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Illustration Dispersion Relation (Graph) of the Lattice Vibrations of a Diatomic Crystal Lattice

Dispersion Relation (Graph) of the Lattice Vibrations of a Diatomic Crystal Lattice
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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 \( k = \frac{\pi}{a} \).