  We present a microscopic calculation of $\alpha$-cluster formation in heavy
nuclei $^{104}$Te ($\alpha$+$^{100}$Sn), $^{212}$Po ($\alpha$+$^{208}$Pb) and
their neighbors $^{102}$Sn, $^{102}$Te, $^{210}$Pb and $^{210}$Po by using the
quartetting wave function approach. Improving the local density approximation,
the shell structure of the core nucleus is considered, and the center-of-mass
(c.o.m.) effective potential for the quartet is obtained self-consistently from
the shell model wavefunctions. The $\alpha$-cluster formation and decay
probabilities are obtained by solving the bound-state of the c.o.m. motion of
the quartet and the scattering state of the formed $\alpha$-cluster in the
Gurvitz approach. Striking shell effects on the $\alpha$-cluster formation
probabilities are analyzed for magic numbers 50, 82 and 126. The computed
$\alpha$-decay half-lives of these special nuclei are compared with the newest
experimental data.
