Why Can't a Lower Energy Photon Remain After Pair Production?

[TheCanadian]

In pair production, if the photon has an energy greater than 1.02 MeV, why can't a lower energy photon remain after creation of the electron-positron pair? For example, if you have a 10 MeV photon interacting with a carbon nucleus, why are the stated products of pair production the carbon nucleus, positron, and electron? Why can't a final photon with energy between $0 < E_\gamma < 8.98$ MeV exist afterwards from this interaction with the carbon nucleus?

 

[Orodruin]

In principle, there is nothing forbidding this. However, the rate of this occurring is going to be suppressed by the fine structure constant and some phase space factor.

 

[TheCanadian]

In principle, there is nothing forbidding this. However, the rate of this occurring is going to be suppressed by the fine structure constant and some phase space factor.

Do you mind providing a resource/explanation discussing this calculation and why it's unlikely by a factor of the fine structure constant/phase space difference?

 

[Orodruin]

It should be pretty obvious at A level. The Feynman diagram needs an extra electromagnetic vertex and it is a 3-body final state.