Question

The $\beta$decay process, discovered around $1900$, is basically the decay of a neutron n. In the laboratory, a proton p and an electron are observed as the decay product of neutrons. Therefore, considering the decay of neutrons 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, Pauli explained the observed electron energy spectrum. Assuming the anti-neutrino to be massless and possessing negligible energy, and the neutrino 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 $3ev{{c}^{-2}}$ (where c is the speed of light ) instead of zero mass, what should be the range of the kinetic energy k of the electron?
$\begin{aligned} & a)0\le K\le 0.8\times {{10}^{6}}eV \\\ & b)3\le K\le 0.8\times {{10}^{6}}eV \\\ & c)3\le K<0.8\times {{10}^{6}}eV \\\ & d)0\le K<0.8\times {{10}^{6}}eV \\\ \end{aligned}$

Answer 1

In the above information, it is given that the anti-neutrino is massless and all others are rest and energy is conserved. But in the question, we are told that the anti-neutrino is having some mass. Any object that has mass will have some energy for sure.

Complete answer:
Initially, when Pauli explained the observed electron energy spectrum it was assumed that antineutrino has no mass, and the energy of the antineutrino is negligible. Also, the momentum and energy principles are applied for all other electrons and protons. In the question we were told that antineutrino had a mass which is not negligible. We know every object that has mass will have some finite energy if the
Antineutrino has some energy; the energy of the electron will not be equal to the maximum kinetic energy but less than the maximum kinetic energy.

Therefore, the correct option is option d.

Additional information:
Beta particles are electrons or positrons. Beta decay occurs, when in a nucleus with too many protons or too many neutrons, one of the protons or neutrons is transformed into the other. In beta minus decay, a neutron decays into a proton, an electron, and an antineutrino whereas in a beta plus decay, a proton decays into a neutron, a positron, and a neutrino. Both reactions occur because in different regions of the chart of the nuclides, one or the other will move the product closer to the region of stability of hydrogen nucleus with or without an electron does not decay however within a nucleus, the beta decay process can change a proton to neutron it will decay with a half time off 10.5 minutes. There are 3 ways in which protons can be changed into neutrons: proton decay comes on neutron decay, electron capture.

Note:
In the above question, we were told that their antineutrino has some amount of mass. As it has mass it contains some energy in it. If there is some amount of energy present in antineutrino the electrons kinetic energy will be less than the maximum kinetic energy for sure. This is the reason the kinetic energy will be greater than or equal to zero, but it can never be equal to the maximum kinetic energy of the electron.