= 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> gives
$$
\omega_\pm^2=K\left(\frac1m+\frac1M\right)\pm K\sqrt{\left(\frac1m+\frac1M\right)^2-\frac4{mM}\sin^2(kd)},\qquad |k|\leq\frac\pi{2d}.
$$
The 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 $\sqrt{2K(1/m+1/M)}$.
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