Chapter 6 Exercises — The Nuclear Shell Model

Section A: Shell Model Basics and Magic Numbers (Problems 1-8)

Problem 1 — Identifying Magic Nuclei

For each of the following nuclei, state whether $Z$, $N$, both, or neither is a magic number. For those with at least one magic number, state which shell closure is involved.

(a) $^{40}$Ca (b) $^{56}$Fe (c) $^{90}$Zr (d) $^{132}$Sn (e) $^{88}$Sr (f) $^{48}$Ca (g) $^{100}$Sn (h) $^{208}$Pb (i) $^{27}$Al (j) $^{56}$Ni

Problem 2 — Separation Energy Discontinuities

The two-neutron separation energy is defined as $S_{2n}(Z,N) = B(Z,N) - B(Z, N-2)$.

(a) Explain why $S_{2n}$ is often preferred over $S_{n}$ for probing shell structure. (Hint: Consider the pairing interaction.)

(b) Use the following experimental binding energies (in MeV) for barium isotopes to compute $S_{2n}$ for $N = 78$ through $N = 86$:

(c) Plot $S_{2n}$ versus $N$ and identify the shell closure. What is the magnitude of the discontinuity?

(d) The "shell gap energy" can be estimated as $\Delta_{2n} = S_{2n}(N) - S_{2n}(N+2)$ evaluated at the magic number. Calculate this for $N = 82$ in barium.

Problem 3 — Harmonic Oscillator Level Counting

(a) Show that the degeneracy of the $N$-th harmonic oscillator shell is $g_N = (N+1)(N+2)$. (Hint: Sum $2(2\ell+1)$ over the allowed $\ell$ values for each $N$.)

(b) Verify that the total number of states up to and including the $N$-th shell is:

$$\sum_{k=0}^{N} g_k = \frac{(N+1)(N+2)(N+3)}{3}$$

(c) Evaluate this for $N = 0$ through $N = 6$ and compare with the harmonic oscillator magic numbers.

(d) Why does the $N = 3$ closure at 40 particles not correspond to an observed magic number?

Problem 4 — Parity from Orbital Angular Momentum

For each of the following single-particle orbits, state the parity $\pi = (-1)^\ell$ and the degeneracy $2j+1$:

(a) $1s_{1/2}$ (b) $1p_{3/2}$ (c) $1d_{5/2}$ (d) $1f_{7/2}$ (e) $1g_{9/2}$ (f) $2p_{1/2}$ (g) $1h_{11/2}$ (h) $1i_{13/2}$ (i) $2d_{3/2}$ (j) $3s_{1/2}$

Problem 5 — Filling the Shells

(a) Write out the complete shell model configuration (listing all filled orbits) for the neutrons in $^{16}$O.

(b) Repeat for the protons in $^{40}$Ca.

(c) Repeat for the neutrons in $^{56}$Ni. Include all orbits up to and including the last filled orbit.

(d) For $^{208}$Pb, list the proton orbits that fill the shell between $Z = 50$ and $Z = 82$, and the neutron orbits that fill the shell between $N = 82$ and $N = 126$.

Problem 6 — Spin-Orbit Splitting Magnitude

The spin-orbit splitting between the $j = \ell + 1/2$ and $j = \ell - 1/2$ partners is proportional to $(2\ell + 1)$.

(a) If the spin-orbit splitting of the $1p$ level ($\ell = 1$) is approximately 6 MeV, estimate the splitting of: - $1d$ ($\ell = 2$) - $1f$ ($\ell = 3$) - $1g$ ($\ell = 4$) - $1h$ ($\ell = 5$)

(b) Compare your $1f$ estimate with the observed splitting $\varepsilon(1f_{5/2}) - \varepsilon(1f_{7/2}) \approx 8$ MeV. Is the linear scaling a good approximation?

(c) Why does the increasing spin-orbit splitting for higher $\ell$ values ensure that the intruder orbital ($j = \ell + 1/2$ from the $N$-shell above) always drops into the lower shell for $\ell \geq 3$?

Problem 7 — Counting States Within a Shell

Verify the degeneracy count for the shell between magic numbers 50 and 82 by listing each orbit, its degeneracy $2j+1$, and the cumulative total:

Confirm that the total is $82 - 50 = 32$.

Problem 8 — The Harmonic Oscillator Parameter

(a) Using the empirical formula $\hbar\omega \approx 41 \, A^{-1/3}$ MeV, calculate $\hbar\omega$ for $^{16}$O, $^{40}$Ca, $^{132}$Sn, and $^{208}$Pb.

(b) For $^{208}$Pb, calculate the oscillator length parameter $b = \sqrt{\hbar/(m\omega)}$, where $m$ is the nucleon mass ($m \approx 939$ MeV/$c^2$). Express your answer in fm.

(c) Compare $b$ with the nuclear radius $R = r_0 A^{1/3}$ for $^{208}$Pb. Are they comparable? Should they be?

Section B: Ground-State Predictions (Problems 9-17)

Problem 9 — Predicting $J^{\pi}$ for Odd-$A$ Nuclei

Use the shell model filling order to predict the ground-state spin and parity $J^{\pi}$ for each of the following nuclei. Show your work by identifying the unpaired nucleon and its orbit.

(a) $^{5}$He ($Z=2$, $N=3$) (b) $^{13}$C ($Z=6$, $N=7$) (c) $^{23}$Na ($Z=11$, $N=12$) (d) $^{29}$Si ($Z=14$, $N=15$) (e) $^{33}$S ($Z=16$, $N=17$) (f) $^{39}$K ($Z=19$, $N=20$) (g) $^{51}$V ($Z=23$, $N=28$) (h) $^{87}$Sr ($Z=38$, $N=49$) (i) $^{91}$Zr ($Z=40$, $N=51$) (j) $^{141}$Pr ($Z=59$, $N=82$)

Compare your predictions with the experimental values (look these up in the NNDC Nuclear Data Sheets or the Evaluated Nuclear Structure Data File).

Problem 10 — Hole States

A "hole" in a closed shell has the same $j^{\pi}$ as the missing nucleon.

(a) Predict $J^{\pi}$ for $^{15}$O ($Z=8$, $N=7$). This is a neutron-hole state relative to $^{16}$O.

(b) Predict $J^{\pi}$ for $^{15}$N ($Z=7$, $N=8$). This is a proton-hole state relative to $^{16}$O.

(c) Predict $J^{\pi}$ for $^{207}$Tl ($Z=81$, $N=126$). This is a proton-hole state relative to $^{208}$Pb.

(d) Predict $J^{\pi}$ for $^{207}$Pb ($Z=82$, $N=125$). This is a neutron-hole state relative to $^{208}$Pb.

(e) The low-lying spectrum of $^{207}$Pb shows states at 0 MeV ($1/2^-$), 0.57 MeV ($5/2^-$), 0.90 MeV ($3/2^-$), 1.63 MeV ($13/2^+$), and 2.34 MeV ($7/2^-$). Identify the single-particle orbit corresponding to each state by interpreting these as neutron-hole states in the $N = 82$-$126$ shell.

Problem 11 — Particle States Beyond $^{208}$Pb

(a) Predict $J^{\pi}$ for $^{209}$Pb ($Z=82$, $N=127$) — one neutron beyond the $N=126$ closure. Which orbit does the extra neutron occupy?

(b) Predict $J^{\pi}$ for $^{209}$Bi ($Z=83$, $N=126$) — one proton beyond the $Z=82$ closure. Which orbit does the extra proton occupy?

(c) The low-lying spectrum of $^{209}$Bi shows states at 0 MeV ($9/2^-$), 0.90 MeV ($7/2^-$), 1.61 MeV ($13/2^+$), and 2.83 MeV ($3/2^-$). Interpret these as single-proton states in the $Z = 82$-$126$ shell. Which orbits do they correspond to?

Problem 12 — Even-Even Nuclei

Explain why every even-even nucleus has $J^{\pi} = 0^+$ in its ground state. Your explanation should address:

(a) Why the pairing interaction favors $J = 0$ coupling of identical nucleons.

(b) Why the parity must be positive even if the nucleons occupy negative-parity orbits.

(c) Whether there are any known exceptions to this rule. (There are not.)

Problem 13 — Mid-Shell Predictions and Their Limitations

(a) Predict $J^{\pi}$ for $^{93}$Nb ($Z=41$, $N=52$) using the naive shell model. The 41st proton goes into which orbit?

(b) The experimental $J^{\pi}$ is $9/2^+$. The naive prediction may or may not agree. If it does not, suggest a reason.

(c) Predict $J^{\pi}$ for $^{121}$Sb ($Z=51$, $N=70$). Compare with the experimental value $5/2^+$. Does this agree? Why is the single-particle picture more reliable when one nucleon species is near a magic number?

Problem 14 — Three Particles in $1f_{7/2}$

Consider a nucleus with 3 identical nucleons in the $1f_{7/2}$ orbit (e.g., the 3 neutrons above the $N = 20$ core in $^{43}$Ca).

(a) What is the maximum possible total angular momentum $J$ for three nucleons in $j = 7/2$, respecting the Pauli principle? (Hint: The maximum $M = m_1 + m_2 + m_3$ with all $m_i$ different.)

(b) The allowed $J$ values for three identical nucleons in $j = 7/2$ are $J = 3/2, 5/2, 7/2, 9/2, 11/2, 15/2$. Verify that $J = 13/2$ and $J = 1/2$ are not allowed.

(c) The ground state of $^{43}$Ca is $7/2^-$. Is this consistent with the shell model? What does it tell you about the pairing of two of the three neutrons?

Section C: Magnetic Moments — Schmidt Values (Problems 15-21)

Problem 15 — Deriving the Schmidt Lines

Starting from the general expression for the magnetic moment of a single nucleon in orbit $n\ell_j$:

$$\mu = \langle j, m_j = j | (g_\ell \ell_z + g_s s_z) | j, m_j = j \rangle \, \mu_N$$

(a) Use the projection theorem to show that:

$$\mu = \frac{1}{2} \left[ g_\ell + g_s + (g_\ell - g_s) \frac{\ell(\ell+1) - s(s+1)}{j(j+1)} \right] j \, \mu_N$$

(b) Evaluate this for $j = \ell + 1/2$ to obtain:

$$\mu = \left[ \left(j - \frac{1}{2}\right) g_\ell + \frac{1}{2} g_s \right] \mu_N$$

(c) Evaluate for $j = \ell - 1/2$ to obtain:

$$\mu = \frac{j}{j+1} \left[ \left(j + \frac{3}{2}\right) g_\ell - \frac{1}{2} g_s \right] \mu_N$$

Problem 16 — Computing Schmidt Values

Calculate the Schmidt magnetic moment (in units of $\mu_N$) for:

(a) An odd proton in $1d_{5/2}$ ($j = 5/2^+$) (b) An odd proton in $1d_{3/2}$ ($j = 3/2^+$) (c) An odd neutron in $1f_{7/2}$ ($j = 7/2^-$) (d) An odd neutron in $2p_{1/2}$ ($j = 1/2^-$) (e) An odd proton in $1g_{9/2}$ ($j = 9/2^+$) (f) An odd neutron in $1h_{11/2}$ ($j = 11/2^-$)

Use $g_s^p = 5.586$, $g_s^n = -3.826$, $g_\ell^p = 1$, $g_\ell^n = 0$.

Problem 17 — Comparison with Experiment

The following experimental magnetic moments are known:

(a) For each nucleus, identify the unpaired nucleon and its orbit. Calculate the Schmidt value.

(b) Compute the ratio $\mu_{\text{exp}} / \mu_{\text{Schmidt}}$ for each.

(c) Which nuclei agree best with the Schmidt prediction? Which deviate most? Discuss the pattern in terms of proximity to closed shells.

(d) Notice that $^{17}$O and $^{17}$F are mirror nuclei (same $A$, same $|T_z|$), as are $^{41}$Ca and $^{41}$Sc. Why are the magnetic moments of mirror pairs related? What does the relationship tell you about the role of orbital and spin contributions?

Problem 18 — Effective $g$-Factors

One approach to improving Schmidt predictions is to use effective $g$-factors that account for core polarization:

$$g_s^{\text{eff}} \approx 0.7 \, g_s^{\text{free}}$$

(a) Recalculate the magnetic moments for $^{209}$Bi and $^{133}$Cs using $g_s^{\text{eff}}$ instead of $g_s^{\text{free}}$. Compare with the experimental values.

(b) Does the quenching of $g_s$ improve agreement? Discuss physically why the spin $g$-factor is quenched in a nuclear medium.

Section D: Advanced Problems (Problems 19-25)

Problem 19 — The Woods-Saxon Potential

Consider the Woods-Saxon potential with $V_0 = 50$ MeV, $R = 1.25 A^{1/3}$ fm, and $a = 0.65$ fm for $^{208}$Pb.

(a) Calculate the potential depth at $r = 0$, $r = R$, $r = R + 2a$, and $r = R + 4a$. Sketch the potential.

(b) Estimate the number of bound states by comparing the well depth to $\hbar\omega \approx 41 A^{-1/3}$ MeV. How many major oscillator shells fit?

(c) The Coulomb potential energy of the 82nd proton at the nuclear center is approximately $V_C(0) = 3 Z e^2 / (8\pi \epsilon_0 R)$. Calculate this value using $e^2/(4\pi\epsilon_0) = 1.44$ MeV$\cdot$fm. How does it compare to $V_0$? What is the effective well depth for protons versus neutrons?

(d) The difference between proton and neutron potential depths means that proton and neutron single-particle energies are shifted relative to each other. Estimate the Coulomb shift at the nuclear surface ($r = R$). Is this shift significant compared to the spin-orbit splittings (typically 3-8 MeV)?

Problem 20 — Isospin and the Shell Model

(a) Explain why $^{48}$Ca (doubly magic, $Z = 20$, $N = 28$) is stable despite having 8 more neutrons than protons. (Hint: Consider both the shell closure and the energetics of beta decay — the $Q$-value for $\beta^-$ decay of $^{48}$Ca is negative.)

(b) The mirror nucleus $^{48}$Ni ($Z = 28$, $N = 20$) should also be doubly magic. However, it is highly unstable and was first observed only in 1999. Explain why the extra stability from shell closure is not enough to bind $^{48}$Ni against proton emission. Estimate the Coulomb energy difference between $^{48}$Ca and $^{48}$Ni using $E_C \approx \frac{3}{5} \frac{Z(Z-1)e^2}{4\pi\epsilon_0 R}$ and comment on the role of the Coulomb interaction in breaking the neutron-proton symmetry.

(c) $^{100}$Sn ($Z = 50$, $N = 50$) is the heaviest known doubly magic $N = Z$ nucleus. Its half-life is approximately 1.16 s, decaying by electron capture / positron emission to $^{100}$In. Why is $^{100}$Sn unstable despite being doubly magic? What does this tell you about the competition between shell closure effects and the symmetry energy?

(d) List all seven doubly magic nuclei discussed in this chapter and indicate which are stable. What trend do you observe regarding the stability of doubly magic nuclei as $A$ increases?

Problem 21 — Energy Levels of $^{17}$O

The low-lying energy spectrum of $^{17}$O shows the following states:

(a) Interpret the $5/2^+$ ground state and the $1/2^+$ state at 0.871 MeV as single-neutron states in the $sd$-shell. Which orbits do they correspond to?

(b) The negative-parity states ($1/2^-, 5/2^-, 3/2^-, 7/2^-$) cannot arise from the $sd$-shell. Propose an interpretation in terms of excitations from the $p$-shell or into the $fp$-shell.

(c) The $3/2^+$ state at 5.085 MeV is the $1d_{3/2}$ single-particle state. The gap between $1d_{5/2}$ (ground state) and $1d_{3/2}$ (5.085 MeV) gives a direct measurement of the $d$-orbit spin-orbit splitting. What is this splitting? Compare with the estimate from Problem 6.

Problem 22 — The Doubly Magic Oxygen-16

$^{16}$O is doubly magic ($Z = 8$, $N = 8$).

(a) Write the complete shell model configuration for protons and neutrons.

(b) What is the predicted $J^{\pi}$? Compare with experiment.

(c) The first excited state of $^{16}$O is at 6.05 MeV ($0^+$). This is not a single-particle excitation. Explain why a $0^+$ excited state requires at least a two-particle-two-hole (2p-2h) excitation.

(d) The second excited state is at 6.13 MeV ($3^-$). Explain how this state can arise from a one-particle-one-hole (1p-1h) excitation (a neutron promoted from $1p_{1/2}$ to $1d_{5/2}$, for example). Check the $J^{\pi}$ from the coupling.

Problem 23 — Semi-Magic Tin Isotopes

Tin ($Z = 50$) is a semi-magic element with an unusually large number of stable isotopes.

(a) List all stable tin isotopes and their neutron numbers. Verify that there are 10.

(b) The first excited $2^+$ energies for several tin isotopes are:

Explain the dramatic jump at $^{132}$Sn. Why does $E(2^+_1)$ drop sharply at $^{134}$Sn?

(c) $^{132}$Sn ($Z = 50$, $N = 82$) is doubly magic but radioactive (half-life 39.7 s). Why is it unstable despite being doubly magic?

Problem 24 — Shell Model with a Computer

Using the code provided in code/shell_model.py (or writing your own):

(a) Generate the energy level diagram for the harmonic oscillator ($N = 0$ through $N = 6$), labeling each sublevel with its spectroscopic notation and degeneracy. Verify that the cumulative particle numbers match the HO magic numbers (2, 8, 20, 40, 70, 112).

(b) Generate the three-panel comparison figure (HO, Woods-Saxon, WS+SO) and identify which orbits move between panels. In particular, trace the $1f_{7/2}$, $1g_{9/2}$, $1h_{11/2}$, and $1i_{13/2}$ orbits and show how each one creates a new magic number.

(c) Use the predict_jp function to predict $J^{\pi}$ for the following nuclei: $^{7}$Li, $^{11}$B, $^{27}$Al, $^{45}$Sc, $^{57}$Ni, $^{89}$Y, $^{137}$Ba, $^{209}$Bi. Compare with experimental values from the NNDC database. For any disagreements, discuss whether the nucleus is near or far from a magic number.

(d) Use the schmidt_moment function to compute Schmidt values for the nuclei in part (c) that have correct $J^{\pi}$ predictions. Plot $\mu_{\text{Schmidt}}$ vs $\mu_{\text{exp}}$ (look up experimental values). Is the correlation positive? Do the points cluster between the Schmidt lines or on them?

Problem 25 — The Island of Inversion: When Magic Numbers Fail

The conventional magic number $N = 20$ appears to break down in the neutron-rich region around $^{32}$Mg ($Z = 12$, $N = 20$).

(a) If $N = 20$ were a good magic number, what would you expect for $E(2^+_1)$ and $B(E2; 0^+ \to 2^+)$ in $^{32}$Mg compared to neighboring isotopes?

(b) Experimentally, $E(2^+_1) = 0.885$ MeV for $^{32}$Mg, and $B(E2) \approx 454 \, e^2 \text{fm}^4$. Compare these with $^{30}$Mg ($E(2^+_1) = 1.482$ MeV) and $^{34}$Mg ($E(2^+_1) = 0.660$ MeV). Is $N = 20$ behaving as a magic number in magnesium?

(c) The explanation involves "intruder states" from the $fp$-shell dropping below the $sd$-shell due to the monopole part of the tensor force. Qualitatively, explain why this shell inversion might occur when the proton $1d_{5/2}$ orbit empties (as $Z$ decreases from 20 toward 10). (Hint: Think about the proton-neutron interaction between specific orbits and its effect on the neutron single-particle energies.)

(d) This problem connects to Chapter 10 (Exotic Nuclei). List two other neutron-rich regions where conventional magic numbers are expected to weaken.

Problem 26 — The Fermi Gas Model and the Shell Model

The Fermi gas model (Chapter 4) treats nucleons as a non-interacting gas of fermions. The shell model also treats nucleons as non-interacting (in the mean-field approximation). What is the essential difference between the two models?

(a) In the Fermi gas model, the level density is treated as continuous. In the shell model, the levels are discrete. For which nuclei — light or heavy — is the Fermi gas approximation better justified? Explain.

(b) The average level spacing for a nucleus with $A$ nucleons in a harmonic oscillator potential is approximately $\hbar\omega / A^{1/3}$. Evaluate this for $^{16}$O and $^{208}$Pb. How does the spacing compare to typical excitation energies (1-10 MeV)?

(c) The Fermi gas model predicts a symmetry energy proportional to $(N-Z)^2 / A$. Show that this can be understood as the cost of filling neutron levels past the point where proton and neutron Fermi energies are equal. How does the shell model modify this picture at magic numbers?

Problem 27 — Pairing Energy and the Even-Odd Staggering

The binding energy of nuclei shows a characteristic even-odd staggering: even-$N$ nuclei are systematically more bound than odd-$N$ neighbors.

(a) Define the empirical pairing gap as:

$$\Delta_n(Z,N) = \frac{(-1)^{N+1}}{2} [S_n(Z,N+1) - S_n(Z,N)]$$

Using the following data for tin isotopes (all values in MeV):

compute $\Delta_n$ for $N = 63$ through $N = 69$.

(b) What is the average pairing gap? Compare with the empirical estimate $\Delta \approx 12 / \sqrt{A}$ MeV.

(c) Explain why the pairing interaction favors $J = 0$ coupling of identical nucleons in a single-$j$ orbit. (Hint: Consider the spatial overlap of the two-body wave function.)

Problem 28 — Comparing Shell Model Predictions Across the Chart

For each of the following nuclei, predict $J^{\pi}$ using the shell model and indicate how confident you are in the prediction (high confidence near closed shells, lower confidence far from them). Then look up the experimental value and assess the model's performance.

Which nuclei were predicted correctly? Is there a pattern relating accuracy to proximity to magic numbers?

Problem 29 — Nuclear Isomers and the Shell Model

A nuclear isomer is a long-lived excited state. Isomers tend to occur when the excited state and the available decay final state differ by a large $\Delta J$ (large spin change), which suppresses the electromagnetic transition rate.

(a) In $^{207}$Pb, the $13/2^+$ state at 1.633 MeV lies above the $3/2^-$ state at 0.898 MeV. If this $13/2^+$ state were to decay by $\gamma$-emission to the $3/2^-$ state, what is the minimum multipolarity? Is this E or M? (Use the selection rules from Chapter 5 or look ahead to Chapter 9.)

(b) The $13/2^+$ isomeric state in $^{207}$Pb corresponds to a hole in the $1i_{13/2}$ orbit. Explain why the shell model predicts that this state should be isomeric, based on the large $\Delta j$ between $1i_{13/2}$ and the nearby negative-parity orbits.

(c) The "islands of isomerism" on the nuclear chart cluster near magic numbers. Why does the shell model predict more isomers near closed shells? (Hint: Consider the spacing and $j$-values of orbits just above and below shell gaps.)

Solutions Notes

Solutions to selected problems (odd-numbered) are available in the instructor's supplement. Key computational results can be verified using the shell_model.py toolkit provided in the code/ directory.

For Problem 2, use the two-neutron separation energy $S_{2n}(Z,N) = B(Z,N) - B(Z,N-2)$; the shell gap manifests as a drop of approximately 6 MeV between $N = 82$ and $N = 84$.

For Problem 9, all nuclei listed can be predicted using the filling order in Section 6.6. The key is identifying the unpaired nucleon and its orbit. For nuclei with one species at or near a magic number, the predictions are typically correct. For mid-shell cases (like $^{25}$Mg), the naive prediction may fail due to configuration mixing.

For Problem 17, the mirror pair relationship $\mu(A, T_z) + \mu(A, -T_z) \approx (g_\ell^p + g_\ell^n) j = j$ (in $\mu_N$) provides a consistency check on the single-particle model.

For Problem 27, the average pairing gap in tin isotopes is approximately 1.1 MeV, consistent with $12/\sqrt{118} \approx 1.1$ MeV.