Nuclear Magnetic Relaxation by Intermolecular Dipole-Dipole Interactions

Abstract

The contribution of intermolecular dipole-dipole interactions to the nuclear magnetic relaxation of identical spin-½ nuclei at equivalent positions in spherical molecules in a liquid is calculated. The assumptions made are that: (1), the effect of cross correlations of different dipole-dipole interactions is negligible, so that the relaxations of the longitudinal and transverse components of the nuclear magnetization are simple exponential decays with relaxation times $T_1$ and $T_2$, respectively; (2), the motions of the molecules can be considered to be translational and rotational diffusion; and (3), the correlation time ${\tau }_0\equiv \left(\frac{2a^2}{D}\right)$, where $a$ is the radius of a molecule and $D$ is the translational diffusion coefficient, is sufficiently short that ${\left({\omega }_0{\tau }_0\right)}^2\ll 1$, where ${\omega }_0$ is the Larmor frequency of the nuclei. As a result of the short correlation time assumption (3), the contributions of the intermolecular interactions to ($\frac{1}{T_1}$) and ($\frac{1}{T_2}$) are found to be the same and are given by an infinite series, the first three terms of which are ${\left(\frac{1}{T_1}\right)}^{\prime \prime }={\left(\frac{1}{T_2}\right)}^{\prime \prime }=\frac{n\pi {\gamma }^4{\hslash }^2}{5aD}\left[1+0.233{\left(\frac{b}{a}\right)}^2+0.15\mathrm{}{\left(\frac{b}{a}\right)}^4+\cdots \right],$ where $n$ is the number of spins per unit volume, $\gamma$ is the gyromagnetic ratio of each nucleus, and $b$ is the distance of each nucleus from the center of the molecule in which it is contained. The first term in the series is the result obtained in previous calculations in which the effect of the rotations of the molecules was neglected. In a typical case in which $\left(\frac{b}{a}\right)\approx \frac{1}{2}$, the second and third terms are 6.8% of the first term.