Question #43704

In a certain $\beta^{-}$ decay process, a parent nucleus at rest emits an electron and an antineutrino. Assume the daughter nucleus is infinitely heavy (negligible recoil) and the antineutrino is massless. The maximum kinetic energy of the emitted electron is observed to be $K_{\text{max}} = 4 m_0 c^2$, where $m_0$ is the rest mass of the electron and $c$ is the speed of light. According to Fermi's theory of beta decay, the transition rate $d\lambda$ into an electron momentum interval $dp$ is proportional to the available phase space volume of the electron-antineutrino pair. What is the ratio of the emission probability per unit momentum interval, $\frac{d\lambda}{dp}$, for an electron emitted with total relativistic energy $E_1 = 3 m_0 c^2$ to that for an electron emitted with total relativistic energy $E_2 = 2 m_0 c^2$?