QB_PH_SEC6_012: Atomic Spectra & Multi-Electron Atoms

According to the Landé interval rule, the fine structure levels of a $^3D$ multiplet have energy separations $\Delta E(J=3 \to J=2)$ and $\Delta E(J=2 \to J=1)$. What is the ratio $\frac{\Delta E(3, 2)}{\Delta E(2, 1)}$? (Give your answer to one decimal place)

Correct Answer

### Step-by-Step Derivation & Verification: 1. **Spin-Orbit Interaction Hamiltonian**: $H_{\text{SO}} = A (\vec{L} \cdot \vec{S}) = \frac{A}{2} [J(J+1) - L(L+1) - S(S+1)]$ 2. **Energy Level Expression**: $E(J) = E_0 + \frac{A}{2} J(J+1)$ 3. **Adjacent Interval Calculation**: $\Delta E(J, J-1) = E(J) - E(J-1) = \frac{A}{2} [J(J+1) - (J-1)J] = \frac{A}{2} J [J + 1 - J + 1] = A \cdot J$ 4. **Ratio Evaluation for $^3D$ Term** ($L=2, S=1 \implies J=3, 2, 1$): $\Delta E(3, 2) = 3A$ $\Delta E(2, 1) = 2A$ $\text{Ratio} = \frac{3A}{2A} = 1.5$ 5. **Official Answer**: 1.5.