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In physics, the dispersion relation is the relation between the energy of a system and its corresponding momentum. For example, for massive particles in free space, the dispersion relation can easily be calculated from the definition of kinetic energy:

i.e. the dispersion relation in this case is a quadratic function. More complicated systems will have different dispersion relations.

Contents 1 Derivation of physical properties 2 Optics 3 Solid State 4 Phonons 5 See also

Many classical physical properties of systems, such as speed, can be extended to other system if they are recast in terms of the dispersion relation. In a classical mechanical system, the speed of the system can be defined as

For electromagnetic waves, the energy is proportional to the frequency of the wave and the momentum to the wavenumber. In this case, Maxwell's equations tell us that the dispersion relation for vacuum is linear:

By using the same reasoning, we can infer the speed of those waves:

This is the speed of light, a constant.

The name "dispersion relation" originally comes from optics. It is possible to make the effective speed of light dependent on wavelength by making light pass through a material which has a non-constant index of refraction, or by using light in a non-uniform medium such as a waveguide. In this case, the waveform will spread over time, such that a narrow pulse will become an extended pulse, i.e. be dispersed. In these materials, is known as the group velocity and correspond to the speed at which the peak propagates, a value different from the phase velocity.

In the study of solids, the study of the dispersion relation of electrons is of paramount importance. The periodicity of crystals mean that for a given momentum, many levels of energy are possible, and that some energies might not be available at any momentum. The collection of all possible energies and momenta is known as the band structure of a material. Properties of the band structure define whether the material is an insulator, semiconductor or conductor.

The dispersion relation of phonons is also important and non-trivial. Most systems will show two separate bands on which phonons live. Phonons on the band that cross the origin are known as acoustic phonons, the others as optical phonons.

• Group velocity