Chapter 6 Quiz — The Nuclear Shell Model

Instructions: Select the best answer for each question. Each question has exactly one correct answer.

Question 1

Which of the following is not a nuclear magic number?

(a) 2 (b) 20 (c) 40 (d) 82

Question 2

The evidence for nuclear shell closures includes all of the following except:

(a) Sharp drops in nucleon separation energies after magic numbers (b) High first-excited-state energies for magic nuclei (c) Enhanced beta-decay rates for magic nuclei (d) Large numbers of stable isotopes/isotones at magic numbers

Question 3

The harmonic oscillator potential gives correct magic numbers up to:

(a) 2 (b) 8 (c) 20 (d) 28

Question 4

The nuclear mean-field concept is physically justified primarily because:

(a) The nuclear force is weak enough that nucleons rarely interact (b) The Pauli exclusion principle blocks most nucleon-nucleon scattering inside the nucleus (c) Nucleons are confined to fixed positions in a crystal-like lattice (d) The nuclear force is exactly equivalent to a harmonic oscillator potential

Question 5

The Woods-Saxon potential differs from the harmonic oscillator potential in that it:

(a) Has a flat bottom and finite depth, going to zero outside the nucleus (b) Rises as $r^2$ for all $r$ (c) Is repulsive at the nuclear center (d) Has no dependence on the nuclear mass number $A$

Question 6

The crucial ingredient that produces the correct magic numbers 28, 50, 82, and 126 is:

(a) The Coulomb potential (b) The pairing interaction (c) The spin-orbit coupling $\boldsymbol{\ell} \cdot \mathbf{s}$ (d) The tensor force

Question 7

In the nuclear spin-orbit interaction, the level with $j = \ell + 1/2$ is:

(a) Pushed up in energy relative to $j = \ell - 1/2$ (b) Pushed down in energy relative to $j = \ell - 1/2$ (c) Degenerate with $j = \ell - 1/2$ (d) Unaffected by the spin-orbit coupling

Question 8

The magic number 28 arises because the spin-orbit interaction pushes which orbital down into the shell below?

(a) $1d_{5/2}$ (b) $1f_{7/2}$ (c) $1g_{9/2}$ (d) $2p_{3/2}$

Question 9

The degeneracy (number of nucleon states) of the $1f_{7/2}$ orbit is:

(a) 6 (b) 7 (c) 8 (d) 14

Question 10

The ground-state spin-parity of every known even-even nucleus is:

(a) $1^+$ (b) $0^+$ (c) $0^-$ (d) $2^+$

Question 11

The ground-state $J^{\pi}$ of $^{17}$O ($Z = 8$, $N = 9$) is predicted to be:

(a) $1/2^+$ (the 9th neutron enters $2s_{1/2}$) (b) $5/2^+$ (the 9th neutron enters $1d_{5/2}$) (c) $3/2^-$ (the 9th neutron enters $2p_{3/2}$) (d) $1/2^-$ (the 9th neutron enters $1p_{1/2}$)

Question 12

The Schmidt magnetic moment for an odd neutron in any $j = \ell + 1/2$ orbit is:

(a) $+1.913 \, \mu_N$ regardless of $j$ (b) $-1.913 \, \mu_N$ regardless of $j$ (c) Proportional to $j$ with a positive slope (d) Zero, because neutrons have no charge

Question 13

Which of the following nuclei is doubly magic?

(a) $^{56}$Fe ($Z = 26$, $N = 30$) (b) $^{90}$Zr ($Z = 40$, $N = 50$) (c) $^{132}$Sn ($Z = 50$, $N = 82$) (d) $^{197}$Au ($Z = 79$, $N = 118$)

Question 14

The nuclear spin-orbit coupling is different from the atomic spin-orbit coupling in that:

(a) The nuclear coupling is much weaker relative to level spacings (b) The nuclear coupling has the opposite sign (the $j = \ell + 1/2$ level is raised rather than lowered) (c) The nuclear coupling arises from the NN interaction and is much stronger relative to level spacings than the atomic (Thomas) coupling (d) The nuclear coupling only affects protons, not neutrons

Question 15

The shell model predictions for ground-state $J^{\pi}$ are most reliable for:

(a) Nuclei in the middle of a shell (many valence nucleons) (b) Nuclei near closed shells (few valence nucleons beyond a magic number) (c) Deformed nuclei in the rare-earth region (d) All nuclei equally, regardless of shell filling

Question 16

$^{208}$Pb has a first excited state at 2.61 MeV. For a typical non-magic heavy nucleus, the first excited state is near:

(a) 0.05 MeV (b) 0.1 - 0.5 MeV (c) 5 - 10 MeV (d) 50 MeV

Question 17

The parity of a single-particle state with orbital angular momentum $\ell$ is:

(a) Always positive (b) Always negative (c) $(-1)^{\ell}$ (d) $(-1)^{j}$

Question 18

Why does the simple shell model fail for nuclei far from closed shells?

(a) The Pauli exclusion principle no longer applies (b) The nuclear force changes character in mid-shell nuclei (c) Many valence nucleons interact through the residual interaction, causing configuration mixing and deformation that the single-particle picture misses (d) The spin-orbit coupling reverses sign for mid-shell nuclei

Question 19

The empirical formula for the harmonic oscillator level spacing is $\hbar\omega \approx 41 \, A^{-1/3}$ MeV. For $^{208}$Pb, this gives approximately:

(a) 3 MeV (b) 7 MeV (c) 14 MeV (d) 41 MeV

Question 20

Which of the following correctly describes the "intruder orbit" mechanism that creates the magic number 50?

(a) The $2d_{5/2}$ orbit drops from the $N = 4$ shell into the $N = 3$ shell (b) The $1g_{9/2}$ orbit drops from the $N = 4$ shell to join the orbits between magic numbers 28 and 50 (c) The $1f_{7/2}$ orbit is pushed up from the $N = 3$ shell into the $N = 4$ shell (d) The $1h_{11/2}$ orbit drops from the $N = 5$ shell into the $N = 4$ shell

Answer Key

Q Answer Brief Explanation
1 (c) 40 is the HO shell closure, not a magic number. The magic numbers are 2, 8, 20, 28, 50, 82, 126.
2 (c) Beta-decay rates are not particularly enhanced at magic numbers. Separation energies, excited state energies, and isotope counts all show magic signatures.
3 (c) The HO gives 2, 8, 20 correctly, but predicts 40, 70, 112... instead of 28, 50, 82, 126.
4 (b) Pauli blocking of final states suppresses NN scattering, giving nucleons long mean free paths.
5 (a) The WS has a flat interior (constant density) and falls to zero at the nuclear surface, unlike the HO.
6 (c) The spin-orbit $\boldsymbol{\ell} \cdot \mathbf{s}$ term splits levels and creates the correct shell gaps.
7 (b) The $j = \ell + 1/2$ (stretched) state is lowered in energy by the attractive spin-orbit interaction.
8 (b) The $1f_{7/2}$ ($j = 3 + 1/2$) is pushed down from the $N=3$ shell, closing at 20 + 8 = 28.
9 (c) Degeneracy = $2j + 1 = 2(7/2) + 1 = 8$.
10 (b) Pairing drives all nucleon pairs to $J = 0$; combined positive parity gives $0^+$. No exceptions.
11 (b) 8 neutrons fill through $1p_{1/2}$; the 9th enters $1d_{5/2}$, giving $J^\pi = 5/2^+$.
12 (b) For odd neutrons with $g_\ell = 0$ and $j = \ell + 1/2$: $\mu = (1/2) g_s^n = (1/2)(-3.826) = -1.913 \, \mu_N$.
13 (c) $^{132}$Sn has $Z = 50$ and $N = 82$, both magic.
14 (c) Nuclear $\boldsymbol{\ell} \cdot \mathbf{s}$ arises from the NN force, is ~$10^3$ times stronger (relative to level spacing) than atomic.
15 (b) Near closed shells, only one or two valence nucleons matter, and the single-particle picture works best.
16 (b) Typical non-magic heavy nuclei have $E(2^+_1) \sim 0.1$-$0.5$ MeV. $^{208}$Pb's 2.61 MeV is anomalously high.
17 (c) Parity $\pi = (-1)^\ell$: even $\ell$ gives positive parity, odd $\ell$ gives negative parity.
18 (c) Multiple valence nucleons bring configuration mixing, deformation, and collective effects beyond single-particle motion.
19 (b) $\hbar\omega \approx 41 \times 208^{-1/3} = 41/5.93 \approx 6.9 \approx 7$ MeV.
20 (b) The $1g_{9/2}$ ($j = 4+1/2$, $\ell=4$) is pushed down from $N=4$ by spin-orbit to join the shell filling between 28 and 50.