Implications of Deformation and Shape Coexistence for the Nuclear Shell Model

Abstract

The successes of the nuclear shell model in explaining the stability properties of magic nuclei are challenged by the observation of rotational bands for which the sequential filling of single-particle energy levels of the spherical shell model are not respected. This Letter proposes criteria for identifying the shell-model configurations appropriate for describing such bands of states.

Symmetry-guided large-scale shell-model theory

“…For a strong spin-orbit splitting, as the one observed for heavy nuclei, l · s energetically separates orbits with the same l but different j, yielding the jj-coupling scheme with single-particle states labeled by η(ls)jmt z , or simply ηljmt z for s = The second of these models, the collective model of Bohr and Mottelson recognizes that deformed shapes dominate the nuclear dynamics. While enhanced deformation has been evident in heavy nuclei and those away from closed shells, deformed configurations are found to be important even in a nucleus such as 16 O, which is commonly treated as spherical in its ground state, but about 40% of the latter is governed by deformed shapes [65]; in addition, the lowest-lying excited 0 + states in 16 O and their rotational bands are dominated by large deformation (see, e.g., [33]). Bohr & Mottelson offered a simple but important description of nuclei in terms of the deformation of the nuclear surface and associated vibrations and rotations.…”

“…In its simplest depiction [33], the symplectic shell model is based on nucleons occupying HO shells with important correlations within each shell and between shells differing by ±2 Ω. The in-shell correlations are dominated by interactions of the quadrupole-quadrupole type, as first introduced by Elliott [39,40], while the inter-shell correlations are of the giant monopole and giant quadrupole type.…”

“…Indeed, models restricted in their interactions and model spaces can mimic simple patterns and can be misleading. One of the most striking recent examples is related to the lowlying 0 + nuclear states in experimental excitation spectra that for a long time have been regarded, and hence, modeled, as vibrations, and only recently the different mechanism of shape coexistence has been suggested [33][34][35]. Shape coexistence has been found to occur in many nuclei across the entire mass surface [36] -it has been argued that it probably occurs in nearly all nuclei [37].…”

The fundamental role of symmetry in nuclear models

“…[25,26,27] to estimate the strength χ =hω/N 0 to leading order in λ /N 0 and µ/N 0 , of the coupling constant for the effective quadrupole-quadrupole interaction in a model Hamiltonian of the formĤ =Ĥ sp − 1 2 χQ · Q. Several important consequences follow from this shape-consistency result.…”

“…In fact, for values of λ and/or µ large compared to the angular momenta of the states of interest, the properties of the SU(3) irreps (λ 0) and (0 λ ) approach those of prolate and oblate rigid rotors, respectively, whereas the properties of a generic SU ( The energy spectrum of basis states for an Sp(3, R) irrep, with respect to the spherical harmonic-oscillator Hamiltonian, Eq. (27), is now obtained as follows. The states of a lowest-grade U(3) irrep { N 0 (λ 0 µ 0 ) } all have the common harmonic-oscillator energy N 0h ω. One-phonon giant-resonance states appear at an energy of (N 0 + 2)hω and because the giant-resonance raising operators are components of an SU(3) (2 0) tensor, they generate the states of a generally reducible SU (3) It is of interest to note that the spectrum of states for an N 0 (0 0) irrep is in 1-1 correspondence with that of a 6-dimensional harmonic oscillator.…”