weakly affected by Coulomb interactions, while the opposite is true for the ratio from LCPs. As discussed earlier, the ratio is found to decease with the incident energy, which is also true here for the isospin asymmetry dependence.

The difference between the soft and stiff symmetry energy results comes from the behavior of free nucleon emission with the isospin asymmetry of the system, i.e., the greater the isospin asymmetry of the system, the greater the contribution of neutrons in the ratio due to the symmetry energy. The soft symmetry energy is stronger at the freeze-out time, which will lead to an increase in the ratio more sharply than the stiff symmetry energy. This effect is again weakly observed in the ratio from LCP’s.

![\[fig:5\] Isospin asymmetry dependence of the single neutron-to-proton ratio for free nucleons at different incident energies. The left panel is at the time of maximum compression, while the right panel is at the freeze-out time. Solid and open circles represent the soft and stiff symmetry energies, respectively. ](figure5.eps){width="90mm"}

![\[fig:6\] Same as in Fig. \[fig:5\] but for the LCPs.](figure6.eps){width="90mm"}

![\[fig:7\](Color online) Excitation function of the double neutron-to-proton ratio from different isotopes of Sn for free nucleons (top) and 