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# The β-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→p+e−+¯¯¯ve, around 1930, Pauli explained the observed electron energy spectrum. Assuming the anti-neutrino (¯¯¯ve) 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 0.8×106eV. The kinetic energy carried by the proton is only the recoil energy.If the anti-neutrino had a mass of 3 eV/c2 (where c is the speed of light) instead of zero mass, what should be the range of the kinetic energy, K, of the electron?0≤K≤0.8×106eV3.0eV≤K≤0.8×106cV3.0eV≤K<0.8×106eV0≤K<0.8×106eV

A
0K0.8×106eV
B
3.0eVK0.8×106cV
C
0K<0.8×106eV
D
3.0eVK<0.8×106eV
Solution
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#### K can't be equal to 0.8×106eV as anti-neutrino must have some energy

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Q1
The β-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. np+e+¯¯¯ve, around 1930, Pauli explained the observed electron energy spectrum. Assuming the anti-neutrino (¯¯¯ve) 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 0.8×106eV. The kinetic energy carried by the proton is only the recoil energy.
If the anti-neutrino had a mass of 3 eV/c2 (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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Q2

Consider the decay of a free neutron at rest: n p+ e

Show that the two-body decay of this type must necessarily give an electron of fixed energy and, therefore, cannot account for the observed continuous energy distribution in the β-decay of a neutron or a nucleus (Fig. 6.19).

[Note: The simple result of this exercise was one among the several arguments advanced by W. Pauli to predict the existence of a third particle in the decay products of β-decay. This particle is known as neutrino. We now know that it is a particle of intrinsic spin ½ (like e, p or n), but is neutral, and either massless or having an extremely small mass (compared to the mass of electron) and which interacts very weakly with matter. The correct decay process of neutron is: n p + e+ ν]

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Q3
If the anti-neutrino had a mass of 3eV/c2 (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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Q4
When subatomic particles undergo reactions, energy is conserved, but mass is not necessarily conserved. However, a particle’s mass “contributes” to its total energy, in accordance with Einstein’s famous equations, E=mc2
In this equation, E denotes the energy a particle carries because of its mass. The particle can also have additional energy due to its motion and its interactions with other particles.
Consider a neutron at rest, and well separated from other particles. It decays into a proton, an electron and an undetected third particle:
Neutron proton + electron + ???
Table 1 summarizes some data from a single neutron decay. An MeV (mega electron volt) is a unit of energy.
Table 1
Data from a single neutron decay
Column-1 shows the rest mass of the particle times the speed of light squared and
Column-2 shows its kinetic energy.
Column-1Column-2particlemass×c2(MeV)Kinetic energy (MeV)Neutron940.970.00Proton939.670.01Electron0.510.39

Given table 1, which properties of the undetected third particle can we calculate?
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Q5

In the decay 64Cu64Ni+e++v, the maximum kinetic energy carried by the positron is found to be 0.650 MeV. (a) What is the energy of the neutrino which was emitted together with a positron of kinetic energy 0.150 MeV? (b) What is the momentum of this neutrino in kg m s1 ? Use the formula applicable to a photon.

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