# Phase Diagram of Nuclear Pastas in Neutron Star Crusts

^{*}

## Abstract

**:**

^{−3}< ρ < 0.085 fm

^{−3}(0.68 × 10

^{14}g/cm

^{3}< ρ < 1.43 × 10

^{14}g/cm

^{3}), and 0.2 MeV < T < 4.0 MeV, where x is proton content, the density is $\rho $, and the temperature is T. The predictions of the two networks were combined to determine the nuclear pasta phase that is thermodynamically stable at a given x, $\rho $, and T, and a three-dimensional phase diagram that extrapolated slightly the regions of existing molecular dynamics data was computed. The jungle gym and anti-jungle gym structures are prevalent at high temperature and low density, while the anti-jungle gym and anti-gnocchi structures dominate at high temperature and high density. A diversity of structures exist at low temperatures and intermediate density and proton content. The trained models used in this work are open access and available at a public repository to promote comparison to pastas obtained with other models.

## 1. Introduction

#### 1.1. Astrophysical Caveats

^{−3}(or about 1.13 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{14}$ g/cm

^{3}) [9], while molecular dynamics simulations put it at about 0.08 fm

^{−3}(or about 1.34 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{14}$ g/cm

^{3}) [5].

^{−3}$\le \rho \le 0.12$ fm

^{−3}(or 0.34 × 10

^{14}g/cm

^{3}$\le \rho \le 2.02\times {10}^{14}$ g/cm

^{3}), temperatures $0.2$ MeV $\le 4$ MeV, and $Z/A$ between 0.5 and 0.3. [By extrapolation these ranges will be extended to $0.015$ fm

^{−3}$\le \rho \le 0.12$ fm

^{−3}, $0.0$ MeV $\le T\le 5$ MeV, and $0\le Z/A\le 0.5$]. This hopefully covers the ranges of interest for both neutron star crusts and supernovae.

#### 1.2. Terminology

^{−3}(2.74 × 10

^{14}g/cm

^{3}). The “neutrino flow” in a neutron star is important for cooling and for transporting energy from the core to the crust. The “pasta” is a variety of exotic structures that can form in NM and NSM, and are called as such after their resemblance to pasta shapes, such as spaghetti, gnocchi, and lasagna. The “Minkowski functionals” of each structure are their respective values of the Euler characteristic, the volume and area occupied by the pastas, and the mean integral curvature. Finally, the “proton content” $x=Z/A$ is the fraction of protons Z to the total number of nucleons A, which includes both neutrons and protons.

#### 1.3. Nuclear Pasta

^{−3}(0.34 $\times {10}^{14}$ g/cm

^{3}) vary from “spaghetti” (tubes) at $T=1.00$ MeV to “gnocchi” (bundles) at $T=0.01$ MeV. At 0.04 fm

^{−3}(0.68 $\times {10}^{14}$ g/cm

^{3}), the pastas go from “anti-gnocchi” (i.e., holes) at $T=1.00$ MeV to “jungle gym” (cross-linked structures similar to a lattice of rods) at $T=0.01$ MeV. These structures vary with the proton content and with the addition of the electron gas for NSM.

#### 1.4. The Importance of This Study

## 2. Materials and Methods

#### 2.1. Research Design

- (i)
- Generate pastas.
- (ii)
- Compute the Minkowski functionals.
- (iii)
- Interpolate and extrapolate the values of the Minkowski functionals using neural networks.
- (iv)
- Plot the Minkowski functionals for the different types of pastas.

#### 2.2. Classical Molecular Dynamics

^{−3}$\phantom{\rule{4pt}{0ex}}<\rho <\phantom{\rule{4pt}{0ex}}0.085$ fm

^{−3}(0.68 $\times {10}^{14}$ g/cm

^{3}$\phantom{\rule{4pt}{0ex}}<\rho <\phantom{\rule{4pt}{0ex}}1.43\times {10}^{14}$ g/cm

^{3}), and temperatures going from T = 4.0 MeV down to 0.2 MeV. Systems were prepared with nucleons placed at random with a minimum distance of 0.01 fm, and with speeds distributed according to a Maxwell-Boltzmann distribution at a given temperature. The evolution of the system at a given density was studied by solving the equations of motion. The position, momenta, and energy of the nuclei were stored at fixed time-steps. The pressure, temperature, and density were also recorded. At the end, the resulting pasta structures at the various combinations of values of x, $\rho $ and T were obtained.

#### 2.3. Minkowski Functionals

#### 2.4. Neural Network Interpolation

## 3. Results

#### Phase Diagram

^{−3}(0.68 $\times {10}^{14}$ g/cm

^{3}), $\rho >0.085$ fm

^{−3}, (1.43 $\times {10}^{14}$ g/cm

^{3}), $x<0.1$, $T<0.2$ MeV, and $T>4$ MeV; are extrapolations.

## 4. Discussion

^{−3}(1.43 × 10

^{14}g/cm

^{3}), in agreement with [9].

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CMD | Classical molecular dynamics |

MD | Molecular dynamics |

ML | Machine learning |

NM | Nuclear matter |

NSM | Neutron star matter |

NMAE | Normalized mean absolute error |

ReLU | Rectified linear unit |

NRMSE | Normalized root mean square error |

## Appendix A. Molecular Dynamics

#### Appendix A.1. Why Use Classical Molecular Dynamics

#### Appendix A.2. The Classical Molecular Dynamics Model

^{−3}(2.7 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{14}$ g/cm

^{3}), a binding energy $E\left({\rho}_{0}\right)=-16$ MeV/nucleon, and a compressibility of about 250 MeV.

^{−3}$\le \rho \le $ 0.085 fm

^{−3}(0.68 × 10

^{14}g/cm

^{3}$\phantom{\rule{4pt}{0ex}}<\rho <\phantom{\rule{4pt}{0ex}}1.43\times {10}^{14}$ g/cm

^{3}). The size of the cell is determined by the number of particles and the density; however, using it with periodic boundary conditions, it is large enough as to avoid finite size effects of smaller lengths than the cell size [54].

Parameter | Value | Parameter | Value |
---|---|---|---|

${V}_{r}$ | 3097.0 MeV | ${\mu}_{r}$ | 1.648 fm^{−1} |

${V}_{a}$ | 2696.0 MeV | ${\mu}_{a}$ | 1.528 fm^{−1} |

${V}_{0}$ | 379.5 MeV | ${\mu}_{0}$ | 1.628 fm^{−1} |

${r}_{c}$ | 5.4/20 fm |

## Appendix B. Minkowski Functionals

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**Figure 1.**Nuclei arrangements (pastas) stable at a given density and temperature when the proportion of protons is 0.5. See text for details. (color online).

**Figure 2.**Constant-density temperature vs. proton proportion cuts of the three-dimensional phase diagram of nuclear pasta structures of neutron star matter predicted by the current methodology. See text for details. (color online).

**Figure 3.**Pasta phases at $T\approx 0$ MeV for given densities (in fm

^{−3}) and proton proportion predicted by the current methodology (

**left**), and static Hartree-Fock from Ref. [36] (

**right**). (color online).

B < 0 | B∼ 0 | B > 0 | |
---|---|---|---|

$\chi >0$ | Anti-Gnocchi | Gnocchi | |

$\chi \sim 0$ | Anti-Spaghetti | Lasagna | Spaghetti |

$\chi <0$ | Anti-Jungle Gym | Jungle Gym |

**Table 2.**Normalized root-mean-square error (NRMSE) and normalized mean absolute error (NMAE) for the neural network models used in this work.

Test | Full | ||
---|---|---|---|

B | NRMSE | 8.8% | 5.5% |

NMAE | 3.0% | 4.4% | |

$\chi $ | NRMSE | 11.8% | 5.4% |

NMAE | 3.9% | 3.8% |

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**MDPI and ACS Style**

Muñoz, J.A.; López, J.A.
Phase Diagram of Nuclear Pastas in Neutron Star Crusts. *Dynamics* **2024**, *4*, 157-169.
https://doi.org/10.3390/dynamics4010009

**AMA Style**

Muñoz JA, López JA.
Phase Diagram of Nuclear Pastas in Neutron Star Crusts. *Dynamics*. 2024; 4(1):157-169.
https://doi.org/10.3390/dynamics4010009

**Chicago/Turabian Style**

Muñoz, Jorge A., and Jorge A. López.
2024. "Phase Diagram of Nuclear Pastas in Neutron Star Crusts" *Dynamics* 4, no. 1: 157-169.
https://doi.org/10.3390/dynamics4010009