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.
Categories: nucl-th nucl-ex