We see that the main difference between equations 13 and 14 is the heavy dependence of \(\gamma\) in the Klein-Nishina formula. If we were to consider the energy of the incident photon to be very low, \(\gamma\) moves towards 0. If we have incident photons with a low enough energy then equation 14 reduces into equation 13.
    To take account, we have two forms of Comptons formula (equations 11 & 12), the classical Thomson differential cross section (equation 13), and the corrected Klein-Nishina differential cross section (equation 14). On top of this, scattering is able to tell us a significant amount of information about the projectiles and targets. Dozens of scattering experiments have been preformed yielding incredible results. The results of some of these experiments are still relevant to both classical and quantum mechanics today \cite{Rutherford_1911}.\cite{Rutherford_1913}.\cite{Geiger_1910}. Using known values along with rigorous statistical analysis, we will also be able to show how this scattering experiment may help us aproximate the fundamental mass of the electron.

Experimental Setup

    We wish to utilize our knowledge of Compton scattering to observe the differential cross section. To do this, we need a gamma ray source, a target, and a detector which may move in an arc around the target. Consider figure 2.