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Summary

Neutron one-particle levels in a spheroidal Woods-Saxon plus spin-orbit potential, of which the radius \(R\) and the depth \(V_{WS}\) are adjusted approximately to those of \(_{12}^{25}\)Mg\(_{13}\). The parameters of the potential at \(\beta\)=0 are taken from p.239 of Ref. #BM69. Positive-parity one-particle levels are denoted by solid curves. One-particle levels with \(\varepsilon_{\Omega}>0\) are obtained as one-particle resonant levels in the potential. The associated resonant widths are not shown, for simplicity. Calculated one-particle resonant energies at \(\beta\)=0 are : \(\varepsilon_{res}\)(p\(_{3/2}\))=0.32, \(\varepsilon_{res}\)(f\(_{7/2}\))=0.33, and \(\varepsilon_{res}\)(f\(_{5/2}\))=10.37 MeV, while the p\(_{1/2}\) level expected just above the \(p_{3/2}\) and \(f_{7/2}\) levels is not obtained as a resonant level for the present potential. Bound one-particle levels are labeled by the asymptotic quantum numbers {[}\(Nn_{z}\Lambda\Omega\){]}, where \(\Omega\) represents the component of the total angular momentum along the symmetry axis and is a constant of the motion for all values of \(\beta\). The additional quantum numbers refer to the structure of the orbits in the limit of large deformations : the total number of nodal surfaces (\(N\)), the number of nodal surfaces along the symmetry axis (\(n_{z}\)), and the component of orbital angular momentum along the symmetry axis (\(\Lambda\)). Each level is doubly degenerate (\(\pm\Omega\)) associated with time reversed invariance of the Hamiltonian. The neutron numbers 16 and 20, which are obtained by filling in all lower-lying levels, are indicated with circles. A pairwise filling of the doubly degenerate (\(\pm\Omega\)) orbits contributes no net angular momentum along the symmetry axis. For example, in \(^{25}\)Mg\(_{13}\) with the neutron number 13 it is seen that the lowest two intrinsic configurations for \(\beta\approx0.4\) are expected to involve the 13th neutron occupying the orbits {[}202 5/2{]} and {[}211 1/2{]}, in agreement with the observed spectroscopic properties shown in . The observed second and third excited bands starting at 2.563 and 3.413 MeV () can be interpreted as having the {[}200 1/2{]} and {[}330 1/2{]} intrinsic configurations, respectively, for the odd neutron.

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