Microscopic description of $^{12}$C+$^{12,13}$C fusion reactions at nuclear astrophysical energies
The $^{12}$C + $^{12}$C fusion reaction plays a key role in several astrophysical phenomena. However, it is difficult to determine its cross sections in the relevant energy region because of both low cross sections and strong resonant structures. On the other hand, the $^{12}$C + $^{13}$C system shows a much smoother energy dependence of fusion cross sections. To simultaneously analyze the $^{12}$C + $^{12,13}$C systems, we here develop a reaction model that explicitly treats the entrance channel and the compound nucleus states. For this purpose, we combine the discrete basis model for the entrance channel and the shell model for the compound nuclei. The coupling strengths between the entrance channel and the compound nucleus states are determined so that the fusion cross sections for these systems match with each other at the resonance energies for the $^{12}$C + $^{12}$C system, as has been observed experimentally. The model successfully reproduces the significantly different behaviors of fusion cross sections in these systems.