Alpha-cluster excited states in $^{32}$S

Excited states having core$+$alpha cluster structure called the alpha-cluster excited state are known to exist in such nuclei as $^{16}$O and $^{20}$Ne. Meanwhile, the existence of alpha-cluster excited states in the middle of sd-shell nuclei is an open problem. Recently, the alpha-cluster excited state in $^{32}$S is suggested by experiments. In order to understand the dynamics of the core-alpha relative motion, we focus on the structure change of the core nuclei and the breaking of the alpha-cluster. In the present work, we construct $^{28}$Si$+$alpha model which has the structure change of the $^{28}$Si core and the alpha-cluster breaking. Using the present model, we calculate the energy expectation value of $^{28}$Si$+$alpha system. We found that the structure change of the core nuclei is energetically rather important while the alpha-cluster breaking is not significant when the alpha-cluster exists at the surface of the $^{28}$Si core. We calculate the ground and excited states with the generator coordinate method. As a result, we suggest the existence of alpha-cluster excited states in $^{32}$S.