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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).
\[\ce{e+ + ^A_Z X -> ^A_{Z - 1}Y + \text{v}}\]
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 Q1. 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 Q2. 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"`
me = 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 Q2 > 0, then Q1 > 0; Also, if Q1> 0, it does not necessarily mean that Q2 > 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.
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