QB_PH_SEC9_011: Nuclear Models & Decays

Which of the following sets of nucleon numbers corresponds strictly to the first seven traditional nuclear magic numbers predicted by the 3D harmonic oscillator with strong spin-orbit coupling?

Correct Answer

### Step-by-Step Derivation & Verification: 1. **Harmonic Oscillator Shell Closures (without Spin-Orbit)**: Yields magic numbers: 2, 8, 20, 40, 70, 112, which fail to match experiment past 20. 2. **Inverted Spin-Orbit Interaction ($V_{\text{SO}}(r) \vec{l} \cdot \vec{s}$)**: - Splitting: $\Delta E_{\text{SO}} = \frac{1}{2}(2l+1) \langle V_{\text{SO}} \rangle$. - Because the nuclear force makes $V_{\text{SO}} < 0$, states with $j = l + 1/2$ are depressed in energy. - At $N_{\text{osc}} = 3$ ($1f, 2p$), the $1f_{7/2}$ subshell is pushed down toward the $N_{\text{osc}}=2$ shell, creating a new gap at: $20 + 8 = 28$. - At $N_{\text{osc}} = 4$, the $1g_{9/2}$ state drops, giving $28 + (6 + 4 + 2) + 10 = 50$. - At $N_{\text{osc}} = 5$, the $1h_{11/2}$ state drops, giving $50 + 32 = 82$. - At $N_{\text{osc}} = 6$, the $1i_{13/2}$ state drops, giving $82 + 44 = 126$. 3. **Authentic Series**: 2, 8, 20, 28, 50, 82, 126. Option (A) is correct.