Transition from subbarrier to deep-subbarrier regimes in heavy-ion fusion reactions

We analyze recent experimental data of heavy-ion fusion cross sections available up to deep-subbarrier energies in order to discuss the threshold incident energy for a deep-subbarrier fusion hindrance phenomenon. To this end, we employ a one-dimensional potential model with a Woods-Saxon internuclear potential. Fitting the experimental data in two different energy regions with different Woods-Saxon potentials, we define the threshold energy as an intersection of the two fusion excitation functions. We show that the threshold energies so extracted are in good agreement with the empirical systematics as well as with the values of the Krappe-Nix-Sierk (KNS) potential at the touching point. We also discuss the asymptotic energy shift of fusion cross sections with respect to the potential model calculations, and show that it decreases with decreasing energies in the deep-subbarrier region, although it takes a constant value at subbarrier energies.

Figure 1

Fusion excitation functions for ${}^{64}$Ni+${}^{64}$Ni [Fig. 1(a)] and ${}^{16}$O+${}^{208}$Pb [Fig. 1(b)]. The solid and the dashed lines are results of single-channel potential model calculations which fit the experimental data in the subbarrier and the deep-subbarier energy regions, respectively. The experimental data are taken from Refs. [4, 8].

Figure 2

The threshold energy $E_s$ for deep-subbarrier hindrance for several systems, determined with the two-slope fit to the experimental fusion cross sections, as a function of the parameter $Z_1{Z}_2\sqrt{A_1{A}_2/(A_1+A_2)}$. The solid curve is the empirical function given by Eq. (1), while the stars denote the “experimental” values defined as the maximum energy of the astrophysical $S$ factors [11].

Figure 3

Fusion excitation functions for the ${}^{64}$Ni+${}^{64}$Ni system. The dashed and the solid lines are results of single-channel potential model calculations which fit the experimental data in the subbarrier region and at energies above the Coulomb barrier, respectively. The experimental data are taken from Ref. [4].

Figure 4

The asymptotic energy shift as a function of fusion cross section for ${}^{28}$Si+${}^{64}$Ni (filled squares), ${}^{16}$O+${}^{208}$Pb (filled triangles), and ${}^{64}$Ni+${}^{64}$Ni (filled circles) systems.