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  • 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).

Q. 13.22  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).

                   e^{+}+_{z}^{A}\textrm{X}\rightarrow _{Z-1}^{A}\textrm{Y}+v

Show that if \beta ^{+}emission is energetically allowed, electron capture is necessarily allowed but not vice–versa.

Answers (1)

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For the electron capture, the reaction would be

_{Z}^{A}\textrm{X}+e^{-}\rightarrow _{Z-1}^{A}\textrm{Y}+\nu +Q_{1}

The mass defect and q value of the above reaction would be

\\\Delta m_{1}=m(_{Z}^{A}\textrm{X})+m_{e}-m(_{Z-1}^{A}\textrm{Y})\\ Q_{1}=([m(_{Z}^{A}\textrm{X})-m(_{Z-1}^{A}\textrm{Y})]+m_{e})c^{2}

where mN(_{Z}^{A}\textrm{X}) and mN(_{Z-1}^{A}\textrm{Y}) are the nuclear masses of elements X and Y respectively

For positron emission, the reaction would be

_{Z}^{A}\textrm{X}\rightarrow _{Z-1}^{A}\textrm{Y}+e^{+}+\bar{\nu }+Q_{2}

The mass defect and q value for the above reaction would be

\\\Delta m_{2}=m(_{Z}^{A}\textrm{X})-m(_{Z-1}^{A}\textrm{Y})-m_{e}\\ Q_{2}=([m(_{Z}^{A}\textrm{X})-m(_{Z-1}^{A}\textrm{Y})]-m_{e})c^{2}

From the above values, we can see that if Q2 is positive Q1 will also be positive but Q1 being positive does not imply that Q2 will also have to positive.

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Sayak

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