QB_PH_SEC6_029: Atomic Spectra & Multi-Electron Atoms

Under what physical condition does the atomic coupling scheme transition from Russell-Saunders ($L-S$) coupling to $j-j$ coupling?

Correct Answer

### Step-by-Step Derivation & Verification: 1. **Two Major Internal Perturbations**: - Residual electrostatic Coulomb repulsion: $H_{\text{Coulomb}} = \sum_{i<j} \frac{e^2}{4\pi\varepsilon_0 r_{ij}}$ - Spin-orbit interaction: $H_{\text{SO}} = \sum_i \xi(r_i) \vec{l}_i \cdot \vec{s}_i \propto Z^4$ 2. **$L-S$ Coupling Limit ($H_{\text{Coulomb}} \gg H_{\text{SO}}$)**: Valid for low-$Z$ atoms (light elements). Orbital angular momenta couple to $\vec{L}$, spins couple to $\vec{S}$, then $\vec{J} = \vec{L} + \vec{S}$. 3. **$j-j$ Coupling Limit ($H_{\text{SO}} \gg H_{\text{Coulomb}}$)**: Valid for high-$Z$ atoms (heavy elements). Individual spin-orbit interaction dominates: $\vec{j}_i = \vec{l}_i + \vec{s}_i$, then $\vec{J} = \sum \vec{j}_i$. 4. **Conclusion**: Option (A) is correct.