Thomson scattering

Description

Thomson scattering describes the classical limit of electromagnetic radiation scattering from a free particle. An incident plane wave accelerates a charged particle which consequently emits radiation of the same frequency. The net effect is to scatter the incident radiation.

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. 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.

In the diagram, everything happens in the plane of the diagram. Electric fields of the incoming and outgoing wave can be divided up into perpendicular components. Those perpendicular to the plane are "tangential" and are not affected. Those components lying in the plane are referred to as "radial". The incoming and outgoing wave directions are also in the plane, and perpendicular to the electric components, as usual. (It is difficult to make these terms seem natural, but it is standard terminology.)

It can be shown that the amplitude of the outgoing wave will be proportional to the cosine of χ, the angle between the incident and scattered outgoing 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 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: $${\displaystyle \begin{array}{cc}{\epsilon }_{\text{t}}& =\frac{3}{16\pi }{\sigma }_{\text{t}}In\\ {\epsilon }_{\text{r}}& =\frac{3}{16\pi }{\sigma }_{\text{t}}In{\cos }^2\chi \end{array}}$$ where n is the density of charged particles at the scattering point, I is incident flux (i.e. energy/time/area/wavelength), χ is the angle between the incident and scattered photons (see figure above) and σ_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): $${\displaystyle \int \epsilon d\mathrm{\Omega }={\int }_0^{2\pi }d\phi {\int }_0^{\pi }d\chi ({\epsilon }_{\text{t}}+{\epsilon }_r)\sin \chi =I\frac{3{\sigma }_{\text{t}}}{16\pi }n2\pi (2+2/3)={\sigma }_{\text{t}}In.}$$

The Thomson differential cross section, related to the sum of the emissivity coefficients, is given by $${\displaystyle \frac{d{\sigma }_{\text{t}}}{d\mathrm{\Omega }}={\left(\frac{q^2}{4\pi {\epsilon }_{0}m{c}^2}\right)}^2\frac{1+{\cos }^2\chi }{2},}$$ where q is the charge of the particle, m is its mass, and ε₀ is the permittivity of free space. Integrating over the solid angle, we obtain the Thomson cross section $${\displaystyle {\sigma }_{\text{t}}=\frac{8\pi }{3}{\left(\frac{q^2}{4\pi {\epsilon }_{0}m{c}^2}\right)}^2=\frac{8\pi }{3}{r}^2=\frac{8\pi }{3}{\left(\alpha \lambda {\overline{}}_{\text{c}}\right)}^2,}$$ where r is the particle's classical radius, ⁠${\displaystyle \lambda {\overline{}}_{\text{c}}}$⁠ is its reduced Compton wavelength, and α is the fine structure constant.

A notable feature is that the cross section is independent of the frequency of the photon.

The value of the Thomson cross-section of the electron is given by:

${\displaystyle {\sigma }_{\text{e}}=\frac{8\pi }{3}{\left(\frac{\alpha \hslash c}{m_{\text{e}}{c}^2}\right)}^2}$= 6.6524587051(62)×10⁻²⁹ m² ≈ 0.665 b.