QB_PH_SEC6_023: Atomic Spectra & Multi-Electron Atoms

For a shell that is MORE than half filled (e.g., $d^8$ configuration), Hund's third rule specifies that the ground-state fine structure level has

Correct Answer

### Step-by-Step Derivation & Verification: 1. **Spin-Orbit Interaction Energy**: $E_{\text{SO}} = \frac{1}{2} A [J(J+1) - L(L+1) - S(S+1)]$ 2. **Sign of Coupling Constant $A$**: - For a less-than-half-filled subshell, electron spin-orbit interaction gives $A > 0$ (regular multiplet), so minimum $J = |L - S|$ lies lowest. - For a more-than-half-filled subshell, the physics is equivalent to vacancies (holes) in an otherwise filled shell. The effective coupling constant changes sign: $A < 0$ (inverted multiplet). 3. **Ground State Selection**: With $A < 0$, maximizing $J(J+1)$ minimizes the total energy $E_{\text{SO}}$. Thus, the ground state corresponds to maximum total angular momentum: $J = L + S$. 4. **Conclusion**: Option (A) is correct.