Comparison of Hindered-Rotor and Tunneling Models for Dipolar Impurities in Cubic Crystals

In alkali-halide crystals, dipolar impurities such as ${\mathrm{CN}}^{-}$ and ${\mathrm{OH}}^{-}$ exhibit energy-level spacings of 0.1-2 ${\mathrm{cm}}^{-1\mathrm{}}$ due to a reorientational type of motion. One calculation of these energy-level spacings is on the basis of a hindered-rotor model, which expresses the rotational motion as a linear combination of spherical-harmonic wave functions. This model is applicable when small or medium potential-energy barriers hinder the rotation. The tunneling model calculates these same energy levels by allowing localized harmonic-oscillator wave functions to overlap the equivalent wave functions in neighboring (but differently oriented) potential wells. The latter model was derived by Gomez et al. only for off-center point masses; furthermore, this model is primarily applicable to cases involving medium or large potential barriers. The present work reformulates the tunneling-model results, so that they are applicable to the cases of rotating polyatomic impurities. It then becomes evident that the two models predict quite similar energy-level spacings over a large range of intermediate barrier heights. An interesting prediction is obtained by utilizing both the hindered-rotor and tunneling models to analyze experimental results from the KCI:${\mathrm{Li}}^{+}$ system. In contrast to the tunneling model alone, the present combination of models provides a straightforward interpretation for the two observed resonance lines. Also it is suggested that a third line may not have been observed because of the limited energy (≤1.2 ${\mathrm{cm}}^{-1\mathrm{}}$) of microwave photons utilized in the resonance experiment.