Dispersion relation
Given the dispersion relation, one can calculate the frequency-dependent phase velocity and group velocity of each sinusoidal component of a wave in the medium, as a function of frequency.
Dispersion may be caused either by geometric boundary conditions (waveguides, shallow water) or by interaction of the waves with the transmitting medium.
Elementary particles, considered as matter waves, have a nontrivial dispersion relation, even in the absence of geometric constraints and other media.
In the presence of dispersion, a wave does not propagate with an unchanging waveform, giving rise to the distinct frequency-dependent phase velocity and group velocity.
Dispersion occurs when sinusoidal waves of different wavelengths have different propagation velocities, so that a wave packet of mixed wavelengths tends to spread out in space.
: The wave's speed, wavelength, and frequency, f, are related by the identity The function
Dispersion relations are more commonly expressed in terms of the angular frequency
Rewriting the relation above in these variables gives where we now view f as a function of k. The use of ω(k) to describe the dispersion relation has become standard because both the phase velocity ω/k and the group velocity dω/dk have convenient representations via this function.
For electromagnetic waves in vacuum, the angular frequency is proportional to the wavenumber: This is a linear dispersion relation, in which case the waves are said to be non-dispersive.
For de Broglie matter waves the frequency dispersion relation is non-linear:
use the de Broglie relations for energy and momentum for matter waves, where ω is the angular frequency and k is the wavevector with magnitude |k| = k, equal to the wave number.
Practical work with matter waves occurs at non-relativistic velocity.
If we start with the non-relativistic Schrödinger equation we will end up without the first, rest mass, term.
This animation portrays the de Broglie phase and group velocities (in slow motion) of three free electrons traveling over a field 0.4 ångströms in width.
Note that as the momentum increases, the phase velocity decreases down to c, whereas the group velocity increases up to c, until the wave packet and its phase maxima move together near the speed of light, whereas the wavelength continues to decrease without bound.
Both transverse and longitudinal coherence widths (packet sizes) of such high energy electrons in the lab may be orders of magnitude larger than the ones shown here.
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.
[3] The dispersion relation for deep water waves is often written as where g is the acceleration due to gravity.
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 where
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 are to sound waves in a solid what photons are to light: they are the quanta that carry it.
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.
With high-energy (e.g., 200 keV, 32 fJ) 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.
[5] This dynamical effect has found application in the precise measurement of lattice parameters, beam energy, and more recently for the electronics industry: lattice strain.
Isaac Newton studied refraction in prisms but failed to recognize the material dependence of the dispersion relation, dismissing the work of another researcher whose measurement of a prism's dispersion did not match Newton's own.
[6] Dispersion of waves on water was studied by Pierre-Simon Laplace in 1776.
[7] The universality of the Kramers–Kronig relations (1926–27) became apparent with subsequent papers on the dispersion relation's connection to causality in the scattering theory of all types of waves and particles.
