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

Normally, the resulting dispersion relation is a periodic function. However, 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 to only the right half of the 1st FC is sufficient.

At the edge of the 1st BZ (i.e. at the point \( \frac{\pi}{a} \)) the group velocity (slope of the function) vanishes and standing waves result for all waves which have the wavenumber \( k = \frac{\pi}{a} \).