# Polarizability and two-quantum excitations in alkali halides and deep impurity states in rare-gas solids

## Abstract

The general theory of the electronic dielectric constant of crystals, developed in a previous paper in the framework of an individual-ion model, is analyzed for diatomic systems. Use is made of an effective two-state model for the individual atoms. An exception is made, however, for certain terms in the polarizability which depend very strongly on the higher excited states, whose effect is thus taken into account, in an average-excitation-energy approximation. The final result of this analysis is a simplified expression for the frequency-dependent polarizability of an individual atom in the crystal, which includes explicitly the quantum correction arising from the induced dipole-dipole interaction. In a first application of these results, we carry out a detailed study of the static polarizability of alkali halide crystals, on the basis of a reasonable assumption for the polarizability of a free ion in an excited state. The dipole-dipole correction in the polarizability is found to be positive and to vary from 2 to 8%. This effect is in good quantitative agreement with a correction which has been derived from an empirical analysis of dielectric-constant data for the alkali halides. For the effects of frequency dispersion in the polarizability, we also find reasonable agreement with similar empirical results for the alkali halide ions. As a second type of application, we consider the problem of deep impurity states in crystals. An explicit expression for the electronic frequency shift of a substitutional impurity is obtained from the study of the singularities of the polarizability. In particular, this leads to a simple expression for the van der Waals constant for an excited impurity interacting with a matrix atom in the ground state. Application of this formula to rare-gas and molecular impurities in rare-gas matrices leads to surprisingly good agreement with empirical results, as obtained from an analysis of experimental shifts in terms of 6-12 potentials. As a last application we study the oscillator strength of two-quantum excitations (double excitons), which arise as a consequence of the quantum dipole fluctuations. For the alkali halides we find that this oscillator strength ranges from 4.5 to 20% of that of the ordinary one-quantum excitations in the NaCl structure, while rising to 30% in the CsCl structure. © 1970 The American Physical Society.