4-momentum conservation (Deuteron + Photon)

1. The problem statement, all variables and given/known data

See attachment.





2. Relevant equations







3. The attempt at a solution



So,



[tex]\gamma + D \rightarrow p + n[/tex]



[tex](E_1,\underline{k}_1) + (E_2,\underline{0}) = 2(E_3,\underline{k}_3)[/tex]



where I have assumed E₃ ≈ E₄ and k₃ ≈ k₄ as m_n≈ m_p and v_p = v_n



then splitting E and K components up and solving for photon energy E₁ I end up with,



[tex]E_1 = \frac{4m_p^2}{2E_2}-\frac{E_2}{2}[/tex]



this is where I start to get confused, so the binding energy of the Deuteron is given in the question, would I need to calculate the rest energy of the Deuteron E₂ as,



[tex]E_2 = [m_p + m_n + m_e]c^2 - B(^2_1D_1)[/tex] (c=1)



if I do this and then sub values in I get E₂ = 0.4236MeV, ok so I realize the question is asking to show that the minimum energy of the photon needed is when v_p=v_n, the way I understand this is that the minimum energy required to break the deuteron apart would be the deuterons binding energy. Therefore my value of E₂ should be equal or at least close to 2.2MeV but it's not so I have obviously interpreted something badly, can anyone help?

Attached Images

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