The level structure of Mg31, which is located close to the region of the N = 20 "island of inversion" in the nuclear chart, has been studied by β-γ spectroscopy with spin-polarized Na31. In Mg31, shape coexistence is expected as a result of subtle competition between the spherical mean field and the nuclear correlation which favors deformed configurations. In the present work, our unique method utilizing the anisotropic β decay of spin-polarized Na31 enables us to firmly assign the spins of all positive-parity excited levels in Mg31 below the neutron separation energy at 2.3 MeV. Furthermore, by constructing a very detailed decay scheme, including two newly found levels, the spins of negative-parity levels are restricted. The examination of the spectroscopic information shows that the deformed rotational bands with Kπ=1/2+ and 1/2-, which have very similar structures to those observed in a higher excitation energy region of Mg25, appear as the ground-state and low-lying bands, respectively, in Mg31. The experimental levels of Mg31 are compared, on the level-by-level basis, with two types of theoretical calculations. These are, first, the antisymmetrized molecular dynamics (AMD) plus generator coordinate method (GCM) and, second, the shell model with the EEdf1 interaction, which is microscopically derived from chiral effective field theory. It is understood that 8 levels among the experimental 11 levels are the members of four types of largely deformed rotational bands and 2 levels are of spherical nature. The 1/2+ 2.244-MeV level is successfully reproduced by the shell-model calculation with a dominant 4p4h configuration. The present work clearly demonstrates that various structures coexist in a low excitation energy region of Mg31.
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