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An analysis of the $\alpha$ + core properties of $^{104}$Te along with aglobal discussion on the $\alpha$-cluster structure above the double-shellclosures is presented from the viewpoint of the local potential model. The$\alpha$ + core interaction is described by a nuclear potential of (1 +Gaussian)$\times$(W.S. + W.S.$^3$) shape with two free parameters, which hasbeen successfully tested in nuclei of different mass regions. The modelproduces $Q_{\alpha}$ values and $\alpha$-decay half-lives for $^{104}$Te inagreement with the 2018 experimental data of Auranen et al. in the energy range$5.13 \; \mathrm{MeV} < Q_{\alpha} < 5.3 \; \mathrm{MeV}$ using an $\alpha$preformation factor $P_{\alpha} = 1$. The comparison of the calculated reduced$\alpha$-widths for the ground state bands of $^{104}$Te, $^{94}$Mo and$^{212}$Po indicates that $^{104}$Te has an $\alpha$-cluster degreesignificantly higher than $^{94}$Mo and much higher than $^{212}$Po. Theseresults point to $^{104}$Te as a preferential nucleus for $\alpha$-clusteringin the $N,Z = 50$ region and corroborate the statement on the superallowed$\alpha$-decay in $^{104}$Te.

$\alpha$-cluster structure above double-shell closures and $\alpha$-decay of $^{104}$Te