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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)
$${ e }^{ + }+_{ Z }^{ A }{ X }\rightarrow _{ Z-1 }^{ A }{ Y }+v\quad $$
Show that if $${\beta}^{+}$$ emission is energetically allowed, electron capture is necessarily allowed but not vice-versa

Solution
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For the nucleus $$_{ Z }^{ A }{ X }$$ the $${\beta}^{+}$$ emission may be represented as
$$_{ Z }^{ A }{ X }\rightarrow _{ Z-1 }^{ A }{ Y }+_{ +1 }^{ 0 }{ e }+v+{ E }_{ 1 }$$
where $${E}_{1}$$ is the energy released
$$\therefore { E }_{ 1 }=\left[ { m }_{ N }\left( _{ Z }^{ A }{ X } \right) -{ m }_{ N }\left( _{ Z-1 }^{ A }{ Y } \right) -{ m }_{ e } \right] \times 931.5MeV\quad $$
$${ m }_{ N }\left( _{ Z }^{ A }{ X } \right) ={ m }_{ N }\left( _{ Z }^{ A }{ X } \right) -z{ m }_{ e }$$
$${ m }_{ N }\left( _{ Z-1 }^{ A }{ Y } \right) ={ m }_{ N }\left( _{ Z-1 }^{ A }{ Y } \right) -(z-1){ m }_{ e }$$
$$\therefore { E }_{ 1 }=\left[ \left\{ { m }_{ N }\left( _{ Z-1 }^{ A }{ Y } \right) -{ m }_{ N }\left( _{ Z-1 }^{ A }{ Y } \right) \right\} -2{ m }_{ e } \right] \times 931.5MeV....(1)\quad $$
For the nucleus $$_{ Z }^{ A }{ X }$$s, the electron capture is represented as
$$\quad _{ -1 }^{ 0 }{ e }+_{ Z }^{ A }{ X }\rightarrow _{ Z-1 }^{ A }{ Y }+v+{ E }_{ 2 }$$ where $${E}_{2}$$ is the energy released
$${ E }_{ 2 }=\left[ { m }_{ e }+{ m }_{ N }\left( _{ Z }^{ A }{ X } \right) -{ m }_{ N }\left( _{ Z-1 }^{ A }{ Y } \right) \right] \times 931.5MeV\quad $$
$${ m }_{ N }\left( _{ Z }^{ A }{ X } \right) ={ m }_{ N }\left( _{ Z }^{ A }{ X } \right) =Z{ m }_{ e }$$
$${ m }_{ N }\left( _{ Z-1 }^{ A }{ Y } \right) ={ m }_{ N }\left( _{ Z-1 }^{ A }{ Y } \right) -(Z-1){ m }_{ e }$$
$${ E }_{ 2 }=\left[ { m }_{ e }+{ m }_{ N }\left( _{ Z }^{ A }{ X } \right) ={ m }_{ N }\left( _{ Z }^{ A }{ X } \right) -Z{ m }_{ e }\left( _{ Z-1 }^{ A }{ Y } \right) \right] \times 931.5MeV$$
$${ E }_{ 2 }=\left[ m\left( _{ Z }^{ A }{ X } \right) -\left( _{ Z-1 }^{ A }{ Y } \right) \right] \times 931.5MeV....(2)$$
From equations (1) and (2) it follows that
if $${E}_{1}> 0$$, then $${E}_{2}> 0$$
ie., if $${\beta}^{+}$$ emission is energetically allowed, then the electron capture is necessarily allowed. On the other hand, if $${E}_{2}> 0$$, then it does not necessarily mean that $${E}_{2}> 0$$. Thus, if the electron capture is allowed, then $${\beta}^{+}$$ emission is not necessarily allowed.

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