# Thomson scattering cross section: free electrons vs atoms

I am trying to find a reference for the Thomson scattering cross section difference between free electrons and atoms. I have always assumed that free electrons have a higher cross sections, but I can not find any sure reference for this.

Please can you address me on this? Physical arguments are very welcome.

If energy band is relevant, then X-rays is the case.

EDIT: the main issue here (for me) is to understand the difference in (Thomson) scattering efficiency between the same amount of electrons but in two "opposite" situations: within an ionized gas (free electrons) and a cold one (atoms).

• Thank you. Do you mean, there is no big difference between the Thomson scattering cross section of a photon hitting a free electron, and a photon hitting an iron atom (or a hydrogen atom, for what it concerns)? Apr 28, 2016 at 10:24
• @RobJeffries, I think what you say is true only at quite high energies, not in the entire X-ray domain. Apr 28, 2016 at 11:54
• Agreed - see below. Apr 28, 2016 at 12:09

The Thomson scattering cross-section from an atom with be $f$ times as big as the scattering cross-section from a free electron, where $f$ is the effective number of free electrons in the atom. The contribution from the nucleus is negligible since the cross-section depends on inversely on the square of the particle mass.
For forward scattering, then $f \simeq Z$, the number of electrons in the atom - each electron contributes (indeed if you are talking about ions, it is just the number of electrons present). However, for other scattering angles there is some destructive interference, $f < Z$ and the scattering cross-section falls with increasing scattering angle. This is quantified by an atomic form factor, which is the ratio between the scattering cross-section of the atom and that of a free electron as a function of scattering angle (i.e. $f= f(\theta)$). This is shown below (e.g. for oxygen (blue $Z=8$); chlorine (green, $Z=17$)) and peaks at a value equal to the number of electrons present.