Octupole correlation effects on two-neutron transfer intensity in rare-earth nuclei

This study utilizes an interacting boson model based on nuclear density functional theory to demonstrate that octupole correlations significantly influence low-lying $0^+$ states and two-neutron transfer intensities in rare-earth nuclei, successfully reproducing the experimental discontinuous changes near $N\approx88$ or 90 associated with shape phase transitions.

The Gist

Imagine an atomic nucleus not as a boring, solid ball of clay, but as a lively, shape-shifting dance troupe. For decades, physicists have studied how these troupes move, spin, and change shape. This paper by Kosuke Nomura explores a specific, somewhat "wobbly" dance move called octupole correlation and how it changes the way these nuclei swap partners.

Here is the story of the paper, broken down into simple concepts and analogies.

1. The Setting: The Nuclear Dance Floor

In the world of rare-earth nuclei (a specific family of heavy atoms), scientists have noticed something strange. These nuclei can change their shape from a round ball (spherical) to a stretched-out football (prolate) very quickly as they gain more neutrons. This is called a Shape Phase Transition.

Think of it like a group of dancers suddenly switching from a slow, round waltz to a fast, stretched-out line dance.

• The Old View: Scientists used a model called the Interacting Boson Model (IBM). Imagine this model as a set of Lego bricks. The standard version only had two types of bricks:

  • s-bricks: Round, smooth, and stable (representing pairs of particles with no spin).
  • d-bricks: Squashed, football-shaped (representing pairs with a little spin).
  • Using just these, scientists could explain the basic dance moves, but they missed some of the wiggles and wobbles.

• The New Ingredient: This paper introduces a third type of brick: the f-brick.

  • f-bricks represent a "pear-shaped" wobble. In physics, this is called octupole correlation. It's like the nucleus isn't just round or football-shaped; it's slightly lopsided, like a pear or a teardrop.

2. The Experiment: The "Two-Neutron Swap"

To understand how these nuclei behave, scientists perform a trick called a two-neutron transfer reaction (specifically $(p, t)$ and $(t, p)$ reactions).

• The Analogy: Imagine two dance troupes. Troupe A has a certain number of dancers. Troupe B wants to swap two dancers with Troupe A.
  • In a $(p, t)$ reaction, Troupe A gives away two dancers (neutrons).
  • In a $(t, p)$ reaction, Troupe A receives two dancers.

By watching how easily this swap happens, scientists can tell if the troupes are dancing in a simple circle or a complex, wobbly pattern. If the swap is easy, the troupes are similar. If it's hard, they are very different.

3. The Discovery: The "Wobble" Matters

The author used a supercomputer to simulate these nuclei. He took the "Lego" model (IBM) and added the f-bricks (octupole/pear-shape) to see what happened.

What he found:

1. The "Ghost" Dancers: Even in the most stable, round-looking nuclei, there are hidden "pear-shaped" vibrations happening underneath. The f-bricks are always there, just in small amounts.
2. The Excited States: When the nucleus gets excited (starts dancing faster), it creates new, higher-energy states. The paper found that these excited states are heavily made up of the "f-bricks." It's like the excited dancers are doing a wild, wobbly pear-shape routine that the old model (without f-bricks) couldn't explain.
3. The "Jump" in the Data: This is the most important part. When the nuclei reach a specific number of neutrons (around 88 or 90), they undergo that shape transition from round to football.
   • Without f-bricks: The model predicted the "swap intensity" (how easy the neutron exchange is) would change slowly and smoothly, like a ramp.
   • With f-bricks: The model predicted a sudden, sharp jump (a discontinuity) in the swap intensity right at that transition point.

Why this is a big deal:
Real-world experiments show that sharp jump. The old model (without the pear-shape) failed to predict it. The new model (with the pear-shape) successfully reproduced the "jump." It turns out that the "wobble" (octupole correlation) is the secret ingredient that explains why the nuclei behave so strangely at that specific moment.

4. The Conclusion: Why We Need the "Pear"

The paper concludes that you cannot fully understand how these heavy nuclei swap neutrons or how they change shape without accounting for the "pear-shaped" wobble.

• The Metaphor: Imagine trying to predict how a rubber ball bounces. If you only look at it as a perfect sphere, you might get the basic bounce right. But if the ball is slightly squashed or pear-shaped, it will bounce differently, especially when it hits the ground at a specific speed.
• The Takeaway: The "pear-shape" (octupole correlation) isn't just a minor detail; it's a crucial part of the physics that dictates how these atomic nuclei dance, change shape, and interact with their neighbors.

In short: By adding a new type of "Lego brick" (the f-brick) to the simulation, the author finally solved a puzzle that had stumped physicists: why the neutron-swap intensity suddenly jumps when these nuclei change shape. The "wobble" is the key.

Technical Summary

1. Problem Statement

The paper addresses a persistent discrepancy in nuclear structure modeling regarding the two-neutron transfer reactions (specifically $(p, t)$ and $(t, p)$) in rare-earth nuclei (Nd, Sm, Gd isotopes).

• The Phenomenon: Experimental data shows a discontinuous change (a "kink" or abrupt drop) in transfer intensities near neutron numbers $N \approx 88$ and $N \approx 90$. This is widely recognized as a signature of a Quantum Phase Transition (QPT) from nearly spherical to strongly prolate deformed shapes.
• The Limitation: Previous theoretical studies using the Interacting Boson Model (IBM) mapped from microscopic Density Functional Theory (DFT) calculations (specifically the "mapped sd-IBM") failed to reproduce this discontinuity. Instead, these models predicted only monotonous changes in transfer intensities as a function of neutron number.
• The Hypothesis: The failure of the standard $sd$-IBM (which includes only $s$ and $d$ bosons) suggests that octupole correlations (involving $f$ bosons, representing $J=3^-$ pairs) play a crucial role not only in negative-parity states but also in low-lying positive-parity states (specifically excited $0^+$ states) and the transfer mechanisms connecting them.

2. Methodology

The author employs a microscopically constrained Interacting Boson Model with $s$, $d$, and $f$ bosons (mapped sdf-IBM).

• Microscopic Input: The study utilizes Constrained Self-Consistent Mean-Field (SCMF) calculations based on the Gogny D1M energy density functional (EDF) within the Hartree-Fock-Bogoliubov (HFB) framework. These calculations generate Potential Energy Surfaces (PES) in terms of quadrupole ($\beta_{20}$) and octupole ($\beta_{30}$) deformations.
• Mapping Procedure: The PES from the microscopic HFB calculations is mapped onto the expectation value of the sdf-IBM Hamiltonian in a coherent state. This determines the Hamiltonian parameters ($\epsilon_d, \epsilon_f, \kappa_2, \kappa_3, \rho, \chi_d, \chi_f, \chi_3$) without phenomenological fitting to spectra.
• Transfer Operators: The $(p, t)$ and $(t, p)$ transfer operators are constructed to include both leading-order terms (involving $s$ bosons) and terms involving the octupole boson number operator ($\hat{n}_f$). The transfer operator form is:
  $$\hat{P} \propto c_1 \hat{A} \tilde{s} + c_2 \hat{n}_f \tilde{s}$$
  Crucially, the coefficient $c_2$ is made dependent on the octupole deformation ($\beta_{30}$) to account for the varying strength of octupole correlations across the isotopic chain.
• Scope: The study focuses on even-even isotopes of Nd, Sm, and Gd with neutron numbers $N = 84$ to $94$, a region known for the shape phase transition near $N=90$.

3. Key Contributions

• Inclusion of Octupole Degrees of Freedom: The paper demonstrates that extending the IBM model space from $sd$ to $sdf$ is essential for reproducing the experimental systematics of two-neutron transfer intensities in this mass region.
• Microscopic Determination: Unlike previous phenomenological IBM studies that fitted parameters to data, this work derives all Hamiltonian parameters strictly from the microscopic Gogny-HFB potential energy surfaces.
• Mechanism for Discontinuity: The study identifies that the inclusion of the $f$-boson number operator in the transfer Hamiltonian, modulated by the octupole deformation, is the key mechanism that generates the observed discontinuous changes in transfer intensity near the phase transition.

4. Key Results

A. Spectroscopic Properties

• Excited $0^+$ States: The mapped $sd$-IBM significantly overestimates the excitation energies of the first and second excited $0^+$ states ($0^+_2, 0^+_3$), particularly for $N \ge 90$. The mapped sdf-IBM lowers these energies, providing a qualitatively better description, though they remain slightly higher than experimental values.
• Octupole Content: Analysis of the wave functions reveals that the excited $0^+_2$ states contain a significant number of $f$-bosons (expectation value $\langle \hat{n}_f \rangle \approx 2$), whereas the ground state ($0^+_1$) has a smaller fraction ($\langle \hat{n}_f \rangle \approx 0.5$). This confirms that excited $0^+$ states in this region are strongly influenced by octupole correlations (interpreted as double-octupole phonon couplings).
• Negative Parity States: The model successfully reproduces the systematics of the $3^-$ state energies and $B(E3)$ transition rates, confirming the relevance of octupole collectivity near $N \approx 88$.

B. Two-Neutron Transfer Intensities

• Reproduction of Discontinuity: The mapped sdf-IBM successfully reproduces the discontinuous change in $(p, t)$ and $(t, p)$ transfer intensities near $N=88$ and $N=90$. In contrast, the mapped sd-IBM (without $f$ bosons) and the phenomenological sd-IBM (CQF) fail to produce this sharp drop when using the same microscopic inputs, producing only smooth variations.
• Role of $f$-boson Terms: The calculation shows that while the $0^+_1 \to 0^+_1$ transfer is dominated by the $s$-boson term, the $0^+_1 \to 0^+_2$ and $0^+_1 \to 0^+_3$ transfers are dominated by the $\hat{n}_f \tilde{s}$ term (the octupole-dependent term).
• Deformation Dependence: The parameter $c_2$, which scales the octupole contribution, varies significantly with neutron number, peaking in magnitude where octupole deformation ($\beta_{30}$) is non-zero or soft. This variation drives the discontinuity in the total transfer intensity.

5. Significance

• Validation of Octupole Correlations: The work provides strong theoretical evidence that octupole correlations are not limited to negative-parity bands but are integral to the structure of low-lying positive-parity excited states and the dynamics of nucleon transfer in rare-earth nuclei.
• Microscopic Explanation of QPT Signatures: It offers a microscopic explanation for the experimental signatures of shape phase transitions. The study suggests that the "kink" in transfer intensities is a direct consequence of the interplay between quadrupole deformation and the onset/cessation of octupole softness/correlations.
• Methodological Advancement: The paper validates the mapped sdf-IBM approach as a powerful tool for connecting microscopic EDF calculations with complex spectroscopic observables, including transfer reactions. It highlights that neglecting octupole degrees of freedom in the model space can lead to incorrect conclusions about phase transitions in nuclear structure.
• Future Directions: The authors suggest that while the current model improves the description, further refinements (such as configuration mixing between normal and intruder states) are needed to fully match experimental $0^+$ energies. The framework is also proposed for application to light actinide nuclei, where static octupole deformation is more pronounced.