Difference between revisions of "Neutron Polarimeter"

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Doing four-vector algebra:
 
Doing four-vector algebra:
  
  <math> p^{\mu}_{\gamma} + p^{\mu}_D = p^{\mu}_p + p^{\mu}_n \Rightarrow </math><br>
+
  <math> p^{\mu}_{\gamma} + p^{\mu}_D = p^{\mu}_p + p^{\mu}_n \Rightarrow </math>
  
 
  <math> p^{\mu\ 2}_p = \left(p^{\mu}_{\gamma} + p^{\mu}_D - p^{\mu}_n\right)^2 =  
 
  <math> p^{\mu\ 2}_p = \left(p^{\mu}_{\gamma} + p^{\mu}_D - p^{\mu}_n\right)^2 =  
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  p^{\mu\ 2}_{\gamma} + p^{\mu\ 2}_D + p^{\mu\ 2}_n + 2\ p^{\mu}_{\gamma}\ p^{\mu}_D - 2\ p^{\mu}_n\left(p^{\mu}_{\gamma} + p^{\mu}_D\right) </math>
 
  p^{\mu\ 2}_{\gamma} + p^{\mu\ 2}_D + p^{\mu\ 2}_n + 2\ p^{\mu}_{\gamma}\ p^{\mu}_D - 2\ p^{\mu}_n\left(p^{\mu}_{\gamma} + p^{\mu}_D\right) </math>
  
  <math> m_p^2 = m_y^2(=0) + m_D^2 + m_n^2 = 2\ T_{\gamma}\ m_D - 2\left( T_{\gamma}\ E_n - T_{\gamma}\ p_n(\cos(\Theta_n)\right) - 2\ m_D\ E_n </math>  
+
  <math> m_p^2 = m_{\gamma}^2(=0) + m_D^2 + m_n^2  
 +
 
 +
            = 2\ T_{\gamma}\ m_D - 2\left( T_{\gamma}\ E_n - T_{\gamma}\ p_n(\cos(\Theta_n)\right) - 2\ m_D\ E_n </math>
 +
 
 +
<math>  2/ T_{\gamma}\left( m_D - E_n + p_n(\cos(\Theta_n) \right) - 2\ m_D\ E_n </math>
  
  
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</math>
 
  
  

Revision as of 21:41, 5 June 2010

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Relativistic kinematic

energy dependence of outcoming neutron versus energy of incoming photons

Collision.png
[math] E = T + m[/math]
[math] E = p^2 + m^2[/math]

writing four-vectors:

[math] p_{\gamma} = \left( T_{\gamma},\ T_{\gamma},\ 0,\ 0  \right) [/math] 
[math] p_D     = \left( m_D,\ 0,\ 0,\ 0  \right) [/math] 
[math] p_{n} = \left( E_n,\ p_n\cos(\Theta_n),\ p_n\sin(\Theta_n),\ 0  \right) [/math] 
[math] p_{p} = \left( E_p,\ p_p\cos(\Theta_p),\ p_p\sin(\Theta_p),\ 0  \right) [/math] 


Doing four-vector algebra:

[math] p^{\mu}_{\gamma} + p^{\mu}_D = p^{\mu}_p + p^{\mu}_n \Rightarrow [/math]
[math] p^{\mu\ 2}_p = \left(p^{\mu}_{\gamma} + p^{\mu}_D - p^{\mu}_n\right)^2 = 

 p^{\mu\ 2}_{\gamma} + p^{\mu\ 2}_D + p^{\mu\ 2}_n + 2\ p^{\mu}_{\gamma}\ p^{\mu}_D - 2\ p^{\mu}_n\left(p^{\mu}_{\gamma} + p^{\mu}_D\right) [/math]
[math] m_p^2 = m_{\gamma}^2(=0) + m_D^2 + m_n^2 

             = 2\ T_{\gamma}\ m_D - 2\left( T_{\gamma}\ E_n - T_{\gamma}\ p_n(\cos(\Theta_n)\right) - 2\ m_D\ E_n [/math] 
[math]  2/ T_{\gamma}\left( m_D - E_n + p_n(\cos(\Theta_n) \right) - 2\ m_D\ E_n [/math]






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