QB_PH_SEC9_013: Nuclear Properties & Nuclear Forces

A nucleus with electric quadrupole moment $Q > 0$ has a non-spherical charge distribution that is shaped as an

Correct Answer

### Step-by-Step Derivation & Verification: 1. **Electric Quadrupole Definition**: $e Q = \int \rho(\vec{r}) (3 z^2 - r^2) d^3 r = \int \rho(\vec{r}) (2 z^2 - x^2 - y^2) d^3 r$ 2. **Geometric Deformation Analysis**: - **Spherically Symmetric** ($\langle x^2 \rangle = \langle y^2 \rangle = \langle z^2 \rangle$): $2\langle z^2 \rangle - 2\langle z^2 \rangle = 0 \implies Q = 0$. - **Elongated along $z$ ($c > a$)**: $\langle z^2 \rangle > \langle x^2 \rangle = \langle y^2 \rangle \implies 3\langle z^2 \rangle - \langle r^2 \rangle > 0 \implies Q > 0$. This geometric shape is a **prolate spheroid** (cigar-shaped). - **Flattened at poles ($c < a$)**: $\langle z^2 \rangle < \langle x^2 \rangle \implies Q < 0$. This is an **oblate spheroid** (pancake-shaped). 3. **Conclusion**: $Q > 0$ corresponds to a prolate spheroid. Option (A) is correct.