Optical phonon (${\omega }_{\mathbf{q}}\to {\omega }_0>0\text{ as }q\to 0$)

An optical phonon is a quantum of vibration on an optical phonon branch, involving relative motion of atoms within a primitive cell. Its zone-center frequency can remain nonzero. Some such modes couple strongly to light when the relative displacement changes electric polarization; the name does not imply that every optical phonon is optically active. Their finite energies produce thresholds for electron emission and thermally suppressed absorption at low temperature.

In a polar insulating crystal, a band with negative real permittivity can occur between the transverse and longitudinal optical resonances. A lossless bulk transverse wave is evanescent there, producing high normal-incidence reflectance. Damping rounds the edges and permits absorption. The reflection minimum beyond the band can occur where the refractive index approaches one; it is not exactly the zero of permittivity.

A rattling mode is a relatively soft, often localized vibration of a guest atom within a more rigid surrounding cage. An approximately harmonic narrow-frequency contribution can be modeled as an Einstein solid term with frequency $\Omega$ and degeneracy $g$, giving the heat capacity in the title. It freezes out exponentially at low temperature and approaches $gR$ per mole at high temperature, so it can bend a plot of $C/T$ versus $T^2$ away from its pure electronic-plus-acoustic straight line.

Calorimetry of Ba4Na2Ge25 shows an additional low-energy contribution fitted by an Einstein energy $\hslash \Omega /k_B\simeq 32\mathrm{K}$, assigned to a barium guest site.^[ref] This is a measured-material example; the narrow-mode description remains an approximation to a crystal's full vibrational spectrum.