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Shape phase transition, coexistence and mixing in the $^{98-106}$Ru isotopes

R. Budaca, P. Buganu, F. El Ouardi, A. Lahbas

Abstract

The deformation properties within the $^{98-106}$Ru even-even isotopic chain, are investigated by means of the Covariant Density Functional Theory with a Density-Dependent Point-Coupling X parametrization. The considered nuclei are found to exhibit very shallow prolate and triaxial ground state deformation. This information is used to ascertain their dynamical behavior within prolate $γ$-stable and $γ$-unstable instances of a phenomenological Bohr-Mottelson Hamiltonian with an octic potential in the axial deformation variable. The comparative study of the low-lying collective states, revealed the presence of a shape phase transition from low to high deformation, as well as evidence of shape coexistence and mixing between spherical vibrator, $γ$-unstable or prolate configurations in ground and excited states. It is also shown that the effect of shape coexistence and mixing on the $γ$-band states can account to some extent for the typical $γ$-unstable staggering even in prolate $γ$-stable conditions.

Shape phase transition, coexistence and mixing in the $^{98-106}$Ru isotopes

Abstract

The deformation properties within the Ru even-even isotopic chain, are investigated by means of the Covariant Density Functional Theory with a Density-Dependent Point-Coupling X parametrization. The considered nuclei are found to exhibit very shallow prolate and triaxial ground state deformation. This information is used to ascertain their dynamical behavior within prolate -stable and -unstable instances of a phenomenological Bohr-Mottelson Hamiltonian with an octic potential in the axial deformation variable. The comparative study of the low-lying collective states, revealed the presence of a shape phase transition from low to high deformation, as well as evidence of shape coexistence and mixing between spherical vibrator, -unstable or prolate configurations in ground and excited states. It is also shown that the effect of shape coexistence and mixing on the -band states can account to some extent for the typical -unstable staggering even in prolate -stable conditions.
Paper Structure (17 sections, 49 equations, 23 figures, 4 tables)

This paper contains 17 sections, 49 equations, 23 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Description of the models
  3. Covariant Density Functional Theory with Density-Dependent Point-Coupling interaction
  4. Point-coupling Lagrangian density
  5. Relativistic Hartree--Bogoliubov formalism and pairing correlations
  6. Quadratic constraint method
  7. The Bohr-Mottelson Hamiltonian with octic potential
  8. Solution of the $\beta$-equation
  9. Solution of the $\gamma$-equation in the $\gamma$-unstable case
  10. Solution of the $\gamma$-equation in the $\gamma$-stable prolate case
  11. Numerical results
  12. Discussion for $^{98}$Ru
  13. Discussion for $^{100}$Ru
  14. Discussion for $^{102}$Ru
  15. Discussion for $^{104}$Ru
  16. ...and 2 more sections

Figures (23)

  • Figure 1: (Color online) Potential energy surfaces in the $(\beta_{2},\gamma)$ plane for $^{98}$Ru and $^{100}$Ru obtained from the CDFT calculations with DD-PCX interaction. The red dot indicates the minimum point, in these cases corresponding to prolate shapes, while the color scale, given in MeV units, shows the relative energy potential depth.
  • Figure 2: (Color online) Potential energy surfaces in the $(\beta_{2},\gamma)$ plane for $^{102}$Ru, $^{104}$Ru and $^{106}$Ru obtained from the CDFT calculations with DD-PCX interaction. The red dot indicates the minimum point, in these cases corresponding to triaxial shapes, while the color scale, given in MeV units, shows the relative energy potential depth.
  • Figure 3: (Color online) The energy spectrum and the electromagnetic transitions for $^{98}$Ru calculated with the Bohr-Mottelson Hamiltonian with octic potential for $\gamma$-unstable system (Octic $\&$$\gamma$-unstable) are compared with the corresponding experimental data (Experiment) Chen. The energies are given in keV units, while the $B(E2)$ transitions in W.u.. The forbidden $B(E2)$s are indicated by $\mathrm{x}$, while the monopole transition between the first excited $0^+$ and the ground state by a dashed arrow.
  • Figure 4: (Color online) The corresponding energies, effective potentials and probability density distributions of deformation for $^{98}$Ru, calculated with Octic $\&$$\gamma$-unstable approach, are given for the states of the ground band in panels (a) and (b), respectively of the $\beta$ band in panels (c) and (d).
  • Figure 5: (Color online) The energy spectrum and the electromagnetic transitions for $^{98}$Ru calculated with the Bohr-Mottelson Hamiltonian with octic potential for $\gamma$-stable prolate system (Octic $\&$ prolate) are compared with the corresponding experimental data (Experiment) Chen. The energies are given in keV units, while the $B(E2)$ transitions in W.u.. The monopole transition between the first excited $0^+$ and the ground state is indicated by a dashed arrow.
  • ...and 18 more figures