Proton and neutron density distributions at supranormal density in low- and medium-energy heavy-ion collisions

Background: The distribution of protons and neutrons in the matter created in heavy-ion collisions is one of the main points of interest for the collision physics, especially at supranormal densities. These distributions are the basis for predictions of the density dependence of the symmetry energy and the density range that can be achieved in a given colliding system. We report results of the first systematic simulation of proton and neutron density distributions in central heavy-ion collisions within the beam energy range of $E_{\mathrm{beam}}\le 800\mathrm{MeV}/\mathrm{nucl}$. The symmetric ${}^{40}\mathrm{Ca}+{}^{40}\mathrm{Ca}, {}^{48}\mathrm{Ca}+{}^{48}\mathrm{Ca}, {}^{100}\mathrm{Sn}+{}^{100}\mathrm{Sn}$, and ${}^{120}\mathrm{Sn}+{}^{120}\mathrm{Sn}$ and asymmetric ${}^{40}\mathrm{Ca}+{}^{48}\mathrm{Ca}$ and ${}^{100}\mathrm{Sn}+{}^{120}\mathrm{Sn}$ systems were chosen for the simulations.

Purpose: We simulate development of proton and neutron densities and asymmetries as a function of initial state, beam energy, and system size in the selected collisions in order to guide further experiments pursuing the density dependence of the symmetry energy.

Methods: The Boltzmann-Uhlenbeck-Uehling (pBUU) transport model with four empirical models for the density dependence of the symmetry energy was employed. Results of simulations using pure Vlasov dynamics were added for completeness. In addition, the time-dependent Hartree-Fock (TDHF) model, with the SV-bas Skyrme interaction, was used to model the heavy-ion collisions at $E_{\mathrm{beam}}\le 40\mathrm{MeV}/\mathrm{nucl}$. Maximum proton and neutron densities ${\rho }_{\mathrm{p}}^{\max }$ and ${\rho }_{\mathrm{n}}^{\max }$, reached in the course of a collision, were determined from the time evolution of ${\rho }_{\mathrm{p}}$ and ${\rho }_{\mathrm{n}}$.

Results: The highest total densities predicted at $E_{\mathrm{beam}}=800\mathrm{MeV}/\mathrm{nucl}.$ were of the order of $\sim 2.5{\rho }_0$ (${\rho }_0=0.16{\text{fm}}^{-3}$) for both Sn and Ca systems. They were found to be only weakly dependent on the initial conditions, beam energy, system size, and a model of the symmetry energy. The proton-neutron asymmetry $\delta =({\rho }_{\mathrm{n}}^{\max }-{\rho }_{\mathrm{p}}^{\max })/({\rho }_{\mathrm{n}}^{\max }+{\rho }_{\mathrm{p}}^{\max })$ at maximum density does depend, though, on these parameters. The highest value of $\delta$ found in all systems and at all investigated beam energies was $\sim 0.17$.

Conclusions: We find that the initial state, beam energy, system size, and a symmetry energy model affect very little the maximum proton and neutron densities, but have a subtle impact on the proton-neutron asymmetry. Most importantly, the variations in the proton-neutron asymmetry at maximum densities are related at most at 50% level to the details in the symmetry energy at supranormal density. The reminder is due to the details in the symmetry energy at subnormal densities and proton and neutron distributions in the initial state. This result brings to the forefront the need for a proper initialization of the nuclei in the simulation, but also brings up the question of microscopy, such as shell effects, that affect initial proton and neutron densities, but cannot be consistently incorporated into semiclassical transport models.

Figure 1

Density dependence of the symmetry energy in the models employed in this work. The values of the symmetry energy $S$ and of its slope $L$, at ${\rho }_0$, are $S\left(L\right)=31.8\left(82.8\right),30.0\left(32.4\right)$, and $30.2\left(32.3\right)\text{MeV}$, for S and SM in pBUU, and for SV-bas, respectively. Models SMS and SSM are added for completeness.

Figure 2

Neutron and proton densities as a function of distance from the center of nucleus, for the nuclei considered in the paper, from static HF and TF calculations.

Figure 3

Time evolution of maximal proton and neutron densities normalized to their static value, in asymmetric collisions, ${}^{40}\mathrm{Ca}+{}^{48}\mathrm{Ca}$ and ${}^{100}\mathrm{Sn}+{}^{120}\mathrm{Sn}$, at different incident energies, within pBUU dynamics for symmetry energy S (left panels) and in TDHF dynamics (right panels).

Figure 4

Contour plots of neutron (dashed lines) and proton density (solid lines) in head-on ${}^{120}\mathrm{Sn}+{}^{100}\mathrm{Sn}$ collisions at $16\mathrm{MeV}/\mathrm{nucl}.$ (top) and $40\mathrm{MeV}/\mathrm{nucl}.$, at different times. The horizontal axis is the collision axis.

Figure 5

Maximum neutron and proton densities in units of ${\rho }_0$ (top panels) and the corresponding values of the asymmetry $\delta$ (bottom panels), vs beam energy for Ca systems, yielded in the pBUU simulations with different indicated forms of symmetry energy.

Figure 6

The same as Fig. 4, but for Sn systems.

Figure 7

Maximum neutron and proton densities in units of ${\rho }_0$ (top panels) and corresponding values of asymmetry $\delta$ (bottom panels), vs beam energy for Ca (left) and Sn (right) systems as calculated in Vlasov simulations with symmetry energy model S.

Figure 8

Maximum neutron and proton densities in units of ${\rho }_0$ (top panels) and the corresponding values of asymmetry $\delta$ (bottom panels), vs beam energy for Ca (left) and Sn (right), from TDHF simulations.