Animated Solution for Physics - Atoms and Nuclei: The decay of a proton to neutron is

Imagine a single, isolated proton floating in the vast emptiness of space. A fundamental question arises: Can this proton spontaneously transform into a neutron? To answer this, we must consult the ultimate ledger of the universe—mass and energy.

When we measure their masses, we find that the mass of a proton ($m_p$) is approximately $1.007276\text{ u}$, while the mass of a neutron ($m_n$) is slightly heavier at $1.008664\text{ u}$. In the realm of physics, mass is simply a concentrated form of energy, as dictated by Einstein's famous equation, $E=m{c}^2$.

For a lighter particle to decay into a heavier one, it would require an injection of external energy. The energy difference required is $\Delta E=(m_n+m_e-m_p)c^2>0$. Without this extra energy, the transformation is strictly forbidden by the law of conservation of energy. Therefore, a free proton is perfectly stable and will never decay into a neutron.

Now, let's change the scenario. What if the proton is not alone but tightly packed inside an atomic nucleus, interacting with other protons and neutrons via the strong nuclear force? Here, the rules of the game change dramatically.

Inside the nucleus, the collection of nucleons possesses something incredibly powerful: Nuclear Binding Energy. This is the energy that holds the nucleus together. Sometimes, a nucleus finds itself in an unstable state with too many protons. If one of these protons were to convert into a neutron, the resulting 'daughter' nucleus might actually be more stable, meaning it would exist at a lower overall energy state.

But wait, doesn't the proton still need extra energy to become a heavier neutron? Yes, it does! However, because the entire nuclear system is transitioning to a more stable state, the nucleus essentially 'pays the energy bill' for the proton.

The available binding energy compensates for the mass difference. This allows the proton to successfully convert into a neutron, emitting a positron ($e^{+}$) and an electron neutrino ($ u_e$) in the process. This phenomenon is known as Positron Emission or ${\beta }^{+}$ decay.

Because this energy compensation can only happen within the bound system of a nucleus, the decay of a proton to a neutron is strictly possible only inside the nucleus. This makes option (b) the correct answer.