Beta Decay — Question 3

Question 3

Electron capture can be allowed when positron emission is forbidden

A parent neutral atom is heavier than the neutral daughter with one fewer proton by .800 MeV in rest-energy units. Take $m_ec^2=.511\,\mathrm{MeV}$ and neglect electronic binding energies initially.

1. Calculate Q for positron emission and for electron capture.

2. If instead the atomic mass-energy difference is 1.800 MeV, find the positron endpoint kinetic energy and the energy available for capture.

Question 3 - Solution

Electron capture can be allowed when positron emission is forbidden

(a) Method and calculation

$$\boxed{Q_{\beta^+}=\Delta Mc^2-2m_ec^2=.800-1.022=-.222\,\mathrm{MeV}}\quad\text{forbidden}.$$$$\boxed{Q_{\mathrm{EC}}=\Delta Mc^2=.800\,\mathrm{MeV}}\quad\text{energetically allowed}.$$

(b) Interpretation and further calculation

$$\boxed{T_{e^+,\max}\simeq1.800-1.022=.778\,\mathrm{MeV}}.$$$$\boxed{Q_{\mathrm{EC}}=1.800\,\mathrm{MeV}}.$$

Physical check

The two electron rest masses in the positron threshold arise from positron creation and the change in neutral-atom electron count. Capture can use an electron already present.