Energy transferred to nucleus in pair-production

[a1234]

Homework Statement

Gamma rays of 22 MeV interact with a sodium sample, resulting in a pair-production reaction, in which an electron and positron are created in the vicinity of a nucleus:
(Photon + nucleus = e- + e+ + nucleus)
What would be the maximum transfer of energy to the sodium atoms in this reaction?

Relevant Equations

E_gamma = 2m_e*c^2 + KE- + KE+ + K_nucleus

In most textbooks, the recoil energy of the nucleus is ignored as it absorbs so little energy, and since its main role in the reaction is to absorb some of the photon's momentum without absorbing much energy.
I'm tempted to say that the nucleus gets the maximum energy when the kinetic energy of the electron and positron are zero, but I don't think we'd ever see an atom with a kinetic energy of 20.978 MeV. Wouldn't some of that energy go towards nuclear excitation?

 

[PeroK]

Are you sure you are not over-thinking this?

 

[a1234]

I feel like I am. I've been staring at the energy and momentum conservation equations to try to find something more "sophisticated," but that doesn't seem to work out since we don't have enough information (e.g. the scattering angle or velocities of the electron-positron pair).

 

[PeroK]

I feel like I am. I've been staring at the energy and momentum conservation equations to try to find something more "sophisticated," but that doesn't seem to work out since we don't have enough information (e.g. the scattering angle or velocities of the electron-positron pair).

You're trying to find the maximum (kinetic) energy of the sodium atom. You've already established the criteria for that.