Theory of the Optical Properties of Quartz in the Infrared

A theoretical study has been carried out of the eight non-degenerate optical vibrations of $\alpha$ quartz including the problem of their infrared and Raman intensities. The investigation consists of three parts: First, the atomic motions and frequencies are calculated on the basis of a valence force model. It is shown that the 207 ${\mathrm{cm}}^{-1\mathrm{}}$ vibration involves atomic motions very similar to those of the $\alpha -\beta$ transformation. Secondly, a general discussion is given of infrared absorption in complex crystals, which shows that in quartz the intensities are determined by the atomic motions through 12 effective charge parameters. The intensities are calculated in good agreement with experiment on the assumption of a 2-charge model suggested by the valence nature of quartz. It is shown that a 1-charge model, the usual model for an ionic crystal, cannot account for the observed intensities. Finally, the relative Raman intensities are calculated with no adjustable parameters in good agreement with experiment on the basis of a simple assumption about the atomic polarizabilities. The calculation accounts for the surprising weakness of the 1082-${\mathrm{cm}}^{-1\mathrm{}}$ stretching vibration in the Raman effect. The three parts of the investigation are mutually dependent, since the infrared and Raman intensities depend in an essential way upon the atomic motions corresponding to each frequency. It is shown that a consideration of the Raman intensities as well as the usual comparison of frequencies is required to determine the bending constants of the valence force model. It is inferred from the success of the calculations that the three principal assumptions of the present work, namely the valence force model for the vibrations, the 2-charge model for the infrared intensities, and the simple Raman model, are all applicable for quartz.