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Here you see the **dispersion relation** \( \omega_{\pm} (k) \) (dependence of the frequency \(\omega_{\pm}\) on the wavenumber \(k\)) for a diatomic crystal lattice consisting of double chains which have a distance \(a\) (called lattice constant) to each other. A diatomic basis has two dispersion branches in 1d: optical and acoustic branches.

Usually the resulting dispersion relation is a periodic function. But since there is no additional information about the lattice vibrations in all the other periods, it is reduced to the *1st Brillouin zone*, which is in the range from \( -\frac{\pi}{a} \) to \( \frac{\pi}{a} \). With this dispersion relation, even a reduction only to the right half of the 1st Brillouin is sufficient.

At the edge of the 1st Brillouin (that is at the point \( \frac{\pi}{a} \)) the group velocity (slope of the function) vanishes and standing waves result for all waves having the wavenumber \( k = \frac{\pi}{a} \).