Microscopic description of the self-conjugate ${}^{108}$Xe and ${}^{104}$Te $\alpha$-decay chain

A microscopic calculation of half-lives for the recently observed ${}^{108}$Xe $\to$ ${}^{104}$Te $\to$ ${}^{100}$Sn $\alpha$-decay chain is performed using a self-consistent framework based on energy density functionals. The relativistic density functional DD-PC1 and a separable pairing interaction of finite range are used to compute axially-symmetric deformation energy surfaces of ${}^{104}$Te and ${}^{108}$Xe as functions of quadrupole, octupole and hexadecupole collective coordinates. Dynamic least-action paths are determined that trace the $\alpha$-particle emission from the equilibrium deformation to the point of scission. The calculated half-lives: 197 ns for ${}^{104}$Te and 50 $\mu$s for ${}^{108}$Xe, are compared to recent experimental values of the half-lives of superallowed $\alpha$-decay of ${}^{104}$Te: $<18$ ns, and ${}^{108}$Xe: 58${}_{-23}^{+106}$ $\mu$s.