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Illustration Dispersion Relation for Lattice Vibrations of a Monatomic Crystal Lattice.

Dispersion relation of lattice vibrations for monatomic crystal lattice
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Here you see the dispersion relation \( \omega (k) \) (dependence of the frequency \(\omega\) on the wavenumber \(k\)) for a monatomic crystal lattice consisting of atomic chains which have a distance \(a\) (called lattice constant) from each other. This relation describes how a crystal lattice oscillates.

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 region between \( -\frac{\pi}{a} \) and \( \frac{\pi}{a} \). With this dispersion relation, even a reduction only to the right half of the 1st Brillouin zone is sufficient.

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