Probing in-medium effect via giant dipole resonance in the extended quantum molecular dynamics model

This study employs a stochastic approach within the extended quantum molecular dynamics model to demonstrate that the peak position and width of the giant dipole resonance in ${}^{208}$Pb are highly sensitive to the symmetry energy and in-medium nucleon-nucleon cross sections, thereby offering a pathway to constrain the nuclear equation of state and confirming that a significant reduction of free cross sections in the medium is necessary to accurately reproduce experimental data.

The Gist

The Big Picture: Listening to the Atomic Heartbeat

Imagine an atom's nucleus (like the heavy lead atom, Lead-208) not as a static ball of clay, but as a giant, squishy water balloon filled with two types of marbles: protons (positive) and neutrons (neutral).

Sometimes, if you give this balloon a little tap, the protons and neutrons start sloshing back and forth against each other. In physics, this rhythmic sloshing is called the Giant Dipole Resonance (GDR). It's like the nucleus is ringing a bell.

The Problem:
Physicists have a mathematical model to predict how this "bell" rings. They want to know two things:

1. The Pitch (Peak Position): How fast does it vibrate?
2. The Damping (Width): How quickly does the sound fade away?

The "fade away" part is tricky. In a real nucleus, the marbles (nucleons) bump into each other constantly. These collisions create friction, which stops the sloshing. The faster they collide, the quicker the sound dies out.

The Old Tool vs. The New Tool

For decades, scientists used a computer model called EQMD to simulate these collisions.

• The Old Way (Geometric Approach): Imagine trying to predict if two people in a crowded room will bump into each other by drawing a rigid circle around them. If the circles touch, they collide. If not, they don't. This is simple, but it's a bit clumsy. It often misses the subtle "bumps" that happen in a crowded, chaotic environment.
• The New Way (Stochastic Approach): The authors of this paper decided to upgrade the model. Instead of rigid circles, they used a probability cloud. Imagine the nucleons are fuzzy ghosts. The chance of them bumping into each other depends on how much their "fuzz" overlaps. This is more like how people actually move in a crowded dance floor—sometimes they brush past, sometimes they collide, and it's all about the odds.

What They Discovered

The team ran simulations of the Lead-208 nucleus using this new "fuzzy ghost" method and compared the results to real-world experiments. Here is what they found:

1. The "Crowded Room" Effect (Medium Effects)

In a vacuum, two marbles might bounce off each other easily. But inside a nucleus, they are surrounded by a dense crowd of other marbles.

• The Finding: The paper confirms that when nucleons are in this dense crowd, they actually bump into each other less often than they would in empty space.
• The Analogy: Think of trying to run through an empty hallway versus a hallway packed with people. In the packed hallway, you might actually move slower or take different paths, effectively reducing your "collision rate" with specific targets because the crowd gets in the way. The authors found that to match the real data, they had to assume the nucleons' "collision cross-section" (their effective size for bumping) shrinks significantly inside the nucleus.

2. The "Spring" Tension (Symmetry Energy)

The nucleus also has a "spring" that tries to keep the protons and neutrons mixed evenly. This is called Symmetry Energy.

• The Finding: The pitch of the nuclear "bell" is very sensitive to how stiff this spring is. By tuning the stiffness in their model, they found the perfect setting (about 33.2 MeV) that matched the real-world pitch of the Lead-208 resonance.

3. The Perfect Match

When they combined the new fuzzy-collision method with the corrected "crowded room" rules and the right spring stiffness, their simulation matched the experimental data almost perfectly.

• The Result: The old model (rigid circles) couldn't reproduce how fast the sound faded. It was too quiet. The new model (fuzzy ghosts + reduced collisions) got the "damping" just right.

Why Does This Matter?

This isn't just about lead atoms. Understanding how particles behave when they are squeezed together in a dense crowd helps us understand:

• The Equation of State (EoS): This is basically the "rulebook" for how nuclear matter behaves under extreme pressure.
• Neutron Stars: These are giant balls of neutrons in space with gravity so strong it crushes matter to densities far higher than anything on Earth. The rules the authors found for the "crowded room" inside a lead atom help us predict what happens inside a neutron star.

Summary in a Nutshell

The authors took a computer model of the atomic nucleus, replaced a clumsy "rigid circle" collision method with a smarter "probability cloud" method, and discovered that particles inside the nucleus actually collide less often than we thought because the crowd gets in the way. By fixing these rules, they could perfectly predict how a heavy atom "rings" and "fades," giving us a better map of the laws that govern the universe's densest matter.

Technical Summary

1. Problem Statement

The study addresses two primary challenges in nuclear transport modeling:

• Limitations of the Original EQMD Model: The original Extended Quantum Molecular Dynamics (EQMD) model, while successful in many areas, has stagnated regarding its collision term. It relies on a geometric approach for binary nucleon-nucleon (NN) collisions, which has known shortcomings, such as spurious collisions and arbitrary threshold parameters (e.g., maximum cross-section $\sigma_{max}$ and collision distance $d_{coll}$).
• In-Medium Effects on NN Cross Sections: In heavy-ion collisions, nucleons are surrounded by nuclear matter, necessitating the use of an in-medium NN cross section ($\sigma^*_{NN}$) rather than the free-space cross section ($\sigma^{free}_{NN}$). While the consensus is that the medium suppresses the cross section, the magnitude of this reduction is not well-constrained, particularly in the low-energy region.
• GDR Width Discrepancy: Previous attempts to reproduce the width of the Isovector Giant Dipole Resonance (GDR) in heavy nuclei (like $^{208}\text{Pb}$) using standard transport models have often failed to match experimental data, suggesting that the collision term and medium effects are not being treated correctly.

2. Methodology

The authors utilized and updated the "soft" EQMD model (recently developed to correct incompressibility issues) with the following key methodological changes:

• Stochastic Collision Approach: Instead of the original geometric method, the authors implemented a stochastic approach (mean free path method) for the collision term.
  • Collision probability is calculated based on the density overlap of Gaussian wave packets rather than a fixed geometric threshold.
  • The probability of scattering is derived from the integral of the density overlap over the time step, ensuring equivalence to the mean free path method in the limit of zero wave packet width.
  • A specific treatment ($P'_{coll} = 1 - e^{-P_{coll}}$) is used to ensure probabilities do not exceed 1.
• Hamiltonian and Mean Field:
  • Nucleons are treated as Gaussian wave packets with dynamical widths (semi-classical/semi-quantum).
  • The mean field is described using the Skyrme energy density functional with "soft" interaction parameters (SLy7 set), which includes bulk, surface, symmetry, and momentum-dependent terms.
• GDR Excitation and Analysis:
  • The GDR is excited by applying a dipole perturbation operator to the Hamiltonian.
  • The strength function $S(E_\gamma)$ is calculated via the Fourier transform of the time-dependent dipole moment response.
  • The GDR width ($\Gamma$) is defined as the Full Width at Half Maximum (FWHM) of the strength function.
• Parameter Variation: The study systematically varied:
  • The symmetry energy coefficient ($c_s$).
  • The in-medium reduction factor ($\alpha_1$) in the parameterization $\sigma^*_{NN} = (1 - \alpha_1 \frac{\rho}{\rho_0}) \sigma^{free}_{NN}$.

3. Key Contributions

• Implementation of Stochastic Collisions in EQMD: This is the first application of the stochastic collision method within the EQMD framework, replacing the outdated geometric approach.
• Decoupling Mean Field and Collision Effects: The study successfully isolates the effects of the symmetry energy (which primarily affects the GDR peak position) from the collisional damping (which affects the GDR width).
• Quantification of In-Medium Suppression: By comparing simulation results with experimental data for $^{208}\text{Pb}$, the authors provide a quantitative constraint on the in-medium reduction of NN cross sections at low energies.

4. Results

• Failure of the Geometric Approach: Simulations using the original geometric collision method (even with low-energy cutoffs) significantly underestimated the GDR width of $^{208}\text{Pb}$ by at least 1 MeV compared to experimental evaluation data.
• Success of the Stochastic Approach: The stochastic method successfully reproduced the GDR width, provided the correct in-medium cross section was used.
• Sensitivity to Symmetry Energy ($c_s$):
  • The GDR peak position is highly sensitive to the symmetry energy coefficient.
  • To reproduce the experimental peak position ($E_\gamma \approx 13.38$ MeV), a symmetry energy coefficient of $c_s \approx 33.2$ MeV was required.
• Sensitivity to In-Medium Cross Section ($\alpha_1$):
  • The GDR width is highly sensitive to the NN cross section.
  • Using the free-space cross section ($\sigma^{free}_{NN}$) resulted in a significant overestimation of the width.
  • Using a constant cross section of 200 mb (twice the original EQMD limit) still underestimated the width.
  • To reproduce the experimental width ($\Gamma \approx 4.08$ MeV), a significant in-medium reduction was necessary. The best fit was achieved with a reduction parameter of $\alpha_1 \approx 0.57$.
• Stability: The stochastic method was verified to maintain the stability of the nuclear ground state over long evolution times (1000 fm/c), with only tiny fluctuations in density distributions.

5. Significance

• Validation of Medium Effects: The results strongly support the hypothesis that the NN elastic cross section is significantly suppressed in the nuclear medium at low energies (low incident momentum). The required reduction factor ($\alpha_1 \approx 0.57$) implies a substantial decrease in collision probability within the nucleus.
• Improved Transport Modeling: The study demonstrates that the stochastic approach is superior to the geometric approach for treating binary collisions in the EQMD model, particularly when dealing with large cross sections or low-energy regimes where collisional damping is critical.
• Constraint on Nuclear EoS: The work provides a new, independent method to constrain the nuclear Equation of State (EoS) and the in-medium modification of NN interactions, complementing other observables like collective flow and nuclear stopping power.
• Consistency with Other Models: The findings align with results from the Lattice BUU (LBUU) model, reinforcing the unified understanding of in-medium effects across different transport frameworks.

In conclusion, this paper establishes that accurate reproduction of the GDR width in $^{208}\text{Pb}$ requires both a "soft" symmetry energy ($c_s \approx 33$ MeV) and a stochastic treatment of collisions that incorporates a significant in-medium reduction of the NN cross section ($\alpha_1 \approx 0.57$).