Exercises — Chapter 8: Collective Motion: Vibrations, Rotations, and Nuclear Deformation

Section A: Vibrational Motion (Problems 1–8)

Problem 1 — Phonon Quantum Numbers

A nucleus has a first excited state $2^+$ at 1.22 MeV, interpreted as a one-phonon quadrupole vibration.

(a) List the $J^\pi$ values expected for the two-phonon multiplet and the predicted energy.

(b) Explain why $J^\pi = 1^+$ and $3^+$ states are absent from the two-phonon multiplet. Your answer should reference the symmetry properties of identical bosons.

(c) For the three-phonon multiplet, derive the allowed $J^\pi$ values by coupling a third phonon ($J = 2$) to the two-phonon states and applying Bose symmetry. Verify that the allowed states are $0^+, 2^+, 3^+, 4^+, 6^+$.

Problem 2 — Vibrational Systematics in $^{112}$Cd

The nucleus $^{112}$Cd ($Z = 48$, $N = 64$) has the following measured states:

(a) Calculate the ratios $E(0^+_2)/E(2^+_1)$, $E(2^+_2)/E(2^+_1)$, and $E(4^+_1)/E(2^+_1)$. Compare with the predictions of the harmonic vibrator model.

(b) The spreading of the two-phonon triplet (from 1224 to 1416 keV, a range of 192 keV) indicates anharmonic effects. Express the anharmonicity as a fraction of the phonon energy $\hbar\omega$.

(c) Would you expect the three-phonon states to show more or less fractional spreading than the two-phonon states? Justify your reasoning.

Problem 3 — Octupole Vibration in $^{208}$Pb

The first $3^-$ state in $^{208}$Pb lies at 2.615 MeV with $B(E3; 3^- \to 0^+) = 34$ W.u. (Weisskopf units).

(a) Explain why the parity of a one-phonon octupole vibration is negative.

(b) The single-particle (Weisskopf) estimate for an $E3$ transition is $B_W(E3) = 1.0 \times (3/4\pi)^2 (r_0 A^{1/3})^6$ in appropriate units. What does the enhancement factor of 34 tell us about the nature of this excitation?

(c) Estimate the phonon energy $\hbar\omega_3$ and mass parameter $B_3$ for the octupole mode, given that the liquid-drop restoring force parameter is approximately $C_3 \approx 0.5 C_2$ for this nucleus. Use the one-phonon energy of 2.615 MeV.

Problem 4 — Vibrational Selection Rules

In the harmonic vibrator model, the electric quadrupole operator is proportional to the phonon creation and annihilation operators: $\hat{T}(E2) \propto (b^\dagger + \tilde{b})$.

(a) Show that this operator changes the phonon number by $\Delta N = \pm 1$, and therefore $E2$ transitions are forbidden between states of the same phonon number.

(b) The $4^+_1 \to 2^+_1$ transition ($N = 2 \to N = 1$) and the $2^+_1 \to 0^+_1$ transition ($N = 1 \to N = 0$) are both allowed. Derive the ratio $B(E2; 4^+_1 \to 2^+_1) / B(E2; 2^+_1 \to 0^+_1)$ in the harmonic limit.

(c) The observed ratio in $^{110}$Cd is approximately 1.7. Compare with your prediction and comment on the agreement.

Problem 5 — Two-Phonon Energies with Anharmonicity

Suppose the vibrational Hamiltonian includes an anharmonic correction:

$$\hat{H} = \hbar\omega \hat{N} + A_2 \hat{N}(\hat{N} - 1),$$

where $\hat{N}$ is the phonon number operator and $A_2$ is the anharmonicity parameter.

(a) Find the energies of the $N = 0$, $N = 1$, and $N = 2$ states.

(b) Define $R = E(N=2)/E(N=1)$ and show that $R = 2 + 2A_2/\hbar\omega$.

(c) For $^{110}$Cd, with $E(2^+_1) = 658$ keV and $E(0^+_2) = 1473$ keV, extract $A_2$.

Problem 6 — Phonon Parity

Prove that a surface vibration of multipolarity $\lambda$ produces a phonon of parity $(-1)^\lambda$ by considering the parity transformation of $Y_{\lambda\mu}(\theta, \phi)$.

Problem 7 — Five-Dimensional Oscillator

The quadrupole vibration is a five-dimensional harmonic oscillator (five components $\alpha_{2\mu}$, $\mu = -2, -1, 0, 1, 2$).

(a) Show that the zero-point energy is $\frac{5}{2}\hbar\omega_2$.

(b) Calculate the degeneracy of the $N$-phonon level for $N = 0, 1, 2, 3$ using the formula for the number of symmetric states of $N$ bosons in 5 dimensions: $g(N) = \binom{N+4}{4} - \binom{N+2}{4}$ for the SO(5) reduction. Verify: $g(0) = 1$, $g(1) = 5$, $g(2) = 14 - 1 = ?$ (Careful: the allowed $J$ values give the actual count of $m$-substates.)

(c) Verify that the total number of $m$-substates for the two-phonon states ($J = 0, 2, 4$) is $(2\cdot0+1) + (2\cdot2+1) + (2\cdot4+1) = 15$. Why does this not equal the naive 5-choose-2-with-replacement = 15? (Hint: it does.)

Problem 8 — Interband Transitions

In a vibrational nucleus, the $B(E2)$ transition strengths from the two-phonon states to the one-phonon state are predicted to be:

$$B(E2; J_i \to 2^+_1) = 2 \times B(E2; 2^+_1 \to 0^+_1) \times C^2(J_i, 2; 2),$$

where $C^2(J_i, 2; 2) = |\langle J_i \, 2 \, 0 \, 0 | 2 \, 0 \rangle|^2$ is a Clebsch-Gordan coefficient squared (here evaluated with specific $m$ values appropriate for the reduced matrix element).

Calculate the relative transition strengths $B(E2; 0^+ \to 2^+_1)$, $B(E2; 2^+ \to 2^+_1)$, and $B(E2; 4^+ \to 2^+_1)$ from the two-phonon triplet to the one-phonon state and show they are in the ratio 1 : 2 : $\frac{50}{7}$.

Section B: Rotational Motion (Problems 9–17)

Problem 9 — Basic Rotational Spectrum

The first excited state of $^{170}$Hf is a $2^+$ state at 100.8 keV.

(a) Assuming a perfect rigid rotor, predict the energies of the $4^+$, $6^+$, $8^+$, and $10^+$ states.

(b) The measured energies are: $4^+$ at 321.3 keV, $6^+$ at 641.5 keV, $8^+$ at 1043 keV, $10^+$ at 1510 keV. Calculate the ratio $E(I)/E(2^+)$ for each and compare with the rigid-rotor prediction.

(c) Extract the moment of inertia $\mathcal{J}$ from the $2^+ \to 0^+$ transition energy in units of $\hbar^2$/MeV.

Problem 10 — Comparing $^{238}$U and $^{164}$Er

Using the data from Section 8.4.3 of the text:

(a) Calculate $\hbar^2/2\mathcal{J}$ for both nuclei from their $E(2^+)$ values.

(b) Compute the rigid-body moment of inertia for each nucleus using $\mathcal{J}_{\text{rigid}} = \frac{2}{5} M_N A R_0^2$ with $R_0 = 1.2 A^{1/3}$ fm. Express the ratio $\mathcal{J}_{\text{exp}}/\mathcal{J}_{\text{rigid}}$ for each.

(c) Why is $\mathcal{J}_{\text{exp}}/\mathcal{J}_{\text{rigid}}$ closer to 1 for $^{238}$U than for $^{164}$Er? Relate your answer to the pairing gap energy.

Problem 11 — Deformation from $B(E2)$

The measured $B(E2; 2^+ \to 0^+)$ for $^{166}$Er is $3.48$ e$^2$b$^2$ (where b = barn = $10^{-24}$ cm$^2$).

(a) Use the formula $Q_0 = \sqrt{16\pi \, B(E2; 2^+ \to 0^+) / 5 \, e^2}$ to extract the intrinsic quadrupole moment $Q_0$.

(b) From $Q_0 = \frac{3}{\sqrt{5\pi}} Z R_0^2 \beta_2 (1 + 0.36\beta_2)$, extract $\beta_2$. Use $R_0 = 1.2 \times 166^{1/3}$ fm.

(c) Is this nucleus prolate or oblate? How do you know?

Problem 12 — Rotational Intensity Rules (Alaga Rules)

For $E2$ transitions within a $K = 0$ band:

$$B(E2; I_i \to I_i - 2) = \frac{5}{16\pi} e^2 Q_0^2 |\langle I_i \, 2 \, 0 \, 0 | (I_i - 2) \, 0 \rangle|^2.$$

(a) Using Clebsch-Gordan coefficient tables, evaluate $|\langle I \, 2 \, 0 \, 0 | (I-2) \, 0 \rangle|^2$ for $I = 2, 4, 6, 8$.

(b) Compute the ratios $B(E2; 4 \to 2)/B(E2; 2 \to 0)$, $B(E2; 6 \to 4)/B(E2; 2 \to 0)$, and $B(E2; 8 \to 6)/B(E2; 2 \to 0)$.

(c) What is the physical reason these ratios are all greater than 1?

Problem 13 — The $K = 1/2$ Decoupling

The ground-state band of $^{183}$W has $K^\pi = 1/2^-$ with a decoupling parameter $a = -0.92$. The rotational energy formula is:

$$E(I) = \frac{\hbar^2}{2\mathcal{J}} \left[ I(I+1) + a(-1)^{I+1/2}(I + \tfrac{1}{2}) \right].$$

(a) Calculate the expected energies (relative to the bandhead) of the $I = 3/2, 5/2, 7/2, 9/2$ states given $\hbar^2/2\mathcal{J} = 11.8$ keV.

(b) Show that for $a \approx -1$, the $I = 1/2$ and $I = 5/2$ levels are nearly degenerate. This signature inversion is a distinctive feature of $K = 1/2$ bands.

(c) Sketch the level scheme and compare with a normal $K = 5/2$ rotational band.

Problem 14 — Moment of Inertia Limits

(a) Derive the rigid-body moment of inertia for a uniformly charged ellipsoid with semi-axes $a = R_0(1 + \epsilon)$ and $b = c = R_0(1 - \epsilon/2)$ (volume conservation) to first order in $\epsilon$, rotating about an axis perpendicular to the symmetry axis. Show that $\mathcal{J}_{\text{rigid}} = \frac{2}{5}MA R_0^2(1 + \epsilon/3 + \ldots)$.

(b) For irrotational flow, the moment of inertia of the same ellipsoid is $\mathcal{J}_{\text{irrot}} = \frac{9}{8\pi}MA R_0^2 \epsilon^2$. Compute the ratio $\mathcal{J}_{\text{irrot}}/\mathcal{J}_{\text{rigid}}$ for $\beta_2 = 0.3$ (where $\epsilon \approx 0.95\beta_2$).

(c) Why is irrotational flow associated with a smaller moment of inertia than rigid-body rotation? Give a physical argument involving the velocity field.

Problem 15 — Centrifugal Stretching

At high spin, the rotational energy can be parameterized as:

$$E(I) = A I(I+1) + B [I(I+1)]^2,$$

where $A = \hbar^2/2\mathcal{J}_0$ and $B < 0$ accounts for centrifugal stretching (the nucleus deforms more as it spins faster, increasing $\mathcal{J}$).

(a) For $^{164}$Er, fit $A$ and $B$ to the $2^+$ (91.4 keV) and $4^+$ (299.5 keV) energies. Predict $E(6^+)$ and $E(8^+)$.

(b) Compare with the measured values (614.4 and 1024.2 keV). Is the two-parameter formula an improvement over the rigid rotor?

(c) Show that the effective spin-dependent moment of inertia is $\mathcal{J}_{\text{eff}}(I) = \mathcal{J}_0 [1 - 2B\mathcal{J}_0 I(I+1)/\hbar^2]^{-1}$.

Problem 16 — Energy Ratios as Structural Indicators

Compute $R_{4/2} = E(4^+_1)/E(2^+_1)$ for the following nuclei and classify each as vibrational, transitional, or rotational:

Problem 17 — Gamma-Band Selection Rules

For the $\gamma$ band ($K = 2$) of an even-even deformed nucleus:

(a) List the allowed spin-parity values of the band members.

(b) Explain why $M1$ transitions between the $\gamma$ band and the ground band ($K = 0$) must change $K$ by 2 and are therefore forbidden for $\Delta K = 2$ (since $M1$ radiation carries at most $\Delta K = 1$). What type of radiation dominates instead?

(c) The branching ratio of the $2^+_\gamma \to 2^+_{gs}$ transition to the $2^+_\gamma \to 0^+_{gs}$ transition is sensitive to the $E2/M1$ mixing ratio. For $^{166}$Er, the measured mixing ratio is $\delta(E2/M1) = -6.5$ for the $2^+_\gamma \to 2^+_{gs}$ transition. What does this large magnitude tell you?

Section C: Backbending and High-Spin Physics (Problems 18–21)

Problem 18 — Backbending Analysis

The ground-state band of $^{158}$Er has the following measured energies:

(a) Calculate the kinematic moment of inertia $2\mathcal{J}^{(1)}/\hbar^2 = (2I - 1)/[E(I) - E(I-2)]$ (in MeV$^{-1}$) for each transition from $4^+ \to 2^+$ through $18^+ \to 16^+$.

(b) Calculate the rotational frequency $\hbar\omega = [E(I) - E(I-2)] / \sqrt{I(I+1) - I_x(I-2)(I_x(I-2)+1)}$ using $I_x(I) = \sqrt{I(I+1)}$ (for $K = 0$).

(c) Plot $2\mathcal{J}^{(1)}/\hbar^2$ vs. $(\hbar\omega)^2$. Identify the backbending region and estimate the critical frequency.

(d) Based on the critical frequency, estimate the pairing gap $\Delta$ using $\hbar\omega_c \approx 2\Delta/(j - 1/2)$ for $i_{13/2}$ neutrons ($j = 13/2$).

Problem 19 — Band Crossing

The backbending in $^{158}$Er can be understood as a crossing between the ground-state band (g-band) and a band built on an aligned $i_{13/2}$ neutron pair (s-band).

(a) If the s-band has a moment of inertia $\mathcal{J}_s = 50\,\hbar^2$/MeV and an alignment of $i_0 = 10\hbar$, write the energy of the s-band as $E_s(I) = E_0 + \hbar^2/(2\mathcal{J}_s) \times (I - i_0)(I - i_0 + 1)$.

(b) For the g-band with $\mathcal{J}_g = 25\,\hbar^2$/MeV, find the spin $I_c$ at which the two bands cross (i.e., $E_g(I_c) = E_s(I_c)$). You may take $E_0 = 1.5$ MeV.

(c) In the absence of an interaction between the bands, the crossing would be sharp. An interaction matrix element $V \approx 50$ keV produces an avoided crossing. Sketch the resulting yrast line.

Problem 20 — Coriolis Matrix Element

The Coriolis force in the rotating frame couples single-particle states through the operator $-\omega \hat{j}_x$.

(a) For a nucleon in the $j = 13/2$ shell, calculate the maximum matrix element $\langle j, \Omega = 1/2 | \hat{j}_x | j, \Omega = 1/2 \rangle$ where $\Omega$ is the projection on the symmetry axis. Use $\langle j_x \rangle_{\text{max}} \approx \sqrt{j(j+1)}$.

(b) At what rotational frequency does the Coriolis energy $\hbar\omega \sqrt{j(j+1)}$ equal the pairing gap $2\Delta = 1.8$ MeV?

(c) Compare this estimate with the observed backbending frequency in the rare-earth region.

Problem 21 — Band Termination

In the $A \sim 110$ mass region, rotational bands in nuclei such as $^{108}$Cd and $^{109}$Sb are observed to terminate — the band reaches a maximum spin and then ceases.

(a) Explain qualitatively why band termination occurs. What happens to the nuclear shape?

(b) If a deformed $^{108}$Cd nucleus in a configuration $[\pi(g_{9/2})^{-2} \nu(h_{11/2})^2]$ has all valence nucleons fully aligned, what is the maximum angular momentum? (The notation indicates two proton holes in $g_{9/2}$ and two neutrons in $h_{11/2}$.)

(c) Why is band termination more common in lighter nuclei than in the rare-earth region?

Section D: Superdeformation and the IBA (Problems 22–27)

Problem 22 — Superdeformed Shell Gaps

In a deformed harmonic oscillator with frequencies $\omega_x = \omega_y \equiv \omega_\perp$ and $\omega_z \equiv \omega_\parallel$:

(a) For the superdeformed ratio $\omega_\perp : \omega_\parallel = 2 : 1$, the single-particle energies are $E = (n_\perp + 1)\hbar\omega_\perp + (n_\parallel + 1/2)\hbar\omega_\parallel$, where $n_\perp = 2n_x + 2n_y + |m|$ (two transverse dimensions). Show that the shell closures (large gaps in the density of states) occur at particle numbers $N = 2, 4, 10, 16, 28, 40, 60, 80$.

(b) Verify that $Z = 66$ (Dy) and $N = 86$ fall near superdeformed shell gaps, helping to explain the discovery of superdeformation in $^{152}$Dy.

(c) Predict other regions of the nuclear chart where superdeformation might be particularly favored.

Problem 23 — Superdeformed Band Properties

A superdeformed band in $^{152}$Dy is observed with the following consecutive $E2$ gamma-ray transitions (in keV): 602, 648, 693, 738, 784, 831, 877, 924, 970, 1016.

(a) These transitions connect states differing by $\Delta I = 2$. If the lowest transition (602 keV) connects the $I$ and $I - 2$ states, and each transition increases by approximately 46 keV, show that this is consistent with $E_\gamma = 4A(I - 1)$ where $A = \hbar^2/2\mathcal{J}$.

(b) Extract $A$ from the data and hence $\mathcal{J}$ in units of $\hbar^2$/MeV.

(c) Compare this $\mathcal{J}$ with the rigid-body value for $\beta_2 = 0.6$. Comment on the ratio.

Problem 24 — IBA Boson Number

(a) Calculate the total boson number $N_B$ for $^{154}$Sm. Use the magic numbers $Z = 50, 82$ for protons and $N = 82, 126$ for neutrons.

(b) Do the same for $^{168}$Er and $^{196}$Pt.

(c) Which of these nuclei do you expect to be closest to the SU(3) (rotational) limit, and why?

Problem 25 — U(5) Spectrum

In the U(5) limit of the IBA, the energy formula (neglecting anharmonic terms) is:

$$E(n_d) = \epsilon \, n_d.$$

(a) Draw the level scheme for $N_B = 6$, showing states up to $n_d = 3$. Label each level with its $J^\pi$.

(b) Calculate $R_{4/2}$. Compare with the harmonic vibrator prediction.

(c) The U(5) limit predicts that $n_d$ can take values $0, 1, 2, \ldots, N_B$. For $N_B = 6$, what is the maximum $n_d$ and what $J^\pi$ values does it generate? This truncation is a key difference between the IBA and the geometric vibrator model.

Problem 26 — SU(3) Spectrum

In the SU(3) limit with $N_B$ bosons, the ground-state band has quantum numbers $(\lambda, \mu) = (2N_B, 0)$ with $L = 0, 2, 4, \ldots, 2N_B$.

(a) For $N_B = 8$ (appropriate for a well-deformed rare-earth nucleus), what is the maximum spin in the ground-state band?

(b) The energy formula is $E(L) = \kappa' L(L+1)$ within a band. Show that $R_{4/2} = 10/3$, independent of $N_B$.

(c) The beta band in SU(3) has $(\lambda, \mu) = (2N_B - 4, 2)$. For $N_B = 8$, list $(\lambda, \mu)$ and the allowed $L$ values of the bandhead.

Problem 27 — Phase Transition in the Samarium Chain

Using the data from Section 8.9.4 of the text (the Sm isotope chain):

(a) Plot $R_{4/2}$ versus neutron number $N$ for $^{148-156}$Sm. Identify the transition region.

(b) The IBA control parameter $\zeta$ ranges from 0 (U(5)) to 1 (deformed). Map each Sm isotope to an approximate $\zeta$ value using $R_{4/2}(\zeta)$ interpolation.

(c) The first-order quantum phase transition at $\zeta_c = N_B / (5N_B + 2)$ predicts coexistence of spherical and deformed shapes near the critical point. For the boson numbers of the Sm isotopes, estimate $\zeta_c$ and identify which isotope is closest to the critical point. Compare with the X(5) identification of $^{152}$Sm.

Bonus Problems

Problem 28 — The Bohr Hamiltonian

The collective Hamiltonian for quadrupole motion in the $(\beta, \gamma)$ variables has the kinetic energy:

$$T = \frac{1}{2} B_2 \left[ \dot{\beta}^2 + \beta^2 \dot{\gamma}^2 + \beta^2 \sum_k \frac{\omega_k^2}{\sin^2(\gamma - 2\pi k/3)} \right],$$

where $\omega_k$ are the angular velocities in the body-fixed frame.

(a) Identify the three terms: which corresponds to vibrations in $\beta$, which to vibrations in $\gamma$, and which to rotation?

(b) Show that for small oscillations about $\beta = \beta_0$, $\gamma = 0$ (axially symmetric equilibrium), the $\beta$ and $\gamma$ vibrations decouple from rotation at lowest order.

(c) The rotational energy for rotation about an axis perpendicular to the symmetry axis (say $k = 1$) involves $1/\sin^2(\gamma)$. Show that for $\gamma \to 0$, this term diverges, which is consistent with the fact that a symmetric rotor cannot rotate about its symmetry axis (the energy cost is infinite).

Problem 29 — Coulomb Excitation Cross Section

The Coulomb excitation cross section for exciting the first $2^+$ state is proportional to $B(E2; 0^+ \to 2^+)$.

(a) For a safe Coulomb excitation experiment (where the projectile stays outside the nuclear surface), the cross section is approximately:

$$\sigma_{\text{CE}} \approx \left(\frac{Z_p Z_t e^2}{4E_{\text{cm}}}\right)^2 \frac{B(E2) \uparrow}{e^2 b^2} \times f(\xi),$$

where $f(\xi)$ is a function of the adiabaticity parameter $\xi$. Why is Coulomb excitation the preferred method for measuring $B(E2)$ values of the first $2^+$ state?

(b) A beam of $^{136}$Xe at 5 MeV/nucleon is used to Coulomb excite a $^{166}$Er target. The measured $2^+$ excitation cross section is 1850 mb. Given that $^{238}$U under similar conditions gives 3200 mb, estimate the ratio of $B(E2)$ values.

(c) Multiple Coulomb excitation (a sequence of successive $E2$ excitations in a single collision) can populate states up to $I \sim 20^+$ in a single pass. Explain how the transition rates along the band, combined with the Alaga rules, allow the intrinsic quadrupole moment $Q_0$ to be extracted from such measurements.

Problem 30 — Summary Calculation: Full Rotational Analysis of $^{176}$Hf

The ground-state band of $^{176}$Hf has the following measured energies:

(a) Extract $\hbar^2/2\mathcal{J}$ from the $2^+$ energy and compute $\mathcal{J}$ in units of $\hbar^2$/MeV.

(b) Calculate $R_{4/2}$, $R_{6/2}$, and $R_{8/2}$ and compare with rigid-rotor predictions.

(c) Fit the data to the two-parameter formula $E(I) = AI(I+1) + B[I(I+1)]^2$. Report $A$ and $B$ and their physical meaning.

(d) The measured $B(E2; 2^+ \to 0^+) = 2.63$ e$^2$b$^2$ for this nucleus. Extract the intrinsic quadrupole moment $Q_0$ and the deformation parameter $\beta_2$. Use $R_0 = 1.2 \times 176^{1/3}$ fm.

(e) Compare $\mathcal{J}_{\text{exp}}$ with $\mathcal{J}_{\text{rigid}}$ for the deformation you found. What fraction of the rigid-body moment of inertia does the nucleus exhibit? Relate this to the pairing gap.

Bonus Problems

Problem B1 — Cranking Model Sum Rule

Show that the cranking moment of inertia satisfies the sum rule:

$$\mathcal{J}_{\text{crank}} = \mathcal{J}_{\text{rigid}} - 2\sum_{i,j \text{ occupied}} \frac{|\langle i | \hat{J}_x | j \rangle|^2}{\epsilon_j - \epsilon_i}.$$

Explain why the second term (which is negative) reduces $\mathcal{J}$ below the rigid-body value and why this reduction vanishes when all single-particle levels are filled (closed shell).

Problem B2 — Connection to Deformed Shell Model

The Nilsson model (Chapter 9 preview) places nucleons in a deformed potential $V = V_0(\beta_2)$. Explain qualitatively why the quadrupole deformation $\beta_2$ that minimizes the total energy of a deformed nucleus depends on which Nilsson orbits are filled, and thus why deformation is largest in mid-shell regions. Use a sketch of Nilsson levels to support your argument.