Particle production and equilibrium properties within a new hadron transport approach for heavy-ion collisions

The microscopic description of heavy-ion reactions at low beam energies is achieved within hadronic transport approaches. In this article a new approach called “Simulating Many Accelerated Strongly interacting Hadrons” (SMASH) is introduced and applied to study the production of nonstrange particles in heavy-ion reactions at $E_{\mathrm{kin}}=0.4A–2A$ GeV. First, the model is described including details about the collision criterion, the initial conditions and the resonance formation and decays. To validate the approach, equilibrium properties such as detailed balance are presented and the results are compared to experimental data for elementary cross sections. Finally results for pion and proton production in $\text{C}+\text{C}$ and $\text{Au}+\text{Au}$ collisions is confronted with data from the high-acceptance dielectron spectrometer (HADES) and FOPI. Predictions for particle production in $\pi +A$ collisions are made.

Figure 1

The time between consecutive actions for a central Cu-Cu collision at ${\sqrt{s}}_{NN}=3$ GeV, averaged over 100 events within the algorithm without fixed time steps. This plot shows the averaged time until the first, second, third,$\cdots ,n\text{th}$ interaction, therefore, the results are still scattered.

Figure 2

Two-dimensional schematic representation of the grid structure for finding collisions between particles. When searching for collisions, particle 1 is checked with particles 2 and 3, but not with particle 4.

Figure 3

“Elastic box” (pion box with constant isotropic cross section). Upper panel shows scattering rate versus density for different time steps. Lower panel shows universal curve: number of collisions per time step $s\mathrm{\Delta }t$ versus $n\sigma \mathrm{\Delta }t$ for 48 possible combinations of time step $\mathrm{\Delta }t\in \{0.1,0.01,0.001\}\text{fm}$, isotropic elastic cross section $\sigma \in \{1,10,30,50\}\text{mb}$ and test-particle numbers $N_{\mathrm{test}}\in \{1,10,50,100\}$.

Figure 4

Coordinate-space distribution of 208 nucleons compared to the Woods–Saxon distribution with the parameters for a lead nucleus.

Figure 5

Initial positions of nuclei such that contracted spheres of radii $(R_P+d_P)$ and $(R_T+d_T)$ will touch at $t=0$ in a central collision.

Figure 6

Momentum-space distribution of neutrons compared to the analytical expectation for a lead nucleus.

Figure 7

Evolution of an average transverse radius $r_{xy}={\left(\langle x^2+y^2\rangle \right)}^{1/2}$ of a nucleus over 200 fm/$c$ with different combinations of Fermi motion (FM) and potentials, (a) ${}^{29}\mathrm{Cu}$ nucleus and (b) ${}^{79}\mathrm{Au}$ nucleus.

Figure 8

Distribution of the ratio of transverse energy of charged particles and the beam energy scaled with the number of nucleons in minimum bias $\text{Au}+\text{Au}$ and $\text{U}+\text{U}$ collisions $\sqrt{s_{\mathrm{NN}}}=3$ GeV.

Figure 9

Comparison of the number of interactions in the same heavy-ion reaction (60 central $\text{Au}+\text{Au}$ collisions at different beam energies) calculated in the center-of-mass and the fixed-target frame. The upper plot (a) shows the absolute numbers of interactions above a cutoff of $\sqrt{s}=2$ GeV per binary collision, while the lower plot (b) indicates the relative difference.

Figure 10

Total and partial decay widths of the $N^{*}{\left(1440\right)}^{+}$ resonance as a function of mass. The vertical and horizontal dashed lines mark the pole mass and width.

Figure 11

Ratio of Pauli-blocked to total found actions in $\text{Cu}+\text{Cu}$ and $\text{Au}+\text{Au}$ collisions at different beam energies. For reference, the total number of found actions per event (both blocked and performed) in an $\text{Au}+\text{Au}$ collision at $E_{\text{kin}}=0.5A\text{GeV}$ is $0.99\times {10}^5$, for $E_{\text{kin}}=5A\text{GeV}$ it constitutes $1.32\times {10}^5$. The number of test particles used in the simulation is $N_{\text{test}}=50$.

Figure 12

Ratio of Pauli-blocked to total found actions in $\text{Cu}+\text{Cu}$ (filled symbols) and $\text{Au}+\text{Au}$ (open symbols) collisions for different numbers of test particles.

Figure 13

(a) ${\pi }^{-}$-proton and (b) ${\pi }^{+}$-proton cross sections compared to data from Ref. [41].

Figure 14

Pion-pion cross section compared to data from Refs. [50, 51].

Figure 15

(a) Proton-proton and (b) proton-neutron cross sections compared with data from Ref. [41].

Figure 16

Cross sections for single-pion production from (left) proton-proton and (right) proton-neutron collisions compared with data from Ref. [52].

Figure 17

Angular distributions for elastic and inelastic $pp$ collisions at two different energies, compared with data from Refs. [53, 54].

Figure 18

Detailed balance for the $\pi \text{−}\rho \text{−}\sigma$ system in a box with periodic boundary conditions. (a) Multiplicities versus time, (c) scaled numbers of forward and backward reactions for $t>20$ fm/$c$, and (b) the same differentially versus the invariant mass of the reaction, which is equal to the resonance mass in this case.

Figure 19

Multiplicities versus time for a $\pi \text{−}N\text{−}\mathrm{\Delta }$ system in a box.

Figure 20

Scaled numbers of forward (triangles right) and backward (triangles left) reactions for (b) $t>80$ fm/$c$ $\pi \text{−}N\text{−}\mathrm{\Delta }$ and (a) the same differentially in the invariant mass of reaction.

Figure 21

Baryon density estimated in SMASH simulation with smearing $\sigma =0.5$ fm (dashed line) and 1.0 fm (dotted line) is compared with the true density profile (solid line). Large $N_{\text{test}}=1000$ for $\sigma =1$ fm and $N_{\text{test}}=10000$ for $\sigma =0.5$ fm is taken to diminish fluctuations.

Figure 22

Eckart rest frame net baryon density ${\rho }_B$ at the target center in central $\text{Au}+\text{Au}$ collision at $E_{\text{kin}}=0.8A\text{GeV}$ in units of the ground-state nuclear density ${\rho }_0$. Time dependence ${\rho }_B\left(t\right)$ of the full SMASH simulation (full line) is compared to ${\rho }_B\left(t\right)$ of the SMASH simulation with all interactions off (dashed line).

Figure 23

Landau rest-frame hadron density $\epsilon$ at the target center in central $\text{Au}+\text{Au}$ collision at $E_{\text{kin}}=0.8A\text{GeV}$ in units of the ground-state nuclear energy density ${\epsilon }_0=0.150$ GeV/fm ${}^3$. Time dependence $\epsilon \left(t\right)$ of the full SMASH simulation (solid line) is compared to $\epsilon \left(t\right)$ of the SMASH simulation with all interactions off (dashed line).

Figure 24

Landau rest-frame energy density $T_L^{00}$ (background color) and velocity of Landau frame (arrows), both for baryons. $\text{Au}+\text{Au}$ collision at $E_{\text{kin}}=0.8A\text{GeV}$ with impact parameter $b=3\text{fm},N_{\text{test}}=20$. Color legend is given above. Velocity is proportional to the arrow length, maximal arrow length corresponds to velocity of $0.55c$.

Figure 25

Pion production in $\text{Au}+\text{Au}$ collisions at kinetic energies ranging from $0.4A\text{GeV}$ to $1.6A\text{GeV}$, as measured by FOPI [58] (markers), in comparison with SMASH (lines). The upper plot (a) shows the excitation function of ${\pi }^{+}$ and ${\pi }^{-}$ multiplicities, the plot in the middle (b) shows the ratio. The lower plot (c) shows the average transverse momentum of the pions. The impact parameter was set to $b=1.33\text{fm}$. The results of the SMASH simulation are shown for the cascade with the following features successively switched on: 20 test particles per real particle, Skyrme and symmetry potentials, Fermi motion, Pauli blocking.

Figure 26

Rapidity spectra for pions measured by FOPI in $\text{Au}+\text{Au}$ collisions at $1.5A$ GeV [58]. The experimental data (markers) are compared with the corresponding SMASH results (lines). $N$ is the total number of pions obtained by integrating the spectrum. The normalized rapidity $y_0=(y-y_{\text{cm}})/y_{\text{cm}}$ was used. The impact parameter for the simulated events was sampled from the distribution given by the $\mathit{ERAT}$ cuts corresponding to the experimental data (see Appendix pp3). The SMASH simulations were performed with potentials, Pauli blocking, and Fermi motion.

Figure 27

Rapidity spectra for protons measured by FOPI in $\text{Au}+\text{Au}$ collisions at $1.5A$ GeV [58]. The experimental data (markers) is compared with the corresponding SMASH results (lines). See Fig. 26 for more details. $N$ is the total number of particles obtained by integrating the spectrum. Spectators (particles only interacting elastically) have been ignored. To distinguish between unbound protons and deuteron or other nuclei, a coalescence afterburner with parameters $p_0=0.3\text{GeV},r_0=0.9\text{fm}$ was used to model the clustering (see Appendix pp4).

Figure 28

Transverse mass spectra for pions measured by HADES in carbon-carbon collisions at $1A$ and $2A\text{GeV}$ [62]. The experimental data (markers) is shown for different longitudinal rapidity bins and compared to the corresponding SMASH results (lines). For readability, the data corresponding to each bin was multiplied with a different power of 10. The impact parameter distribution provided by HADES was used for sampling the events with SMASH.

Figure 29

Transverse mass and rapidity spectra for charged pions and nucleons in ${\pi }^{-}$-carbon collisions at $E_{\text{kin}}=1.7\text{GeV}$. They were obtained from a SMASH simulation with 20 test particles per real particle, including potentials, Fermi motion, and Pauli blocking. Spectators (particles only interacting elastically) were ignored. Data for this scenario have been measured by HADES, but are not yet published. The legend shows the total multiplicity $N$ obtained from integrating the rapidity spectrum.

Figure 30

Number of reactions in a ${\pi }^{-}$-carbon collision at $E_{\text{kin}}=1.7\text{GeV}$, averaged over 25 000 events. The upper plot (a) shows the forward rates (right arrows, solid lines) and backward rates (left arrows, dashed lines) per event for the most important reactions as a function of time. The lower plot (b) shows the total number of forward and backward reactions per event for various reactions. The same SMASH simulation results as in Fig. 29 were used.