Origin of elliptic flow and its dependence on the equation of state in heavy ion reactions at intermediate energies

Recently it has been discovered that the elliptic flow, $v_2$, of composite charged particles emitted at midrapidity in heavy-ion collisions at intermediate energies shows the strongest sensitivity to the nuclear equation of state (EoS), which has been observed up to now within a microscopic model. This dependence on the nuclear EoS is predicted by quantum molecular dynamics (QMD) calculations [A. Le Fèvre et al.. Nucl. Phys. A 945, 112 (2016)], which show as well that the absorption or rescattering of in-plane emitted particles by the spectator matter is not the main reason for the EoS dependence of the elliptic flow at midrapidity but different density gradients (and therefore different forces) in the direction of the impact parameter ($x$ direction) as compared to the direction perpendicular to the reaction plan ($y$ direction), caused by the presence of the spectator matter. The stronger density gradient in the $y$ direction accelerates the particles more and creates therefore a negative $v_2$. When using a soft momentum-dependent EoS, the QMD calculations reproduce the experimental results.

Figure 1

Elliptic flow $v_2$ of $Z=1$ particles at midrapidity as a function of incident beam energy in semicentral ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ collisions as measured by various experiments, indicated by the different symbols. Data are extracted from Refs. [6, 8, 10, 11].

Figure 2

Time evolution of the proton density profile (${\rho }_{ij}$ in ${\text{event}}^{-1}{\text{fm}}^{-2}$) in the reaction ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ at $1.5A$ GeV incident energy with an impact parameter $b=6$ fm. A SM EoS is applied in the calculations. Different projections are shown: projections onto the zx plane in the top row, onto the $zy$ plane in the middle, and onto the xy plane in the bottom. The density profile are shown at different times: 2, 8, and 16 fm/c (0.1, 0.5, 1.0 $t_{\text{pass}}$) in the left, central and right columns, respectively.

Figure 3

Difference of the proton density profiles, $\delta {\rho }_{x,y}={\rho }_{xy}{}^{\text{SM}}-{\rho }_{xy}{}^{\text{HM}}$ in ${\text{event}}^{-1}{\text{fm}}^{-2}$) between a SM and a HM EoS in the (from top to bottom) $xy$, $zx$, and $zy$ planes. Predictions for ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ at $0.6\mathrm{A}$ and $1.5\mathrm{A}$ GeV incident energies are shown on the left and columns, respectively. The impact parameter is 6 fm and the model results are for $t=t_{\text{pass}}$. The red color stands for positive values, the blue color for negative ones. Positive values are emphasized with black contour lines in addition.

Figure 4

Same as Fig. 3 but expressed in the planes of $(u_{x0},u_{y0})$, $(u_{x0},y_0)$, $(u_{y0},y_0)$ scaled velocities (see text).

Figure 5

Mean reduced baryonic density ($\rho /{\rho }_0$) in coordinate space as perceived by protons in ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ collisions with a SM EoS, $b=6$ fm, at $0.6\mathrm{A}$ (top) and $1.5A$ (bottom) GeV incident energies, at two different times: at full overlap of the system $0.5t_{\text{pass}}$ (left) and at the passing time $t_{\text{pass}}$ (right). Black lines and colored contours correspond to all protons and to those finally emitted at midrapidity ($|y_0|<0.2$) with a high transverse velocity ($u_{t0}>0.4$), respectively. The top four-panel groups show projections on $xy$ plane, and the lower ones show projections on the $xz$ planes.

Figure 6

Time evolution of the average elliptic flow $v_2\left(t\right)$ of protons at midrapidity in the ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ collisions at $0.6A$ (top) and $1.5A$ (bottom) GeV incident energies, $b=6\text{fm}$. We show results obtained with a HM (red lines) and a SM (black lines) EoS, and with (dashed lines) or without (full lines) excluding the protons having finally a low transverse velocity $u_{t0}\le 0.4$. The dashed vertical lines indicate the passing time, and the grayed areas the statistical uncertainties.

Figure 7

IQMD (with SM EoS) predictions for midcentral ($b=6$ fm) collisions of ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ at $0.6A$ GeV incident energy, at two times: $t_{\text{pass}}/2$ (maximal overlap) and $t_{\text{pass}}$, top and bottom panels respectively. The panels display $\mathrm{\Delta }{P}_t^o\left(t\right)$ in $\text{MeV}/{\text{fm}}^2/\text{event}$ defined in the text as a function of the ($x,y$) positions of protons at the respective times. Only protons finally at midrapidity ($|y_0|<0.2$) are selected. The left and right panels show the momentum transfer due to collisions and to the mean field, respectively. As a reference, the superimposed circles show the spatial extension of the incoming projectile and target in this plane. Positive values are marked by black contour lines.

Figure 8

Same as described in the caption of Fig. 7 for $1.5A$ GeV incident energy.

Figure 9

IQMD predictions of the time evolution of various observables. Protons are selected which are finally observed at midrapidity ($|y_0|<0.2$) in the midperipheral ($b=6$ fm) collisions of ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ . Left and right panels show results at $0.6A$ and $1.5A$ GeV incident energies, respectively. Panels (a) and (b) show for the SM EoS the integrated momentum transfer in $x$ (red symbols) and $y$ (black symbols) directions, caused by the mean field (m.f., open circles) and by collisions (coll., diamonds). The momentum transfer due to collisions is divided by a factor of 10. Panels (c) and (d): idem for the HM EoS. Panels (e) and (f) show for the SM EoS the integrated difference between the out-of-plane ($y$) and in-plane ($x$) contribution of the momentum transfer from the mean field (green open circles) and from the collisions with (orange full diamonds) or without (orange empty diamonds) Pauli blocking. Panels (g) and (h): idem for the HM EoS. Panels (i) and (j) show the number of collisions suffered by the selected protons comparing the SM (black circles) and HM (red squares) EoSs, with (full symbols) or without (open symbols) Pauli blocking. Panels (k) and (l): idem with the maximal force due to the mean field. The vertical dashed lines indicate the passing time.

Figure 10

Time evolution of the average elliptic flow, $v_2$, of protons finally emitted at midrapidity ($|y_0|<0.2$) with a large transverse velocity $u_{t0}>0.4$ in ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ collisions at $b=6$ fm and at $0.6\mathrm{A}$ (top) and $1.5\mathrm{A}$ (bottom) GeV incident energy, with a SM (right) and a HM (left) EoS. The protons situated, at the passing time, transversally close (radial distance to the center of the collision on the transversal plane $R_{xy}<4\text{fm}$) or far ($R_{xy}>4\text{fm}$) to/from the main axis of the collision are distinguished, respectively, by red and black lines. The overall $v_2$ (symbols) is detailed into its two contributions: the $v_2$ developed by the momentum transfer due to the mean field and due to the collisions are depicted by dashed and dotted lines, respectively. Black vertical dashed lines indicate the passing time.

Figure 11

Excitation function of the elliptic flow $v_2$ of protons at midrapidity. The experimental data (black circles) are from the FOPI collaboration published in Fig. 29 of Ref. [7]. The data is measured in the impact parameter range $3.1\text{fm}<b<5.6\text{fm}$ and a cut on $u_{t0}>0.8$ is applied. IQMD Model results are presented for two different nuclear EoS (HM with red lines and SM with black lines) for $b=4\text{fm}$ and with an additional cut on $u_{t0}>0.8$ (full lines) and without any cut (dashed lines).