Thermal properties of asymmetric nuclear matter

The thermal properties of asymmetric nuclear matter are investigated in a relativistic mean-field approach. We start from free-space $NN$ interactions and derive in-medium self-energies by the Dirac-Brueckner theory. By the density-dependent relativistic hadron procedure we derive in a self-consistent approach density-dependent meson-baryon vertices. At the mean-field level, we include isoscalar and isovector-scalar and vector interactions. The nuclear equation of state is investigated for a large range of total baryon densities up to the neutron star regime, the full range of asymmetries $\xi =Z/A$ from symmetric nuclear matter to pure neutron matter, and temperatures up to $T\sim 100$ MeV. The isovector-scalar self-energies are found to modify strongly the thermal properties of asymmetric nuclear matter. A striking result is the change of phase transitions when isovector-scalar self-energies are included.

Figure 1

Density dependence of the DDRH nucleon-meson vertices at $T=0$ [18].

Figure 2

Comparison of $k_S$ versus the Fermi momentum $k_F$ for $T=10$, 20, and 30 MeV.

Figure 3

The real part of the scalar (a) and vector (b) DB self-energies at different temperatures and densities [8], compared to the quadratic approximation discussed in the text.

Figure 4

Temperature dependence of the scalar coupling functions $I_C^{\sigma }$ (a) and $I_s^{\sigma }$ (b) at different densities: $\rho =0.5{\rho }_0$ (blue line), $\rho ={\rho }_0$ (orange dashed line), $\rho =2{\rho }_0$ (green dotted line), where ${\rho }_0$ is the saturation density at $T=0$. Lower curves belong to lower densities.

Figure 5

Dependence of the Hartree mean-field vertices on chemical potential and temperature in symmetric nuclear matter. Results for the vector vertices ($\omega ,\rho$) and the scalar vertices ($\sigma ,\delta$), respectively, are displayed in the left and the right columns, respectively. The dashed isolines indicate the constant density values of $0.5{\rho }_0,{\rho }_0$, and $2{\rho }_0$.

Figure 6

(a) Density dependence of the kinetic (upper curves), potential (lower curves). (b) Binding energy per particle within the DDRH approach. Isotherms for temperatures $T=0$ MeV to $T=30$ MeV in equidistant steps of 5 MeV are displayed. Note that lower curves correspond to lower temperatures.

Figure 7

The rearrangement self-energy ${\Sigma }^R$ as a function of baryon density $\rho$ at various values of temperature $T$ (in MeV). Lower curves correspond to higher temperatures.

Figure 8

The rearrangement contributions to the pressure. (a) Low-density region for $T=10$, 15, and 20 MeV (lower curves correspond to lower $T$). (b) High-density region for $T=20$ MeV.

Figure 9

Comparison of the EOS $P\left[\varepsilon \right(\rho \left)\right]=P\left(\rho \right)$ for symmetric nuclear matter at various temperatures. The numbers denote the temperature $T$ in MeV.

Figure 10

Functional behavior of the pressure $P$ in the high-density region for symmetric nuclear matter. The graph shows the results at $T=0$ MeV (lower curves) and $T=20$ MeV (upper curves) for DDRH and QHD models. The gray region corresponds to the region of pressures consistent with the experimental flow data. The upper blue shaded area, labeled by $P>\varepsilon$, exhibits the causality limit.

Figure 11

Entropy per particle. (Left) As a function of $\rho$ for fixed $T$ from 5 to 30 MeV in steps of 5 MeV. The dashed line shows the classical limit at $T=30$ MeV. (Right) As a function of $T$ for fixed density values from $0.5{\rho }_0$ to $2{\rho }_0$ in steps of $\frac{1}{4}{\rho }_0$. The dashed line shows the limit of a Fermi liquid as explained in the text.

Figure 12

Comparing the entropy per particle from the DDRH calculations to experimental results [47]. The theoretical band corresponds to $0.047\le \rho \le 0.053$ $\text{fm}{}^{-3}$, accounting for the uncertainties in the experimental analysis.

Figure 13

The free energy per particle. Isotherms are chosen the same as in Fig. 11.

Figure 14

Effective nucleon masses $M_q^{*}$ ($q=p,n$) for protons (upper blue line) and neutrons (lower orange line) at $T=0$ as functions of the total baryon density for pure neutron matter (left) and as functions of the asymmetry parameter $\xi =\frac{Z}{A}$ at saturation density $\rho ={\rho }_0$ (right). The thin dashed curve in the left panel represents the result in case of symmetric nuclear matter.

Figure 15

The binding energy (left) and the pressure (right) at $T=0$ for values of $\xi$ from 0 to $0.5$ in steps of 0.1. Lower curves belong to lower values of $\xi$.

Figure 16

(Left) The entropy per particle as a function of the proton fraction $\xi$ at $T=10$ and $\rho =0.16$ $\text{fm}{}^{-3}$. The results for the DDRH, DD-ME2 [24], and QHD parametrization are shown. (Right) Comparison of the temperature dependence of the entropy per particle between symmetric (upper curve) and pure neutron matter (lower curve) in the DDRH model at fixed density $\rho =0.16$ $\text{fm}{}^{-3}$.

Figure 17

Free energy per particle in neutron matter for temperatures from 0 to 30 MeV in equidistant steps of 5 MeV (lower curves correspond to higher temperatures). The results of the DDRH (solid lines) and the DD-ME2 (dashed lines) model are compared.

Figure 18

Symmetry and free symmetry energy as functions of the density for fixed values of $T$ from 0 to 50 MeV. At $T=0$ (blue curve) $F_{\mathrm{sym}}=E_{\mathrm{sym}}$. Curves above (below) the zero temperature line correspond to $F_{\mathrm{sym}}$ ($E_{\mathrm{sym}}$). The right and the left panels show the results for the DDRH and DD-ME2 models, respectively. The points indicate experimental results for the symmetry energy at $T=0$. The data points are taken from [53] (CE), [54] (NS), and [55] (HI).

Figure 19

The phase transition diagrams of symmetric nuclear matter. (Left) The $T-\rho$ phase diagram in the DDRH model. The binodal and spinodal curves are indicated by thick and thin lines, respectively. (Right) The $T-P$ phase diagram for the DDRH and DD-ME2 model calculations.

Figure 20

The chemical potential isobars of nuclear matter in the DDRH model as a function of $\xi$ at $T=10$ MeV. The left panel shows the neutron (solid blue line) and proton (dashed orange line) isobars for values of $P$ from 0.1 to 0.25 MeV $\text{fm}{}^{-3}$. The right panel indicates the geometrical construction in case of $P=0.15$ MeV $\text{fm}{}^{-3}$ as described in the text.

Figure 21

The binodal section at $T=10$ MeV in the DDRH model. The right panel shows the projection of the points $A_1$ and $A_2$ on the pressure at fixed proton concentration $\xi =0.2$.

Figure 22

(a) Comparison of the binodal surface at $T=10$ MeV among the DDRH, DD-ME2, QHD, and SLy230a [10] models. (b) Results of calculations without $\delta$ meson interaction (dashed line) and without momentum correction (short dashed line) are compared to the full DDRH calculation.

Figure 23

The critical temperature of mechanical instability as a function of the proton fraction $\xi$. The results are compared between different models. The good agreement between our DDRH and the ChPT results of Ref. [48] is remarkable.