Beta Decay — Question 5

Question 5

Daughter recoil slightly lowers the beta endpoint

For a beta-minus decay let Q=1.00 MeV and daughter rest energy $Mc^2=100000\,\mathrm{MeV}$. At the electron endpoint the massless antineutrino energy tends to zero. Use $p_e^2c^2=T_e(T_e+2m_ec^2)$ and $m_ec^2=.511\,\mathrm{MeV}$.

1. Derive the endpoint energy balance including nonrelativistic daughter recoil.

2. Evaluate the leading recoil correction by inserting $T_e\simeq Q$ in the recoil term.

Question 5 - Solution

Daughter recoil slightly lowers the beta endpoint

(a) Method and calculation

$$\mathbf p_D=-\mathbf p_e\Rightarrow T_D=p_e^2/(2M).$$$$\boxed{Q=T_{e,\max}+\frac{T_{e,\max}(T_{e,\max}+2m_ec^2)}{2Mc^2}}.$$

(b) Interpretation and further calculation

$$\boxed{\Delta T\simeq\frac{Q(Q+2m_ec^2)}{2Mc^2}=1.011\times10^{-5}\,\mathrm{MeV}=10.11\,\mathrm{eV}}.$$$$\boxed{T_{e,\max}\simeq.99998989\,\mathrm{MeV}}.$$

Physical check

The endpoint is approached as neutrino energy vanishes. Recoil is small on a MeV scale but can matter in precision endpoint measurements.