For the $\beta^{+}$ (positron) emission from a nucleus,there is another competing process known as electron capture (an electron from an inner orbit,say,the $K$-shell,is captured by the nucleus and a neutrino is emitted).
$_{z}^{A} X + e^{-} \rightarrow _{z-1}^{A} Y + \nu$
Show that if $\beta^{+}$ emission is energetically allowed,electron capture is necessarily allowed,but not vice-versa.

(N/A) Let the energy released during the electron capture process be $Q_{1}$. The nuclear reaction is:
$_{z}^{A} X + e^{-} \rightarrow _{z-1}^{A} Y + \nu + Q_{1}$
Using nuclear masses,$Q_{1} = [m_{N}(_{z}^{A} X) + m_{e} - m_{N}(_{z-1}^{A} Y)] c^{2}$.
In terms of atomic masses $m(_{z}^{A} X) = m_{N}(_{z}^{A} X) + Zm_{e}$ and $m(_{z-1}^{A} Y) = m_{N}(_{z-1}^{A} Y) + (Z-1)m_{e}$,we get:
$Q_{1} = [m(_{z}^{A} X) - Zm_{e} + m_{e} - (m(_{z-1}^{A} Y) - (Z-1)m_{e})] c^{2} = [m(_{z}^{A} X) - m(_{z-1}^{A} Y)] c^{2}$.
Let the energy released during the $\beta^{+}$ emission be $Q_{2}$. The reaction is:
$_{z}^{A} X \rightarrow _{z-1}^{A} Y + e^{+} + \nu + Q_{2}$
$Q_{2} = [m_{N}(_{z}^{A} X) - m_{N}(_{z-1}^{A} Y) - m_{e}] c^{2}$.
Substituting atomic masses:
$Q_{2} = [m(_{z}^{A} X) - Zm_{e} - (m(_{z-1}^{A} Y) - (Z-1)m_{e}) - m_{e}] c^{2} = [m(_{z}^{A} X) - m(_{z-1}^{A} Y) - 2m_{e}] c^{2}$.
Comparing the two:
$Q_{1} = Q_{2} + 2m_{e}c^{2}$.
Since $2m_{e}c^{2} > 0$,it follows that if $Q_{2} > 0$ ($\beta^{+}$ emission allowed),then $Q_{1} > 0$ (electron capture allowed). However,if $Q_{1} > 0$,$Q_{2}$ may be negative,meaning electron capture can occur even when $\beta^{+}$ emission is not energetically allowed.