# Thomson scattering

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Thomson scattering is the elastic scattering of electromagnetic radiation by a free charged particle, as described by classical electromagnetism. It is the low-energy limit of Compton scattering: the particle's kinetic energy and photon frequency do not change as a result of the scattering. [1] This limit is valid as long as the photon energy is much smaller than the mass energy of the particle: ${\displaystyle \nu \ll mc^{2}/h}$, or equivalently, if the wavelength of the light is much greater than the Compton wavelength of the particle (e.g., for electrons, longer wavelengths than hard x-rays).

## Description of the phenomenon

In the low-energy limit, the electric field of the incident wave (photon) accelerates the charged particle, causing it, in turn, to emit radiation at the same frequency as the incident wave, and thus the wave is scattered. Thomson scattering is an important phenomenon in plasma physics and was first explained by the physicist J. J. Thomson. As long as the motion of the particle is non-relativistic (i.e. its speed is much less than the speed of light), the main cause of the acceleration of the particle will be due to the electric field component of the incident wave. In a first approximation, the influence of the magnetic field can be neglected.[ citation needed ] The particle will move in the direction of the oscillating electric field, resulting in electromagnetic dipole radiation. The moving particle radiates most strongly in a direction perpendicular to its acceleration and that radiation will be polarized along the direction of its motion. Therefore, depending on where an observer is located, the light scattered from a small volume element may appear to be more or less polarized.

The electric fields of the incoming and observed wave (i.e. the outgoing wave) can be divided up into those components lying in the plane of observation (formed by the incoming and observed waves) and those components perpendicular to that plane. Those components lying in the plane are referred to as "radial" and those perpendicular to the plane are "tangential". (It is difficult to make these terms seem natural, but it is standard terminology.)

The diagram on the right depicts the plane of observation. It shows the radial component of the incident electric field, which causes the charged particles at the scattering point to exhibit a radial component of acceleration (i.e., a component tangent to the plane of observation). It can be shown that the amplitude of the observed wave will be proportional to the cosine of χ, the angle between the incident and observed waves. The intensity, which is the square of the amplitude, will then be diminished by a factor of cos2(χ). It can be seen that the tangential components (perpendicular to the plane of the diagram) will not be affected in this way.

The scattering is best described by an emission coefficient which is defined as ε where ε dt dV dΩ dλ is the energy scattered by a volume element ${\displaystyle dV}$ in time dt into solid angle dΩ between wavelengths λ and λ+dλ. From the point of view of an observer, there are two emission coefficients, εr corresponding to radially polarized light and εt corresponding to tangentially polarized light. For unpolarized incident light, these are given by:

${\displaystyle \varepsilon _{t}={\frac {3}{16\pi }}\sigma _{t}In}$
${\displaystyle \varepsilon _{r}={\frac {3}{16\pi }}\sigma _{t}In\cos ^{2}\chi }$

where ${\displaystyle n}$ is the density of charged particles at the scattering point, ${\displaystyle I}$ is incident flux (i.e. energy/time/area/wavelength) and ${\displaystyle \sigma _{t}}$ is the Thomson cross section for the charged particle, defined below. The total energy radiated by a volume element ${\displaystyle dV}$ in time dt between wavelengths λ and λ+dλ is found by integrating the sum of the emission coefficients over all directions (solid angle):

${\displaystyle \int \varepsilon \,d\Omega =\int _{0}^{2\pi }d\varphi \int _{0}^{\pi }d\chi (\varepsilon _{t}+\varepsilon _{r})\sin \chi =I{\frac {3\sigma _{t}}{16\pi }}n2\pi (2+2/3)=\sigma _{t}In.}$

The Thomson differential cross section, related to the sum of the emissivity coefficients, is given by

${\displaystyle {\frac {d\sigma _{t}}{d\Omega }}=\left({\frac {q^{2}}{4\pi \varepsilon _{0}mc^{2}}}\right)^{2}{\frac {1+\cos ^{2}\chi }{2}}}$

expressed in SI units; q is the charge per particle, m the mass of particle, and ${\displaystyle \varepsilon _{0}}$ a constant, the permittivity of free space. (To obtain an expression in cgs units, drop the factor of 4πε0.) Integrating over the solid angle, we obtain the Thomson cross section

${\displaystyle \sigma _{t}={\frac {8\pi }{3}}\left({\frac {q^{2}}{4\pi \varepsilon _{0}mc^{2}}}\right)^{2}}$

in SI units.

The important feature is that the cross section is independent of photon frequency. The cross section is proportional by a simple numerical factor to the square of the classical radius of a point particle of mass m and charge q, namely

${\displaystyle \sigma _{t}={\frac {8\pi }{3}}r_{e}^{2}}$

Alternatively, this can be expressed in terms of ${\displaystyle \lambda _{c}}$, the Compton wavelength, and the fine structure constant:

${\displaystyle \sigma _{t}={\frac {8\pi }{3}}\left({\frac {\alpha \lambda _{c}}{2\pi }}\right)^{2}}$

For an electron, the Thomson cross-section is numerically given by: [2]

${\displaystyle \sigma _{t}={\frac {8\pi }{3}}\left({\frac {\alpha \hbar c}{mc^{2}}}\right)^{2}=6.6524587158\ldots \times 10^{-29}{\text{ m}}^{2}=66.524587158\ldots {\text{ (fm)}}^{2}}$

## Examples of Thomson scattering

The cosmic microwave background contains a small linearly-polarized component attributed to Thomson scattering. That polarized component mapping out the so-called E-modes was first detected by DASI in 2002.

The solar K-corona is the result of the Thomson scattering of solar radiation from solar coronal electrons. The ESA and NASA SOHO mission and the NASA STEREO mission generate three-dimensional images of the electron density around the sun by measuring this K-corona from three separate satellites.

In tokamaks, corona of ICF targets and other experimental fusion devices, the electron temperatures and densities in the plasma can be measured with high accuracy by detecting the effect of Thomson scattering of a high-intensity laser beam.

In the Sunyaev-Zeldovich effect, where the photon energy is much less than the electron rest mass, the inverse-Compton scattering can be approximated as Thomson scattering in the rest frame of the electron. [3]

X-ray crystallography is based on Thomson scattering.

## Related Research Articles

The Beer–Lambert law, also known as Beer's law, the Lambert–Beer law, or the Beer–Lambert–Bouguer law relates the attenuation of light to the properties of the material through which the light is travelling. The law is commonly applied to chemical analysis measurements and used in understanding attenuation in physical optics, for photons, neutrons, or rarefied gases. In mathematical physics, this law arises as a solution of the BGK equation.

In physics, the cross section is a measure of the probability that a specific process will take place when some kind of radiant excitation intersects a localized phenomenon. For example, the Rutherford cross-section is a measure of probability that an alpha particle will be deflected by a given angle during an interaction with an atomic nucleus. Cross section is typically denoted σ (sigma) and is expressed in units of area, more specifically in barns. In a way, it can be thought of as the size of the object that the excitation must hit in order for the process to occur, but more exactly, it is a parameter of a stochastic process.

Rutherford scattering is the elastic scattering of charged particles by the Coulomb interaction. It is a physical phenomenon explained by Ernest Rutherford in 1911 that led to the development of the planetary Rutherford model of the atom and eventually the Bohr model. Rutherford scattering was first referred to as Coulomb scattering because it relies only upon the static electric (Coulomb) potential, and the minimum distance between particles is set entirely by this potential. The classical Rutherford scattering process of alpha particles against gold nuclei is an example of "elastic scattering" because neither the alpha particles nor the gold nuclei are internally excited. The Rutherford formula further neglects the recoil kinetic energy of the massive target nucleus.

Rayleigh scattering, named after the 19th-century British physicist Lord Rayleigh, is the predominantly elastic scattering of light or other electromagnetic radiation by particles much smaller than the wavelength of the radiation. For light frequencies well below the resonance frequency of the scattering particle, the amount of scattering is inversely proportional to the fourth power of the wavelength.

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In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ε (epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in response to an applied electric field than a material with low permittivity, thereby storing more energy in the material. In electrostatics, the permittivity plays an important role in determining the capacitance of a capacitor.

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Compton scattering, discovered by Arthur Holly Compton, is the scattering of a high frequency photon after an interaction with a stationary charged particle, usually an electron. If it results in a decrease in energy of the photon, it is called the Compton effect. Part of the energy of the photon is transferred to the recoiling electron. Inverse Compton scattering occurs when a charged particle transfers part of its energy to a photon.

In the physical sciences, the wavenumber is the spatial frequency of a wave, measured in cycles per unit distance or radians per unit distance. It is analogous to temporal frequency, which is defined as the number of wave cycles per unit time or radians per unit time.

In particle physics, the Klein–Nishina formula gives the differential cross section of photons scattered from a single free electron, calculated in the lowest order of quantum electrodynamics. It was first derived in 1928 by Oskar Klein and Yoshio Nishina, constituting one of the first successful applications of the Dirac equation. The formula describes both the Thompson scattering of low energy photons and the Compton scattering of high energy photons, showing that the total cross section and expected deflection angle decrease with increasing photon energy.

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In radiometry, radiant exitance or radiant emittance is the radiant flux emitted by a surface per unit area, whereas spectral exitance or spectral emittance is the radiant exitance of a surface per unit frequency or wavelength, depending on whether the spectrum is taken as a function of frequency or of wavelength. This is the emitted component of radiosity. The SI unit of radiant exitance is the watt per square metre, while that of spectral exitance in frequency is the watt per square metre per hertz (W·m−2·Hz−1) and that of spectral exitance in wavelength is the watt per square metre per metre (W·m−3)—commonly the watt per square metre per nanometre. The CGS unit erg per square centimeter per second is often used in astronomy. Radiant exitance is often called "intensity" in branches of physics other than radiometry, but in radiometry this usage leads to confusion with radiant intensity.

Dielectric loss quantifies a dielectric material's inherent dissipation of electromagnetic energy. It can be parameterized in terms of either the loss angleδ or the corresponding loss tangent tan δ. Both refer to the phasor in the complex plane whose real and imaginary parts are the resistive (lossy) component of an electromagnetic field and its reactive (lossless) counterpart.

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Surface-extended X-ray absorption fine structure (SEXAFS) is the surface-sensitive equivalent of the EXAFS technique. This technique involves the illumination of the sample by high-intensity X-ray beams from a synchrotron and monitoring their photoabsorption by detecting in the intensity of Auger electrons as a function of the incident photon energy. Surface sensitivity is achieved by the interpretation of data depending on the intensity of the Auger electrons instead of looking at the relative absorption of the X-rays as in the parent method, EXAFS.

Free carrier absorption occurs when a material absorbs a photon, and a carrier is excited from an already-excited state to another, unoccupied state in the same band. This intraband absorption is different from interband absorption because the excited carrier is already in an excited band, such as an electron in the conduction band or a hole in the valence band, where it is free to move. In interband absorption, the carrier starts in a fixed, nonconducting band and is excited to a conducting one.

Surface plasmon polaritons (SPPs) are electromagnetic waves that travel along a metal–dielectric or metal–air interface, practically in the infrared or visible-frequency. The term "surface plasmon polariton" explains that the wave involves both charge motion in the metal and electromagnetic waves in the air or dielectric ("polariton").

## References

1. Chen, Szu-yuan; Maksimchuk, Anatoly; Umstadter, Donald (December 17, 1998). "Experimental observation of relativistic nonlinear Thomson scattering". Nature. 396 (6712): 653–655. arXiv:. Bibcode:1998Natur.396..653C. doi:10.1038/25303. S2CID   16080209.
2. "National Institute of Standards and Technology" . Retrieved 3 February 2015.
3. Birkinshaw, Mark (1999). "The Sunyaev–Zel'dovich effect". Physics Reports. 310 (2–3): 97–195. arXiv:. Bibcode:1999PhR...310...97B. doi:10.1016/s0370-1573(98)00080-5. hdl:1983/5d24f14a-26e0-44d3-8496-5843b108fec5. S2CID   119330362 . Retrieved 4 November 2021.

Johnson W.R.; Nielsen J.; Cheng K.T. (2012). "Thomson scattering in the average-atom approximation". Physical Review. 86 (3): 036410. arXiv:. Bibcode:2012PhRvE..86c6410J. doi:10.1103/PhysRevE.86.036410. PMID   23031036. S2CID   10413904.