Quantum Theory of the Polarizability of an Idealized Crystal

The quantum-mechanical polarizability of an idealized insulating crystal is calculated by the semiclassical method, in which the electromagnetic radiation of frequency small compared with electronic resonance absorption frequencies is treated classically. A generalization of the Lorentz "internal field" effect emerges naturally from the treatment. Correction terms to the Lorentz value for the polarization of a crystal, $n_0\left[\frac{{\alpha }_a}{(1-\frac{4\pi n_0{\alpha }_a}{3})}\right]\mathcal{E}$, where ${\alpha }_a$ is the polarizability of an isolated atom, $n_0$ the density of atoms, and $\mathcal{E}$ the applied field, arise from three main sources: (1) exchange and overlap effects, (2) higher order terms in the atomic interactions than dipole-dipole terms, and (3) second and higher powers of the small parameter $\frac{4\pi n_0{\alpha }_a}{3}$. These correction terms, which are shown to be important in real crystals, are briefly discussed, although more must be known about specific wave functions in order to make quantitative calculations. The connection is pointed out between the propagation of excitons and the Lorentz effect.