 Diatomic-chain dispersion relation with equal springs

For alternating masses $m,M$ at spacing $d$, equal nearest-neighbour spring constants $K$, and cell length $2d$, a two-component normal mode givesThe minus sign is the acoustic phonon branch and the plus sign is the optical phonon branch. For $M>m$ the edge frequencies are $\sqrt{2K/M}$ and $\sqrt{2K/m}$, leaving a gap. At $k=0$ they are zero and $\sqrt{2K(1/m+1/M)}$.