Thomson scattering

Last updated June 28, 2024

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: $\nu \ll mc^{2}/h$ , 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).^[2]

Description of the phenomenon

Thomson scattering is a model for the effect of electromagnetic fields on electrons when the field energy $h\nu$ is much less than the rest mass of the electron $m_{0}c^{2}$ . In the model the electric field of the incident wave 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.^[2]^: 15 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 εdtdVdΩ dλ is the energy scattered by a volume element $dV$ 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, ε_r corresponding to radially polarized light and ε_t corresponding to tangentially polarized light. For unpolarized incident light, these are given by: ${\begin{aligned}\varepsilon _{t}&={\frac {3}{16\pi }}\sigma _{t}In\\[1ex]\varepsilon _{r}&={\frac {3}{16\pi }}\sigma _{t}In\cos ^{2}\chi \end{aligned}}$

where $n$ is the density of charged particles at the scattering point, $I$ is incident flux (i.e. energy/time/area/wavelength), $\chi$ is the angle between the incident and scattered photons (see figure above) and $\sigma _{t}$ is the Thomson cross section for the charged particle, defined below. The total energy radiated by a volume element $dV$ in time dt between wavelengths λ and λ+dλ is found by integrating the sum of the emission coefficients over all directions (solid angle): $\int \varepsilon \,d\Omega =\int _{0}^{2\pi }d\varphi \int _{0}^{\pi }d\chi (\varepsilon _{t}+\varepsilon _{r})\sin \chi =I{\frac {3\sigma _{t}}{16\pi }}n2\pi (2+2/3)=\sigma _{t}In.$

The Thomson differential cross section, related to the sum of the emissivity coefficients, is given by ${\frac {d\sigma _{t}}{d\Omega }}=\left({\frac {q^{2}}{4\pi \varepsilon _{0}mc^{2}}}\right)^{2}{\frac {1+\cos ^{2}\chi }{2}}$ expressed in SI units; q is the charge per particle, m the mass of particle, and $\varepsilon _{0}$ a constant, the permittivity of free space. (To obtain an expression in cgs units, drop the factor of 4 $π$ ε₀.) Integrating over the solid angle, we obtain the Thomson cross section $\sigma _{t}={\frac {8\pi }{3}}\left({\frac {q^{2}}{4\pi \varepsilon _{0}mc^{2}}}\right)^{2}$ in SI units.

The important feature is that the cross section is independent of light 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^[2]^: 17 $\sigma _{t}={\frac {8\pi }{3}}r_{e}^{2}$

Alternatively, this can be expressed in terms of $\lambda _{c}$ , the Compton wavelength, and the fine structure constant: $\sigma _{t}={\frac {8\pi }{3}}\left({\frac {\alpha \lambda _{c}}{2\pi }}\right)^{2}$

For an electron, the Thomson cross-section is numerically given by:^[3] $\sigma _{t}={\frac {8\pi }{3}}\left({\frac {\alpha \hbar c}{mc^{2}}}\right)^{2}=6.6524587321(60)\times 10^{-29}{\text{ m}}^{2}\approx 66.5{\text{ fm}}^{2}=0.665{\text{ b}}$

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. An upgraded Thomson scattering system in the Wendelstein 7-X stellarator uses Nd:YAG lasers to emit multiple pulses in quick succession. The intervals within each burst can range from 2 ms to 33.3 ms, permitting up to twelve consecutive measurements. Synchronization with plasma events is made possible by a newly added trigger system that facilitates real-time analysis of transient plasma events.^[4]

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.^[5]

Models for X-ray crystallography are based on Thomson scattering.

Related Research Articles

In physics, the cross section is a measure of the probability that a specific process will take place in a collision of two particles. 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.

Rayleigh scattering, named after the 19th-century British physicist Lord Rayleigh, is the predominantly elastic scattering of light, or other electromagnetic radiation, by particles with a size much smaller than the wavelength of the radiation. For light frequencies well below the resonance frequency of the scattering medium, the amount of scattering is inversely proportional to the fourth power of the wavelength, e.g., a blue color is scattered much more than a red color as light propagates through air.

The Stefan–Boltzmann law, also known as Stefan's law, describes the intensity of the thermal radiation emitted by matter in terms of that matter's temperature. It is named for Josef Stefan, who empirically derived the relationship, and Ludwig Boltzmann who derived the law theoretically.

<span class="mw-page-title-main">Permittivity</span> Measure of the electric polarizability of a dielectric material

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 material. 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.

In the physical sciences, the wavenumber, also known as repetency, 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.

Thermal radiation is electromagnetic radiation emitted by the thermal motion of particles in matter. Thermal radiation transmits as an electromagnetic wave through both matter and vacuum. When matter absorbs thermal radiation its temperature will tend to rise. All matter with a temperature greater than absolute zero emits thermal radiation. The emission of energy arises from a combination of electronic, molecular, and lattice oscillations in a material. Kinetic energy is converted to electromagnetism due to charge-acceleration or dipole oscillation. At room temperature, most of the emission is in the infrared (IR) spectrum. Thermal radiation is one of the fundamental mechanisms of heat transfer, along with conduction and convection.

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 Thomson 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.

The classical electron radius is a combination of fundamental physical quantities that define a length scale for problems involving an electron interacting with electromagnetic radiation. It links the classical electrostatic self-interaction energy of a homogeneous charge distribution to the electron's relativistic mass-energy. According to modern understanding, the electron is a point particle with a point charge and no spatial extent. Nevertheless, it is useful to define a length that characterizes electron interactions in atomic-scale problems. The classical electron radius is given as

The Compton wavelength is a quantum mechanical property of a particle, defined as the wavelength of a photon whose energy is the same as the rest energy of that particle. It was introduced by Arthur Compton in 1923 in his explanation of the scattering of photons by electrons.

<span class="mw-page-title-main">Beam emittance</span> Property of a charged particle beam

In accelerator physics, emittance is a property of a charged particle beam. It refers to the area occupied by the beam in a position-and-momentum phase space.

In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density is the quantity of charge per unit volume, measured in the SI system in coulombs per cubic meter (C⋅m⁻³), at any point in a volume. Surface charge density (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m⁻²), at any point on a surface charge distribution on a two dimensional surface. Linear charge density (λ) is the quantity of charge per unit length, measured in coulombs per meter (C⋅m⁻¹), at any point on a line charge distribution. Charge density can be either positive or negative, since electric charge can be either positive or negative.

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.

In electrical engineering, dielectric loss quantifies a dielectric material's inherent dissipation of electromagnetic energy. It can be parameterized in terms of either the loss angle $δ$ 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.

The Frank–Tamm formula yields the amount of Cherenkov radiation emitted on a given frequency as a charged particle moves through a medium at superluminal velocity. It is named for Russian physicists Ilya Frank and Igor Tamm who developed the theory of the Cherenkov effect in 1937, for which they were awarded a Nobel Prize in Physics in 1958.

The word electricity refers generally to the movement of electrons, or other charge carriers, through a conductor in the presence of a potential difference or an electric field. The speed of this flow has multiple meanings. In everyday electrical and electronic devices, the signals travel as electromagnetic waves typically at 50%–99% of the speed of light in vacuum. The electrons themselves move much more slowly. See drift velocity and electron mobility.

When an electromagnetic wave travels through a medium in which it gets attenuated, it undergoes exponential decay as described by the Beer–Lambert law. However, there are many possible ways to characterize the wave and how quickly it is attenuated. This article describes the mathematical relationships among:

<span class="mw-page-title-main">Surface plasmon polariton</span> Electromagnetic waves that travel along an interface

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").

Plasmonic nanoparticles are particles whose electron density can couple with electromagnetic radiation of wavelengths that are far larger than the particle due to the nature of the dielectric-metal interface between the medium and the particles: unlike in a pure metal where there is a maximum limit on what size wavelength can be effectively coupled based on the material size.

<span class="mw-page-title-main">Lorentz oscillator model</span> Theoretical model describing the optical response of bound charges

The Lorentz oscillator model describes the optical response of bound charges. The model is named after the Dutch physicist Hendrik Antoon Lorentz. It is a classical, phenomenological model for materials with characteristic resonance frequencies for optical absorption, e.g. ionic and molecular vibrations, interband transitions (semiconductors), phonons, and collective excitations.

Shneider-Miles scattering is the quasi-elastic scattering of electromagnetic radiation by charged particles in a small-scale medium with frequent particle collisions. Collisional scattering typically occurs in coherent microwave scattering of high neutral density, low ionization degree microplasmas such as atmospheric pressure laser-induced plasmas. Shneider-Miles scattering is characterized by a 90° phase shift between the incident and scattered waves and a scattering cross section proportional to the square of the incident driving frequency. Scattered waves are emitted in a short dipole radiation pattern. The variable phase shift present in semi-collisional scattering regimes allows for determination of a plasma's collisional frequency through coherent microwave scattering.

References

↑ Chen, Szu-yuan; Maksimchuk, Anatoly; Umstadter, Donald (December 17, 1998). "Experimental observation of relativistic nonlinear Thomson scattering". Nature. 396 (6712): 653–655. arXiv: physics/9810036 . Bibcode:1998Natur.396..653C. doi:10.1038/25303. S2CID 16080209.
1 2 3 Froula, Dustin H. Plasma scattering of electromagnetic radiation. Academic Press is an imprint of Elsevier, 2011.
↑ "National Institute of Standards and Technology" . Retrieved 3 February 2015.
↑ Damm, H.; Pasch, E.; Dinklage, A.; et al. (2019). "First results from an event synchronized—high repetition Thomson scattering system at Wendelstein 7-X". Journal of Instrumentation. 14 (9): C09037. arXiv: 1907.00492 . Bibcode:2019JInst..14C9037D. doi:10.1088/1748-0221/14/09/C09037. S2CID 195767387.
↑ Birkinshaw, Mark (1999). "The Sunyaev–Zel'dovich effect". Physics Reports. 310 (2–3): 97–195. arXiv: astro-ph/9808050 . 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.

External links

Thomson scattering notes Archived 2020-06-29 at the Wayback Machine
Thomson scattering: principle and measurements

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[1] Chen, Szu-yuan; Maksimchuk, Anatoly; Umstadter, Donald (December 17, 1998). "Experimental observation of relativistic nonlinear Thomson scattering". Nature. 396 (6712): 653–655. arXiv: physics/9810036 . Bibcode:1998Natur.396..653C. doi:10.1038/25303. S2CID 16080209.

[Froula-2] 1 2 3 Froula, Dustin H. Plasma scattering of electromagnetic radiation. Academic Press is an imprint of Elsevier, 2011.

[NIST-3] "National Institute of Standards and Technology" . Retrieved 3 February 2015.

[4] Damm, H.; Pasch, E.; Dinklage, A.; et al. (2019). "First results from an event synchronized—high repetition Thomson scattering system at Wendelstein 7-X". Journal of Instrumentation. 14 (9): C09037. arXiv: 1907.00492 . Bibcode:2019JInst..14C9037D. doi:10.1088/1748-0221/14/09/C09037. S2CID 195767387.

[5] Birkinshaw, Mark (1999). "The Sunyaev–Zel'dovich effect". Physics Reports. 310 (2–3): 97–195. arXiv: astro-ph/9808050 . 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.

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Thomson scattering

Contents

Description of the phenomenon

Examples of Thomson scattering

See also

Related Research Articles

References

Further reading

External links

Light–matter interaction

Low-energy phenomena:
Photoelectric effect
Mid-energy phenomena:
Thomson scattering
Compton scattering
High-energy phenomena:
Pair production
Photodisintegration
Photofission
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