Ncert Exercises & Solutions

13.22 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).

\(e^{+}+{ }_{Z}^{A} X \rightarrow {}_{z-1}^{A} Y+\nu\)

Show that if β⁺ emission is energetically allowed, electron capture is necessarily allowed but not vice–versa.

Hint: Q-value must be positive for an energetically-allowed nuclear reaction
Step 1: Find the nuclear reactions for β⁺ emission and electron capture.
Let the amount of energy released during the electron capture process be

Q_{1}

. The nuclear reaction can be written as:
\(e^{+}+{ }_{Z}^{A} X \rightarrow{ }_{Z-1}^{A} Y+\nu+Q_{1}\)

Let the amount of energy released during the positron capture process be

Q_{2}

. The nuclear reaction can be written as:
\({ }_{Z}^{A} X \rightarrow{ }_{Z-1}^{A} Y+e^{+}+\nu+Q_{2}\)

where,

m_{N} ({}_{Z}^{A}X)

=Nuclear mass of

{}_{Z}^{A}X

m ({}_{Z}^{A}X)

=Atomic mass of

{}_{Z}^{A}X

m_{N} ({}_{Z - 1}^{A}Y)

=Nuclear mass of

{}_{Z - 1}^{A}Y

m ({}_{Z - 1}^{A}Y)

=Atomic mass of

{}_{Z - 1}^{A}Y

m_{e}

=Mass of an electron

c=Speed of light
Step 2: Find the Q-values for the two reactions.

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) - Z m_{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)] c^{2}

= [m ({}_{Z}^{A}X) - Z m_{e} - m ({}_{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)

Step 3: Compare the Q-values for the two reactions.
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

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.

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