Converting to barns

Using the equation from Landau and Lifshitz(Quantum Electrodynamics Volume 4)

where

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

Figure 5a: The theoretical Moller electron differential cross-section for an incident 11 GeV(Lab) electron in the Center of Mass frame of reference.

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:

and

with

and

Where

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,

Figure 5b: The Moller electron scattering angle theta distribution for an incident 11 GeV(Lab) electron in the Center of Mass frame of reference.

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);