Systematic study of superheavy nuclei within a microscopic collective Hamiltonian: Impact of quantum shape fluctuations

Imagine the atomic nucleus not as a rigid, static ball of clay, but as a wobbly, dancing jelly. For decades, scientists have tried to map the "island of stability" in the superheavy elements—those massive atoms with huge numbers of protons that usually fall apart instantly.

This paper is like a new, high-definition weather forecast for that island. It uses a sophisticated computer model to predict how these giant, unstable atoms behave, focusing on a specific phenomenon: Quantum Shape Fluctuations (QSFs).

Here is the breakdown in simple terms:

1. The Old Way vs. The New Way

The Old Way (Mean-Field): Imagine trying to predict the shape of a jelly by freezing it in a block of ice. You get a clear, static picture. In physics, this is called the "mean-field" approximation. It tells you the nucleus is either a perfect sphere or a stretched football (prolate). It's a good snapshot, but it misses the movement.
The New Way (5DCH): This paper uses a "Microscopic Collective Hamiltonian." Think of this as putting the jelly in a warm room and filming it in slow motion. The nucleus isn't just sitting there; it's vibrating, wobbling, and changing shape constantly due to quantum mechanics. The authors call these "Quantum Shape Fluctuations."

2. The Main Discovery: The "Unstable" Islands

The researchers looked at 585 different superheavy nuclei (atoms with 104 to 126 protons). They found something surprising:

The "Trap" Analogy: Imagine a valley (a stable energy state) surrounded by a hill (a barrier that keeps the atom from falling apart/fissioning).
The Finding: In many of these superheavy atoms, the "valley" is so shallow, and the "wobbling" (fluctuations) is so violent, that the atom doesn't actually stay in the valley. It's like trying to balance a marble on a very shallow dish while someone shakes the table. The marble (the nucleus) rolls right off the edge.
The Result: For about 175 of these nuclei, the model predicts they are "unbound." Even though the static calculations said they should exist, the quantum wobbling makes them fall apart (fission) almost instantly. They might not be realizable in a lab at all.

3. Smoothing Out the "Cliffs"

In the old static models, the energy of these atoms changed in sharp, jagged steps, like walking up a staircase with huge, steep cliffs. This happened at specific numbers of neutrons (184 and 258), which were thought to be "magic numbers" (super stable shells).

The Analogy: Imagine walking up a staircase where every third step is a 10-foot drop.
The New View: When you add the "wobbling" (QSFs) into the mix, those steep cliffs turn into gentle ramps. The sharp drops in energy are smoothed out.
The Shift: Because of this smoothing, the "magic numbers" (where atoms are most stable) shift slightly. Instead of being stable at neutron number 184, they are actually most stable at 182. Instead of 258, it's 256. It's like the map of the island shifted a few miles to the left.

4. The Shape-Shifting Dance

The paper also describes how these atoms change their shape as they get heavier:

Lighter Superheavies: They look like stretched American footballs (prolate).
Middle Weight: They become squishy and floppy, changing shape easily (gamma-soft).
Heavy/Shell-Closed: They try to be perfect spheres.
Very Heavy (Z ≥ 120): They prefer to be flattened discs (oblate), like a pancake.

The new model shows that this transition isn't a sudden switch; it's a gradual, fluid dance.

5. Why Does This Matter?

For Chemists and Physicists: If you try to build these atoms in a lab (like at CERN or in Russia), you need to know if they will survive long enough to be studied. This paper says, "Don't bother trying to make these specific ones; they will vanish before you can blink."
For the Future: It suggests that to find the next stable elements, we need to look slightly differently at the numbers of neutrons we are aiming for (shifting from 184 to 182).
The "Good News": The elements everyone is excited about right now (Z=119 and 120) are predicted to be stable enough to be made, even with all this wobbling.

The Bottom Line

This paper is a reality check for the superheavy element hunt. It tells us that the universe is wobblier than we thought. By accounting for the quantum "jitter" of the nucleus, we get a more accurate map of which superheavy atoms can actually exist and which are just fleeting illusions that fall apart the moment they are born.

Here is a detailed technical summary of the paper "Systematic study of superheavy nuclei within a microscopic collective Hamiltonian: Impact of quantum shape fluctuations."

1. Problem Statement

The synthesis and characterization of superheavy nuclei (SHN) remain a frontier challenge in nuclear physics. While experimental facilities have extended the periodic table to $Z=118$ , theoretical understanding of the structure of even heavier elements ( $Z \le 126$ ) is limited by the scarcity of experimental data.

Limitations of Mean-Field Approximations: Traditional studies based on the static mean-field approximation (e.g., Hartree-Bogoliubov) often fail to capture dynamic correlations arising from Quantum Shape Fluctuations (QSFs). These fluctuations are crucial for accurately predicting binding energies, decay properties, and the stability of nuclei near shell closures.
Gap in Systematic Studies: Previous investigations incorporating QSFs were often limited to specific isotopic chains or specific regions. There was a need for a systematic, large-scale study covering the entire even-even superheavy landscape ($104 \le Z \le 126 $,$ N \le 258$) to understand how QSFs influence shape transitions, shell closures, and the existence of bound ground states.

2. Methodology

The authors employed a Microscopic Five-Dimensional Collective Hamiltonian (5DCH) framework based on the Relativistic Density Functional Theory (DFT).

Underlying Model: The calculations were grounded in Constrained Triaxial Relativistic Hartree-Bogoliubov (TRHB) calculations using the PC-PK1 relativistic density functional.
The 5DCH Approach:
- The Hamiltonian includes collective degrees of freedom for quadrupole vibrations and rotations in both $\beta$ (axial deformation) and $\gamma$ (triaxiality) coordinates, plus the Euler angles ( $\Omega$ ).
- Parameters: Mass parameters ( $B_{\beta\beta}, B_{\beta\gamma}, B_{\gamma\gamma}$ ) and moments of inertia ( $I_k$ ) were derived self-consistently using the perturbative cranking formula.
- Potential: The collective potential ( $V_{coll}$ ) was calculated by subtracting zero-point energy (ZPE) corrections from the total mean-field energy.
Scope: The study systematically analyzed 585 even-even superheavy nuclei with proton numbers $104 \le Z \le 126 $and neutron numbers$ N \le 258$.
Observables Calculated:
- Binding energies ( $B$ ) and comparison with AME2020 data.
- Collective potentials and probability density distributions in the $(\beta, \gamma)$ plane.
- Excitation energies of the first $2^+ $state ($ E(2^+1) $), energy ratios ($ R{42} = E(4^+_1)/E(2^+_1) $), and$ B(E2)$ transition probabilities.
- Two-nucleon separation energies ( $S_{2n}, S_{2p}$ ) and $\alpha$ -decay energies ( $Q_\alpha$ ).
- Fission saddle heights ( $B_f$ ) vs. ground state energies to determine nuclear stability.

3. Key Contributions

Systematic Application of 5DCH: This is one of the first comprehensive studies applying the triaxial 5DCH model across the entire superheavy region ( $Z=104-126$ ), moving beyond limited case studies.
Discovery of "Unbound" Ground States: The study identified a critical phenomenon where, due to QSFs, the zero-point vibrational energy in transitional regions exceeds the height of the fission barrier. Consequently, no bound $0^+$ ground states are predicted for a significant number of nuclei, despite the mean-field potential showing minima.
Shift of Shell Closures: The research demonstrated that QSFs significantly alter the predicted locations of neutron shell closures. The sharp shell effects predicted at $N=184$ and $N=258$ in mean-field models are smoothed out and shifted to $N=182$ and $N=256$ , respectively, due to the evolution of dynamical correlation energies.
Shape Evolution Mapping: The paper provides a detailed map of shape transitions from well-prolate to $\gamma$ -soft and finally to spherical shapes, highlighting the role of triaxiality and the emergence of oblate deformations in very heavy isotopes ( $Z \ge 120$ ).

4. Key Results

A. Binding Energies and Predictive Power

The 5DCH model significantly improved the description of binding energies compared to the static TRHB mean-field model.
Root-Mean-Square (RMS) Deviation: Reduced from 2.30 MeV (TRHB) to 0.57 MeV (5DCH) when compared to experimental data. For the 10 measured masses, the deviation dropped to 0.35 MeV.
The model effectively captures the isospin dependence of binding energies across isotopic chains, demonstrating superior predictive power compared to macroscopic-microscopic models like WS4 and FRDM.

B. Shape Transitions and Deformations

Prolate Region: Lighter SHN ( $Z=104-118$ ) with $N \approx 140-160$ and $188-220 $exhibit significant prolate deformations ($ \beta \approx 0.25-0.3$).
Transitional Regions: Around $N \approx 176$ and $N \approx 246$ , nuclei transition to medium-deformed, $\gamma$ -soft shapes.
Spherical Regions: Near the predicted shell closures $N=184$ and $N=258$ , nuclei approach spherical shapes.
Oblate Region: For isotopes with $Z \ge 120$ around $N \approx 178$ , oblate deformations are favored.
Impact of QSFs: The inclusion of QSFs leads to a smoother shape evolution. The sharp prolate-oblate phase transitions predicted by mean-field theory are "smoothed out" in the 5DCH results.

C. The "Unbound" Phenomenon

Critical Finding: In transitional regions with $N \gtrsim 184$ and $N \gtrsim 240$ , the collective potential wells are very shallow. The zero-point vibrational energy (induced by QSFs) pushes the calculated $0^+$ ground state energy above the fission saddle height.
Consequence: 175 nuclei (out of 585 calculated) are predicted to have no bound ground states. They may exist only as resonances or undergo immediate fission.
Exceptions: The highly anticipated new elements $Z=119$ and $Z=120$ (with $N \approx 178-180$ ) remain bound even when QSFs are included, offering hope for future synthesis via fusion reactions.

D. Shell Closures and Separation Energies

Shifted Shells: The sharp drops in two-neutron separation energies ( $S_{2n}$ ) and $\alpha$ -decay energies ( $Q_\alpha$ ) predicted at $N=184$ and $N=258$ by mean-field theory are significantly reduced and shifted to $N=182$ and $N=256$ in the 5DCH calculations.
Mechanism: This shift is caused by the rapid evolution of dynamical correlation energies associated with QSFs around the spherical shells. The spherical nuclei possess only vibrational modes, while deformed nuclei have both vibrational and rotational modes, altering the correlation energy landscape.

E. Collective Observables

Rotational vs. Spherical: Nuclei with $N \lesssim 170$ and $196 \lesssim N \lesssim 240 $exhibit rotor-like characteristics ($ R_{42} \gtrsim 3.2 $). Nuclei near$ N=184 $and$ 258 $are spherical ($ R_{42} \lesssim 2.0$).
Agreement with Data: The calculated $R_{42}$ ratios and $B(E2)$ values show good agreement with available experimental data (e.g., for $^{254,256}\text{Rf}$ ), validating the model's ability to describe collective properties.

5. Significance and Outlook

Theoretical Advancement: This work underscores the necessity of going beyond the static mean-field approximation to understand the stability limits of superheavy elements. It highlights that QSFs are not a minor correction but a dominant factor determining whether a nucleus is bound or unbound.
Experimental Guidance: The prediction of "unbound" regions suggests that synthesizing certain isotopes in the $N \gtrsim 184$ region may be impossible via current methods, while the confirmation of bound states for $Z=119/120$ guides future experimental targets.
Future Directions: The authors note that while the 5DCH captures triaxial quadrupole fluctuations, octupole deformations (which can lower ground state minima) are not fully included. They propose future work using a Seven-Dimensional Collective Hamiltonian (7DCH) to incorporate octupole degrees of freedom to refine the stability predictions for these extreme nuclei.

In summary, this paper provides a rigorous microscopic framework that redefines the landscape of superheavy nuclei, revealing that quantum shape fluctuations can destabilize nuclei previously thought to be stable and shift the location of the "island of stability."