Classical Mechanics · Small oscillations and normal modes
A diatomic lattice opens an acoustic-optical gap
Problem
An infinite one-dimensional lattice alternates masses $m$ and $M>m$, with every nearest-neighbor spring equal to $k$. A unit cell of length $a$ contains displacements $u_n,v_n$. Derive the Bloch eigenproblem and both branches $\omega_\pm(K)$. Evaluate the zone-edge frequencies for $m=1.00\,\mathrm{kg}$, $M=4.00\,\mathrm{kg}$, and $k=100\,\mathrm{N/m}$, and identify the band gap. Also check the translational mode at $K=0$.
Hint
The determinant is $(2k-m\omega^2)(2k-M\omega^2)-4k^2\cos^2(Ka/2)=0$.
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