Parameter-free predictions for the collective deformation variables beta and gamma within the pseudo-SU(3) scheme

The consequences of the short range nature of the nucleon-nucleon interaction, which forces the spatial part of the nuclear wave function to be as symmetric as possible, on the pseudo-SU(3) scheme are examined through a study of the collective deformation parameters beta and gamma in the rare earth region. It turns out that beyond the middle of each harmonic oscillator shell possessing an SU(3) subalgebra, the highest weight irreducible representation (the hw irrep) of SU(3) has to be used, instead of the irrep with the highest eigenvalue of the second order Casimir operator of SU(3) (the hC irrep), while in the first half of each shell the two choices are identical. The choice of the hw irrep predicts a transition from prolate to oblate shapes just below the upper end of the rare earth region, between the neutron numbers N=114 and 116 in the W, Os, and Pt series of isotopes, in agreement with available experimental information, while the choice of the hC irrep leads to a prolate to oblate transition in the middle of the shell, which is not seen experimentally. The prolate over oblate dominance in the ground states of even-even nuclei is obtained as a by-product.Comment: 18 pages, 4 tables, 8 figures. arXiv admin note: text overlap with arXiv:1909.0196

Signatures for shape coexistence and shape/phase transitions in even-even nuclei

Systematics of B(E2) transition rates connecting the first excited 0+ state to the first excited 2+ state of the ground state band in even-even nuclei indicates that shape coexistence of the ground state band and the first excited K=0 band should be expected in nuclei lying within the stripes of nucleon numbers 7-8, 17-20, 34-40, 59-70, 96-112 predicted by the dual shell mechanism of the proxy-SU(3) model, avoiding their junctions, within which high deformation is expected. Systematics of the excitation energies of the first excited 0+ states in even-even nuclei show that shape coexistence due to proton-induced neutron particle-hole excitations is related to a first-order shape/phase transition from spherical to deformed shapes, while shape coexistence due to neutron-induced proton particle-hole excitations is observed along major proton shell closures.Comment: 13 pages, 4 figures, 4 table

Why nuclear forces favor the highest weight irreducible representations of the fermionic SU(3) symmetry

The consequences of the attractive, short-range nucleon-nucleon (NN) interaction on the wave functions of the Elliott SU(3) and the proxy-SU(3) symmetry are discussed. The NN interaction favors the most symmetric spatial SU(3) irreducible representation, which corresponds to the maximal spatial overlap among the fermions. The percentage of the symmetric components out of the total in an SU(3) wave function is introduced, through which it is found, that no SU(3) irrep is more symmetric than the highest weight irrep for a certain number of valence particles in a three dimensional, isotropic, harmonic oscillator shell. The consideration of the highest weight irreps in nuclei and in alkali metal clusters, leads to the prediction of a prolate to oblate shape transition beyond the mid-shell region.Comment: 16 pages, 1 figure, 10 table

Shape coexistence in even-even nuclei: A theoretical overview

The last decade has seen a rapid growth of our understanding of the microscopic origins of shape coexistence, assisted by the new data provided by the modern radioactive ion beam facilities built worldwide. Islands of the nuclear chart in which shape coexistence can occur have been identified, and the different microscopic particle-hole excitation mechanisms leading to neutron-induced or proton-induced shape coexistence have been clarified. The relation of shape coexistence to the islands of inversion, appearing in light nuclei, to the new spin-aligned phase appearing in N=Z nuclei, as well as to shape/phase transitions occurring in medium mass and heavy nuclei, has been understood. In the present review, these developments are considered within the shell model and mean field approaches, as well as by symmetry methods. In addition, based on systematics of data, as well as on symmetry considerations, quantitative rules are developed, predicting regions in which shape coexistence can appear, as a possible guide for further experimental efforts, which can help in improving our understanding of the details of the nucleon-nucleon interaction, as well as of its modifications occurring far from stability.Comment: 80 pages, 14 figures, 837 reference

Islands of shape coexistence: theoretical predictions and experimental evidence

Parameter-free theoretical predictions based on a dual shell mechanism within the proxy-SU(3) symmetry of atomic nuclei, as well as covariant density functional theory calculations using the DDME2 functional indicate that shape coexistence (SC) based on the particle-hole excitation mechanism cannot occur everywhere on the nuclear chart, but is restricted on islands lying within regions of 7-8, 17-20, 34-40, 59-70, 96-112, 146-168 protons or neutrons. Systematics of data for even-even nuclei possessing K=0 (beta) and K=2 (gamma) bands support the existence of these islands, on which shape coexistence appears whenever the K=0 bandhead 0_2^+ and the first excited state of the ground state band 2_1^+ lie close in energy, with nuclei characterized by 0_2^+ lying below the 2_1^+ found in the center of these islands. In addition a simple theoretical mechanism leading to multiple shape coexistence is briefly discussed.Comment: 14 pages, 3 tables, 5 figure

Proxy-SU(3) symmetry in the shell model basis

The proxy-SU(3) symmetry has been proposed for spin-orbit like nuclear shells using the asymptotic deformed oscillator basis for the single particle orbitals, in which the restoration of the symmetry of the harmonic oscillator shells is achieved by a change of the number of quanta in the z-direction by one unit for the intruder parity orbitals. The same definition suffices within the cartesian basis of the Elliott SU(3) model. Through a mapping of the cartesian Elliott basis onto the spherical shell model basis, we translate the proxy-SU(3) approximation into spherical coordinates, proving, that in the spherical shell model basis the proxy-SU(3) approximation corresponds to the replacement of the intruder parity orbitals by their de Shalit--Goldhaber partners. Furthermore it is shown, that the proxy-SU(3) approximation in the cartesian Elliott basis is equivalent to a unitary transformation in the z-coordinate, leaving the x-y plane intact, a result which in the asymptotic deformed oscillator coordinates implies, that the z-projections of angular momenta and spin remain unchanged. The present work offers a microscopic justification of the proxy-SU(3) approximation and in addition paves the way, for taking advantage of the proxy-SU(3) symmetry in shell model calculations.Comment: 15 pages, 7 tables, 1 figur

The islands of shape coexistence within the Elliott and the proxy-SU(3) Models

A novel dual-shell mechanism for the phenomenon of shape coexistence in nuclei within the Elliott SU(3) and the proxy-SU(3) symmetry is proposed for all mass regions. It is supposed, that shape coexistence is activated by large quadrupole-quadrupole interaction and involves the interchange among the spin-orbit (SO) like shells within nucleon numbers 6-14, 14-28, 28-50, 50-82, 82-126, 126-184, which are being described by the proxy-SU(3) symmetry, and the harmonic oscillator (HO) shells within nucleon numbers 2-8, 8-20, 20-40, 40-70, 70-112, 112-168 of the Elliott SU(3) symmetry. The outcome is, that shape coexistence may occur in certain islands on the nuclear map. The dual-shell mechanism predicts without any free parameters, that nuclei with proton number (Z) or neutron number (N) between 7-8, 17-20, 34-40, 59-70, 96-112, 146-168 are possible candidates for shape coexistence. In the light nuclei the nucleons flip from the HO shell to the neighboring SO-like shell, which means, that particle excitations occur. For this mass region, the predicted islands of shape coexistence, coincide with the islands of inversion. But in medium mass and heavy nuclei, in which the nucleons inhabit the SO-like shells, shape coexistence is accompanied by a merging of the SO-like shell with the open HO shell. The shell merging can be accomplished by the outer product of the SU(3) irreps of the two shells and represents the unification of the HO shell with the SO-like shell.Comment: 31 pages, 25 figures, 4 table