Time dependence of partition into spectators and participants in relativistic heavy-ion collisions

The process of formation of the participant system in heavy-ion collisions is investigated in the framework of a simplified analytic Glauber-like model, which is based on the relativistic Boltzmann transport equation. The key point lies in the time-dependent partition of the nucleon system into two groups: nucleons, which did not take part in any interaction before a given time, and nucleons, which already have interacted. In the framework of the proposed model we introduce a natural energy-dependent temporal scale $t_c$, which allows us to remove all dependencies of the model on the collision energy except for the energy dependence of the nucleon-nucleon cross section. By investigating the time dependence of the total number of participants we conclude that the formation process of the participant system becomes complete at $t\simeq 1.5t_c$. Time dependencies of participant total angular momentum and vorticity are also considered and used to describe the emergence of rotation in the reaction plane.

Figure 1

Schematic drawing of the system evolution in the presented model. Open blue circles indicate nucleons which have not interacted before present time moment while solid red circles indicate nucleons which already have interacted.

Figure 2

The time dependence of the total number of participant nucleons in $\mathrm{Pb}+\mathrm{Pb}$ collisions at (a) SPS and RHIC energies (${\sigma }_{{}_{\mathrm{NN}}}=33$ mb) and (b) LHC energy (${\sigma }_{{}_{\mathrm{NN}}}=70$ mb) for different values of impact parameter. Solid lines depict calculations in the proposed model while dashed lines in panel (a) correspond to calculations from the UrQMD model at $\sqrt{s_{{}_{\mathrm{NN}}}}=17.3$ GeV.

Figure 3

The dependence of the total angular momentum of the participant system on impact parameter at different times for $\mathrm{Pb}+\mathrm{Pb}$ collisions at $\sqrt{s}=2.76$ TeV.

Figure 4

The dependence of the participant angular momentum per participant on impact parameter at different times for $\mathrm{Pb}+\mathrm{Pb}$ collisions at $\sqrt{s}=2.76$ TeV.

Figure 5

The time dependence of the total number of participant nucleons and of the total angular momentum of participants divided by their final values in $\mathrm{Pb}+\mathrm{Pb}$ collisions.

Figure 6

The (a) classical and (b) relativistic weighted participant vorticity ${\Omega }_{zx}$ in units of $c$/fm, calculated in the reaction plane, i.e., $\left(xz\right)$ plane, at time moment $t=0.5t_c$ in $\mathrm{Pb}+\mathrm{Pb}$ collisions. The collision energy is $\sqrt{s_{{}_{NN}}}=2.76$ TeV and $b=0.7b_{\max }$. The collision axis $z$ is scaled with the $\gamma$ factor ${\gamma }_0$, which corresponds to the collision energy.

Figure 7

Same as Fig. 6, but for $t=t_c$.

Figure 8

Same as Fig. 6, but for $t=1.5t_c$.

Figure 9

The binary reaction density ${\tilde{\Gamma }}_{\mathrm{coll}}(t,z)$ in coordinates $(t,z)$ in $\mathrm{Pb}+\mathrm{Pb}$ collisions. The collision energy is $\sqrt{s_{{}_{NN}}}=2.76$ TeV and $b=0.7b_{\max }$. The collision axis $z$ is scaled with the $\gamma$ factor ${\gamma }_0$, which corresponds to the collision energy.

Figure 10

The frequency of binary collisions ${\nu }_{\mathrm{coll}}\left(t\right)$ in $\mathrm{Pb}+\mathrm{Pb}$ collisions calculated in our model and in Glauber Monte Carlo. The collision energy is $\sqrt{s_{{}_{NN}}}=2.76$ TeV and $b=0.7b_{\max }$.