Dynamical calculations of the above-barrier heavy-ion fusion cross sections using Hartree-Fock nuclear densities with the SKX coefficient set

Background: In our previous paper [Gontchar et al., Phys. Rev. C 89, 034601 (2014)] we have calculated the capture (fusion) excitation functions for several reactions with ${}^{16}\mathrm{O},{}^{28}\mathrm{Si}$, and ${}^{32}\mathrm{S}$ nuclei as the projectiles and ${}^{92}\mathrm{Zr},{}^{144}\mathrm{Sm}$, and ${}^{208}\mathrm{Pb}$ nuclei as the targets. These calculations were performed by using our fluctuation-dissipation trajectory model based on the double-folding approach with the density-dependent M3Y NN forces that include the finite range exchange part. For the nuclear matter density the Hartree-Fock approach with the SKP coefficient set that includes the tensor interaction was applied. It was found that for most of the reactions induced by ${}^{16}\mathrm{O}$ the calculated cross sections cannot be brought into agreement with the data. This suggested that the deviation in the calculated nuclear density for ${}^{16}\mathrm{O}$ from the experimental one was crucial.

Method: The SKX parameter set is used to obtain the nuclear densities. Reactions with ${}^{12}\mathrm{C}$ and ${}^{36}\mathrm{S}$ as the projectiles and ${}^{204}\mathrm{Pb}$ as the target are included in the analysis in addition to those of the previous paper. Only data that correspond to the collision energy $E_{\mathrm{c}.\mathrm{m}.}>1.1U_{B0}$ $(U_{B0}$ is the $s$-wave fusion barrier height) are included in the analysis. The radial friction strength $K_R$ is used as the individual adjustable parameter for each reaction.

Results: For all 13 reactions (91 points) it is possible to reach an agreement with the experimental fusion cross sections within 10%. Only at ten points does the deviation exceed 5%. The value of $K_R$, which provides the best agreement with the data in general, decreases as the system gets heavier in accord with the previous paper [Gontchar et al., Phys. Rev. C 89, 034601 (2014)]. A universal analytical approximation for the dependence of $K_R$ upon the Coulomb barrier height is found.

Conclusions: The developed model is able to reproduce the above-barrier portion of the fusion excitation function within 5% with a probability of 90%. Only one fitting parameter per excitation function $K_R$ is used. The model can be used to predict the results of relevant measurements. The universal analytical approximation of the $K_R$ dependence upon the Coulomb barrier height helps to find the starting value of $K_R$ for a more accurate description.

Figure 1

The ratio $\xi ={\sigma }_{\mathrm{th}}/{\sigma }_{\mathrm{expt}.}$ as the function of $U_{B0}/E_{\mathrm{c}.\mathrm{m}.}$ for 13 reactions listed in Table 2. All reactions are split into four groups according to the projectile nucleus. These calculations were performed with the values of $K_R$ (shown in the figures in units of ${\mathrm{zs}\mathrm{GeV}}^{-1})$, which provide the minimum value of ${\chi }_{v\sigma }^2$ for each reaction.

Figure 2

The value of the radial friction strength that provides the minimal ${\chi }_{v\sigma }^2$ versus $B_Z$ (symbols). The line represents $K_{\mathrm{Re}}\left(B_Z\right)$ defined by Eq. (3).