Nuclear Magnetic Relaxation of Three Spin Systems Undergoing Hindered Rotations

Abstract

The longitudinal nuclear magnetic relaxation of an ensemble of spin systems exposed to a constant magnetic field $H_0\mathrm{k}$ is calculated. Each spin system consists of three identical spin-½ nuclei located at the vertices of an equilateral triangle. Each spin system undergoes hindered rotation about an axis that is perpendicular to the plane of the three spins, and is oriented at an angle $\beta$ with the constant field. Two models for the rotation are studied: In one model there are only three possible equilibrium orientations of each system about its rotation axis and the group makes random jumps between these orientations; in the other model each group performs stochastic rotational diffusion about its rotation axis. Both models lead to results of the same form, which differ only in the definition of the correlation time ${\tau }_c$ of the motion. It is assumed that the relaxation is due to the magnetic dipole-dipole interactions between the nuclei within each group. The calculation is performed by use of the semiclassical form of the density operator theory of relaxation. The treatment includes terms arising from the cross correlation of different dipole-dipole interactions with one another, from the nonzero average of the dipole-dipole interactions, and from the second-order correction to the Zeeman energy due to the dipole-dipole interactions. The relaxation is, in general, the sum of four decaying exponentials. For $\beta =0\mathrm{}$, the relaxation does not decay to zero. By use of an electronic computer, explicit solutions have been calculated for $cos\beta =0, \pm 0.1, \pm 0.2, \cdots , \pm 1.0$ for many values of the correlation time. Also, explicit solutions have been calculated for situations in which the axes of hindered rotation are isotropically oriented. All results are compared with the results of a calculation in which cross correlations are omitted, and are shown to differ significantly. The results are presented in a form which can be compared with experimental data, and used to determine the correlation time of the hindered rotations.