Finite-crystal phonon freeze-out ($N_{\mathrm{ph}}\sim g_{\min }{e}^{-\hslash {\omega }_{\min }/(k_{B}T)}$)

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.