The formation region of emitted $\alpha$ and heavier particles inside radioactive nuclei

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.