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In this article we investigate the beam energy dependence of relativistic retardation effects of electrical fields on the single and double $��^{-}/��^{+}$ ratios in three heavy-ion reactions with an isospin- and momentum-dependent transport model IBUU11. With the beam energy increasing from 200 to 400 MeV/nucleon, effects of the relativistically retarded electrical fields on the $��^{-}/��^{+}$ ratio are found to increase gradually from negligibly to considerably significant as expectedly; it is however, the interesting observation is the relativistic retardation effects of electrical fields on the $��^{-}/��^{+}$ ratio are becoming gradually insignificant as the beam energy further increasing from 400 to 800 MeV/nucleon. Moreover, we also investigate the isospin dependence of relativistic retardation effects of electrical fields on the $��^{-}/��^{+}$ ratio in two isobar reaction systems of $^{96}$Ru+$^{96}$Ru and $^{96}$Zr+$^{96}$Zr at the beam energies from 200 to 800 MeV/nucleon. It is shown that the relativistic retardation effects of electrical fields on the $��^{-}/��^{+}$ ratio are independent of the isospin of reaction. Furthermore, we also examine the double $��^{-}/��^{+}$ ratio in reactions of $^{96}$Zr+$^{96}$Zr over $^{96}$Ru+$^{96}$Ru at the beam energies from 200 to 800 MeV/nucleon with the static field and retarded field, respectively. It is shown the double $��^{-}/��^{+}$ ratio from two reactions is still an effective observable of symmetry energy without the interference of electrical field due to using the relativistic calculation compared to the nonrelativistic calculation., 8 pages, 7 figures. Abbreviated abstract to meet the criterion of arXiv platform. Accepted for publication in Physical Review C. An extended but necessary complementary study to arXiv:1709.09127

The studies, from several independent methods, consistently show that $��^-/��^+$ ratio in the $^{132}$Sn+$^{124}$Sn reaction at 300 MeV/nucleon, which is being carried out at Radioactive Isotope Beam Facility (RIBF) in Japan, just probes the symmetry energy around saturation density., 6 pages, 6 figures, accepted by Journal of Physics G: Nuclear and Particle Physics

It is theoretically and experimentally argued that there may exist bubble or toroid-shaped configurations in some nucleus systems. Based on the nuclear transport model, it is shown that compared with the collision of normal nuclei, there is a depletion of central density of compression with bubble configurations in projectile and target nuclei. This depletion of central compression density may affect some observables in heavy-ion collisions., 5 pages, 6 figures, accepted by European Physical Journal A

Using the isospin- and momentum-dependent hadronic transport model IBUU04, the effects of symmetry energy on the π−/π+ ratio are studied. Our investigations are based on the calculations of the 132Sn+124Sn semi-central collisions at beam energies of 400/A MeV, 600/A MeV and 800/A MeV. It is found that both the transverse momentum and kinetic energy distributions of the π−/π+ ratio are rather sensitive to the symmetry energy, especially around the Coulomb peaks. The position of the Coulomb peak is shown to be nearly independent of beam energy. The sensitivity of the π−/π+ ratio to the symmetry energy decreases as the beam energy increases from 600/A MeV to 800/A MeV.

Using an isospin- and momentum-dependent hadronic transport model, we investigate effects of the symmetry energy on several collective flows in heavy-ion collisions induced by radioactive beams at intermediate energies. It is found that the neutron-proton differential directed flow and the neutron-proton differential elliptic flow are strongly correlated with the symmetry energy, while the position averaged radial flow is weakly correlated with the symmetry energy.

The equation of state of asymmetric nuclear matter at finite temperature is studied in the framework of the Brueckner–Hartree–Fock approximation which is extended to include three-body force correlations. Due to the additional repulsion produced by three-body force contribution, the proton or neutron potential energy and effective mass are correspondingly enhanced. The symmetry energy gradually becomes more sensitive to the temperature with the increasing density compared with the result that only includes the two hole-line contribution. The asymmetry parameter dependence of symmetry energy is also indicated. It is shown that there is the same linear relation between symmetry energy and β2 as the zero temperature case with β being the asymmetry parameter.

Within the framework of the isospin-dependent Boltzmann-Uehling-Uhlenbeck(IBUU) transport model, we studied the effects of the pion potential and the symmetry energy on the pion production in the central $^{197}Au+^{197}Au$ collisions around the pion production threshold. It is found that the pion potential has opposite effects on the value of $��^-/��^+$ ratio at low and high pion energies. The effect of the pion potential on the total $��^-/��^+$ ratio becomes larger in heavy-ion collisions at beam energies below the pion production threshold. And at beam energies below the pion production threshold, with the pion potential, the effect of the symmetry energy on the $��^-/��^+$ ratio becomes smaller compared with that above the pion production threshold., 13 pages, 10 figures, PRC in production

A systematic study of the $��^-/��^+$ ratio in heavy-ion collisions with the same neutron/proton ratio but different masses can help single out effects of the nuclear mean field on pion production. Based on simulations using the IBUU04 transport model, it is found that the $��^-$ / $��^+$ ratio in head-on collisions of $^{48}$Ca+$^{48}$Ca, $^{124}$Sn +$^{124}$Sn and $^{197}$Au+$^{197}$Au at beam energies from 0.25 to 0.6 \agev increases with increasing the system size or decreasing the beam energies. A comprehensive analysis of the dynamical isospin fractionation and the \rpi ratio as well as their time evolution and spatial distributions demonstrates clearly that the \rpi ratio is an effective probe of the high-density behavior of the nuclear symmetry energy., New results and discussions added. Version to appear in PRC (2009)