= Rattling mode
{title2=$C_E=gR\,x^2e^x/(e^x-1)^2,\quad x=\hbar\Omega/(k_BT)$}
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 $\hbar\Omega/k_B\simeq32\ \mathrm K$, assigned to a barium guest site.<Superconductivity in the Filled Cage Compounds Ba6Ge25 and Ba4Na2Ge25 by Friedrich Malte Grosche and others>[Page 247003-3, Figure 3]{ref} This is a measured-material example; the narrow-mode description remains an approximation to a crystal's full vibrational spectrum.
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