Calculations of 4-momentum components

Setup

Since we want to run for a evenly spaced energy range for Moller electrons, we will need to use some of the scattered electrons to help cover this range. A Moller scattering data file of 4E8 events has no Moller electrons with momentum over 5500 MeV.

awk 'NR == 1 {line = $0; max = $15} NR >1 && $15 > max {line =$0; max =$15} END{print line}' MollerScattering_NH3_4e8incidnet.dat

11000    0     0     11000.5     0     0     -510     5502.94     -14.7812     -50.9086     5503.19     0     0     -505.319     5495.8     14.7812     50.9086     5496.06     0     0     -505.319

Since momentum is conserved, and the data is verified kinematicly, we cannot simply "switch" the data. This data can be altered to have a certain number of different phi values for each energy to match the Moller cross section. This data can then be written to a LUND file, and compared to the previous calculations which did not factor in loss of initial energy.

Prepare Data

Using the existing Moller scattering data from a GEANT simulation of 4E8 incident electrons, a file of just scattered momentum components can be constructed using:

awk '{print $9, $10, $11, $16, $17, $18}' MollerScattering_NH3_4e8incident.dat > Just_Scattered_Momentum.dat

Transfer to CM Frame

Reading in the data from the dat file, we use a C++ program to read the momentum components for the Scattered and Moller electrons into 4-momentum vectors defined as the Lab_final frame of reference.

Performing a Lorentz boost to a Center of Mass frame for the two 4-vectors from the Lab_final frame of reference, we move to a frame where the energies are equal and the momentum are equal but opposite.

Relativistic Kinematics

For Moller Electron energies above 500 MeV, in the Lab frame, histograms of momentum, and theta as well as a 2-D histogram of Energy vs. Theta for the Moller Electron in the CM frame will be filled.

Using the histogram for Theta in the CM frame, we can determine the relative number of events that occur at a given angle. This information will be used to keep the relative number of particles having the same Theta angle, but multiple Psi angles to evenly cover the detector area

Run for Necessary Amount to match Cross Section

Using the above plot for the target material, we can find the relative amount that each Theta angle should observe for this process which gives a known Moller differential cross section.

Theta (degrees)	Differential Cross-Section (barns)	Number of Events
90.000000	0.000017	17
91.000000	0.000017	17
92.000000	0.000017	17
93.000000	0.000017	17
94.000000	0.000017	17
95.000000	0.000017	17
96.000000	0.000017	17
97.000000	0.000017	17
98.000000	0.000017	17
99.000000	0.000018	18
100.000000	0.000018	18
101.000000	0.000018	18
102.000000	0.000019	19
103.000000	0.000019	19
104.000000	0.000019	19
105.000000	0.000020	20
106.000000	0.000020	20
107.000000	0.000021	21
108.000000	0.000022	22
109.000000	0.000022	22
110.000000	0.000023	23
111.000000	0.000024	24
112.000000	0.000025	25
113.000000	0.000025	25
114.000000	0.000026	26
115.000000	0.000028	28
116.000000	0.000029	29
117.000000	0.000030	30
118.000000	0.000031	31
119.000000	0.000033	33
120.000000	0.000035	35
121.000000	0.000036	36
122.000000	0.000038	38
123.000000	0.000040	40
124.000000	0.000043	43
125.000000	0.000045	45
126.000000	0.000048	48
127.000000	0.000051	51
128.000000	0.000054	54
129.000000	0.000058	58
130.000000	0.000062	62
131.000000	0.000067	67
132.000000	0.000072	72
133.000000	0.000077	77
134.000000	0.000083	83
135.000000	0.000090	90
136.000000	0.000097	97
137.000000	0.000106	106
138.000000	0.000115	115
139.000000	0.000126	126
140.000000	0.000138	138
141.000000	0.000152	152
142.000000	0.000167	167
143.000000	0.000185	185
144.000000	0.000205	205
145.000000	0.000228	228
146.000000	0.000254	254
147.000000	0.000285	285
148.000000	0.000321	321
149.000000	0.000362	362
150.000000	0.000411	411
151.000000	0.000468	468
152.000000	0.000537	537
153.000000	0.000618	618
154.000000	0.000716	716
155.000000	0.000834	834
156.000000	0.000978	978

We can set up conditional statements to check what range the Theta angle falls in, then by dividing

[math]\Delta \phi=\frac {2\pi}{number\ of\ events}[/math]

we should find the change in phi needed to give an evenly distributed distribution around the xy plane for a given Theta angle.

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