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Can gamma particles with sufficient energy for pair production gain energy by recombining their positron with a more energetic beta particle than the one they produced? If yes, is this something that would happen to any significant degree in a sample of radioactive material decaying with high energy gammas and betas eg. Na24?

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Pair production with a gamma ray

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Feynman diagram of electron–positron pair production. One must calculate multiple diagrams to get the net cross section

needs a field to interact with, in the diagram the charged field of a nucleus, for momentum conservation in the center of mass of the final electron-positron pair. A gamma has mass zero, the pair at least two electron masses, so a single gamma cannot decay to a pair.

Once the pair is produced the original gamma disappears. The positron of the interaction will have an energy momentum vector certainly smaller than the incoming gamma. If the positron interacts with an electron two fresh gammas will be produced (at the low nuclear energy levels your question assumes). If the electron has high energy, it will be distributed according to the probability of the interaction, on the two new gammas.

At the level of nuclear physics it is not possible energetically to get one of the two outgoing gammas to have higher energy than the original incoming one.

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