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

Abstract

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 ${\ensuremath{\tau}}_{2}$ 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 $\ensuremath{\eta}$. The dynamic NMR frequency shifts are described by the functional form $f({\ensuremath{\tau}}_{2})={{(\frac{{\ensuremath{\tau}}_{2}}{{T}_{n}})[1 \ensuremath{-}\mathrm{exp}(\frac{{\ensuremath{-}T}_{n}}{{\ensuremath{\tau}}_{2}})]}}^{\ensuremath{\alpha}}$, where $\ensuremath{\alpha}=1$ for the first-order and $\ensuremath{\alpha}=2$ for the second-order quadrupole splittings. In the slow-motional region, $\frac{{T}_{n}}{{\ensuremath{\tau}}_{2}}\ensuremath{\ll}1$, $f({\ensuremath{\tau}}_{2})$ depends on ${\ensuremath{\tau}}_{2}$ as $1\ensuremath{-}\ensuremath{\alpha}\frac{{T}_{n}}{2{\ensuremath{\tau}}_{2}}$, which approaches unity as ${\ensuremath{\tau}}_{2}\ensuremath{\rightarrow}\ensuremath{\infty}$, corresponding to the rigid limit. In the fast-motional region, $\frac{{T}_{n}}{{\ensuremath{\tau}}_{2}}\ensuremath{\gg}1$, $f({\ensuremath{\tau}}_{2})$ depends on ${\ensuremath{\tau}}_{2}$ as ${(\frac{{\ensuremath{\tau}}_{2}}{{T}_{n}})}^{\ensuremath{\alpha}}$, which approaches zero as ${\ensuremath{\tau}}_{2}\ensuremath{\rightarrow}0$, corresponding to the completely motional averaged limit.