Signature of Smooth Transition from Sudden to Adiabatic States in Heavy-Ion Fusion Reactions at Deep Sub-Barrier Energies

We propose a novel extension of the standard coupled-channels framework for heavy-ion reactions in order to analyze fusion reactions at deep-sub-barrier incident energies. This extension simulates a smooth transition between the two-body and the adiabatic one-body states. To this end, we damp gradually the off-diagonal part of the coupling potential, for which the position of the onset of the damping varies for each eigenchannel. We show that this model accounts well for the steep falloff of the fusion cross sections for the ${}^{16}\mathrm{O}+{}^{208}\mathrm{Pb}$, ${}^{64}\mathrm{Ni}+{}^{64}\mathrm{Ni}$, and ${}^{58}\mathrm{Ni}+{}^{58}\mathrm{Ni}$ reactions.

Figure 1

Fusion cross sections for the ${}^{64}\mathrm{Ni}+{}^{64}\mathrm{Ni}$ and ${}^{16}\mathrm{O}+{}^{208}\mathrm{Pb}$ systems versus the incident energies. The solid and the dashed lines are the calculated result with and without the damping factor, respectively. The dotted line is the result of no coupling with the YPE potential.

Figure 2

Astrophysical $S$ factor for the ${}^{64}\mathrm{Ni}+{}^{64}\mathrm{Ni}$ and ${}^{16}\mathrm{O}+{}^{208}\mathrm{Pb}$ systems versus the incident energies. The meaning of each line is the same as in Fig. 1. The dot-dashed line is the result of the sudden model taken from Refs. 8, 10.

Figure 3

Logarithmic derivatives of fusion cross sections, $L(E)=d\ln (E_{\mathrm{c}.\mathrm{m}.}{\sigma }_{\mathrm{fus}})/d{E}_{\mathrm{c}.\mathrm{m}.}$, for the ${}^{64}\mathrm{Ni}+{}^{64}\mathrm{Ni}$ and ${}^{16}\mathrm{O}+{}^{208}\mathrm{Pb}$ systems versus the incident energies. The meaning of each line is the same as in Fig. 1.

Figure 4

The adiabatic potential for the ${}^{16}\mathrm{O}+{}^{208}\mathrm{Pb}$ system versus the center-of-mass distance. The solid line is the adiabatic potential obtained with the damping factor. The dashed line is the result obtained with the conventional CC approach. The dotted line and the solid circle are the potential and the touching point for the uncoupled case, respectively. The gray region denotes the adiabatic potential obtained with the potential inversion method, taken from Ref. 13.