Converting to barns
Using the equation from Landau and Lifshitz(Quantum Electrodynamics Volume 4)
For ultra-relativistic particles, this can be simplified to the form
Plugging in the values expected for a scattering electron in the CM frame:
Using unit analysis on the term outside the parantheses, we find that the differential cross section for an electron at this momentum should be around
Using the conversion of
The trigonometric function part of the equation comes out to it's minimum of 9 at 90 degrees.
We find that the differential cross section scale is
Converting the number of electrons scattered per angle theta to barns, we can use the relation
where ρtarget is the density of the target material, ltarget is the length of the target, and Iscattered is the number of incident particles scattered per time.
This gives, for LH2 in a 5cm long target in the Lab Frame, or in the CM frame:
Using the number of incident electrons, for 1 Moller electron:
We can use this number to scale the number of electrons per angle to a differential cross-section in barns. Using the plot of the Moller electron scattering angle theta in the Center of Mass frame,
We can rescale and combine the theoretical differential cross-section for one electron.
TH1F *Combo=new TH1F("TheoryExperiment","Theoretical and Experimental Differential Cross-Section CM Frame",360,90,180); Combo->Add(MollerThetaCM,7.92e-7); Combo->Draw(); Theory->Draw("same");