Question

For the ​β⁺ (positron) emission from a nucleus, there is another competing process known as electron capture (electron from an inner orbit, say, the K−shell, is captured by the nucleus and a neutrino is emitted).

Show that if ​β⁺ emission is energetically allowed, electron capture is necessarily allowed but not vice−versa.

Answer 1

Let the amount of energy released during the electron capture process be Q_1. The nuclear reaction can be written as:

Let the amount of energy released during the positron capture process be Q₂. The nuclear reaction can be written as:

In other words, this means that if ​β⁺ emission is energetically allowed, then the electron capture process is necessarily allowed, but not vice versa. This is because the Q-value must be positive for an energetically-allowed nuclear reaction.