Editing Dispersion relation (section)

==Frequency versus wavenumber==
As mentioned above, when the focus in a medium is on refraction rather than absorption—that is, on the real part of the [[refractive index]]—it is common to refer to the functional dependence of angular frequency on wavenumber as the ''dispersion relation''. For particles, this translates to a knowledge of energy as a function of momentum.

===Waves and optics===
{{Further|Dispersion (optics)}}
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, <math>\frac{\partial \omega}{\partial k}</math> is known as the [[group velocity]]<ref>{{cite book| author=F. A. Jenkins and H. E. White |date=1957|title=Fundamentals of optics| url=https://archive.org/details/fundamentalsofop00jenk | url-access=registration |publisher=McGraw-Hill|location=New York| page=[https://archive.org/details/fundamentalsofop00jenk/page/223 223]| isbn=0-07-032330-5}}</ref> and corresponds to the speed at which the peak of the pulse  propagates, a value different from the [[phase velocity]].<ref>{{cite book|author= R. A. Serway, C. J. Moses and C. A. Moyer |year=1989| title=Modern Physics| publisher=Saunders|location=Philadelphia|page= 118|isbn= 0-534-49340-8}}</ref>

===Deep water waves===
{{Further|Dispersion (water waves)|Airy wave theory}}
[[Image:Wave group.gif|frame|right|Frequency dispersion of surface gravity waves on deep water. The {{colorbull|red|square}} red square moves with the phase velocity, and the {{colorbull|limegreen|circle}} green dots propagate with the group velocity. In this deep-water case, the phase velocity is twice the group velocity. The {{colorbull|red|square}} red square traverses the figure in the time it takes the {{colorbull|limegreen|circle}} green dot to traverse half.]]

The dispersion relation for deep [[ocean surface wave|water waves]] is often written as

: <math>\omega = \sqrt{gk},</math>

where ''g'' is the acceleration due to gravity. Deep water, in this respect, is commonly denoted as the case where the water depth is larger than half the wavelength.<ref>{{cite book | title=Water wave mechanics for engineers and scientists | author=R. G. Dean and R. A. Dalrymple | year=1991 | series=Advanced Series on Ocean Engineering | volume=2 | publisher=World Scientific, Singapore | isbn=978-981-02-0420-4 }} See page 64–66.</ref> In this case the phase velocity is

: <math>v_p = \frac{\omega}{k} = \sqrt{\frac{g}{k}},</math>

and the group velocity is

: <math>v_g = \frac{d\omega}{dk} = \frac{1}{2} v_p.</math>

===Waves on a string===
{{Further|Vibrating string}}
[[Image:Two-frequency beats of a non-dispersive transverse wave (animated).gif|frame|right|Two-frequency beats of a non-dispersive transverse wave. Since the wave is non-dispersive, {{colorbull|red|circle}} phase and {{colorbull|limegreen|circle}} group velocities are equal.]]

For an ideal string, the dispersion relation can be written as

: <math>\omega = k \sqrt{\frac{T}{\mu}},</math>

where ''T'' is the tension force in the string, and ''μ'' is the string's mass per unit length. As for the case of electromagnetic waves in vacuum, ideal strings are thus a non-dispersive medium, i.e. the phase and group velocities are equal and independent (to first order) of vibration frequency.

For a nonideal string, where stiffness is taken into account, the dispersion relation is written as

: <math>\omega^2 = \frac{T}{\mu} k^2 + \alpha k^4,</math>

where <math>\alpha</math> is a constant that depends on the string.

===Electron band structure===
In the study of solids, the study of the dispersion relation of electrons is of paramount importance. The periodicity of crystals means that many [[Fermi surface|levels of energy]] are possible for a given momentum 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 [[Electrical insulation|insulator]], [[semiconductor]] or [[Conductor (material)|conductor]].

===Phonons===
{{further|Phonon#Dispersion relation}}Phonons are to sound waves in a solid what photons are to light: they are the quanta that carry it. The dispersion relation of [[phonon]]s is also non-trivial and important, being directly related to the acoustic and thermal properties of a material. For most systems, the phonons can be categorized into two main types: those whose bands become zero at the center of the [[Brillouin zone]] are called [[acoustic phonon]]s, since they correspond to classical sound in the limit of long wavelengths. The others are [[optical phonon]]s, since they can be excited by electromagnetic radiation.

===Electron optics===
With high-energy (e.g., {{convert|200|keV|abbr=on|disp=comma}}) electrons in a [[transmission electron microscope]], the energy dependence of higher-order [[Laue zone]] (HOLZ) lines in convergent beam [[electron diffraction]] (CBED) patterns allows one, in effect, to ''directly image'' cross-sections of a crystal's three-dimensional [[Brillouin zone|dispersion surface]].<ref>{{cite journal| author=P. M. Jones, G. M. Rackham and J. W. Steeds | year=1977|title= Higher order Laue zone effects in electron diffraction and their use in lattice parameter determination| journal=Proceedings of the Royal Society| volume=A 354
| issue=1677| page=197| doi=10.1098/rspa.1977.0064| bibcode=1977RSPSA.354..197J| s2cid=98158162}}</ref> This [[Dynamical theory of diffraction|dynamical effect]] has found application in the precise measurement of lattice parameters, beam energy, and more recently for the electronics industry: lattice strain.