Dynamic frequency shifts of NMR fine-structure splittings under orientational diffusion motion in condensed matter

By employing the restricted ensemble-averaging process developed for completely anisotropic systems based on the solution of an isotropic rotational diffusion equation, analytical expressions have been calculated for dynamic nuclear-magnetic-resonance (NMR) frequency shifts of NMR fine-structure splittings under orientational diffusion motion. The calculations are so formulated that the experimental orientational motional correlation time ${\tau }_2$ at a given sample temperature can be determined in terms of the dominant or shortest nuclear relaxation time $T_n$ at that temperature and the rigid-limit NMR fine-structure coupling constant $Q$ and asymmetry parameter $\eta$. The dynamic NMR frequency shifts are described by the functional form $f\left({\tau }_2\right)={\left\{\right(\frac{{\tau }_2}{T_n}\left)\right[1\mathrm{} -\exp \left(\frac{{-T}_n}{{\tau }_2}\right)\left]\right\}}^{\alpha }$, where $\alpha =1\mathrm{}$ for the first-order and $\alpha =2\mathrm{}$ for the second-order quadrupole splittings. In the slow-motional region, $\frac{T_n}{{\tau }_2}\ll 1$, $f\left({\tau }_2\right)$ depends on ${\tau }_2$ as $1-\alpha \frac{T_n}{2{\tau }_2}$, which approaches unity as ${\tau }_2\to \infty$, corresponding to the rigid limit. In the fast-motional region, $\frac{T_n}{{\tau }_2}\gg 1$, $f\left({\tau }_2\right)$ depends on ${\tau }_2$ as ${\left(\frac{{\tau }_2}{T_n}\right)}^{\alpha }$, which approaches zero as ${\tau }_2\to 0$, corresponding to the completely motional averaged limit.