Latest revision as of 20:44, 15 May 2018

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4.1 Uniform Distribution in Energy and Theta LUND files

To ensure that all the electrons simulated in GEMC are Moller electrons, electrons are first creating in the center of mass frame after scattering has occured. In this frame, the Moller differential cross section is well defined, and can be used to ensure that all particles to be simulated in other frames of reference will be Moller electrons. Due to the fact that GEMC simulations will utilize particle momentum in the lab frame, this imlies the LUND files must contain lab frame information. The transformation from CM to lab frame can be done using Lorentz boosts. In addition, all the data given is for 11GeV electrons that have not yet had the chance to interact with the target material. This implies that all electrons have the same incident energy in the lab, and as such have a constant value in the CM frame.

In the lab frame there are 2 degrees of freedom, i.e. [math]P_{\theta}[/math] and [math]P_{\phi}[/math] . The number of degrees of freedom can be reduced to 1 degrees of freedom in the center of mass frame of reference since the Moller differential cross section does not rely on [math]\phi[/math] and all energies are considered equal in the CM frame. The simplicity of the center of mass frame for Moller electrons is that the interacting particles have equal energies and equal but opposite momentum vectors. In such a system where these conditions must be met, the directions of the momentum vectors must be opposite, which reduces the degrees of freedom to one direction. Since the Moller differential cross-section only relies on [math]\theta[/math] in all frames it implies for a given angle [math]\theta[/math] we should find the same results for any given angle [math]\phi[/math].

To ensure that we can produce the Moller differential cross section in the center of mass frame we start with a uniform isotropic distribution of Moller electrons with respect to the scattering angle theta. From this distribution, each particle can be applied an appropriate weight that will result in the differential cross section being reproduced.

Figure 4.1.1: An Isotropic CM frame distribution of final scattering angle theta for Moller electrons. 90-180 degrees in the center of mass frame, with a spacing of .0001 degree increments.
Figure 4.1.2: An CM frame distribution for an isotropic distribution in final scattering angle theta for Moller electrons. All Moller electrons in the center of mass frame have the same energy, and equal but opposite momentum vectors.

The LUND file is created by creating an isotropic distribution of electrons within the Moller center of mass frame of reference after scattering. These particles are uniformly distributed through the angle theta with respect to the beam line in the range 90-180 in the center of mass frame. This is initially done at a set angle phi (0 degrees) with respect to the perpendicular components with respect to the beam line. In the CM frame, the spacing of the angle theta bins are 0.0001 degrees wide. A Lorenz contraction occurs for a particle's momentum component that is parallel to the beam line. As the angle of the particle approaches a direction perpendicular to the beam line, the Lorentz contraction decreases. As a result, a uniform angular distribution in the Lab frame will not be uniform in the CM frame.

Figure 2: A Mankowski diagram demonstrating the Lorentz contraction increasing as the z component approaches the speed of light. Taking the perpendicular axis as the lab frame, the spacing between the arbitary measurements is equal when viewed from within the specific frame, but unequal as shown by the dots viewed from the lab frame.

This implies that the number of events that occur in the lab frame of reference near the beam line is larger than the number as it approaches a perpendicular direction. To understand the "density" of the number of events per bins in the lab frame, a study of 1000 events in the center of mass frame per 0.01 degree in the lab frame is investigated. A weighting factor, used to reproduce the Moller cross section, appears in the LUND file but not the GEMC evio output file thereby requiring both the LUND and evio files to be read simultaneously.

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Difference between revisions of "Uniform distribution in Energy and Theta LUND files"

Latest revision as of 20:44, 15 May 2018

4.1 Uniform Distribution in Energy and Theta LUND files

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@@ Line 1: / Line 1: @@
-=[[File:LUND_Spread.C|Creating uniform LUND files]]=
+<center><math>\underline{\textbf{Navigation}}</math>
-The LUND file is created by creating an isotropic distribution of particles within the Moller Center of Mass frame of reference.  These particles are also uniformly distributed through the angle theta with respect to the beam line in the range 90-180 in the Center of Mass frame.  This is done at a set angle phi (10 degrees) with respect to the perpendicular components with respect to the beam line.
+[[Preparing_Drift_Chamber_Efficiency_Tests|<math>\vartriangleleft </math>]]
+[[VanWasshenova_Thesis#Preparing_Drift_Chamber_Efficiency_Tests|<math>\triangle </math>]]
+[[1000_Events_per_degree_in_the_range_5_to_40_degrees_for_Lab_Frame|<math>\vartriangleright </math>]]
-<center>[[File:MolThetaCM_spread.png]][[File:MolMomCM_spread.png]]</center>
+</center>
-==Center of Mass for Stationary Target==
-For an [[DV_Calculations_of_4-momentum_components#Center_of_Mass_Frame|incoming electron of 11GeV striking a stationary electron]] we would expect:
-[[File:Screen_Shot_2016-05-12_at_6.42.46_PM.png]]
+=4.1 Uniform Distribution in Energy and Theta LUND files=
+[[File:LUND_Spread.C|Creating uniform LUND files]]
+To ensure that all the electrons simulated in GEMC are Moller electrons, electrons are first creating in the center of mass frame after scattering has occured.  In this frame, the Moller differential cross section is well defined, and can be used to ensure that all particles to be simulated in other frames of reference will be Moller electrons.  Due to the fact that GEMC simulations will utilize particle momentum in the lab frame, this imlies the LUND files must contain lab frame information.  The transformation from CM to lab frame can be done using Lorentz boosts.  In addition, all the data given is for 11GeV electrons that have not yet had the chance to interact with the target material.  This implies that all electrons have the same incident energy in the lab, and as such have a constant value in the CM frame.
-Boosting to the Center of Mass Frame:
-[[File:Screen_Shot_2016-05-12_at_6.44.44_PM.png]]
+In the lab frame there are 2 degrees of freedom, i.e. <math>P_{\theta}</math> and <math>P_{\phi}</math> .  The number of degrees of freedom can be reduced to 1 degrees of freedom in the center of mass frame of reference since the Moller differential cross section does not rely on <math>\phi</math> and all energies are considered equal in the CM frame.  The simplicity of the center of mass frame for Moller electrons is that the interacting particles have equal energies and equal but opposite momentum vectors.  In such a system where these conditions must be met, the directions of the momentum vectors must be opposite, which reduces the degrees of freedom to one direction.  Since the Moller differential cross-section only relies on <math>\theta</math> in all frames it implies for a given angle <math>\theta</math> we should find the same results for any given angle <math>\phi</math>.
-<center>[[File:Init_e_Mom_CM.png|600 px]][[File:Init_Mol_Mom_CM.png|600 px]]</center>
+To ensure that we can produce the Moller differential cross section in the center of mass frame we start with a uniform isotropic distribution of Moller electrons with respect to the scattering angle theta.  From this distribution, each particle can be applied an appropriate weight that will result in the differential cross section being reproduced.
-<center>[[File:Init_e_Theta_CM.png|600 px]][[File:Init_Mol_Theta_CM.png|600 px]]</center>
+<center><gallery widths=500px heights=400px>
+File:MolThetaCM_spread.png|'''Figure 4.1.1:''' An Isotropic CM frame distribution of final scattering angle theta for Moller electrons.  90-180 degrees in the center of mass frame, with a spacing of .0001 degree increments.
+File:MolMomCM_spread.png|'''Figure 4.1.2:''' An CM frame distribution for an isotropic distribution in final scattering angle theta for Moller electrons.  All Moller electrons in the center of mass frame have the same energy, and equal but opposite momentum vectors.
+</gallery></center>
-==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:
-<center><math>p^2=p_x^2+p_y^2+p_z^2</math></center>
-In the Center of Mass frame, this becomes:
+The LUND file is created by creating an isotropic distribution of electrons within the Moller center of mass frame of reference after scattering.  These particles are uniformly distributed through the angle theta with respect to the beam line in the range 90-180 in the center of mass frame.  This is initially done at a set angle phi (0 degrees) with respect to the perpendicular components with respect to the beam line.  In the CM frame, the spacing of the angle theta bins are 0.0001 degrees wide.  A Lorenz contraction occurs for a particle's momentum component that is parallel to the beam line.  As the angle of the particle approaches a direction perpendicular to the beam line, the Lorentz contraction decreases.  As a result, a uniform angular distribution in the Lab frame will not be uniform in the CM frame.
-<center><math>p_x^{*2}+p_y^{*2} = p^{*2}-p_z^{*2}</math></center>
+[[File: Mankowski_Diagram.png |thumb | border | center |600 px |alt=Mankowski Diagram demonstrating Lorentz contraction|'''Figure 2:''' A Mankowski diagram demonstrating the Lorentz contraction increasing as the z component approaches the speed of light.  Taking the perpendicular axis as the lab frame, the spacing between the arbitary measurements is equal when viewed from within the specific frame, but unequal as shown by the dots viewed from the lab frame.]]
+This implies that the number of events that occur in the lab frame of reference near the beam line is larger than the number as it approaches a perpendicular direction.  To understand the "density" of the number of events per bins in the lab frame, a study of 1000 events in the center of mass frame per 0.01 degree in the lab frame is investigated.  A weighting factor, used to reproduce the Moller cross section, appears in the LUND file but not the GEMC evio output file thereby requiring both the LUND and evio files to be read simultaneously.
-Since the momentum in the CM frame is a constant, this implies that pz must decrease.  For phi=10 degrees:
-This is repeated for rotations of 60 degrees in phi in the Lab frame.
+----
-We can use the variable rapidity:
-<center><math>y \equiv \frac {1}{2} \ln \left(\frac{E+p_z}{E-p_z}\right)</math></center>
+<center><math>\underline{\textbf{Navigation}}</math>
-where
+[[Preparing_Drift_Chamber_Efficiency_Tests|<math>\vartriangleleft </math>]]
+[[VanWasshenova_Thesis#Preparing_Drift_Chamber_Efficiency_Tests|<math>\triangle </math>]]
+[[1000_Events_per_degree_in_the_range_5_to_40_degrees_for_Lab_Frame|<math>\vartriangleright </math>]]
-<center><math> P^+ \equiv E+p_z</math></center>
+</center>
-<center><math> P^- \equiv E-p_z</math></center>
-this implies that as
-<center><math>p_z \rightarrow 0 \Rrightarrow \frac{E+p_z}{E-p_z} \rightarrow 1 \Rrightarrow \ln 1 \rightarrow 0 \Rrightarrow y=0</math></center>
-For forward travel in the light cone:
-<center><math>p_z \rightarrow E \Rrightarrow \frac{E+p_z}{E-p_z} \rightarrow \infin \Rrightarrow \ln \infin \rightarrow \infin \Rrightarrow y \rightarrow \infin </math></center>
-For backward travel in the light cone:
-<center><math>p_z \rightarrow -E \Rrightarrow \frac{E+p_z}{E-p_z} \rightarrow 0 \Rrightarrow \ln 0 \rightarrow -\infin \Rrightarrow y \rightarrow -\infin </math></center>
-For a particle that transforms from the Lab frame to the CM frame where the particle is not within the light cone:
-<center><math>p_x^2+p_y^2=52.589054^2+9.272868^2=53.400MeV > 53.015 MeV (E) \therefore p_z \rightarrow imaginary</math></center>
-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.
-==Determining Momentum Components After Collision in CM Frame==
-The energy and total momentum of the Moller and scattered electron remain the same under a rotation in any frame of reference.  After the collision, these quantities remain the same, but the x, y, z components change.
-In this frame, we can cycle through values of theta from 90 to 180 degrees which physically correspond to a stationary electron being impinged by an electron.
-===Theta Dependent Components===
-<center>[[File:xz_lab.png | 400 px]]</center>
-<center>'''Figure 3: Definition of Moller electron variables in the CM Frame in the x-z plane.'''</center>
-<center>Using <math>\theta '_2=\arccos \left(\frac{p^'_{2(z)}}{p^'_{2}}\right)</math></center>
-{| class="wikitable" align="center"
-| style="background: gray"      | <math>\Longrightarrow {p^'_{2(z)}=p^'_{2}\cos(\theta '_2)}</math>
-|}
-Checking on the sign resulting from the cosine function, we are limited to:
-{| class="wikitable" align="center"
-| style="background: red"      | <math>90^\circ \le \theta '_2 \le 180^\circ \equiv \frac{\pi}{2} \le \theta '_2 \le \pi Radians</math>
-|}
-Since,
-<center><math>\frac{p^'_{2(z)}}{p^'_{2}}=cos(\theta '_2)</math></center>
-<center><math>\Longrightarrow p^'_{2(z)}\ should\ always\ be\ negative</math></center>
-===Phi Dependent Components===
-Since only the z direction is considered to be the relativistic direction of motion, this implies that the x and y components are not effected by a Lorentz transformation and remain the same in the CM and Lab frame.  Holding the angle Phi constant at an initial value of 10 degrees, allows us to find the x and y components.
-<center>[[File:xy_lab.png | 400 px]]</center>
-<center>'''Figure 4: Definition of Moller electron variables in the Lab Frame in the x-y plane.'''</center>
-<center>Similarly, <math>\phi '_2=\arccos \left( \frac{p^'_{2(x) Lab}}{p^'_{2(xy)}} \right)</math></center>
-<center>where <math>p_{2(xy)}^'=\sqrt{(p_{2(x)}^')^2+(p^'_{2(y)})^2}</math></center>
-<center><math>(p^'_{2(xy)})^2=(p^'_{2(x)})^2+(p^'_{2(y)})^2</math></center>
-<center>and using <math>p^2=p_{(x)}^2+p_{(y)}^2+p_{(z)}^2</math></center>
-<center>this gives <math>(p^'_{2})^2=(p^'_{2(xy)})^2+(p^'_{2(z)})^2</math></center>
-<center><math>\Longrightarrow (p'_{2})^2-(p'_{2(z)})^2 = (p'_{2(xy)})^2</math></center>
-<center><math>\Longrightarrow p_{2(xy)}^'=\sqrt{(p^'_{2})^2-(p^'_{2(z)})^2}</math></center>
-<center>which gives<math>\phi '_2 = \arccos \left( \frac{p_{2(x)}'}{\sqrt{p_{2}^{'\ 2}-p_{2(z)}^{'\ 2}}}\right)</math></center>
-{| class="wikitable" align="center"
-| style="background: gray"      | <math>\Longrightarrow p_{2(x)}'=\sqrt{p_{2}^{'\ 2}-p_{2(z)}^{'\ 2}} \cos(\phi)</math>
-|}
-<center>Similarly, using <math>p_{2}^2=p_{2(x)}^2+p_{2(y)}^2+p_{2(z)}^2</math></center>
-<center><math>\Longrightarrow p_{2}^{'\ 2}-p_{2(x)}^{'\ 2}-p_{2(z)}^{'\ 2}=p_{2(y)}^{'\ 2}</math></center>
-{| class="wikitable" align="center"
-| style="background: gray"      | <math>p_{2(y)}'=\sqrt{p_{2}^{'\ 2}-p_{2(x)}^{'\ 2}-p_{2(z)}^{'\ 2}}</math>
-|}
-Checking on the sign from the cosine results for <math>\phi '_2</math>
-We have the limiting range that <math>\phi</math> must fall within:
-{| class="wikitable" align="center"
-| style="background: grey"      | <math>-\pi \le \phi '_2 \le \pi\ Radians</math>
-|}
-<center>[[File:xy_plane.png | 400px]]</center>
-Examining the signs of the components which make up the angle <math>\phi</math> in the 4 quadrants which make up the xy plane:
-{| class="wikitable" align="center"
-| style="background: red"      | <math>For\ 0 \ge \phi '_2 \ge \frac{-\pi}{2}\ Radians</math>
-|-
-| <center>p<sub>x</sub>=POSITIVE</center>
-|-
-| <center>p<sub>y</sub>=NEGATIVE</center>
-|}
-{| class="wikitable" align="center"
-| style="background: red"      | <math>For\ 0 \le \phi '_2 \le \frac{\pi}{2}\ Radians</math>
-|-
-| <center>p<sub>x</sub>=POSITIVE</center>
-|-
-| <center>p<sub>y</sub>=POSITIVE</center>
-|}
-{| class="wikitable" align="center"
-| style="background: red"      | <math>For\ \frac{-\pi}{2} \ge \phi '_2 \ge -\pi\ Radians</math>
-|-
-| <center>p<sub>x</sub>=NEGATIVE</center>
-|-
-| <center>p<sub>y</sub>=NEGATIVE</center>
-|}
-{| class="wikitable" align="center"
-| style="background: red"      | <math>For\ \frac{\pi}{2} \le \phi '_2 \le \pi\ Radians</math>
-|-
-|<center>p<sub>x</sub>=NEGATIVE</center>
-|-
-| <center>p<sub>y</sub>=POSITIVE</center>
-|}
-===Electron Center of Mass Frame===
-Relativistically, the x, y, and z components have the same magnitude, but opposite direction, in the conversion from the Moller electron's Center of Mass frame to the electron's Center of Mass frame.
-{| class="wikitable" align="center"
-| style="background: gray"      | <math>p^*_{2(x)}= -p^*_{1(x)}</math>
-|}
-{| class="wikitable" align="center"
-| style="background: gray"  | <math>p^*_{2(y)}= -p^*_{1(y)}</math>
-|}
-{| class="wikitable" align="center"
-| style="background: gray"  | <math>p^*_{2(z)}=-p^*_{1(z)}</math>
-|}
-where previously it was shown
-{| class="wikitable" align="center"
-| style="background: gray"      | <math>|p^*_{1}|\equiv |p^*_{2}|</math>
-|}
-{| class="wikitable" align="center"
-| style="background: gray"      | <math>E^*_{1}\equiv E^*_{2}</math>
-|}
-{| class="wikitable" align="center"
-| style="background: gray"      | <math>\theta_{1}^*=\pi-\theta_{1}^*</math>
-|}
-==Lorentz Transformation to Lab Frame==
-\begin{bmatrix}
-\gamma & 0 & 0 & -\gamma \beta \\
-& 1 & 0 & 0 \\
-& 0 & 1 & 0 \\
--\gamma \beta & 0 & 0 & \gamma
-\end{bmatrix}
-==Weight==
-[[File:MolThetaCMweightedTheory.png]]
-[[File:MolThetaLab_weighted.png]]