The Leptons — Question 5

Question 5

Inverse beta decay has a recoil threshold beyond the mass difference

Consider $\bar\nu_e+p\to n+e^+$ with the proton initially at rest and negligible neutrino mass. Use proton, neutron and electron rest energies 938.27, 939.57 and .511 MeV.

1. Derive the antineutrino threshold from invariant energy.

2. Evaluate the threshold and compare with the final-minus-initial rest-energy difference.

Question 5 - Solution

Inverse beta decay has a recoil threshold beyond the mass difference

(a) Method and calculation

$$sc^4=m_p^2c^4+2m_pc^2E_\nu,\quad s_{\mathrm{th}}c^4=(m_n+m_e)^2c^4.$$$$\boxed{E_{\nu,\mathrm{th}}=\frac{(m_n+m_e)^2-m_p^2}{2m_p}c^2}.$$

(b) Interpretation and further calculation

$$\Delta=(939.57+.511)-938.27=1.811\,\mathrm{MeV},\quad\boxed{E_{\nu,\mathrm{th}}=\Delta+\frac{\Delta^2}{2(938.27)}=1.812748\,\mathrm{MeV}}.$$$$\boxed{E_{\nu,\mathrm{th}}-\Delta=1.748\,\mathrm{keV}}>0.$$

Physical check

Final particles must share a common center-of-mass motion at threshold. A stationary target cannot absorb incident momentum without a recoil-energy cost.