for the `beta^(+)` (positron) emission from a nucleus, there is another competing process known as electron capture. Electron from an inner orbit (say K shell) is captured by the nucleus and neutrino is emitted. Show that if `beta^(+)` emission is energetically allowed, electron capture is necessarily allowed but not vice -versa.

Explanation

Consider the two competing processes:
(i) Case of positron emission.
`""_(Z)X^(A) to ""_(z-1)Y^(A) +e^(+) + v +Q_(1)` (positron emission)
Energy released in this case is given by
`Q_(1)=(m_(N)(""_(z-1)Y^(A)) -m_(e))c^(2)`
`[m(""_(z)X^(A))-m(""_(z-1)Y^(A))-2m_(e)]c^(2)`........(i)
(ii) Case of electron capture
`e^(-) +""_(z)X^(A) to ""_(z-1)Y^(A) +v+Q_(2)` (electron capture)
`Q_(2)=[m(""_(Z)X^(A))-m(""_(z-1)Y^(A))]c^(2)`.............(ii)
If `Q_(1)` is greater than zero, then `Q_(2)` will necessarily be greater than zero. It means that if positron emission is allowed, then electron capture will necessarily be allowed.
If `Q_(2)` is greater than zero, then `Q_(1)` , may not be greater than zero. It means that if electron capture is allowed, positron emission may not be allowed.