Rattling mode ($C_E=gR{x}^2{e}^x/(e^x-1{)}^2,x=\hslash \Omega /(k_{B}T)$)

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.