Shear viscosity from nuclear stopping

Within a Boltzmann transport model, we demonstrate correlation between stopping observables and shear viscosity in central nuclear collisions at intermediate energies (on the order of 10–1000 MeV/nucleon). The correlation allows us to assess the viscosity of nuclear matter, by tuning the in-medium nucleon-nucleon cross section in our transport model to agree with nuclear stopping data. We also calculate the ratio of shear viscosity to entropy density to determine how close the system is to the universal quantum lower limit proposed in the context of ultrarelativistic heavy-ion collisions.

Figure 1

Schematic of asymmetric collision. Projectile transfers momentum to the target. To assess linear momentum transfer, the longitudinal velocity of the targetlike fragment is compared to the velocity of the center of mass of the collision.

Figure 2

Linear momentum transfer for ${}^{40}\mathrm{Ar}+\mathrm{Cu}$. Lines represent the theoretical results incorporating different in-medium $NN$ cross sections. The “tempered” reductions are marked with their adjustable parameter $\nu$. Symbols represent experimental data [38].

Figure 3

Linear momentum transfer for ${}^{40}\mathrm{Ar}+\mathrm{Ag}$. Lines represent the theoretical results incorporating different in-medium $NN$ cross sections. The “tempered” reductions are marked with their adjustable parameter $\nu$. Symbols represent experimental data [38].

Figure 4

Linear momentum transfer for ${}^{40}\mathrm{Ar}+\mathrm{Au}$. Lines represent the theoretical results incorporating different in-medium $NN$ cross sections. The “tempered” reductions are marked with their adjustable parameter $\nu$. Symbols represent experimental data [38].

Figure 5

Stopping observable varxz for protons in $\mathrm{Au}+\mathrm{Au}$ collisions at different beam energies at $b_{\text{red}}<0.15$. Lines show the effects of different in-medium reductions of the $NN$ cross section. The “tempered” cross-section reductions are marked with their tunable parameter $\nu$. Symbols are experimental data from the FOPI Collaboration [40].

Figure 6

Analogous to Fig. 5, except that $A=2,3$ composite particle formation is enabled in the simulations, so a comparison to results for different particle species in the experiment [40] becomes possible.

Figure 7

Isospin tracer observable for central collisions of ${}^{96}\mathrm{Zr}+{}^{96}\mathrm{Ru}$ (bottom) and ${}^{96}\mathrm{Ru}+{}^{96}\mathrm{Zr}$ (top) at $400\mathrm{MeV}$/nucleon vs scaled center-of-mass rapidity, such that $y_{z0}=-1$ is the initial projectile rapidity. Data are from the FOPI Collaboration [42]. A tendency of $R_z$ to stay closer to zero at finite rapidities indicates a higher degree of mixing and thus stopping.

Figure 8

Mean-field sensitivity of stopping observables. S (H) refers to a soft (hard) compressibility, while M refers to the inclusion of momentum dependence in the mean field. SM and H are limiting combinations that give similar predictions for flow in semicentral collisions [8]. (a) Linear momentum transfer in $\mathrm{Ar}+\mathrm{Ag}$, (b) linear momentum transfer in $\mathrm{Ar}+\mathrm{Au}$, (c) varxz, and (d) isospin tracer.

Figure 9

Cumulative number of elastic $NN$ collisions vs elapsed time in central $\mathrm{Au}+\mathrm{Au}$ collisions at $400\mathrm{MeV}$/nucleon, for three different in-medium cross sections. Panels (a) and (b) show, respectively, net number of collisions and number of collisions weighted with the viscous weight.

Figure 10

Correlation between the number of nucleon-nucleon collisions that took place up until $100\mathrm{fm}/c$ and the stopping observable $varxz$ for the central $400\mathrm{MeV}$/nucleon collision. The top panel utilizes the unweighted collision number, and the bottom panel utilizes the number weighted with the viscous weight. The weighted collision number rises monotonically with the stopping, unlike the unweighted number.

Figure 11

Shear viscosity for symmetric nuclear matter at different densities and temperatures, deduced from Boltzmann equation with in-medium cross sections.

Figure 12

This work's estimate for $\eta /s$ values in selected reactions (open circles), alongside the estimates arrived at Relativistic Heavy Ion Collider (RHIC) energies following the KLN model (square) [49] and the Glauber model (triangle) [50] for the initial conditions. The ratio is given in units of $\hslash /k_{\mathrm{B}}$. The conjectured lower quantum limit, $1/4\pi$ [2], is shown in a dotted line.