On the relation of the shell, collective and cluster models

Abstract

Abstract. The intersection of the shell, collective and cluster models is described for multimajor-shell problems. IntroductionThe fundamental models of nuclear structure are based on different physical pictures. The shell model indicates that the atomic nucleus is something like a small atom, the cluster model suggests that it is similar to a molecule, while the collective model says that it is a microscopic liquid drop. Therefore, in order to understand the nuclear structure we need to study (among others) the interrelation of these models, find their common intersection, etc.The basic connections were found in the fifties. Elliott [1] showed how the quadrupole deformation and collective rotation can be derived from the spherical shell model: the states belonging to a collective band are determined by their specific SU(3) symmetry. Wildermuth and Kanellopoulos [2] established the relation between the shell and cluster models. They proved that the Hamiltonians of the two models can be rewritten into each other exactly in the harmonic oscillator approximation. This relation results in a close connection between the corresponding eigenvectors, too: the wavefunction of one model is a linear combination of those of the other, which belong to the same energy. Later on this relation was interpreted by Bayman and Bohr [3] in terms of the SU(3) symmetry. As a consequence, the cluster states are also selected from the shell model space by their specific SU(3) symmetries. (In fact only one kind of cluster states have this feature, and there are other kinds, too, as discussed later.)We will refer to this interrelation among the three basic structure models as the SU(3) connection. It was established in 1958 for a single major shell problem. Here we consider its extension to multi-major-shells.The connection of the shell model and the cluster model is especially interesting due to the fact that both models have a complete set of basis states, i.e. any nuclear states can be expanded in both bases. Depending on the simple or complicated nature of these expansions we can distinguish four different cases. i) If it is simple in the shell basis, but complicated in the cluster one, then we can speak about a simple shell state, which is a poor cluster state. ii) If it is simple in the cluster basis, but complicated in the shell basis, one has a good cluster state, which is a poor shell state. (Later on we will refer to these states as rigid molecule-like cluster

Dual breaking of symmetries in algebraic models

“…In fact, it is able to describe the spectra of different cluster configurations and the shell (or quartet) model in a unified way [33,12]. Furthermore, since U(3) symmetry defines the quadrupole deformation, the spectrum of the dynamical symmetry of group-chain (17) and energy-operator (18) represents the common intersection of the shell, cluster and (quadrupole) collective model [13] for a multi-shell problem. A detailed example on the performance of this Hamiltonian was discussed in [14].…”

“…In this paper we show how it takes place first in the Elliott-model [8][9][10][11], which is the prototype of the algebraic structure models, then we mention a few other cases. Finally a detailed example is presented in relation with the multiconfigurational dynamical symmetry (MUSY) [12], which is the connecting symmetry of the shell, collective and cluster models for the multi-major-shell problem [13,14].…”

Some new chapters of the long history of SU(3)

“…It is very remarkable that these three extensions of the Elliott model, i.e. the symplectic shell model, the contracted symplectic (collective) model and the cluster model have basis states characterized by the groupchain [10]…”

“…(6). Therefore, the common intersection of the three fundamental structure models of the multi-major-shell problem is again a dynamical symmetry [10]. In particular, the basis sates are defined by the representation labels of the groups in chain, Eq.…”

Quartet excitations and cluster spectra in light nuclei

“…The first term is the harmonic oscillator Hamiltonian (linear invariant of the U(3)), with a strength obtained from the systematics [18] SU3 ) distinguishes between the prolate and oblate shapes. θ is the moment of inertia calculated classically for the rigid shape determined by the U(3) quantum numbers (for a rotor with axial symmetry) [19], and the a, b and d parameters were fitted to the experimental data: a = −0.133 MeV, b = 0.000444 MeV d = 1.003 MeV. The B(E2) value is given as [1]:…”