Low density nuclear matter with light clusters in a generalized nonlinear relativistic mean-field model

We systematically investigate the thermodynamic properties of homogeneous nuclear matter with light clusters at low densities and finite temperatures using a generalized nonlinear relativistic mean-field (gNL-RMF) model, in which light clusters up to $\alpha \left(1\le A\le 4\right)$ are included as explicit degrees of freedom and treated as pointlike particles, with their interactions described by meson exchanges. The medium effects on the cluster binding energies are described by density- and temperature-dependent energy shifts with the parameters obtained by fitting the experimental cluster Mott densities. We find that the composition of low density nuclear matter with light clusters is essentially determined by the density and temperature dependence of the cluster binding energy shifts. Compared with the values of the conventional (second-order) symmetry energy, symmetry free energy, and symmetry entropy, their fourth-order values are found to be significant at low densities $\left(n\sim {10}^{-3}{\mathrm{fm}}^{-3}\right)$ and low temperatures $\left(T\lesssim 3\mathrm{MeV}\right)$, indicating the invalidity of the empirical parabolic law for the isospin asymmetry dependence of these nuclear matter properties. Our results indicate that, in the density region of $n\gtrsim 0.02{\mathrm{fm}}^{-3}$, the clustering effects become insignificant and the nuclear matter is dominated by the nucleon degree of freedom. In addition, we compare the gNL-RMF model predictions with the corresponding experimental data on the symmetry energy and symmetry free energy at low densities and finite temperatures extracted from heavy-ion collisions, and reasonable agreement is found.

Figure 1

The Mott density vs temperature in symmetric nuclear matter for light clusters $d$, $t$, $h$, and $\alpha$. The full symbols with error bars are experimental data from heavy-ion collisions by Hagel et al. [47] while the lines represent the predictions from the gNL-RMF model with original [panel (a)] and revised [panel (b)] parameters for the in-medium light cluster binding energy shifts.

Figure 2

Binding energies of light clusters $d$, $t$, $h$, and $\alpha$ as functions of the total baryon density $n_B$ at $T=10\mathrm{MeV}$ (a), 5 MeV (b), and 3 MeV (c). The results with the original [11] and revised parameters (see Table 2) for the binding energy shifts are indicated by solid and dotted lines, respectively.

Figure 3

The number fraction of nucleons and light clusters $d$, $t$, $h$, and $\alpha$ as a function of the total baryon density $n_B$ with FSU parameter set for $T=10\mathrm{MeV}$ and $Y_p=0.5$ (a), $T=10\mathrm{MeV}$ and $Y_p=0.1$ (b), $T=3\mathrm{MeV}$ and $Y_p=0.5$ (c), and $T=3\mathrm{MeV}$ and $Y_p=0.1$ (d).

Figure 4

The number fraction of light clusters $d$, $t$, $h$, and $\alpha$ as a function of the total baryon density $n_B$ in nuclear matter with $Y_p=0.5$ and $Y_p=0.1$ at $T=3$ and 10 MeV using the NL-RMF parameter sets NL3, FSU, FSUGold5, and FSU-II.

Figure 5

The number fraction of light clusters $d$, $t$, $h$, and $\alpha$ as a function of the total baryon density $n_B$ in nuclear matter with $Y_p=0.5$ and $Y_p=0.1$ at $T=3$ and 10 MeV using the original [11] and revised parameters (see Table 2) for the binding energy shifts.

Figure 6

Density dependence of $E_{\mathrm{sym}}$ (left panels) and $E_{\mathrm{sym},4}$ (right panels) at $T=3$, 5, and 10 MeV in the gNL-RMF model with FSU, FSU-II, and FSUGold5. For comparison, the results for $E_{\mathrm{sym}}$ in the NL-RMF model without considering clusters are also included (thin lines).

Figure 7

Density dependence of $E_{\mathrm{sym}}$ (left panels) and $E_{\mathrm{sym},4}$ (right panels) at $T=3$, 5, and 10 MeV in the gNL-RMF model using the original [11] and revised parameters (see Table 2) for the binding energy shifts. For comparison, the results for $E_{\mathrm{sym}}$ in the NL-RMF model without considering clusters are also included (dotted lines).

Figure 8

Density dependence of $E_{\mathrm{sym}}$, $E_{\mathrm{sym},4}$, and $E_{\mathrm{para}}$ in the gNL-RMF model with FSU at $T=3\mathrm{MeV}$ (a), 5 MeV (b), and 10 MeV (c). The results of $E_{\mathrm{sym}}$ in the NL-RMF model without considering clusters are also included for comparison.

Figure 9

Same as Fig. 8, but for the symmetry free energy.

Figure 10

Same as Fig. 8, but for the symmetry entropy.

Figure 11

$E_{\mathrm{int}}$ (a), $F$ (b), and $S$ (c) vs isospin asymmetry squared $[{\delta }^2={\left(1-2Y_p\right)}^2]$ in the gNL-RMF model with the FSU parameter set at $n_B=0.002{\mathrm{fm}}^{-3}$ and $T=3$, 5, and 10 MeV.

Figure 12

The gNL-RMF model predictions for the density dependence of the symmetry energy (a)–(e) and symmetry free energy (f)–(j) with FSU at different temperature intervals. The experimental data from Refs. [46] and [44] as well as the results in the NL-RMF model without considering clusters are also included for comparison.