= Finite-crystal phonon freeze-out
{title2=$N_{\mathrm{ph}}\sim g_{\min}e^{-\hbar\omega_{\min}/(k_BT)}$}
An internal harmonic vibration spectrum with lowest positive frequency $\omega_{\min}$ has the displayed low-<temperature> thermal <phonon> population, where $g_{\min}$ is its degeneracy. The continuum bulk law fails when thermally occupied wavelengths reach the sample size. Mechanical boundary conditions determine $\omega_{\min}$; rigid translation of the entire solid is not an internal <phonon> mode.
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