Kinetic Energy Range of Electron with Massive Anti Neutrino

To determine the valid range of the electron's kinetic energy $K$ when the anti-neutrino has a non-zero rest mass, we analyze the decay process:

$n \rightarrow p + e^- + \bar{\nu}_e$

1. Lower Bound of Kinetic Energy ($K_{\text{min}}$): The electron can be produced nearly at rest, with its momentum balanced by the proton and the anti-neutrino. Therefore, the minimum possible kinetic energy for the electron remains: $K_{\text{min}} = 0$

2. Upper Bound of Kinetic Energy ($K_{\text{max}}$): When the anti-neutrino is assumed to be massless ($m_{\bar{\nu}} = 0$), its minimum total energy can approach zero ($E_{\bar{\nu},\text{min}} \to 0$), allowing the electron to achieve a maximum kinetic energy of: $K_{\text{max, massless}} = 0.8 \times 10^6 \text{ eV}$

   However, if the anti-neutrino has a finite rest mass $m_{\bar{\nu}} = 3\text{ eV}/c^2$, its total energy $E_{\bar{\nu}}$ must always be greater than or equal to its rest mass energy: $E_{\bar{\nu}} = \sqrt{p_{\bar{\nu}}^2 c^2 + (m_{\bar{\nu}} c^2)^2} \ge m_{\bar{\nu}} c^2 = 3\text{ eV}$

   Since the anti-neutrino carries away at least $3\text{ eV}$ of energy from the available decay energy $Q$, the maximum kinetic energy that the electron can acquire is reduced by $3\text{ eV}$: $K_{\text{max}} = 0.8 \times 10^6\text{ eV} - 3\text{ eV} < 0.8 \times 10^6\text{ eV}$

3. Conclusion: Combining the lower and upper bounds, the valid range for the electron's kinetic energy $K$ is: $0 \le K < 0.8 \times 10^6 \text{ eV}$

This corresponds to Option (D).