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

Let the amount of energy relased during the electron capure process be `Q_(1)` the nuclear reaction can be writen as :
`e^(+)+""_(Z)^(A)Xto ""_(Z-1)^(A)Y +v+Q_(1) …….(i) `
Let th eemount of energy released during the positron capture process be `Q_(2)` the nuclear reaction can be written as:
`""_(Z)^(A)X to ""_(Z-1)^(A)Y +e^(+)+v+Q_(2)`
`m_(N)(""_(Z)^(A)X)=` Nucleae mass of `""_(Z)^(A)X`
`m_(N)(""_(Z-1)^(A)Y)=` Nuclear mass of `""_(Z-1)^(A)Y`
`m(""_(z)^(A)X)-`atomic mass of `""_(Z)^(A)X`
` m_(e) = ` mass ofan electron
C= Speed of light
Q- value of the electrons capture reaction is given as:
`Q_(1)[m_(N)(""_(Z)^(A)X)+m_(e)-m_(N)(""_(Z-1)^(A)Y)]c^(2)`
`=[m(""_(Z)^(A)X)-Zm_(e)+m_(e)-m(""_(Z-1)^(A)Y)+(Z-1)m(e)]c^(2)`
`=[m(""_(z)^(A)X)-m_(N)(""_(Z-1)^(A)Y)]c^(2)`
Q- value of the positrion capture reaction if givren as:
`Q_(2)=[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) ...(4)`
it can be inferrred that `Q_(2)gt 0 then Q_(1)gt 0` also if `Q_(1)gt0,`it does necessearlly mean that `Q_(2)gt0.`