Aspects of the momentum dependence of the equation of state and of the residual $NN$ cross section, and their effects on nuclear stopping

With the semiclassical Landau-Vlasov transport model we studied the stopping observable $R_E$, the energy-based isotropy ratio, for the ${}^{129}\mathrm{Xe}+{}^{120}\mathrm{Sn}$ reaction at beam energies spanning $12A$ to $100A$ MeV. We investigated the impacts of the nonlocality of the nuclear mean field, of the in-medium modified nucleon-nucleon ($NN$) cross section, and of the reaction centrality. A fixed set of model parameters yields $R_E$ values that favorably compare with the experimental ones, but only for energies below the Fermi energy $E_F$. Above $E_F$ agreement is readily possible, but by a smooth evolution with energy of the parameter that controls the in-medium modification of $NN$ cross section. By comparing the simulation correction factor $\mathcal{F}$ applied to the free $NN$ cross section with the one deduced from experimental data [Phys. Rev. C 90, 064602 (2014)], we infer that the zero-range mean field almost entirely reproduces it. Also, in accordance with what has been deduced from experimental data, around $E_F$ a strong reduction of the free $NN$ cross section is found. In order to test the impact of sampling central collisions by multiplicity, an event generator (hipse) was used. We obtain that high multiplicity events are spread over a broad impact parameter range, but it turns out that this has a small effect on the observable $R_E$ and, thus, on $\mathcal{F}$ as well.

Figure 1

Landau-Vlasov simulation results on the dependence of the transverse-to-longitudinal energy ratio $R_E^{\mathrm{sim}}$ of Eq. (5) as a function of incident energy for the central ${}^{129}\mathrm{Xe}+{}^{120}\mathrm{Sn}$ reaction and several parametrizations of ${\sigma }_{NN}^m$. Panel (a) displays the results with the momentum dependent D1-G1 EOS and panel (b) those with the zero-range Zamick EOS. Heavy dotted curves represent the $R_E^{\mathrm{sim}}$ excitation function obtained with the ${\sigma }_{NN}^{\mathrm{free}}$ due to Chen et al. [24]. The gray zones show the range of the $R_E^{\mathrm{sim}}$ values limited by the impact parameter $b=1$ fm (dashed bordering curves) and $b=2$ fm (full bordering curves), while the heavy curve in each zone represents the $b$-weighted $R_E^{\mathrm{sim}}$ values in the range $b=0–2$ fm for ${\sigma }_{NN}^m$ due to Cugnon et al. [25] (red curves and zones) and Li and Machleidt [16] (blue curves and zones). Thin dashed curves are obtained with the parametrization of ${\sigma }_{NN}^m$ due to Klakow et al. [27] where the results for the different values of the parameter $\alpha$ of Eq. (6) are distinguished by the varied dash size and color. The filled circles and associated errors stand for the $Z=1$ experimental $R_E^{\exp }$ values [7]. The thin dotted horizontal line denotes the full stopping value. The entrance channel values of $R_E$ are shown by (i) the thin dashed curve resulting from the two Fermi spheres ($E_F=38$ MeV) displaced for the entrance channel relative momentum and (ii) by the thin dash-dotted curves for the LV model values at the contact of colliding nuclei for each of the two EOS used.

Figure 2

$R_E^{\mathrm{sim}}$ as a function of incident energy for the central Xe + Sn reaction and several values of the ${\sigma }_{NN}^{\mathrm{free}}$ scaling factor $\mathcal{F}$ of Eq. (7). Upper (lower) panel shows results obtained with the D1-G1 (Zamick) EOS. The colored zones and curves have the same meaning as in Fig. 1, but here for the scaled ${\sigma }_{NN}$ of Chen et al. [24]. For more details see the caption of Fig. 1 and the text.

Figure 3

Cross section correction factor $\mathcal{F}$ of Eq. (7) as a function of $E_{\mathrm{inc}}$. Open circles and squares interpolated by the dashed curve display ${\mathcal{F}}_{\mathrm{sim}}$ values obtained for $R_E^{\mathrm{sim}}$ evaluated in the $b=0–2$ fm range using the D1-G1 EOS and Zamick EOS, respectively. Open triangles, upright and reversed, interpolated by the dash-dotted curve denote the Gaussian weighted $R_E^{\mathrm{sim}}$ evaluated in the interval $b=0–5$ fm for the D1-G1 EOS and Zamick EOS, respectively. The symbols (but not the curves) are slightly shifted in $E_{\mathrm{inc}}$ to avoid overlapping error bars. Full dots, full curve, and gray zone represent ${\mathcal{F}}_{\exp }$ values and their uncertainty deduced from the experimental $R_E^{\exp }$ [7]. All curves are merely intended to guide the eye. The inset explains the procedure used to extract the values of ${\mathcal{F}}_{\mathrm{sim}}$. For more details see text.

Figure 4

Simulation of the ${}^{129}\mathrm{Xe}+{}^{120}\mathrm{Sn}$ reaction with the hipse code [29]. Shown are the $b$ distributions of the full data set (black line) and of the high multiplicity subset (filled and hollow red-line histograms), which is by number of events equivalent to the one of $b=0–2$ fm (hatched part of the black histograms) for four $E_{\mathrm{inc}}$. The curves are the best fits by a Gaussian function. For more details see text.

Figure 5

Dimensionless weighting factor $f(E,\theta )$ which modulates total elastic cross section as a function of polar angle ${\theta }_{\mathrm{c}.\mathrm{m}.}$ and nucleon incident energy $E$ for the scattering of neutron and proton (upper panel) and between protons (lower panel), where the applicate axis is in logaritmic scale.