QB_PH_SEC6_026: Atomic Spectra & Multi-Electron Atoms

For a hydrogen-like atom with nuclear charge $Z$, the spin-orbit splitting $\Delta E_{\text{SO}}$ between the $2p_{3/2}$ and $2p_{1/2}$ fine-structure states scales with atomic number $Z$ as

Correct Answer

### Step-by-Step Derivation & Verification: 1. **Spin-Orbit Interaction Operator**: $H_{\text{SO}} = \frac{1}{2 m^2 c^2} \frac{1}{r} \frac{dV}{dr} (\vec{L} \cdot \vec{S})$ 2. **Coulomb Potential & Gradient**: $V(r) = -\frac{Z e^2}{4\pi\varepsilon_0 r} \implies \frac{1}{r} \frac{dV}{dr} = \frac{Z e^2}{4\pi\varepsilon_0 r^3}$ 3. **Radial Expectation Value in Hydrogen-like Ions**: From the radial wavefunctions $R_{nl}(r)$, distances scale inversely with $Z$ ($r \propto 1/Z$): $\left\langle \frac{1}{r^3} \right\rangle_{nl} = \frac{Z^3}{a_0^3 n^3 l(l+1/2)(l+1)} \propto Z^3$ 4. **Total Scaling**: $\Delta E_{\text{SO}} \propto Z \times \left\langle \frac{1}{r^3} \right\rangle \propto Z \times Z^3 = Z^4$ 5. **Conclusion**: Scales as $Z^4$. Option (A) is correct.