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 tro-body dcay process, it was observed that the electron kinetic energy has a continuous spectrum. Considering a three-body decay process i.e., `n rarr p + e^(-)+overset(-)v_(e )`, around `1930`, Pauli explained the observed electron energy spectrum. Assuming the anti-neutrino `(overset(-)V_(e ))` to be massless and possessing negligible energy, and neutron to be at rest, momentum and energy conservation principles are applied. From this calculation, the maximum kinetic energy of the electron is `0.8xx10^(6)eV`. The kinetic energy carried by the proton is only the recoil energy.
What is the maximum energy of the anti-neutrino?

Options

A. zero
B. Much less than `0.8xx10^(6)eV`
C. Nearly `0.8xx10^(6)eV`
D. Much larger than `0.8xx10^(6)eV`

Explanation

To solve the problem of finding the maximum energy of the anti-neutrino in the beta decay process, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Decay Process**:
The beta decay process involves a neutron decaying into a proton, an electron, and an anti-neutrino. The reaction can be represented as:
\[ n \rightarrow p + e^- + \overline{\nu}_e \]

2. **Conservation of Energy**:
In this decay process, the total energy before and after the decay must be conserved. Initially, the neutron is at rest, so its energy is its rest mass energy, \(E_n = m_n c^2\).

3. **Kinetic Energy of Products**:
After the decay, the energy is distributed among the proton, electron, and anti-neutrino. The maximum kinetic energy of the electron is given as:
\[ KE_e^{max} = 0.8 \times 10^6 \, \text{eV} \]

4. **Recoil Energy of the Proton**:
The proton will have a recoil energy due to the conservation of momentum. However, in this scenario, we are interested in the maximum energy of the anti-neutrino.

5. **Energy Distribution**:
The total energy after decay can be expressed as:
\[ E_n = KE_p + KE_e + E_{\overline{\nu}_e} \]
where \(KE_p\) is the recoil energy of the proton, \(KE_e\) is the kinetic energy of the electron, and \(E_{\overline{\nu}_e}\) is the energy of the anti-neutrino.

6. **Maximum Energy of the Anti-neutrino**:
To find the maximum energy of the anti-neutrino, we can consider the scenario where the electron has its maximum kinetic energy and the proton has negligible recoil energy. Thus, we can set:
\[ E_{\overline{\nu}_e} = E_n - KE_e^{max} - KE_p \]
If we assume the recoil energy of the proton is negligible (or very small), we can simplify this to:
\[ E_{\overline{\nu}_e} \approx E_n - KE_e^{max} \]

7. **Conclusion**:
Since the neutron is at rest, the maximum energy of the anti-neutrino will be equal to the maximum kinetic energy of the electron when the proton's recoil energy is negligible. Therefore, the maximum energy of the anti-neutrino is:
\[ E_{\overline{\nu}_e} = 0.8 \times 10^6 \, \text{eV} \]

### Final Answer:
The maximum energy of the anti-neutrino is approximately \(0.8 \times 10^6 \, \text{eV}\).

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