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).
`e^(+) ._(Z)^(A)X to ._(Z-1)^(A)Y+v`
Show that if `beta^(+)` emission is energetically allowed, electron capture is necessarily allowed but not vice-versa.

Answer 1

The `beta^(+)` emission from a nucleus `._(Z)X^(A)` may be represented as
`._(Z)X^(A)= ._(Z-1)Y^(A)+ ._(1)e^(0)+ v + Q_(1)` ______ (i)
The other competing process of electron capture may be represented as
`._(-1)e^(0)+ ._(Z)X^(A)= ._(Z-1)Y^(A)+ v + Q_(2)` _______ (ii)
The energy released `Q_(1)` in 1. is given by
`Q_(1)=[m_(N)(._(Z)X^(A))-m_(N)(._(Z-1)Y^(A))-m_(e)]c^(2)`
`=[m_(N)(._(Z)X^(A))+Zm_(e)-m_(N)(._(Z-1)Y^(A))-(Z-1)m_(e)-2m_(e)]c^(2)`
`Q_(1)=[m(._(Z)X^(A))-m(._(Z-1)Y^(A))-2m_(e)]c^(2)` _______ (iii)
Note that `m_(N)` here denotes mass of nucleus and m denotes the mass of atom similarly from (ii)
`Q_(2)=[m_(N)(._(Z)X^(A))+m_(e)-m_(N)(._(Z-1)Y^(A))-m_(e)]c^(2)`
`= [m_(N)(._(Z)X^(A))+Zm_(e)-m_(N)(._(Z-1)Y^(A))-(Z-1)m_(e)-m_(e)]c^(2)`
`Q_(2)=[m(._(X)Z^(A))+m-(._(Z-1)Y^(A))]c^(2)`
`= [m_(N)(._(Z)X^(A))+Zm_(e)+m_(e)-m_(N) (._(Z-1)Y^(A))-(Z-1)m_(e)-m_(e)]c^(2)`
`Q_(2)=[m(._(Z)X^(A))-m-(._(Z-1)Y^(A))]c^(2)`
Ir `Q_(1)gt 0` then `Q_(2) gt 0`.
i.e., If positron emission is energetically allowed electron capture is necessarily allowed. But `Q_(2)gt 0` does not necessarily mean `Q_(1)gt 0`. Hence the reverse is not true.