Question

A plot of the number of neutrons \((N)\) against the number of protons \((P)\) of stable nuclei exhibits upward deviation from linearity for atomic number, \(Z>20\). For an unstable nucleus having \(N / P\) ratio less than 1, the possible mode(s) of decay is(are) (A) \(\beta^{-}\) -decay \((\beta\) emission) (B) orbital or \(K\) -electron capture (C) neutron emission (D) \(\beta^{+}\) -decay (positron emission)

Short answer

(A) \(\beta^{-}\)-decay, (D) \(\beta^{+}\)-decay, and (B) orbital or \(K\)-electron capture are possible modes of decay for a nucleus with \(N/P < 1\).

Step-by-step solution

Understand the Stability of Nuclei and the N/Z Ratio

For a nucleus to be stable, it generally needs a certain number of neutrons to protons. Lighter nuclei typically have near-equal numbers of protons and neutrons (\(N/Z \rightarrow 1\)), while heavier nuclei require more neutrons than protons to remain stable due to the increasing electrostatic repulsion among protons. An \(N/P < 1\) suggests that there are fewer neutrons compared to the number of protons.

Determine Decay Modes for Proton-rich Nuclei

A nucleus with an \(N/P < 1\) ratio is proton-rich. To reach stability, such a nucleus needs to convert protons into neutrons, or reduce the number of protons. Modes of decay that can accomplish this are \textbf{(A) \(\beta^{-}\)-decay} and \textbf{(D) Positron emission (or \(\beta^{+}\)-decay)}, since both processes essentially increase the neutron count relative to protons, either through the conversion of a proton into a neutron (\textbf{(D)}) or the emission of an electron with a corresponding increase in neutron number (\textbf{(A)}).

Eliminate Inappropriate Decay Modes

Orbital or \textbf{(B) \(K\)-electron capture} is also a process that reduces the number of protons and increases neutrons and thus can occur in proton-rich nuclei. However, \textbf{(C) Neutron emission} increases the \(N/P\) ratio by reducing the neutron count and would not occur in a proton-rich nucleus, because it would further unbalance the ratio of neutrons to protons.

Key concepts

N/Z Ratio in Nuclei

The nuclear stability of an atom is fundamentally influenced by the ratio of neutrons () to protons () in its nucleus, commonly referred to as the ratio. This ratio plays a crucial role in determining whether an atom is stable or radioactive. For lighter elements, those with an atomic number less than 20, a 1:1 neutron to proton ratio often coincides with stability. However, as we move beyond this point in the periodic table, stable nuclei tend to have more neutrons than protons.

The need for a higher neutron count in heavy elements is due to the electrostatic repulsion between protons; neutrons help to mitigate this repulsion by providing an attractive nuclear force that offsets the proton-proton repulsion. A deviation from the ideal neutron to proton ratio can lead to instability and give rise to various modes of radioactive decay, which the nucleus undergoes in an attempt to achieve a more stable state.

Beta Decay (eta^- and eta^+)

Beta decay is a common mode of radioactive decay that occurs in two primary forms: decay (electron emission) and decay (positron emission). In decay, a neutron within an unstable nucleus is transformed into a proton while emitting an electron and an antineutrino. This process increases the number of protons and decreases the number of neutrons, altering the ratio towards stability for neutron-rich nuclei.

Conversely, in decay, a proton is converted into a neutron with the emission of a positron and a neutrino. This type of decay is characteristic of proton-rich nuclei, as it serves to lower the number of protons and increase neutrons, striving for a more favorable balance. Both of these processes are mediated by the weak nuclear force and contribute to the transmutation of elements within the nucleus.

Electron Capture

Electron capture is another decay route for proton-rich nuclei, where an electron, often from the innermost shell of an atom (called the K-shell), is captured by the nucleus and combines with a proton to form a neutron and emit a neutrino. This process decreases the atomic number by one and is an alternate way for atoms with a low neutron to proton ratio to stabilize. Unlike beta decay which emits particles, electron capture involves the direct absorption of an electron, resulting in a change of the atomic nucleus without releasing any particles into the surrounding environment.

Neutron Emission

Neutron emission is a less common type of radioactive decay where an unstable nucleus releases a neutron. It typically occurs in extremely neutron-rich nuclei where the ratio is significantly above the optimal range for stability. Neutron emission directly reduces the number of neutrons, thereby increasing the ratio and moving the nucleus closer to the band of stability. It is primarily observed in heavy elements that have undergone fission or been artificially produced in a reactor or particle accelerator.