Probing the momentum dependence of the symmetry potential by the free $n/p$ ratio of pre-equilibrium emission

Based on an isospin and momentum-dependent transport model, we studied the effect of the momentum-dependent symmetry potential on the free neutron-to-proton ratio of pre-equilibrium nucleon emission. It is found that, for the ${}^{132}\text{Sn}{+}^{124}\text{Sn}$ reaction at 400 MeV/nucleon incident beam energy, the free $n/p$ ratio of pre-equilibrium nucleon emission mainly probes the momentum dependence of the symmetry potential at a nucleon momentum around 400–600 MeV/$c$, whereas for 200 MeV/nucleon incident beam energy this observable mainly probes the momentum dependence of the symmetry potential at a nucleon momentum around 200–400 MeV/$c$. To probe the symmetry energy/potential using free $n/p$ ratio, not all the details of the momentum dependence of the symmetry potential are important: the values of symmetry potential in only a certain momentum range are crucial for an observable. It is important to input reasonable density and momentum dependence of the symmetry potential according to the magnitude of incident beam energy of heavy-ion collisions. The present experimental data on the symmetry potential are not enough for probing the density-dependent symmetry energy. More experimental data (such as nucleon and nuclei scattering experiments at different nucleonic momenta and densities) on the symmetry potential are therefore needed to pin down the density-dependent symmetry energy.

Figure 1

Momentum-dependence of the symmetry potential $(U_n-U_p)/2\delta$ at different densities using the MDI interaction with $x=0$; the shaded region indicates the experimental constraints.

Figure 2

Evolutions of pre-equilibrium free $n/p$ ratio $R_i$ (see text for details) by turning off the symmetry potential in momentum regions $0<p<200$ MeV/$c$, $200<p<400$ MeV/$c$, $400<p<600$ MeV/$c$, $600<p<800$ MeV/$c$, and comparison with the standard calculation $R_0$.

Figure 3

Relative sensitivity of pre-equilibrium free $n/p$ ratio to the symmetry potential in different momentum regions at a beam energy of 400 MeV/nucleon in ${}^{132}\text{Sn}{+}^{124}\text{Sn}$ collisions with an impact parameter $b=1$ fm.

Figure 4

Same as Fig. 3, but at a beam energy of 200 MeV/nucleon.

Figure 5

Location of baryon number in momentum space as a function of time in ${}^{132}\text{Sn}{+}^{124}\text{Sn}$ reactions at incident beam energies of 400 (a) and 200 (b) MeV/nucleon, respectively.