Probing the density dependence of the symmetry potential at low and high densities

We have investigated the influence of different forms of symmetry energies on various observables proposed to be sensitive to the symmetry energy at subnormal and supranormal densities for reactions ${}^{208}\mathrm{Pb}$ +${}^{208}\mathrm{Pb}$, ${}^{132}\mathrm{Sn}$ +${}^{124}\mathrm{Sn}$, and ${}^{96}\mathrm{Zr}$ $+{}^{96}\mathrm{zR}$ at $E_b=0.4A\mathrm{GeV}$ within the microscopic transport model—ultrarelativistic quantum molecular dynamics model. The same systems are adopted for testing the probes of the symmetry potential at both subnormal and supranormal densities. It is shown that the differences between the predicted values of investigated quantities by different symmetry potentials have a close correspondence with the different behavior of the density dependence of the interaction part of the different symmetry energies. We also find that the comprehensive study with multiple probes provides a possibility for gaining the density dependence of the symmetry potential in a broad density region, which allows us to extract the information of the isospin-dependent part of the effective interaction.

Figure 1

(Color online) Parametrizations of the nuclear symmetry potential energy DDH${\rho }^{*}$, DDH3$\rho {\delta }^{*}$, F15, Fa3, and the linear one as a function of the reduced density u.

Figure 2

(Color online) Time evolution of central baryon densities ($R_{c.m.}<3$ fm) with various symmetry potentials in the head-on reaction ${}^{208}\mathrm{Pb}$ +${}^{208}\mathrm{Pb}$ at $0.4A$ GeV.

Figure 3

(Color online) The rapidity distributions of ${\pi }^{-}$ and ${\pi }^{+}$ for central ${}^{208}\mathrm{Pb}$ +${}^{208}\mathrm{Pb}$ collisions with different symmetry potentials (see text).

Figure 4

(Color online) Transverse-momentum distributions of ${\pi }^{-}$ and ${\pi }^{+}$ from central ${}^{208}\mathrm{Pb}$ +${}^{208}\mathrm{Pb}$ collisions at $E_b=0.4A\mathrm{GeV}$ for different symmetry potentials. The ${\pi }^{-}/{\pi }^{+}$ ratios are also shown as functions of transverse momentum.

Figure 5

(Color online) Transverse-momentum distributions of free neutron-proton ratios for different density-dependent symmetry potentials.

Figure 6

(Color online) Transverse-momentum distributions of free neutrons and protons (upper plot) as well as the corresponding $n/p$ ratios (lower plot) at mid-rapidity ($|(y_c^{(0)}|<0.3$) and at reaction times $t=50$ and $150\hspace{1em}\mathrm{fm}/c$. The F15 and DDH${\rho }^{*}$ symmetry potentials are chosen. The arrows in the upper plot show the maxima of the distributions.

Figure 7

(Color online) Isotope distributions of H, He, Li, Be, B, and C with the symmetry potentials F15 and Fa3 (see text).

Figure 8

(Color online) The ratios of the triton and ${}^3\mathrm{He}$ yields with and without a rapidity cut as calculated with four different forms of the symmetry potentials.