Systematics of heavy-ion fusion hindrance at extreme sub-barrier energies

The recent discovery of hindrance in heavy-ion induced fusion reactions at extreme sub-barrier energies represents a challenge for theoretical models. Previously, it has been shown that in medium-heavy systems, the onset of fusion hindrance depends strongly on the “stiffness” of the nuclei in the entrance channel. In this work, we explore its dependence on the total mass and the Q-value of the fusing systems and find that the fusion hindrance depends in a systematic way on the entrance channel properties over a wide range of systems.

Figure 1

Comparison of the logarithmic derivative and S-factor representations of the fusion cross section for the systems, ${}^{58}\mathrm{Ni}+{}^{58}\mathrm{Ni}$ [14] and ${}^{64}\mathrm{Ni}+{}^{64}\mathrm{Ni}$ [12]. The dashed curves correspond to a constant S-factor, whereas the solid curves display results of coupled-channels calculations. The $L(E)$ data were obtained from a fit to the cross sections at three consecutive beam energies.

Figure 2

Logarithmic derivative representations for a range of systems. The dashed curves correspond to a constant S-factor, whereas the dashed-dotted curves display results of coupled-channels calculations, and the solid curves represent a fit to the data using the function $a+b/E^{3/2}$. The range of $E_s$ values are indicated by vertical line segments for heavy systems. For the four lightest systems only lower limits of $L_s$ can be derived (shown as arrows). The data are taken from Refs. ${}^{10}\mathrm{B}+{}^{10}\mathrm{B}$ [15], ${}^{11}\mathrm{B}+{}^{12}\mathrm{C}$ , ${}^{12}\mathrm{C}+{}^{13}\mathrm{C}$, ${}^{12}\mathrm{C}+{}^{16}\mathrm{O}$, ${}^{16}\mathrm{O}+{}^{16}\mathrm{O}$ [16], ${}^{48}\mathrm{Ca}+{}^{48}\mathrm{Ca}$ [17], ${}^{60}\mathrm{Ni}+{}^{89}\mathrm{Y}$ [10], and ${}^{90}\mathrm{Zr}+{}^{92}\mathrm{Zr}$ [18].

Figure 3

(a) Experimental values (symbols) and limits (horizontal bars) are shown as a function of $Z_1{Z}_2\sqrt{\mu }$ for $E_s$ (panel a), $L_s$ (panel b), logarithmic slope ratio, $R=(\mathit{dL}/\mathit{dE}){}_{\exp }/(\mathit{dL}/\mathit{dE}){}_{\mathit{cs}}$ at $E_s$ (panel c), ratio $E_s/V_{\mathrm{Bass}}$ (panel d), and $E^{*}=E_s+Q$ (panel e), which also shows effective particle emission thresholds (histogram). Solid lines in panels (a) and (b) correspond to $L_s=2.33$ MeV${}^{-1}$, whereas dashed curves represent empirical trends of the data. Open diamonds in panel (c) represents the slope ratio at the crossing point obtained from the fit to the data.