# Symmetry Energy and the Pauli Exclusion Principle

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## Abstract

**:**

## 1. Introduction

#### 1.1. Antecedents

#### 1.2. Quantum Caveats

#### 1.3. Pauli Blocking

## 2. The Model

## 3. Finite Nuclei

## 4. Infinite Nuclear Matter

#### 4.1. Nuclear Matter

^{−3}for σ = 1.625 fm in Equation (2). The flattening of the energy-density isotherms at sub-saturation densities indicates a departure from the liquid-like phase that exists around saturation density to a mixed liquid-gas phase; at $T=0.10$ MeV the state corresponds to a frozen medium, and to a pasta structure at low densities.

^{−3}. Notice that the system with ρ = 0.12 fm experiences a sharp change around T ≈ 1.5 MeV, denoting a change of phase at that density; previous studies with other potentials [31] connect these discontinuities with a liquid-to-solid change of phase.

#### Pastas of Nuclear Matter

^{−3}depict continuous liquid-like arrangements, while that at $\rho =$ 0.08 fm

^{−3}show pasta-like structures. Figure 8 shows the structure obtained with 5832 nucleons, equal number of protons and neutrons, at $\rho =$0.04 fm

^{−3}and T = 0.8 MeV; the structure on the left shows the simulation cell and its replicas.

#### 4.2. Neutron Star Matter

^{−3}and $0.19$ fm

^{−3}. The resulting compressibility for neutron star matter turns out to be $\kappa =523.3$ MeV and $572.3$ MeV (not to be confused with that for nuclear matter which is about 288 MeV).

^{−3}, for a system with 5832 nucleons with equal numbers of protons and neutrons (with the corresponding spins). Notice that the energy at $\rho =0.12$ fm

^{−3}experiences a sharp change around T ≈ 1.5 MeV, denoting a change of phase. It should be remarked that, at a difference from molecular dynamics calculations, MMC calculations do not explore the energy landscape profusely, and arrive at configurations of minimum energy that correspond to meta-stable liquid states which, at slightly lighter densities drop in internal energy corresponding to non-uniform states; these changes appear as sudden drops in the $E-\rho $ plots at around $\rho \approx $ 0.13 fm

^{−3}, and tend to diminish for $T\lesssim $ 0.1 MeV, and are less noticeable in nuclear matter (cf. Figure 5).

^{−3}. Figure 11 also shows that the potential energy changes its trend at T ≈ 3 MeV, presumably due to the Pauli potential at low and high momentum transfer.

^{−3}≲ ρ ≲ 0.12 fm

^{−3}, denoting the changes from uniformity to pasta-like structures at low densities, but not at $\rho \ge $ 0.13 fm

^{−3}.

^{−3}, with a “bubble” appearing in the simulation cell appearing at $T=1.4$ MeV.

#### Pastas of Neutron Star Matter

^{−3}, while the first bubble can be seen at 0.12 fm

^{−3}. At $\rho =$ 0.08 fm

^{−3}a “pasta-like” structure is formed. Examining the low density region more closely, Figure 17 shows the pseudo-pasta structures obtained at $\rho =$ 0.04, 0.08, and 0.10 fm

^{−3}for a system with 2916 protons and 2916 neutrons.

^{−3}; to clarify the structures, the position of the protons is also presented.

## 5. Symmetry Energy

^{−3}.

## 6. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Metropolis Monte Carlo

^{−3}with 1000 nucleons. All the simulations ran over 1000 nucleons for nuclear matter and 6000 for neutron star matter.

## Appendix B. Symmetry Energy

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**Figure 1.**The upper blue line corresponds to the spatial factor of the Pauli potential, ${V}_{q}={V}_{Pauli}(r,p=0)$, plotted as a function of r in units of ${q}_{0}=6$ fm; notice the truncation at $r=9$ fm. The lower orange curve is the exponential reduction imposed by the momentum dependent factor, ${V}_{p}=exp(-{p}^{2}/2{p}_{0}^{2})$, as it goes from $p=0$ to $p=2{p}_{0}$.

**Figure 2.**Radius of nuclei at the ground state (T = 0.01 MeV). The bottom orange dots corresponds to the MMC simulation. The data points in blue correspond to commonly accepted experimental values. The dashed curve in gray is a least square fit of the MMC data of the type $R=c\phantom{\rule{0.166667em}{0ex}}{A}^{1/3}$, where c is 0.835.

**Figure 3.**Binding energy obtained for the nuclei of Figure 2.

**Figure 4.**Average distance between the nuclei of Figure 2. The insets show the un-normalized probability of finding a nucleon at a distance d, for the pairs $np$, $nn$, and $pp$ in the nuclei ${}^{4}He$ and ${}^{137}Cs$.

**Figure 5.**Average energy per nucleon as a function of the density for nuclear matter at $T=0.10$ and $1.0$ MeV. The systems were constructed with 500 protons and 500 neutrons. The dashed lines correspond to quadratic fittings of the energies in the density range $\rho $ = 0.13–0.18 fm

^{−3}. The estimates of the compressibility $\kappa $ can be seen in the legend.

**Figure 7.**Structures produced in nuclear matter at densities $\rho =$0.08, 0.12, and 0.16 fm

^{−3}and $T=0.01$ MeV. Pseudo-pastas form at sub-saturation densities.

**Figure 8.**Structure produced in nuclear matter at a density of $\rho =$ 0.04 fm

^{−3}and $T=0.8$ MeV. The snapshot on the left shows the simulation cell and its images. Detailed snapshots of the simulation cell can be seen on the right.

**Figure 9.**Average energy per nucleon as a function of the density for neutron star matter. The dashed lines correspond to the fitting of data ($\rho =$ 0.13–0.19 fm

^{−3}) into a quadratic function. The estimates of the compressibility $\kappa $ are shown in the legend.

**Figure 11.**Average kinetic energy per nucleon as a function of the temperature for neutron star matter.

**Figure 12.**Average potential energy per nucleon as a function of the temperature for neutron star matter.

**Figure 16.**Structures formed in neutron star matter at densities $\rho =$ 0.08, 0.12, and 0.16 fm

^{−3}and $T=0.1$ MeV. The system has equal number of protons and neutrons (5832 total nucleons). Notice the formation of pasta-like structures at sub-saturation densities.

**Figure 17.**Pasta structures formed in neutron star matter at densities $\rho =$ 0.04, 0.08, and 0.10 fm

^{−3}and $T=0.1$ MeV for a 5832-nucleon system with a similar number of neutrons and protons.

**Figure 18.**Pasta structures formed in isospin asymmetric neutron star matter at densities $\rho =0.04$ and 0.08 fm

^{−3}and $T=0.1$ MeV for a 5832-nucleon system with 30% of protons and the remainder as neutrons.

**Figure 19.**Energy per nucleon as a function of x for $T=$ 0.5, 1.0, and 2.0 MeV, and $\rho =$ 0.04, 0.06, 0.08, and 0.10 fm

^{−3}.

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Dorso, C.O.; Frank, G.; López, J.A.
Symmetry Energy and the Pauli Exclusion Principle. *Symmetry* **2021**, *13*, 2116.
https://doi.org/10.3390/sym13112116

**AMA Style**

Dorso CO, Frank G, López JA.
Symmetry Energy and the Pauli Exclusion Principle. *Symmetry*. 2021; 13(11):2116.
https://doi.org/10.3390/sym13112116

**Chicago/Turabian Style**

Dorso, Claudio O., Guillermo Frank, and Jorge A. López.
2021. "Symmetry Energy and the Pauli Exclusion Principle" *Symmetry* 13, no. 11: 2116.
https://doi.org/10.3390/sym13112116