Microscopic description of triaxiality in Ru isotopes with covariant energy density functional theory

Background: Triaxiality in nuclear low-lying states has attracted great interest for many years. Recently, reduced transition probabilities for levels near the ground state in ${}^{110}\mathrm{Ru}$ have been measured and provided strong evidence of a triaxial shape of this nucleus.

Purpose: The aim of this work is to provide a microscopic study of low-lying states for Ru isotopes with $A\approx 100$ and to examine in detail the role of triaxiality and the evolution of quadrupole shapes with the isospin and spin degrees of freedom.

Method: Low-lying excitation spectra and transition probabilities of even-even Ru isotopes are described at the beyond-mean-field level by solving a five-dimensional collective Hamiltonian with parameters determined by constrained self-consistent mean-field calculations based on the relativistic energy density functional PC-PK1.

Results: The calculated energy surfaces, low-energy spectra, and intraband and interband transition rates, as well as some characteristic collective observables, such as $E\left(4_{\mathrm{g}.\mathrm{s}.}^{+}\right)/E\left(2_{\mathrm{g}.\mathrm{s}.}^{+}\right),E\left(2_{\gamma }^{+}\right)/E\left(4_{\mathrm{g}.\mathrm{s}.}^{+}\right)$, and $B(E2;2_{\mathrm{g}.\mathrm{s}.}^{+}\to 0_{\mathrm{g}.\mathrm{s}.}^{+})$ and $\gamma$-band staggerings, are in good agreement with the available experimental data.

Conclusions: The main features of the experimental low-lying excitation spectra and electric transition rates are well reproduced and, thus, strongly support the onset of triaxiality in the low-lying excited states of Ru isotopes around ${}^{110}\mathrm{Ru}$.

Figure 1

Potential energy surfaces of even-even ${}^{100-114}\mathrm{Ru}$ isotopes in the $\beta \text{−}\gamma$ plane calculated by the constrained triaxial RMF + BCS with the PC-PK1 functional. All energies are normalized with respect to the binding energy of the absolute minimum (indicated by a red circle). The energy difference between the neighboring contour lines is 0.25 MeV.

Figure 2

(a) Deviations of the binding energies calculated by the triaxial CDFT and 5DCH from the data for Ru isotopes. Evolution of (b) the theoretical two-neutron separation energies $S_{2n}$, (c) the root-mean-square charge radii $r_c$, and (d) the isotope shifts of the ground-state charge radii ${\langle r_c^2\rangle }_{A+2}-{\langle r_c^2\rangle }_A$ as functions of the mass number in Ru isotopes, in comparison with available data [82, 83].

Figure 3

Calculated excitation energies (in MeV) and intraband and interband $B\left(E2\right)$ values (in W.u.) for ground-state bands and $\gamma$ bands in even-even ${}^{100-114}\mathrm{Ru}$ isotopes by 5DCH-CDFT, in comparison with the experimental data (see [86]) [23, 84].

Figure 4

Energy spectra for ${}^{110}\mathrm{Ru}$ calculated by 5DCH-CDFT with PC-PK1 [45] and DD-PC1 [85], in comparison with the available mapped IBM results [15] and experimental data [23, 84]. In the calculation with DD-PC1, a separable pairing [87] is adopted.

Figure 5

Evolution of (a) $E\left(4_{\mathrm{g}.\mathrm{s}.}^{+}\right)/E\left(2_{\mathrm{g}.\mathrm{s}.}^{+}\right)$, (b) $E\left(2_{\gamma }^{+}\right)/E\left(4_{\mathrm{g}.\mathrm{s}.}^{+}\right)$, and (c) $B(E2;2_{\mathrm{g}.\mathrm{s}.}^{+}\to 0_{\mathrm{g}.\mathrm{s}.}^{+})$ values (in W.u.) with the mass number for Ru isotopes calculated by 5DCH-CDFT, in comparison with the available data [23, 84] and theoretical results from the mapped IBM [15].

Figure 6

Staggering parameters $S\left(I\right)$ of even-even ${}^{100-114}\mathrm{Ru}$ isotopes calculated by 5DCH-CDFT in comparison with the available data [84].

Figure 7

Distributions of $K$ components in the collective wave functions for the $2^{+},4^{+},6^{+},8^{+}$, and ${10}^{+}$ states in the ground-state and $\gamma$ bands of ${}^{100,104,110,112}\mathrm{Ru}$.

Figure 8

Probability density distributions in the $\beta \text{−}\gamma$ plane for the $0_{\mathrm{g}.\mathrm{s}.}^{+},2_{\mathrm{g}.\mathrm{s}.}^{+}$, and $2_{\gamma }^{+}$ states in ${}^{100,104,110,112}\mathrm{Ru}$.