Both the valence nucleons (holes) and the isospin asymmetry dependencies of the -cluster inside parents radioactive nuclei are the preformation probability of an investigated. The calculations are employed in the framework of the density-dependent cluster model of ?-decay process for the even-even spherical parents nuclei with protons number around the closed shell Z0= 82 and neutrons number around the closed shells Z0=82 and Z0=126. The microscopic ?-daughter nuclear interaction potential is calculated in the framework of the Hamiltonian energy density approach based on the SLy4 Skyrme-like effective interaction. Also, the calculations based on the realistic effective M3Y-Paris nucleon-nucleon, NN, force have been used to confirm the results. The calculations then proceed to find the assault frequency and the ? penetration probability within the WKB approximation. The half-lives of the different mentioned ? decays are then determined and have been used in turn to find the spectroscopic factor. We found that the spectroscopic factor increases with increasing the isospin asymmetry of the parent nuclei if they have valence protons and neutrons. When the parent nuclei have neutron or proton holes in addition to the valence protons or neutrons then the spectroscopic factor is found to decrease with increasing the isospin asymmetry. The obtained results show also that the deduced spectroscopic factors follow individual linear behaviors as a function of the multiplication of the valence protons (Np) and neutron (Nn) numbers. These linear dependencies are correlated with the closed shells core (Z0, N0 ). The same individual linear behaviors are obtained as a as a function of the multiplication of Np Nn and the isospin asymmetry parameter, Np Nn I. Moreover, the whole deduced spectroscopic factors are found to exhibit a nearly general linear trend with the function Np Nn/ (Z0+N0).

}, author = {W. M. Seif and M. Shalaby and M. F. Alrakshy} }