---
title: Octupole Excitations in 208Pb
url: https://www.emergentmind.com/topics/octupole-excitations-in-208-pb
type: topic
---

# Octupole Excitations in 208Pb

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 [2203.13541, 2008.13579].

## 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 [2203.13541]. 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 [2008.13579]. 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 [2508.14217].

### 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% [1107.3581].

### 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 [2601.17822].

## 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 [2601.17822]. 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) [1607.02126].
- 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 [2203.13541, 2008.13579]. 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 [2508.14217].

## 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 [1607.02126]. 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 [2601.17822].

## 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 [2601.17822, 1107.3581, 2203.13541, 2008.13579, 2508.14217, 1607.02126].

Source: https://www.emergentmind.com/topics/octupole-excitations-in-208-pb