Extended analytical solutions of the Bohr Hamiltonian with the sextic oscillator: Pt-Os isotopes

Abstract

The sextic oscillator adapted to the Bohr Hamiltonian has been used to describe even Pt and Os isotopes from $A=$188 to 198 and $A=$186 to 192, respectively. The purpose of this study was investigating the possible transition from the $\gamma$-unstable to the spherical vibrator shape phases. In this setup the potential appearing in the Bohr Hamiltonian is independent from the $\gamma$ shape variable, and the physical observables (energy eigenvalues, $B(E2)$) can be obtained in closed analytical form within the quasi-exactly solvable (QES) formalism for the model space containing 30 of the lowest-lying levels. Experimental energy levels have been associated to the theoretical ones. The available electric quadrupole transition data ($B(E2)$, decay preferences) have been taken into account in matching the experimental and theoretical levels. Special attention has been paid to transitions from the first two excited $0^+$ levels to the $2^+_1$ and $2^+_2$ levels, as these indicate the change of shape phases with spherical and deformed potential minimum. The three parameters of the Hamiltonian have been determined by a weighted least square fit procedure. Trends in the location of states belonging to the ground-state, the $K^{\pi}=2^+$ and two excited $K^{\pi}=0^+$ bands have been analysed. The trajectory determined by the fitted parameters in the two-dimensional phase space has also been plotted, and it has been found that all the nuclei are characterized by a deformed potential minimum, except for the heaviest Pt isotope ($^{198}$Pt), for which the transition to the spherical shape phase is realised. Although the spectroscopic information on the next isotopes of the chains ($^{200}$Pt and $^{194}$Os) is far less complete, there are indications that these nuclei are also close to or fall within the domain of spherical potential minimum.

Nuclear Shape-Phase Transitions and the Sextic Oscillator

“…The mass parameter B appears as a moment of inertia, and as such, it has the physical dimension mass × length 2 . Its chosen value sets the energy scale [23].…”

“…It is obvious that c π differs for τ-even (c + ) and τ-odd (c − ) combinations, which also introduces a slight difference in the coefficient of the quadratic potential component. This effect, which proved to be marginal in the applications in [20][21][22][23], was handled by prescribing that the minima of the two potentials coincide. This can be achieved by applying different constants in potential (11).…”

“…[19]). Our contribution to these endeavors includes the proposal of the sextic oscillator potential for the β shape variable, making the potential energy surface independent of γ [20][21][22][23]. This potential exhibits remarkable flexibility: its characteristics can vary depending on the parameters, featuring either a spherical minimum, a deformed minimum, or both.…”

“…Fortunately, these few lowest energy levels generally correspond to the set of nuclear states for which a sufficient amount of experimental spectroscopic information is available, allowing for a quantitative comparison of experiment and theory. We have also derived electromagnetic transition rates (B(E2), B(E0)) for this potential in closed form and computed benchmark values associated with the transitions between different shape phases [21][22][23].…”

Proxy-SU(4) symmetry in A = 60–90 region

“…More importantly, the subject of ʼsymmetries in nuclei' is undergoing a second renaissance in the last 5-10 years with several new directions in this subject are being explored. Some of these are (i) point group symmetries in heavy nuclei such as 152 Sm, 156 Gd, 236 U [8][9][10] and in lighter nuclei such as 12 C, 13 C and 16 O [11][12][13], (ii) symmetries for shape coexistence [14], (iii) multiconfiguration or composite symmetries for cluster states [15], (iv) symmetry adopted no-core-shell model based on SU(3) and Sp(6, R) algebras [16][17][18][19], (v) proton-neutron Sp(12, R) model [20][21][22], (vi) multiple algebras in shell model and IBM spaces giving for example multiple pairing and SU(3) algebras [23,24], (vii) pairing algebras with higher order interactions, generalized seniority and seniority isomers [24][25][26][27], (viii) proxy-SU(3) scheme within shell model [28,29], (ix) symmetry restoration methods in mean-field theories [30], (x) new techniques for obtaining Wigner coefficients involving SU(3) ⊃ SO(3), SO(5) ⊃ SO(3) and spin (S)-isospin(T) SU ST (4) ⊃ SU S (2) ⊗ SU T (2) [31][32][33] and (xi) studies involving Bohr Hamiltonian with sextic and other types of potentials for shape phase transitions and shape coexistence pointing out the need for three and higher-body terms in IBM and perhaps also in shell model [34][35][36][37][38][39]. (xii) random matrix ensembles with Lie algebraic ...…”