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 1E7 events has no Moller electrons with momentum over 5500 MeV. Since momentum is conserved, and the data is verified kinematicly verified, we can simply "switch" the data. This data can then 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 4E7 incident electrons, a file of just scattered momentum components can be constructed using:
awk '{print $9, $10, $11, $16, $17, $18}' MollerScattering_NH3_Large.dat > Just_Scattered_Momentum.dat
Transfer to CM Frame
We can perform a Lorentz transformation from the Center of Mass frame, with zero total momentum, to the Lab frame.
[math]\left( \begin{matrix}E^*_{1}+E^*_{2}\\ p^*_{1(x)}+p^*_{2(x)} \\p^*_{1(y)}+ p^*_{2(y)} \\ p^*_{1(z)}+p^*_{2(z)}\end{matrix} \right)=\left(\begin{matrix}\gamma & 0 & 0 & -\beta \gamma \\0 & 1 & 0 & 0 \\ 0 & 0 & 1 &0 \\ -\beta \gamma & 0 & 0 & \gamma \end{matrix} \right) . \left( \begin{matrix}E^'_{1}+E^'_{2}\\ p^'_{1(x)}+p^'_{2(x)} \\ p^'_{(1(y)}+p^'_{2(y)} \\ p^'_{1(z)}+p^'_{2(z)}\end{matrix} \right)[/math]
[math]\left( \begin{matrix}E^*\\ p^*_{x}\\p^*_{y}\\ p^*_{z}\end{matrix} \right)=\left(\begin{matrix}\gamma & 0 & 0 & -\beta \gamma\\0 & 1 & 0 & 0 \\ 0 & 0 & 1 &0 \\ -\beta \gamma & 0 & 0 & \gamma \end{matrix} \right) . \left( \begin{matrix}E^'\\ p^'_{x} \\ p^'_{y} \\ p^'_{z}\end{matrix} \right)[/math]
[math]\Longrightarrow\begin{cases}
E^*=\gamma E^'+\beta \gamma p^'_{z} \\
p^*_{z}=\beta \gamma E^'+ \gamma p^'_{z}
\end{cases}[/math]
Since in the CM Frame, p*=0
[math]\beta \gamma E^'=- \gamma p^'_{z}[/math]
[math]\Longrightarrow E^'=- \frac{ p^'_{z}}{\beta}[/math]
[math]\Longrightarrow\begin{cases}
E^*=-\gamma \frac{ p^'_{z}}{\beta}+\beta \gamma p^'_{z} \\
\beta \gamma E^'=- \gamma p^'_{z}
\end{cases}[/math]
Alter Phi Angles
Run for Necessary Amount to match Cross Section