Correlation between the12C+12C,12C+13C, and13C+13C fusion cross sections

The fusion cross section for ${}^{12}$C+${}^{13}$C has been measured down to ${E}_{\mathrm{c}.\mathrm{m}.}=2.6$ MeV, at which the cross section is of the order of 20 nb. By comparing the cross sections for the three carbon isotope systems, ${}^{12}$C+${}^{12}$C, ${}^{12}$C+${}^{13}$C, and ${}^{13}$C+${}^{13}$C, it is found that the cross sections for ${}^{12}$C+${}^{13}$C and ${}^{13}$C+${}^{13}$C provide an upper limit for the fusion cross section of ${}^{12}$C+${}^{12}$C over a wide energy range. After calibrating the effective nuclear potential for ${}^{12}$C+${}^{12}$C using the ${}^{12}$C+${}^{13}$C and ${}^{13}$C+${}^{13}$C fusion cross sections, it is found that a coupled-channels calculation with the ingoing wave boundary condition (IWBC) is capable of predicting the major peak cross sections in ${}^{12}$C+${}^{12}$C. A qualitative explanation for this upper limit is provided by the Nogami-Imanishi model and by level density differences among the compound nuclei. It is found that the strong resonance found at 2.14 MeV in ${}^{12}$C+${}^{12}$C exceeds this upper limit by a factor of more than 20. The preliminary result from the most recent measurement shows a much smaller cross section at this energy, which agrees with our predicted upper limit.