$\alpha$-decay systematics for superheavy nucleus: the effect of deformation of daughter nucleus

This study extends V.Yu. Denisov's deformation-based empirical formula to the DUR, AKRA, and New Geiger-Nuttall models, demonstrating that the modified AKRA model best reproduces experimental $\alpha$-decay half-lives for 400 isotopes and provides consistent predictions for superheavy nuclei (Z = 118–124), particularly outperforming Denisov's original formula at high neutron numbers due to the inclusion of higher-order deformation effects.

The Gist

Imagine the atomic nucleus as a tiny, bustling city made of protons and neutrons. Sometimes, this city gets so crowded and unstable that it needs to get rid of a small, heavy package: a helium nucleus (two protons and two neutrons). This process is called alpha decay.

Scientists have been trying to predict exactly how long it takes for a city to get rid of this package (its "half-life") for over a century. It's like trying to predict exactly when a specific person will leave a crowded party.

The Problem: The City Isn't a Perfect Ball

For a long time, scientists treated these atomic cities as if they were perfect, smooth spheres (like billiard balls). They used formulas to guess the exit time based on the city's size and energy.

But here's the catch: Superheavy atoms are not perfect spheres. They are more like squashed balls, rugby balls, or even weirdly shaped potatoes. They have bumps, bulges, and stretches.

In 2024, a scientist named V. Yu. Denisov realized that if you account for the fact that the "daughter" city (the one left behind after the package leaves) is squashed or stretched, your predictions get much better. He added a "deformation" factor to his formula, which improved accuracy significantly.

The New Study: Adding More Detail to the Map

The paper you provided, written by Jinyu Hu and Chen Wu, asks a simple question: "Is Denisov's map detailed enough?"

Denisov only looked at the big, main bulge (called quadrupole deformation). Hu and Wu thought, "What if there are smaller, more subtle bumps and wiggles that also matter?"

They decided to add two more layers of detail to the map:

1. Hexadecapole ($\beta_4$): Think of this as the "corners" or the next level of squashing.
2. Hexacontatetrapole ($\beta_6$): This is the "fine print"—tiny, high-frequency ripples on the surface of the nucleus.

The Experiment: Testing Five Different GPS Apps

To test their idea, the authors took three existing "GPS apps" (mathematical formulas) used to predict decay times:

1. DUR: A formula based on the shape of the nuclear "barrier."
2. AKRA: A formula that pays extra attention to the balance between protons and neutrons (isospin).
3. NGN: A modern update to the classic Geiger-Nuttall law.

They then created "Pro Versions" of these apps by adding the new, detailed deformation data (the $\beta_4$ and $\beta_6$ bumps). So, they had:

• DUR vs. DUR+D (Pro)
• AKRA vs. AKRA+D (Pro)
• NGN vs. NGN+D (Pro)

They ran these six apps against data from 400 different atomic nuclei to see which one predicted the "exit time" most accurately.

The Results: The "AKRA+D" App Wins

The results were clear:

• The "Pro" versions were better: Adding those extra deformation details ($\beta_4$ and $\beta_6$) made all the formulas more accurate.
• The Winner: The AKRA+D model was the champion. It was the only model that combined two powerful ideas:
  1. The "Pro" Deformation: It accounted for the big squashes and the tiny ripples on the nucleus.
  2. The "Proton-Neutron Balance": It paid close attention to how the number of protons and neutrons differed, which is crucial for heavy atoms.

By combining these factors, the AKRA+D model reduced its prediction errors by about 22% for the most common type of atoms (even-even nuclei). That's a huge improvement in the world of physics!

Why Does This Matter? The Search for New Elements

Why do we care about predicting how long an atom lasts? Because scientists are currently trying to build new, superheavy elements (like Element 118, 120, 122, and 124) in laboratories.

These new elements are incredibly unstable. They might exist for only a fraction of a second before they decay. To find them, scientists need to know exactly what to look for and how long to wait.

The authors used their new, super-accurate AKRA+D formula to predict the behavior of 71 potential new superheavy nuclei.

• The Prediction: They found that at certain specific sizes (neutron numbers 178 and 184), the atoms might be slightly more stable, like a "magic number" that makes the nucleus hold together a bit longer.
• The Difference: When they compared their new "Pro" models against the older "Standard" model (Denisov's), they saw that the new models predicted slightly longer lifetimes for the heaviest atoms. This suggests that ignoring those tiny ripples ($\beta_4$ and $\beta_6$) was causing scientists to underestimate how long these new elements might survive.

The Bottom Line

Think of this paper as upgrading a weather forecast.

• Old Model: "It's going to rain because the sky is cloudy." (Good, but vague).
• Denisov's Model: "It's going to rain because the sky is cloudy and the wind is blowing from the west." (Better).
• Hu and Wu's Model: "It's going to rain because the sky is cloudy, the wind is from the west, and there are tiny pressure changes in the air that we can now measure." (Best).

By accounting for the complex, wobbly shapes of superheavy atoms, this research gives scientists a much sharper tool to predict the existence and behavior of the next generation of elements on the periodic table.

Technical Summary

1. Problem Statement

While α-decay is the dominant decay mode for superheavy nuclei and a crucial tool for identifying new elements, predicting α-decay half-lives with high accuracy remains a significant challenge.

• Limitations of Existing Models: Traditional empirical formulas (e.g., Geiger-Nuttall, Royer, Viola-Seaborg) often neglect the specific geometric deformation of the daughter nucleus. While semi-classical theoretical models (like WKB-based approaches) account for deformation, many empirical formulas rely on spherical approximations.
• Recent Advances: V. Yu. Denisov (2024) recently improved empirical accuracy by incorporating quadrupole deformation ($\beta_2$) into the formula, reducing deviations by ~23%.
• The Gap: Denisov's work considered only the quadrupole term. However, higher-order deformations (hexadecapole $\beta_4$ and hexacontatetrapole $\beta_6$) are known to influence the Coulomb barrier and preformation probability, particularly in heavy and superheavy nuclei. There is a need to determine if extending empirical models to include these higher-order terms further enhances predictive accuracy.

2. Methodology

The authors extended the deformation-dependent framework proposed by Denisov to three established empirical models:

1. DUR: The Deng et al. formula (based on Royer).
2. AKRA: The Akrawy and Poenaru formula (incorporating isospin asymmetry).
3. NGN: The New Geiger-Nuttall law by Ren and Ren.

Key Technical Modifications:

• Deformation Term Integration: The authors introduced a unified deformation term into the logarithmic half-life equations. This term accounts for the change in the Coulomb interaction caused by the non-spherical shape of the daughter nucleus.
• Higher-Order Terms: Unlike Denisov's model (which used only $\beta_2$), the new models (denoted as DUR+D, AKRA+D, and NGN+D) incorporate quadrupole ($\beta_2$), hexadecapole ($\beta_4$), and hexacontatetrapole ($\beta_6$) deformations.
• Radius Calculation: The effective radius of the deformed daughter nucleus ($R_L$) was calculated using spherical harmonics up to the hexacontatetrapole order:
  $$R(\theta) = R_0 [1 + \beta_2 Y_{20}(\theta) + \beta_4 Y_{40}(\theta) + \beta_6 Y_{60}(\theta)]$$
  The maximum radius ($R_L$) determines the minimum Coulomb potential, which is then used to derive the correction term in the half-life formula.
• Data Set:
  • Training/Validation: 400 nuclei were analyzed, categorized into even-even, even-odd, odd-even, and odd-odd groups.
  • Prediction: 71 even-even superheavy nuclei with proton numbers $Z = 118, 120, 122, 124$ were predicted.
• Parameters: Deformation parameters ($\beta_2, \beta_4, \beta_6$) and decay energies ($Q_\alpha$) were sourced from the WS4 mass table.
• Evaluation Metric: Root-Mean-Square (RMS) deviation between calculated and experimental $\log_{10} T_{1/2}$ values.

3. Key Contributions

• Extension of Empirical Frameworks: Successfully generalized the deformation correction method to three distinct empirical models (DUR, AKRA, NGN), creating six total models for comparison.
• Inclusion of Higher-Order Deformations: Demonstrated that incorporating $\beta_4$ and $\beta_6$ terms, in addition to $\beta_2$, provides a more realistic description of the nuclear shape's effect on the Coulomb barrier.
• Identification of the Optimal Model: Systematically proved that the AKRA+D model (which combines isospin asymmetry with multi-pole deformation) yields the highest accuracy.
• Superheavy Element Predictions: Provided a comprehensive set of predicted α-decay half-lives for 71 undiscovered or unmeasured superheavy isotopes, serving as a guide for future synthesis experiments.

4. Results

• Model Performance (400 Nuclei):
  • The AKRA+D model achieved the best agreement with experimental data across all nuclear parity types.
  • RMS Reduction: Compared to the original AKRA model, AKRA+D reduced the RMS deviation by:
    • 22% for even-even nuclei.
    • 14.4% for odd-odd nuclei.
    • 5% for even-odd nuclei.
    • 4.7% for odd-even nuclei.
  • The DUR+D and NGN+D models showed improvements but were less effective than AKRA+D, suggesting that isospin asymmetry (included in AKRA) plays a more critical role than spin-parity blocking (included in DUR/NGN) when combined with deformation.
• Shell Effects: The analysis confirmed discontinuities in half-life predictions at neutron shell closures $N=126$ and $N=152$, supporting $N=152$ as a potential neutron magic number.
• Superheavy Predictions (Z=118-124):
  • Predictions from AKRA+D, DUR+D, and other advanced models (ND, Improved+UL, Improved+EF) showed strong consistency.
  • Divergence at High Neutron Numbers: For $N > 190$, the AKRA+D and DUR+D models predicted longer half-lives than the Denisov (ND) model. This divergence is attributed to the inclusion of $\beta_4$ and $\beta_6$ terms, which are absent in the ND model.
  • Magic Number Indication: The systematic behavior of predicted half-lives at parent neutron numbers $N=180$ and $N=186$ suggests corresponding daughter neutron numbers $N=178$ and $N=184$ may represent a neutron magic number and a submagic number, respectively.

5. Significance

• Enhanced Predictive Power: The study establishes that the AKRA+D model is currently the most reliable empirical tool for α-decay half-life calculations, particularly for heavy and superheavy nuclei where deformation is significant.
• Experimental Guidance: The provided predictions for $Z=118, 120, 122, 124$ offer critical guidance for experimentalists attempting to synthesize and identify new superheavy elements, helping to optimize detection strategies based on expected half-lives.
• Physical Insight: The results reinforce the physical understanding that higher-order multipole deformations ($\beta_4, \beta_6$) significantly influence the Coulomb barrier and, consequently, the tunneling probability of α-particles.
• Future Directions: The authors note that while the model is robust, future work should focus on coupling deformation effects on both the Coulomb barrier and the nuclear mass (which determines $Q_\alpha$) simultaneously, and refining the input deformation parameters for superheavy regions.