[Physics] When does electron capture occur and when does positron emission occur

Question

I’ve been told that electron capture occurs when there isn’t enough energy to produce a positron by beta plus decay. Exactly why is this the case? Why does it take more energy for positron emission than electron capture?

Answer 1

Electron capture: $p+e^-\rightarrow n+\nu_e$
Beta plus decay: $p\rightarrow n+e^++\bar{\nu_e}$

Let's check the masses of both sides of the processes:
Electron capture, initial state: $m_p+m_e=938.78 \frac{MeV}{c^2}$
Final state: $m_n=939.56 \frac{MeV}{c^2}$
The difference: (Final minus initial) $0.78 \frac{MeV}{c^2}$

Beta plus decay: (Positron emission)
Intial state: $m_p=938.27 \frac{MeV}{c^2}$
Final state: $m_n+m_e=940.07 \frac{MeV}{c^2}$
The difference: (Final minus initial) $1.8 \frac{MeV}{c^2}$

You see that you need to add more energy in order to do beta plus decay than to capture an electron.

Also note: A free proton can't emit a positron and turn into a neutron, but a free proton can capture an electron.