Abstract:
Generalized symmetry energy coefficients of asymmetric nuclear matter are obtained as screening functions. The dependence of the isospin symmetry energy coefficient on the neutron proton (n-p) asymmetry may be determined unless by a constant (exponent) $Z$ which depend on microscopic properties. The dependence of the generalized symmetry energy coefficients with Skyrme forces on the n-p asymmetry and on the density, only from .5 up to 1.5 $\rho_0$, are investigated in the isospin and scalar channels. The use of Skyrme-type effective forces allows us to obtain analytical expressions for these parameters as well as their dependences on the neutron-proton (n-p) asymmetry, density and even temperature. Whereas the density dependence of these coefficients obtained with Skyrme forces is not necessarily realistic the dependence on the n-p asymmetry exhibit a more consistent behaviour. The isospin symmetry energy coefficient (s.e.c.) increases as the n-p asymmetry acquires higher values whereas the isoscalar s.e.c. decreases. Some consequences for the Supernovae mechanism are discussed.

Abstract:
In the framework of the isospin-dependent Boltzmann-Uehling-Uhlenbeck transport model, effect of the momentum dependence of nuclear symmetry potential on pion-/pion+ ratio in the neutron-rich reaction 132Sn+124Sn at a beam energy of 400 MeV/nucleon is studied. We find that the momentum dependence of nuclear symmetry potential affects the compressed density of colliding nuclei, numbers of produced pion- and pion+, as well as the value of pion-/pion+ ratio. The momentum dependent nuclear symmetry potential increases the compressed density of colliding nuclei, numbers of produced resonances delta(1232), N*(1440), pion- and pion+, as well as the value of pion-/pion+ ratio.

Abstract:
The relativistic mean field theory in combination with the analytic continuation in the coupling constant method is used to determine the energies and widths of single-particle resonant states in Sn isotopes. It is shown that there exists clear shell structure in the resonant levels as appearing in the bound levels. In particular, the isospin dependence of pseudospin symmetry is clearly shown in the resonant states, is consistent with that in the bound states, where the splittings of energies and widths between pseudospin doublets are found in correlation with the quantum numbers of single-particle states, as well as the nuclear mass number. The similar phenomenon also emerges in the spin partners.

Abstract:
The density dependence of the nuclear symmetry energy is inspected using the Statistical Multifragmentation Model with Skyrme effective interactions. The model consistently considers the expansion of the fragments' volumes at finite temperature at the freeze-out stage. By selecting parameterizations of the Skyrme force that lead to very different equations of state for the symmetry energy, we investigate the sensitivity of different observables to the properties of the effective forces. Our results suggest that, in spite of being sensitive to the thermal dilation of the fragments' volumes, it is difficult to distinguish among the Skyrme forces from the isoscaling analysis. On the other hand, the isotopic distribution of the emitted fragments turns out to be very sensitive to the force employed in the calculation.

Abstract:
The density dependence of the symmetry energy in the equation of state of isospin asymmetric nuclear matter is studied using the isoscaling of the fragment yields and the antisymmetrized molecular dynamic calculation. It is observed that the experimental data at low densities are consistent with the form of symmetry energy,E$_{sym}$ $\approx$ 31.6 ($\rho/\rho_{\circ})^{0.69}$, in close agreement with those predicted by the results of variational many-body calculation. A comparison of the present result with those reported recently using the NSCL-MSU data suggests that the heavy ion studies favor a dependence of the form, E$_{sym}$ $\approx$ 31.6 ($\rho/\rho_{\circ})^{\gamma}$, where $\gamma$ = 0.6 - 1.05. This constraints the form of the density dependence of the symmetry energy at higher densities, ruling out an extremely " stiff " and " soft " dependences.

Abstract:
Based on the UrQMD model, we have investigated the influence of the symmetry potential on the negatively and positively charged $\pi$ and $\Sigma$ hyperon production ratios in heavy ion collisions at the SIS energies. We find that, in addition to $\pi^-/\pi^+$ ratio, the $\Sigma^-/\Sigma^+$ ratio can be taken as a sensitive probe for investigating the density dependence of the symmetry potential of nuclear matter at high densities (1-4 times of normal baryon density). This sensitivity of the symmetry potential to both the $\pi^-/\pi^+$ and $\Sigma^-/\Sigma^+$ ratios is found to depend strongly on the incident beam energy. Furthermore, the $\Sigma^-/\Sigma^+$ ratio is shown to carry the information about the isospin-dependent part of the $\Sigma$ hyperon single-particle potential.

Abstract:
Quantitative information on the tensor force induced isospin-dependence of short-range nucleon-nucleon correlation (SRC) extracted from recent J-Lab experiments was used in constraining the high-momentum tails of single nucleon momentum distributions in both symmetric nuclear matter (SNM) and pure neutron matter (PNM). Its effects on the Equations of State of SNM and PNM as well as the nuclear symmetry energy are investigated. It is found that the tensor force induced isospin-dependence of SRC softens significantly the nuclear symmetry energy especially at supra-saturation densities.

Abstract:
Within a transport model we search for potential probes of the isospin dependence of the in-medium nucleon-nucleon (NN) cross sections. Traditional measures of the nuclear stopping power are found sensitive to the magnitude but they are ambiguous for determining the isospin dependence of the in-medium NN cross sections. It is shown that isospin tracers, such as the neutron/proton ratio of free nucleons, at backward rapidities/angles in nuclear reactions induced by radioactive beams in inverse kinematics is a sensitive probe of the isospin dependence of the in-medium NN cross sections. At forward rapidities/angles, on the other hand, they are more sensitive to the density dependence of the symmetry energy. Measurements of the rapidity/angular dependence of the isospin transport in nuclear reactions will enable a better understanding of the isospin dependence of in-medium nuclear effective interactions.

Abstract:
Within two different frameworks of isospin-dependent transport model, i.e., Boltzmann-Uehling-Uhlenbeck (IBUU04) and Ultrarelativistic Quantum Molecular Dynamics (UrQMD) transport models, sensitive probes of nuclear symmetry energy are simulated and compared. It is shown that neutron to proton ratio of free nucleons, pi-/pi+ ratio as well as isospin-sensitive transverse and elliptic flows given by the two transport models with their "best settings", all have obvious differences. Discrepancy of numerical value of isospin-sensitive n/p ratio of free nucleon from the two models mainly originates from different symmetry potentials used and discrepancies of numerical value of charged pi-/pi+ ratio and isospin-sensitive flows mainly originate from different isospin-dependent nucleon-nucleon cross sections. These demonstrations call for more detailed studies on the model inputs (i.e., the density- and momentum-dependent symmetry potential and the isospin-dependent nucleon-nucleon cross section in medium) of isospin-dependent transport model used. The studies of model dependence of isospin sensitive observables can help nuclear physicists to pin down the density dependence of nuclear symmetry energy through comparison between experiments and theoretical simulations scientifically.

Abstract:
In the framework of the isospin-dependent Boltzmann-Uehling-Uhlenbeck transport model, effect of the momentum dependence of nuclear symmetry potential on nuclear transverse and elliptic flows in the neutron-rich reaction $^{132}$Sn+$^{124}$Sn at a beam energy of 400 MeV/nucleon is studied. We find that the momentum dependence of nuclear symmetry potential affects the rapidity distribution of the free neutron to proton ratio, the neutron and the proton transverse flows as a function of rapidity. The momentum dependence of nuclear symmetry potential affects the neutron-proton differential transverse flow more evidently than the difference of neutron and proton transverse flows as well as the difference of proton and neutron elliptic flows. It is thus better to probe the symmetry energy by using the difference of neutron and proton flows since the momentum dependence of nuclear symmetry potential is still an open question. And it is better to probe the momentum dependence of nuclear symmetry potential by using the neutron-proton differential transverse flow and the rapidity distribution of the free neutron to proton ratio.