Compton-like scattering for protons Why don't we usually calculate scattering cross sections for Compton scattering of photons (gamma rays) off protons? After all both electrons and protons are fermions. Hence shouldn't the amplitude be identical for both type of scattering?
 A: Two reasons.
Firstly the proton has a much higher mass than an electron. What this means in practice is that for a given photon energy in the fermion rest frame,  the proton has a lower recoil speed after energy-momentum conservation is considered, which means the photon loses much less energy. So basically the cross section stays in the low energy Thomson limit. And the Thomson cross section is proportional to $\frac{1}{m} $ so actually it is lower for a proton tan for an electron
Secondly, and more importantly, an electron is a point charge (as far as the calculations go), while a proton is made up of other stuff and has a spatial extent. The internal structure means that the photon may be exciting internal states of the proton, which is a completely different proposition. Or interacting with the partially charged quarks inside individually. Obviously there is some threshold below which this becomes less relevant, but then you're back to dealing with the Thomson limit worth a cross section reduced by a factor of $\frac {m_e}{m_p}$ compared to am electron. 
