Nuclear “pasta” formation

The formation of complex nonuniform phases of nuclear matter, known as nuclear pasta, is studied with molecular dynamics (MD) simulations containing 51200 nucleons. A phenomenological nuclear interaction is used that reproduces the saturation binding energy and density of nuclear matter. Systems are prepared at an initial density of $0.10{\mathrm{fm}}^{-3}$ and then the density is decreased by expanding the simulation volume at different rates to densities of $0.01{\mathrm{fm}}^{-3}$ or less. An originally uniform system of nuclear matter is observed to form spherical bubbles (“swiss cheese”), hollow tubes, flat plates (“lasagna”), thin rods (“spaghetti”) and, finally, nearly spherical nuclei with decreasing density. We explicitly observe nucleation mechanisms, with decreasing density, for these different pasta phase transitions. Topological quantities known as Minkowski functionals are obtained to characterize the pasta shapes. Different pasta shapes are observed depending on the expansion rate. This indicates nonequilibrium effects. We use this to determine the best ways to obtain lower energy states of the pasta system from MD simulations and to place constraints on the equilibration time of the system.

Figure 1

Normalized Minkowski functionals as a function of the threshold charge density $n_{\text{th}}$ for systems at ten different nucleon densities $n$ indicated by the lines at the top. Configurations analyzed were obtained by stretching the pasta system from $0.10$ to $0.01{\mathrm{fm}}^{-3}$ at a rate of $\dot{\xi }=1.0\times {10}^{-7}$ $c$/fm. From top to bottom the figures show (a) occupied volume fraction $V/V_{\mathrm{tot}}$, (b) surface area per unit volume $A/V_{\mathrm{tot}}$, (c) average mean curvature $B/A$, and (d) average Gaussian curvature $\chi /A$.

Figure 2

Nucleon pair-pair autocorrelation function as a function of pair distance for our five constant-density runs. See Sec. 3a for description of the five runs. From top to bottom the plots show (a) proton-proton correlations $g_{pp}\left(r\right)$, (b) neutron-proton correlations $g_{np}\left(r\right)$, and (c) neutron-neutron correlations $g_{nn}\left(r\right)$.

Figure 3

Side by side comparison of charge density isosurfaces of configurations with densities of 0.010, 0.025, 0.050, 0.075, and 0.090 ${\mathrm{fm}}^{-3}$ obtained from four different runs. The golden surfaces represent isosurfaces of charge density with $n_{\mathrm{ch}}=0.03{\mathrm{fm}}^{-3}$ while the cream color shows regions such that $n_{\mathrm{ch}}>0.03{\mathrm{fm}}^{-3}$. The first “Runs” column shows configurations obtained neglecting Coulomb interactions while the second Runs column shows constant density runs. Finally, the third and fourth Runs columns show respectively configurations obtained with the fastest stretching rate, $\dot{\xi }=1.0\times {10}^{-5}$ fm/$c$, and the slowest stretching rates, $\dot{\xi }=1.0\times {10}^{-7}$ fm/$c$. This figure was generated using paraview [41].

Figure 4

Mass fraction $W_A$ obtained from the MST algorithm discussed in Sec. 2c. $W_A$ is the mass fraction of the system composed of nuclei with mass number $A$ at a densities of $0.010{\mathrm{fm}}^{-3}$ and less for three different runs: one from (a) stretching the box at a rate $\dot{\xi }=1.0\times {10}^{-7}$ $c$/fm, one obtained from (b) stretching the box at $\dot{\xi }=1.0\times {10}^{-5}$ $c$/fm, and, finally, one obtained from (c) letting the system equilibrate at a constant density.

Figure 5

System at a density of $0.010{\mathrm{fm}}^{-3}$ obtained from stretching the box at a rate of $\dot{\xi }=1.0\times {10}^{-7}$ $c$/fm from $0.10{\mathrm{fm}}^{-3}$. The golden surfaces represent isosurfaces of charge density with $n_{\mathrm{ch}}=0.03{\mathrm{fm}}^{-3}$ while the cream color shows regions such that $n_{\mathrm{ch}}>0.03{\mathrm{fm}}^{-3}$. The system is shown at an angle that makes it easier to see that the nuclei form some type of lattice. This figure was generated using paraview [41].

Figure 6

System at a density of $0.010{\mathrm{fm}}^{-3}$ obtained from an initial random configuration of protons and neutrons at the same density. The golden surfaces represent isosurfaces of charge density with $n_{\mathrm{ch}}=0.03{\mathrm{fm}}^{-3}$ while the cream color shows regions such that $n_{\mathrm{ch}}>0.03{\mathrm{fm}}^{-3}$. The system is shown from the same angle as Fig. 5. This figure was generated using paraview [41].

Figure 7

(a) The mass fractions $W_A^{\mathrm{MST}}$ and $W_A^{\mathrm{MSTE}}$ obtained from the MST and MSTE algorithms discussed in Sec. 2c. $W_A$ is the percentage of the system that is composed of nuclei with mass number $A$ for the system stretched at a rate of $\dot{\xi }=1.0\times {10}^{-7}$ $c$/fm when at a density of $0.010{\mathrm{fm}}^{-3}$ or less. (b) The same as (a) with the mass fraction $W_A^{\mathrm{MST}}$ shifted by 3 nucleon units, $A\to A-3$. (c) The ratio between $W_A^{\mathrm{MSTE}}$ and $W_{A-3}^{\mathrm{MST}}$.

Figure 8

Normalized mean breadth $B/A$ as a function of the density $n$ for three complete calculations using stretch rates $\dot{\xi }={10}^{-5}$, ${10}^{-6}$, and ${10}^{-7}$ $c$/fm and one calculation with stretch rate ${\dot{\xi }}^{*}={10}^{-6}$ $c$/fm ignoring Coulomb interactions. Results are compared to five computations done at constant densities of $0.010$, $0.025$, $0.050$, $0.075$, and $0.090{\mathrm{fm}}^{-3}$.

Figure 9

Lines represent the normalized Euler characteristic $\chi /A$ as a function of the density $n$ for three complete calculations using stretch rates $\dot{\xi }={10}^{-5}$, ${10}^{-6}$, and ${10}^{-7}$ $c$/fm and one calculation with stretch rate ${\dot{\xi }}^{*}={10}^{-6}$ $c$/fm ignoring Coulomb interactions. Results are compared to five computations done at constant densities of $0.010$, $0.025$, $0.050$, $0.075$, and $0.090{\mathrm{fm}}^{-3}$.