Equilibrium nuclear ensembles taking into account vaporization of hot nuclei in dense stellar matter

We investigate the high-temperature effect on the nuclear matter that consists of mixture of nucleons and all nuclei in the dense and hot stellar environment. The individual nuclei are described within the compressible-liquid-drop model that is based on Skyrme interactions for bulk energies and that takes into account modifications of the surface and Coulomb energies at finite temperatures and densities. The free-energy density is minimized with respect to the individual equilibrium densities of all heavy nuclei and the nuclear composition. We find that their optimized equilibrium densities become smaller and smaller at high temperatures because of the increase in thermal contributions to bulk free energies and the reduction of surface energies. The neutron-rich nuclei become unstable and disappear one after another at given temperatures. The calculations are performed for two sets of model parameters leading to different values of the slope parameter in the nuclear-symmetry energy. It is found that the larger slope parameter reduces the equilibrium densities and the melting temperatures. We also compare the proposed model with some other approaches and find that the mass fractions of heavy nuclei in the previous calculations that omit vaporization are underestimated at $T\lesssim 10$ MeV and overestimated at $T\gtrsim 10$ MeV. The further sophistication of calculations of nuclear vaporization and of light clusters would be required to construct the equation of state for explosive astrophysical phenomena.

Figure 1

Free energies per baryon of symmetric (left panel, $x=0.5$) and asymmetric (right panel, $x=0.1$) uniform nuclear matter for parameter sets B (blue solid lines) and E (red dashed lines) at $T=0$ (thin lines), 4, 8, 12, and 16 MeV (medium lines) and $T_c$ (thick lines). The critical temperatures are 18.1 MeV for both parameter sets at $x=0.5$ and are 5.7 and 5.5 MeV for the parameter sets B and E, respectively, at $x=0.1$ as shown in Fig. 2.

Figure 2

The critical temperatures for bulk nuclear matter, above which the bulk pressure has no local minimum, as a function of charge fraction for parameter sets B (blue solid lines) and E (red dashed lines).

Figure 3

Mass fraction $X_i$ (left column) and equilibrium density $n_{\mathrm{eqi}}$ (right column) in $(N,Z)$ plane for the parameter set E at $n_B=0.3n_0$, $Y_p=0.2$, and $T=1$, 3, 5, and 10 MeV (from top to bottom).

Figure 4

Mass fraction $X_i$ (left column) and equilibrium density $n_{\mathrm{eqi}}$ (right column) for the nuclei with $A_i=50$ as functions of charge fraction $Z_i/A_i$ for parameter sets B (blue solid lines) and E (red dashed lines) at $T=1$ MeV (thin lines), 5 MeV (medium lines), and 10 MeV(thick lines) and at $(n_B,Y_p)=(0.1n_0,0.2)$, $(0.1n_0,0.4)$, $(0.3n_0,0.2)$, and $(0.3n_0,0.4)$ from top to bottom.

Figure 5

Equilibrium density of ${}^{50}\mathrm{Ca}$ (thin lines) and average denisty over all the heavy nuclei, $\langle n_{\mathrm{eq}}\rangle$, (thick lines) as functions of $T$ for parameter sets B (blue solid lines) and E (red dashed lines) at $n_B=0.1n_0$ (top row) and $0.3n_0$ (bottom row) and $Y_p=0.2$ (left column) and 0.4 (right column).

Figure 6

Average mass and proton numbers (thick and thin lines) of heavy nuclei as functions of $T$ in the self-consistent model for parameter sets B (blue solid lines) and E (red dashed lines) at $n_B=0.1n_0$ (top row) and $0.3n_0$ (bottom row) and $Y_p=0.2$ (left column) and 0.4 (right column).

Figure 7

Mass fractions of heavy nuclei (thick lines), light clusters (medium lines), and free nucleons (thin lines) as functions of $T$ for parameter sets B (blue solid lines) and E (red dashed lines) at $n_B=0.1n_0$ (top row) and $0.3n_0$ (bottom row) and $Y_p=0.2$ (left column) and 0.4 (right column).

Figure 8

Comparison of mass fractions of heavy nuclei among different nuclear models listed on Table 2, the self-consistent calculation with the compressible-liquid-drop model (Model I, red solid lines), the incompressible-liquid-drop models with the same $T_c(Z,N)$ of surface tensions as in Model I (Model II, blue dashed lines) and with $T_c=18$ MeV for the all nuclei (Model III, green dashed-dotted lines) and the incompressible one with $T_c=\infty$ and the nuclear dissolution factor [43], Eq. (20), (Model IV, magenta dashed double-dotted lines) at $n_B=0.1n_0$ (top row) and $0.3n_0$ (bottom row) and $Y_p=0.2$ (left column) and 0.4 (right column). All the four models are calculated with the parameter set B.

Figure 9

Mass fractions of deuteron (top row), triton and helion (middle row), and alpha particle (bottom row) for Model I (red solid lines), Model III (green dashed-dotted lines), and Model V (orange dashed lines) with a quantum approach to light clusters and Model VI (black dashed double-dotted lines) with a level-density approach to light clusters at $n_B=0.1n_0$ and $Y_p=0.2$ (left column) and 0.4 (right column). The three models other than Model I employ the same incompressible-liquid-drop models for the heavy nuclei.