Optical Absorption in Ionic Crystals Involving Small Polarons

Abstract

A theory of optical absorption by electrons in crystals is given for two cases: (1) for transitions from a wide valence band to a narrow conduction band; (2) for transitions from one narrow band to another. All coupling of electrons in the wide valence band with the phonons is ignored while for the narrow bands coupling with the longitudinal optical phonons is treated by use of the wave functions of small polaron theory. Dispersion of phonon frequencies, interaction with acoustical modes, and all effects involving spin are neglected. Transition probabilities between states are obtained by standard time-dependent perturbation theory with the electron-radiation interaction as a perturbation. Indirect transitions via a third band and exciton absorption are qualitatively discussed. Consideration of the effects of an interaction quadratic in the normal coordinates is given for the case of electronic interaction with a single mode of lattice vibration, and Urbach's rule is discussed using this model. The applicability of the theory to the transition metal oxides and alkali halides is considered.