NCERT Solutions
Class 12 Physics
Nuclei
Q.22
For the β+ (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+ + AZX à AZ-1Y + ν
Show that if β+ emission is energetically allowed, electron capture is necessarily allowed but not vice–versa.
Given,
Let the amount of energy released during the electron capture process be Q1. The nuclear reaction can be written as:
e+ + AZX à A(Z-1) Y + ν + Q1 (equation(1))
Let the amount of energy released during the positron capture process be Q2. The nuclear reaction can be written as:
AZX à A (Z-1) Y + e+ + ν + Q2 (equation (2))
mN (AZX) =Nuclear mass of (AZX)
mN (A (Z-1) Y) = Nuclear mass of (A (Z-1) Y)
m(AZX) = Atomic mass of (AZX)
m (A (Z-1) Y) = Atomic mass of (A (Z-1) Y)
me = Mass of an electron
c = Speed of light
Q-value of the electron capture reaction is given as:
Q1= [mN (AZX) + me -mN (A (Z-1) Y)] c2
= [m (AZX) -Zme + me –m (A (Z-1) Y) + (Z-1) me] c2
= [m (AZX) -m A (Z-1) Y] c2 (equation (3))
Q-value of the positron capture reaction is given as:
Q2= mN (AZX) -mN (A (Z-1) Y) - me] c2
= [m (AZX) -Zme – m (A (Z-1) Y) + (Z-1) me - me] c2
= [m (AZX) -m A (Z-1) Y – 2me] c2 (equation (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 β+ 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.