Chapter 7 Quiz: Beyond the Single Particle — Residual Interactions and Nuclear Correlations

Instructions: Select the best answer for each question. Some questions may have more than one correct answer where indicated.

Q1. The universal $J^\pi = 0^+$ ground state of even-even nuclei is a consequence of:

(a) The Pauli exclusion principle alone (b) The long-range Coulomb repulsion between protons (c) The short-range attractive nucleon-nucleon force preferentially coupling pairs to $J = 0$ (d) Angular momentum conservation requiring the total to vanish

Q2. The empirical pairing gap formula $\Delta \approx 12/\sqrt{A}$ MeV predicts a pairing gap for $^{144}$Sm ($A = 144$) of approximately:

(a) 0.5 MeV (b) 1.0 MeV (c) 1.5 MeV (d) 2.0 MeV

Q3. In the BCS pairing model, the occupation probability $v_k^2$ of a single-particle level well below the Fermi energy ($\epsilon_k \ll \lambda$) is approximately:

(a) 0 (b) 0.5 (c) 1 (d) $\Delta / (\epsilon_k - \lambda)$

Q4. The seniority quantum number $\nu$ represents:

(a) The total number of nucleons in a shell (b) The number of nucleons coupled in $J = 0$ pairs (c) The number of nucleons NOT coupled in $J = 0$ pairs (d) The number of available orbits in the shell

Q5. In the seniority scheme for identical nucleons in a $j$-shell, the $B(E2; 0^+ \to 2^+)$ follows what dependence on the number of valence nucleons $n$?

(a) Linear increase with $n$ (b) Constant across the shell (c) Parabolic, peaking at mid-shell (d) Exponential decrease with $n$

Q6. Two identical nucleons in a $j = 7/2$ shell can couple to which total angular momentum values?

(a) $J = 0, 1, 2, 3, 4, 5, 6, 7$ (b) $J = 0, 2, 4, 6$ (c) $J = 1, 3, 5, 7$ (d) $J = 0, 7$

Q7. In the two-neutron spectrum of $^{210}$Pb (two neutrons outside the $^{208}$Pb core), the $0^+$ state is observed to lie significantly below the other members of the $(g_{9/2})^2$ multiplet. This is primarily due to:

(a) The Coulomb interaction (b) The spin-orbit force (c) The short-range pairing component of the residual interaction (d) Three-body forces

Q8. The residual interaction $V_{\text{res}}$ is defined as:

(a) The total nucleon-nucleon interaction (b) The mean-field potential (c) The difference between the full two-body interaction and the mean-field potential (d) The spin-orbit potential

Q9. Configuration mixing means that the true nuclear ground state is:

(a) A single Slater determinant (b) A superposition of many shell-model configurations (c) Always dominated by a single configuration with no admixtures (d) Degenerate with the first excited state

Q10. The dimension of the shell-model Hamiltonian matrix for mid-$pf$-shell nuclei (e.g., $^{56}$Ni) is approximately:

(a) $10^3$ (b) $10^6$ (c) $10^9$ (d) $10^{15}$

Q11. Which computational method is most commonly used to extract the lowest eigenvalues of large shell-model matrices without full diagonalization?

(a) Gaussian elimination (b) The Lanczos iterative algorithm (c) Monte Carlo integration (d) Fourier transform methods

Q12. A nuclear isomer is best defined as:

(a) A nucleus with the same mass number but different proton number (b) A metastable excited state with an unusually long half-life (c) A nucleus at the ground state (d) A nucleus with a permanent electric dipole moment

Q13. The medical imaging isotope $^{99m}$Tc is isomeric because:

(a) It is a shape isomer with a different deformation from the ground state (b) Its decay requires a large change in angular momentum ($\Delta J = 4$, $M4$ transition) (c) It is stabilized by the Coulomb barrier (d) It has a $K$-forbidden transition

Q14. The $K^\pi = 16^+$ isomer in $^{178}$Hf stores an excitation energy of 2.446 MeV per nucleus. This isomer survives for 31 years because:

(a) The excitation energy is below the neutron separation threshold (b) The decay is $K$-forbidden with $|\Delta K| - \lambda \gg 0$ (c) The Coulomb barrier prevents alpha emission (d) The nucleus is doubly magic

Q15. In the Nilsson model, the good quantum number that replaces $j$ for a deformed nucleus is:

(a) The principal quantum number $N$ (b) The orbital angular momentum projection $\Lambda$ (c) The total angular momentum projection on the symmetry axis, $\Omega$ (d) The spin projection $\Sigma$

Q16. For a prolate (cigar-shaped) deformation, orbits with large values of $n_z$ (many quanta along the symmetry axis):

(a) Increase in energy (b) Decrease in energy (c) Are unaffected (d) Become degenerate

Q17. The Nilsson quantum number labeling $\Omega^\pi[Nn_z\Lambda]$ for the orbital $7/2^-[514]$ indicates:

(a) $\Omega = 7/2$, positive parity, $N = 5$, $n_z = 1$, $\Lambda = 4$ (b) $\Omega = 7/2$, negative parity, $N = 5$, $n_z = 1$, $\Lambda = 4$ (c) $\Omega = 7/2$, negative parity, $N = 1$, $n_z = 5$, $\Lambda = 4$ (d) $\Omega = 5$, negative parity, $N = 7/2$, $n_z = 1$, $\Lambda = 4$

Q18. Particle-vibration coupling in nuclei leads to:

(a) Fragmentation of single-particle strength over several states (b) Enhancement of the pairing gap (c) Vanishing of the spin-orbit splitting (d) Perfect spherical symmetry

Q19. Which statement about the hierarchy of nuclear models is correct?

(a) The liquid drop model is more accurate than the shell model for all observables (b) The independent-particle shell model already includes pairing correlations (c) The interacting shell model includes configuration mixing and can describe detailed spectroscopy (d) The Nilsson model is only valid for spherical nuclei

Q20. The "islands of isomerism" on the chart of nuclides are located:

(a) Near the neutron drip line (b) Near doubly-magic nuclei, where high-$j$ intruder orbits create large spin differences (c) Only in the lightest nuclei ($A < 20$) (d) Uniformly distributed across all mass numbers

Answer Key

Q Answer Brief Explanation
1 (c) The short-range $NN$ force maximizes overlap for time-reversed pairs, giving the $J=0$ state the most binding energy.
2 (b) $12/\sqrt{144} = 12/12 = 1.0$ MeV.
3 (c) Far below the Fermi energy, the level is nearly fully occupied: $v_k^2 \to 1$.
4 (c) Seniority counts unpaired nucleons — those not in $J = 0$ pairs.
5 (c) The seniority prediction is $B(E2) \propto n(\Omega - n + 1)$, a parabola peaking at mid-shell.
6 (b) Identical fermions in the same $j$-orbit: Pauli principle requires antisymmetric spatial wave function, allowing only even $J$.
7 (c) The short-range residual interaction gives the strongest matrix element at $J = 0$.
8 (c) $V_{\text{res}} = \sum V_{ij} - \sum U_i$, the difference between the full interaction and the mean field.
9 (b) Configuration mixing means the true state is $|\Psi\rangle = \sum c_\alpha |\Phi_\alpha\rangle$, a superposition.
10 (c) $^{56}$Ni in the full $pf$-shell has matrix dimensions $\sim 10^9$.
11 (b) The Lanczos algorithm iteratively extracts extremal eigenvalues without storing or diagonalizing the full matrix.
12 (b) A nuclear isomer is a metastable excited state, long-lived due to hindrance of electromagnetic decay.
13 (b) The $1/2^- \to 9/2^+$ transition requires $\Delta J = 4$ with parity change, an $M4$ transition.
14 (b) The $K$-forbiddenness ($\nu = |\Delta K| - \lambda = 14$ for the $E2$ step) causes extreme hindrance.
15 (c) In axial symmetry, $\Omega$ (projection of $\mathbf{j}$ on the symmetry axis) is conserved.
16 (b) Prolate deformation means $\omega_z < \omega_\perp$; orbits elongated along $z$ (large $n_z$) have lower energy.
17 (b) $\Omega = 7/2$, $\pi = (-)$, $N = 5$, $n_z = 1$, $\Lambda = 4$. Negative parity because $\ell$ must include an odd-$\ell$ component for $N = 5$.
18 (a) Coupling to phonons distributes the single-particle strength across multiple nuclear states.
19 (c) The interacting shell model diagonalizes $V_{\text{res}}$ in the valence space, providing detailed spectroscopic predictions.
20 (b) Isomers cluster near magic numbers where high-$j$ intruder orbits create large angular momentum differences.