# 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.



### 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

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