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).
`e^+ + ""_Z^AX to ""_(Z - 1)^(A)Y + upsilon`
Show that if `beta^+` emission is energetically allowed, electron capture is necesarily allowed but not vice - versa.

Explanation

Consider the competing processes :
`""(Z)^(A)rarr""(Z-1)^(A)Y+e^(+)+v_(e)+Q_(1)" (poitron emission)"`
`e^(-)+""(Z)^(A)Xrarr""(Z-1)^(A)Y+v_(e)+Q_(2)" (electron capture)"`
`Q_(1)=[m_(N)(""(Z)^(A)X)-m(N)(""(Z-1)^(A)Y)-m(e)]c^(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)`
`Q_(2)=[m_(N)(""(Z)^(A)X)+m(e)-m_(N)(""(Z-1)^(A)Y)]c^(2)=[m(""(Z)^(A)X)-m(""(Z-1)^(A)Y)]^(2)`
This means `Q(1) gt 0` implies `Q_(2) gt0` but `Q_(2) gt0` does not necessarily mean `Q_(1)gt0`. Hence the result.