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Register a new userProxy-SU(3) symmetry for heavy deformed nuclei: nuclear spectra
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The systematics of experimental energy differences between the levels of the ground state band and the gamma-1 band in even-even nuclei are studied as a function of the angular momentum L, demonstrating a decrease of the energy differences with increasing L, in contrast to what is seen in vibrational, gamma-unstable, and triaxial nuclei. After a short review of the relevant predictions of several simple collective models, it is shown that this decrease is caused in the framework of the proxy-SU(3) scheme by the same three-body and/or four body operators which break the degeneracy between the ground state band and the gamma-1 band, predicting in parallel the correct form of odd-even staggering within the gamma-1 bands.
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The rapid increase of computational power over the last several years has allowed detailed microscopic investigations of the structure of many nuclei in terms of Relativistic Mean Field theories as well as in the framework of the no-core Shell Model. In heavy deformed nuclei, in which microscopic calculations remain a challenge, algebraic models based on the SU(3) symmetry offer specific predictions, parameter-independent in several cases, directly comparable to experimental data. Two different approximate models for heavy deformed nuclei based on the SU(3) symmetry, the pseudo-SU(3) and the proxy-SU(3) schemes will be discussed and the compatibility between their predictions for the nuclear deformation parameters will be shown. In particular, the dominance of prolate over oblate shapes in the ground states of even-even nuclei and the prolate to oblate shape phase transition occurring in heavy rare earths will be considered.
Symmetries are manifested in nature through degeneracies in the spectra of physical systems. In the case of heavy deformed nuclei, when described in the framework of the Interacting Boson Model, within which correlated proton (neutron) pairs are approximated as bosons, the ground state band has no symmetry partner, while the degeneracy between the first excited beta and gamma bands is broken through the use of three-body and/or four-body terms. In the framework of the proxy-SU(3) model, in which an approximate SU(3) symmetry of fermions is present, the same three-body and/or four-body operators are used for breaking the degeneracy between the ground state band and the first excited gamma band. Experimentally accessible quantities being independent of any free parameters are pointed out in the latter case.
We present a review of the pseudo-SU(3) shell model and its application to heavy deformed nuclei. The model have been applied to describe the low energy spectra, B(E2) and B(M1) values. A systematic study of each part of the interaction within the Hamiltonian was carried out. The study leads us to a consistent method of choosing the parameters in the model. A systematic application of the model for a sequence of rare earth nuclei demonstrates that an overarching symmetry can be used to predict the onset of deformation as manifested through low-lying collective bands.The scheme utilizes an overarching sp(4,R) algebraic framework.
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.
We present an analysis based on the deformed Quasi Particle Random Phase Approximation, on top of a deformed Hartree-Fock-Bogoliubov description of the ground state, aimed at studying the isoscalar monopole and quadrupole response in a deformed nucleus. This analysis is motivated by the need of understanding the coupling between the two modes and how it might affect the extraction of the nuclear incompressibility from the monopole distribution. After discussing this motivation, we present the main ingredients of our theoretical framework, and we show some results obtained with the SLy4 and SkM$^{*}$ interactions for the nucleus ${}^{24}$Mg.
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