Question
For the $\beta^{+} ($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$).$
$\text{e}^{+}+^{\text{A}}_{\text{Z}}\text{X}\rightarrow\ ^{\text{A}}_{\text{Z}-1}\text{Y}+\text{v}$
Show that if $\beta^{+}$ emission is energetically allowed, electron capture is necessarily allowed but not vice $–$ versa.
$\text{e}^{+}+^{\text{A}}_{\text{Z}}\text{X}\rightarrow\ ^{\text{A}}_{\text{Z}-1}\text{Y}+\text{v}$
Show that if $\beta^{+}$ emission is energetically allowed, electron capture is necessarily allowed but not vice $–$ versa.
