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$_{36}$.Comment: 2 figure

Interplay between one-particle and collective degrees of freedom in nuclei

Some developments of nuclear-structure physics uniquely related to Copenhagen School are sketched based on theoretical considerations versus experimental findings and one-particle versus collective aspects. Based on my personal overview I pick up the following topics; (1) Study of vibration in terms of particle-vibration coupling; (2) One-particle motion in deformed and rotating potentials, and yrast spectroscopy in high-spin physics; (3) Triaxial shape in nuclei: wobbling motion and chiral bands; (4) Nuclear structure of drip line nuclei: in particular, shell-structure (or magic numbers) change and spherical or deformed halo phenomena; (5) shell structure in oblate deformation.Comment: 19 pages, 9 figure

Possible Presence and Properties of Multi Chiral Pair-Bands in Odd-Odd Nuclei with the Same Intrinsic Configuration

Applying a relatively simple particle-rotor model to odd-odd nuclei, possible presence of multi chiral pair-bands is looked for, where chiral pair-bands are defined not only by near-degeneracy of the levels of two bands but also by almost the same expectation values of squared components of three angular-momenta that define chirality. In the angular-momentum region where two pairs of chiral pair-bands are obtained the possible interband M1/E2 decay from the second-lowest chiral pair-bands to the lowest chiral pair-bands is studied, with the intention of finding how to experimentally identify the multi chiral pair-bands. It is found that up till almost band-head the intraband M1/E2 decay within the second chiral pair-bands is preferred rather than the interband M1/E2 decay to the lowest chiral pair-bands, though the decay possibility depends on the ratio of actual decay energies. It is also found that chiral pair-bands in our model and definition are hardly obtained for $\gamma$ values outside the range $25^{\circ} < \gamma < 35^{\circ}$, although either a near-degeneracy or a constant energy-difference of several hundreds keV between the two levels for a given angular-momentum $I$ in "a pair bands" is sometimes obtained in some limited region of $I$. In the present model calculations the energy difference between chiral pair-bands is always one or two orders of magnitude smaller than a few hundreds keV, and no chiral pair-bands are obtained, which have an almost constant energy difference of the order of a few hundreds keV in a reasonable range of $I$.Comment: 15 pages, 14 figure

Shell structure of weakly-bound and resonant neutrons

The systematic change of shell structure in both weakly bound and resonant neutron one-particle levels in nuclei towards the neutron drip line is exhibited, solving the coupled equations derived from the Schr\"{o}dinger equation in coordinate space with the correct asymptotic behaviour of wave functions for $r \rightarrow \infty$. The change comes from the behaviour unique in the one-particle motion with low orbital angular momenta compared with that with high orbital angular momenta. The observed deformation of very neutron-rich nuclei with N \simgeq 20 in the island of inversion is a natural result of this changed shell structure, while a possible deformation of neutron-drip-line nuclei with $N \approx 51$, which are not yet observed, is suggested.Comment: Paper presented at the 10th International Spring Seminar on Nuclear Physics. Vietri sul Mare, May 21-25, 201

Neutron shell structure and deformation in neutron-drip-line nuclei

Neutron shell-structure and the resulting possible deformation in the neighborhood of neutron-drip-line nuclei are systematically discussed, based on both bound and resonant neutron one-particle energies obtained from spherical and deformed Woods-Saxon potentials. Due to the unique behavior of weakly-bound and resonant neutron one-particle levels with smaller orbital angular-momenta $\ell$, a systematic change of the shell structure and thereby the change of neutron magic-numbers are pointed out, compared with those of stable nuclei expected from the conventional j-j shell-model. For spherical shape with the operator of the spin-orbit potential conventionally used, the $\ell_{j}$ levels belonging to a given oscillator major shell with parallel spin- and orbital-angular-momenta tend to gather together in the energetically lower half of the major shell, while those levels with anti-parallel spin- and orbital-angular-momenta gather in the upper half. The tendency leads to a unique shell structure and possible deformation when neutrons start to occupy the orbits in the lower half of the major shell. Among others, the neutron magic-number N=28 disappears and N=50 may disappear, while the magic number N=82 may presumably survive due to the large $\ell =5$ spin-orbit splitting for the $1h_{11/2}$ orbit. On the other hand, an appreciable amount of energy gap may appear at N=16 and 40 for spherical shape, while neutron-drip-line nuclei in the region of neutron number above N=20, 40 and 82, namely N $\approx$ 21-28, N $\approx$ 41-54, and N $\approx$ 83-90, may be quadrupole-deformed though the possible deformation depends also on the proton number of respective nuclei.Comment: 16 pages, 4 figure

Shell-structure of one-particle resonances in deformed potentials

Shell structure of low-lying neutron resonant levels in axially-symmetric quadrupole-deformed potentials is discussed, which seems analogous to that of weakly-bound neutrons. As numerical examples, nuclei slightly outside the neutron-drip-line, $^{39}_{12}$Mg$_{27}$ and $^{21}_{6}$C$_{15}$, are studied. For the lowest resonance I obtain $I^{\pi}$ = $\Omega^{\pi}$ = 5/2$^{-}$ for $^{39}$Mg which is likely to be prolately deformed, while $I^{\pi}$ = $\Omega^{\pi}$ = 1/2$^{+}$ may be assigned to the nucleus $^{21}$C which may be oblately deformed. Consequently, $^{21}$C will not be observed as a recognizable resonant state, in agreement with the experimental information.Comment: 16 pages and 3 figure

Deformed Halo of ^{29}_{9}F_{20}

Using a simple model based on the knowledge of spherical and deformed Woods-Saxon potentials, it is shown that the recent observation of halo phenomena in $^{29}$F can be interpreted as an evidence for the prolate deformation of the ground state of $^{29}$F. The prolate deformation is the result of the shell structure, which is unique in one-neutron resonant levels, in particular near degeneracy of the neutron 1$f_{7/2}$ and 2$p_{3/2}$ resonant levels, together with the strong preference of prolate shape by the proton number $Z$ = 9. On the other hand, in oxygen isotopes spherical shape is so much favored by the proton number $Z$ = 8 that the presence of possible neutron shell-structure may not make the system deformed. Thus, the strong preference of particular shape by the proton numbers 8 and 9, respectively, together with a considerable amount of the energy difference between the neutron $1d_{3/2}$ and $2s_{1/2}$ orbits in oxygen isotopes seems to play an important role in the phenomena of oxygen neutron drip line anomaly, as was suggested by H. Sakurai {\it et al.} in 1999.Comment: 11 pages, 2 figures

Interpretation of Coulomb breakup of 31Ne in terms of deformation

The recent experimental data on Coulomb breakup of the nucleus $^{31}$Ne are interpreted in terms of deformation. The measured large one-neutron removal cross-section indicates that the ground state of $^{31}$Ne is either s- or p-halo. The data can be most easily interpreted as the spin of the ground state being 3/2$^-$ coming from either the Nilsson level [330 1/2] or [321 3/2] depending on the neutron separation energy $S_n$. However, the possibility of 1/2$^{+}$ coming from [200 1/2] is not excluded. It is suggested that if the large ambiguity in the measured value of $S_n$ of $^{31}$Ne, 0.29$\pm1.64$ MeV, can be reduced by an order of magnitude, say to be $\pm$100 keV, one may get a clear picture of the spin-parity of the halo ground state.Comment: 8 pages, 4 figure

Shape Deformations in Atomic Nuclei

The ground states of some nuclei are described by densities and mean fields that are spherical, while others are deformed. The existence of non-spherical shape in nuclei represents a spontaneous symmetry breaking.Comment: 20 pages, 10 figures, submitted to scholarpedi

Change of shell structure and magnetic moments of odd-N deformed nuclei towards neutron drip line

Examples of the change of neutron shell-structure in both weakly-bound and resonant neutron one-particle levels in nuclei towards the neutron drip line are exhibited. It is shown that the shell-structure change due to the weak binding may lead to the deformation of those nuclei with the neutron numbers $N \approx$ 8, 20, 28 and 40, which are known to be magic numbers in stable nuclei. Nuclei in the "island of inversion" are most easily and in a simple manner understood in terms of deformation. As an example of spectroscopic properties other than single-particle energies, magnetic moments of some weakly-bound possibly deformed odd-N nuclei with neutron numbers close to those traditional magic numbers are given, which are calculated using the wave function of the last odd particle in deformed Woods-Saxon potentials.Comment: 21 pages, 6 figure