Difference between revisions of "Occupancy for Sector 1"

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=Finding the Cross Section=
 
=Finding the Cross Section=
==Total cross section==
+
==Total cross section over φ==
 
[[File:CrossSectionMathematicaProof.png]]
 
[[File:CrossSectionMathematicaProof.png]]
  
 +
==Total cross section over DC limits==
  
 +
 +
If we make the assumption that the beam of incoming electrons is a flux over an area for a given time,
 +
 +
<center><math>N_{incident}=\Phi\ A_{beam}\ t_{run} \rightarrow dN_{incident}=\Phi\ dA_{beam}\ t_{run}\rightarrow\  \frac{dN_{incident}}{ dA_{beam}}=\Phi\ t_{run}</math></center>
 +
 +
 +
Using the definition  of the differential cross section:
 +
 +
<center><math>\frac{d\sigma}{d\Omega}\equiv \frac{ \Biggl(\frac{dN_{scattered}}{d\Omega} \Biggr)}{\Biggl(\frac{dN_{incident}}{dA}\Biggr)}\rightarrow \frac{d\sigma}{d\Omega}\Biggl(\frac{dN_{incident}}{dA}\Biggr)=\Biggl(\frac{dN_{scattered}}{d\Omega} \Biggr)</math></center>
 +
 +
 +
Substituting using the flux
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 +
<center><math> \frac{d\sigma}{d\Omega}\Biggl(\frac{dN_{incident}}{dA}\Biggr)=\Biggl(\frac{dN_{scattered}}{d\Omega} \Biggr)\rightarrow  \frac{d\sigma}{d\Omega}\Phi\ t_{run}=\Biggl(\frac{dN_{scattered}}{d\Omega} \Biggr)</math></center>
 +
 +
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<center><math>\rightarrow dN_{scattered}= \frac{d\sigma}{d\Omega}\Phi d\Omega= \frac{d\sigma}{d\Omega}\Phi\ t_{run}\ \sin \theta\ d\theta\ d\phi</math></center>
 +
 +
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Since the differential cross section is known in the Center of Mass frame of reference, but measurements are taken in the Lab Frame, a transformation must occur.
 +
 +
<center><math>\rightarrow dN_{scattered}= \frac{d\sigma}{d\Omega_{Lab}}\Phi\ t\ \sin \theta_{Lab}\ d\theta_{Lab}\ d\phi_{Lab}</math></center>
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 +
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<center><math>\frac{d\sigma}{d\Omega_{Lab}}\sin \theta_{Lab}\ d\theta_{Lab}\ d\phi_{Lab}=\frac{d\sigma}{d\Omega_{CM}}\sin \theta_{CM}\ d\theta_{CM}\ d\phi_{CM}</math></center>
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<center><math>\frac{d\sigma}{d\Omega_{Lab}}=\frac{d\sigma}{d\Omega_{CM}}\frac{\sin \theta_{CM}\ d\theta_{CM}\ d\phi_{CM}}{\sin \theta_{Lab}\ d\theta_{Lab}\ d\phi_{Lab}}</math></center>
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<center><math>\rightarrow dN_{scattered}=\frac{d\sigma}{d\Omega_{CM}}\frac{\sin \theta_{CM}\ d\theta_{CM}\ d\phi_{CM}}{\sin \theta_{Lab}\ d\theta_{Lab}\ d\phi_{Lab}}\Phi\ t_{run}\ \sin \theta_{Lab}\ d\theta_{Lab}\ d\phi_{Lab}</math></center>
 +
 +
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If we divide both sides by time
 +
 +
 +
<center><math>\rightarrow \frac{dN_{scattered}}{t_{run}}=\frac{d\sigma}{d\Omega_{CM}}\frac{\sin \theta_{CM}\ d\theta_{CM}\ d\phi_{CM}}{\sin \theta_{Lab}\ d\theta_{Lab}\ d\phi_{Lab}}\Phi \sin \theta_{Lab}\ d\theta_{Lab}\ d\phi_{Lab}</math></center>
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 +
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<center><math>\rightarrow \frac{dN_{scattered}}{t_{run}}=\frac{d\sigma}{d\Omega_{CM}}\frac{\sin \theta_{CM}\ d\theta_{CM}\ d\phi_{CM}}{\sin \theta_{Lab}\ d\theta_{Lab}\ d\phi_{Lab}}\frac{N_{incident}}{t_{run}} \sin \theta_{Lab}\ d\theta_{Lab}\ d\phi_{Lab}</math></center>
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<center><math>\rightarrow \frac{dN_{scattered}}{N_{incident}}=\frac{d\sigma}{d\Omega_{CM}}\sin \theta_{CM}\ d\theta_{CM}\ d\phi_{CM}</math></center>
 
Performing a Riemann sum for <math>-30^{\circ} \lt \phi \lt 30^{\circ}</math>
 
Performing a Riemann sum for <math>-30^{\circ} \lt \phi \lt 30^{\circ}</math>
  
Line 105: Line 154:
 
===Bin Spacing of 0.1 degrees for θ in Lab Frame===
 
===Bin Spacing of 0.1 degrees for θ in Lab Frame===
  
=Using the Cross Section=
 
  
If we make the assumption that the beam of incoming electrons is a flux over an area for a given time,
 
 
<center><math>N_{incident}=\Phi\ A_{beam}\ t_{run} \rightarrow dN_{incident}=\Phi\ dA_{beam}\ t_{run}\rightarrow\  \frac{dN_{incident}}{ dA_{beam}}=\Phi\ t_{run}</math></center>
 
 
 
Using the definition  of the differential cross section:
 
 
<center><math>\frac{d\sigma}{d\Omega}\equiv \frac{ \Biggl(\frac{dN_{scattered}}{d\Omega} \Biggr)}{\Biggl(\frac{dN_{incident}}{dA}\Biggr)}\rightarrow \frac{d\sigma}{d\Omega}\Biggl(\frac{dN_{incident}}{dA}\Biggr)=\Biggl(\frac{dN_{scattered}}{d\Omega} \Biggr)</math></center>
 
 
 
Substituting using the flux
 
 
<center><math> \frac{d\sigma}{d\Omega}\Biggl(\frac{dN_{incident}}{dA}\Biggr)=\Biggl(\frac{dN_{scattered}}{d\Omega} \Biggr)\rightarrow  \frac{d\sigma}{d\Omega}\Phi\ t_{run}=\Biggl(\frac{dN_{scattered}}{d\Omega} \Biggr)</math></center>
 
 
 
<center><math>\rightarrow dN_{scattered}= \frac{d\sigma}{d\Omega}\Phi d\Omega= \frac{d\sigma}{d\Omega}\Phi\ t_{run}\ \sin \theta\ d\theta\ d\phi</math></center>
 
 
 
 
Since the differential cross section is known in the Center of Mass frame of reference, but measurements are taken in the Lab Frame, a transformation must occur.
 
 
<center><math>\rightarrow dN_{scattered}= \frac{d\sigma}{d\Omega_{Lab}}\Phi\ t\ \sin \theta_{Lab}\ d\theta_{Lab}\ d\phi_{Lab}</math></center>
 
 
 
 
<center><math>\frac{d\sigma}{d\Omega_{Lab}}\sin \theta_{Lab}\ d\theta_{Lab}\ d\phi_{Lab}=\frac{d\sigma}{d\Omega_{CM}}\sin \theta_{CM}\ d\theta_{CM}\ d\phi_{CM}</math></center>
 
 
 
 
<center><math>\frac{d\sigma}{d\Omega_{Lab}}=\frac{d\sigma}{d\Omega_{CM}}\frac{\sin \theta_{CM}\ d\theta_{CM}\ d\phi_{CM}}{\sin \theta_{Lab}\ d\theta_{Lab}\ d\phi_{Lab}}</math></center>
 
 
 
<center><math>\rightarrow dN_{scattered}=\frac{d\sigma}{d\Omega_{CM}}\frac{\sin \theta_{CM}\ d\theta_{CM}\ d\phi_{CM}}{\sin \theta_{Lab}\ d\theta_{Lab}\ d\phi_{Lab}}\Phi\ t_{run}\ \sin \theta_{Lab}\ d\theta_{Lab}\ d\phi_{Lab}</math></center>
 
 
 
If we divide both sides by time
 
 
 
<center><math>\rightarrow \frac{dN_{scattered}}{t_{run}}=\frac{d\sigma}{d\Omega_{CM}}\frac{\sin \theta_{CM}\ d\theta_{CM}\ d\phi_{CM}}{\sin \theta_{Lab}\ d\theta_{Lab}\ d\phi_{Lab}}\Phi \sin \theta_{Lab}\ d\theta_{Lab}\ d\phi_{Lab}</math></center>
 
 
 
 
<center><math>\rightarrow \frac{dN_{scattered}}{t_{run}}=\frac{d\sigma}{d\Omega_{CM}}\frac{\sin \theta_{CM}\ d\theta_{CM}\ d\phi_{CM}}{\sin \theta_{Lab}\ d\theta_{Lab}\ d\phi_{Lab}}\frac{N_{incident}}{t_{run}} \sin \theta_{Lab}\ d\theta_{Lab}\ d\phi_{Lab}</math></center>
 
 
 
 
<center><math>\rightarrow \frac{dN_{scattered}}{N_{incident}}=\frac{d\sigma}{d\Omega_{CM}}\sin \theta_{CM}\ d\theta_{CM}\ d\phi_{CM}</math></center>
 
  
 
=Number of Hits on Wires=
 
=Number of Hits on Wires=

Revision as of 15:15, 16 May 2018

[math]\underline{\textbf{Navigation}}[/math]

[math]\vartriangleleft [/math] [math]\triangle [/math] [math]\vartriangleright [/math]


A bash script to run the GEMC simulations is created. tcsh scripts to run root2evio on lds2 is called using sshpass. The lds2 scripts use sshfs The main script on lds3:

BUILD_GEMC_SIMULATION.sh


The 3 scripts on lds2:

first_commands.tcsh

second_commands.tcsh

last_commands.tcsh


LUND File Output

0.1 degree spacing in the Lab frame. CM Frame is not evenly spaced.


MolThetaLab LUND DC limits.pngMolThetaCM LUND DC limits.png

Finding the Cross Section

Total cross section over φ

CrossSectionMathematicaProof.png

Total cross section over DC limits

If we make the assumption that the beam of incoming electrons is a flux over an area for a given time,

[math]N_{incident}=\Phi\ A_{beam}\ t_{run} \rightarrow dN_{incident}=\Phi\ dA_{beam}\ t_{run}\rightarrow\ \frac{dN_{incident}}{ dA_{beam}}=\Phi\ t_{run}[/math]


Using the definition of the differential cross section:

[math]\frac{d\sigma}{d\Omega}\equiv \frac{ \Biggl(\frac{dN_{scattered}}{d\Omega} \Biggr)}{\Biggl(\frac{dN_{incident}}{dA}\Biggr)}\rightarrow \frac{d\sigma}{d\Omega}\Biggl(\frac{dN_{incident}}{dA}\Biggr)=\Biggl(\frac{dN_{scattered}}{d\Omega} \Biggr)[/math]


Substituting using the flux

[math] \frac{d\sigma}{d\Omega}\Biggl(\frac{dN_{incident}}{dA}\Biggr)=\Biggl(\frac{dN_{scattered}}{d\Omega} \Biggr)\rightarrow \frac{d\sigma}{d\Omega}\Phi\ t_{run}=\Biggl(\frac{dN_{scattered}}{d\Omega} \Biggr)[/math]


[math]\rightarrow dN_{scattered}= \frac{d\sigma}{d\Omega}\Phi d\Omega= \frac{d\sigma}{d\Omega}\Phi\ t_{run}\ \sin \theta\ d\theta\ d\phi[/math]


Since the differential cross section is known in the Center of Mass frame of reference, but measurements are taken in the Lab Frame, a transformation must occur.

[math]\rightarrow dN_{scattered}= \frac{d\sigma}{d\Omega_{Lab}}\Phi\ t\ \sin \theta_{Lab}\ d\theta_{Lab}\ d\phi_{Lab}[/math]


[math]\frac{d\sigma}{d\Omega_{Lab}}\sin \theta_{Lab}\ d\theta_{Lab}\ d\phi_{Lab}=\frac{d\sigma}{d\Omega_{CM}}\sin \theta_{CM}\ d\theta_{CM}\ d\phi_{CM}[/math]


[math]\frac{d\sigma}{d\Omega_{Lab}}=\frac{d\sigma}{d\Omega_{CM}}\frac{\sin \theta_{CM}\ d\theta_{CM}\ d\phi_{CM}}{\sin \theta_{Lab}\ d\theta_{Lab}\ d\phi_{Lab}}[/math]


[math]\rightarrow dN_{scattered}=\frac{d\sigma}{d\Omega_{CM}}\frac{\sin \theta_{CM}\ d\theta_{CM}\ d\phi_{CM}}{\sin \theta_{Lab}\ d\theta_{Lab}\ d\phi_{Lab}}\Phi\ t_{run}\ \sin \theta_{Lab}\ d\theta_{Lab}\ d\phi_{Lab}[/math]


If we divide both sides by time


[math]\rightarrow \frac{dN_{scattered}}{t_{run}}=\frac{d\sigma}{d\Omega_{CM}}\frac{\sin \theta_{CM}\ d\theta_{CM}\ d\phi_{CM}}{\sin \theta_{Lab}\ d\theta_{Lab}\ d\phi_{Lab}}\Phi \sin \theta_{Lab}\ d\theta_{Lab}\ d\phi_{Lab}[/math]


[math]\rightarrow \frac{dN_{scattered}}{t_{run}}=\frac{d\sigma}{d\Omega_{CM}}\frac{\sin \theta_{CM}\ d\theta_{CM}\ d\phi_{CM}}{\sin \theta_{Lab}\ d\theta_{Lab}\ d\phi_{Lab}}\frac{N_{incident}}{t_{run}} \sin \theta_{Lab}\ d\theta_{Lab}\ d\phi_{Lab}[/math]


[math]\rightarrow \frac{dN_{scattered}}{N_{incident}}=\frac{d\sigma}{d\Omega_{CM}}\sin \theta_{CM}\ d\theta_{CM}\ d\phi_{CM}[/math]

Performing a Riemann sum for [math]-30^{\circ} \lt \phi \lt 30^{\circ}[/math]


CrossSection60deg.png


The cross section should be equal between both frames since the number of particles is an invariant. The differential cross section must differ between frames since the solid angle does vary.

[math]\sigma_{(CM)}=\sigma{(Lab)}[/math]


[math]\frac{d\sigma}{d\Omega}_{(CM)} d\Omega_{(CM)}=\frac{d\sigma}{d\Omega}_{(Lab)} d\Omega_{(Lab)}[/math]



[math]\frac{d\sigma}{d\Omega}_{(CM)} \sin \theta_{(CM)}\ d\theta_{(CM)}\ d\phi=\frac{d\sigma}{d\Omega}_{(Lab)} \sin \theta_{(Lab)}\ d\theta_{(Lab)}\ d\phi[/math]


[math]\rightarrow \frac{d\sigma}{d\Omega}_{(Lab)}=\frac{d\sigma}{d\Omega}_{(CM)} \frac{\sin \theta_{(CM)}\ d\theta_{(CM)}\ d\phi}{ \sin \theta_{(Lab)}\ d\theta_{(Lab)}\ d\phi}[/math]


[math]\rightarrow d\sigma_{(Lab)}=\frac{d\sigma}{d\Omega}_{(CM)} \frac{\sin \theta_{(CM)}\ d\theta_{(CM)}\ d\phi}{ \sin \theta_{(Lab)}\ d\theta_{(Lab)}\ d\phi}\sin \theta_{(Lab)} d\theta_{(Lab)}\ d\phi[/math]


MolThetaCMdsigmaIntegral.pngMolThetaLabdSigmaIntegral.png

AssociatedWeights2.pngDSigmaCMLab.png

Adjust for DC Sector 1 Limits

IntegralDCLimitsdSigmaCM.pngIntegralDCLimitsdSigmaLab.png

GEMC Cross Section

CORRELATED HITS

CORRELATED conditions
GEMC conditions Meaning
k=0 1st registered hit
dpid[k]=11 Electron
tid[k]=2 Moller electron from LUND file
mpid[k]=0 The mother particle implied from LUND file
sector[k]=1 Hit is in sector 1


Bin Spacing of 0.05 degrees for θ in Lab Frame

Bin Spacing of 0.1 degrees for θ in Lab Frame

Number of Hits on Wires

Not all 1st hits are on layer 1. Using the correlated theoretical wire number associated with the LUND Theta and Phi values:


WireBinsDCLimits.pngDSigmaVsWireBins.png


The theoretical model has events which are detected by physically impossible valued wires. If we limit the lowest wire value to 0.5 and the highest to less than 112.5


TheoreticalWireBinsCorrected.pngDSigmaVsWireBinCorrected.png

Using the histogram integral function we find the sum of the values for the wire 1 bin. Collecting the individual [math]d\sigma[/math] for each theoretical and physical hits on DC wires.

Occupancy

LH2_NOSol_0Tor_11GeV_IsotropicPhi_v2_6_ShieldOut

Run

./BUILD_GEMC_SIMULATION.sh 

DVMacro

Clas12Mon

Create hipo file


Move hipo file to clas12mon folder

mv LH2_NOSol_0Tor_11GeV_IsotropicPhi_v2_6_ShieldOut.hipo ~/clas12mon

Run monitor program

./README

Load hipo file

"Press H for hipo"
"Press play"
"Switch to 


For [math]5^{\circ}\gt \theta\lt 40^{\circ}\ -30^{\circ}\gt \phi\lt 30^{\circ}[/math]

Clas12monNoSolNoShield.png


FOR DC Limits

OccupancyDCLimits Unweighted.png

Calculating

[math]N_0=\Delta t \cdot R_{events}=\Delta t \cdot \frac{N_{events}}{t_{simulated}}=250\times 10^{-9}\ s \cdot \frac{98181}{9.3\times 10^{-6}\ s}=2639[/math]


[math]Occupancy=\frac{N_{hits}}{N_0}=\frac{N_{hits}}{\Delta t \cdot R_{events}}=\frac{t_{simulated}\cdot N_{hits}}{N_{events}\cdot \Delta t}=[/math]