Sequence of potentials lying between the U(5) and X(5) symmetries

Abstract

Starting from the original collective Hamiltonian of Bohr and separating the β and γ variables as in the X(5) model of Iachello, an exactly soluble model corresponding to a harmonic oscillator potential in the β-variable (to be called X(5)-β 2 ) is constructed. Furthermore, it is proved that the potentials of the form β 2n (with n being integer) provide a "bridge" between this new X(5)-β 2 model (occuring for n = 1) and the X(5) model (corresponding to an infinite well potential in the β-variable, materialized for n → ∞). Parameter-free (up to overall scale factors) predictions for spectra and B(E2) transition rates are given for the potentials β 2 , β 4 , β 6 , β 8 , corresponding to R 4 = E(4)/E(2) ratios of 2.646, 2.769, 2.824, and 2.852 respectively, compared to the R 4 ratios of 2.000 for U(5) and 2.904 for X(5). Hints about nuclei showing this behaviour, as well as about potentials "bridging" the X(5) symmetry with SU(3) are briefly discussed.PACS numbers: 21.60.Ev, 21.60.Fw, 21.10.Re 1. Introduction Models providing parameter-independent predictions for nuclear spectra and electromagnetic transition rates serve as useful benchmarks in nuclear theory. The recently introduced E(5) [1] and X(5) [2] models belong to this category, since their predictions for nuclear spectra (normalized to the excitation energy of the first excited state) and B(E2) transition rates (normalized to the B(E2) transition rate connecting the first excited state to the ground state) do not contain any free parameters. The E(5) model appears to be related to a phase transition from U(5) (vibrational) to O(6) (γ-unstable) nuclei [1], while X(5) is related to a phase transition from U(5) (vibrational) to SU(3) (prolate deformed) nuclei [2]. Both models originate (under certain simplifying assumptions) from the Bohr collective Hamiltonian [3], which is known to possess the U(5) symmetry of the five-dimensional (5-D) harmonic oscillator [4].In the present paper we study a sequence of potentials lying between the U(5) symmetry of the Bohr Hamiltonian and the X(5) model. The potentials, which are of the form 1

Parameter-free solution of the Bohr Hamiltonian for actinides critical in the octupole mode

“…[28]). Indeed the present model can be considered as an extension of X (5), in which the octupole degree of freedom is taken into account in order to account for the low-lying negative parity bands, while in parallel the γ degree of freedom is left out in order to keep the problem tractable.…”

Relation betweenE(5)models and the interacting boson model

“…Lévai and Arias [31] solved the Bohr equation with a sextic potential with a centrifugal barrier [32], arriving to almost closed analytical formulas for the energies and wave functions. Immediately after, Bonatsos and collaborators explored the possibility of getting numerical solutions for the γ -independent Bohr Hamiltonian with potentials of the type β 2n , with n 1 [33]. These sequences of potentials allow to go from the vibrational limit, n = 1, to E(5), n → ∞.…”

“…In particular, in Ref. [33] spectra and transition rates for the potentials β 4 , β 6 , and β 8 are given explicitly and compared with the original E(5) (infinite square well potential) case. As mentioned above, all these models are produced in the BM scheme and a natural question is to ask for the corresponding equivalence in the IBM.…”

Alternative solvable description of the E(5) critical point symmetry in the interacting boson model

“…The levels of the nuclei fitted have been confirmed in experiment with relatively abundant B(E2) data except 128 Xe, of which only four B(E2) values are available. These nuclei have also been well studied in the original E(5) model [5][6][7][8], the model with the sextic type potential in β [9], and the confined γ-soft rotor model [11]. The fitting results of the EXT and the CQ to the level energies and experimentally deduced B(E2) values for these nuclei are shown in Tables VII and VIII, in which the fitting results of the model with the sextic potential in β (SET) shown in [9] for 102 Pd, 104 Ru, 108 Pd, 116 Cd, and 114 Cd are also provided for comparison.…”

“…It has been shown that there are many nuclei with the E(5) critical point symmetry, such as 134 Ba [5], 104 Ru [6], 102 Pd [7], 108 Pd [8], and 116 Cd [9]. Inspired by the E(5) model, Lévai and Arias studied the Bohr Hamiltonian with a sextic potential and a centrifugal barrier, of which quasi-exact solutions can be derived [10], while Bonatsos et al explored numerical solutions for the γ-independent Bohr Hamiltonian with β 2n potentials for n ≥ 1 called the confined γ-soft rotor model [11], in which the spectra and transition rates for the β 2n potentials for 2 ≤ n ≤ 4 are given explicitly and compared with the original E(5) model.…”