Effect of deformation and vibration on the $\alpha$ decay half-life

In this work, we expand upon our previous study of the effect of surface vibrations (low-lying vibrational states) on the calculation of penetration probability in $\alpha$ decay of spherical isotopes [Phys. Rev. C 91, 034611 (2015)]. For this pupose, the Coulomb and proximity potential model, taking into account the ground state deformations of the involved nuclei along with the surface vibrations in the daughter nucleus, is used to evaluate the $\alpha$ decay probability. The results are compared with those obtained with a spherical potential barrier, which shows the dramatic effect of employing the ground state deformations in case of deformed nuclei. As well, inclusion of surface vibrations gives rise to an increase in the value of tunneling probability in better agreement with experimental data.