= Diatomic-chain zone-edge polarizations
{title2=$\omega_{\rm ac}^2=2\alpha/M,\quad\omega_{\rm op}^2=2\alpha/m$}
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.
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