Time-dependent Hartree-Fock approach to nuclear “pasta” at finite temperature

We present simulations of neutron-rich matter at subnuclear densities, like supernova matter, with the time-dependent Hartree-Fock approximation at temperatures of several MeV. The initial state consists of $\alpha$ particles randomly distributed in space that have a Maxwell-Boltzmann distribution in momentum space. Adding a neutron background initialized with Fermi distributed plane waves the calculations reflect a reasonable approximation of astrophysical matter. This matter evolves into spherical, rod-like, and slab-like shapes and mixtures thereof. The simulations employ a full Skyrme interaction in a periodic three-dimensional grid. By an improved morphological analysis based on Minkowski functionals, all eight pasta shapes can be uniquely identified by the sign of only two valuations, namely the Euler characteristic and the integral mean curvature. In addition, we propose the variance in the cell density distribution as a measure to distinguish pasta matter from uniform matter.

Figure 1

Typical shapes of pasta structures at the lowest temperatures. These continue to evolve with time but do not change their morphological character anymore. Bubble shape illustrations show gas phase, which is indicated by the color-scale [from $0.03{\mathrm{fm}}^{-3}$ (blue/light gray) to $0.12{\mathrm{fm}}^{-3}$ (red/dark gray)]. (a) Sphere. (b) Rod. (c) Rod(2). (d) Rod(3). (e) Slab. (f) Rod(2) bubble. (g) Rod bubble. (h) Sphere bubble.

Figure 2

Map of pasta shapes achieved in a TDHF calculation starting from a gas of $\alpha$ particles with neutron background for various real temperatures and mean densities. The proton fraction is 1/3. Each dot represents two calculations. The solid black line shows a phase separation line discussed in Sec. 6.

Figure 3

$W_2$, proportional to the integral mean curvature, and $W_3$, proportional to the Euler characteristic as function of threshold density for the samples shown in Fig. 1.

Figure 4

Threshold density profiles for calculations varying in density and temperature. On each single plot the number of grid points $n$ corresponding to a certain density value is plotted. The dashed lines show the mean density.