Shape transitions and ground-state properties of tungsten isotopes in covariant density functional theory

This study employs covariant density functional theory to investigate the structural evolution of even-even tungsten isotopes from $^{154}$W to $^{264}$W, revealing dynamic shape transitions, identifying a potential subshell closure at $N=118$, predicting a neutron drip line at $N=184$, and validating these findings against experimental data and other theoretical models to enhance understanding of nuclear structure and r-process nucleosynthesis.

The Gist

Imagine the atomic nucleus not as a static, hard marble, but as a drop of liquid that can stretch, squish, and change shape depending on how many "guests" (neutrons) are inside it. This is the story of Tungsten, a heavy metal element, and how its various "families" (isotopes) behave when you add more and more neutrons to them.

This paper is like a detailed architectural blueprint for these nuclear families, drawn by a physicist named Usuf Rahaman. He used a powerful computer simulation tool called Covariant Density Functional Theory (CDFT) to predict how these atoms look, how stable they are, and when they might fall apart.

Here is the breakdown of the study in simple terms:

1. The Playground: The Tungsten Family

Think of Tungsten (element 74) as a large family. Some members have very few neutrons (the "neutron-poor" cousins), and some have a huge number of them (the "neutron-rich" cousins).

• The Goal: The researchers wanted to see how the shape of these atoms changes as you add more neutrons, all the way until the family gets so crowded that it can't hold any more (the "drip line," where neutrons start dripping out).

2. The Tools: Four Different Lenses

To get the most accurate picture, the author didn't just use one rulebook. He used four different mathematical models (DD-ME1, DD-ME2, DD-PC1, and DD-PCX).

• The Analogy: Imagine trying to predict the weather. You might check a satellite image, a radar, a computer model, and a local forecaster. If they all say "it's going to rain," you can be pretty sure it will. By using four different models, the author ensured his predictions were robust and not just a fluke of one specific math trick.

3. The Shape-Shifting Dance

The most exciting part of the paper is how the atoms change their shape.

• The Ball (Spherical): At certain "magic numbers" of neutrons (82, 126, and 184), the nucleus is perfectly round, like a billiard ball. It's very stable and happy.
• The Football (Prolate): In the middle of the family, the atoms stretch out into an American football shape. This is the most common shape for Tungsten.
• The Pancake (Oblate): Sometimes, instead of stretching, they flatten out like a pancake.
• The "Double Life" (Shape Coexistence): This is the coolest part. In some specific isotopes (like 158W or 194W), the nucleus is undecided. It's like a person who can't decide between wearing a suit or a tuxedo; it can exist in both shapes at almost the same energy level. The paper found that these atoms are constantly switching between being round, football-shaped, and pancake-shaped.

4. The "Magic" Numbers and the Mystery of 118

In nuclear physics, certain numbers of particles make the nucleus extra stable, like a full shelf in a bookcase. We know 82 and 126 are "magic numbers."

• The Discovery: The study found strong evidence for a new "sub-magic" number at 118.
• The Clue: At 118 neutrons, the atoms suddenly become harder to pull apart, and the "glue" holding the neutrons together (pairing energy) almost disappears. It's like a sudden pause in a dance where everyone freezes for a second. This suggests 118 is a special, stable configuration, even if it's not as famous as 82 or 126.
• The "Almost" Number: They also looked at 112, but it seemed less special, more like a "maybe" number that depends on which math model you use.

5. The Limit: The Neutron Drip Line

How many neutrons can you stuff into a Tungsten atom before it breaks?

• The Prediction: The study predicts the limit is at 184 neutrons.
• The Analogy: Imagine a backpack. You keep adding books (neutrons). At first, it's fine. Then it gets heavy. Finally, at 184 books, the zipper bursts, and the extra books fall out. The atom at 184 neutrons is the last one that can hold its shape; the next one immediately loses a neutron.
• The Return to Roundness: Interestingly, right at this breaking point (184), the atom snaps back into a perfect sphere before it falls apart.

6. Why Does This Matter?

You might ask, "Why do we care about the shape of a heavy metal atom?"

• Cosmic Cooking: These heavy atoms are created in the most violent events in the universe, like colliding neutron stars (the "r-process"). Understanding how Tungsten behaves helps astrophysicists understand how the universe creates heavy elements like gold and uranium.
• Earth Science: Tungsten isotopes are used by geologists to study the Earth's mantle and volcanoes. Knowing their properties helps us understand our own planet's history.
• Future Experiments: This paper acts as a map for scientists building new particle accelerators. It tells them, "Go look at these specific isotopes; you might find something surprising there."

Summary

In short, this paper is a comprehensive tour of the Tungsten family. It tells us that these atoms are not static spheres but dynamic, shape-shifting entities. They dance between round, football, and pancake shapes, have a secret stable "sub-magic" number at 118, and finally hit a hard limit at 184 neutrons where they can't hold on any longer. The study confirms that our current mathematical tools are excellent at predicting these behaviors, giving us a clearer picture of the building blocks of our universe.

Technical Summary

1. Problem Statement

The study addresses the complex structural evolution of even-even tungsten isotopes ($^{154-264}\text{W}$), specifically focusing on the transition from neutron-deficient to neutron-rich regimes up to the predicted neutron drip line. Key challenges in this mass region ($Z=74$) include:

• Shape Dynamics: Understanding the interplay between spherical shell effects and deformation-driving correlations, particularly the phenomenon of shape coexistence (where spherical, prolate, and oblate shapes exist at similar energies).
• Shell Structure: Identifying traditional magic numbers ($N=82, 126$) and potential new subshell closures (specifically around $N=112$ and $N=118$) in neutron-rich nuclei.
• Drip Line Prediction: Determining the precise location of the neutron drip line and the associated return to spherical symmetry.
• Model Robustness: Validating the predictive power of Covariant Density Functional Theory (CDFT) against experimental data and other theoretical frameworks (Skyrme HFB, FRDM, RMF).

2. Methodology

The authors employed Covariant Density Functional Theory (CDFT) within the Relativistic Hartree-Bogoliubov (RHB) framework to treat both mean-field and pairing correlations self-consistently.

• Functionals: Four distinct relativistic energy density functionals were utilized to assess model dependence:
  • Meson-Exchange Models: DD-ME1 and DD-ME2 (finite-range interactions).
  • Point-Coupling Models: DD-PC1 and DD-PCX (zero-range contact interactions).
• Pairing: A separable finite-range pairing force was used, calibrated to reproduce pairing gaps in symmetric nuclear matter.
• Computational Setup:
  • Calculations were performed in a harmonic oscillator (HO) basis with $N_{sh} = 20$ major shells to ensure convergence for bound and continuum states.
  • Constrained Calculations: Potential Energy Curves (PECs) were generated by fixing the axial mass quadrupole moment ($\beta_2$) using the quadratic constraint method to map energy surfaces and identify shape coexistence.
• Benchmarking: Results were compared against:
  • Experimental data (binding energies, deformation, charge radii).
  • Other theoretical models: Deformed HFB (Skyrme SLy4), Finite Range Droplet Model (FRDM), DRHBc (PC-PK1), and RMF (NL3).

3. Key Contributions

• Comprehensive Systematic Study: Provided a unified analysis of the entire even-even tungsten isotopic chain ($N=80$ to the drip line) using four modern relativistic functionals.
• Identification of Subshell Closures: Systematically evaluated candidates for subshell closures in the neutron-rich region, distinguishing between robust and tentative features based on multiple observables ($S_{2n}$, $\delta S_{2n}$, and pairing energy).
• Shape Evolution Mapping: Detailed the transition from spherical to prolate/oblate shapes and back to spherical, highlighting specific isotopes exhibiting strong shape coexistence.
• Drip Line Determination: Predicted the neutron drip line for tungsten and analyzed the structural changes (return to sphericity) occurring at this limit.

4. Key Results

A. Binding Energies and Stability

• Binding energy per nucleon ($BE/A$) peaks around $A \approx 174$ ($N=100$) and decreases thereafter.
• CDFT predictions show excellent agreement with FRDM and Skyrme SLy4 HFB models, particularly on the neutron-rich side, validating the robustness of the relativistic approach.

B. Quadrupole Deformation ($\beta_2$)

• Neutron-Deficient Region: Isotopes exhibit predominantly prolate shapes.
• Transitional Region ($N \approx 116-128$): A transition toward oblate and spherical configurations is observed.
• Neutron-Rich Region: A return to prolate deformation occurs, followed by a return to spherical symmetry at the drip line.
• Shape Coexistence: Pronounced coexistence of prolate and oblate shapes was identified in $^{158,160}\text{W}$, $^{194,196}\text{W}$, $^{206}\text{W}$, and near $^{244-248}\text{W}$. In these cases, the energy difference between competing minima is often $< 1$ MeV.

C. Shell Closures and Subshells

• Magic Numbers: Clear signatures of shell closures were confirmed at $N=82$, $N=126$, and $N=184$ via sharp drops in two-neutron separation energies ($S_{2n}$), peaks in shell gaps ($\delta S_{2n}$), and vanishing neutron pairing energies ($E_{pair,n}$).
• Subshell Candidates:
  • $N=118$: Identified as a robust subshell closure. It exhibits a pronounced peak in $\delta S_{2n}$ and a near-vanishing pairing energy across all four functionals.
  • $N=112$: Shows a weaker, model-dependent structure. The pairing energy does not vanish, suggesting it is a tentative subshell feature rather than a robust closure.

D. Radii and Neutron Skin

• Neutron radii ($r_n$) increase steadily with neutron number, while proton radii ($r_p$) show only a gradual increase.
• Neutron Skin Thickness ($\Delta r = r_n - r_p$): Increases nearly linearly, particularly beyond $N=126$. The slope depends on the isovector properties of the functional, with RMF (NL3) predicting the thickest skins.
• Drip Line: Predicted at $N=184$ ($^{258}\text{W}$), where $S_{2n}$ becomes negative. At this point, the nucleus returns to a spherical configuration due to the $N=184$ shell closure.

E. Potential Energy Curves (PECs)

• PECs reveal shallow competing minima in transitional nuclei, indicating $\gamma$-softness and the potential presence of triaxial correlations, although the study was restricted to axial symmetry.
• The energy surfaces confirm that shape coexistence is a recurring feature in the neutron-rich domain, driven by the competition between shell effects and collective correlations.

5. Significance

• Nuclear Structure Theory: The study reinforces CDFT as a reliable framework for describing heavy, neutron-rich nuclei, demonstrating consistency with non-relativistic models and experimental data.
• Astrophysics: The findings provide critical inputs for r-process nucleosynthesis modeling. The location of the neutron drip line and the stability of isotopes near $N=118$ and $N=184$ influence the abundance patterns of heavy elements in the universe.
• Experimental Guidance: The prediction of shape coexistence in specific neutron-rich isotopes (e.g., $^{244-248}\text{W}$) and the identification of $N=118$ as a subshell closure offer clear targets for future experiments at radioactive ion beam facilities.
• Equation of State: The analysis of neutron skin thickness contributes to constraining the isovector part of the nuclear equation of state, which is relevant for understanding neutron star properties.

In conclusion, this work provides a definitive map of the structural landscape of tungsten isotopes, resolving ambiguities regarding subshell closures and shape transitions, and establishing a benchmark for theoretical models in the medium-to-heavy mass region.