Abstract
It is one of the important subjects in nuclear physics and astrophysics to determine the nuclear symmetry energy at various densities. In particular, intermediate-energy heavy-ion collisions are believed to be useful to investigate the symmetry energy at high density around 2$\rho_0$. The $\pi^{-}/\pi^{+}$ ratio is considered to be a good probe [B. A. Li, Phys. Rev. Lett. 88, 192701 (2002)]. Some theoretical studies have been performed by different transport models to investigate the sensitivity of pion observables, but at present these results are contradicting to each other even qualitatively. Therefore, it is necessary to understand the mechanism how pions are produced reflecting the dynamics of neutrons and protons. We study the pion production using a new approach by combining an antisymmetrized molecular dynamics (AMD) [A. Ono et al., Prog. Theor. Phys. 87, 1185 (1992)] and a hadronic cascade model (JAM) [Y. Nara et al., Phys. Rev. C 61, 024901 (2000)]. The dynamics of neutrons and protons is solved by AMD, and then pions and $\Delta$ resonances in the reaction process are handled by JAM. This work is a collaboration with A. Ono, Y. Nara, and A. Ohnishi. In this presentation, we will show results for the central $^{132}$Sn+$^{124}$Sn collision at 300 MeV/nucleon which corresponds to a proposed experiment at RIBF/RIKEN. AMD calculations were performed for some cases with and without cluster correlations, and with two effective interactions corresponding to different density dependences of the symmetry energy. We first confirm that the relation $\pi^{-}/\pi^{+} \approx \Delta^{-}/\Delta^{++} $ actually holds between the final pion ratio and the ratio of the $\Delta$ production rates $NN \rightarrow N \Delta$, even though many of produced $\Delta$ resonances are absorbed by $N\Delta \rightarrow N N$ reactions. However, the $\Delta^{-}/\Delta^{++}$ ratio (and therefore $\pi^{-}/\pi^{+}$ ratio) is greater than $(N/Z)^2$ of the system in some cases. We find that an agreement $\Delta^{-}/\Delta^{++} = N^2 /Z ^2$ is obtained when $N$ and $Z$ are chosen as the numbers of neutrons and protons, respectively, that are in the high density region $\rho \gtrsim \rho_0$ and with a sufficiently high momentum $p \gtrsim 500$~MeV/c for $\Delta$ production. Calculations show that stiff symmetry energy makes the high density region less neutron-rich and thus the $\Delta^{-}/\Delta^{++}$ and ${\pi}^{-}/{\pi}^{+}$ ratios get reduced. On the other hand, both stiff symmetry energy and cluster correlations have a effect to make the high momentum part more neutron-rich and thus to increase the pion ratio. In conclusion, the pion ratio certainly carries the information on neutrons and protons at the dynamical stage of collisions. However, at the incident energy 300 MeV/nucleon, the neutron and proton distributions in momentum space play an important role to determine the pion ratio. It is essential to understand the effects of the symmetry energy and cluster correlations on the nucleon dynamics.