= Solution
A <phonon> is one quantum of a <normal mode>[normal vibrational mode] of a <crystal>. In the <harmonic approximation>, each mode with <wavevector> $\mathbf q$, branch index $s$ and <angular frequency> $\omega_s(\mathbf q)$ is an <oscillator> with energies $\hbar\omega_s(n+1/2)$. A <phonon> changes the occupation $n$ by one; it is not a particular atom moving through the solid.
For a three-dimensional <primitive unit cell>[primitive cell] containing $r$ atoms there are $3r$ branches. Three <acoustic phonon branches>[acoustic branches] approach zero frequency as $\mathbf q\to0$: at zero <wavevector> they represent rigid translations, with all <crystal motif>[basis] atoms displaced together. Their small-<wavevector> frequencies usually grow linearly with <wavevector>. The remaining $3r-3$ <optical phonon branches>[optical branches] have internal relative motion of <crystal motif>[basis] atoms and generally nonzero zone-centre frequencies. The names reflect their coupling to radiation in suitable polar <crystals>, not a rule that every optical mode must be infrared active. Acoustic and <optical phonon branches>[optical branches] can each have longitudinal or transverse polarizations.
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