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$β$ decay and evolution of low-lying structure in Ge and As nuclei

A simultaneous calculation for the shape evolution and the related spectroscopic properties of the low-lying states, and the $β$-decay properties in the even- and odd-mass Ge and As nuclei in the mass $A\approx70-80$ region, within the framework of the nuclear density functional theory and the particle-core coupling scheme, is presented. The constrained self-consistent mean-field calculations using a universal energy density functional (EDF) and a pairing interaction determines the interacting-boson Hamiltonian for the even-even core nuclei, and the essential ingredients of the particle-boson interactions for the odd-nucleon systems, and of the Gamow-Teller and Fermi transition operators. A rapid structural evolution from $γ$-soft oblate to prolate shapes, as well as the spherical-oblate shape coexistence around the neutron sub-shell closure $N=40$, is suggested to occur in the even-even Ge nuclei. The predicted low-energy spectra, electromagnetic transition rates, and $β$-decay $\log{ft}$ values are in a reasonable agreement with experiment. The predicted $\log{ft}$ values reflect the structures of the wave functions for the initial and final nuclei of $β$ decay, which are, to a large extent, determined by the microscopic input provided by the underlying EDF calculation.