Difference between revisions of "Pair Production Rate Calculation"

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(Created page with '[http://wiki.iac.isu.edu/index.php/PhotoFission_with_Polarized_Photons_from_HRRL Go Back] ==LINAC parameters used in calculations== 1) pulse width 50 ps <br> 2) pulse current 50…')
 
 
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[http://wiki.iac.isu.edu/index.php/PhotoFission_with_Polarized_Photons_from_HRRL Go Back]
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[https://wiki.iac.isu.edu/index.php/Roman_calculation Go Back]
  
 
==LINAC parameters used in calculations==
 
==LINAC parameters used in calculations==
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==Number of photons/sec out of radiator==
 
==Number of photons/sec out of radiator==
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===1/2 mil of Ti===
  
 
#<math>\sigma_{brems}=0.1\ \mbox{photons/electrons/MeV/r.l}</math>
 
#<math>\sigma_{brems}=0.1\ \mbox{photons/electrons/MeV/r.l}</math>
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  <math>\frac{12.5\ \mu m}{3.59\ cm} = 3.48 \cdot 10^{-4}\ r.l.</math><br>
 
  <math>\frac{12.5\ \mu m}{3.59\ cm} = 3.48 \cdot 10^{-4}\ r.l.</math><br>
  
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  <math>0.47 \cdot 10^{13} \frac{e^-}{sec} \times 0.1\ \frac{\gamma 's}{(e^- \cdot MeV \cdot r.l.)} \times 3.48 \cdot 10^{-4}\ r.l. \times 10\ MeV =1.63 \cdot 10^{9} \frac{\gamma}{sec}</math><br><br>
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  <math>0.47 \cdot 10^{13}\ \frac{e^-}{sec} \times 0.1\ \frac{\gamma 's}{(e^- \cdot MeV \cdot r.l.)} \times 3.48 \cdot 10^{-4}\ r.l. \times 10\ MeV =1.63 \cdot 10^{9} \frac{\gamma}{sec}</math><br><br>
  
 
'''Alex factor is 6.85 %'''
 
'''Alex factor is 6.85 %'''
  
 
  <math>1.63 \cdot 10^{9} \frac{\gamma}{sec} \cdot 6.85\ % = 1.12 \cdot 10^{8} \frac{\gamma}{sec}</math><br><br>
 
  <math>1.63 \cdot 10^{9} \frac{\gamma}{sec} \cdot 6.85\ % = 1.12 \cdot 10^{8} \frac{\gamma}{sec}</math><br><br>
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===1/2 mil of Al===
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#<math>\sigma_{brems}=0.1\ \mbox{photons/electrons/MeV/r.l}</math>
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#<math>\mbox{r.l.(Al)} = 8.89\ \mbox{cm}</math>
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#<math>\mbox{radiator}\ \mbox{thickness} = 12.5\ \mu m</math>
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<math>\frac{12.5\ \mu m}{8.89\ cm} = 1.41 \cdot 10^{-4}\ r.l.</math><br>
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<math>0.47 \cdot 10^{13}\ \frac{e^-}{sec} \times 0.1\ \frac{\gamma 's}{(e^- \cdot MeV \cdot r.l.)} \times 1.41 \cdot 10^{-4}\ r.l. \times 10\ MeV =0.66 \cdot 10^{9} \frac{\gamma}{sec}</math><br><br>
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'''Alex factor is 6.85 %'''
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<math>0.66 \cdot 10^{9} \frac{\gamma}{sec} \cdot 6.85\ % = 0.45 \cdot 10^{8} \frac{\gamma}{sec}</math><br><br>
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===Conversion factor from Ti to Al===
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All my following calculation for pair production rate are based on (1/2) mil of Ti radiator. If we want to recalculate for (1/2) mil of Al converter we need to use the conversion factor:
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<math>\frac{0.45\ (1/2\ mil\ of\ Al)}{1.12\ (1/2\ mil\ of\ Ti)} = 0.40 </math>
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===Appendix===
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[[File:bremss44MeV.png | 500 px]]
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[[File:Al 44MeV.png | 500 px]]
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in (10,20) MeV region we have about '''0.1 photons/electrons/MeV/r.l''' both for Ti and Al radiators
  
 
==Pair production rate==
 
==Pair production rate==
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  <math>N_{\mbox{Nitrogen}} = \frac{0.00125\ \frac{g}{cm^3} \times 6.02 \cdot 10^{23}\ \frac{atoms}{mol} \times 1.0\ \mbox{m}} {14.01\ \frac{g}{mol}} = 5.37 \cdot 10^{25}\ \frac{\mbox{atoms}}{m^2}</math>
 
  <math>N_{\mbox{Nitrogen}} = \frac{0.00125\ \frac{g}{cm^3} \times 6.02 \cdot 10^{23}\ \frac{atoms}{mol} \times 1.0\ \mbox{m}} {14.01\ \frac{g}{mol}} = 5.37 \cdot 10^{25}\ \frac{\mbox{atoms}}{m^2}</math>
  
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  <math>\frac {1.12 \cdot 10^{8}\ \frac{\gamma}{sec} \times \sigma_{pairs} \times N_{Al}} {f} = 300\ \frac{\mbox{pairs}}{\mbox{pulse}} </math>
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  <math>\frac {1.12 \cdot 10^{8}\ \frac{\gamma}{sec} \times \sigma_{\mbox{pairs}} \times N_{\mbox{Nitrogen}}} {f} = 300\ \frac{\mbox{pairs}}{\mbox{pulse}} </math>
  
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==Appendix==
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===1 m of air vs. 3.0 um of Al converter===
  
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===pair production cross sections in an Al target===
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<math>\frac{300\ \frac{\mbox{pairs}}{\mbox{pulse}}} {3.38\ \frac{\mbox{pairs}}{\mbox{pulse}}} = 88.8\ \mbox{times!}</math>
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===Appendix===
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====pair production cross sections in an Al target====
  
 
'''Ref.''' ''Geant4 and Theoretical Pair Production Cross Sections for 1 MeV - 100 GeV photons in Aluminum. Vakho Makarashvili, December 18, 2007''
 
'''Ref.''' ''Geant4 and Theoretical Pair Production Cross Sections for 1 MeV - 100 GeV photons in Aluminum. Vakho Makarashvili, December 18, 2007''
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[[File:pair_production_Al.png | 800 px]]
 
[[File:pair_production_Al.png | 800 px]]
  
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===pair production cross sections in a Nitrogen===
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====pair production cross sections in a Nitrogen====
  
 
'''Ref.''' ''Photon Cross Section, Attenuation Coefficients, and Energy Absorption Coefficients From 10 keV to 100 Gev. J.H.Hubbell. Center for Radiation Research.National Bureau of Standards. Washington, D.C. 20234''
 
'''Ref.''' ''Photon Cross Section, Attenuation Coefficients, and Energy Absorption Coefficients From 10 keV to 100 Gev. J.H.Hubbell. Center for Radiation Research.National Bureau of Standards. Washington, D.C. 20234''
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[[File:Sigma nitrogen fitted m2.png | 800 px]]
 
[[File:Sigma nitrogen fitted m2.png | 800 px]]
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[http://wiki.iac.isu.edu/index.php/PhotoFission_with_Polarized_Photons_from_HRRL Go Back]
 
[http://wiki.iac.isu.edu/index.php/PhotoFission_with_Polarized_Photons_from_HRRL Go Back]

Latest revision as of 19:04, 24 May 2012

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LINAC parameters used in calculations

1) pulse width 50 ps
2) pulse current 50 A
3) repetition rate 300 Hz
4) energy 44 MeV

Number of electrons/sec on radiator

[math] 50\ \frac{C}{sec} \times \frac{1\cdot e^-}{1.6\cdot 10^{-19}\ C} \times 50\ \mbox{ps} \times 300\ \mbox{Hz} = 0.47 \cdot 10^{13}\ \frac{e^-}{sec}[/math]

Number of photons/sec out of radiator

1/2 mil of Ti

  1. [math]\sigma_{brems}=0.1\ \mbox{photons/electrons/MeV/r.l}[/math]
  2. [math]\mbox{r.l.(Ti)} = 3.59\ \mbox{cm}[/math]
  3. [math]\mbox{radiator}\ \mbox{thickness} = 12.5\ \mu m[/math]


[math]\frac{12.5\ \mu m}{3.59\ cm} = 3.48 \cdot 10^{-4}\ r.l.[/math]


[math]0.47 \cdot 10^{13}\ \frac{e^-}{sec} \times 0.1\ \frac{\gamma 's}{(e^- \cdot MeV \cdot r.l.)} \times 3.48 \cdot 10^{-4}\ r.l. \times 10\ MeV =1.63 \cdot 10^{9} \frac{\gamma}{sec}[/math]

Alex factor is 6.85 % 

[math]1.63 \cdot 10^{9} \frac{\gamma}{sec} \cdot 6.85\ % = 1.12 \cdot 10^{8} \frac{\gamma}{sec}[/math]

1/2 mil of Al

  1. [math]\sigma_{brems}=0.1\ \mbox{photons/electrons/MeV/r.l}[/math]
  2. [math]\mbox{r.l.(Al)} = 8.89\ \mbox{cm}[/math]
  3. [math]\mbox{radiator}\ \mbox{thickness} = 12.5\ \mu m[/math]


[math]\frac{12.5\ \mu m}{8.89\ cm} = 1.41 \cdot 10^{-4}\ r.l.[/math]


[math]0.47 \cdot 10^{13}\ \frac{e^-}{sec} \times 0.1\ \frac{\gamma 's}{(e^- \cdot MeV \cdot r.l.)} \times 1.41 \cdot 10^{-4}\ r.l. \times 10\ MeV =0.66 \cdot 10^{9} \frac{\gamma}{sec}[/math]

Alex factor is 6.85 % 

[math]0.66 \cdot 10^{9} \frac{\gamma}{sec} \cdot 6.85\ % = 0.45 \cdot 10^{8} \frac{\gamma}{sec}[/math]

Conversion factor from Ti to Al

All my following calculation for pair production rate are based on (1/2) mil of Ti radiator. If we want to recalculate for (1/2) mil of Al converter we need to use the conversion factor: 

[math]\frac{0.45\ (1/2\ mil\ of\ Al)}{1.12\ (1/2\ mil\ of\ Ti)} = 0.40 [/math]

Appendix

Bremss44MeV.png Al 44MeV.png

in (10,20) MeV region we have about 0.1 photons/electrons/MeV/r.l both for Ti and Al radiators

Pair production rate

out of Al converter

[math]\sigma_{pairs} = 0.5\ \frac{\mbox{barns}}{\mbox{atom}}[/math]

[math]l = 3.0\ \mu m[/math] (by varying width we can vary the yield)

[math]N_{Al} = \frac{2.70\ \frac{g}{cm^3} \times 6.02 \cdot 10^{23}\ \frac{atoms}{mol} \times 3.0\ \mu m} {26.98\ \frac{g}{mol}} = 1.81 \cdot 10^{23}\ \frac{\mbox{atoms}}{m^2}[/math]

[math]\frac {1.12 \cdot 10^{8}\ \frac{\gamma}{sec} \times \sigma_{pairs} \times N_{Al}} {f} = 3.38\ \frac{\mbox{pairs}}{\mbox{pulse}} [/math]

through 1 m of air

Assume air consists entirely from Nitrogen: 

[math]\sigma_{pairs}\ (\mbox{Nitrogen}) = 0.15\ \frac{\mbox{barns}}{\mbox{atom}}[/math] 

[math]l = 1.0\ \mbox{m}[/math]

[math]N_{\mbox{Nitrogen}} = \frac{0.00125\ \frac{g}{cm^3} \times 6.02 \cdot 10^{23}\ \frac{atoms}{mol} \times 1.0\ \mbox{m}} {14.01\ \frac{g}{mol}} = 5.37 \cdot 10^{25}\ \frac{\mbox{atoms}}{m^2}[/math]

[math]\frac {1.12 \cdot 10^{8}\ \frac{\gamma}{sec} \times \sigma_{\mbox{pairs}} \times N_{\mbox{Nitrogen}}} {f} = 300\ \frac{\mbox{pairs}}{\mbox{pulse}} [/math]

1 m of air vs. 3.0 um of Al converter

[math]\frac{300\ \frac{\mbox{pairs}}{\mbox{pulse}}} {3.38\ \frac{\mbox{pairs}}{\mbox{pulse}}} = 88.8\ \mbox{times!}[/math]

Appendix

pair production cross sections in an Al target

Ref. Geant4 and Theoretical Pair Production Cross Sections for 1 MeV - 100 GeV photons in Aluminum. Vakho Makarashvili, December 18, 2007


Pair production Al.png

pair production cross sections in a Nitrogen

Ref. Photon Cross Section, Attenuation Coefficients, and Energy Absorption Coefficients From 10 keV to 100 Gev. J.H.Hubbell. Center for Radiation Research.National Bureau of Standards. Washington, D.C. 20234

Pair prodiction nitrogen.png


Here I just plotted the table above for pair production cross section in (0, 40) MeV energy region


Sigma nitrogen fitted m2.png



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