Super- and hyper-deformation in $^{60}$Zn, $^{62}$Zn, and $^{64}$Ge at high spins

Background: The observation of the superdeformed (SD) bands in $^{60,62}$Zn indicates that the particle number 30 is a magic particle number, where two and four neutron single-particles are considered to be promoted to the intruder $1g_{9/2}$ shell. However, the SD-yrast band in $^{62}$Zn is assigned negative parity. Purpose: I investigate various SD configurations in the rapidly rotating $^{60,62}$Zn and $^{64}$Ge, and attempt elucidating the different roles of the energy gaps at particle numbers 30 and 32. Method: I employ a nuclear energy-density functional (EDF) method: the configuration-constrained cranked Skyrme-Kohn-Sham approach is used to describe the rotational bands near the yrast line. Results: The negative-parity SD bands appear higher in energy than the positive-parity SD-yrast band in $^{60}$Zn by about 4 MeV, which is indicative of the SD doubly-magic nucleus. However, the energy gap in $^{64}$Ge is smaller $\sim 2\text{-}3$ MeV, though the quadrupole deformation of the SD states in $^{64}$Ge is greater than that of $^{60}$Zn. The present calculation predicts the occurrence of the hyperdeformed state in $^{60}$Zn and $^{64}$Ge at a high rotational frequency $\sim 2.0$ MeV$/\hbar$ due to the occupation of the $h_{11/2}$ shell. Conclusions: An SD-shell gap at particle number 30 and 32 appears at different deformations and the energy gap at particle number 32 is low, which make the SD structures of $^{62}$Zn unique, where the negative-parity SD states appear lower in energy than the positive-parity one.