# Dispersion relation

In physics , the **dispersion relation** describes the **relationship** between the *course of* a physical process ( frequency , energy ) and the *properties of* the quantities describing it (wave number, refractive index, speed of propagation, momentum).

Mathematically, the dispersion relation is the relationship between the angular frequency and the angular wave number . It is obtained from the linear wave equation by a Fourier transformation in space and time and has the form

- .

In the simplest case, the angular frequency and angular wave number are always proportional

- ,

with the constant phase velocity . In this case there is *no *dispersion .

The speed of a wave packet , on the other hand, is the group speed

A wave packet consists of waves of different frequencies that can have different phase velocities. Therefore, a wave packet generally diverges. Wave packets that *do not* diverge due to non-linear effects despite dispersion are called solitons .

## optics

In the dispersion relation of optics , the ( complex ) refractive index appears as a function of the angular frequency:

With

- the phase velocity of light in a medium
- the speed of light in vacuum .

## Particle physics

Because the frequency is always related to the energy

and the wave number (or wave vector ) with the momentum

The energy-momentum relationships in particle physics are also referred to as the dispersion relationship (or dispersion relationship), e.g. B. for free electrons in the non- relativistic limit case:

where the reduced Planck constant and the mass of the particle denotes.

## Solid state physics

In solid-state physics , dispersion is given as the relationship between the energy or angular frequency and the wave number of a particle or quasiparticle . In solids , on the one hand the phonons (lattice vibrations of the atomic lattice ) are assigned a phonon dispersion relation , on the other hand an electron dispersion relation can be assigned to the electrons, which is described with the aid of the band structure .

## literature

- Dieter Meschede: Optics, light and laser . Springer-Verlag, 2015, ISBN 3-663-10954-2 , pp. 29 f .