Self-consistent calculation of the nuclear composition in hot and dense stellar matter

We investigate the mass fractions and in-medium properties of heavy nuclei in stellar matter at characteristic densities and temperatures for supernova (SN) explosions. The individual nuclei are described within the compressible liquid-drop model taking into account modifications of bulk, surface, and Coulomb energies. The equilibrium properties of nuclei and the full ensemble of heavy nuclei are calculated self-consistently. It is found that heavy nuclei in the ensemble are either compressed or decompressed depending on the isospin asymmetry of the system. The compression or decompression has a little influence on the binding energies, total mass fractions, and average mass numbers of heavy nuclei, although the equilibrium densities of individual nuclei themselves are changed appreciably above one-hundredth of normal nuclear density. We find that nuclear structure in the single-nucleus approximation deviates from the actual one obtained in the multinucleus description, since the density of free nucleons is different between these two descriptions. This study indicates that a multinucleus description is required to realistically account for in-medium effects on the nuclear structure in supernova matter.

Figure 1

Average equilibrium density of heavy nuclei for the compressible liquid drop model (Model MC, red solid thick lines) and the incompressible one (Model MI, blue dashed-dotted thick lines) in the multinucleus descriptions, and the equilibrium density of a representative nucleus for the single nucleus EOS with the compressible liquid drop model (Model SC, green dashed lines) as a function of $n_B$ at $T=1$ MeV (top row) and 3 MeV (bottom row) and $Y_p=0.2$ (left column) and 0.4 (right column). Thin lines indicate individual equilibrium densities of ${}_{20}{}^{50}\mathrm{Ca}$ at $(T,Y_p)=(3\mathrm{MeV},0.2)$ and of ${}_{100}{}^{300}\mathrm{Fm}$ at the other conditions for Models MC (red solid lines) and MI (blue dashed-dotted lines). Green dashed thin lines show results for the single-nucleus approximation, in which the representative nucleus is fixed as ${}^{50}\mathrm{Ca}$ or ${}^{300}\mathrm{Fm}$ at any density.

Figure 2

Energy shifts per baryon in bulk (solid lines), surface (dashed lines), and Coulomb (dotted lines) terms, $\{F_i^{B/S/C}(T,n_e,n_p^{\prime },n_n^{\prime })-F_i^{B/S/C}(T,0,0,0)\}/A_i$, of ${}^{50}\mathrm{Ca}$ at $(T,Y_p)=(3\mathrm{MeV},0.2)$ (top) and of ${}^{300}\mathrm{Fm}$ at $(T,Y_p)=(1\mathrm{MeV},0.4)$ (bottom) for Models MC (red lines), MI (blue lines), and SC (green lines).

Figure 3

Internal free energies per baryon, $(F_i^B+F_i^S+F_i^C)/A_i$, of ${}^{50}\mathrm{Ca}$ at $(T,Y_p)=(3\mathrm{MeV},0.2)$ (thick lines) and of ${}^{300}\mathrm{Fm}$ at $(T,Y_p)=(1\mathrm{MeV},0.4)$ (thin lines), as functions of $n_B$ for Models MC (red solid lines), MI (blue dashed-dotted lines), and SC (green dashed lines).

Figure 4

Equilibrium densities of the nuclei with $A=50$ (thin) and 300 (thick) as a function of proton fraction in nuclei, $Z/A$, at $n_B=0.3n_0$ for $T=1$ MeV (top row) and 3 MeV (bottom row) and $Y_p=0.2$ (left column) and 0.4 (right column). Red solid lines are those obtained in Model MC, $n_{si}(T,n_e,n_p^{\prime },n_n^{\prime })$. Blue dotted and green dashed lines indicate those in vacuum (or in Model MI), $n_{si}(0,0,0,0)$, and those at finite temperature and zero density, $n_{si}(T,0,0,0)$, respectively. Black dashed-dotted lines are the density corresponding to the minimum of bulk free energy $\omega$ as a function of charge fraction $x=Z/A$, $n_{s,\mathrm{bulk}}(Z/A)$.

Figure 5

Mass fractions of elements as a function of mass number for Models MC (red solid lines) and MI (blue dashed-dotted lines) in the multinucleus description at $T=1$ MeV (top row) and 3 MeV (bottom row) and $Y_p=0.2$ (left column) and 0.4 (right column). Vertical green dashed lines display the mass numbers of representative nuclei in the single-nucleus description.

Figure 6

Mass fractions of heavy nuclei (thick lines), $\alpha$ particles (middle lines), and free nucleons (thin lines) for Models MC (red solid lines), SC (green dashed lines), and MI (blue dashed-dotted lines), respectively, as a function of $n_B$ at $T=1$ MeV (top row) and 3 MeV (bottom row) and $Y_p=0.2$ (left column) and 0.4 (right column). Those of alpha particles and/or free nucleons are scaled for best display.

Figure 7

Average mass and proton numbers (thick and thin lines) of heavy nuclei with $Z\ge 6$ for Models MC (red solid lines) and MI (blue dashed-dotted lines) and those of a representative nucleus for Model SC (green dashed lines) as a function of $n_B$ at $T=1$ MeV (top row) and 3 MeV (bottom row) and $Y_p=0.2$ (left column) and 0.4 (right column).

Figure 8

Critical lines for the compression (top) and mass fraction of free nucleons (bottom) in Model MC for $Y_p=0.25$ (cyan dotted line), 0.30 (blue dashed-dotted line), 0.35 (green dashed line), and 0.40 (red solid line). They are defined as the compression parameter $\xi ={\sum }_i{n}_i(d{n}_{si}/d{n}_B)/\left({\sum }_i{n}_i\right)=0$ and the mass fraction of free nucleons $X_p+X_n=0.05$. The positive value of $\xi$ means that in general the nuclei are compressed. Note that the line of mass fraction for $Y_p=0.25$ is not displayed, since $X_p+X_n>0.05$ at all conditions.

Figure 9

Average equilibrium density (thick lines) of heavy nuclei as a function of $n_B$ at $T=1$ MeV (top row) and 3 MeV (bottom row) and $Y_p=0.2$ (left column) and 0.4 (right column) for MC models with bulk parameter sets $B$ (black dashed lines) and $E$ (red thick lines), the latter of which is identical to the MC model shown in Fig. 1. Thin lines indicate those of ${}_{20}{}^{50}\mathrm{Ca}$ at $(T,Y_p)=(3\mathrm{MeV},0.2)$ and ${}_{100}{}^{300}\mathrm{Fm}$ at the other conditions.

Figure 10

Equilibrium densities of the nuclei with $A=50$ (thin) at $(T,Y_p)=(3\mathrm{MeV},0.2)$ (top panel) and 300 (thick) at $(T,Y_p)=(1\mathrm{MeV},0.4)$ (bottom panel) as a function of proton fraction in nuclei, $Z/A$, for MC models with bulk parameter sets $B$ (black lines) and $E$ (red lines). Solid, dashed, and dashed-dotted lines are those obtained at $n_B=0.3n_0$, those at finite temperature and zero density, and those at the density corresponding to the minimum of bulk free energy, $n_{s,\mathrm{bulk}}(Z/A)$, respectively.