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

Section A: Pairing Interaction and the Pairing Gap

Exercise 7.1 — Pairing Gap from Mass Data (Computational)

Using the Atomic Mass Evaluation (AME2020) data, compute the three-point pairing gap $\Delta^{(3)}(N)$ for the tin isotopes ($Z = 50$) from $^{100}$Sn to $^{134}$Sn.

(a) Plot $\Delta^{(3)}(N)$ as a function of neutron number $N$. Verify the odd-even staggering.

(b) Compute the average pairing gap $\bar{\Delta}$ for even-$N$ tin isotopes in the range $52 \leq N \leq 82$ (between the two magic numbers). Compare with the formula $\Delta = 12/\sqrt{A}$ MeV.

(c) What happens to the pairing gap at $N = 50$ and $N = 82$? Explain in terms of shell structure.

(d) Repeat using the five-point formula $\Delta^{(5)}(N)$. How does the result compare?

Exercise 7.2 — Pairing Energy of the Delta Interaction

Consider two identical nucleons in a $j = 7/2$ orbit interacting through a surface delta interaction $V = -4\pi V_0 \delta(\cos\theta_1 - \cos\theta_2)\delta(\phi_1 - \phi_2) \delta(r_1 - R) \delta(r_2 - R)$.

(a) Show that the matrix elements for the allowed $J$ values ($J = 0, 2, 4, 6$) are proportional to:

$$\langle (7/2)^2; J | V | (7/2)^2; J \rangle \propto (2J+1) \begin{pmatrix} 7/2 & 7/2 & J \\ 1/2 & -1/2 & 0 \end{pmatrix}^2$$

(b) Evaluate the $3j$-symbol numerically for each $J$. Verify that $J = 0$ has by far the largest matrix element.

(c) Define the ratio $R_J = \langle J | V | J \rangle / \langle J=0 | V | J=0 \rangle$. Compute $R_J$ for $J = 2, 4, 6$.

(d) How does this result depend on $j$? Repeat the calculation for $j = 11/2$ and compare.

Exercise 7.3 — BCS Gap Equation

Consider a schematic model with $\Omega = 10$ doubly-degenerate levels uniformly spaced by $d = 1$ MeV, centered on the Fermi energy (i.e., single-particle energies $\epsilon_k = (k - 5.5)d$ for $k = 1, \ldots, 10$). Assume half-filling ($n = 10$ particles).

(a) Write down the BCS gap equation for this system with pairing strength $G$.

(b) Solve the gap equation numerically (or graphically) for $G/d = 0.3$, $0.5$, and $1.0$. Find $\Delta$ in each case.

(c) Plot the occupation probabilities $v_k^2$ as a function of $\epsilon_k$ for each value of $G/d$. Compare with the step-function Fermi distribution.

(d) At what critical value $(G/d)_{\text{crit}}$ does the pairing gap vanish? What is the physical significance of this transition?

Exercise 7.4 — BCS Occupation Probabilities

For a nucleus with $\Omega = 8$ levels uniformly spaced by $d = 1$ MeV, centered on the Fermi energy, and a pairing gap $\Delta = 1.2$ MeV:

(a) Compute the BCS occupation probability $v_k^2$ for each level using:

$$v_k^2 = \frac{1}{2}\left(1 - \frac{\epsilon_k - \lambda}{\sqrt{(\epsilon_k - \lambda)^2 + \Delta^2}}\right)$$

with $\lambda = 0$ (at the center of the level scheme) and $\epsilon_k = (k - 4.5) \times 1$ MeV for $k = 1, \ldots, 8$.

(b) Compare $v_k^2$ with the step-function distribution of the independent-particle model (all levels below $\lambda$ fully occupied, all above empty). Sketch both on the same graph.

(c) Compute the average particle number $\langle N \rangle = 2 \sum_{k=1}^{8} v_k^2$. Does it equal 8 (half-filling)? If not, what adjustment to $\lambda$ would be needed?

(d) Compute the quasiparticle energies $E_k = \sqrt{(\epsilon_k - \lambda)^2 + \Delta^2}$ for each level. What is the minimum quasiparticle energy? Where does it occur?

(e) What is the energy cost to create the lowest two-quasiparticle excitation? How does this compare to the lowest particle-hole excitation energy in the independent-particle model?

Exercise 7.5 — Even-Odd Mass Staggering

The binding energies of the calcium isotopes $^{40}$Ca through $^{48}$Ca are:

(a) Compute $S_n(N)$, the one-neutron separation energy, for $N = 21$-$28$.

(b) Plot $S_n$ vs $N$. Identify the odd-even staggering.

(c) Extract the neutron pairing gap $\Delta_n$ for this region. Compare with $12/\sqrt{A}$.

(d) What is special about $S_n$ at $N = 28$? Why?

Section B: Seniority and Two-Particle Configurations

Exercise 7.5 — Seniority Classification

For $n$ identical nucleons in a $j = 9/2$ shell ($\Omega = 5$):

(a) List all allowed seniority values $\nu$ and their corresponding angular momenta $J$ for $n = 2$.

(b) How many states exist for $n = 4$ with $\nu = 0$, $\nu = 2$, and $\nu = 4$? What are the allowed $J$ values in each case?

(c) Using the pairing energy formula $E(n, \nu) = -\frac{G}{4}(n - \nu)(\Omega - n - \nu + 2) + E_\nu$, calculate the ground-state pairing energy for $n = 2, 4, 6, 8$ (all with $\nu = 0$). Plot the pairing energy as a function of $n$.

(d) Show that the energy of the first excited state ($\nu = 2$) relative to the ground state ($\nu = 0$) is independent of $n$. What is this excitation energy in terms of $G$ and $\Omega$?

Exercise 7.6 — $B(E2)$ Parabola in the Tin Isotopes

The seniority model predicts $B(E2; 0^+ \to 2^+_1) \propto n(\Omega - n + 1)$ for the even tin isotopes filling the $N = 50$-$82$ shell.

(a) Calculate the predicted relative $B(E2)$ values for $n = 2, 4, 6, \ldots, 32$ (corresponding to $^{102}$Sn through $^{132}$Sn). Normalize to the $n = 16$ value.

(b) The experimental $B(E2; 0^+ \to 2^+_1)$ values (in $e^2 \text{fm}^4$) are approximately: $^{104}$Sn: 1050, $^{108}$Sn: 1620, $^{112}$Sn: 2410, $^{114}$Sn: 2430, $^{116}$Sn: 2090, $^{118}$Sn: 2050, $^{120}$Sn: 2000, $^{122}$Sn: 1960, $^{124}$Sn: 1600, $^{130}$Sn: 300. Plot the experimental values (with the available data) alongside your seniority prediction.

(c) Where does the seniority model work best? Where does it fail? What physical effects cause the deviations?

Exercise 7.7 — Two-Neutron Spectrum of $^{210}$Pb

Two neutrons outside the $^{208}$Pb core occupy the $g_{9/2}$ orbit.

(a) What values of $J$ are allowed by the Pauli principle? Explain why odd-$J$ states are forbidden.

(b) The experimental energies of the $(g_{9/2})^2$ multiplet are: $0^+$ (0.000 MeV), $2^+$ (0.800), $4^+$ (1.098), $6^+$ (1.195), $8^+$ (1.278). Verify that the $J = 0$ state is anomalously depressed compared to the higher-$J$ states.

(c) Assuming a surface delta interaction, the energy splitting is proportional to $(2J+1) \begin{pmatrix} j & j & J \\ 1/2 & -1/2 & 0 \end{pmatrix}^2$. Using $j = 9/2$, compute the predicted ratio $[E(J) - E(8^+)] / [E(0^+) - E(8^+)]$ for each $J$ and compare with experiment.

(d) Now consider the mixed configuration. The $0^+$ state of $^{210}$Pb is not pure $(g_{9/2})^2$ but also contains admixtures of $(i_{11/2})^2$, $(j_{15/2})^2$, etc. If the residual interaction matrix element connecting $(g_{9/2})^2$ and $(i_{11/2})^2$ at $J = 0$ is $V_{\text{mix}} = -0.3$ MeV, and the unperturbed energy difference is $\Delta E = 2 \times 0.779 = 1.558$ MeV, estimate the mixing amplitude and the energy shift using first-order perturbation theory.

Exercise 7.8 — Particle-Hole Symmetry

(a) Show that the two-body matrix elements for two particles in orbit $j$ are the same as those for two holes:

$$\langle j^{-2}; J | V | j^{-2}; J \rangle = \langle j^2; J | V | j^2; J \rangle$$

(Hint: Use the particle-hole transformation and the properties of the interaction under time reversal.)

(b) Compare the experimental spectra of $^{210}$Pb (two neutrons above $N = 126$) and $^{206}$Pb (two neutron holes in $N = 126$). The $(p_{1/2})^{-2}$ multiplet in $^{206}$Pb has states at: $0^+$ (0.000), with the first excited $0^+$ at 1.166 MeV and $2^+$ at 0.803 MeV. How does this compare to the $^{210}$Pb spectrum? Discuss the similarities and differences.

Section C: Configuration Mixing and the Interacting Shell Model

Exercise 7.9 — Configuration Interaction for Two Levels

Consider a simple two-level mixing problem: two configurations $|\Phi_1\rangle$ and $|\Phi_2\rangle$ with unperturbed energies $E_1$ and $E_2$ ($E_1 < E_2$), connected by a residual interaction $V_{12} = \langle \Phi_1 | V | \Phi_2 \rangle$.

(a) Write down the $2 \times 2$ Hamiltonian matrix and solve for the eigenvalues $E_\pm$.

(b) Show that the energy shift of the lower level is always downward (increased binding), regardless of the sign of $V_{12}$.

(c) Show that the level repulsion $E_+ - E_-$ is always greater than or equal to $|E_2 - E_1|$.

(d) For $|V_{12}| = 0.5$ MeV and $E_2 - E_1 = 2.0$ MeV, compute the eigenvalues and the mixing amplitudes $c_1$, $c_2$. What fraction of the ground-state wave function is $|\Phi_2\rangle$?

(e) Repeat for $E_2 - E_1 = 0.5$ MeV (near-degenerate case). How does strong mixing change the picture?

Exercise 7.10 — Monopole Interaction and Shell Evolution

The monopole component of the residual interaction shifts single-particle energies as orbits fill. The monopole matrix element is defined as:

$$\bar{V}_{j_1 j_2} = \frac{\sum_J (2J+1) \langle j_1 j_2; J T | V | j_1 j_2; J T \rangle}{\sum_J (2J+1)}$$

(a) Show that the effective single-particle energy of orbit $j_1$ shifts as orbit $j_2$ fills:

$$\epsilon_{j_1}^{\text{eff}} = \epsilon_{j_1}^{(0)} + n_{j_2} \bar{V}_{j_1 j_2}$$

where $n_{j_2}$ is the number of nucleons in orbit $j_2$.

(b) If $\bar{V}_{\pi(d_{5/2}), \nu(d_{3/2})} = -0.8$ MeV, how much does the proton $d_{5/2}$ level shift when the neutron $d_{3/2}$ orbit fills from 0 to 4 neutrons? In which direction?

(c) This monopole shift is responsible for the "collapse of the $N = 20$ shell gap" in neutron-rich nuclei (the island of inversion, Chapter 10). Explain qualitatively how filling neutrons in the $d_{3/2}$ orbit can reduce the gap between the proton $d_{5/2}$ and $s_{1/2}$ orbits.

Exercise 7.11 — Shell-Model Dimensions

(a) Calculate the $m$-scheme dimension for $^{28}$Si (6 protons and 6 neutrons in the $sd$-shell: $d_{5/2}$, $s_{1/2}$, $d_{3/2}$ with $\Omega_\pi = \Omega_\nu = 12$).

(b) Calculate the dimension for $^{48}$Cr (4 protons and 4 neutrons in the $pf$-shell: $f_{7/2}$, $p_{3/2}$, $f_{5/2}$, $p_{1/2}$ with $\Omega_\pi = \Omega_\nu = 20$).

(c) Estimate the memory required to store one eigenvector for $^{48}$Cr in double precision (8 bytes per component). Is this feasible on a modern computer with 1 TB of RAM?

(d) For $^{60}$Zn (10 protons and 10 neutrons in the $pf$-shell), estimate the dimension and the memory requirement. What does this tell you about the need for truncation schemes?

Exercise 7.12 — Effective Charges

In the shell model, electromagnetic transitions are computed using effective charges $e_p^{\text{eff}}$ and $e_n^{\text{eff}}$ that differ from the bare proton charge $e$ and neutron charge 0. Typical values are $e_p^{\text{eff}} \approx 1.5e$ and $e_n^{\text{eff}} \approx 0.5e$.

(a) Why are effective charges necessary? What physics do they capture that is missing from the valence-space shell model?

(b) The $B(E2; 0^+ \to 2^+_1)$ for $^{18}$O (two neutrons in the $sd$-shell) is measured to be $\approx 46 \, e^2 \text{fm}^4$. If the dominant configuration is $(d_{5/2})^2$, estimate $B(E2)$ using $e_n^{\text{eff}} = 0$ (bare charge) and $e_n^{\text{eff}} = 0.5e$. Which agrees better with experiment?

(c) Discuss the physical origin of the neutron effective charge. How is it related to core polarization?

Section D: Nuclear Isomers

Exercise 7.13 — Weisskopf Estimates for Isomeric Transitions

The isomeric state in $^{99m}$Tc decays by a 140.5 keV $M4$ transition from $1/2^-$ to $9/2^+$.

(a) Compute the Weisskopf single-particle estimate for the $M4$ half-life of this transition, using $R = 1.2 A^{1/3}$ fm. The Weisskopf estimate for $M\lambda$ transitions is:

$$T_W(M\lambda) = \frac{3.5 \times 10^{21}}{\hbar} \frac{10}{(\lambda+3)^2} \left(\frac{3}{\lambda+3}\right)^2 \left(\frac{E_\gamma}{\hbar c}\right)^{2\lambda+1} R^{2\lambda-2} \text{ s}^{-1}$$

Use this or an equivalent formulation from your reference to compute $t_{1/2}^W$.

(b) The experimental half-life is 6.01 hours. Calculate the hindrance factor $F_W = t_{1/2}^{\text{exp}} / t_{1/2}^W$.

(c) Discuss why the $M4$ transition dominates over the (in principle lower-multipolarity) $E3$ transition to the $7/2^+$ excited state at 181 keV. Consider the transition energy, the matrix element structure, and the branching ratios.

(d) What would happen to the half-life if the transition energy were doubled (from 140 keV to 280 keV)? How sensitive is the isomeric lifetime to the transition energy for high-multipolarity transitions?

Exercise 7.14 — $K$-Isomers

The $K^\pi = 16^+$ isomer in $^{178}$Hf has $E^* = 2.446$ MeV and $t_{1/2} = 31$ years.

(a) The isomer decays primarily through a cascade, but the first step involves a transition with $\Delta K = 8$ and multipolarity $E2$. Calculate the $K$-forbiddenness $\nu = |\Delta K| - \lambda$. How many orders of magnitude of hindrance does this imply if each degree of $K$-forbiddenness contributes roughly a factor of 100?

(b) Estimate the total energy stored in 1 gram of pure $^{178m2}$Hf, in joules. Compare with the chemical energy in 1 gram of TNT ($\approx 4200$ J).

(c) Why is the "hafnium bomb" concept considered unphysical? What would be required to release the stored energy rapidly?

Exercise 7.15 — $^{180m}$Ta: Nature's Most Stable Isomer

(a) The isomeric state $^{180m}$Ta has $K^\pi = 9^-$ at 77 keV, while the ground state is $1^+$. What is the $K$-forbiddenness for an electromagnetic transition connecting them?

(b) The ground state has $t_{1/2} = 8.15$ hours and decays by electron capture and $\beta^-$. The isomer has $t_{1/2} > 1.2 \times 10^{15}$ years. Explain why the isomer does not simply decay through the ground state.

(c) What fraction of natural tantalum is $^{180m}$Ta? (Answer: 0.0120%) Why is its continued existence after 4.6 billion years of solar system history remarkable?

(d) Discuss the proposed astrophysical production mechanism: the $\nu$-process in core-collapse supernovae, where neutrino-induced excitation of $^{180}$Hf produces $^{180}$Ta in the isomeric state via the $^{180}$Hf$(\nu_e, e^-)^{180m}$Ta reaction.

Section E: The Nilsson Model

Exercise 7.16 — Nilsson Level Splittings

Consider the $1d_{5/2}$ level ($\ell = 2$, $j = 5/2$) in a deformed harmonic oscillator potential.

(a) List all possible $\Omega$ values for this orbit. Which $\Omega$ values have which parities?

(b) For prolate deformation, which $\Omega$ component is lowest in energy? Explain in terms of the classical orbit orientation relative to the symmetry axis.

(c) Using the first-order perturbation theory result for a quadrupole deformation $\epsilon$:

$$\Delta E(\Omega) \propto -\epsilon \left[ 3\Omega^2 - j(j+1) \right]$$

compute the energy splitting of the five $\Omega$ substates for $j = 5/2$ at $\epsilon = 0.3$. Verify that the center of gravity is preserved.

Exercise 7.17 — Nilsson Asymptotic Quantum Numbers

For the Nilsson orbital $3/2^+[411]$:

(a) Identify the values of $\Omega$, parity $\pi$, $N$, $n_z$, and $\Lambda$.

(b) Calculate $n_\perp = N - n_z$ and verify that $\Omega = \Lambda + \Sigma$ with $\Sigma = \pm 1/2$.

(c) From which spherical orbit does this level originate? (Hint: $N = 4$ and positive parity imply $\ell$ is even.)

(d) At what deformation does this level cross the $1/2^+[420]$ orbital? Sketch the level diagram qualitatively.

Exercise 7.18 — Ground-State Predictions of Deformed Nuclei

Using a Nilsson diagram (available in standard nuclear data tables or textbooks):

(a) Predict the ground-state spin and parity of $^{173}$Yb ($Z = 70$, $N = 103$). The equilibrium deformation is $\epsilon \approx 0.30$. Identify the Nilsson orbital of the unpaired neutron. Compare with the experimental value $J^\pi = 5/2^-$.

(b) Predict $J^\pi$ for $^{153}$Eu ($Z = 63$, $N = 90$). The deformation is $\epsilon \approx 0.32$. Compare with experiment ($J^\pi = 5/2^+$).

(c) Explain why the spherical shell model would fail for these nuclei. What spin and parity would the spherical model predict, and why is it wrong?

Exercise 7.19 — Volume Conservation in the Deformed Oscillator

The Nilsson model parameterizes the deformed potential using frequencies $\omega_z$ and $\omega_\perp$ with the constraint of volume conservation: $\omega_\perp^2 \omega_z = \omega_0^3$.

(a) Show that this constraint follows from requiring the volume enclosed by the equipotential surface $V = \text{const}$ to be independent of deformation.

(b) Express $\omega_z$ and $\omega_\perp$ in terms of $\omega_0$ and the deformation parameter $\epsilon$, defined by $\omega_z = \omega_0(1 - 2\epsilon/3)$ and $\omega_\perp = \omega_0(1 + \epsilon/3)$. Verify that the volume-conservation condition is satisfied to second order in $\epsilon$.

(c) For a prolate deformation of $\epsilon = 0.3$, compute the axis ratio $\omega_\perp / \omega_z$ and the corresponding ratio of semi-axes $c/a$ (where $c$ is the symmetry axis and $a$ is the perpendicular axis).

Section F: Synthesis and Advanced Problems

Exercise 7.20 — From Pairing to Superconductivity (Essay)

Write a short essay (1-2 pages) comparing nuclear pairing with superconductivity in metals. Address the following points:

(a) What plays the role of the Debye frequency cutoff in the nuclear case?

(b) In metals, the number of Cooper pairs is $\sim 10^{20}$; in nuclei, it is $\sim 5$–$15$. What are the consequences of this difference for the validity of the BCS approximation?

(c) What is the nuclear analog of the Meissner effect (expulsion of magnetic flux)?

(d) Why is particle-number projection important in the nuclear case but not in condensed matter?

Exercise 7.21 — Moment of Inertia Reduction

The rigid-body moment of inertia of a nucleus is $\mathcal{J}_{\text{rigid}} = \frac{2}{5} m_N A R^2 (1 + 0.31 \beta_2)$, where $\beta_2$ is the quadrupole deformation and $R = 1.2 A^{1/3}$ fm.

(a) Calculate $\mathcal{J}_{\text{rigid}}$ for $^{168}$Er ($A = 168$, $\beta_2 = 0.34$).

(b) The experimental moment of inertia, extracted from the $2^+$ to $0^+$ energy spacing $E(2^+) = 79.8$ keV using $E(2^+) = \hbar^2 \times 6 / (2\mathcal{J})$, gives $\mathcal{J}_{\text{exp}}$. Compute it.

(c) Form the ratio $\mathcal{J}_{\text{exp}} / \mathcal{J}_{\text{rigid}}$. Is it less than 1? Explain in terms of pairing and the nuclear superfluid.

(d) What would happen to the moment of inertia if pairing were suddenly turned off? Relate to the "backbending" phenomenon (Chapter 8).

Exercise 7.22 — Comprehensive: $^{134}$Te as a Two-Proton System ★

The nucleus $^{134}$Te has $Z = 52$ (two protons above $Z = 50$) and $N = 82$ (neutron shell closed). It is an excellent two-proton-particle system.

(a) List the proton single-particle orbits above $Z = 50$ and their approximate energies (use $^{133}$Sb single-particle spectrum).

(b) Construct the expected $(g_{7/2})^2$ multiplet ($J = 0, 2, 4, 6$) and the mixed configurations with $(d_{5/2})^2$, $(g_{7/2} d_{5/2})$, etc.

(c) The experimental low-lying spectrum of $^{134}$Te shows: $0^+$ (0.000), $2^+$ (1.279), $4^+$ (1.576), $6^+$ (1.692). Analyze this spectrum: Is it consistent with a seniority-like picture? What is the effective pairing gap?

(d) Compare with the two-proton-hole nucleus $^{130}$Sn ($Z = 50$, $N = 80$ — wait, this has two neutron holes). Instead, consider $^{130}$Cd ($Z = 48$, two proton holes below $Z = 50$). What would you predict for its spectrum based on particle-hole symmetry? The experimental $2^+$ energy is 1.325 MeV — is this consistent?

Exercise 7.23 — Nilsson Model and Rotational Bands ★

In the Nilsson model, a deformed odd-$A$ nucleus has a ground-state rotational band built on the Nilsson orbital with quantum numbers $\Omega^\pi$.

(a) The rotational energy spectrum is $E(I) = \frac{\hbar^2}{2\mathcal{J}} [I(I+1) - \Omega^2]$ for $I = \Omega, \Omega+1, \Omega+2, \ldots$ Show that the minimum spin in the band is $I = \Omega$.

(b) For $^{177}$Hf, with ground state $\Omega^\pi = 7/2^-$, list the expected spin values of the first five members of the ground-state rotational band.

(c) The experimental energies of these states are: $7/2^-$ (0.000), $9/2^-$ (0.112), $11/2^-$ (0.250), $13/2^-$ (0.411), $15/2^-$ (0.594 MeV). Fit these to the rotational formula and extract the moment of inertia $\mathcal{J}$. How does it compare with the rigid-body value?

(d) If the deformation were increased, how would the moment of inertia change? What physical limit does $\mathcal{J}$ approach as pairing weakens?

Exercise 7.24 — Synthesis: Why Is Lead-208 Special? ★

Bring together material from Chapters 4, 6, and 7 to explain why $^{208}$Pb is the most important nucleus in nuclear structure physics.

(a) From the SEMF (Chapter 4): Where does $^{208}$Pb sit in the valley of stability? What is its predicted binding energy, and how does it compare with the measured value?

(b) From the shell model (Chapter 6): Why are both $Z = 82$ and $N = 126$ magic numbers? Which orbits close each shell?

(c) From this chapter: Why is $^{208}$Pb the ideal "laboratory" for testing residual interactions? What makes two-particle ($^{210}$Pb) and two-hole ($^{206}$Pb) spectra so clean?

(d) What is the connection between the doubly-magic nature of $^{208}$Pb and its role as the end-product of natural radioactive decay chains?

(e) In what ways is $^{208}$Pb used as a benchmark in modern nuclear theory (shell model, density functional theory, ab initio approaches)?

Exercise 7.25 — Proton-Neutron Pairing and the $np$ Interaction ★

So far we have discussed pairing between identical nucleons (neutron-neutron or proton-proton). The proton-neutron ($np$) interaction is different because it is not restricted by the Pauli principle to even $J$.

(a) For a proton in the $g_{9/2}$ orbit and a neutron in the $g_{9/2}$ orbit, what values of $J$ are allowed? (Compare with the identical-nucleon case.)

(b) The nucleus $^{210}$Bi has one proton and one neutron outside the $^{208}$Pb core. The low-lying spectrum shows states at: $1^-$ (0.000 MeV), $0^-$ (0.047), $9^-$ (0.271), $7^-$ (0.320), $2^-$ (0.348), $8^-$ (0.396), ... Discuss the differences from the identical-nucleon $(g_{9/2})^2$ spectrum of $^{210}$Pb.

(c) Why is the ground state of $^{210}$Bi $1^-$ rather than $0^-$? What does this tell you about the proton-neutron residual interaction?

(d) In light $N = Z$ nuclei, proton-neutron pairing in the isospin $T = 0$ channel may produce a different kind of condensate than the $T = 1$ pairing discussed in this chapter. Why is $T = 0$ pairing suppressed in heavy nuclei?

Exercise 7.26 — Deformation Energy and the Competition of Shell Effects ★

The total energy of a nucleus as a function of deformation can be written schematically as:

$$E(\epsilon) = E_{\text{LDM}}(\epsilon) + E_{\text{shell}}(\epsilon)$$

where $E_{\text{LDM}}$ is the liquid-drop energy (smooth function, minimum at $\epsilon = 0$ for most nuclei) and $E_{\text{shell}}$ is the shell correction energy (oscillatory function of $\epsilon$).

(a) Explain why the liquid-drop model predicts spherical ground states for most nuclei. What is the physical origin of the surface energy cost of deformation?

(b) The shell correction energy can be negative (increased binding) at certain deformations where shell gaps align with the Fermi surface. Using the Nilsson diagram, explain why $\epsilon \approx 0.3$ is energetically favorable for nuclei with $N \approx 90$-$100$.

(c) The Strutinsky shell-correction method combines the smooth liquid-drop energy with the oscillatory shell correction. Describe qualitatively how this method works and why it is more successful than either the liquid-drop model or the shell model alone for predicting nuclear ground-state deformations.

(d) At what nucleon numbers would you expect shape coexistence — the situation where two or more shapes (spherical and deformed, or prolate and oblate) coexist at similar energies? Give an example from experimental data.