The disagreement on ratio at 45 degree seems to stem from the assumption that the electrons inside the scatterer are free. At low scattering angles, the difference is negligible since the momentum necessary for the electrons to be transmitted to that angle is lower. As the prediction fails at higher scattering angle, the number of free electrons, \(N_e\), can be considered not equal between the two scatterer. By the experimental data, there should be 1.75 times the number of free electrons in copper than the aluminum.