Animated Solution for Physics - Atoms and Nuclei: Comprehension Passage

Imagine you are a physicist in the 1920s. You are observing beta decay, where a neutron decays into a proton and an electron. According to the laws of physics, if a particle at rest decays into two particles, they must fly apart with specific, constant energies to conserve both momentum and energy. But experiments showed something baffling: the emitted electrons had a continuous range of energies! It seemed as if energy was vanishing into thin air.

To save the sacred law of energy conservation, Wolfgang Pauli proposed a radical idea in 1930: a "ghost" particle was carrying away the missing energy. This particle, later named the neutrino (or anti-neutrino in this specific case), was neutral, incredibly light, and interacted so weakly with matter that it was invisible to detectors of that era.

Let's break down the physics. When a neutron decays, the total energy released is called the $Q$-value. In our problem, this is given as $0.8\times 10^6\text{ eV}$. This energy must be shared among the products: the proton, the electron, and the anti-neutrino.

Because the proton is thousands of times more massive than the electron and the anti-neutrino, it barely moves. Its recoil kinetic energy ($K_p$) is practically zero. Thus, the $Q$-value is essentially shared between the electron and the anti-neutrino:

Now, let's tackle the first question. What if the anti-neutrino isn't perfectly massless? Suppose it has a tiny mass of $3\text{ eV}/c^2$. According to Einstein's famous equation, $E=m{c}^2$, any particle with mass has an intrinsic rest mass energy. For our anti-neutrino, this is $3\text{ eV}$.

This means that even if the anti-neutrino is completely stationary, it must consume $3\text{ eV}$ of the available $Q$-value just to exist!

Because the anti-neutrino is stealing at least $3\text{ eV}$, the electron can never have the full $0.8\times 10^6\text{ eV}$. Its maximum possible kinetic energy is strictly less than the $Q$-value:

Since kinetic energy cannot be negative, the electron's kinetic energy must fall in the range:

Moving to the second question, we want to find the absolute maximum energy the anti-neutrino can possess. Since the total energy is fixed, the anti-neutrino gets the most energy when the electron gets the least.

The minimum kinetic energy the electron can have is zero (it just pops into existence and sits there). If $K_e\approx 0$, then the anti-neutrino runs away with almost all the available energy:

This beautiful interplay of energy and momentum conservation not only solves our problem but also highlights how physicists deduce the properties of the invisible quantum world!