Cluster-shell competition and its effect on the $E0$ transition probability in $^{20}$Ne

$^{20}$Ne has been known as a typical example of a nucleus with $\alpha$ cluster structure ($^{16}$O+$\alpha$ structure). However according to the spherical shell model, the spin-orbit interaction acts attractively for four nucleons outside of the $^{16}$O core, and this spin-orbit effect cannot be taken into account in the simple $\alpha$ cluster models. We investigate how the $\alpha$ cluster structure competes with independent particle motions of these four nucleons. The antisymmetrized quasi-cluster model (AQCM) is a method to describe a transition from the $\alpha$ cluster wave function to the $jj$-coupling shell model wave function. In this model, the cluster-shell transition is characterized by only two parameters; $R$ representing the distance between clusters and $\Lambda$ describing the breaking of $\alpha$ clusters, and the contribution of the spin-orbit interaction, very important in the $jj$-coupling shell model, can be taken into account by changing $\alpha$ clusters to quasi clusters. In this article, based on AQCM, we apply $^{16}$O plus one quasi cluster model for $^{20}$Ne. Here we focus on the $E0$ transition matrix element, which has been known as the quantity characterizing the cluster structure. The $E0$ transition matrix elements are sensitive to the change of the wave functions from $\alpha$ cluster to $jj$-coupling shell model.