As can be seen from figure 7, both equations have the same differential cross section when the scattering angle is near 0. Furthermore, the K-N equation approaches the same function as Thompson's formula when \(\gamma\) approaches 0. Therefore, K-N formula reduces to Thompson's formula only with low \(\gamma\) or low scattering angle. It is evident that Klein-Nishina formula describes the quantum mechanical system due to the plank's constant hidden in equation 1.
The two formulas also predicts that the only dependence of the differential cross section is the incident energy and the scattering angle when the electrons in the scatterer are considered free. Therefore, the ratio of the differential cross section between copper and aluminum scatterer should be 1 to 1.
Comparison of Data and Theoretical Model
The experimental data of E are plotted over scattering angle with uncertainty, \(\Delta E = 0.0016 MeV\) due to the channel assignment, such that the inverse of the rest mass of electron can be determined from the slope. Figure 7shows the linear graph with the slope of \(1.53\ MeV^{-1}\). Using solver fit in excel, the \(\chi ^{2}\) is set to the minimum which is 0.04 with allowed value of the slope, \(1.84 MeV^{-1}\). Then, \(\chi^{2}\) was deteriorated to 1.04 with the slope, \(0.95 MeV^{-1}\). Therefore, the uncertainty of the slope is determined to be \(0.45 MeV^{-1}\). Therefore, the experimentally measured rest mass energy of electron is \(m_ec^{2}=0.65 \pm 0.19 MeV\). As the uncertainty is quite significant compared to the measurement, the only confirmation the data can provide is on the order of the magnitude.