QB_PH_SEC9_003: Nuclear Models & Decays

According to the extreme single-particle nuclear shell model with strong spin-orbit coupling, the ground-state spin and parity ($J^P$) of the $_{8}^{17}\text{O}$ nucleus is

Correct Answer

### Step-by-Step Derivation & Verification: 1. **Nucleon Allocation**: - Protons: $Z = 8$ (even, magic closed shell: $1s_{1/2}^2 1p_{3/2}^4 1p_{1/2}^2$). Net contribution to spin and parity is $0^+$. - Neutrons: $N = 17 - 8 = 9$ (odd neutron outside closed shell $N = 8$). 2. **Shell Hierarchy with Spin-Orbit Splitting**: - $1s_{1/2}$ capacity 2 (fills $N=1,2$) - $1p_{3/2}$ capacity 4 (fills $N=3,4,5,6$) - $1p_{1/2}$ capacity 2 (fills $N=7,8$) - The 9th neutron occupies the lowest subshell of the next oscillator shell ($N_{\text{osc}} = 2$): $1d_{5/2}$ ($l=2, j=5/2$). 3. **Parity Determination**: $\pi = (-1)^l = (-1)^2 = +1$ 4. **Ground State $J^P$**: $J^P = \frac{5}{2}^+$ Option (A) is correct.