# Range of kinetic energy

The $$\beta$$-decay process, discovered around 1900, is basically the decay of a neutron $$(n)$$. In the laboratory, a proton $$(p)$$ and an electron $$( e^- )$$ are observed as the decay products of the neutron. Therefore, considering the decay of a neutron as a two-body decay process, it was predicted theoretically that the kinetic energy of the electron should be a constant. But experimentally, it was observed that the electron kinetic energy has a continuous spectrum. Considering a three-body decay process, i.e. $$n \rightarrow p + e^- + (\overline{v_e})$$, around 1930, Pauli explained the observed electron energy spectrum. Assuming the anti-neutrino $$( \overline{v_e} )$$ to be massless and possessing negligible energy, and the neutron to be at rest, momentum and energy conservation principles are applied. From this calculation, the maximum kinetic energy of the electron is $$0.8 \times 10^6$$ eV. The kinetic energy carried by the proton is only the recoil energy.

If the anti-neutrino had a mass of $$3 eV/c^2$$ (where $$c$$ is the speed of light) instead of zero mass, what should be the range of the kinetic energy, $$K$$, of the electron?

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