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 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, \frac{\partial \omega}{\partial k} is known as the
group velocity and corresponds to the speed at which the peak of the pulse propagates, a value different from the
phase velocity.
Deep water waves The dispersion relation for deep
water waves is often written as \omega = \sqrt{gk}, 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. In this case the phase velocity is v_p = \frac{\omega}{k} = \sqrt{\frac{g}{k}}, and the group velocity is v_g = \frac{d\omega}{dk} = \frac{1}{2} v_p.
Waves on a string For an ideal string, the dispersion relation can be written as \omega = k \sqrt{\frac{T}{\mu}}, 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 \omega^2 = \frac{T}{\mu} k^2 + \alpha k^4, where \alpha 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
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
insulator,
semiconductor or
conductor.
Phonons Phonons are to sound waves in a solid what photons are to light: they are the quanta that carry it. The dispersion relation of
phonons 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 phonons, since they correspond to classical sound in the limit of long wavelengths. The others are
optical phonons, since they can be excited by electromagnetic radiation.
Electron optics With high-energy (e.g., ) 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
dispersion surface.{{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 == History ==