Effects of the symmetry energy in the ${}^{132}\mathrm{Sn}+{}^{124}\mathrm{Sn}$ reaction at 300 MeV/nucleon

Based on the recently updated isospin-dependent Boltzmann-Uehling-Uhlenbeck (IBUU) transport model, we studied the effects of symmetry energy on the neutron to proton ratio $n/p$ and the ${\pi }^{-}/{\pi }^{+}$ ratio in the central ${}^{132}\mathrm{Sn}+{}^{124}\mathrm{Sn}$ reaction at 300 MeV/nucleon. It is found that the $n/p$ ratio and the ${\pi }^{-}/{\pi }^{+}$ ratio in the central ${}^{132}\mathrm{Sn}+{}^{124}\mathrm{Sn}$ reaction at 300 MeV/nucleon mainly probe the symmetry energy in the density region 1–1.5 times saturation density. However, the ${\pi }^{-}/{\pi }^{+}$ ratio may be able to probe the density-dependent symmetry energy above 1.5 times saturation density by making some kinematic restrictions such as the azimuthal angle and kinetic energy cuts of emitting pions.

Figure 1

Comparisons of average neutron/proton ratio with densities higher than the normal nuclear matter density as a function of time (left panel) and ${\pi }^{-}/{\pi }^{+}$ ratio as a function of time (right panel) calculated with the IBUU model used in 2003 from Ref. [24] (shown with lines) and that with the IBUU model used in 2016 from Ref. [21] (shown with symbols). The solid (dashed) lines or solid (hollow) symbols are the results using almost the same stiff (soft) symmetry energy.

Figure 2

Maximal baryon density reached in the central ${}^{132}\mathrm{Sn}+{}^{124}\mathrm{Sn}$ reaction at 300 MeV/nucleon with stiff ($x=0$) and soft ($x=1$) symmetry energies. The corresponding symmetry energy functionals with $x=0$ (stiff) and 1 (soft) are depicted in the inset. To check whether or not the symmetry energy at densities above $1.5{\rho }_0$ affects symmetry-energy-sensitive observables, we also calculated the case $x_{\rho <1.5\rho {}_0}=1$, $x_{\rho \ge 1.5\rho {}_0}=0$ for comparison.

Figure 3

Evolution of neutron to proton ratio $n/p$ in isospin fractionation. The $n/p$ ratio of the gas phase is shown in panel (a). Panel (b) shows the $n/p$ ratio of the liquid phase, and panel (c) denotes the strength of isospin fractionation. We identify a free nucleon by its local density $\rho \le 1/8{\rho }_0$.

Figure 4

Evolution of the ${\pi }^{-}/{\pi }^{+}$ ratio in the central ${}^{132}\mathrm{Sn}+{}^{124}\mathrm{Sn}$ reaction at 300 MeV/nucleon with different symmetry energy settings.

Figure 5

Average multiplicity of ${\pi }^{-}$, ${\pi }^{+}$ as a function of $x$ parameter in the reaction ${}^{132}\mathrm{Sn}+{}^{124}\mathrm{Sn}$ at a beam energy of 300 MeV/nucleon with an impact parameter of 1 fm.

Figure 6

Production of $\mathrm{\Delta }$ resonances as a function of time with different $x$ parameters in the reaction ${}^{132}\mathrm{Sn}+{}^{124}\mathrm{Sn}$ at a beam energy of 300 MeV/nucleon and an impact parameter of 1 fm.

Figure 7

Kinetic energy spectra of ${\pi }^{-}$, ${\pi }^{+}$ with different symmetry energies in the reaction of ${}^{132}\mathrm{Sn}+{}^{124}\mathrm{Sn}$ at a beam energy of 300 MeV/nucleon and an impact parameter of 1 fm.

Figure 8

${\pi }^{-}/{\pi }^{+}$ ratio as a function of pion kinetic energy in the central ${}^{132}\mathrm{Sn}+{}^{124}\mathrm{Sn}$ reaction at 300 MeV/nucleon with (b) and without (a) azimuthal angle cuts.