for the `beta^(+)` (positron) emission form a nucleus, there is another competing process known as electron capture. Electron form 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

The `beta^(+)` emission form a nucleus `._ZX^A` may be represented as
`._ZX^A = ._(Z-1)Y^A+._1e^0+v+Q_1 .....(i)`
The other competing process of electron capture may be represented as
`._(-1)e^0+._ZX^A= ._(Z-1)Y^(A)+v+Q_2 .......(i)`
The energy released `Q_1` in (i) is given by
`Q_1=[m_N(._ZX^A)-m_N(._(Z-1)Y^A)-m_e]c^2=[m_N(._ZX^A)+Zm_e-m_N(._(Z-1)Y^A)-(Z-1)m_e-2m_e]c^2`
`Q_1=[m_N(._ZX^A)-m_N(._(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, form (ii)
`Q_2=[m_N(._ZX^A)+m_e-m_N(._(Z-1)Y^A)]c^2`
`=[m_N(._ZX^A)+Zm_e+m_e-m_N(._(Z-1)Y^A)-(Z-1)m_e-m_e]c^2`
`Q_2=[m(._ZX^A)-m(._(Z-1)Y^(A))]c^2.......(iv)`
If `Q_1gt0`, then `Q_2gt0`
i.e., if positron emission is energetically allowed, electron capture is necessarily allowed. But `Q_2gt0` does not necessarily mean `Q_1gt0`. Hence the reverse is not true.