Probing the momentum-dependent symmetry potential via nuclear collective flows

The dependence of the momentum-dependent symmetry potential on the nuclear collective flows produced in semicentral ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ collisions at incident energies from 250 to 800A MeV has been investigated within the isospin-dependent quantum molecular dynamics model. It is found that the theoretical results overestimate the values of the experimental data on directed flow of protons in the domain of large rapidities but can reproduce well the one of mass-three cluster. Neutron-proton differential flows and difference of neutron-proton collective flows are sensitive to the momentum-dependent symmetry potential, especially in the domain of larger rapidities and momenta. This sensitivity becomes stronger with increasing cut of the transverse velocity and becomes smaller with increasing incident energies.

Figure 1

Density dependence of the nuclear symmetry energy after including the momentum-dependent part for hard, linear, soft, and supersoft cases in the present work [48, 57].

Figure 2

Symmetry potential as a function of momentum for the hard and supersoft symmetry energies at $0.3{\rho }_0$, $0.5{\rho }_0$, ${\rho }_0$, and $2{\rho }_0$ with $\delta =0.2$, respectively. The solid lines show the momentum-dependent contributions and the dashed lines show the density-dependent ones.

Figure 3

Directed flow $v_1$ of protons produced in ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ collisions at 250, 400, 600, and 800A MeV for centrality $0.25<b_0<0.45$. The results are calculated by using the hard (full symbols) and supersoft (hollow symbols) symmetry energies and the mass splittings of $m_n^{*}>m_p^{*}$ (squares) and $m_n^{*}<m_p^{*}$ (circles) with the transverse 4-velocities cut of $u_{t0}>0.8$ for 250A MeV and of $u_{t0}>0.4$ for the other energies. The data are from Ref. [19].

Figure 4

The same as in Fig. 3, but for fragments with mass number of $A=3$.

Figure 5

The same as in Fig. 3, but for elliptic flow of protons.

Figure 6

Rapidity dependence of the neutron-proton differential directed flow produced in ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ collisions at 250, 600, and 800A MeV for centrality $0.25<b_0<0.45$. The results are calculated by using the hard (full symbols) and supersoft (hollow symbols) symmetry energies and the mass splittings $m_n^{*}>m_p^{*}$ (squares) and $m_n^{*}<m_p^{*}$ (circles) with the transverse 4-velocities cut of $u_{t0}>0.4$.

Figure 7

The same as in Fig. 6, but for transverse 4-velocities cut of $u_{t0}>0.8$.

Figure 8

The same as in Fig. 6, but for neutron-proton differential elliptic flow.

Figure 9

The same as in Fig. 7, but for neutron-proton differential elliptic flow.

Figure 10

Transverse momentum dependence of the neutron-proton differential directed flow produced in ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ collisions at 250, 600, and 800A MeV for centrality $0.25<b_0<0.45$. The results are calculated by using the hard (full symbols) and supersoft (hollow symbols) symmetry energies and the mass splittings $m_n^{*}>m_p^{*}$ (squares) and $m_n^{*}<m_p^{*}$ (circles) with the rapidity cut of $0.4\le y_0\le 0.8$.

Figure 11

Transverse momentum dependence of the difference of neutron-proton directed flow in ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ collisions at 250, 600, and 800A MeV for centrality $0.25<b_0<0.45$. The results are calculated by using the hard symmetry energy and the mass splittings $m_n^{*}>m_p^{*}$ (squares) and $m_n^{*}<m_p^{*}$ (circles) with the rapidity cut of $|y_0|\le 0.4$.

Figure 12

The same as in Fig. 11, but for the supersoft symmetry energy.

Figure 13

Transverse momentum dependence of the difference of neutron-proton elliptic flow in ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ collisions at 250, 600, and 800A MeV for centrality $0.25<b_0<0.45$. The results are calculated by using the hard symmetry energy and the mass splittings $m_n^{*}>m_p^{*}$ (squares) and $m_n^{*}<m_p^{*}$ (circles) with the rapidity cut of $|y_0|\le 0.4.$

Figure 14

The same as in Fig. 13, but for the supersoft symmetry energy.