$\alpha$-$\alpha$ Double Folding Cluster Potential Description of the $^{12}$C+$^{24}$Mg System

We present a simultaneous analysis of the elastic scattering and fusion cross-section data of the $^{12}$C+$^{24}$Mg system around the Coulomb barrier and over energies by using the microscopic $\alpha$-$\alpha$ double folding cluster potential within the framework of the optical model and the coupled-channels formalism. The $\alpha$-$\alpha$ double folding cluster potential is obtained by using the $\alpha$-cluster distribution densities of the nuclei in the usual double folding procedure. The microscopic potential results are compared with the findings of the phenomenological deep and shallow potentials. It is subsequently shown that only phenomenological deep real, microscopic nucleon-nucleon and $\alpha$-$\alpha$ double folding cluster potentials provide a consistent description of the angular distributions and fusion cross-section data simultaneously. The effect of the inclusion of the excited states of the target nucleus $^{24}$Mg on the fusion cross-section predictions is also determined by the coupled-channels calculations, which are shown to improve the agreement.