Dynamics of fragment formation in neutron-rich matter

Background: Neutron stars are astronomical systems with nucleons subjected to extreme conditions. Due to the longer range Coulomb repulsion between protons, the system has structural inhomogeneities. Several interactions tailored to reproduce nuclear matter plus a screened Coulomb term reproduce these inhomogeneities known as nuclear pasta. These structural inhomogeneities, located in the crusts of neutron stars, can also arise in expanding systems depending on the thermodynamic conditions (temperature, proton fraction, etc.) and the expansion velocity.

Purpose: We aim to find the dynamics of the fragment formation for expanding systems simulated according to the little big bang model. This expansion resembles the evolution of merging neutron stars.

Method: We study the dynamics of the nucleons with semiclassical molecular dynamics models. Starting with an equilibrium configuration, we expand the system homogeneously until we arrive at an asymptotic configuration (i.e., very low final densities). We study, with four different cluster recognition algorithms, the fragment distribution throughout this expansion and the dynamics of the cluster formation.

Results: Studying the topology of the equilibrium states, before the expansion, we reproduced the known pasta phases plus a novel phase we called pregnocchi, consisting of proton aggregates embedded in a neutron sea. We have identified different fragmentation regimes, depending on the initial temperature and fragment velocity. In particular, for the already mentioned pregnocchi, a neutron cloud surrounds the clusters during the early stages of the expansion, resulting in systems that give rise to configurations compatible with the emergence of the $r$ process.

Conclusions: We showed that a proper identification of the cluster distribution is highly dependent on the cluster recognition algorithm chosen, and found that the early cluster recognition algorithm (ECRA) was the most stable one. This approach allowed us to identify the dynamics of the fragment formation. These calculations pave the way to a comparison between Earth experiments and neutron star studies.

Figure 1

Potential energy of the proton-neutron interaction of different models: SSP, CMD, and new medium.

Figure 2

Binding energies of ground-state nuclei obtained with CMD, SSP, and new medium models. Note that the new medium model yields results much closer to the experimental ones.

Figure 3

Schematic representation of 2D clusters, recognized only in the cell and not through the periodic walls, labeled as N, S, W, E. The clusters inside the cell are labeled from 1 to 6.

Figure 4

Graph of the clusters with connections labeled by the wall of the boundary they connect through. The graph can be divided in two subgraphs that do not connect: 1–2–3–4 and 5–6. Each of these subgraphs is a cluster when periodic boundary conditions are considered.

Figure 5

Snapshots of configurations for different parametrizations of the nuclear interaction, all with the same thermodynamic conditions: $x=0.1, \rho =0.05{\text{fm}}^{-3}$, and $T=0.1\text{MeV}$. The qualitative differences between the CMD medium potential and the other two parametrizations (new medium and SSP) are evident. We call the structures shown in new medium and SSP models pregnocchi. Note that neutrons are represented by points to avoid hindering the visualization of the proton structure.

Figure 6

Snapshots of configurations for different parametrizations of the nuclear interaction, all with the same thermodynamic conditions: $x=0.4, \rho =0.05{\text{fm}}^{-3}$, and $T=0.1\text{MeV}$. The qualitative differences between the CMD medium potential and the other two parametrizations (new medium and SSP) are evident. While the CMD potential shows a jungle gym structure, both new medium and SSP show lasagna structures that are slightly different from each other.

Figure 7

Asymptotic mass distribution for $x=0.1, \rho =0.05{\text{fm}}^{-3}$, and $T=0.8\text{MeV}$ and two different expansion velocities: fast $\dot{\eta }=0.01c/\text{fm}$ and slow $\dot{\eta }=0.0001c/\text{fm}$.

Figure 8

Asymptotic mass distribution for $x=0.4, \rho =0.05{\text{fm}}^{-3}$, and $T=0.1\text{MeV}$ and two different expansion velocities: fast $\dot{\eta }=0.01c/\text{fm}$ and slow $\dot{\eta }=0.0001c/\text{fm}$.

Figure 9

Three different expansions of neutron star matter: lasagna (fast expansion): $x=0.4, \dot{\eta }=0.01c/\text{fm}, T=0.8\text{MeV}$; lasagna (slow expansion): $x=0.4, \dot{\eta }=0.0001c/\text{fm}, T=0.8\text{MeV}$; pregnocchi: $x=0.1, \dot{\eta }=0.0001c/\text{fm}, T=0.1\text{MeV}$.

Figure 10

Initial and asymptotic mass distributions for a system with $x=0.1, \rho =0.05{\text{fm}}^{-3}$ and $T=0.1\text{MeV}$, for a slow expansion ($\dot{\eta }=0.0001c/\text{fm}$), with the MSTE cluster recognition.

Figure 11

Initial and asymptotic mass distributions for a system with $x=0.1, \rho =0.05{\text{fm}}^{-3}$, and $T=0.1\text{MeV}$, for a slow expansion ($\dot{\eta }=0.0001c/\text{fm}$), with the proton-MST cluster recognition.

Figure 12

Initial and asymptotic mass distributions for a system with $x=0.1, \rho =0.05{\text{fm}}^{-3}$, and $T=0.1\text{MeV}$, for a slow expansion ($\dot{\eta }=0.0001c/\text{fm}$), with the ECRA cluster recognition. In comparing with Fig. 11, notice the difference in the $y$ scale.

Figure 13

Mass of the largest cluster for MSTE, MSTpC, and ECRA for the early stages of the evolution, for the pregnocchi configuration. We can see that the ECRA fragment remains relatively stable and stabilizes quickly, while the other two algorithms yield fragments that are always larger and stabilize more slowly.

Figure 14

Mass of the largest cluster for MSTE, MSTpC, and ECRA, for the early stages of the evolution, for the lasagna configuration. Note that, as in the previous case, the ECRA algorithm recognizes very early in the expansion the fracture of the clusters. In the asymptotic regime (for very large times, not shown in the figure) all three algorithms yield the same result.