Difference between revisions of "Calculations of 4-momentum components"

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! Theta (degrees)
 
! Theta (degrees)
! Differential Cross-Section
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! Differential Cross-Section (barns)
 
! Number of Events
 
! Number of Events
  

Revision as of 00:27, 16 April 2016

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 4E7 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_4e7incidnet.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.

MolEThetaCM 500.png

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

Filling Histogram

Run for Necessary Amount to match Cross Section

XSect above550MeV.png

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 5
91.000000 0.000017 5
92.000000 0.000017 5
93.000000 0.000017 5
94.000000 0.000017 5
95.000000 0.000017 5
96.000000 0.000017 5
97.000000 0.000017 5
98.000000 0.000017 5
99.000000 0.000018 6
100.000000 0.000018 6
101.000000 0.000018 6
102.000000 0.000019 7
103.000000 0.000019 7
104.000000 0.000019 7
105.000000 0.000020 8
106.000000 0.000020 8
107.000000 0.000021 9
108.000000 0.000022 10
109.000000 0.000022 10
110.000000 0.000023 11
111.000000 0.000024 12
112.000000 0.000025 13
113.000000 0.000025 13
114.000000 0.000026 14
115.000000 0.000028 16
116.000000 0.000029 17
117.000000 0.000030 18
118.000000 0.000031 19
119.000000 0.000033 21
120.000000 0.000035 23
121.000000 0.000036 24
122.000000 0.000038 26
123.000000 0.000040 28
124.000000 0.000043 31
125.000000 0.000045 33
126.000000 0.000048 36
127.000000 0.000051 39
128.000000 0.000054 42
129.000000 0.000058 46
130.000000 0.000062 50
131.000000 0.000067 55
132.000000 0.000072 60
133.000000 0.000077 65
134.000000 0.000083 71
135.000000 0.000090 78
136.000000 0.000097 85
137.000000 0.000106 94
138.000000 0.000115 103
139.000000 0.000126 114
140.000000 0.000138 126
141.000000 0.000152 140
142.000000 0.000167 155
143.000000 0.000185 173
144.000000 0.000205 193
145.000000 0.000228 216
146.000000 0.000254 242
147.000000 0.000285 273
148.000000 0.000321 309
149.000000 0.000362 350
150.000000 0.000411 399
151.000000 0.000468 456
152.000000 0.000537 525
153.000000 0.000618 606
154.000000 0.000716 704
155.000000 0.000834 822
156.000000 0.000978 966

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.

UniformPhi.png



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