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)X to ""_(Z - 1)^(A)Y + e^+ + upsilon_e + Q_1 ` (positron capture)
`e^- + ""_(Z)^(A)X to ""_(Z- 1)^(A)Y + upsilon_(e ) + Q_2` (electron capture)
`Q_1 = [m_N(""_Z^AX) - m_N(""_(Z - 1)^(A)Y) - m_e]c^2`
`= [m_N(""_Z^AX) - 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^AX) + m_e - m_N(""_(Z-1)^(A)Y)]c^2 = [m(""_Z^AX) - m(""_(Z-1)^(A)Y]c^2`
This means `Q_1 > 0` implies `Q_2 > 0` but `Q_2 > 0` does not necessarily mean `Q_1 > 0`. Hence the result.