Nuclear “waffles”

Background: The dense neutron-rich matter found in supernovae and inside neutron stars is expected to form complex nonuniform phases, often referred to as nuclear pasta. The pasta shapes depend on density, temperature and proton fraction and determine many transport properties in supernovae and neutron star crusts.

Purpose: To characterize the topology and compute two observables, the radial distribution function (RDF) $g\left(r\right)$ and the structure factor $S\left(q\right)$, for systems with proton fractions $Y_p=0.10,0.20,0.30$, and $0.40$ at about one-third of nuclear saturation density, $n=0.050{\mathrm{fm}}^{-3}$, and temperatures near $kT=1\mathrm{MeV}$.

Methods: We use two recently developed hybrid CPU/GPU codes to perform large scale molecular dynamics (MD) simulations with $51200$ and $409600$ nucleons. From the output of the MD simulations we obtain the two desired observables.

Results: We compute and discuss the differences in topology and observables for each simulation. We observe that the two lowest proton fraction systems simulated, $Y_p=0.10$ and $0.20$, equilibrate quickly and form liquidlike structures. Meanwhile, the two higher proton fraction systems, $Y_p=0.30$ and $0.40$, take a longer time to equilibrate and organize themselves in solidlike periodic structures. Furthermore, the $Y_p=0.40$ system is made up of slabs, lasagna phase, interconnected by defects while the $Y_p=0.30$ systems consist of a stack of perforated plates, the nuclear waffle phase.

Conclusions: The periodic configurations observed in our MD simulations for proton fractions $Y_p\ge 0.30$ have important consequences for the structure factors $S\left(q\right)$ of protons and neutrons, which relate to many transport properties of supernovae and neutron star crust. A detailed study of the waffle phase and how its structure depends on temperature, size of the simulation, and the screening length showed that finite-size effects appear to be under control and, also, that the plates in the waffle phase merge at temperatures slightly above $1.0\mathrm{MeV}$ and the holes in the plates form a hexagonal lattice at temperatures slightly lower than $1.0\mathrm{MeV}$.

Figure 1

Charge density isosurfaces of runs with $51200$ nucleons, mean density $n=0.05{\mathrm{fm}}^{-3}$, temperature $kT=1.00$ MeV, and proton fractions $Y_p=0.10,0.20,0.30$, and 0.40 after ${10}^7$ fm/$c$ evolution time. In this figure, and all similar ones throughout this paper, the golden surfaces represent isosurfaces of charge density $n_{\mathrm{ch}}=0.03{\mathrm{fm}}^{-3}$, while the cream color shows regions such that $n_{\mathrm{ch}}>0.03{\mathrm{fm}}^{-3}$. All such figures were generated using paraview [47].

Figure 2

Plots of (a) normalized mean curvature $B/A$ and (b) normalized mean Gaussian curvature $\chi /A$ as a function of simulation time $t$ for four simulations with $Y_p=0.30,n=0.050{\mathrm{fm}}^{-3}$, and $kT=1.00$ MeV.

Figure 3

Charge density isosurfaces of runs with 51 200 (top) and 409 600 nucleons (bottom). The two leftmost (rightmost) figures show the final configurations that used a screening length of $10\mathrm{fm}$ $\left(13.6\mathrm{fm}\right)$ from different angles.

Figure 4

Projection along one axis of the charge density isosurfaces of run with 51 200 and $\lambda =13.6\mathrm{fm}$ as the temperature is increased from $kT=1.0\mathrm{MeV}$ to $1.5\mathrm{MeV}$.

Figure 5

Plots of (a) normalized mean and (b) Gaussian curvatures as a function of temperature $kT$ for a $Y_p=0.30$ simulation started at a temperature $kT=1.0\mathrm{MeV}$ and increased or decreased at a rate of $d\left(kT\right)/dt={10}^{-7}\mathrm{MeV}/(\mathrm{fm}/\mathrm{c})$. The circles (squares) represent the value for the initial mean curvature for the simulations with $\lambda =10.0\mathrm{fm}$ $\left(\lambda =13.6\mathrm{fm}\right)$.

Figure 6

Final projection along the $z$ direction of two of the six perforated plates formed in the run with 51 200 nucleon with proton fraction $Y_p=0.30$ and screening length $\lambda =13.6\mathrm{fm}$. The simulation was started at $kT=1.0\mathrm{MeV}$ and cooled to $0.5\mathrm{MeV}$ (shown). Panel (a) shows two next-nearest neighboring plates, plate 3 (blue) and plate 5 (red), separated by a third plate which is not shown. Panel (b) shows two nearest neighboring plates, plate 4 (blue) and plate 5 (red). The opacity of the plates was decreased so the holes on the blue plates in the back could also be seen.

Figure 7

Charge density isosurfaces of run with 51 200 nucleons and $\lambda =13.6$ fm, cooled from $kT=2.5$ to 0.5 MeV.

Figure 8

Charge density isosurfaces of run with 51 200 nucleons and $\lambda =10.0$ fm, cooled from $kT=2.5$ to 0.5 MeV.

Figure 9

Plots of (a) normalized mean and (b) Gaussian curvatures as a function of temperature $kT$ for 51 200-nucleon systems with $Y_p=0.30$, cooled at rate $d\left(kT\right)/dt=-{10}^{-7}$ MeV/(fm/$c)$ from $kT=2.5$ to 0.5 MeV. One simulation used $\lambda =10$ fm, while the other used $\lambda =13.6$ fm. The circles (squares) represent the initial mean curvatures for the constant $kT=1.0\mathrm{MeV}$ runs with $\lambda =10.0\mathrm{fm}$ $\left(\lambda =13.6\mathrm{fm}\right)$.

Figure 10

Radial distribution functions (RDFs) $g\left(r\right)$ for (a) proton-proton, (b) neutron-proton, and (c) neutron-neutron pairs for 51 200 nucleon simulations with a density of $n=0.050{\mathrm{fm}}^{-3}$, temperature $kT=1.0\mathrm{MeV}$, screening length $\lambda =10\mathrm{fm}$, and proton fractions $Y_p=0.10,0.20,0.30$, and $0.40$.

Figure 11

Angle averaged structure factors $S\left(q\right)$ for (a) protons and (b) neutrons for simulations with density $n=0.050{\mathrm{fm}}^{-3}$, temperature $kT=1.0\mathrm{MeV}$, screening length $\lambda =10\mathrm{fm}$, and proton fractions $Y_p=0.10,0.20,0.30$, and $0.40$.

Figure 12

Angle averaged structure factors $S\left(q\right)$ for (a) protons and (b) neutrons for simulations with 51 200 and 409 600 with density $n=0.050{\mathrm{fm}}^{-3}$, temperature $kT=1.0\mathrm{MeV}$, and proton fraction $Y_p=0.30$.