Thomson scattering is the elastic scattering of radiation from a free electron. The cross section for this process, known as the Thomson cross section is often encountered in various radiative processes.

The cross section is given by

where is the electron's classical radius.

Here we derive this result, to within an order of magnitude.

Consider an incoming electromagnetic wave, with angular frequency , and an electric field amplitude . Assuming that the charge is moving at sub-relativistic velocities, we neglect the effect of the corresponding magnetic field. The Lorentz force will be smaller by a factor of compared with the electric force, and is hence negligible.

The electron will thus experience an acceleration of order

where is the electron mass, and hence the second time-derivative of its dipole moment is roughly .

Using the Larmor formula, the power of the radiation emitted by the accelerating charge is given by

where the acceleration was substituted.

Finally, the flux of incoming radiation is roughly - the Electric field energy density is . The scattering cross section is given by

In a strong magnetic field, Thompson scattering in certain polarisations in suppressed. To understand this effect, let us consider a particle in a static magnetic field and an electromagnetic wave moving parallel to the magnetic field. The equation of motion (neglecting the magnetic field of the wave is)

In the absence of a magnetic field , the amplitude of the acceleration is and the amplitude of the velocity is where is the wave frequency. When the magnetic field is very strong we can disregard the dynamic term in the equation of motion, so the velocity amplitude in this case is

where is the classical cyclotron frequency. The velocity still changes on a timescale and so the acceleration amplitude is

We found that the acceleration in this case is smaller than the previous case by a factor of . Since the luminosity is quadratic in the acceleration , the cross section is also quadratic in the acceleration, meaning that the cross section in the magnetic case is smaller by the same factor compared to the non magnetic case