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Dispersion relations describe the ways that wave propagation varies with the wavelength or frequency of a wave. This variation has long explained how white light is dispersed into different colors, thus making rainbows possible. 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.

Important clues to the wide-ranging utility of dispersion relations came from work in the early 20th century by H. Kramers and R. Kronig. Their relations take the form of integrals relating the real and imaginary parts of a property, called the complex refractive index, of any medium in which waves travel. 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. The imaginary part of the index describes how the wave is absorbed in the medium.

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. 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 vector k, or equivalently through de Broglie relations to the dependence of energy E=ħω on momentum p=ħk. 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.g. v = dω/dk = dE/dp.

Kramers–Kronig relations and waves

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.

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.

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.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.

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. In this case, Maxwell's equations tell us that the dispersion relation for vacuum is linear:

${\displaystyle \omega =ck.\,}$

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

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

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, ${\displaystyle {\frac {\partial \omega }{\partial k}}}$ is known as the group velocity and correspond to the speed at which the peak propagates, a value different from the phase velocity.

Deep water waves

The dispersion relation for deep water waves is often written as

${\displaystyle \omega ={\sqrt {gk}},}$

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

${\displaystyle v_{p}={\frac {\omega }{k}}={\sqrt {\frac {g}{k}}}}$

and the group velocity is v_g = dω/dk = ½ v_p.

Waves on a string

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

${\displaystyle \omega =k{\sqrt {\frac {T}{\mu }}}}$

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.

Dispersion and propagation of general waveforms

A traveling wave of fixed shape in one dimension is described by a an amplitude varying in space x and time t as f(x − vt). with v the wave speed. If the traveling wave repeats itself, f is a periodic function of its argument.

Under rather general conditions, a function f(x) can be expressed as a sum of basis functions {φ_n(x)} in the form:

${\displaystyle f(x)=\sum _{n=1}^{\infty }c_{n}\varphi _{n}(x)\ ,}$

known variously as Fourier series, Fourier-Bessel series, generalized Fourier series, and so forth, depending upon the basis used.

For a periodic function f with spatial periodicity λ, the basis functions satisfy φ_n(x + λ) = φ_n(x). This condition can be satisfied by basis functions that repeat more often in space than does f itself, and so have wavelengths shorter than the function f.

In particular, for such a periodic function f , the basis may be chosen as a set of sinusoidal functions, selected with wavelengths λ/n (n an integer) to ensure φ_n(x + λ) = φ_n(x). For a sine wave sin(kx) the implication is kλ = 2nπ (n an integer), or k = 2πn/λ, where k is called the wave vector and n is called the wavenumber. The wavelength of sin(kx) = sin(2πn x /λ) is λ/n. In this case, the basis function with wavelength λ is referred to as the fundamental and the other basis functions as harmonics. Many examples of such representations are found in books on Fourier series. For example, application to a number of sawtooth waves is presented by Puckette.

Whatever the basis functions, the wave becomes:

${\displaystyle y=f(x-vt)=\sum _{n=1}^{\infty }c_{n}\varphi _{n}(x-vt)\ .}$

Thus, all the components must travel at the same rate to insure that the waveform remains unchanged as it moves. To turn this discussion upside down, because a general waveform may be viewed as a superposition of shorter wavelength basis functions, a requirement upon the physical medium propagating a traveling wave of fixed shape is that the medium must be capable of propagating disturbances of different wavelengths at the same wave speed. This requirement is met in many simple wave propagating mediums, but is not a general property of all media. More commonly, a medium has a non-linear dispersion relation connecting wave vector to frequency of oscillation, and the medium is dispersive, which means propagation of rigid waveforms is not possible in general, but requires very particular circumstances.

Such circumstances sometimes do occur in nonlinear media. For example, in large-amplitude ocean waves, due to properties of the nonlinear surface-wave medium, wave shapes can propagate unchanged. A related phenomenon is the cnoidal wave, a periodic traveling wave named because it is described by the Jacobian elliptic function of m-th order, usually denoted as cn (x; m). Another is the wave motion in an inviscid incompressible fluid, where the wave shape is given by:

${\displaystyle y=A\ \mathrm {sech} ^{2}\ ,}$

one of the early solutions to the Korteweg–de Vries equation of 1895. Here β is a constant related to height of the wave and depth of the water. The Korteweg-de Vries equation is an exactly solvable model, that is, a non-linear partial differential equation whose solutions can be exactly and precisely specified. The solutions include examples of solitons, traveling waves without a wavelength because they are not periodically recurring, but still capable of representation as sums of functions with definite wavelengths using the Fourier integral.

Application to particles

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

${\displaystyle E={\frac {1}{2}}mv^{2}={\frac {p^{2}}{2m}}}$

i.e. the dispersion relation in this case is a quadratic function. 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 ${\displaystyle {\sqrt {m^{2}c^{4}+p^{2}c^{2}}}-mc^{2}}$, which for particles with momentum per unit mass much greater than c (including photons) yields a kinetic energy of pc, i.e. proportional to p instead of p. 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:

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

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.

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

${\displaystyle \omega ={\frac {\sqrt {(mc^{2})^{2}+(c\hbar k)^{2}}}{\hbar }}}$

so that group velocity

${\displaystyle v_{g}={\frac {d\omega }{dk}}={\frac {c}{\sqrt {1+\left(\displaystyle {\frac {mc}{\hbar k}}\right)^{2}}}}}$

and phase velocity v_p = ω/k = c/v_g. 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.

Animation: phase and group velocity of electrons

This animation portrays the de Broglie phase and group velocities (in slow motion) of three free electrons traveling over a field 0.4 Ångstroms in width. The momentum per unit mass (proper velocity) of the middle electron is lightspeed so that its group velocity is 0.707 c. The top electron has twice the momentum, while the bottom electron has half. Note that wavelength and phase velocity decrease as the group velocity increases, until the wave packet and its phase maxima move together near the speed of light and only wavelength noticeably decreases past that. 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.

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. 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 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.

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. This dynamical effect has found application in the precise measurement of lattice parameters, beam energy, and more recently for the electronics industry: lattice strain.

References

1. A.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.
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. R. G. Dean and R. A. Dalrymple (1991). Water wave mechanics for engineers and scientists. Advanced Series on Ocean Engineering. Vol. 2. World Scientific, Singapore. ISBN 978-9810204204. See page 64–66.
10. Stanley J Farlow (1993). Partial differential equations for scientists and engineers (Reprint of Wiley 1982 ed.). Courier Dover Publications. p. 82. ISBN 048667620X.
11. See for example, Gerald B Folland (2009). "Convergence and completeness". Fourier Analysis and its Applications (Reprint of Wadsworth & Brooks/Cole 1992 ed.). American Mathematical Society. pp. 77 ff. ISBN 0821847902.
12. Roger W. Hockney, James W. Eastwood (1988). Computer simulation using particles (Paperback reprint of 1981 ed.). CRC Press. p. 501. ISBN 0852743920. If B is a periodic function of the continuous variable x with a period length L then ... the wavenumber k takes only those values permitting integral numbers of wavelengths to fit in period length L.
13. Miller Puckette (2007). The theory and technique of electronic music. World Scientific. ISBN 9812700773.
14. Ken'ichi Okamoto (2001). Global environment remote sensing. IOS Press. p. 263. ISBN 9781586031015.
15. Roger Grimshaw (2007). "Solitary waves propagating over variable topography". In Anjan Kundu (ed.). Tsunami and Nonlinear Waves. Springer. pp. 52 ff. ISBN 3540712550.
16. P. G. Drazin, R. S. Johnson (1996). Solitons: an introduction. Cambridge University Press. p. 8. ISBN 0521336554.
17. Louis-Victor de Broglie (1925) Recherches sur la Théorie des Quanta, Ann. de Phys. 10 série, t. III (translation)
18. 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

See also

• Group velocity
• Dispersion (optics)
• Dispersion (water waves)
• Ellipsometry

External links

• Poster on CBED simulations to help visualize dispersion surfaces, by Andrey Chuvilin and Ute Kaiser

• Fundamental physics concepts