Skip to main content

Illustration Dispersion relation of the lattice vibration (graph) - diatomic base (1d)

Dispersion relation of the lattice vibration (graph) - diatomic base (1d)
Dispersion relation of the lattice vibration (graph) - diatomic base (1d)
Download illustration

Share — copy and redistribute the material in any medium or format

Adapt — remix, transform, and build upon the material for any purpose, even commercially.

Sharing and adapting of the illustration is allowed with indication of the link to the illustration.

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 \( k = \frac{\pi}{a} \).