Ground-state properties of superheavy $Z=122$ isotopes within the deformed relativistic Hartree-Bogoliubov theory in continuum

This paper investigates the ground-state properties, drip lines, and potential magic numbers of superheavy $Z=122$ isotopes using the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc), while comparing results with the RCHB framework and proposing a strategy for determining ground states through the analysis of deformation effects and angular-momentum cutoffs.

The Gist

Imagine the periodic table of elements as a vast, crowded city. Most of the buildings (elements) are stable and well-known. But as you move to the very edge of the city, into the "Superheavy District," the buildings become incredibly unstable, wobbling and collapsing almost instantly. Scientists have been trying to build new, heavier buildings in this district, but they are struggling to find the blueprints.

This paper is like a team of theoretical architects using a super-powerful computer simulation to design the blueprints for a brand-new, ultra-heavy building: Element 122 (which they've nicknamed "Unbibium" or "Ubb").

Here is a breakdown of what they did, using simple analogies:

1. The Challenge: Building on Shaky Ground

In the real world, building a skyscraper requires a solid foundation. In the atomic world, the "foundation" is the balance between protons (positive charge) and neutrons (neutral glue).

• The Problem: Element 122 has so many protons pushing against each other that the nucleus wants to fly apart.
• The Solution: The scientists used a sophisticated mathematical model called DRHBc. Think of this model as a "3D virtual reality simulator" that doesn't just look at the building from the outside; it simulates the internal vibrations, the shape-shifting, and even the "leaking" of particles from the edges.

2. The Shape-Shifting Dance

Most people think of an atom as a perfect, hard sphere (like a billiard ball). But the scientists found that for these super-heavy atoms, the shape is much more dynamic.

• The Analogy: Imagine a blob of jelly. Sometimes it's round, sometimes it squashes flat (like a pancake), and sometimes it stretches out (like a rugby ball).
• The Discovery: The team found that Element 122 isotopes (versions with different numbers of neutrons) love to change shape. They found a "sweet spot" where the nucleus flattens out significantly (oblate deformation) to stay stable. If they tried to force it to stay round, it would fall apart.

3. Finding the "Magic Numbers" (The Safe Zones)

In this atomic city, there are specific numbers of neutrons that act like "safety locks." If you have exactly that many neutrons, the nucleus becomes extra stable, like a building reinforced with steel beams.

• The Magic Numbers: The study suggests three new "safe zones" for neutrons: 184, 258, and 350.
• The "Island of Stability": If you can build an atom with Element 122 and exactly 258 or 350 neutrons, it might live much longer than its neighbors. It's like finding a calm, sunny island in the middle of a stormy ocean.

4. The Edge of the Map (Drip Lines)

Every element has a limit. If you add too many neutrons, they start to "drip" off the nucleus because the glue isn't strong enough to hold them.

• The Proton Drip Line: The team calculated that if you have fewer than 182 neutrons, the atom is so unstable it can't even hold onto its protons. It's like trying to build a tower with too few bricks at the base; it collapses immediately.
• The Neutron Drip Line: If you add too many neutrons (past 320), they start to fall off. However, they found a strange "peninsula" of stability—a few weird islands where the atom is stable even though it's surrounded by unstable water.

5. The "Cut-Off" Strategy

One of the paper's technical contributions was figuring out how to run the simulation efficiently.

• The Analogy: Imagine trying to listen to a symphony. Do you need to hear every single instrument to know the song? Or just the main melody?
• The Finding: The team tested different "cutoffs" (how much detail to include). They found that they didn't need to simulate every tiny detail to get the right answer. They could stop at a certain level of complexity (an angular momentum cutoff of 31/2) and still get a perfect picture of the atom's ground state. This saves a massive amount of computer time.

6. The Big Picture: Why Does This Matter?

Even though we haven't built Element 122 in a lab yet, this paper is crucial because:

• It's a Map: It tells experimentalists exactly where to look. Instead of guessing, they now know which combinations of protons and neutrons might create a stable atom.
• It Tests Physics: It pushes our understanding of the laws of physics to the absolute limit. If these atoms exist, they prove that our current theories about how matter holds together are correct, even in extreme conditions.

In summary: This paper is a theoretical roadmap for the future of nuclear physics. It tells us that Element 122 is likely to be a shape-shifting, jelly-like giant that only survives if we give it the exact right amount of "neutron glue." It identifies the safe zones where these monsters might actually live, guiding scientists on their quest to build the heaviest elements in the universe.

Technical Summary

1. Problem Statement

The synthesis of superheavy elements (SHEs) remains a frontier in nuclear physics, with experimental efforts currently focused on elements with atomic numbers $Z = 119$ and $120$. However, no experiment has yet successfully synthesized elements with $Z \ge 119$. Theoretical models are essential for guiding these experiments, particularly for predicting the stability limits (drip lines) and identifying potential "islands of stability" defined by magic numbers.

While previous studies using the Deformed Relativistic Hartree-Bogoliubov theory in continuum (DRHBc) have covered nuclei up to $Z=120$, there is a critical need to extend these investigations to $Z=122$ (temporarily named Unbibium, Ubb). Key challenges include:

• Determining the ground-state configurations in the presence of strong Coulomb repulsion and complex deformation.
• Accurately predicting proton and neutron drip lines where nuclei become unbound.
• Identifying new magic numbers in the superheavy region ($N > 184$).
• Establishing a robust strategy to distinguish true ground states from local minima in potential energy surfaces (PES), especially given the sensitivity of large deformations to numerical cutoffs and symmetry constraints.

2. Methodology

The authors employed the Deformed Relativistic Hartree-Bogoliubov theory in continuum (DRHBc), a self-consistent mean-field model that simultaneously accounts for:

• Relativistic effects: Using the Dirac equation.
• Deformation: Allowing for axial and reflection symmetry breaking (though the primary DRHBc calculations assumed axial and reflection symmetry).
• Pairing correlations: Treated via the Bogoliubov transformation.
• Continuum effects: Crucial for weakly bound nuclei near the drip lines.

Key Computational Details:

• Functional: The relativistic density functional PC-PK1 was used.
• Basis: The Dirac Woods-Saxon (DWS) basis was constructed in a box of $R_{box} = 20$ fm with a mesh of $\Delta r = 0.1$ fm.
• Cutoffs: Angular momentum cutoff $J_{max} = 31/2 \hbar$ and energy cutoff in the Fermi sea $E_{cut} = 300$ MeV.
• Pairing: A density-dependent zero-range pairing force was adopted with strength $V_0 = -300$ MeV$\cdot$fm$^3$.
• Comparison: Results were benchmarked against the Relativistic Continuum Hartree-Bogoliubov (RCHB) theory, which assumes spherical symmetry, to quantify the impact of deformation.
• Strategy Validation: To determine the true ground state among multiple local minima, the authors:
  1. Examined the convergence of Potential Energy Curves (PECs) with respect to $J_{max}$.
  2. Utilized Multi-Dimensionally Constrained Covariant Density Functional Theory (MDC-CDFT) to test the stability of minima against triaxial and octupole deformations (breaking axial and reflection symmetries).

3. Key Contributions

• Ground-State Determination Strategy: The paper proposes a rigorous strategy for identifying ground states in superheavy nuclei. It demonstrates that while a cutoff of $J_{max} = 31/2 \hbar$ is sufficient, one must carefully distinguish between small-deformation minima ($|\beta_2| < 0.3$) and large-deformation minima. The study shows that large prolate minima ($\beta_2 > 0.5$) are often artifacts of symmetry constraints and disappear when octupole deformation is included, whereas large oblate minima ($\beta_2 \sim -0.45$) remain stable.
• Extension to Z=122: This work provides the first systematic DRHBc calculation for the entire $Z=122$ isotopic chain ($N=171$ to $352$), extending the DRHBc Mass Table Collaboration's scope.
• Drip Line Prediction: It establishes precise proton and neutron drip lines for $Z=122$ by analyzing Fermi energies and separation energies, highlighting significant differences between spherical (RCHB) and deformed (DRHBc) predictions.

4. Key Results

A. Ground-State Properties and Deformation

• Deformation Evolution: The ground states of $Z=122$ isotopes exhibit a complex evolution. Near magic numbers ($N=184, 258, 350$), nuclei are spherical. Between these magic numbers, they rapidly develop large oblate deformations ($\beta_2 \sim -0.4$ to $-0.5$) before transitioning to moderate prolate or weakly oblate shapes.
• Stability of Minima: The study confirms that the large oblate minima identified in DRHBc are the true ground states. In contrast, large prolate minima found in axially symmetric calculations are often eliminated when octupole deformation is considered (via MDC-CDFT), indicating they are not physical ground states.

B. Magic Numbers

• Neutron Magic Numbers: The analysis of Fermi energies, two-neutron separation energies ($S_{2n}$), and pairing gaps identifies $N = 184, 258,$ and $350$ as robust neutron magic numbers.
  • $N=184$ and $N=258$ show clear shell gaps.
  • $N=350$ is identified as a magic number, though the associated shell gap resides in the positive energy continuum.
• Proton Magic Numbers: $Z=120$ is confirmed as a magic number, but $Z=122$ does not exhibit a significant shell gap, leading to pairing collapse in certain regions.

C. Drip Lines

• Proton Drip Line:
  • DRHBc: Predicts the proton drip line at $^{304}\text{Ubb}$ ($N=182$). Isotopes with $N \le 181$ are unbound.
  • RCHB: Predicts the drip line at $^{303}\text{Ubb}$ ($N=181$).
• Neutron Drip Line:
  • DRHBc: Predicts the neutron drip line at $^{442}\text{Ubb}$ ($N=320$). However, a "stability peninsula" is observed where nuclei with $N=321-340$ are unbound by two-neutron separation ($S_{2n} < 0$) but bound by one-neutron separation ($S_n > 0$) and have negative Fermi energies.
  • RCHB: Predicts a much further neutron drip line at $^{472}\text{Ubb}$ ($N=350$), suggesting that neglecting deformation and continuum effects significantly overestimates the stability of neutron-rich superheavy nuclei.

D. Radii and Pairing

• Radii: Charge ($R_{ch}$) and neutron ($R_n$) radii calculated by DRHBc are generally larger than RCHB results, particularly for nuclei with large oblate deformations.
• Pairing Gaps: The neutron pairing gap ($\Delta_n$) drops to zero at the magic numbers ($184, 258, 350$) and specific subshell closures. The proton pairing gap ($\Delta_p$) collapses in regions of large oblate deformation, correlating with the shape transitions.

5. Significance

• Theoretical Validation: The study validates the DRHBc framework as a superior tool for superheavy nuclei compared to spherical approximations (RCHB), particularly in predicting drip lines and deformation-driven stability.
• Experimental Guidance: By pinpointing the proton drip line at $N=182$ and identifying $N=258$ and $N=350$ as potential islands of stability, the paper provides crucial targets for future synthesis experiments of elements $Z=119-122$.
• Methodological Advancement: The proposed strategy for handling angular momentum cutoffs and symmetry constraints (triaxial/octupole) sets a new standard for determining ground states in the superheavy region, preventing the misidentification of local minima as ground states.
• Insight into "Stability Peninsulas": The discovery of bound nuclei beyond the two-neutron drip line (where $S_{2n} < 0$ but $S_n > 0$) in the $Z=122$ chain offers new insights into the limits of nuclear existence and the role of pairing correlations in extreme neutron-rich environments.