Low-spin states in $^{80}$Ge populated in the $\beta$ decay of the $^{80}$Ga $3^-$ isomer

The structure of $^{80}$Ge has been investigated at the ISOLDE facility at CERN. A previous study reported for the first time a low-lying $0^+_2$ intruder state at 639 keV, based on the coincidence with a previously unobserved 1764-keV $γ$ ray, and suggested it as evidence for shape coexistence in $^{80}$Ge. We used the $\beta$ decay from the $3^-$ 22.4-keV state in $^{80}$Ga to enhance the population of low-spin states in $^{80}$Ge, including any excited $0^+$ level, and $\gamma \gamma$ coincidences to investigate it. We observed a 1764-keV $\gamma$ ray in coincidence with strong transitions in $^{80}$Ge, thus not feeding the proposed 639-keV $0^+_2$. No connecting transitions from previously known levels to the 639-keV and 2403-keV $2^+_3$ states could be established either. Shell-model calculations for Ge isotopes and $N=48$ isotones were performed. They succeed to explain most of the experimental levels, but fail to reproduce the presence of a $0^+_2$ state below $≈1200$ keV in $^{80}$Ge. Our experimental findings and shell-model calculations are difficult to reconcile with a very low-lying $0^+_2$ state in $^{80}$Ge.