= Zone folding in an equal-mass diatomic chain
Putting $M=m$ in the <diatomic-chain dispersion relation with equal springs> gives $\omega_-=2\sqrt{K/m}|\sin(kd/2)|$ and $\omega_+=2\sqrt{K/m}|\cos(kd/2)|$ in the reduced zone $|k|\leq\pi/(2d)$. The branches touch at its edge. If the sites are actually identical, the true <primitive unit cell> has length $d$; the apparent <optical phonon branch> is the outer part of the <monatomic-chain dispersion relation> folded by an unnecessarily doubled cell. A gap does not appear merely from redrawing the zone.
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