Let's suppose we can form ^3 He or ^3 H by adding a proton or a neutron to ^2 H, which has sp

step 1 What are the possible values of the total angular momentum? Okay, so let's think about this firs

Let's suppose we can form ^3 He or ^3 H by adding a proton...

Let's suppose we can form ${ }^{3} \mathrm{He}$ or ${ }^{3} \mathrm{H}$ by adding a proton or a neutron to ${ }^{2} \mathrm{H}$, which has spin equal to 1 and even parity. Let $\ell$ be the orbital angular momentum of the added nucleon relative to the ${ }^{2} \mathrm{H}$ center of mass. What are the possible values of the total angular momentum of ${ }^{3} \mathrm{H}$ or ${ }^{3} \mathrm{He}$ ? Given that the ground-state parity of ${ }^{3} \mathrm{H}$ and ${ }^{3} \mathrm{He}$ is even, which of these can be eliminated? What is the most likely value of the ground-state angular momentum of ${ }^{3} \mathrm{H}$ or ${ }^{3} \mathrm{He}$ ? Can you make a similar argument based on removing a proton or a neutron from ${ }^{4} \mathrm{He}$ ? (What is the ground-state spin-parity of ${ }^{4} \mathrm{He}$ ?) How would you account for the spin-parity of ${ }^{3} \mathrm{Li}$ and ${ }^{5} \mathrm{He}\left(\frac{2}{2}{ }^{-}\right)$?

Key Concepts

Nuclear Shell Model Configuration

The nuclear shell model is used to describe the arrangement of nucleons within discrete energy levels or shells, much like electrons in atomic orbitals. It explains the observed magic numbers and spin-parity assignments of nuclei. For example, the ground state of a doubly magic nucleus like 4He is 0+, and when a single nucleon is removed or added, the quantum numbers of the resultant state are largely determined by the properties of the single particle in the topmost occupied shell. This same reasoning is applied to account for the spin-parity of nearby nuclei such as 3Li and 5He.

Single-Particle Model and Core Nucleus

The single-particle model considers the addition or removal of one nucleon from a core nucleus, treating the remaining nucleons as an inert core with a fixed spin and parity. This framework allows one to assign quantum numbers to the extra nucleon (or the hole in the core) by considering its orbital and intrinsic spins as well as its coupling to the core. It provides a straightforward way to understand the leading configurations in light nuclei and predict their spin-parity assignments.

Orbital Angular Momentum and Parity

Orbital angular momentum, denoted by ?, is crucial in nuclear structure because it not only contributes to the total angular momentum of the system but also determines the parity of the nuclear state through the factor (–1)^?. A change in ? will change the parity of the state, so when experimental evidence (such as an even ground?state parity) is known, this can be used to eliminate possibilities with incompatible ? values, thereby simplifying the determination of the ground-state angular momentum.

Angular Momentum Coupling

This concept deals with the way in which the intrinsic spin of a nucleon, its orbital angular momentum relative to a core, and the spin of the core nucleus are combined to give the total angular momentum of the composite system. In nuclear physics, one uses the angular momentum addition rules (as given by the Clebsch–Gordan series) to determine all possibilities for the total spin of a nucleus formed by adding a nucleon to a core with a known spin.

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