Oblate deformation of light neutron-rich even-even nuclei

Light neutron-rich even-even nuclei, of which the ground state is oblately deformed, are looked for, examining the Nilsson diagram based on realistic Woods-Saxon potentials. One-particle energies of the Nilsson diagram are calculated by solving the coupled differential equations obtained from the Schr\"{o}dinger equation in coordinate space with the proper asymptotic behavior for $r \rightarrow \infty$ for both one-particle bound and resonant levels. The eigenphase formalism is used in the calculation of one-particle resonant energies. Large energy gaps on the oblate side of the Nilsson diagrams are found to be related to the magic numbers for the oblate deformation of the harmonic-oscillator potential where the frequency ratios ($\omega_{\perp} : \omega_{z}$) are simple rational numbers. In contrast, for the prolate deformation the magic numbers obtained from simple rational ratios of ($\omega_{\perp} : \omega_{z}$) of the harmonic-oscillator potential are hardly related to the particle numbers, at which large energy gaps appear in the Nilsson diagrams based on realistic Woods-Saxon potentials. The argument for an oblate shape of $^{42}_{14}$Si$_{28}$ is given. Among light nuclei the nucleus $^{20}_{6}$C$_{14}$ is found to be a good candidate for having the oblate ground state. In the region of the mass number $A \approx 70$ the oblate ground state may be found in the nuclei around $^{76}_{28}$Ni$_{48}$ in addition to $^{64}_{28}$Ni$_{36}$.