We want to derive the how the energy of detected neutron depends on energy of incoming photons
[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_{\gamma} = \left( T_{\gamma},\ T_{\gamma},\ 0,\ 0 \right) [/math]
[math] p^{\mu}_{\gamma} + p^{\mu}_{D} = p^{\mu}_{p} + p^{\mu}_{n} [/math] [math] p^{\mu}_{\gamma} + p^{\mu}_{D} = p^{\mu}_{p} + p^{\mu}_{n} [/math]