Class 12

Biology

Cell: Structure and Functions

Nuclei

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).

$e_{+}+_{Z}X→_{Z−1}Y+v$

Show that if $β_{+}$ emission is energetically allowed, electron capture is necessarily allowed but not vice versa.

$_{Z}W→_{Z−1}Y+e_{+}+v_{e}+Q_{1}$ (positron cature)

$e_{−}+_{Z}W→_{Z−1}Y+v_{e}+Q_{2}$ (electron capture)

$Q_{1}=[m_{N}(_{Z}X)−m_{N}(_{Z−1}Y)−m_{e}]c_{2}$

$=[m_{N}(_{Z}X)−Zm_{e}−m(_{Z−1}Y)−(Z−1)m_{e}−m_{e}]c_{2}$

$=[m_{N}(_{Z}X)−m_{N}(A_{Z−1}Y)−2m_{e}]c_{2}$

$Q_{2}=[m_{N}(_{Z}X)+m_{e}−m_{N}(_{Z−1}Y)]c_{2}=[m(_{Z}X)−m(_{Z−1}Y)]c_{2}$

This means $Q_{1}>0$ implies $Q_{2}>0$, but $Q_{2}>0$ does not necessarily mean $Q_{1}>0$.