  The Woods-Saxon-Gaussian (WSG) potential is proposed as a new
phenomenological potential to describe systematically the level scheme,
electromagnetic transitions, and alpha-decay half-lives of the alpha-cluster
structures in various alpha+closed shell nuclei. It modifies the original
Woods-Saxon (WS) potential with a shifted Gaussian factor centered at the
nuclear surface. We determine the free parameters in the WSG potential by
reproducing the correct level scheme of
$^{212}\text{Po}={}^{208}\text{Pb}+\alpha$. It is found that the resulting WSG
potential matches with the M3Y double-folding potential at the surface region
and makes corrections to the inner part of the cluster-core potential. We also
find that the WSG potential with the almost same parameters determined for
$^{212}$Po (except for a rescaled radius) could also be used to describe
alpha-cluster structures in $^{20}\text{Ne}={}^{16}\text{O}+\alpha$ and
$^{44}\text{Ti}={}^{40}\text{Ca}+\alpha$. In all three cases, the calculated
values of the level schemes, electromagnetic transitions, and alpha-decay
half-lives agree with the experimental data, which shows that the WSG potential
could indeed grasp many important features of the alpha-cluster structures in
alpha+closed shell nuclei. The study here is a useful complement to the
existing cluster-core potentials in literature. The Gaussian form factor
centered at the nuclear surface might also help deepen our understanding on the
alpha-cluster formation taking place around the same place.
