The O(6) description of the nuclear rotation

Abstract

A recently developed algebraic version of the collective model has shown that a full range of Bohr model calculations can be executed in bases that range continuously from those of a spherical vibrator to a beta-vibrational Wilets-Jean limit. Thus, the establishment of close relationships between the algebraic structure of this model and the IBM is of special importance because one can learn from the complementary perspectives they afford. In this paper, we show by calculations that the familiar rotor-gamma vibrational spectra of the Bohr model can be obtained in the IBM by the addition of a scalar cubic in the quadrupole moment operators, of the type considered recently by Van Isacker, to an 0(6) Hamiltonian. Simple fits of the low-lying spectra, electromagnetic transition rates and moments of inertia of the ground and gamma bands of 162Dy and 16SEr are presented. PACS: 21.60.Ev, 21.60.Fw The relationship between the simple interacting boson model (IBM) [1] (by which we mean the so-called IBM-1 version of the model) and the Bohr model (BM) [2] was studied in Ref. [3, 4] and subsequently explored by many authors [5]. Recently, we have investigated the mappings between the related algebraic structures of the IBM and BM [6].The value of determining the explicit relationship between the two models is enhanced by the recent development of an algebraic version of the BM [7, 8] which makes it possible to execute a full range of BM calculations in bases that range continuously from those of a spherical vibrator to a beta-vibrational Wilets-Jean limit. Our objective in pursuing the relationships between these two models is to discover what can be learned about one model from the other and the extent to which the results of one model can be reproduced by the other. In Ref.[6] we show that there are mappings of IBM states into the Hilbert space of the Bohr model, based on group contractions, which define the appropriate correspondence between the states of the two models. However, different mappings are required for the different symmetry limits of the IBM. We also show that, with only minor modification, the new computational tools developed for the Bohr model can be applied also in the IBM.

Analytical solutions of the Bohr Hamiltonian with the Morse potential

“…Refs. [36,37]. Using this numerical method, the Bohr Hamiltonian has been solved [38] for the same potentials used in X(5), but avoiding the approximate separation of variables, resulting in evidence for strong beta-gamma mixing.…”

“…The relations between the algebraic collective model and the different limiting symmetries of the Interacting Boson Model [12] have been studied in Refs. [36,37]. Using this numerical method, the Bohr Hamiltonian has been solved [38] for the same potentials used in X( 5), but avoiding the approximate separation of variables, resulting in evidence for strong beta-gamma mixing.…”

Exactly separable version of the Bohr Hamiltonian with the Davidson potential

“…A more detailed study of this issue has been recently carried out [37], leading to the use of spheroidal or Mathieu functions. Periodic γ-potentials involving cos 3γ have been used in an early solution involving a harmonic oscillator for the β-potential [38], as well as more recently in the framework of the algebraic collective model [21,32,33], where their treatment is tractable because cos 3γ is, within a constant, an SO(5) spherical harmonic with v = 3 and L = 0 (where v the seniority and L the angular momentum) [21]. The potential csc 2 3γ, which is the partner of the infinite well potential in supersymmetric quantum mechanics [39], has also been used recently [40] in the Bohr Hamiltonian for triaxial nuclei.…”

“…The correspondence between this approach, called the algebraic collective model [23], and the different limiting symmetries of the Interacting Boson Approximation (IBA) model [29] has been studied in Refs. [32,33]. This powerful method has been recently extended [34] to the SU(1,1)×SO(N) case.…”

Regularities and symmetries of subsets of collective0+states

“…The relationship between the algebraic collective model and the dif-ferent limiting symmetries of the IBA has been studied in Refs. [24,25]. The connection between geometrical models spanning structure near E(5) and the IBA has also been investigated [26].…”