  We investigate the formation distance ($R_0$) from the center of the
radioactive parent nucleus at which the emitted cluster is most probably
formed. The calculations are microscopically performed starting from solving
the time-independent Schr\"{o}dinger wave equation for the $\alpha$-core
system, using nuclear potential based on the Skyrme-SLy4 nucleon-nucleon
interaction and folding Coulomb potential, to determine the incident and
transmitted wave functions of the system. Our results advocate that the emitted
cluster is mostly formed in the pre-surface region of the nucleus, under the
effect of Pauli blocking from the saturated core density. The deeper
$\alpha$-formation distance inside the nucleus gives rise to less preformation
probability, and indicates more stable nucleus of longer half-life. Also, the
$\alpha$-particle tends to be formed at a bit deeper region inside the nuclei
having larger isospin asymmetry and in the closed shell nuclei. Regarding the
emitted nuclei heavier than $\alpha$-particle, we find that the formation
distance of the emitted clusters heavier than $\alpha$-particle increases with
increasing the isospin asymmetry of the formed cluster, rather than with
increasing its mass number. The partial half-life of a certain cluster-decay
mode increases upon increasing either the mass number or the isospin asymmetry
of the emitted cluster.
