Fragment productions in DJBUU and SQMD: comparative study

This paper presents a comparative study of fragment production in $^{208}$Pb+$^{40,48}$Ca reactions at 50 and 100 AMeV using DJBUU and SQMD transport codes, revealing generally similar outcomes but highlighting specific discrepancies at 100 AMeV for central collisions attributed to differences in the equation of state and model stability.

The Gist

Imagine you are trying to predict what happens when two giant, complex Lego structures crash into each other at high speed. Do they shatter into tiny dust? Do they bounce off? Or do they smash together to form a new, weirdly shaped giant block?

This is essentially what the scientists in this paper are doing, but instead of Legos, they are smashing atomic nuclei (the cores of atoms) together. Specifically, they are crashing a massive Lead nucleus into Calcium nuclei.

Here is the story of their research, broken down into simple concepts:

1. The Goal: Building a "Rare Isotope" Factory

The scientists are working toward a future project called RAON in Korea. Think of RAON as a massive, high-tech "particle factory." Its job is to smash atoms together to create Rare Isotopes—unstable, exotic versions of elements that don't exist naturally on Earth but are crucial for understanding how stars explode and how the universe was made.

To build this factory, you need to know exactly what will happen when you smash things together. You can't just guess; you need a simulation.

2. The Two Competing Simulators (The "Video Games")

To predict the crash, the team used two different computer programs (models) to simulate the collision. Think of these as two different video game engines trying to render the same crash scene:

• DJBUU (The "Smooth Flow" Engine): This model treats the atomic particles like a smooth, flowing fluid. It's very stable and good at predicting the average behavior of the crowd. It's like watching a river flow; you see the general direction, but you don't track every single water molecule individually.
• SQMD (The "Individual Chaos" Engine): This model treats every single particle as an individual actor with its own personality. It tracks every particle's movement and how they bump into each other. It's like a chaotic mosh pit where everyone is jostling individually. It's great for seeing the "fluctuations" or the wild, unpredictable moments.

3. The Experiment: The Crash Test

The team ran simulations of crashing a Lead atom into a Calcium atom at two different speeds:

• Speed 1: A moderate crash (50 AMeV).
• Speed 2: A high-speed, violent crash (100 AMeV).

They also changed the "aim" of the crash:

• Head-on (b=0): A direct, full-force collision.
• Glancing (b=6): A sideswipe where they barely touch.

4. The Results: What Did They Find?

The Good News:
For most crashes, both computer programs agreed! Whether they used the "Smooth Flow" engine or the "Individual Chaos" engine, they produced very similar results. The big chunks of debris (fragments) they created were roughly the same size and shape. This gives scientists confidence that their predictions for the RAON factory are reliable.

The Interesting Disagreement:
However, when they did a head-on crash at the highest speed, the two programs started to disagree noticeably.

• DJBUU predicted the crash would leave behind a slightly larger, more stable chunk.
• SQMD predicted the chunk would be smaller and more broken up.

Why did they disagree?
The authors explain this using two main reasons:

1. The "Rules of the Road" (Equation of State): Both programs have slightly different rules for how "squishy" or "stiff" nuclear matter is. At high speeds, the matter gets squeezed very hard. Because the rules are different, the "squish" results in different outcomes.
2. Stability vs. Chaos: The "Smooth Flow" engine (DJBUU) is naturally more stable, so it resists breaking apart. The "Individual Chaos" engine (SQMD) is more sensitive to small fluctuations, so it breaks things apart more easily.

5. The "Neutron" Twist

They also noticed something weird with the Neutron-Rich Calcium (Calcium-48). Even though this version has more neutrons (which usually makes things heavier), the resulting fragments were actually smaller than when they used the normal Calcium (Calcium-40).

The Analogy:
Imagine a group of people trying to huddle together to stay warm. If you add a bunch of people who are constantly pushing everyone else away (the neutrons acting due to "symmetry energy"), the group can't huddle as tightly. They get pushed apart, resulting in a smaller, looser cluster. The "neutron-rich" version pushes the group apart more effectively, preventing the formation of a giant fragment.

The Bottom Line

This paper is a "quality control" check. The scientists compared their two main tools to make sure they are working correctly.

• Overall: The tools agree, which is great news for the future Rare Isotope factory.
• Specifics: They found a specific scenario (high-speed, head-on crash) where the tools differ. This isn't a failure; it's a discovery! It tells scientists exactly where they need to refine their rules to get a perfect prediction.

In short, they are tuning the "physics engines" of their simulation so that when the real RAON factory opens, they know exactly what rare treasures they will find in the debris.

Technical Summary

1. Problem Statement

The study addresses the need to validate and compare transport models used to simulate heavy-ion collisions, specifically for the production of rare isotopes at facilities like RAON (Rare Isotope Accelerator complex for ON-line experiments) in Korea.

• Context: Rare isotope (RI) science relies on the interplay between experiment, theory, and simulation. Understanding nuclear symmetry energy and the properties of dense nuclear matter requires accurate modeling of heavy-ion collisions.
• The Challenge: Two distinct classes of transport models exist:
  • BUU-type (Boltzmann-Uehling-Uhlenbeck): Deterministic, mean-field based, uses test particles, and generally lacks event-by-event fluctuations.
  • QMD-type (Quantum Molecular Dynamics): Stochastic, based on an N-body Hamiltonian, and better suited for event-by-event analysis of fragment formation.
• Objective: To perform a direct comparison of the DJBUU (DaeJeon Boltzmann-Uehling-Uhlenbeck) and SQMD (Sindong Quantum Molecular Dynamics) codes by analyzing the production of primary fragments in $^{208}\text{Pb} + ^{40,48}\text{Ca}$ reactions. The goal is to identify discrepancies arising from different equations of state (EoS) and inherent model stabilities.

2. Methodology

The authors simulated central and peripheral collisions using two specific reaction systems at two beam energies.

• Reaction Systems:
  • Projectile: $^{208}\text{Pb}$
  • Targets: $^{40}\text{Ca}$ and $^{48}\text{Ca}$ (to study isospin asymmetry effects).
  • Beam Energies ($E_{\text{beam}}$): 50 and 100 AMeV.
  • Impact Parameters ($b$): 0 fm (central), 3 fm, and 6 fm (peripheral).
• Simulation Parameters:
  • DJBUU: Uses a test-particle method with 100 test particles per nucleon. Simulations were run for 10 events. The model employs a specific profile function for test particles and a relativistic mean-field potential derived from an extended parity doublet model.
  • SQMD: Uses Gaussian wave packets for nucleons ($\sigma_r = 1.3$ fm) and a Skyrme interaction potential. Simulations were run for 10,000 events (verified with 20,000 for robustness).
  • Duration: Simulations ran until $t = 300$ fm/c.
• Fragment Identification:
  • DJBUU: Identifies clusters by scanning a 3D grid ($160^3$ cells) for regions where baryon density $\rho \geq 0.1 \rho_0$ (where $\rho_0 = 0.17 \text{ fm}^{-3}$). Test particles within these cells are counted to determine proton ($Z$) and neutron ($N$) numbers.
  • SQMD: Uses the Minimum Spanning Tree (MST) algorithm. Nucleons are connected if their distance is $< 3.5$ fm. Connected groups are identified as primary fragments.
• Metric: The study focuses on the "Biggest Fragment" (BF)—the fragment with the largest mass number ($A$) in each event—and aggregates these results.

3. Key Contributions

• Direct Model Comparison: Provides a rare, direct comparison of primary fragment production between a BUU-type and a QMD-type model without the use of "afterburners" (statistical decay models), isolating the dynamics of the transport phase.
• Isospin Asymmetry Analysis: Explicitly compares $^{40}\text{Ca}$ and $^{48}\text{Ca}$ targets to isolate the effects of nuclear symmetry energy on fragment formation.
• Equation of State (EoS) Sensitivity: Demonstrates how differences in the symmetry energy parameterization ($\sigma$) between the two models affect outcomes at higher densities and energies.
• Stability Analysis: Highlights the inherent stability differences between deterministic (BUU) and stochastic (QMD) approaches, particularly in central collisions.

4. Results

• General Agreement: At lower energies ($50$ AMeV) and peripheral collisions ($b=6$ fm), both models produce similar fragment distributions (typically $A \approx 162-168$, $Z \approx 70-74$).
• Significant Discrepancies at High Energy/Central Collisions:
  • At $E_{\text{beam}} = 100$ AMeV and $b = 0$ fm, a noticeable difference emerges.
  • DJBUU produces significantly larger fragments (e.g., $A \approx 123$ for $^{40}\text{Ca}$) compared to SQMD (e.g., $A \approx 78$ for $^{40}\text{Ca}$).
  • Cause: This is attributed to:
    1. Equation of State: The symmetry energy term ($E_{\text{sym}} \propto \rho^\sigma$) differs between models ($\sigma=1$ for DJBUU vs. $\sigma=0.7$ for SQMD). At higher beam energies, maximum density increases, amplifying the effect of this difference.
    2. Model Stability: BUU models are generally more stable against fragmentation than QMD models, which naturally include fluctuations that can lead to earlier or more extensive breakup.
• Isospin Effects:
  • In $^{208}\text{Pb} + ^{48}\text{Ca}$ reactions (neutron-rich), the resulting fragments are smaller than in $^{208}\text{Pb} + ^{40}\text{Ca}$ reactions, even though the system has more neutrons.
  • Mechanism: The nuclear symmetry energy pushes excess neutrons out of the dense core, disrupting the formation of large clusters. This effect is more pronounced in the neutron-rich $^{48}\text{Ca}$ system, confirmed by higher isospin asymmetry ($\alpha_I$) observed in the collision center.
• Impact Parameter Dependence: The difference between models increases as the impact parameter decreases (from 6 fm to 0 fm). Central collisions involve the bulk of the system and are highly sensitive to the EoS, whereas peripheral collisions involve fewer nucleons and show less sensitivity.

5. Significance

• Validation for RAON: The study validates the use of both DJBUU and SQMD for predicting rare isotope production at the RAON facility, while highlighting the specific conditions (high energy, central collisions) where model choice significantly impacts predictions.
• Understanding Symmetry Energy: The results underscore the critical role of the density dependence of the symmetry energy in determining fragment yields in heavy-ion collisions.
• Future Improvements: The authors suggest that incorporating surface terms (time-dependent nucleon widths) into DJBUU could further align the two models.
• Methodological Insight: The paper clarifies that discrepancies between BUU and QMD models are not merely numerical but stem from fundamental differences in how fluctuations and the equation of state are treated, which is crucial for interpreting experimental data from future rare isotope facilities.