Optical Spectra of Orientationally Disordered Crystals. III. Infrared Spectra of the Sound Waves

The theory of the absorption of light by the translational vibrations of orientationally disordered crystals has been extended to low frequencies where the vibrations are short‐wavelength sound waves whose density of states can be represented by the Debye model. Both the integrated intensity of a normal vibration and the density of states are proportional to the frequency squared, so the absorptivity is proportional to the fourth power of the frequency. The proportionality constant can be related to the difference between the limiting low‐ and high‐frequency permittivities of the translation band. The absorptivity of ice has been measured below 45 cm⁻¹ in order to test the theory. As predicted, it is approximately proportional to the fourth power of the wavenumber. At low frequencies the absorptivity becomes approximately proportional to the square of the wavenumber and is probably caused by two‐phonon difference bands involving high‐frequency acoustic phonons. The microwave absorptivity reported by others appears to be consistent, at least approximately, with this square line extrapolated, which suggests that it also is due mainly to two‐phonon difference bands. The absorptivity is less than predicted due to two approximations in the theory, namely, the neglect of the strong orientational correlation in ice and the assumption that $R$ defined by Eq. (20) is independent of the type of internal coordinate.