Beta Decay — Question 9

Question 9

Electron capture leaves energy in both a neutrino and an atomic vacancy

For a specified electron-capture branch the neutral-atom Q value is .900 MeV. The daughter nuclear excitation is .200 MeV and the atomic vacancy carries excitation energy .0300 MeV. Neglect recoil initially.

1. Find the neutrino energy and the total energy later released in nuclear and atomic relaxation.

2. For daughter rest energy $Mc^2=100000\,\mathrm{MeV}$, estimate the neutrino recoil correction and corrected neutrino energy.

Question 9 - Solution

Electron capture leaves energy in both a neutrino and an atomic vacancy

(a) Method and calculation

$$\boxed{E_\nu\simeq Q-E_*-E_{\mathrm{vac}}=.900-.200-.0300=.670\,\mathrm{MeV}}.$$$$\boxed{E_{\mathrm{relax}}=E_*+E_{\mathrm{vac}}=.230\,\mathrm{MeV}}.$$

(b) Interpretation and further calculation

$$\boxed{T_R\simeq E_\nu^2/(2Mc^2)=.670^2/200000=2.2445\,\mathrm{eV}}.$$$$\boxed{E_{\nu,\mathrm{corrected}}\simeq.66999776\,\mathrm{MeV}}.$$

Physical check

Capture from a defined atomic shell gives a nearly discrete neutrino energy for a specified nuclear branch. Atomic relaxation can produce X rays or Auger electrons.