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Bound clusters on top of doubly magic nuclei

An effective $α$ particle equation is derived for cases where an $α$ particle is formed on top of a doubly magic nucleus. As an example, we consider $^{212}$Po with the $α$ on top of the $^{208}$ Pb core. We will consider the core nucleus infinitely heavy, so that the $α$ particle moves with respect to a fixed center, i.e., recoil effects are neglected. The fully quantal solution of the problem is discussed. The approach is inspired by the THSR (Tohsaki-Horiuchi-Schuck-Röpke) wave function concept that has been successfully applied to light nuclei. Shell model calculations are improved by including four-particle ($α$-like) correlations that are of relevance when the matter density becomes low. In the region where the $α$-like cluster penetrates the core nucleus, the intrinsic bound state wave function transforms at a critical density into an unbound four-nucleon shell model state. Exploratory calculations for $^{212}$Po are presented. Such preformed cluster states are only hardly described by shell model calculations. Reasons for different physics behavior of an $α$-like cluster with respect to a deuteron-like cluster are discussed.