Production probability of super-heavy nuclei in fusion

This paper proposes a new analytical model, EBD3, which successfully reproduces experimental evaporation residue cross sections for super-heavy nuclei (Z≥110) within one order of magnitude and provides predictions for synthesizing element 119, identifying promising reaction pathways such as $^{45}$Sc + $^{249}$Cf.

The Gist

Imagine you are trying to build a brand-new, incredibly fragile castle out of sand. You want to build the tallest, most complex tower possible (a "super-heavy nucleus"), but the moment you add one too many grains of sand, the whole thing collapses into a pile of rubble (fission).

This is the daily challenge for nuclear physicists trying to create super-heavy elements (like Element 119 or 120) that don't exist in nature. They do this by smashing two smaller atoms together at high speeds, hoping they stick to form a new, heavier atom.

This paper, written by Ning Wang, introduces a new "recipe book" (a mathematical model called EBD3) to predict how likely it is that these atomic collisions will actually succeed.

Here is the breakdown of the paper using simple analogies:

1. The Problem: Why is this so hard?

Think of smashing two atoms together like trying to merge two spinning, sticky, electrically charged balls of dough.

• The Challenge: When you smash them, they might bounce off (capture fails), they might stick for a split second and then immediately fall apart (quasi-fission), or they might stick, cool down, and survive (success).
• The Uncertainty: Scientists have been trying to predict the success rate for years, but their guesses have been wildly off—sometimes by a factor of 10 or even 100. It's like trying to predict the weather a month in advance; the tiny details (like how the dough is shaped or how fast it's spinning) make a huge difference.

2. The Solution: The New "Recipe" (EBD3)

The author proposes a new, simpler formula to calculate the odds of success. Instead of trying to simulate every single particle inside the atoms (which is like trying to count every grain of sand in the castle), this model looks at the big picture using three main steps:

• Step A: Getting Through the Gate (Capture): The two atoms have to overcome their natural repulsion (like two magnets pushing apart) to touch. The model uses a "tunneling" concept, imagining the atoms as ghosts that can sometimes walk through a wall they shouldn't be able to cross.
• Step B: Sticking Together (Compound Nucleus): Once they touch, do they fuse, or do they bounce apart? The model looks at how "lopsided" the collision is. If you smash a tiny pebble into a giant boulder, they are more likely to stick than if you smash two giant boulders together.
• Step C: Surviving the Heat (Evaporation): The new atom is born super-hot and excited. It needs to cool down by shooting off tiny particles (like steam) without falling apart. The model calculates how strong the "walls" of the castle are to see if it can hold together while cooling.

3. How Good is the New Recipe?

The author tested this new formula against 64 real-world experiments where scientists actually tried to make these elements.

• The Result: The new model was surprisingly accurate. It predicted the success rates within a factor of 10 (which is a huge improvement over previous methods that were often off by 100 or 1,000).
• The Analogy: If previous models were like guessing the score of a football game by looking at the clouds, this new model is like looking at the team's stats and the weather report. It's not perfect, but it's reliable enough to plan the next game.

4. The Future: Hunting for Element 119

The most exciting part of the paper is the prediction for the next big discovery: Element 119.

The model acts like a GPS for scientists, telling them which "roads" (combinations of atoms) are most likely to lead to a new element.

• The Best Route: The model suggests that smashing Scandium-45 into Californium-249 is the most promising path. It predicts a decent chance of success.
• The Harder Routes: It also warns that using heavier projectiles like Chromium-54 or Titanium-50 will be much harder, with success rates dropping to almost zero (like trying to build a sandcastle during a hurricane).

5. Why Does This Matter?

Creating these elements isn't just about adding a new name to the Periodic Table.

• The "Island of Stability": Physicists believe there is a hidden "island" in the ocean of atoms where super-heavy elements might actually be stable and last a long time. Finding Element 119 is a step toward finding that island.
• Understanding the Universe: These elements help us understand how heavy elements in the universe (like gold and uranium) were formed in exploding stars.

Summary

Ning Wang has created a new, simpler, and more accurate calculator for nuclear physicists. Instead of getting lost in complex quantum mechanics, this model uses a few key "rules of thumb" to predict how likely it is to build a new, super-heavy atom. It tells scientists exactly which atomic collisions to try next, saving them time and money on experiments that are unlikely to work, and guiding them toward the ones that might create history.

Technical Summary

1. Problem Statement

The synthesis of super-heavy nuclei (SHN) with atomic numbers $Z \ge 110$ via heavy-ion fusion reactions is a critical frontier in nuclear physics, essential for testing nuclear models and exploring the "island of stability." However, theoretical predictions of the evaporation residue cross sections ($\sigma_{ER}$) suffer from large uncertainties (typically 1–2 orders of magnitude).

These uncertainties stem from the complexity of the reaction chain, which is traditionally modeled as the product of three factors:
$$\sigma_{ER} = \sigma_{cap} \times P_{CN} \times W_{sur}$$
Where:

• $\sigma_{cap}$: Capture cross-section (overcoming the Coulomb barrier).
• $P_{CN}$: Probability of compound nucleus formation (surviving quasi-fission).
• $W_{sur}$: Survival probability of the excited compound nucleus (resisting fission during de-excitation).

Current models struggle because:

1. Complexity: They rely on simplified frameworks to describe highly complex quantum many-body processes (nucleon interactions, collective excitations, dissipation).
2. Parameter Sensitivity: Results are heavily dependent on nuclear structure parameters (deformation, fission barriers) that are difficult to determine precisely. A 1 MeV uncertainty in fission barrier height can alter survival probabilities by an order of magnitude.
3. Data Scarcity: Experimental data for SHN is limited (cross sections often $< 10$ pb), making it difficult to constrain all model parameters simultaneously.

2. Methodology: The EBD3 Model

The author proposes a new, improved analytical model named EBD3 (Empirical Barrier Distribution version 3) to systematically describe $\sigma_{ER}$ for SHN. The model combines an empirical barrier distribution method for capture with a new analytical formula for production probability based on barrier tunneling concepts.

Key Components of the Model:

• Capture Cross-Section ($\sigma_{cap}$): Calculated using the established EBD2.2 formula, which utilizes an empirical barrier distribution method. This part has been validated against 426 datasets with high accuracy.
• Production Probability ($P$): Defined as $P = P_{CN} \times W_{sur}$. The model factorizes this into:
  $$P = s \cdot P_{mac}(Z, \eta) \cdot P_1(E^*) \cdot P_2(E^*)$$
  • $s$ (Scale Factor): An empirical factor accounting for the relative height of the fission barrier, differing for "hot" (actinide targets) and "cold" (Pb/Bi targets) fusion.
  • $P_{mac}$ (Macroscopic Part): Represents the survival probability of the dinuclear system (DNS) against quasi-fission. It is modeled as a Fermi function dependent on the charge number ($Z$) and mass asymmetry ($\eta$), capturing the transition from fusion-dominant (light/asymmetric) to fission-dominant (heavy/symmetric) systems.
  • $P_1$ (Energy-dependent DNS Survival): Describes the probability of the DNS surviving the fast fission-like process, modeled via barrier penetration concepts involving an effective tunneling barrier (ETB).
  • $P_2$ (Compound Nucleus Survival): Based on the Bohr-Wheeler statistical decay width, describing the survival of the excited compound nucleus against fission. It incorporates the fission barrier height ($B_f$) and a pre-formation factor ($Y$) that accounts for "extra-push" effects in hot fusion and shell effects in cold fusion.

Specific Innovations:

• Fission Barrier Handling: For extremely neutron-deficient nuclei, the model uses the minimum fission barrier ($B_f^{min}$) among neighboring nuclei to account for isospin effects and model uncertainties.
• Effective Fusion Barrier ($V_{eff}$): The model defines an effective barrier $V_{eff} = V_B + \delta B_f + \Delta$ (where $\delta=1$ for hot fusion, $0$ for cold; $\Delta$ is the shell gap). This quantity is used to predict optimal incident energies.

3. Key Contributions

1. Unified Analytical Framework: Developed a single, parameter-efficient analytical formula (EBD3) that describes both capture and production probabilities for SHN ($Z \ge 110$) across both hot and cold fusion regimes.
2. Systematic Description of Quasi-Fission: Introduced a macroscopic term ($P_{mac}$) based on charge and mass asymmetry that successfully reproduces the systematic decrease in survival probability as systems become heavier and more symmetric.
3. High Accuracy with Few Parameters: The model reproduces 64 measured evaporation residue cross sections within one order of magnitude, achieving a root-mean-square deviation (RMSD) of 0.351.
4. Predictive Capability for Element 119: Provided specific predictions for the synthesis of Element 119, identifying optimal projectile-target combinations and incident energies.

4. Results

• Validation: The EBD3 model was tested against 64 experimental datasets (including hot fusion with $^{48}$Ca and cold fusion with $^{64}$Ni, $^{70}$Zn). The predicted cross sections match experimental data remarkably well, with deviations generally within one order of magnitude.
• Systematic Trends:
  • The model correctly predicts that the production probability drops drastically (by factors of $\sim 10^{14}$) as the system moves from light-asymmetric (e.g., $^{48}$Ca + $^{237}$Np) to heavy-symmetric (e.g., $^{138}$Ba + $^{154}$Sm) combinations.
  • It successfully explains why reactions like $^{50}$Ti + $^{242}$Pu have significantly higher cross sections than $^{54}$Cr + $^{238}$U (due to higher mass asymmetry and capture pocket depth).
• Predictions for Element 119:
  • $^{45}$Sc + $^{249}$Cf: Identified as the most promising reaction with a maximum cross section of $107.5^{+120}_{-56.7}$ fb at an excitation energy of 45.6 MeV. The high asymmetry and longer half-life of the $^{249}$Cf target make it favorable.
  • $^{50}$Ti + $^{249}$Bk: Predicted maximum cross section of $54.9$ fb at 228 MeV.
  • $^{54}$Cr + $^{243}$Am: Predicted maximum cross section of $3.2$ fb at 244 MeV.
• Optimal Energies: The study confirms that for hot fusion, the optimal incident energy requires an "extra-push" energy roughly equal to the sum of the fission barrier height and the shell gap ($V_{eff} \approx V_B + B_f + \Delta$).

5. Significance

• Experimental Guidance: The EBD3 model serves as a practical, validated tool for guiding future experiments aimed at synthesizing elements 119 and 120. It helps prioritize projectile-target combinations (favoring highly asymmetric systems like Sc+Cf) and suggests optimal beam energies.
• Theoretical Insight: By successfully reproducing data with a simplified analytical approach, the model highlights that the systematic behavior of SHN production is largely governed by macroscopic factors (barrier heights, mass asymmetry) and shell effects, rather than requiring complex microscopic transport calculations for initial estimates.
• Uncertainty Quantification: The model provides a robust framework for estimating uncertainties (dominated by fission barrier height uncertainties), offering a reliability comparable to the experimental errors themselves.
• Machine Learning Potential: Due to its high accuracy and simplicity, the EBD3 model is proposed as a suitable dataset generator or baseline for future machine-learning analyses in nuclear reaction modeling.

In conclusion, the paper presents a significant advancement in the theoretical description of super-heavy element synthesis, offering a reliable, low-parameter model that bridges the gap between complex reaction mechanisms and experimental observables.