Nuclear clusters bound to doubly magic nuclei: The case of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mmultiscripts><mml:mi mathvariant="normal">Po</mml:mi><mml:mprescripts/><mml:none/><mml:mrow><mml:mn>212</mml:mn></mml:mrow></mml:mmultiscripts></mml:math>

An effective $\ensuremath{\alpha}$-particle equation is derived for cases where an $\ensuremath{\alpha}$ particle is bound to a doubly magic nucleus. As an example, we consider $^{212}\mathrm{Po}$ with the $\ensuremath{\alpha}$ on top of the $^{208}\mathrm{Pb}$ core. We consider the core nucleus infinitely heavy, so that the $\ensuremath{\alpha}$ particle moves with respect to a fixed center; that is, recoil effects are neglected. The fully quantal solution of the problem is discussed. The approach is inspired by the Tohsaki-Horiuchi-Schuck-R\"opke wave function concept that has been successfully applied to light nuclei. Shell-model calculations are improved by including four-particle ($\ensuremath{\alpha}$-like) correlations that are of relevance when the matter density becomes low. In the region where the $\ensuremath{\alpha}$-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 $^{212}\mathrm{Po}$ are presented. Such preformed cluster states are very difficult to describe with shell-model calculations. Reasons for the different physics behavior of an $\ensuremath{\alpha}$-like cluster with respect to a deuteron-like cluster are discussed.