Sidebands of theU-Center-Induced Infrared Absorption Spectra of Alkali-Halide Crystals

A $U$ center (substitutional ${\mathrm{H}}^{-}$ or ${\mathrm{D}}^{-}$ ion) in an alkali-halide crystal is known to give rise to triply degenerate localized vibrational modes that are infrared-active. The $U$-center-induced infrared lattice absorption spectrum shows a characteristic structure consisting of a central prominent peak flanked on both sides by broad sidebands. The central peak arises from one-phonon absorption processes due to localized-mode phonons, and recently it has been studied extensively. The present paper deals with a theory of the sidebands based on two-phonon absorption processes involving one localized-mode phonon and one band-mode phonon, taking account of phonon coupling mechanisms due to the cubic anharmonicity of the crystal and to the crystal second-order electric dipole moment. Formal expressions are derived for the contributions from these two mechanisms to the sideband absorption coefficient, in terms of the phonon Green's-function matrix for the harmonic perfect-host crystal, the cubic anharmonic force constants, and the second-order electric-dipole-moment coefficients. The phonon spectrum of the perturbed crystal is actually taken into account. A simplified expression for the absorption coefficient is obtained for the case where only that part of the anharmonicity and the electric dipole moment which arises from short-range overlap forces between the impurity and its nearest neighbors is considered. Numerical calculations are carried out for the higher-frequency sideband of a KI crystal containing 6×${10}^{17}$ $U$ centers (${\mathrm{H}}^{-}$ or ${\mathrm{D}}^{-}$) per ${\mathrm{cm}}^3$, at 4.3°K, using Hardy's deformation-dipole model for the perfect-host crystal. The $U$ center is described by a phenomenological "effective mass defect" which takes into account approximately the change in force constants at the impurity site, using the experimental value of the localized-mode frequency. The calculated spectrum is in satisfactory agreement with Timusk's experimental data. It is found that the anharmonicity is the dominant phonon-coupling mechanism. It is about ${10}^4$ times stronger than the second-order electric-dipole-moment mechanism near the localized-mode frequency, and about 10 times stronger in the high-frequency limit of the sideband. Some discrepancies in the position of the phonon frequency gap and the relative intensities of the main lines are tentatively attributed to the use of inaccurate phonon data, the effective-mass-defect model for the $U$ center, and the approximate numerical values for the overlap forces.