Question

I am delving into the nuclear spin characteristics of Boron-10, which notably has a nuclear spin of 3. While I have a foundational understanding of some quantum mechanical principles, the specific interaction of these principles leading to this particular spin state in Boron-10 remains elusive.

I am acquainted with the Pauli Exclusion Principle, which stipulates that no two identical fermions can occupy the same quantum state simultaneously. Also, I understand Hund's Rule, which suggests a maximization of total spin in a system by aligning the spins of individual particles. Moreover, I am familiar with the role of Clebsch-Gordan coefficients in the coupling of angular momenta.

Given the 5 protons and 5 neutrons in Boron-10:

How does the Pauli Exclusion Principle manifest in the arrangement of nucleons in Boron-10 to achieve a nuclear spin of 3?
How does Hund's Rule contribute to the alignment and coupling of the spins of these nucleons, and how does this alignment lead to a nuclear spin of 3?
In the coupling of angular momenta to achieve a nuclear spin of 3, how are Clebsch-Gordan coefficients utilized in Boron-10?
Are there specific nuclear models or theories that provide a detailed explanation on how Boron-10 achieves a nuclear spin of 3 through the interaction of the above-mentioned principles?

Answer 1

In Boron-10, the Pauli Exclusion Principle dictates that nucleons must occupy unique quantum states, while Hund's Rule suggests spins align to maximize total spin. Clebsch-Gordan coefficients then describe the combinations of these individual spins to yield a nuclear spin of 3.

Step-by-step explanation:

The question pertains to the nuclear spin of Boron-10 and how principles like the Pauli Exclusion Principle, Hund's Rule, and the use of Clebsch-Gordan coefficients interact to give it a nuclear spin of 3. Boron-10 has 5 protons and 5 neutrons. According to the Pauli Exclusion Principle, no two identical fermions (which protons and neutrons are) can occupy the same quantum state simultaneously.

Thus, within a nucleus, protons and neutrons will fill available nuclear shell states with their spins aligned to maximize total spin, adhering to nuclear shell models. When particles combine their angular momenta, such as spins in a nucleus, the resultant possible states are given by Clebsch-Gordan coefficients, which help to predict the resulting spins after the combination of two angular momenta. Specifically, these coefficients are used in quantum mechanics to construct the spin state of a composite system like a nucleus from the individual spins of the particles.