Analysis of heavy-ion fusion reactions at extreme sub-barrier energies

A coupled-channels analysis has been carried out for fusion reactions in the system ${}^{60}\mathrm{Ni}+{}^{89}\mathrm{Y}$. It demonstrates that conventional coupled-channels calculations are unable to reproduce the unexpected steep falloff of the recently measured cross sections at extreme sub-barrier energies. Heavy-ion fusion excitation functions are also analyzed in terms of the $S$ factor, as this offers a pragmatic way to study fusion behavior in the energy regime of interest. It is shown that the steep falloff in cross section observed in several heavy-ion systems translates into a maximum of the $S$ factor. The energies where the maximum occurs can be parametrized with a simple empirical formula. The parametrization, which is derived here for rather stiff heavy-ion systems, provides an upper limit for reactions involving softer nuclei.

Figure 1

(a) Fusion excitation function for ${}^{60}\mathrm{Ni}+{}^{89}\mathrm{Y}$. The data are from Ref. 1. The arrow represents an upper limit; (b) logarithmic derivatives for the same system, derived from the data of (a). The solid and dotted curves are the results of the coupled-channels calculations discussed in the text. Also shown are the results for a one-dimensional barrier penetration calculation based on an adjusted interaction (see text for details).

Figure 2

(a) Ion-ion potentials used in the coupled-channels calculations for ${}^{60}\mathrm{Ni}+{}^{89}\mathrm{Y}$. (b) Logarithmic derivatives for the same system. The solid curve is the same as in Fig. 1b. The dashed curve is the coupled-channels calculations with $a_i=2a$. The dotted-dashed line corresponds to a constant $S$ factor (see text for details).

Figure 3

Plots of the $S$ factors vs $E-E_{off}$. The parameters ${\eta }_0$ and $E_{off}\left(\text{MeV}\right)$ are used to conveniently place many colliding systems on the same plot. The solid curves are results of coupled-channels calculations, and the dashed curves are fits to the fusion data based on Wong’s formula. (a) For systems that have a clear maximum in the $S$ factor at low energy, values of ${\eta }_0$ and $E_{off}$ are 69.26, 26 for ${}^{58}\mathrm{Ni}+{}^{58}\mathrm{Ni}$, 92.98, 50 for ${}^{60}\mathrm{Ni}+{}^{89}\mathrm{Y}$, 126.02, 92 for ${}^{90}\mathrm{Zr}+{}^{89}\mathrm{Y}$, 130.00, 87 for ${}^{90}\mathrm{Zr}+{}^{92}\mathrm{Zr}$, 128.24, 84 for ${}^{90}\mathrm{Zr}+{}^{90}\mathrm{Zr}$; (b) for systems where the $S$ factor has not quite developed a clear maximum, values of ${\eta }_0$ and $E_{off}$ are 40.10, $-11$ for ${}^{16}\mathrm{O}+{}^{144}\mathrm{Sm}$, 48.41, $-3$ for ${}^{16}\mathrm{O}+{}^{208}\mathrm{Pb}$, 56.30, $-3$ for ${}^{19}\mathrm{F}+{}^{208}\mathrm{Pb}$, 74.20, 7 for ${}^{64}\mathrm{Ni}+{}^{64}\mathrm{Ni}$, 126.80, 92 for ${}^{50}\mathrm{Ti}+{}^{208}\mathrm{Pb}$, respectively.

Figure 4

Systematics of the energy $E_s$ where the $S$ factor has a maximum as a function of the parameter, $Z_1{Z}_2\sqrt{A_1{A}_2∕(A_1+A_2)}$. The solid curve is calculated with the empirical expression given by Eq. (11). The solid points were obtained for systems that exhibit a clear maximum in the $S$ factor (category I in Table ). The triangles were obtained by extrapolating the logarithmic derivative to the value for a constant $S$ factor, Eq. (9). The open circles show the lowest measured energy $E_{\mathit{min}}$ for those systems where no sign of a maximum in the $S$ factor has been found so far.