DC hits to Moller XSection

Verification using LUND and evio files

The LUND files are broken into 1000 events per file to produce manageable GEMC output file sizes. The kinematics of these Moller events is shown fin Figure 1.

Figure 1a: A plot of Moller electron momentum in the center of mass frame.

Figure 1b:A plot of Moller electron scattering angle theta in the center of mass frame.

A Lorenz contraction occurs for a particle's momentum component that is parallel to the beam line. As the angle of the particle approaches a direction perpendicular to the beam line, the Lorentz contraction decreases. As a result, a uniform angular distribution in the Lab frame will not be uniform in the CM frame.

Figure 2: A Mankowski diagram demonstrating the Lorentz contraction increasing as the z component approaches the speed of light. Taking the perpendicular axis as the lab frame, the spacing between the arbitary measurements is equal when viewed from within the specific frame, but unequal as shown by the dots viewed from the lab frame.

This implies that the number of events that occur in the lab frame of reference near the beam line is larger than the number as it approaches a perpendicular direction. To understand the "density" of the number of events per bins in the lab frame, a study of 1000 events in the center of mass frame per 0.5 degree in the lab frame is investigated. A weighting factor, used to reproduce the moller cross section, appears in the LUND file but not the GEMC evio output file thereby requiring both the LUND and evio files to be read simultaneously.

1000 events per degree of 5 to 40 degrees in the Lab Frame

Examining the CM Frame Theta angles which correspond to Lab frame angles within 5-40 degrees (with bin widths of 0.5 degree), for only on degree in Phi (0 degrees). For each degree in Theta in the Lab frame, the correct kinematic variables for a specific events is written to a LUND file 1000 times. File:LUND Spread LH2 5to40Lab.C

Figure 3a: A plot of the number of Moller scattering angle theta in the lab frame. The width of the bins is 0.5 degrees for the angles 5 to 40 in the lab frame.

Figure 3b: A plot of the number of Moller scattering angle theta in the center of mass frame. The width of the bins is 0.001 degrees for the angles in the center of mass frame corresponding to angles of 5 to 40 degrees in the lab frame.

These events are read from the evio file using File:Evio2root.c

The Moller Scattering angle Theta is read from the LUND file, it's weight applied, and it's weight adjusted by dividing by the number it was multiplied by (1000) to give a differential cross-section for one electron. This is plotted against the theoretical differential cross-section for one Moller electron. File:MollerDiffXSect.c

Figure 4a: A plot of the number of Moller scattering angle theta in the center of mass frame versus the theoretical differential cross section. The width of the bins is 0.001 degrees for the angles in the center of mass frame corresponding to angles of 5 to 40 degrees in the lab frame. A weight has been assigned for each value in theta which will give the theoretical differential cross section when applied.

Figure 4b: A plot of the number of Moller scattering angle theta in the lab frame versus the theoretical differential cross section. The width of the bins is 0.5 degrees for the angles in the lab frame. A weight has been assigned for each value in theta which will give the theoretical differential cross section when applied.

From the evio file, a histogram can be constructed that will only record the Moller events which result in hits in the drift chamber. A hit for this setting is the procID value of 90 which corresponds to a transportation process (i.e. if the particle was a primary particle). Dividing DC hits by the procID variable, we find for the hits which are not transportation process for the primary parent particle:

Figure 5: A plot showing the physical process that is responsible for registered hits in the DC which are not transportation of the primary parent particle.

Process ID

Process Name	ID
1	e ionization
2	compton scattering
3	e bremsstrahlung
4	Photo Electric Effect
5	Gamma Conversion
6	e+ Annihilation
90	na

Plotting the Moller scattering angle Theta in the Center of Mass frame just for on occurance of the CM angle gives the Moller Differential Cross-section.

Figure 6: A plot showing a plot of the Moller scattering angle theta that result in DC hits. Each angle theta is only plotted once, and not repeated for multiple hits using the same angle.

Adjusting the weight by the number of times the specific angle caused hits. We should recover the Moller differential cross-section as found previously.

Figure 7: A plot showing a plot of the Moller scattering angle theta that result in DC hits. Each angle theta is plotted by as many occurances that result in hits, and renormalized to account for only one angle as would be found from one electron.

Once the validity of the Moller differential cross-section is established, we can transform from the center of mass to the lab frame as shown earlier. Since the detector is in the lab frame, the hits collected will have to be used to compose a histogram for the Moller scattering angle in the lab frame. We can show that the Moller events that register as hits in the lab matches the Moller differential cross-section in the lab.

Isotropic Spread in CM Frame for 5-40 degrees in Lab

As was done for the situation of 1000 events per degree in the lab frame of reference, the isotropic distribution of scattering angle theta in the center of mass frame can be weighted to reproduce a Moller differential cross-section.

Similarly, as was done earlier, the center of mass frame is transfered to the lab frame.

Since the dimension parallel to the direction of motion "compresses" with speeds approaching the speed of light, the number of events that occur at 5 degrees are different than 40 degrees. Collecting the number of events that occur within 0.5 degree bin width.

This can be plotted by dividing each entry into the bin by the number of events per that bin and multiplying by the corresponding weight factor.

Applying weight to DC hits

Ideally, we would like to be able to recover the Moller differential cross section from DC hits. Since this occurs in the lab frame, we will use the differential cross section found in the previous section in this frame. Each hit in the DC is associated with a generated particle's angle theta, plotting this we find:

Plot of Theta from GenPart if moller hits any one DC Layer (event is only counted once no matter how many layers register a DC hit)

Figure 13: A plot of Moller electron scattering angle theta in the lab frame that have an associated hit in the DC. Each occurrence of a DC hit associated with a generated particle's angle theta is recorded.

Limiting the number of times the angle theta is counted, regardless of the amount of hits registered in the DC, we can apply a weight and renormalize the bins to obtain the Moller differential cross-section.

Figure 13b: A plot of Moller electron scattering angle theta in the lab frame that have an associated hit in the DC. Each occurrence of a generated particle's angle theta is counted once, regardless of the number of hits associated with it. A weight has been applied to each occurrence of the angle theta that yields the Moller scattering differential cross section. The theoretical Moller differential cross-section is plotted for comparison.

Figure 1a: A plot of Moller electron momentum in the center of mass frame.

Figure 1a: A plot of Moller electron momentum in the center of mass frame.

Figure 1a: A plot of Moller electron momentum in the center of mass frame.