Role of internal kinetic energy in the surface density-overlap region of fusing nuclei

The density overlap between two interacting nuclei results in a change of the internal kinetic energy, which magnifies a repulsive contribution. We investigated to what extent the internal kinetic energy can be calculated using the extended Thomas-Fermi approximation (ETF) with inhomogeneous terms and with surface corrections. The complete fusion reactions involving light and heavy nuclei from $^{16}\mathrm{O}$ to $^{208}\mathrm{Pb}$ are considered as examples. The fusion excitation function is calculated in the framework of the extended quantum diffusion model, using a nucleus-nucleus interaction potential based on the Skyrme energy-density functional. We found that both the fixed strengths of the inhomogeneous terms of the ETF approximation and its semiclassical expansion up to ${\ensuremath{\hbar}}^{2}$ (ETF2) fail to correctly describe the complete fusion process and, thus, the Coulomb barrier parameters. The density and mass-number dependencies for the strengths of these inhomogeneous terms of the internal kinetic-energy density functions are suggested, with which the fusion cross sections are successfully reproduced.