  Recently, experimental researches on the $\alpha$ decay with long lifetime
are one of hot topics in the contemporary nuclear physics [e.g. N. Kinoshita
{\sl et al.} (2012) and J. W. Beeman {\sl et al.} (2012) ]. In this study, we
have systematically investigated the extremely long-lived $\alpha$-decaying
nuclei within a generalized density-dependent cluster model involving the
experimental nuclear charge radii. In detail, the important density
distribution of daughter nuclei is deduced from the corresponding experimental
charge radii, leading to an improved $\alpha$-core potential in the quantum
tunneling calculation of $\alpha$-decay width. Besides the excellent agreement
between theory and experiment, predictions on half-lives of possible candidates
for natural $\alpha$ emitters are made for future experimental detections. In
addition, the recently confirmed $\alpha$-decay chain from $^{294}$117 is well
described, including the attractive long-lived $\alpha$-decaying $^{270}$Db,
i.e., a positive step towards the "island of stability" in the superheavy mass
region.
