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$^{+106}_{-23}$ $mu$s.