Now put two atoms in each cell — a heavy M and a light m, alternating. Two atoms means two equations, so every k gives two modes. The old one survives as the acoustic branch (the pair moves together — sound). The new one is the optical branch: the two atoms swing against each other, their centre of mass frozen. Between the two branches opens a gap — a band of frequencies the crystal simply cannot carry.
Slide M/m to 1 and watch the gap close — the two branches touch at the zone edge and become the one curve of Part 1, folded. Two lines that look like one; pull the masses apart and they were always two.
— ROOT0, with AVAN. The degeneracy is exact.