Question

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.

Answer 1

Given: The equation of electron capture process is e + + X Z A → Y Z−1 A +ν

The nuclear reaction can be written as,

e + + X Z A → Y Z−1 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 Z−1 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 Z−1 A ) ] c 2 =[ m A ( X Z A )−Z m e + m e − m A ( Y Z−1 A )+( Z−1 ) m e ] c 2 =[ m A ( X Z A )− m A ( Y Z−1 A ) ] c 2 (1)

where, the nuclear mass of X Z A is m N ( X Z A ), the nuclear mass of Y Z−1 A is m N ( Y Z−1 A ), the atomic mass of X Z A is m A ( X Z A ), the atomic mass of Y Z−1 A is m A ( Y Z−1 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 Z−1 A )− m e ] c 2 =[ m A ( X Z A )−Z m e − m A ( Y Z−1 A )+( Z−1 ) m e − m e ] c 2 =[ m A ( X Z A )− m A ( Y Z−1 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.