  An effective $\alpha$ particle equation is derived for cases where an
$\alpha$ particle is formed on top of a doubly magic nucleus. As an example, we
consider $^{212}$Po with the $\alpha$ on top of the $^{208}$ Pb core. We will
consider the core nucleus infinitely heavy, so that the $\alpha$ 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-R\"{o}pke) wave function concept that has been
successfully applied to light nuclei. Shell model calculations are improved by
including four-particle ($\alpha$-like) correlations that are of relevance when
the matter density becomes low. In the region where the $\alpha$-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 $^{212}$Po are presented. Such preformed cluster
states are only hardly described by shell model calculations. Reasons for
different physics behavior of an $\alpha$-like cluster with respect to a
deuteron-like cluster are discussed.
