Nonlocality Effect in the Tunneling of Alpha Radioactivity with the Aid of Machine Learning

This study extends the alpha-particle non-locality effect within the two-potential approach and employs machine learning models, particularly decision tree regression and XGBRegressor, to significantly improve the accuracy of alpha-decay half-life predictions for known and superheavy nuclei compared to traditional benchmarks.

The Gist

Imagine you are trying to predict how long a specific type of unstable atom (an "alpha emitter") will last before it breaks apart and shoots out a tiny particle (an alpha particle). This is like trying to guess exactly when a very shaky tower of blocks will topple over.

For decades, physicists have used a set of rules called the Two-Potential Approach (TPA) to make these guesses. Think of this method as a standard map. It's pretty good, but it has a flaw: it assumes the "runner" (the alpha particle) has a constant weight the whole time it's trying to escape the atom.

The Problem: The "Shape-Shifting" Runner

In reality, as the alpha particle tries to tunnel through the energy barrier holding it inside the nucleus, its "effective weight" changes slightly due to a weird quantum phenomenon called nonlocality. It's like a runner who suddenly gets heavier or lighter depending on exactly where they are on the track. The old map didn't account for this; it treated the runner as if they were a solid, unchanging rock.

The Solution: Teaching a Computer to See the Details

The authors of this paper, Jinyu Hu and Chen Wu, decided to fix this map using Machine Learning. Instead of manually calculating the weight changes for every single atom (which would take forever and might still be wrong), they taught three different computer "students" to learn the pattern:

1. Decision Tree (DT): Like a game of "20 Questions" where the computer asks yes/no questions to narrow down the answer.
2. Random Forest (RF): Like asking a whole crowd of people (a forest of trees) for their opinion and taking the average.
3. XGBRegressor (XG): A super-smart, high-speed student that learns from its mistakes very quickly.

These computers were fed data from 196 different atoms (from element 52 to 118). Their job was to figure out the exact "weight adjustment" needed for each specific atom to make the prediction match reality.

The Results: A Much Sharper Map

When they tested these new computer-optimized maps against real-world experiments, the results were impressive:

• The old map had a certain amount of error (like missing the target by a few inches).
• The new maps, especially the Decision Tree and XG models, reduced that error by about 54%.
• It's as if they took a blurry, low-resolution photo of the atom and turned it into a crisp, 4K high-definition image.

The Big Leap: Predicting the Future

Once the computers learned the rules, the authors asked them to predict the future for 20 super-heavy, man-made elements (specifically elements 118 and 120) that haven't been studied in as much detail yet.

They compared their computer predictions with two other famous formulas used by physicists.

• The Verdict: The Decision Tree model agreed almost perfectly with one of the best existing formulas (called "New+D").
• The Discovery: Their predictions suggest that these super-heavy atoms might have "magic numbers" of neutrons (like 178 and 184) that make them surprisingly stable, much like how a full shell of electrons makes a noble gas stable.

Why Does This Matter?

Think of this research as upgrading the GPS for nuclear physicists.

1. Better Navigation: It helps scientists know exactly where to look when they are trying to synthesize new, heavier elements in the lab.
2. Understanding the Rules: It proves that the "shape-shifting" weight of the alpha particle (nonlocality) is a real, important factor that we can now measure and predict accurately using AI.
3. The Future: By understanding these heavy atoms better, we get closer to understanding the limits of the periodic table and how the universe builds matter.

In short, the authors took a complex physics problem, handed it to a team of AI students, and found that the students could learn the hidden rules of the universe better than the old textbooks could.

Technical Summary

1. Problem Statement

Alpha decay ($\alpha$-decay) is a fundamental process in nuclear physics, particularly critical for understanding the stability and synthesis of superheavy elements. While the Two-Potential Approach (TPA) is a established theoretical framework for calculating $\alpha$-decay half-lives, traditional implementations often rely on simplified assumptions regarding the effective mass of the $\alpha$-particle.

• The Gap: Standard models frequently neglect the nonlocality effect (spatially variable effective mass) or treat the mass parameters ($\rho_S$) via manual tuning, which lacks systematic optimization and physical insight.
• The Challenge: Accurately predicting $\alpha$-decay half-lives for superheavy nuclei (e.g., $Z=118, 120$) is difficult due to complex nuclear interactions, deformation effects, and the sensitivity of half-lives to small variations in decay energy ($Q_\alpha$) and effective mass.

2. Methodology

The authors propose an improved TPA framework enhanced by Supervised Machine Learning (ML) to optimize the coordinate-dependent effective mass parameters.

A. Theoretical Framework (TPA with Nonlocality)

• Decay Width: The half-life ($T_{1/2}$) is derived from the decay width $\Gamma$, calculated using the Gamow formula modified with a pre-exponential factor.
• Nonlocality Effect: The effective mass of the $\alpha$-particle ($m^*$) is defined as a function of the radial distance $r$:
  $$m^* = \frac{m}{1 - \rho(r)}$$
  where $\rho(r)$ is an isotropic function dependent on a parameter $\rho_S$. This parameter accounts for the change in the incident particle's mass due to nonlocal interactions.
• Potential: The nuclear potential is modeled using a cosh-parametrized form, combined with Coulomb and centrifugal potentials.

B. Machine Learning Integration

Instead of manually tuning $\rho_S$ for each nucleus, the authors treat $\rho_S$ as the target output variable for a regression task.

• Input Features: Based on Royer's empirical formula, the inputs are $Z/\sqrt{Q_\alpha}$ and the mass number $A$.
• Training Data: A dataset of 196 even-even nuclei ($Z=52$ to $Z=118$) was used. The target values for $\rho_S$ were pre-optimized manually to minimize the deviation between calculated and experimental half-lives (constrained to $<0.001$).
• Models Evaluated: Three tree-based supervised learning algorithms were employed:
  1. Decision Tree Regression (DT)
  2. Random Forest Regression (RF)
  3. XGBoost Regressor (XG)
• Optimization: Hyperparameters were automatically tuned using Python 3.7 strategies to maximize prediction accuracy.

3. Key Contributions

1. Integration of Nonlocality and ML: The paper successfully extends the nonlocality effect concept (previously applied by Medeiros et al.) into the TPA framework, using ML to systematically determine the optimal $\rho_S$ parameter for individual nuclei.
2. Model Comparison: A rigorous comparison of three ML models reveals that while Random Forest performed best in raw parameter regression, Decision Tree (DT) and XGBoost (XG) yielded superior accuracy in predicting the final $\alpha$-decay half-lives.
3. Superheavy Nuclei Prediction: The optimized models were applied to predict the $\alpha$-decay half-lives of 20 even-even superheavy nuclei with proton numbers $Z=118$ and $Z=120$, regions where experimental data is scarce or non-existent.
4. Benchmarking: The predictions were validated against two established benchmarks: the improved Deng-Zhang-Royer (DZR) formula and the New+D empirical expression (which includes nuclear deformation).

4. Results

• Accuracy Improvement:
  • Compared to the standard TPA without nonlocality effects, the ML-optimized models significantly reduced the standard deviation ($\sigma$) between theoretical and experimental half-lives.
  • XGBoost: Improved $\sigma$ by 53.7% (reduced to 0.259).
  • Decision Tree: Improved $\sigma$ by 54.5% (reduced to 0.264).
  • Random Forest: Showed inferior performance compared to DT and XG in terms of final half-life prediction accuracy ($\sigma = 0.306$).
• Specific Nuclei Analysis:
  • For specific isotopes like $^{108}\text{Xe}$, $^{114}\text{Ba}$, $^{184}\text{Hg}$, and $^{290}\text{Fl}$, the inclusion of the nonlocality effect (optimized by ML) reduced deviations by 50% to 100% compared to the free-mass model.
• Superheavy Predictions ($Z=118, 120$):
  • The Decision Tree model showed high consistency with the New+D expression.
  • The XGBoost model showed some deviation from the other two but still provided physically reasonable trends.
  • Shell Effects: The predicted half-life trends suggest a neutron magic number at $N=186$ and a submagic number at $N=180$, evidenced by local minima in half-life values at these neutron numbers.

5. Significance

• Theoretical Advancement: This work demonstrates that machine learning is not just a "black box" predictor but can effectively learn and optimize complex physical parameters (like the nonlocal mass parameter $\rho_S$) that are difficult to derive analytically.
• Predictive Power for New Elements: The ability to reliably predict half-lives for $Z=118$ and $Z=120$ isotopes provides crucial guidance for experimental facilities (like GSI, RIKEN, JINR) attempting to synthesize and detect new superheavy elements.
• Physical Insight: By linking ML-optimized parameters to physical observables, the study confirms that nonlocal interactions play a critical role in the tunneling process of $\alpha$-decay, improving the theoretical description of nuclear dynamics.
• Methodological Blueprint: The approach of using ML to tune specific parameters within a physics-based model (hybrid modeling) offers a robust strategy for future nuclear structure research, potentially applicable to other decay modes or nuclear properties.