Let's break down how to calculate the electric quadrupole moment in the extreme single-particle model.

Understanding the Basics

* Electric Quadrupole Moment: This quantity measures the deviation of a charge distribution from spherical symmetry. A positive quadrupole moment indicates a prolate (football-like) shape, while a negative moment indicates an oblate (pancake-like) shape.

* Extreme Single-Particle Model: This model simplifies the nucleus by assuming that all nucleons (protons and neutrons) except one are in a spherically symmetric core. The single particle outside the core contributes the entire quadrupole moment.

Calculation

1. Consider the Single Particle: We need to focus on the single particle outside the core. Let's assume it has a charge *e* and is in an orbital with angular momentum *l*.

2. Quantize the Angular Momentum: In quantum mechanics, the *z*-component of angular momentum is quantized, meaning it can only take on discrete values: *m*ħ, where *m* ranges from -*l* to +*l*.

3. Define the Quadrupole Moment Operator: The quadrupole moment operator, *Q*, is given by:

*Q* = (2/e) Σ *i* (3*z_i² - *r_i²)

* *i* denotes each particle in the nucleus.

* *z_i* is the *z*-coordinate of the *i*-th particle.

* *r_i* is the radial distance of the *i*-th particle from the nucleus's center.

4. Evaluate for the Single Particle: Since we're dealing with the extreme single-particle model, we only need to consider the single particle's contribution:

*Q* = (2/e) (3*z² - *r²)

5. Express in Spherical Coordinates: Convert *z* and *r* to spherical coordinates (r, θ, φ):

* *z* = *r* cos(θ)

* *r²* = *r²*

6. Simplify: Substitute into the quadrupole moment equation:

*Q* = (2/e) *r² (3 cos²(θ) - 1)

7. Average over Angular Coordinates: The quadrupole moment is an expectation value. To find it, we need to average over all possible angles:

*Q* = (2/e) *r² ∫₀^2π dφ ∫₀^π sin(θ) (3 cos²(θ) - 1) dθ

8. Evaluate the Integrals: The integral evaluates to:

*Q* = (4/5) *e* *r²

9. Final Expression: The electric quadrupole moment for a single particle in the extreme single-particle model is:

*Q* = (4/5) *e* *r²

Interpretation

* The quadrupole moment depends on the charge (*e*) and the radial distance squared (*r²*) of the single particle.

* A larger *r* (particle farther from the core) leads to a larger quadrupole moment.

* The sign of the quadrupole moment (positive in this case) indicates a prolate shape, consistent with a single particle sitting outside a spherically symmetric core.

Note: This calculation assumes a single particle in the nucleus. For real nuclei, multiple particles contribute, and more sophisticated models are needed to accurately calculate the quadrupole moment.