= Acoustic and optical branches
The minus solution is the <acoustic phonon branch>. As $q\to0$, $\omega_-$ tends to zero and $U=V$: adjacent atoms move together in a long-wavelength compression. The plus solution is the <optical phonon branch>, with nonzero zone-centre <frequency> and $m_AU+m_BV=0$: the two sublattices move against one another with fixed cell centre of <mass>. For oppositely charged ions, this relative displacement can couple to an electric field, motivating the optical name.
At the zone boundary the frequencies are $\sqrt{2\alpha/m_{\rm heavy}}$ and $\sqrt{2\alpha/m_{\rm light}}$. Different masses produce a gap; equal masses make the two branches meet, recovering a folded monatomic dispersion.
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