Octupole deformation properties in the actinides region using Fayans functionals

This study presents the first survey of octupole deformation and other ground-state properties in heavy actinide nuclei using Fayans energy density functionals, demonstrating their accuracy against experimental data and showing trends consistent with previous Skyrme-based models.

The Gist

Imagine the atomic nucleus not as a perfect, smooth marble, but as a blob of dough that can be squished, stretched, and twisted into all sorts of weird shapes. Most of the time, these blobs are round or slightly oval (like a rugby ball). But in the heavy, "actinide" family of elements (like Uranium and Plutonium), some of these nuclear blobs get even stranger. They start to look like pears—wider on one side and narrower on the other. This is called octupole deformation.

This paper is a massive survey conducted by two physicists, Gauthier Danneaux and Markus Kortelainen, to see if a new, advanced computer model can predict where these "pear-shaped" nuclei exist and how they behave.

Here is the breakdown of their work using simple analogies:

1. The Problem: The "Map" of Nuclear Shapes

For a long time, scientists have used a specific set of mathematical rules (called Skyrme functionals) to map out the nuclear world. These rules are like an old, reliable GPS. They are good at telling us where the "oval" nuclei are, but they sometimes struggle with the finer details, like the exact size of the nucleus or how it wobbles.

The authors wanted to test a newer, more sophisticated GPS system called Fayans functionals. Think of the Fayans model as a high-definition, 3D map that pays extra attention to the "texture" of the nuclear dough, specifically how it pairs up its internal particles.

2. The Experiment: Searching for the "Pear"

The researchers zoomed in on the heavy end of the periodic table (the actinides). They knew from previous studies that there is a specific "cluster" of elements here that likes to turn into pears.

• The Method: They ran millions of computer simulations. Imagine taking a piece of clay (a nucleus) and slowly squishing it in different directions. They asked the computer: "If I squish this nucleus into a pear shape, does it become more stable (lower energy) than if I leave it round?"
• The Scope: They looked at 13 different elements (from Polonium to Hassium) and dozens of their isotopes (versions with different numbers of neutrons).

3. The Findings: The New Model Works!

The results were exciting. The new Fayans model successfully found the same "pear-shaped" islands that the old Skyrme models found.

• The "Pear" Zone: They confirmed that nuclei with specific numbers of protons and neutrons (roughly between Uranium and Plutonium) naturally want to be pear-shaped.
• The Energy Gain: When these nuclei turn into pears, they release a tiny bit of energy (about 1 MeV). It's like a spring snapping into a more comfortable position. The Fayans model calculated this energy drop very accurately.

4. The Special Sauce: Why Fayans is Different

The paper highlights a specific "secret ingredient" in the Fayans model that makes it special: The Gradient Term.

• The Analogy: Imagine the old Skyrme model is like a blanket that covers a bed evenly. It's good, but it doesn't notice if there's a lump under the sheet. The Fayans model is like a smart blanket that feels the lumps. It pays attention to how the density of the nuclear "dough" changes at the edges (the surface).
• The Result: Because of this sensitivity, the Fayans model is much better at predicting the charge radius (the size of the nucleus). It correctly predicts a "staggering" effect where nuclei with odd numbers of neutrons are slightly different in size than their even-numbered neighbors. The old models often missed this subtle wobble.

5. The Real-World Impact: Why Should We Care?

You might ask, "Why do we care if a nucleus is a pear?"

• The "Schiff Moment": Pear-shaped nuclei are like tiny magnets for a very specific kind of physics. They are crucial for experiments trying to find Time-Reversal Violation (basically, looking for evidence of why the universe has more matter than antimatter). If the nucleus isn't a pear, these experiments can't work.
• Nuclear Fission: Understanding these shapes helps us understand how heavy atoms split (fission), which is vital for nuclear power and understanding how heavy elements are created in exploding stars (neutron star mergers).

The Bottom Line

This paper is a "dress rehearsal" for the next generation of nuclear physics. The authors showed that the Fayans functionals are a powerful new tool. They can predict the existence of pear-shaped nuclei just as well as the old tools, but they do a better job at the fine details, like the exact size of the nucleus.

It's like upgrading from a standard-definition TV to 4K: you see the same picture (the pear-shaped nuclei are there), but now you can see the texture and the subtle movements clearly. This gives scientists a better foundation to explore the deepest mysteries of the universe, from the stability of nuclear reactors to the origins of the elements.

Technical Summary

1. Problem Statement and Motivation

The paper addresses the need to understand static octupole deformation (reflection-asymmetric shapes) in heavy nuclei, specifically within the actinide region ($Z \approx 84-108$, $N \approx 120-150$). While Skyrme-based Energy Density Functionals (EDFs), such as UNEDF0, have successfully mapped regions of octupole deformation, they often struggle to reproduce subtle local variations in nuclear observables, particularly the odd-even staggering of charge radii.

The authors aim to:

• Conduct the first systematic survey of octupole deformation properties using Fayans EDFs, a newer class of functionals known for their improved description of pairing and charge radii via gradient terms.
• Determine if Fayans functionals can reproduce the known "pear-shaped" nuclear landscape found in Skyrme models.
• Assess the predictive power of Fayans functionals for ground state properties (binding energy, deformation, charge radii, separation energies) in heavy, deformed nuclei compared to experimental data and Skyrme benchmarks.

2. Methodology

The study employs the Hartree-Fock-Bogoliubov (HFB) framework, utilizing the HFBTHO code (version 2.00d-fayans) with an axially symmetric harmonic oscillator basis.

• Functionals Used: Two specific Fayans parameterizations were tested:
  • Fy(std): A standard re-optimized set based on semi-magic and double-magic nuclei.
  • Fy($\Delta r$,HFB): A version further adjusted using differential charge radii of Calcium isotopes within the HFB framework.
• Computational Strategy:
  • Even-Even Nuclei: Systematic constrained calculations were performed across the actinide cluster ($Z=84-108$, $N=120-150$). The potential energy surfaces (PES) were mapped by constraining both quadrupole ($Q_2$) and octupole ($Q_3$) moments. Unconstrained calculations were subsequently performed to find the global energy minima.
  • Odd-A Nuclei: Calculations utilized the quasiparticle blocking procedure (equal filling approximation) to handle odd-mass nuclei. Only unconstrained calculations were performed for these to determine ground state properties, as extensive constrained calculations were deemed inconsistent due to the sensitivity of blocked orbitals to initial deformation.
• Observables Calculated: Ground state energy, quadrupole ($\beta_2$) and octupole ($\beta_3$) deformation parameters, root-mean-square (rms) charge radii, and one/two-neutron separation energies ($S_n$, $S_{2n}$).
• Comparison: Results were benchmarked against:
  • Experimental data (charge radii from laser spectroscopy/electron scattering; separation energies from AME2020).
  • The UNEDF0 Skyrme functional.

3. Key Contributions

• First Systematic Fayans Survey: This work provides the first comprehensive mapping of octupole deformation in the actinide region using Fayans functionals.
• Validation of Fayans EDFs for Heavy Nuclei: It demonstrates that Fayans functionals, originally tuned for lighter nuclei and charge radii, are robust enough to describe the complex deformation landscapes of heavy actinides.
• Gradient Term Analysis: The study highlights the specific role of the gradient term in the pairing channel (a unique feature of Fayans functionals) in reproducing the odd-even staggering of charge radii, a feature where Skyrme functionals often fail.
• Benchmarking: It establishes a direct comparison between Fayans and Skyrme (UNEDF0) predictions for octupole deformability, showing that while the general "pear-shaped" cluster is consistent, the detailed behavior of the deformation energy surfaces differs.

4. Key Results

• Octupole Deformation Landscape:
  • Both Fayans functionals predict a distinct cluster of octupole-deformed nuclei centered around $^{226}$U ($Z=92, N=134$).
  • The region of non-zero octupole deformation ($\beta_3 > 0.02$) extends roughly from $Z=84$ to $108$ and $N=126$ to $148$.
  • The inclusion of octupole deformation lowers the ground state energy by 0.2 to 1.25 MeV, confirming its necessity for accurate modeling.
  • The shape and location of the deformed cluster align closely with previous Skyrme-based surveys (e.g., UNEDF0), validating the Fayans approach.
• Charge Radii and Odd-Even Staggering:
  • Fayans functionals successfully reproduce the general trend of charge radii across isotopic chains.
  • Fy($\Delta r$,HFB) exhibits a stronger odd-even staggering effect compared to Fy(std).
  • Interestingly, the calculated staggering in some chains (e.g., Thorium) appears inverted compared to the standard expectation (odd-$N$ radii larger than even-$N$ neighbors). The authors note this matches specific experimental hints from resonance ionization spectroscopy in actinides, suggesting the Fayans gradient term captures physics related to reflection asymmetry that Skyrme models miss.
  • Fy(std) showed exceptional accuracy for Uranium charge radii, reproducing both values and staggering almost perfectly.
• Separation Energies:
  • Both functionals reproduce experimental neutron separation energies ($S_n$) and two-neutron separation energies ($S_{2n}$) with high fidelity, particularly in the Uranium chain ($135 \le N \le 144$).
  • Minor deviations were observed in specific regions, potentially linked to deficiencies in the pairing interaction or the lack of a time-odd channel in the current EDF formulation.
• Comparison with UNEDF0:
  • Fayans functionals reproduce the UNEDF0 results for bulk properties (radii, separation energies, quadrupole moments) within a reasonable margin.
  • Fayans results for octupole deformation tend to be smoother, with fewer abrupt "kinks" along isotopic chains compared to UNEDF0, particularly for Fy($\Delta r$,HFB).

5. Significance and Conclusion

The study concludes that Fayans EDFs are a viable and powerful alternative to Skyrme-based functionals for describing heavy, deformed nuclei.

• Predictive Power: They successfully replicate the known octupole deformation "island" in the actinides while offering superior descriptions of local charge radius variations and odd-even effects.
• Future Applications: The ability to accurately model octupole deformation is crucial for understanding phenomena such as the nuclear Schiff moment (relevant for CP-violation searches), fission fragment mass distributions, and r-process nucleosynthesis.
• Path Forward: The authors suggest that while Fayans functionals are excellent starting points, further refinements are needed. These include adjusting parameters at the deformed HFB level, incorporating different pairing strengths for protons and neutrons, and extending surveys to odd-odd nuclei to fully exploit the potential of the next generation of EDFs.

In summary, this paper establishes that Fayans functionals can handle the complex interplay of quadrupole and octupole deformations in the actinide region, providing a more nuanced description of nuclear structure properties than traditional Skyrme models, particularly regarding charge radii.