Dispersion relation

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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. 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[1].

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]. 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 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[5]. 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. 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.

Contents

[edit] 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.

[edit] 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 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.

[edit] 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.

[edit] 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:

\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.

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[7] and correspond to the speed at which the peak propagates, a value different from the phase velocity[8].

[edit] 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.

[edit] 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.

[edit] 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. 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.e. proportional to p instead of p2. This transition shows up as a slope change in the log-log dispersion plot at right.

[edit] 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}.

[edit] 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[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.

[edit] 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.

[edit] 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.

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

[edit] See also

[edit] 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. ^ 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

[edit] External links