Thomson Cross Section

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 $\sigma _{T}={\frac {8\pi }{3}}r_{0}^{2}$

where $r_{0}=q^{2}/m_{e}c^{2}$ 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 $\omega$, and an electric field amplitude $E_{0}$. 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 $v_{e}/c$ compared with the electric force, and is hence negligible.

The electron will thus experience an acceleration of order $a=qE_{0}/m_{e}$

where $m_{e}$ is the electron mass, and hence the second time-derivative of its dipole moment is roughly ${\ddot {d}}\approx ea$.

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

$P={\frac {({\ddot {d}})^{2}}{c^{3}}}\approx a^{2}q^{2}/c^{3}={\frac {q^{4}E_{0}^{2}}{m_{q}^{2}c^{3}}}$

where the acceleration $a$ was substituted.

Finally, the flux of incoming radiation is roughly $F=E_{0}^{2}c$ - the Electric field energy density is $E_{0}^{2}/8\pi$. The scattering cross section is given by

$\sigma _{T}={\frac {P}{cE_{0}^{2}}}\approx {\frac {q^{4}}{m_{e}^{2}c^{4}}}=r_{0}^{2}$

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 $B$ and an electromagnetic wave moving parallel to the magnetic field. The equation of motion (neglecting the magnetic field of the wave is)

$m{\frac {dv}{dt}}=qE+q{\frac {v}{c}}\times B$

In the absence of a magnetic field $B=0$, the amplitude of the acceleration is $|a|\approx q|E|/m$ and the amplitude of the velocity is $|v|\approx q|E|/m\omega _{E}$ where $\omega _{E}$ 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

$|v|\approx c{\frac {E}{B}}\approx {\frac {qE}{m\omega _{B}}}$

where $\omega _{B}$ is the classical cyclotron frequency. The velocity still changes on a timescale $1/\omega _{E}$ and so the acceleration amplitude is

$|a|\approx {\frac {qE}{m}}{\frac {\omega _{E}}{\omega _{B}}}$

We found that the acceleration in this case is smaller than the previous case by a factor of $\left(\omega _{E}/\omega _{B}\right)^{2}$. Since the luminosity is quadratic in the acceleration $L\propto |a|^{2}$, 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

$\sigma \approx \sigma _{T}\left({\frac {\omega _{E}}{\omega _{B}}}\right)^{2}$