 Diatomic-chain zone-edge polarizations

In an alternating-mass chain with nearest-neighbour spacing $a$ and cell length $2a$, the inter-sublattice Bloch coupling is proportional to $\cos qa$. At the first-zone edge $q=\pi /(2a)$ it vanishes. For $M>m$, the acoustic edge mode moves only the heavy sublattice and the optical edge mode only the light one. Their frequency difference is $\sqrt{2\alpha /m}-\sqrt{2\alpha /M}$. At long wavelength the acoustic displacements are in phase, whereas the optical mode has zero centre-of-mass displacement.