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Dispersion of a light beam in a prism.
Dispersion of a light beam in a prism.

Dispersion relations describe the ways that wave propagation varies with the wavelength or frequency of a wave. In Physics wavelength is the distance between repeating units of a propagating Wave of a given Frequency. Frequency is a measure of the number of occurrences of a repeating event per unit Time. A wave is a disturbance that propagates through Space and Time, usually with transference of Energy. This variation has long explained how white light is dispersed into different colors, thus making rainbows possible. A rainbow is an optical and meteorological phenomenon that causes a spectrum of Light to appear in the Sky when the Sun It turns out, thanks to the wave nature of all traveling objects, that dispersion relations are key to understand how energy and objects are transported from point to point in any medium. This story likely began, however, with interest in the dispersion of waves on water for example by Pierre-Simon Laplace in 1776[1]. Year 1776 ( MDCCLXXVI) was a Leap year starting on Monday (link will display the full calendar of the Gregorian calendar (or a

Important clues to the wide-ranging utility of dispersion relations came from work in the early 20th century by H. Kramers[2] and R. Kronig[3]. Hendrik Anthony Kramers ( Rotterdam, February 2, 1894 &ndash Oegstgeest, April 24, 1952) was a Dutch Physicist Ralph Kronig was a German-American Physicist ( March 10, 1904 — November 16, 1995) Their relations take the form of integrals relating the real and imaginary parts of a property, called the complex refractive index[4], of any medium in which waves travel. The Kramers–Kronig relations are mathematical properties connecting the real and imaginary parts of any complex function which is analytic Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted The refractive index (or index of Refraction) of a medium is a measure for how much the speed of light (or other waves such as sound waves is reduced inside the medium The real part of this index describes how waves of different frequency refract (change speed and hence bend or disperse) through different angles on entering the medium. In Mathematics, the real numbers may be described informally in several different ways Refraction is the change in direction of a Wave due to a change in its Speed. In Optics, dispersion is the phenomenon in which the Phase velocity of a wave depends on its frequency The imaginary part of the index describes how the wave is absorbed in the medium. Geometric interpretation Geometrically imaginary numbers are found on the vertical axis of the complex number plane In Physics, absorption of electromagnetic radiation is the process by which the Energy of a Photon is taken up by matter typically the electrons of an

The universality of the concept became apparent with subsequent papers, on the dispersion relation's connection to causality in the scattering theory of all types of waves and particles[5]. In Mathematics and Physics, scattering theory is a framework for studying and understanding the Scattering of Waves and particles. For scattering processes where absorption can be ignored (i. e. attention focuses on the real refractive index), the term dispersion relation has also been applied to the dependence of wave frequency ω on wave number k, or equivalently through de Broglie's relations to the dependence of energy E=ħω on momentum p=ħk. Frequency is a measure of the number of occurrences of a repeating event per unit Time. Wavenumber in most physical sciences is a Wave property inversely related to Wavelength, having SI units of reciprocal meters In Physics, the de Broglie hypothesis (pronounced /brœj/ as French breuil close to "broy" is the statement that all Matter (any object has a Wave In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός The Planck constant (denoted h\ is a Physical constant used to describe the sizes of quanta. In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product The Planck constant (denoted h\ is a Physical constant used to describe the sizes of quanta. From dispersion relations in this form, the refractive index and the wave's "particle" or group velocity v are obtained by taking the derivative e. The group velocity of a Wave is the Velocity with which the variations in the shape of the wave's amplitude (known as the modulation or envelope g. v = dω/dk = dE/dp.

Contents

Kramers–Kronig relations and waves

Main article: Kramers–Kronig relation

This is an overview of applications for the Kramers–Kronig integral dispersion relations that connect real and imaginary parts of a medium's index of refraction. The Kramers–Kronig relations are mathematical properties connecting the real and imaginary parts of any complex function which is analytic The Kramers–Kronig relations are mathematical properties connecting the real and imaginary parts of any complex function which is analytic

Electron spectroscopy

In electron energy loss spectroscopy, Kramers–Kronig analysis allows one to calculate the energy dependence of both real and imaginary parts of a specimen's light optical permittivity, together with other optical properties such as the absorption coefficient and reflectivity[6]. In electron energy loss spectroscopy (EELS a material is exposed to a beam of Electrons with a known narrow range of kinetic energies. Permittivity is a Physical quantity that describes how an Electric field affects and is affected by a Dielectric medium and is determined by the ability The absorption coefficient α is a property of a material It defines the extent to which a material absorbs energy for example that of Sound waves or Electromagnetic In photometry and Heat transfer, reflectivity is the fraction of incident radiation reflected by a surface

In short, by measuring the number of high energy (e. g. 200 keV) electrons which lose energy ΔE over a range of energy losses in traversing a very thin specimen (single scattering approximation), one can calculate the energy dependence of permittivity's imaginary part. The dispersion relations allow one to then calculate the energy dependence of the real part.

This measurement is made with electrons, rather than with light, and can be done with very high spatial resolution! One might thereby, for example, look for ultraviolet (UV) absorption bands in a laboratory specimen of interstellar dust less than a 100 nm across, i. Presolar grains are isotopically-distinct clusters of material found in the fine-grained matrix of primitive Meteorites, whose differences from the surrounding meteorite e. too small for UV spectroscopy. Although electron spectroscopy has poorer energy resolution than light spectroscopy, data on properties in visible, ultraviolet and soft x-ray spectral ranges may be recorded in the same experiment. Spectroscopy was originally the study of the interaction between Radiation and Matter as a function of Wavelength (λ The electromagnetic (EM spectrum is the range of all possible Electromagnetic radiation frequencies

Frequency versus wavenumber

As mentioned above, when the focus in a medium is on refraction rather than absorption i. e. on the real part of the refractive index, it is common to refer to the functional dependence of frequency on wavenumber as the dispersion relation. For particles, this translates to a knowledge of energy as a function of momentum.

Waves and optics

For electromagnetic waves, the energy is proportional to the frequency of the wave and the momentum to the wavenumber. Electromagnetic radiation takes the form of self-propagating Waves in a Vacuum or in Matter. Frequency is a measure of the number of occurrences of a repeating event per unit Time. Wavenumber in most physical sciences is a Wave property inversely related to Wavelength, having SI units of reciprocal meters In this case, Maxwell's equations tell us that the dispersion relation for vacuum is linear:

\omega = c k.\,

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

v = \frac{\partial E}{\partial p} = \frac{\partial \omega}{\partial k} = c.

This is the speed of light, a constant. In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric

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. The refractive index (or index of Refraction) of a medium is a measure for how much the speed of light (or other waves such as sound waves is reduced inside the medium A waveguide is a structure which guides waves such as Electromagnetic waves Light, or Sound waves 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[7] and correspond to the speed at which the peak propagates, a value different from the phase velocity[8]. The group velocity of a Wave is the Velocity with which the variations in the shape of the wave's amplitude (known as the modulation or envelope The phase velocity (or phase speed) of a Wave is the rate at which the phase of the wave propagates in space

Deep water waves

The dispersion relation for deep water waves is often written as

\omega = \sqrt{g k}

where g is the acceleration due to gravity. In this case the phase velocity

v_p = \frac{\omega}{k} = \sqrt{\frac{g}{k}}

and the group velocity is vg = dω/dk = vp/2.

Frequency dispersion of gravity surface-waves on deep water. The red dot moves with the phase velocity, and the green dots propagate with the group velocity. In this deep-water case, the phase velocity is twice the group velocity. The red dot overtakes two green dots, when moving from the left to the right of the figure.
Frequency dispersion of gravity surface-waves on deep water. The red dot moves with the phase velocity, and the green dots propagate with the group velocity. In this deep-water case, the phase velocity is twice the group velocity. The red dot overtakes two green dots, when moving from the left to the right of the figure.

Waves on a string

The dispersion relation for an ideal string is often 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 a 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.

Two-frequency beats of a non-dispersive transverse wave. Since the wave is non-dispersive, phase (red) and group (green) velocities are equal.
Two-frequency beats of a non-dispersive transverse wave. Since the wave is non-dispersive, phase (red) and group (green) velocities are equal.

Application to particles

The free-space dispersion plot of kinetic energy versus momentum, for many objects of everyday life.
The free-space dispersion plot of kinetic energy versus momentum, for many objects of everyday life.

With classical particles in free space the dispersion relation follows from the expression for kinetic energy:

E = \frac{1}{2} m v^{2} = \frac{p^{2}}{2m}

i. e. the dispersion relation in this case is a quadratic function. A quadratic function, in Mathematics, is a Polynomial function of the form f(x=ax^2+bx+c \\! where a \ne 0 \\! Note that derivatives of E are not affected by changes in the energy zero e. g. by addition of a constant rest-energy term. More complicated systems will have different dispersion relations.

To illustrate this, note that the above equation works only for particles whose momentum per unit mass is much less than lightspeed c. Kinetic energy is more generally \sqrt{m^2 c^4 + p^2 c^2}-m c^2, which for particles with momentum per unit mass much greater than c (including photons) yields a kinetic energy of pc, i. In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena e. proportional to p instead of p2. This transition shows up as a slope change in the log-log dispersion plot at right.

Derivation of physical properties

Many classical physical properties of systems, such as speed, can be extended to other systems if they are recast in terms of the dispersion relation for frequency as a function of wavenumber, or for energy as a function of momentum. For example, in classical mechanical systems the particle velocity follows from:

v = \frac{\partial E}{\partial p} = \frac{p}{m}.

Application to quanta

By quanta, here we refer to particulate excitations like electrons, photons, plasmons and phonons whose dual particle-wave and/or quantum mechanical nature is not easy to ignore. The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena In Physics, a plasmon is a quantum of a plasma oscillation The plasmon is the Quasiparticle resulting from the Quantization of Plasma oscillations In Physics, a phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the Atomic lattice of a Solid

For example, the total energy dispersion relation for de Broglie matter waves[9] of mass m in free space may be written:

\omega = \frac{\sqrt{(m c^2)^2+(c \hbar k)^2}}{\hbar}

so that group velocity

v_g = \frac{d \omega}{d k} = \frac{c}{\sqrt{1+\left(\displaystyle\frac{m c}{\hbar k}\right)^2}}

and phase velocity vp = ω/k = c2/vg. The relationship between momentum and wavelength that this predicts (i. e. p = h/λ) has since been verified in practical application for atoms and small molecules as well as for elementary particles.

Solid state

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. In Condensed matter physics, the Fermi surface is an abstract boundary useful for predicting the thermal electrical magnetic and optical properties of Metals The collection of all possible energies and momenta is known as the band structure of a material. In Solid-state physics, the electronic band structure (or simply band structure) of a Solid describes ranges of Energy that an Electron Properties of the band structure define whether the material is an insulator, semiconductor or conductor. An insulator, also called a Dielectric, is a material that resists the flow of Electric current. A semiconductor' is a Solid material that has Electrical conductivity in between a conductor and an insulator; it can vary over that In Science and engineering, a conductor is a material which contains movable Electric charges.

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 important and non-trivial. In Physics, a phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the Atomic lattice of a Solid 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. In Physics, a phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the Atomic lattice of a Solid Optical phonons are Phonon polarization modes with a minimum Frequency, regardless of Wavelength, which occur in Crystals with more than one atom per

Electron optics

With high energy (e. g. 200 keV) 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[10]. Electron diffraction is a technique used to study matter by firing Electrons at a sample and observing the resulting Interference pattern In Mathematics and Solid state physics, the first Brillouin zone is a uniquely defined Primitive cell of the Reciprocal lattice in the This dynamical effect has found application in the precise measurement of lattice parameters, beam energy, and more recently for the electronics industry: lattice strain. The dynamical theory of diffraction describes the interaction of Waves with a regular lattice

See also

References

  1. ^ A. The group velocity of a Wave is the Velocity with which the variations in the shape of the wave's amplitude (known as the modulation or envelope In Optics, dispersion is the phenomenon in which the Phase velocity of a wave depends on its frequency In Fluid dynamics, dispersion of water waves generally refers to Frequency dispersion Ellipsometry is a versatile and powerful Optical technique for the investigation of the Dielectric properties (complex Refractive index or Dielectric D. D. Craik (2004). "The origins of water wave theory". Annual Review of Fluid Mechanics 36: 1–28. doi:10.1146/annurev.fluid.36.050802.122118. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.  
  2. ^ H. A. Kramers (1927) Estratto dagli Atti del Congresso Internazionale de Fisici Como (Nicolo Zonichelli, Bologna)
  3. ^ R. de L. Kronig (1926) On the theory of the dispersion of X-rays, J. Opt. Soc. Am. 12:547-557
  4. ^ H. Cohen (2003) Fundamentals and applications of complex analysis (Springer, Amsterdam) ISBN 0306477483
  5. ^ cf. John S. Toll (1956) Causality and the dispersion relation: Logical foundations, Phys. Rev. 104:1760-1770
  6. ^ R. F. Egerton (1996) Electron energy-loss spectroscopy in the electron microscope (Second Edition, Plenum Press, NY) ISBN 0-306-45223-5
  7. ^ cf. F. A. Jenkins and H. E. White (1957) Fundamentals of optics (McGraw-Hill, NY), page 223
  8. ^ cf. R. A. Serway, C. J. Moses and C. A. Moyer (1989) Modern Physics (Saunders, Philadelphia), page 118
  9. ^ Louis-Victor de Broglie (1925) Recherches sur la Théorie des Quanta, Ann. de Phys. 10e série, t. III (translation)
  10. ^ P. M. Jones, G. M. Rackham and J. W. Steeds (1977) Higher order Laue zone effects in electron diffraction and their use in lattice parameter determination, Proc. Roy. Soc. (London) A 354:197

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