When dealing with scattering of free electrons in a plasma you can basically define two main regimes of interest.
i) Using the classical dipole treatment for the electron we get the Thomson scattering with a cross section of (8*Pi/3)*Re^2. Notice that the cross section is frequency independent.
ii) For more energetic radiation you have to use the Klein-Nishina formula for calculating the cross section. It's lengthy so Google it, I'm sure there are many web pages with it on.
We now have 2 regimes within this case:
a) Non-relativistic electrons doing the scattering.
Cross section is (8*Pi/3)*Re^2 *(1-2A+(26/5)A^2+....) where A is h*v/Me*c^2 where v is frequency.
b)Extreme relativistic regime where A>>1 which means the cross section is now (8*Pi/3)*Re^2*(3/8A)*(Ln(2A)+1/2).
For the in between regime then you use the full Klein-Nishina formula.
** Note that I remembered this out of my head so I might have an error above as I was too lazy to Google **
Also note that Thomson scattering, which is also when the recoil of the electron can be neglected is also called coherent scattering because the frequency of the incoming and outgoing photon is unchanged. Compton scattering is where the recoil has to be taken into account, thats what the Klein-Nishina formula does, is not coherent as the frequency is changed.
I remember in school having to derive the Klein-Nishina formula. A task I would not like to do tonight from memory!
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