Modes of Nuclear Decay for Low Ratio of Neutrons to Protons

To determine the possible modes of decay for an unstable nucleus with a neutron-to-proton ratio ($N/P$) less than $1$, we analyze how each nuclear decay process affects the number of neutrons ($N$) and protons ($P$).

For a nucleus to attain stability, its $N/P$ ratio must increase toward the stable region (where $N/P \ge 1$). This requires a process that either increases the number of neutrons ($N$) or decreases the number of protons ($P$), or both.

Let us evaluate each decay mode:

1. $\beta^+$-decay (Positron emission): In this process, a proton inside the nucleus transforms into a neutron, emitting a positron ($e^+$) and an electron neutrino ($\nu_e$): $p \longrightarrow n + e^+ + \nu_e$

   • Number of neutrons changes from $N \to N + 1$
   • Number of protons changes from $P \to P - 1$

   The new ratio becomes: $\left(\frac{N}{P}\right)_{\text{new}} = \frac{N + 1}{P - 1} > \frac{N}{P}$ Since this increases the $N/P$ ratio toward the band of stability, $\beta^+$-decay is a possible mode of decay.

2. Orbital or $K$-electron capture: In this process, the nucleus captures an inner atomic electron (usually from the $K$-shell), which reacts with a nuclear proton to form a neutron and emit a neutrino ($\nu_e$): $p + e^- \longrightarrow n + \nu_e$

   • Number of neutrons changes from $N \to N + 1$
   • Number of protons changes from $P \to P - 1$

   The new ratio becomes: $\left(\frac{N}{P}\right)_{\text{new}} = \frac{N + 1}{P - 1} > \frac{N}{P}$ Since this increases the $N/P$ ratio toward stability, orbital electron capture is a possible mode of decay.

3. $\beta^-$-decay ($\beta$ emission): In this process, a neutron transforms into a proton, emitting an electron ($e^-$) and an antineutrino ($\bar{\nu}_e$): $n \longrightarrow p + e^- + \bar{\nu}_e$

   • Number of neutrons changes from $N \to N - 1$
   • Number of protons changes from $P \to P + 1$

   The new ratio becomes: $\left(\frac{N}{P}\right)_{\text{new}} = \frac{N - 1}{P + 1} < \frac{N}{P}$ This decreases the $N/P$ ratio even further away from stability, so $\beta^-$-decay is not possible for a proton-rich nucleus.

4. Neutron emission: In this process, a neutron is ejected directly from the nucleus:

   • Number of neutrons changes from $N \to N - 1$
   • Number of protons remains unchanged ($P$)

   The new ratio becomes: $\left(\frac{N}{P}\right)_{\text{new}} = \frac{N - 1}{P} < \frac{N}{P}$ This also decreases the $N/P$ ratio, so neutron emission is not possible for a proton-rich nucleus.

Thus, the possible modes of decay are orbital or $K$-electron capture and $\beta^+$-decay.

Correct Options: (B) orbital or $K$-electron capture (D) $\beta^+$-decay (positron emission)