Theory of the Dielectric Constant in the Individual-Ion Model of Ionic and Rare-Gas Crystals

We present an exact formal treatment of the effect of quantum dipolar fluctuations on the dielectric constant, in the framework of the individual-ion (atom) model of ionic (rare-gas) crystals. Since we concentrate primarily on the study of fluctuations, which manifest themselves in the dipole-dipole interaction, we disregard any overlap effects at this stage. The formalism is first developed in the case of the static dielectric constant, and afterwards it is generalized to the problem of the refractive index. In a first step, the classical Lorentz field effect is derived exactly in a nonperturbative way, whereby the approximate validity of the usual classical treatment is established. After the classical effect has been diagonalized, the remaining perturbation, which describes the dipolar fluctuations only is studied in a second step. For this purpose perturbation theory is used, and the lowest-order corrections which are considered are proportional to $R^{-6\mathrm{}}$ ($R$ denotes a typical interatomic distance). In addition to the ordinary Van der Waals energy, this leads to a set of new field-dependent terms which are expressed in terms of various anharmonicity parameters of the isolated atoms. Finally, the expression for the correction to the polarizability of the isolated atoms which results from this effect is obtained.