= Solution
An <acoustic phonon branch> approaches zero frequency linearly, $\omega_{\rm ac}\simeq v_s|q|$, 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 $N_q=[e^{\hbar\omega_q/(k_BT)}-1]^{-1}$. At low temperature, appreciably populated acoustic modes have
$$
\boxed{|q|\lesssim q_T\sim\frac{k_BT}{\hbar v_s}.}
$$
As temperature falls, the available phonon wavevectors concentrate near the zone centre and optical modes freeze out when $k_BT\ll\hbar\omega_{\rm op}(0)$. These are thermal-occupation restrictions, not a change in the allowed geometric <Brillouin zone>.
\Image[/past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2019-q4-phonons-and-umklapp.png]
{title=Thermally available <phonons> and the minimum momentum for electron-phonon Umklapp scattering}
{description=Original schematic acoustic and optical dispersions, alongside a repeated-zone Fermi-surface construction. In the right panel wavevectors are in units of $\pi/a$: two radius-0.7 Fermi circles have centres separated by 2, leaving a minimum phonon wavevector of 0.6 between them. The optical branch in the left panel illustrates a crystal with an atomic basis.}
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