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).

  • $\begingroup$ 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)? $\endgroup$
    – Py-ser
    Apr 28, 2016 at 10:24
  • $\begingroup$ @RobJeffries, I think what you say is true only at quite high energies, not in the entire X-ray domain. $\endgroup$
    – Py-ser
    Apr 28, 2016 at 11:54
  • $\begingroup$ Agreed - see below. $\endgroup$
    – ProfRob
    Apr 28, 2016 at 12:09

1 Answer 1


By definition, Thomson scattering is the elastic scattering of light by a free charged particle. Atoms cannot be described as such, but the electrons in an atom may approximate to free electrons if their binding energy is much lower than the photon energy. This might be true for X-ray wavelengths, although if the photon energy gets too high then elastic scattering will not occur and you would have to consider Compton scattering instead.

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.

Atomic form factors

Edit: In conclusion For a fixed number of electrons in a gas, then the X-ray Thomson scattering cross section is maximised if the gas is completely ionised. If the gas is atomic, the cross-section will be similar for directly forward scattering, but smaller in all other directions, by an amount given by the form factor.

  • $\begingroup$ Thank you @RobJeffries. In conclusion, if you have to compare the scattering efficiency from the same amount of electrons, either in an ionized gas or in an atomic gas, which one is the more efficient? I can update my original question if this point was not clear from the beginning. $\endgroup$
    – Py-ser
    May 2, 2016 at 8:00
  • $\begingroup$ @Py-ser see edit. $\endgroup$
    – ProfRob
    May 2, 2016 at 8:23
  • $\begingroup$ I think now everything is clear. The bounty will be yours. If by any chance you know a reference book please let me know (I do not see any suitable - to my background - reference in the Wikipedia link). $\endgroup$
    – Py-ser
    May 4, 2016 at 8:53
  • 1
    $\begingroup$ @Py-ser Perhap "Essentials of crystallography" onlinelibrary.wiley.com/doi/10.1002/crat.2170280607/abstract or even better (you can see the relevant section for free) "X-ray diffraction". books.google.co.uk/… $\endgroup$
    – ProfRob
    May 5, 2016 at 6:01

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