interacting unlike spins, it is shown that the longitudinal components of the magnetic moments do not have simple exponential decays. This gives rise to a steady and transient Overhauser effect. The transverse components, however, have in all cases, simple exponential decay defined by a single relaxation time ${T}_{2}$. A set of modified Bloch's equations is found, giving the correct equation of motion of the macroscopic magnetic moments of such a system of pairs of unlike spins.The equality of ${T}_{1}$ and ${T}_{2}$ has been verified in paramagnetic solutions, and a nuclear Overhauser effect has been observed in anhydrous hydrofluoric acid. If one assumes that the extreme narrow case corresponds to the actual motion, the experimental results are not consistent with the picture of a pure dipole-dipole interaction between the hydrogen and fluorine nuclei of a molecule without taking into account the effect of the other molecules.