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For the β+(positron) emission from a nucleus, there is anothercompeting process known as electron capture (electron from an innerorbit, say, the K–shell, is captured by the nucleus and a neutrino isemitted).1A A e X Y Z Z ν++ → + −Show that if β+emission is energetically allowed, electron captureis necessarily allowed but not vice–versa.

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Solution

Given: The equation of electron capture process is e + + X Z A Y Z1 A +ν

The nuclear reaction can be written as,

e + + X Z A Y Z1 A +ν+ Q 1

where, the amount of energy released during the electron capture process is Q 1 .

The nuclear reaction during the positron capture process can be written as,

X Z A Y Z1 A +ν+ e + + Q 2

where, the amount of energy released during the positron capture process is Q 2 .

The Q value of the electron capture reaction can be written as,

Q 1 =[ m N ( X Z A )+ m e m N ( Y Z1 A ) ] c 2 =[ m A ( X Z A )Z m e + m e m A ( Y Z1 A )+( Z1 ) m e ] c 2 =[ m A ( X Z A ) m A ( Y Z1 A ) ] c 2 (1)

where, the nuclear mass of X Z A is m N ( X Z A ), the nuclear mass of Y Z1 A is m N ( Y Z1 A ), the atomic mass of X Z A is m A ( X Z A ), the atomic mass of Y Z1 A is m A ( Y Z1 A ), the speed of light is c, and mass of an electron is m e

The Q value of the positron capture reaction can be written as,

Q 2 =[ m N ( X Z A ) m N ( Y Z1 A ) m e ] c 2 =[ m A ( X Z A )Z m e m A ( Y Z1 A )+( Z1 ) m e m e ] c 2 =[ m A ( X Z A ) m A ( Y Z1 A )2 m e ] c 2 (2)

From equation (1) and (2),

Q 2 >0 , then Q 1 >0

Thus, it is shown that, if Q 2 >0 then Q 1 >0 and, if Q 1 >0 then it is not necessary that Q 2 >0.


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