Difference between revisions of "Known Moller differential cross section"

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<center><math>\frac{5.3279\times 10^{-5}}{4\times 2.81\times 10^{15}eV^2}=4.74\times 10^{-21} eV^{-2}=\frac{4.74\times 10^{-21}}{1eV^2}\times \frac{1\times 10^{18}eV^2 }{1\ GeV^2}=\frac{.0047}{GeV^2}</math></center>
 
<center><math>\frac{5.3279\times 10^{-5}}{4\times 2.81\times 10^{15}eV^2}=4.74\times 10^{-21} eV^{-2}=\frac{4.74\times 10^{-21}}{1eV^2}\times \frac{1\times 10^{18}eV^2 }{1\ GeV^2}=\frac{.0047}{GeV^2}</math></center>
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 +
 +
The trigonometric function part of the equation comes out to it's minimum of 9 at 90 degrees.
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 +
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<center><math>\frac{(3+Cos^2(90))^2}{Sin^4(90)}=9</math></center>
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Using the conversion of  
 
Using the conversion of  
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<center><math>\frac{1\ GeV^{-2}}{.3894\ mb}=\frac{.0047\ GeV^{-2}}{x\ mb}</math></center>
 
<center><math>\frac{1\ GeV^{-2}}{.3894\ mb}=\frac{.0047\ GeV^{-2}}{x\ mb}</math></center>
  
We find that the differential cross section scale is <math>\frac{d\sigma}{d\Omega}\approx 1.8\times 10^{-3}mb=1.8\mu b</math>
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We find that the differential cross section scale is <math>\frac{d\sigma}{d\Omega}\approx 1.8\times 10^{-3}mb=1.8\mu b\times 9=17\mu b</math>
  
  
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Converting the number of electrons to barns,
 
Converting the number of electrons to barns,
  
:::::<math>L=\frac{i_{scattered}}{\sigma} \approx i_{scattered}\times \rho_{target}\times l_{target}</math>
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<center><math>L=\frac{i_{scattered}}{\sigma} \approx i_{scattered}\times \rho_{target}\times l_{target}</math></center>
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 +
 
 +
where ρ<sub>target</sub> is the density of the target material, l<sub>target</sub> is the length of the target, and i<sub>scattered</sub> is the number of incident particles scattered.
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 +
 
 +
For a ammonia target,
 +
 
 +
Ammonia density:
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 +
<math>\rho = \frac{.8 g}{cm^3}</math>
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 +
 
 +
Molecular weight calculation:
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<math>\frac{14.0067 g}{1 mole} + \frac{1.00794 g}{1 mole}\times 3=\frac{17 g}{1 mole}</math>
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 +
 
 +
:::::<math>\rho_{target}\times l_{target}=\frac{.8 g}{1 cm^3}\times \frac{1 mole}{17 g} \times  \frac{6\times10^{23} atoms}{1 mole} \times \frac{1 cm}{ } \times \frac{10^{-24} cm^2}{barn} =2.82\times 10^{-2} barns</math>
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 +
 
 +
 
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For 400 million incoming electrons
 +
 
 +
 
 +
:::::<math>\frac{1}{\rho_{target}\times l_{target} \times 4 \times 10^8}=8.87\times 10^{-8} barns</math>
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where ρ<sub>target</sub> is the density of the target material, l<sub>target</sub> is the length of the target, and i<sub>scattered</sub> is the number of incident particles scattered.  For a ammonia target,
 
  
 +
[[File: MolThetaCM_E500Lab.png]][[File:NH3TheoryDiffXSect.png]]
  
:::::<math>\rho_{target}\times l_{target}=\frac{.8 g}{1 cm^3}\times \frac{1 mole}{2.02 g} \times \frac{1000g}{1 kg} \times \frac{6\times10^{23} atoms}{1 mole} \times \frac{1cm}{100 cm} \times \frac{1 m}{ } \times \frac{10^{-23} m^2}{barn} =2.10\times 10^{-2} barns</math>
 
  
 +
Combining these plots, and rescaling the Final Theta in the Center of Mass for barns:
  
:::::<math>\frac{1}{\rho_{target}\times l_{target} \times 4\times 10^7}=1.19\times 10^{-6} barns</math>
 
  
[[File:FnlThetaCM.png]][[File:Theory_new.png]]
 
  
  
Combining these plots, and rescaling the Final Theta in the Center of Mass for micro-barns, we find
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<pre>
 +
TH1F *Combo=new TH1F("TheoryExperiment","Theoretical and Experimental Differential Cross-Section CM Frame",360,90,180);
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TH1F *Combo2=new TH1F("TheoryExperiment2","Theoretical and Experimental Differential Cross-Section CM Frame",360,90,180);
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Combo->Add(MolThetaCM,8.87e-8);
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Combo2->Add(Theory,1e-1);
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Combo->Draw();
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Combo2->Draw("same");
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</pre>
  
  
  
<center>[[File:XSect_new_zoom.png]]</center>
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<center>[[File:Adjusted_MollerXSect_NH3.png]]</center>

Latest revision as of 21:03, 15 April 2016

Comparing experimental vs. theoretical for Møller differential cross section 11GeV

Using the equation from [1]

[math]\frac{d\sigma}{d\Omega}=\frac{ e^4 }{8E^2}\left \{\frac{1+cos^4\frac{\theta}{2}}{sin^4\frac{\theta}{2}}+\frac{1+sin^4\frac{\theta}{2}}{cos^4\frac{\theta}{2}}+\frac{2}{sin^2\frac{\theta}{2}cos^2\frac{\theta}{2}} \right \}[/math]


where [math]\alpha=\frac{e^2}{\hbar c}\quad with\quad \hbar = c =1[/math]


This can be simplified to the form


[math]\frac{d\sigma}{d\Omega}=\frac{ \alpha^2 }{4E^2}\frac{ (3+cos^2\theta)^2}{sin^4\theta}[/math]

Plugging in the values expected for a scattering electron:



[math]\alpha ^2=5.3279\times 10^{-5}[/math]


[math]E\approx 53 MeV[/math]


Using unit analysis on the term outside the parantheses, we find that the differential cross section for an electron at this momentum should be around

[math]\frac{5.3279\times 10^{-5}}{4\times 2.81\times 10^{15}eV^2}=4.74\times 10^{-21} eV^{-2}=\frac{4.74\times 10^{-21}}{1eV^2}\times \frac{1\times 10^{18}eV^2 }{1\ GeV^2}=\frac{.0047}{GeV^2}[/math]


The trigonometric function part of the equation comes out to it's minimum of 9 at 90 degrees.


[math]\frac{(3+Cos^2(90))^2}{Sin^4(90)}=9[/math]


Using the conversion of


[math]\frac{1}{1GeV^2}=.3894 mb[/math]


[math]\frac{1\ GeV^{-2}}{.3894\ mb}=\frac{.0047\ GeV^{-2}}{x\ mb}[/math]

We find that the differential cross section scale is [math]\frac{d\sigma}{d\Omega}\approx 1.8\times 10^{-3}mb=1.8\mu b\times 9=17\mu b[/math]


Converting the number of electrons to barns,

[math]L=\frac{i_{scattered}}{\sigma} \approx i_{scattered}\times \rho_{target}\times l_{target}[/math]


where ρtarget is the density of the target material, ltarget is the length of the target, and iscattered is the number of incident particles scattered.


For a ammonia target,

Ammonia density:

[math]\rho = \frac{.8 g}{cm^3}[/math]


Molecular weight calculation:

[math]\frac{14.0067 g}{1 mole} + \frac{1.00794 g}{1 mole}\times 3=\frac{17 g}{1 mole}[/math]


[math]\rho_{target}\times l_{target}=\frac{.8 g}{1 cm^3}\times \frac{1 mole}{17 g} \times \frac{6\times10^{23} atoms}{1 mole} \times \frac{1 cm}{ } \times \frac{10^{-24} cm^2}{barn} =2.82\times 10^{-2} barns[/math]


For 400 million incoming electrons


[math]\frac{1}{\rho_{target}\times l_{target} \times 4 \times 10^8}=8.87\times 10^{-8} barns[/math]



MolThetaCM E500Lab.pngNH3TheoryDiffXSect.png


Combining these plots, and rescaling the Final Theta in the Center of Mass for barns:



TH1F *Combo=new TH1F("TheoryExperiment","Theoretical and Experimental Differential Cross-Section CM Frame",360,90,180);
TH1F *Combo2=new TH1F("TheoryExperiment2","Theoretical and Experimental Differential Cross-Section CM Frame",360,90,180);
Combo->Add(MolThetaCM,8.87e-8);
Combo2->Add(Theory,1e-1);
Combo->Draw();
Combo2->Draw("same");


Adjusted MollerXSect NH3.png