Three-Dimensional Approach Applied to Quasi-stationary States of Deformed $\alpha$-Emitters

We apply a three-dimensional (3D) approach to investigate the quasi-stationary states of well-deformed $\alpha$-emitters. With a splitting of the anisotropic 3D potential into internal and external parts at a separation surface, the 3D $\alpha$-cluster decay width is determined by the initial wave function of a true bound state of an anisotropic harmonic oscillator potential and a non-resonance scattering wave function of Coulomb potential. Substantial difference between the 1D and 3D decay width is found for typical $\alpha$-emitters with large quadrupole and hexadecapole deformations.