Effect of the spin-orbit interaction on flows in heavy-ion collisions at intermediate energies

The effect of the spin-orbit coupling in heavy ion collisions is investigated based on an updated version of the ultrarelativistic quantum molecular dynamics (UrQMD) model, in which the Skyrme potential energy density functional is employed. And in particular, the spin-orbit coupling effects on the directed and elliptic flows of free nucleons emitted from ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ collisions as functions of both the beam energy and the impact parameter are studied. Our results show that the net contribution of the spin-orbit term to flows of nucleons is negligible, whereas a directed flow splitting between spin-up and spin-down nucleons is visible, especially at large impact parameters, and a peak of the splitting is found at the beam energy around 150 $\mathrm{MeV}/\mathrm{nucleon}$. We also found that the directed flow splitting between spin-up and spin-down neutrons is comparable with the neutron directed flow difference calculated by a soft and a stiff symmetry energy, indicating that the directed flow of neutrons cannot be used to pin down the stiffness of symmetry energy any more without considering the spin degree of freedom in models in the case of spin polarization.

Figure 1

Directed flow $v_1$ (a) and elliptic flow $v_2$ (b) of free protons from Au + Au collisions at $E_{\text{lab}}$=150 MeV/nucleon and centrality $0.25<b_0<0.45$ as a function of the normalized rapidity $y_0$. Calculations with (solid lines) and without (dashed lines) spin-orbit potential are compared with the FOPI data (solid stars) reported in Ref. [29].

Figure 2

Rapidity distribution of the directed flow $v_1$ [upper panels (a)–(c)] and the elliptic flow $v_2$ [lower panels (d)–(f)] of free protons from Au + Au collisions at 150 $\mathrm{MeV}/\mathrm{nucleon}$ for three centralities. Lines with up-triangles, solid lines, and lines with down-triangles represent flows of spin-up protons, all free protons, and spin-down protons, respectively.

Figure 3

Time evolution of ${\left(\nabla \rho \right)}_x{p}_z$ in the reaction x-z plane produced from Au + Au collisions at 150 $\mathrm{MeV}/\mathrm{nucleon}$ with impact parameters $b$ = 2 fm [upper panels, (a)–(c)], $b$ = 7 fm [middle panels, (d)–(f)], and $b$ = 12 fm [lower panels, (g)–(i)]. Panel (c) is blank because the values of ${\left(\nabla \rho \right)}_x{p}_z$ lie between 0.01 and $-0.01$ ${\text{fm}}^{-5}$.

Figure 4

Centrality dependence (a) and excitation function (b) of the slope difference ${\kappa }_{\text{down}}-{\kappa }_{\text{up}}$ between spin-up and spin-down protons. The dashed line in (a) gives a linear fit to the calculation. The inset in each panel shows the corresponding (${\kappa }_{\text{down}}-{\kappa }_{\text{up}}$)/${\kappa }_{\text{all}}$ ratio.

Figure 5

Slopes of the directed flow of spin-up (“$\uparrow$”) and spin-down (“$\downarrow$”) protons (a) and neutrons (b) calculated with the soft (“S”, $\gamma$ = 0.5) and stiff (“H”, $\gamma$ = 1.5) symmetry potential energies.