 Solution

An acoustic phonon branch approaches zero frequency linearly, ${\omega }_{\mathrm{ac}}\simeq v_s|q|$, because a long-wavelength translation has no restoring force. An optical phonon branch has nonzero frequency at $q=0$ and represents relative motion of atoms within a primitive unit cell. Such an optical branch requires more than one independent atom per primitive cell; a truly monatomic simple-cubic model has only the acoustic branches.

The Bose occupation is $N_q=[e^{\hslash {\omega }_q/(k_{B}T)}-1{]}^{-1}$. At low temperature, appreciably populated acoustic modes haveAs temperature falls, the available phonon wavevectors concentrate near the zone centre and optical modes freeze out when $k_{B}T\ll \hslash {\omega }_{\mathrm{op}}(0)$. These are thermal-occupation restrictions, not a change in the allowed geometric Brillouin zone.