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 ${\displaystyle \omega }$ ${\displaystyle k}$

${\displaystyle \omega =f(k)}$.

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

${\displaystyle \omega =v_{\text{Phase}}\cdot k}$,

with the constant phase velocity . In this case there is no dispersion . ${\displaystyle v_{\text{Phase}}={\frac {\omega }{k}}}$

The speed of a wave packet , on the other hand, is the group speed ${\displaystyle v_{\text{Gruppe}}={\frac {\mathrm {d} \omega }{\mathrm {d} k}}.}$

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

Band structure of a one-dimensional photonic crystal . The dispersion relation can be read off directly from the slope of the bands ${\displaystyle \omega (k_{\mathrm {z} })}$

In the dispersion relation of optics , the ( complex ) refractive index appears as a function of the angular frequency: ${\displaystyle n}$

${\displaystyle \omega =c_{\text{M}}\,k={\frac {c}{n(\omega )}}\,k\quad \Rightarrow \quad n={\frac {c}{c_{\text{M}}}}={\frac {c\,k}{\omega }}=f(\omega )}$

With

• the phase velocity of light in a medium ${\displaystyle c_{\text{M}}}$
• the speed of light in vacuum . ${\displaystyle c}$

Particle physics

Because the frequency is always related to the energy

${\displaystyle \omega ={\frac {E}{\hbar }}}$

and the wave number (or wave vector ) with the momentum ${\displaystyle p}$

${\displaystyle {\vec {k}}={\frac {\vec {p}}{\hbar }},}$

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:

${\displaystyle {\begin{aligned}&&E&={\frac {p^{2}}{2m}}\\\Rightarrow &&\hbar \,\omega &={\frac {\hbar ^{2}k^{2}}{2m}}\\\Leftrightarrow &&\omega &={\frac {\hbar }{2m}}k^{2},\end{aligned}}}$

where the reduced Planck constant and the mass of the particle denotes. ${\displaystyle \hbar }$ ${\displaystyle m}$

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 .