Neutron Polarimeter

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Relativistic kinematic

energy dependence of outcoming neutron versus energy of incoming photons

[math]\displaystyle{  E = T + m }[/math]
[math]\displaystyle{  E = p^2 + m^2 }[/math]

writing four-vectors:

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

Doing four-vector algebra:

[math]\displaystyle{  p^{\mu}_{\gamma} + p^{\mu}_D = p^{\mu}_p + p^{\mu}_n \Rightarrow  }[/math]

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

 (p^{\mu}_{\gamma})^2 + (p^{\mu}_D)^2 + (p^{\mu}_n)^2 + 2p^{\mu}_{\gamma}p^{\mu}_D - 2p^{\mu}_n(p^{\mu}_{\gamma} + p^{\mu}_D)

 \Rightarrow

 m_p^2 = m_y^2(=0) + m_D^2 + m_n^2  


 }[/math]

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