Octupole Excitations in 208Pb

• The paper demonstrates that the first 3⁻ state in 208Pb, at 2.615 MeV with B(E3) ≈34 W.u., exemplifies a coherent, collective octupole vibration.
• Microscopic models, including shell-model and ab initio approaches, capture the coherent 1p–1h excitations and crucial neutron–proton interactions underlying the state’s collectivity.
• The analysis reveals that the 3⁻ energy’s strong sensitivity to nuclear surface energy can constrain energy density functionals for improved theoretical predictions.

Octupole excitations in $^{208}$Pb are prototypical examples of low-lying collective vibrational modes in doubly magic, spherical nuclei. The first $3^-$ state in $^{208}$Pb, at $E_{\rm exp}(3^-)=2.615$ MeV with a large $B(E3)\approx34$ W.u., represents a small-amplitude surface vibration with pronounced collectivity arising from coherent particle-hole excitations across major shells. The near-ideality of $^{208}$Pb as a harmonic octupole vibrator makes it a cornerstone for benchmarking theoretical frameworks ranging from ab initio methods to energy-density functionals and shell-model approaches.

1. Phenomenology and Physical Characterization

The first $3^-$ state of $^{208}$Pb is a textbook example of a collective excitation in a closed-shell system. Empirically, it lies at 2.615 MeV, and the associated transition probability $B(E3;3^-_1\to0^+_1)$ is measured at 34–36 W.u. This strong collectivity signals that the excitation is not a simple single-particle transition but a coherent superposition of many particle–hole configurations. The collective motion corresponds to an octupolar distortion of the surface, as captured in both geometric collective models and microscopic shell-model calculations (Isacker et al., 2022, Isacker, 2020).

2. Microscopic Theories and Model Implementations

Shell Model and Solvable Models

The shell model expresses the collective $3^-$ state as a coherent sum of neutron and proton 1p–1h excitations across the $N=126$ and $Z=82$ gaps:

$$|3^-_{\rm c}\rangle = \sum_{\rho=\nu,\pi}\sum_{k<\mathrm{hole}}\sum_{k'>\mathrm{particle}} c^\rho_{k'k}|j_{\rho k'}j_{\rho k}^{-1}; 3^-\rangle.$$

Here, $c^\rho_{k'k}$ are the amplitudes determined by diagonalizing a realistic Hamiltonian containing empirical single-particle energies and neutron–proton two-body interactions (Isacker et al., 2022). The collectivity is achieved primarily by constructive interference of all components, in particular due to the neutron–proton interaction; eliminating this coupling destroys the coherence and drastically suppresses $B(E3)$.

A solvable model using the surface-delta interaction (SDI) with degenerate orbitals provides analytic expressions for the energy and wave function of the octupole phonon in terms of Wigner $3j$ coefficients (Isacker, 2020). The resulting collective energy takes the form

$$E(3^-_{c,\rho}) \approx \Delta\epsilon_\rho - \frac{a_{1\rho}}{2} S_\rho,$$

where $S_\rho$ denotes an angular-momentum algebraic sum over particle–hole configurations. The, (normalized) $3^-$ state amplitudes are proportional to $$(-1)^{j-\frac12}\sqrt{(2j+1)(2j'+1)/S_\rho}\begin{pmatrix}j'&j&3\\frac12&-\frac12&0\end{pmatrix}$$. In $^{208}$Pb this analytic structure remains an excellent approximation, with full shell-model and SDI amplitudes matching to within a few percent.

Ab Initio Approaches

Coupled-cluster theory, using chiral effective field theory interactions and including singles, doubles, and perturbative triples, enables direct computation of the first $3^-$ state in $^{208}$Pb without empirical adjustments. Calculations yield $E_{\rm th}(3^-)\approx1.8$–2.1 MeV, within 0.5 MeV of experiment, with the 3$^-$ state dominantly single-phonon in character (≳85% 1p–1h content). This confirms the collective nature as a coherent superposition, largely dictated by neutron 1$i_{13/2}\to2g_{9/2}$ transitions (Bonaiti et al., 19 Aug 2025).

Mean-Field and Generator-Coordinate Methods

The generator-coordinate method (GCM) built atop Hartree–Fock–Bogoliubov (HFB) theory, often using the Gogny D1S interaction, offers a global framework for octupole excitations. In $^{208}$Pb, the unconstrained mean field is spherical; the GCM projects and mixes configurations with constrained octupole moments, yielding a pure vibrational $3^-$ state.

Table: Theoretical and Experimental Observables for $3^-$ in $^{208}$Pb

Approach	$E_{3^-}$ (MeV)	$B(E3;\,0^+\to3^-)$ (W.u.)	$E_{3^-,\,{\rm exp}}$ (MeV)
Shell Model	2.46	36	2.615
GCM/HFB (raw)	4.0	53	2.62
Ab Initio CCSD(T)	1.8–2.1	–	2.614
Experimental	–	34	2.615

GCM raw excitation energies typically overshoot experiment by ≈60%, but once globally rescaled, reproduce energies within ≈20%; $B(E3)$ values are captured within 50% (Robledo et al., 2011).

Time-Dependent Density Functional Theory

Small-amplitude TDDFT (corresponding to QRPA/RPA limits) with systematically varied Skyrme functionals has established a nearly perfect linear correlation between the $3^-$ energy and the microscopic surface energy coefficient $a_s$ of the energy density functional. Each 1 MeV increase in $a_s$ raises $E_{3^-}$ by ≃0.15 MeV; all considered functionals place $E_{3^-}$ ≳3.1–3.3 MeV, overestimating experiment and highlighting the sensitivity of octupole vibrations to surface-energy parameters (Alharthi et al., 25 Jan 2026).

3. Surface Energy Dependence and Macroscopic Insights

In the macroscopic liquid-drop framework, the surface energy $a_{\rm surf}$ represents the energetic cost of generating a nuclear surface; its analog in Skyrme functionals is $a_s$. TDDFT calculations using the SLy5sX family—each with distinct $a_s$ but identical bulk properties—reveal:

$$E_{3^-}[\mathrm{MeV}] = (0.1506 \pm 0.0010)a_s[\mathrm{MeV}] + (0.4812 \pm 0.020),$$

with $a_s$ spanning 17.55–18.89 MeV giving $E_{3^-}=3.122$–3.323 MeV (Alharthi et al., 25 Jan 2026). The experimental energy 2.615 MeV would require extrapolating $a_s$ to $\sim$14.1 MeV, well below standard fit intervals. This underscores the extreme sensitivity of the $3^-$ vibrational energy to the nuclear surface energy and suggests that empirical input from octupole spectra could provide stringent constraints on EDF parametrizations.

4. Anharmonicity, Phonon Structure, and Quadrupole–Octupole Coupling

While the level scheme in $^{208}$Pb superficially resembles a harmonic vibrator (energies of two- and three-octupole-phonon excitations nearly scale as $2E_1$, $3E_1$), the transition matrix elements and wave functions display significant anharmonicity. Multi-reference covariant DFT (MR-CDFT) studies combining quadrupole and octupole shape mixing show:

• Excitation energies of the one-, two-, and three-phonon states deviate modestly from the harmonic limit (e.g., $E_{2ph}=9.6$ MeV vs. $2\times4.3=8.6$ MeV) (Yao et al., 2016).
• Transition strengths B(E3) between the multi-phonon bands are strongly enhanced (e.g., $B(E3;3_1^-\to[3_1^-\otimes3_1^-])\sim60$–70 W.u., far exceeding the $2\times B(E3;0_1^+\to3_1^-)$ harmonic-vibrator prediction).
• Inclusion of quadrupole-shape fluctuations fragments the two-phonon band over several states and weakens strong, isolated transitions, reproducing observed fragmentation patterns.
• In $^{208}$Pb, the underlying collective wave functions, particularly for two-phonon states, become widely spread in the quadrupole–octupole plane, in contrast to the sharply defined nodes predicted by a pure $\beta_3$-vibrator model.

5. Empirical and Theoretical Constraints from Transition Strengths

Comprehensive shell model and GCM calculations reproduce the large experimental $B(E3;0^+\to 3^-)\approx34$–36 W.u. only if full proton-neutron correlations are included and configuration mixing is allowed. In the absence of the neutron–proton interaction, $B(E3)$ drops precipitously and collectivity is lost (Isacker et al., 2022, Isacker, 2020). Both shell-model and geometric-collective interpretations yield the same reduction formula:

$$B(\mathrm{E3};3^-\to 0^+) = \left(\frac{3}{4\pi}ZeR^3\sqrt{\frac{\hbar\omega_3}{2C_3}}\right)^2,$$

identifying the collective amplitude $\sqrt{\hbar\omega_3/2C_3}$ with the net coherence of all microscopic components.

Ab initio methods, while less developed for transitions, confirm that the $3^-$ state is largely single-phonon in character, with over 85% norm in the 1p–1h sector and minor correction from higher-order correlations (Bonaiti et al., 19 Aug 2025).

6. Impact on Reaction Dynamics and Nuclear Structure Constraints

Microscopically derived octupole transition strengths have consequential impact in nuclear reaction theory. In sub-barrier fusion of $^{16}$O+$^{208}$Pb, the strong anharmonicity and fragmentation of the octupole (and coupled quadrupole) channels lead to a reduction of the calculated fusion barrier distribution peak, aligning theory with experimental barrier data (Yao et al., 2016). This demonstrates the importance of detailed multiphonon structure—not just single-phonon level energies—for the accurate description of large-amplitude reaction observables.

A key implication is that future energy-density functional fits would benefit from explicit inclusion of low-lying octupole vibrational energies as pseudodata, tightly correlating surface energy and collective spectra (Alharthi et al., 25 Jan 2026).

7. Summary and Context

Octupole excitations in $^{208}$Pb represent a nearly ideal realization of the nuclear octupole phonon, serving as a benchmark for microscopic and macroscopic theories. Modern shell-model and GCM approaches, especially those which account for neutron–proton interactions and configuration mixing, reproduce both energies and transition strengths in close agreement with experiment. Ab initio many-body methods continue to approach quantitative accuracy for the $3^-$ energy, with further improvement anticipated from expanded model spaces and refined three-nucleon treatments. The extreme sensitivity of the $3^-$ energy to the nuclear surface energy makes this mode a powerful probe of the underlying energy density functional, highlighting its importance for nuclear structure and reactions at both the phenomenological and microscopic levels (Alharthi et al., 25 Jan 2026, Robledo et al., 2011, Isacker et al., 2022, Isacker, 2020, Bonaiti et al., 19 Aug 2025, Yao et al., 2016).