Sensitivity of the fusion cross section to the density dependence of the symmetry energy

Background: The study of the nuclear equation of state (EOS) and the behavior of nuclear matter under extreme conditions is crucial to our understanding of many nuclear and astrophysical phenomena. Nuclear reactions serve as one of the means for studying the EOS.

Purpose: It is the aim of this paper to discuss the impact of nuclear fusion on the EOS. This is a timely subject given the expected availability of increasingly exotic beams at rare isotope facilities [A. B. Balantekin et al., Mod. Phys. Lett. A 29, 1430010 (2014)]. In practice, we focus on ${}^{48}\mathrm{Ca}+{}^{48}\mathrm{Ca}$ fusion.

Method: We employ three different approaches to calculate fusion cross sections for a set of energy density functionals with systematically varying nuclear matter properties. Fusion calculations are performed using frozen densities, using a dynamic microscopic method based on density-constrained time-dependent Hartree-Fock (DC-TDHF) approach, as well as direct TDHF study of above barrier cross sections. For these studies, we employ a family of Skyrme parametrizations with systematically varied nuclear matter properties.

Results: The folding-potential model provides a reasonable first estimate of cross sections. DC-TDHF, which includes dynamical polarization, reduces the fusion barriers and delivers much better cross sections. Full TDHF near the barrier agrees nicely with DC-TDHF. Most of the Skyrme forces which we used deliver, on the average, fusion cross sections in good agreement with the data. Trying to read off a trend in the results, we find a slight preference for forces which deliver a slope of symmetry energy of $L\approx 50$ MeV that corresponds to a neutron-skin thickness of ${}^{48}\mathrm{Ca}$ of $R_{\mathrm{skin}}=(0.180–0.210)$ fm.

Conclusions: Fusion reactions in the barrier and sub-barrier region can be a tool to study the EOS and the neutron skin of nuclei. The success of the approach will depend on reduced experimental uncertainties of fusion data as well as the development of fusion theories that closely couple to the microscopic structure and dynamics.

Figure 1

Neutron root-mean-square radius (r.m.s.), neutron diffraction radius, and neutron halo of ${}^{48}\mathrm{Ca}$ obtained from a set of Skyrme parametrizations with systematically varied nuclear matter parameters (incompressibility $K$, effective mass $m^{*}/m$, symmetry energy $J$, and ${\kappa }_{\mathrm{TRK}})$ [60] for the SV density functionals used in this study.

Figure 2

Folding model ion-ion interaction potentials, $V_F\left(R\right)$, for ${}^{48}\mathrm{Ca}+{}^{48}\mathrm{Ca}$ system using the Skyrme SV EDFs; (a) potentials for SV-mas07, SV-mas08, SV-mas10, SV-sym28, SV-sym30, SV-sym32, and SV-sym34. Also shown is the folding potential corresponding to SV-bas. (b) potentials for SV-K218, SV-K226, SV-K241, SV-kap00, SV-kap20, and SV-kap60.

Figure 3

${}^{48}\mathrm{Ca}+{}^{48}\mathrm{Ca}$ fusion cross-sections corresponding to the folding-model ion-ion interaction potentials $V_F\left(R\right)$ for Skyrme SV EDFs, scaled by the fusion cross-section corresponding to the SV-bas barrier. (a) Cross-section ratios for SV-mas07, SV-mas08, SV-mas10, SV-sym28, SV-sym30, SV-sym32, and SV-sym34. (b) Cross-section ratios for SV-K218, SV-K226, SV-K241, SV-kap00, SV-kap20, and SV-kap60. The shaded area indicates the range of folding-model potential barrier heights obtained using the SV forces.

Figure 4

${}^{48}\mathrm{Ca}+{}^{48}\mathrm{Ca}$ fusion cross section using an ion-ion interaction $V_F\left(R\right)$ obtained in the double-folding approximation with densities computed from the set of systematically varied Skyrme forces. Results are plotted on a logarithmic scale for (a) SV-mas07, SV-mas08, SV-mas10, SV-sym28, SV-sym30, SV-sym32, and SV-sym34, and for (b) SV-K218, SV-K226, SV-K241, SV-kap00, SV-kap20, and SV-kap60. Also shown is the cross section corresponding to SV-bas, and the experimental data (filled circles) are taken from Ref. [79].

Figure 5

DC-TDHF ion-ion interaction potentials $V\left(R\right)$ for Skyrme SV EDFs. (a) Potentials for SV-mas07, SV-mas10, SV-sym28, and SV-sym34. (b) Potentials for SV-K218, SV-K241, SV-kap00, and SV-kap60.

Figure 6

Fusion cross-sections corresponding to the DC-TDHF potentials $V\left(R\right)$ for Skyrme SV EDFs scaled with the cross sections corresponding to the SV-bas EDF. The shaded area indicates the range of DC-TDHF potential barrier heights obtained using the SV forces. The experimental data, also scaled with the SV-bas cross sections (filled circles), are taken from Ref. [79] and include statistical as well as $\pm 7$% systematic error.

Figure 7

Fusion cross-sections corresponding to the DC-TDHF potentials $V\left(R\right)$ for Skyrme SV EDFs plotted on a logarithmic scale. (a) Cross sections for SV-sym28, SV-sym34, SV-mas07, and SV-mas10. (b) Cross sections for SV-K218, SV-K241, SV-kap00, and SV-kap60. The experimental data (filled circles) are taken from Ref. [79].

Figure 8

The maximum height of the potential energy barrier $V_B$ as a function of the neutron skin $R_{\mathrm{skin}}$, as predicted by the set of systematically varied Skyrme parametrizations used in this paper. The dashed lines show the same quantities for the folding potentials.

Figure 9

Fusion cross sections in the vicinity of barrier-top energies plotted for SV parametrizations that are most compatible with the fusion data in this energy regime.

Figure 10

Cross sections calculated directly from TDHF with SV-bas, SV-sym28, and SV-sym34. Panel (a) shows the data on a plot with a linear $y$ axis and panel (b) shows the same data with a logarithmic $y$ axis. The experimental data (filled circles) are taken from Ref. [79].