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).

\[\ce{e+ + ^A_Z X -> ^A_{Z - 1}Y + \text{v}}\]

Show that if `beta^+` emission is energetically allowed, electron capture is necessarily allowed but not vice−versa.

#### Solution

Let the amount of energy released during the electron capture process be Q_{1}. The nuclear reaction can be written as:

\[\ce{e+ + ^A_Z X -> ^A_{Z - 1}Y + \text{v} + Q_1}\] .....(1)

Let the amount of energy released during the positron capture process be Q_{2}. The nuclear reaction can be written as:

\[\ce{e+ + ^A_Z X -> ^A_{Z - 1}Y + \text{v} + Q_1}\] ...(2)

`"m"_"N"(""_"Z"^"A" "X")` = Nuclear 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"(""_("Z" - 1)^"A" "X")` = Atomic mass of `""_("Z" -1)^"A" "X"`

m_{e} = Mass of an electron

c = Speed of light

Q-value of the electron 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"(""_("z" - 1)^"A" "Y")]"c"^2` ....(3)

Q-value of the positron capture reaction is given as:

`"Q"_2 = ["m"_"N" (""_"Z"^"A" "X") - "m"_"N"(""_("z"-1)^"A" "Y") - "m"_"e"]"c"^2`

`= ["m"_"N"(""_"Z"^"A" "X") - "m"_"N" (""_("z"-1)^"A" "Y") + ("Z" - 1)"m"_"e" - "m"_"e"]"c"^2`

`= ["m"(""_"Z"^"A" "X") - "m"(""_("z" - 1)^"A" "Y") - 2"m"_"e"]"c"^2` ...(4)

It can be inferred that if Q_{2} > 0, then Q_{1 }> 0; Also, if Q_{1}> 0, it does not necessarily mean that Q_{2} > 0.

In other words, this means that if `beta^+` emission is energetically allowed, then the electron capture process is necessarily allowed, but not vice-versa. This is because the Q-value must be positive for an energetically allowed nuclear reaction.