Effective nucleus-nucleus potentials for heavy-ion fusion reactions

Based on the Skyrme energy density functional and the reaction $Q$-value, we propose an effective nucleus-nucleus potential for describing the capture barrier in heavy-ion fusion processes. The 443 extracted barrier heights are well reproduced with a root-mean-square (rms) error of 1.53 MeV and the rms deviations with respect to 144 TDHF capture barrier heights is only 1.05 MeV. Together with the Siwek-Wilczy\'{n}ski formula in which the three parameters are determined by the proposed effective potentials, the measured capture cross sections at energies around the barriers can be reasonably well reproduced for a series of fusion reactions induced by not only nearly spherical nuclei but also the nuclei with large deformations such as $^{154}$Sm and $^{238}$U. The shallow capture pockets and small values of the average barrier radii play a role in the reduction of the capture cross sections for $^{52,54}$Cr and $^{64}$Ni induced reactions which are related to the synthesis of new super-heavy nuclei.