Results of the ASY-EOS experiment at GSI: The symmetry energy at suprasaturation density

Directed and elliptic flows of neutrons and light-charged particles were measured for the reaction ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ at 400 MeV/nucleon incident energy within the ASY-EOS experimental campaign at the GSI laboratory. The detection system consisted of the Large Area Neutron Detector (LAND), combined with parts of the CHIMERA multidetector, the ALADIN Time-of-flight Wall, and the Washington University Microball detector. The latter three arrays were used for the event characterization and reaction-plane reconstruction. In addition, an array of triple telescopes, KRATTA was used for complementary measurements of the isotopic composition and flows of light-charged particles. From the comparison of the elliptic-flow ratio of neutrons with respect to charged particles with UrQMD predictions, a value $\gamma =0.72\pm 0.19$ is obtained for the power-law coefficient describing the density dependence of the potential part in the parametrization of the symmetry energy. It represents a new and more stringent constraint for the regime of suprasaturation density and confirms, with a considerably smaller uncertainty, the moderately soft to linear density dependence deduced from the earlier FOPI-LAND data. The densities probed are shown to reach beyond twice saturation.

Figure 1

Schematic view of the experimental setup of the ASY-EOS experiment S394 at GSI. The chosen coordinate system is indicated; the $y$ direction points upwards in the laboratory. The target area with the Microball is not to scale in the main drawing but shown with a scale factor of approximately 5:1 in the lower left corner (see Sec. 2b for coverage and dimensions).

Figure 2

Kinematic acceptance in the transverse-velocity vs rapidity plane of the detector systems used in the S394 experiment. The contour lines refer to the specified values of the kinetic energy of protons in the laboratory, ranging from 10 MeV to 1 GeV. The indicated lower and upper limits in energy are for protons (stopped protons for KRATTA and CHIMERA) and were calculated for the specific detector thresholds and configurations. Average values were chosen for the four types of detector elements of the Microball (labeled $\mu$Ball in the figure).

Figure 3

Measured invariant hit distribution for ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ collisions at 400 MeV/nucleon incident energy in the transverse-velocity vs rapidity plane for charged particles detected with the three systems—Microball, CHIMERA, and ATOF Wall—with full azimuthal coverage and for neutrons detected with LAND. The velocities of particles detected with the Microball are not measured and shown here with an arbitrarily chosen homogeneous kinetic-energy distribution in the interval $0\le E_{\mathrm{kin}}\le 100\mathrm{MeV}$. The apparent angular variation may be influenced by ring-dependent thresholds. The arrows indicate the rapidities of the projectile $y_p=0.896$ and of the c.m. system.

Figure 4

Identification plot of CsI(Tl) signals recorded with a CHIMERA module of ring 7 (${\theta }_{\mathrm{lab}}\approx {17}^{\circ }$) from ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ collisions at 400 MeV/nucleon displaying the ratio of fast-over-slow vs the slow signal components. The loci of hydrogen and helium ions punching through the full length of the detector are labeled as ${\mathrm{H}}_{\mathrm{pt}}$ and ${\mathrm{He}}_{\mathrm{pt}}$. An expanded view of the area within the rectangular box is shown in the inset. Besides the punch-through groups, also the loci of mass-identified light ions are indicated there.

Figure 5

ATOF identification plots of calibrated time of flight vs recorded energy loss $\mathrm{\Delta }E$ for two slats approximately 30 and 35 cm to the right of the beam direction (${\theta }_{\mathrm{lab}}\approx 4.7^{\circ }$ and $\approx 5.4^{\circ }$, respectively, at their central parts). The groups of light elements are clearly recognized up to atomic number $Z\approx 10$ as shown in the insets on logarithmic scales.

Figure 6

Experimental correlation of the maximum atomic number, $Z_{\max }$, of the fragments within an event and the quantity $Z_{\mathrm{bound}}$ as deduced from the fragments detected with the ATOF Wall. The dots represent the mean values of $Z_{\max }$ over the intervals of $Z_{\mathrm{bound}}$ indicated by the horizontal error bars.

Figure 7

Experimental correlation of the quantity $Z_{\mathrm{bound}}$ as deduced from the fragments detected with the ATOF Wall with the charged-particle multiplicity measured with CHIMERA. Events within the hatched area were excluded from the analysis; the high-intensity group near multiplicity 27 with $Z_{\mathrm{bound}}\approx 0$ is caused by central collisions; the symbols represent the mean $Z_{\mathrm{bound}}$ of the remaining distribution as a function of the CHIMERA multiplicity.

Figure 8

(Top row) Inclusive $Z_{\mathrm{bound}}$ vs ZRAT correlation for data sets taken with (a) and without (b) a target foil in place (EF stands for empty frame). (Bottom row) UrQMD calculations for the correlation of $Z_{\mathrm{bound}}$ vs ZRAT for ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ collisions at 400 MeV/nucleon and impact parameter $b<10.0$ fm, filtered to match the experimental conditions (d), and for impact-parameter distributions $d\sigma /db$ obtained under various conditions (c). The unbiased distribution for the full reaction for $b<10.0$ fm is given by the black (solid) histogram while the blue, green, and red lines show impact-parameter distributions obtained when selecting very central, semicentral, and peripheral event classes, respectively, by gating either on $Z_{\mathrm{bound}}$ (dashed) or on ZRAT (dotted, see Table 1). The dashed horizontal lines in (a) represent the corresponding gates for the $Z_{\mathrm{bound}}$ selection. The line of $Z_{\mathrm{bound}}$ centroids as a function of ZRAT of the UrQMD distribution of (d) is drawn into the experimental distribution (a).

Figure 9

Bidimensional representations of the difference of the azimuthal reaction-plane orientations individually obtained from CHIMERA and the ATOF Wall under the condition that the CHIMERA and Microball orientations are within the adopted anticorrelation gate and shown for measurements with (a) and without (b) a gold foil in the target frame. Panel (c) shows yield curves for these two cases, $\mathrm{Au}+\mathrm{Au}$ and $\mathrm{Au}+\mathrm{EF}$ (EF stands for empty frame), normalized with respect to the integrated beam intensity.

Figure 10

(Top row) Inclusive distributions of the angle ${\mathrm{\Phi }}_{\mathrm{RP}}$ representing the reaction-plane orientation obtained with the $Q$-vector method from the combined CHIMERA and AToOF data for peripheral (a), semicentral (b), and very central (c) impact-parameter intervals (see Table 1 for their definitions). (Bottom row) Distributions of the difference of orientations of the subevent reaction planes for the same three event classes, peripheral (d), semicentral (e), and very central (f), obtained with the mixing technique of Refs. [52, 53] and by using the weight $Z{\beta }_t\gamma$ (see text and Table 2). The corresponding values of the reaction-plane dispersion parameter $\chi$ are indicated.

Figure 11

(Top row) Correlation between the $Q$-vector orientations determined with CHIMERA (abscissa) and with the Microball (ordinate) for data sets taken with (a) and without (b) a target foil in place (EF stands for empty frame). (Bottom row) Difference of the $Q$-vector orientations for $\mathrm{Au}+\mathrm{Au}$ and for $\mathrm{Au}+\mathrm{EF}$ data (c), normalized with respect to the integrated beam intensity (BP stands for beam particles), and the raw hit multiplicities (d) registered with LAND for $\mathrm{Au}+\mathrm{Au}$ (solid line) and for $\mathrm{Au}+\mathrm{EF}$ data sets (dotted line). The hatched area in (c) indicates the range of events rejected by the required anticorrelation of the CHIMERA and Microball $Q$-vector orientations (Sec. 3d).

Figure 12

Measured directed (top) and elliptic flows (bottom) of charged particles as determined with different timing corrections in comparison with FOPI results (solid triangles, from [48]) for the same ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ reaction at 400 MeV/nucleon in the interval of impact parameters $3.35\le b\le 6$ fm. The percentages of cases to which the so-called second step of the timing corrections was applied are given in the legend (see text). The solid and dashed black lines indicate the limits 40% and 100%, respectively, of the studied probability interval.

Figure 13

Measured flow parameters $v_1$ (top) and $v_2$ (bottom) for the central event class ($b<7.5$ fm) in ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ collisions at 400 MeV/nucleon for neutrons (solid circles) and charged particles (solid triangles) as a function of the transverse momentum $p_t/A$. The UrQMD predictions for neutrons and charged particles obtained with a stiff ($\gamma =1.5$, red solid and dotted lines, respectively) and a soft ($\gamma =0.5$, blue dashed and dash-dotted lines, respectively) density dependence of the symmetry term have been filtered to correspond to the geometrical acceptance of the experiment. The experimental data are corrected for the dispersion of the reaction-plane orientation. Where not shown, the statistical errors are smaller than the size of the symbols.

Figure 14

Elliptic flow ratio of neutrons over all charged particles for central ($b<7.5$ fm) collisions of ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ at 400 MeV/nucleon as a function of the transverse momentum per nucleon $p_t/A$, evaluated with a fraction of 80% for the second step of timing corrections (see Sec. 4a). The black squares represent the experimental data; the green triangles and purple circles represent the UrQMD predictions for stiff ($\gamma =1.5$) and soft ($\gamma =0.5$) power-law exponents of the potential term, respectively. The solid line is the result of a linear interpolation between the predictions, weighted according to the experimental errors of the included four bins in $p_t/A$ and leading to the indicated $\gamma =0.75\pm 0.10$.

Figure 15

Potential-term coefficient $\gamma$ deduced by interpolating between the UrQMD predictions shown in Fig. 14 as a function of the fraction chosen for the second step of timing corrections (see Sec. 4a).

Figure 16

Detection probability of the first plane of the LAND detector, preceded by the veto wall, for neutrons (dots), protons (solid squares), deuterons (solid triangles), tritons (solid tip-down triangles), ${}^3\mathrm{He}$ (open circles), ${}^4\mathrm{He}$ (open squares), and ${}^7\mathrm{Li}$ (open triangles) as a function of the particle kinetic energy per nucleon $E_{\mathrm{kin}}/A$.

Figure 17

Measured elliptic-flow ratio for central ($b<7.5$ fm) collisions of ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ at 400 MeV/nucleon in comparison with stiff and soft UrQMD predictions (top) and deduced symmetry-term coefficient $\gamma$ (bottom) as a function of the upper limit of the time-of-flight interval used to obtain time-integrated results. The dashed horizontal lines mark the upper and lower limits of the $1\sigma$ statistical error margin $\mathrm{\Delta }\gamma =\pm 0.10$ within the time interval $60<\mathrm{TOF}<90$ ns.

Figure 18

Constraints deduced for the density dependence of the symmetry energy from the present data in comparison with the FOPI-LAND result of Ref. [5] as a function of the reduced density $\rho /{\rho }_0$. The low-density results of Refs. [78, 79, 80, 81] as reported in Ref. [82] are given by the symbols, the gray area (HIC), and the dashed contour (IAS). For clarity, the FOPI-LAND and ASY-EOS results are not displayed in the interval $0.3<\rho /{\rho }_0<1.0$.

Figure 19

Density dependence of the difference of the elliptic-flow ratio (DEFR) of neutrons over charged particles, defined by Eq. (9), for ${}^{197}\mathrm{Au}+{}^{197}\mathrm{Au}$ collisions at 400 MeV/nucleon obtained with the TüQMD transport model [14] and the FOPI-LAND acceptance filter (top) and the corresponding sensitivity density (bottom, solid line) together with sensitivity densities obtained from elliptic-flow ratios of neutrons over all hydrogen isotopes (dashed line) and neutrons over protons (dash-dotted line).

Figure 20

(a),(b) Observed correlations of $t_{17}$ vs calibrated time $t_{\mathrm{cal}}$ for the two signals $t_{\mathrm{cal}\_1}$ and $t_{\mathrm{cal}\_2}$ of a module of the first plane of LAND, respectively. (c),(d) Observed correlations of the position signals $Y_{\mathrm{time}}$ vs $Y_{\mathrm{amp}}$ deduced from the time and amplitude information of these signals, respectively, before (c) and after (d), the first correction step. (e) The same correlation after the correction step $1^{\mathrm{st}}$ bis. The significance of the marked regions in panels (a) through (c) is explained in the text.

Figure 21

Hit distributions for a module of the first plane of LAND as a function of the position $Y_{\mathrm{time}}$ (top) and of the arrival time $t_{\mathrm{hit}}$ (bottom). Black solid lines denote the uncorrected distributions, the colored lines represent the distributions after the $1^{\mathrm{st}}$ (red, small dots), the $1^{\mathrm{st}}$ bis (green, thick dots), and the $2^{\mathrm{nd}}$ (blue, thick dashed) corrections. “Bad peaks” refers to artificial sharp peaks at 25-ns intervals (solid purple areas) generated by the electronic readout.

Figure 22

High-velocity tail of normalized velocity spectra for several assumptions on the time-zero value $t_0$ in comparison with the corresponding spectrum measured in the FOPI-LAND experiment.