Uniform distribution in Energy and Theta LUND files
File:LUND Spread.C
The LUND file is created by creating an isotropic distribution of particles within the energy range of 2MeV-5.5GeV as is found through GEANT simulation. These particles are also uniformly distributed through the angle theta with respect to the beam line in the range 5-40 degrees. This is done at a set angle phi (10 degrees) with respect to the perpendicular components with respect to the beam line.
![Init Mol E Lab.png](/./images/b/ba/Init_Mol_E_Lab.png)
![Init Mol Theta Lab.png](/./images/b/b3/Init_Mol_Theta_Lab.png)
![Init Mol Mom Lab.png](/./images/d/de/Init_Mol_Mom_Lab.png)
Center of Mass for Stationary Target
For an incoming electron of 11GeV striking a stationary electron we would expect:
Boosting to the Center of Mass Frame:
![Init e Mom CM.png](/./images/thumb/5/59/Init_e_Mom_CM.png/600px-Init_e_Mom_CM.png)
![Init Mol Mom CM.png](/./images/thumb/a/a3/Init_Mol_Mom_CM.png/600px-Init_Mol_Mom_CM.png)
![Init e Theta CM.png](/./images/4/45/Init_e_Theta_CM.png)
![Init Mol Theta CM.png](/./images/e/e1/Init_Mol_Theta_CM.png)
Phase space Limiting Particles
Since the angle phi has been constrained to remain constant, the x and y components of the momentum will increase in the positive first quadrant. This implies that the z component of the momentum must decrease by the relation:
In the Center of Mass frame, this becomes:
Since the momentum in the CM frame is a constant, this implies that pz must decrease. For phi=10 degrees:
![MolPxPyLab.png](/./images/3/3d/MolPxPyLab.png)
![MolPxPyCM.png](/./images/3/35/MolPxPyCM.png)
This is repeated for rotations of 60 degrees in phi in the Lab frame.
We can use the variable rapidity:
where
this implies that as
For forward travel in the light cone:
For backward travel in the light cone:
For a particle that transforms from the Lab frame to the CM frame where the particle is not within the light cone:
These particles are outside the light cone and are more timelike, thus not visible in normal space. This will reduce the number of particles that will be detected.
![MolEThetaRapidityCM.png](/./images/b/b4/MolEThetaRapidityCM.png)
Using Initial Condition based on After Collision in Lab Frame
Starting with the Energy, and a Phi angle of 10 degrees, the 4-momentum vector can be kinematically determined:
Using the fact that phi is defined in the xy plane to be: