Phases of hot nuclear matter at subnuclear densities

Structure of hot dense matter at subnuclear densities is investigated by quantum molecular dynamics (QMD) simulations. We analyze nucleon distributions and nuclear shapes using two-point correlation functions and Minkowski functionals to determine the phase-separation line and to classify the phase of nuclear matter in terms of the nuclear structure. Obtained phase diagrams show that the density of the phase boundaries between the different nuclear structures decreases with increasing temperature due to the thermal expansion of nuclear matter region. The critical temperature for the phase separation is $\gtrsim 6\text{MeV}$ for the proton fraction $x=0.5$ and $\gtrsim 5\text{MeV}$ for $x=0.3$. Our result suggests the existence of “spongelike” phases with negative Euler characteristic in addition to the simple “pasta” phases in supernova cores until $T\lesssim 3\text{MeV}$.

Figure 1

Sampled values of the instantaneous effective temperature $T_{\text{eff}}$ and their long-time averages $⟨T_{\text{eff}}⟩$ for $\rho =0.1,0.3,0.5$, and $0.8{\rho }_0$.

Figure 2

(Color online) The nucleon distributions for $x=0.5$, $\rho =0.225{\rho }_0$ at the temperatures of 1, 2, 3 and $4\text{MeV}$. 2048 nucleons are contained in the simulation box of size $L_{\text{box}}=38.07\text{fm}$. These figures show the top views along the axis of the rodlike nuclei. Protons are represented by the red particles, and neutrons by the green ones.

Figure 3

(Color online) The same as Fig. 2 at the temperatures of 1, 2, and $3\text{MeV}$ for the system with 16384 nucleons. The box size $L_{\text{box}}$ is $76.14\text{fm}$. The upper panels show the top views along the axis of the cylindrical nuclei at $T=0$, and the lower ones show the side views.

Figure 4

(Color online) The nucleon distributions for $x=0.5$, $\rho =0.4{\rho }_0$ at the temperatures of 1,2,3, and $4\text{MeV}$. 2048 nucleons are contained in the simulation box of size $L_{\text{box}}=31.42\text{fm}$. Protons are represented by the red particles, and neutrons by the green ones.

Figure 5

(Color online) The same as Fig. 4 at the temperatures of 1,2, and $3\text{MeV}$ for the system with 16384 nucleons. The box size $L_{\text{box}}$ is $62.84\text{fm}$. These figures are shown in the direction parallel to the plane of the slablike nuclei at $T=0$.

Figure 6

Comparison of the two-point correlation functions ${\xi }_{NN}$ for the systems with 2048 and 16384 nucleons. These are calculated for $x=0.5$ and $\rho =0.225{\rho }_0$, where the system is in the phase with rodlike nuclei at zero temperature. The error bars are the standard deviations in the long-time average.

Figure 7

The same as Fig. 6 for $x=0.5$ and $\rho =0.4{\rho }_0$, where the system is in the phase with rodlike nuclei at zero temperature.

Figure 8

(Color online) The nucleon distributions for $x=0.3$, $\rho =0.175{\rho }_0$ at the temperatures of 1, 2, and $3\text{MeV}$. 16384 nucleons are contained in the simulation box of size $L_{\text{box}}=82.788\text{fm}$. Protons are represented by the red particles, and neutrons by the green ones. The upper panels show the top views along the axis of the cylindrical nuclei at $T=0$, and the lower ones show the side views.

Figure 9

(Color online) The nucleon distributions for $x=0.3$, $\rho =0.34{\rho }_0$ at the temperatures of 1, 2, and $3\text{MeV}$. 16384 nucleons are contained in the simulation box of size $L_{\text{box}}=66.34\text{fm}$. Protons are represented by the red particles, and neutrons by the green ones. These figures are shown in the direction parallel to the plane of the slablike nuclei at $T=0$.

Figure 10

Two-point correlation function of the density fluctuation calculated for $N=2048$, $x=0.5$ and $\rho =0.225{\rho }_0$, where the system is in the phase with rodlike nuclei at zero temperature. The error bars are the standard deviations in the long-time average.

Figure 11

Two-point correlation function of the density fluctuation calculated for $N=2048$, $x=0.5$ and $\rho =0.4{\rho }_0$, where the system is in the phase with slablike nuclei at zero temperature. The error bars are the standard deviations in the long-time average.

Figure 12

Two-point correlation function of the density fluctuation calculated for $N=2048$, $x=0.3$ and $\rho =0.175{\rho }_0$, where the system is in the phase with rodlike nuclei at zero temperature. The error bars are the standard deviations in the long-time average.

Figure 13

Two-point correlation function of the density fluctuation calculated for $N=2048$, $x=0.3$ and $\rho =0.35{\rho }_0$, where the system is in the phase with slablike nuclei at zero temperature. The error bars are the standard deviations in the long-time average.

Figure 14

Euler characteristic density for various temperatures plotted as a function of the threshold density ${\rho }_{\text{th}}$. The data shown here are calculated for $x=0.5$ and $\rho =0.225{\rho }_0$, where the system is in the phase with rodlike nuclei at zero temperature. The error bars are the standard deviations in the long-time average. For $T=2\text{MeV}$, the plateau region lies between ${\rho }_{\text{th}}\simeq 0.45{\rho }_0$ and $\simeq 0.6{\rho }_0$, where $\chi ∕V$ is negative.

Figure 15

The same as Fig. 14 for $\rho =0.4{\rho }_0$, where the system is in the phase with slablike nuclei at zero temperature.

Figure 16

Euler characteristic density for various temperatures plotted as a function of the threshold density ${\rho }_{\text{th}}$. The data shown here are calculated for $x=0.3$ and $\rho =0.175{\rho }_0$, where the system is in the phase with rodlike nuclei at zero temperature. The error bars are the standard deviations in the long-time average.

Figure 17

The same as Fig. 16 for $0.35{\rho }_0$, where the system is in the phase with slablike nuclei at zero temperature.

Figure 18

Phase diagram of matter at $x=0.5$ plotted in the $\rho \text{−}T$ plane. The dashed and the dotted lines on the diagram show the phase separation line and the limit below which the nuclear surface can be identified, respectively; thus the regions above the dotted line and the dashed line show the nonuniform regions, which are difficult to be classified into specific phases in terms of the nuclear structure. The dash-dotted lines are the phase boundaries between the different nuclear shapes. The symbols SP, C, S, CH, SH, U stand for nuclear shapes, i.e., sphere, cylinder, slab, cylindrical hole, spherical hole, and uniform, respectively. The parentheses (A,B) show intermediate phases between A and B phases. The regions (a)–(g) correspond to the nuclear shapes characterized by $⟨H⟩$ and $\chi ∕V$ as follows: (a) $⟨H⟩>0,\chi ∕V>0$; (b) $⟨H⟩>0,\chi ∕V=0$; (c) $⟨H⟩>0,\chi ∕V<0$; (d) $⟨H⟩=0,\chi ∕V=0$; (e) $⟨H⟩<0,\chi ∕V<0$; (f) $⟨H⟩<0,\chi ∕V=0$; (g) $⟨H⟩<0,\chi ∕V>0$. Simulations have been carried out at points denoted by circles.

Figure 19

Phase diagram of matter at $x=0.3$ plotted in the$\rho \text{−}T$ plane.