Simplified modeling of cluster-shell competition in $^{20}$Ne and $^{24}$Mg

We investigate properties of the Generator Coordinate Method (GCM) on a collective basis of Antisymmetrized Quasi-Cluster (AQC) states to describe the cluster-shell competition in $^{20}$Ne and $^{24}$Mg due to the spin-orbit interaction. By introducing a single additional parameter in the antisymmetrized-cluster basis function, a continuous transformation of the $\alpha$ cluster(s) into independent nucleons can be described. We apply this GCM trial wave function to study in details the transition from cluster states $[ ^{16}$O+$\alpha ]$ and $[ ^{16}$O+$\alpha + \alpha]$ to shell model (SM) states $[ ^{16}$O+${\rm 4N} ]$ and $[ ^{16}$O+${\rm 8N} ]$, respectively. An optimal value of the strength of the spin-orbit interaction is deduced by reproducing level spacing in $^{20}$Ne. A possible connection to the group theoretical understanding of the cluster-shell configuration transition is also discussed.