Diversity of shapes and rotations in the gamma-soft 130Ba nucleus: first observation of a t-band in the A=130 mass region

TL;DR

This paper reports the first observation of a two-quasiparticle t-band in 130Ba, revealing diverse shapes and rotational behaviors in a gamma-soft nucleus at medium and high spins, supported by theoretical models.

Contribution

It presents the first detection of a t-band in the A=130 region and provides comprehensive experimental and theoretical analysis of shape coexistence and rotational modes.

Findings

First observation of a two-quasiparticle t-band in 130Ba

Identification of multiple new bands including odd-spin partners

Evidence of shape coexistence and diverse rotational behaviors

Abstract

Several new bands have been identified in 130Ba, among which there is one with band-head spin 8+. Its properties are in agreement with the Fermi-aligned \nu h11/2^2 , 7/2+[523] \otimes 9/2-[514] Nilsson configuration. This is the first observation of a two-quasiparticle t-band in the A=130 mass region. The t-band is fed by a dipole band involving two additional h11/2 protons. The odd-spin partners of the proton and neutron S-bands and the ground-state band at high spins are also newly identified. The observed bands are discussed using several theoretical models, which strongly suggest the coexistence of prolate and oblate shapes polarized by rotation aligned two-proton and two-neutron configurations, as well as prolate collective rotations around axes with different orientations. With the new results, 130Ba presents one of the best and most complete sets of collective excitations that a gamma-soft nucleus can manifest at medium and high spins, revealing a diversity of shapes and rotations for the nuclei in the A = 130 mass region.

Figures

Tables

Table 1. Table 1: The deformation parameters β 𝛽 \beta and γ 𝛾 \gamma , moments of inertia 𝒥 0 subscript 𝒥 0 \mathcal{J}_{0} (in units of ℏ 2 / MeV superscript Planck-constant-over-2-pi 2 MeV \hbar^{2}/\textrm{MeV} ), ξ 𝜉 \xi parameters, and their corresponding active nucleon configurations in the configuration-fixed constrained triaxial covariant density functional theory calculations.

$(\beta ,\gamma )$	${𝒥}_0$	$\xi$	$\mathrm{Active}\mathrm{nucleon}\mathrm{configuration}$
	(${\hslash }^2/$MeV)
$(0.23,{13.9}^{\circ })$			Nucleons paired
$(0.23,{11.7}^{\circ })$	21	1.00	$\nu \left[514\right]9/2^{-}\left[404\right]7/2^{+}$
$(0.21,{16.8}^{\circ })$	22	1.00	$\nu h_{11/2}^{-2}$
$(0.23,{24.2}^{\circ })$	18	0.96	$\nu h_{11/2}^{-2}\otimes \pi h_{11/2}^2$
$(0.25,{30.1}^{\circ })$	24	0.95	$\nu \left[514\right]9/2^{-}\left[404\right]7/2^{+}\otimes \pi h_{11/2}^2$

Full text

Nuclear reaction: 122Sn(13C,$5n$)130Ba; E= 65 MeV; Measured $\gamma\gamma\gamma\gamma$-coincidences; Eγ; Iγ; anisotropy ratios; angular distributions; 130Ba deduced levels; spin and parity; model calculation

I Introduction

The nuclei of the $A=130$ mass region around $N=76$ often have properties in agreement with a $\gamma$-soft triaxial shape at low spins, while at medium and high spins the shape can change to nearly axially symmetric as for the high-$K$ isomers Qiang et al. (2019), and highly-deformed or superdeformed bands (see e. g. Wyss et al. (1988a); Petrache et al. (2012)). States with spins higher than $10^{+}$ in even-even nuclei are built by breaking one nucleon pair with alignment of the spins of the particles or holes along the rotation axis. The spins of the low-$\Omega$ protons at the prolate (or nearly prolate) Fermi surface are rapidly aligned parallel to the rotation axis under the influence of the Coriolis force, giving rise to rotation-aligned (RAL) bands called $S$-bands, while the corresponding high-$\Omega$ neutrons (i. e. for prolate shape) are strongly coupled and their spins remain parallel to the long axis, giving rise to deformation-aligned (DAL) rotational bands called $K$-bands. However, the shape of $\gamma$-soft nuclei can become oblate under the polarizing effect of the neutron holes, and in such a case one can also obtain an $S$-band by rotation alignment of the neutron holes. As the structure and orientation axis in the two-neutron high-$K$ configurations are very different from those of the ground-state low-$K$ band, the band-heads have often long lifetimes and can become isomeric. This situation has been recently discussed in 130Ba Qiang et al. (2019), in which a dipole band built on the long-lived $K^{\pi}=8^{-}$ isomer has been observed, having properties in agreement with nearly axially symmetric prolate shape. Such high-$K$ isomers are also known in the deformed rare-earth nuclei with A $\approx 160-180$, like the Os and W nuclei Dracoulis et al. (2016). The 130Ba nucleus also exhibits a $\gamma$-band with strong energy staggering between the odd and even spins, a fingerprint of the $\gamma$-softness or $O(6)$ symmetry (see Sevrin et al. (1987); Kirch et al. (1995); Caprio (2011)). This softness facilitates the coexistence of different shapes, which can change depending on the specific configuration, ranging from prolate to oblate. In addition to the RAL $S$-bands and DAL $K$-bands, there can exist a special type of bands, called $t$-bands, intermediate between the DAL and RAL regimes, in which the cranking axis is tilted away from the principal axes of the spheroidal core. Such bands originate from high-$j$ quasiparticles with the Fermi level in the middle of the shells, which were called Fermi-aligned (FAL) by Frauendorf Frauendorf (1993, 2000). One example is 180W, in which a rotational band built on the $8^{+}$, 2132-keV state has been interpreted as a $\nu i^{2}_{13/2}$ FAL $t$-band Walker et al. (1993), and another is 182Os, in which two bands built on $8^{+}$ and $9^{+}$ states are also interpreted as a $\nu i^{2}_{13/2}$ FAL $t$-band Kutsarova et al. (1995). No two-quasiparticle $t$-bands were reported until now in the $A\approx 130$ mass region, even though the Fermi level for $N\approx 74$ can be mid-way among the $h_{11/2}$ orbitals and favor $t$-band configurations.

In the present letter we report for the first time in the $A\approx 130$ mass region the observation in 130Ba of a band with characteristics similar to those of the FAL configurations assigned to the $t$-bands. New experimental information is also reported on several medium-spin bands related to the $t$-band, in particular odd-spin partners of the $S$-bands and the continuation at high spins of the ground state band. Several theoretical models have been used to interpret the observed structures: Total Routhian Surface (TRS) Wyss et al. (1988a), tilted axis cranking (TAC) Frauendorf (1993), particle rotor model (PRM) Chen et al. (2018), and projected shell model (PSM) Hara and Sun (1995); Sun et al. (2001).

II Experimental results

The 130Ba nucleus has been populated via the 122Sn(13C,5n) reaction at a beam energy of 65 MeV. The target consisted of a stack of two self-supporting 122Sn foils with a thickness of 0.5 mg/cm2 each. The 13C beam of 5 pnA was provided by the XTU Tandem accelerator of the Laboratori Nazionali di Legnaro. The $\gamma$-rays were detected by the GALILEO spectrometer Valiente-Dobón et al. (2014); Testov et al. (2018), which consisted of 25 Compton-suppressed Ge detectors placed on four rings at 90*∘* (10 detectors), 119*∘* (5 detectors), 129*∘* (5 detectors) and 152*∘* (5 detectors). To distinguish different reaction channels, charged particles and neutrons were detected by the EUCLIDES silicon apparatus Testov (2019) and the Neutron Wall array Skeppstedt et al. (1999); Ljungvall et al. (2004), respectively.Data were recorded by the GALILEO data acquisition system which was designed for the GALILEO-EUCLIDES-NWALL Experiment Berti et al. (2014). More details of the experimental setup and data analysis can be found in Ref. Qiang et al. (2019) and a forthcoming paper Guo et al. .

In the present work, five new bands with positive parity were identified and the previously known bands built on the $10^{+}$ states were extended at higher spins, as shown in Fig. 1. Representative double-gated spectra are shown in Fig. 2 and SM . The $11^{+}$ and $12^{+}$ levels of the $t$-band were previously established by means of the decaying 441-, 540- and 981-keV transitions Kaur et al (2014), whereas the levels with spins $8^{+}$, $9^{+}$, $10^{+}$ and $13^{+}$ are new. Several new transitions linking the $t$-band to low-lying states are identified. Band $D1$ built on the $12^{+}$ state is completely new. It decays directly and through intermediate states, which will be published in a forthcoming paper Guo et al. , to the $t$-band, $\gamma$-band, ground state band ($GSB$), and to bands $S1$ and $S2o$. Interestingly, there is an accidental degeneracy between the $18^{+}$ states of band $D1$ and $S2o$ lying 14 keV apart, which gives rise to the 446-, 473- and 1201-keV connecting transitions. Also the $15^{+}$ states of the bands $D1$ and $S2o$ are separated by only 40 keV, which can explain the observation of the 977-keV connecting transition. These accidental degeneracies are important in understanding the connecting transitions between the bands $D1$ and $S2o$, which are interpreted as based on prolate and oblate shapes, respectively (see the following discussion subsection). For the bands populated with sufficient intensity, we extracted the mixing ratios $\delta$ from the analysis of the angular distributions, and deduced the $B(M1)/B(E2)$ and $B(E2)_{out}/B(E2)_{in}$ ratios of reduced transition probabilities (see the supplemental material SM ).

The bands $S1$ and $S2o$ previously reported in Ref. Kaur et al (2014) are confirmed only up to spin $18^{+}$ and $16^{+}$, respectively. The previously reported 927-keV transition of band $S1$, as well as 1027- and 1040-keV transitions of band $S2o$ are not observed and therefore not confirmed by the present data. Five transitions of 961, 1050, 1117, 1130 and 1163 keV in band $S1$, and four transitions of 927, 1087, 1247 and 1414 keV composing the band $S2o$-high, which decays to band $S2o$ through the 918-keV transition, are newly identified at high spins. The odd-spin bands $S1^{\prime}$ and $S2o^{\prime}$, as well as their connecting transitions to the even-spin partners are completely new. Detailed experimental information on the transitions of bands $S1$, $S1^{\prime}$, $S2o$ and $S2o^{\prime}$ will be published in a forthcoming paper Guo et al. .

We also identified the high-spin part of the ground-state band, $S2p$, consisting of the 962-, 1091-keV transitions, which decays to the $12^{+}$ state of $GSB$ via the 862-keV transition. Three new levels with spins $15^{-}$, $17^{-}$ and $19^{-}$ on top of the odd-spin cascade built on the $8^{-}$ isomer are newly identified, which can be the signature partner of the even-spin cascade reported previously Qiang et al. (2019).

III Discussion

In order to understand the nature of the observed positive-parity bands we performed a detailed theoretical investigation, employing several theoretical models. It appears that the medium-spin states of 130Ba constitute one of the most complete sets of bands built on different shapes rotating around axes with different orientations, offering thus a fertile field of investigation of multi-quasiparticle collective excitations. In fact, we already had indications that 130Ba, a nucleus with a pronounced $\gamma$-softness at low spin, can acquire a stable axial shape in two-quasiparticle configurations, as in the $\nu h_{11/2}^{-1}g_{7/2}^{-1}$ configuration assigned to the $8^{-}$ isomer Qiang et al. (2019).

In order to understand the observed structures and to assign them configurations, we first extracted the effective moments of inertia (MOI) from the slope of the linear part of the $I-\omega$ curves, which were subsequently used to determine the single-particle aligned angular momenta $i_{x}$ following the procedure of Ref. Bengtsson and Frauendorf (1979); Bengtsson et al. (1986) (see the supplemental material SM ).

The first important result of the present work is the observation of the odd-spin bands $S1^{\prime}$ and $S2o^{\prime}$, of the even-spin bands $S1$ and $S2o$. There is very scarce experimental information on such odd-spin states decaying to the states of the $S$-bands, typically only a few levels being observed Orce et al. (2006); Frauendorf (1991); Vogel et al (1999); Juutinen et al. (1995); Petrache et al. (2016); Lv et al. (2018); Petrache et al. (2012). Their interpretation is often handwaving, invoking the $\gamma$-band built on the $S$-bands, or is completely missing. These weakly populated odd-spin bands are however essential for the characterization of the configurations on which the $S$-bands are built, in particular the $K$-composition and the nuclear shape. To our knowledge, the present bands $S1^{\prime}$ and $S2o^{\prime}$ are among the most extended cascades composed of odd-spin states decaying to the $S$-bands observed in the $A=130$ mass region. We therefore paid special attention to the analysis of these bands and tried to globally understand the ensemble of bands denoted $S1$, $S1^{\prime}$, $S2o$, $S2o^{\prime}$ and $S2p$ in Fig. 1.

The second important result of the present work is the observation of the $t$-band, which is for the first time identified in the $A=130$ mass region. The assigned $\nu h_{11/2}^{2}$ configuration of the $t$-band is composed of the two FAL quasineutron orbitals, which leads to rotation around a tilted axis, as schematically drawn in Fig. 1. No such band has been observed in the neighboring even-even nuclei, in particular in the isotone 132Ce, which has a similar level scheme Paul et al. (1997). This structure can, however, be compared with the $K^{\pi}=23/2^{+}$ band in 129Ba Byrne et al. (1992), as well as with the recently reported high-$K$ band in 127Xe Chakraborty et al. (2006), both being built on isomeric states that support their high-$K$ assignments. In 130Ba the band-head of the $t$-band is not isomeric, but has a significant decay branch to the $K^{\pi}=8^{-}$ isomer, indicating its high-$K$ character. Such $t$-bands, built on $\nu i^{2}_{13/2}$ configurations, have also been identified in the A $\approx$180 region Walker and Xu (2016). A specific comparison can be made with 180W Walker et al. (1993), in which a band built on a $8^{+}$ state interpreted as having a $\nu i_{13/2}^{2}$ configuration has a prompt decay branch to a $K^{\pi}=8^{-}$ isomer, like in the case of 130Ba. We therefore suggest that the newly identified $t$-band of 130Ba provides the first evidence of a two-quasiparticle $t$-band in the $A=130$ mass region.

We also identified band $D1$, composed of two degenerate signature partner cascades connected by intense $\Delta I=1$ transitions, which strongly suggests the presence in its configuration of two high-$\Omega$ neutron orbitals, which can be the same as in the $t$-band ($\nu h_{11/2}^{2}$ FAL) and two low-$\Omega$ $\pi h_{11/2}$ protons. We therefore assign the $\pi h^{2}_{11/2}\otimes\nu h_{11/2}^{2}$FAL configuration to band $D1$, which is obtained by coupling the prolate configurations $\pi h^{2}_{11/2}$ assigned to band $S1$ and $\nu h_{11/2}^{2}$ assigned to the $t$-band.

III.1 TRS calculations

The bands $S1$ and $S2o$ have been reported previously in Ref. Sun et al. (1983), and interpreted, based on cranking calculations, as two-quasiparticle proton and neutron aligned bands, respectively. Similar conclusions have been also drawn in other theoretical papers devoted to the structure of the Xe, Ba, Ce nuclei Rohozinski et al. (1977); Faessler et al. (1985); Hammaren et al. (1986); Wyss et al. (1988b); Wyss et al. (1989); Granderath and Mantica (1996).

In order to check this interpretation, we first performed TRS calculations Wyss et al. (1988a), to understand which are the favored two-quasiparticle alignments (see supplemental material SM ). The TRS calculations confirm that a pair of $h_{11/2}$ neutrons decouples first and aligns to the rotation axis at a rotational frequency of 0.35 MeV/$\hbar$, driving the nuclear shape towards a near-oblate shape with $(\beta_{2},\gamma)\approx(0.17,-45^{\circ})$. Coexisting are near-prolate minima with $(\beta_{2},\gamma)\approx(0.20,+15^{\circ})$ and $(\beta_{2},\gamma)\approx(0.20,-15^{\circ})$ which support the bands S1, t, 8- and D1. The assignment of S2p will be discussed below.

III.2 TAC calculations

In order to investigate the orientation of the rotation axis, the components of the angular momenta on the three axes of the intrinsic system and the transition probabilities we carried out TAC calculations Frauendorf (2000) for the two axial deformations $\varepsilon_{2}=0.20$ and -0.19, and $\varepsilon_{4}=0.02$, which are close to the coexisting minima found in the TRS calculations and the deformations used in the PRM and PSM calculations. A schematic drawing of the different angular momentum coupling of the $\nu h_{11/2}^{2}$ quasineutron pair in the FAL $t$-band, and the RAL bands $S2o$ and $S2p$ with oblate and prolate shapes, respectively, is shown in Fig. 3. The angles of the total angular momentum $I$ with respect to the symmetry axis (z) for the $t$-band and the $K^{\pi}=8^{-}$ band have similar behavior. They are much smaller than that of band $D1$ (see Fig. 4), because the low-$\Omega$ quasiproton pair drives $\vec{J}$ toward 90∘. The energies and $B(M1)/B(E2)$ ratios of reduced transition probabilities are shown in Figs. 6 and 7 of the following subsection on PSM calculations. One can see a good agreement with the measured values of all three bands, $K^{\pi}=8^{-}$, $t$-band, and band $D1$.

The two FAL can be alternatively combined to form the configuration denoted $S2p$ in Fig. 3, which is observed in 180W and 182Os Walker et al. (1993); Kutsarova et al. (1995) as an up-bend of the even-$I$ yrast sequence. In analogy, we suggest assigning a configuration of the $S2p$ type to the band that crosses the $GSB$ at $I$=12 in Fig. 1, which is based on the following observations. The gain of angular momentum at the crossing with the $GSB$ indicates that it must be caused by the alignment of a pair of $h_{11/2}$ quasiparticles. At oblate shape the RAL quasineutrons $e,f,g$ have increasing energy and signatures $\alpha=-1/2,1/2,-1/2$, respectively. The band $S2o$ is assigned to $\nu[ef$] and $S2o$’ to $\nu[eg$]. We have discarded the possible assignment of $S2p$ to $\nu[fg$] because there are no connecting transitions to $S2o^{\prime}$ like the ones observed from $S2o^{\prime}$ to $S2o$. At prolate shape the analogous structure appears for the RAL quasiprotons. We have discarded the possible assignment $\pi[fg$] because there are no connecting transitions observed from $S2p$ to $S1^{\prime}$. Ruling out the two possibilities leaves the combination of the two FAL quasineutrons to $S2p$ as alternative.

For the $\nu h_{11/2}^{2}$ configuration the TRS calculations show a minimum at $\gamma\approx-60^{\circ}$ and a kind of plateau around $\gamma=0^{\circ}$. 130Ba is very $\gamma$-soft (close to the O(6) limit Kirch et al. (1995)), and large-scale collective motion in the $\gamma$-degree of freedom is expected. The collective wave functions for a Bohr Hamiltonian with a qualitatively similar potential are shown in Ref. Caprio (2011) (Fig. 8, case $\chi=50$, change $\gamma\rightarrow 60^{\circ}-\gamma$. The presence of such a state at 1179 keV was demonstrated in Ref. Kirch et al. (1995).) The collective ground state ($S2o$) is centered near oblate shape. The first exited spin-zero state ($\gamma\gamma_{0}$) has a node and two maxima, one on the oblate side and one (with larger probability) near $\gamma=30^{\circ}$. The calculation assumes a constant mass parameter. It is possible that a $\gamma$-dependent mass parameter gives more weight to the prolate side. With increasing $\gamma$ the FAL states in $S2p$ develop toward RAL, i. e. the $z$-component decreases and the $x$-component increases, which means that the total $x$-component increases while the $z$-component remains zero. We keep the label $S2p$ because the structure is the same as for prolate shape.

III.3 PRM calculations

The active nucleon configurations that can be assigned to the bands observed in 130Ba have been calculated using the configuration-fixed constrained triaxial covariant density functional theory (CDFT) framework Meng () (Ed.) and those calculated using PC-PK1 effective interaction Zhao et al. (2010) are given in Table 1. With the obtained deformation parameters, quantal PRM calculations Qi et al. (2009); Streck et al. (2018); Chen et al. (2018) have been carried out to describe the experimental energy spectra and electromagnetic transition probabilities. Both the CDFT and PRM calculations were performed without pairing.

The used moments of inertia $\mathcal{J}_{0}$ and Coriolis attenuation factors $\xi$ in the PRM calculations are also listed in Table 1. The obtained calculated results in comparison with the experimental data are shown in Fig. 5. It is seen that the experimental energy spectra, $B(M1)/B(E2)$, as well as $B(E2)_{\rm{out}}/B(E2)_{\rm{in}}$ of bands $K^{\pi}=8^{-}$, $t$-band, and $D1$ are described reasonably well by the PRM. This gives strong support for the configurations assigned to these three bands.

One notes that the experimental $B(M1)/B(E2)$ of the $K^{\pi}=8^{-}$ band and the $t$-band are similar. The $B(M1)/B(E2)$ values of the $t$-band are overall a bit larger that those of the $K^{\pi}=8^{-}$ band, because the calculated $B(M1)$ values of the $t$-band are larger that those of the $K^{\pi}=8^{-}$ band. This can be attributed to the different effective $g$-factors $(g_{\nu}-g_{R})$ (with $g_{R}=Z/A$) of the $\nu h_{11/2}$ and $\nu g_{7/2}$ orbitals. For the $\nu h_{11/2}$ orbital, the $(g_{\nu}-g_{R})=-0.64$, while for the $\nu g_{7/2}$ orbital $(g_{\nu}-g_{R})=-0.18$. The two quasineutrons in the two bands have a large angular momentum component $j_{3}$ to generate the band-head spin $I=8$ (see Figs. 1 and 5). Hence, the $M1$ operator and consequently the $B(M1)$ values of the $t$-band, which are proportional to the square of the magnetic moment, are larger than those of the $K^{\pi}=8^{-}$ band.

Moreover, the $B(E2)$ values are determined by the Clebsch-Gordan coefficients $\langle I_{i}K20|I_{f}K\rangle$ for a band with good $K$ Bohr and Mottelson . When the spin $I_{i}$ is small, the Clebsch-Gordan coefficients for $I_{f}=I_{i}-1$ is larger than that for $I_{f}=I_{i}-2$. As a consequence, the $B(E2)_{\rm{out}}$ is larger than the $B(E2)_{\rm{in}}$. This feature is seen in Fig. 5 for the $t$-band and the $K^{\pi}=8^{-}$ band. Furthermore, due to the same nominal $K=8$ value in the two bands, their $B(E2)_{\rm{out}}/B(E2)_{\rm{in}}$ values are also similar, though there is more $K$-mixing in the $t$-band.

Fig. 4 shows the angle of the total angular momentum with the (near-) symmetry axis 3. It is obtained as $\cos\theta=\sqrt{\langle J_{3}^{2}\rangle}/(I+1/2)$. The results are in good agreement with the TAC calculations.

III.4 PSM calculations

PSM Hara and Sun (1995); Sun (2016) has been successfully applied for studying the structure of high-spin states, such as tilted rotation in the A$\approx 180$ mass region Sheikh et al. (1998), multi-quasiparticle configurations Wang et al. (2014); Liu et al. (2011) and multiple dipole bands Petrache et al. (1996, 1997). Angular-momentum projection is performed for each K configuration and the mixing among states with different K values is calculated by diagonalizing the shell model Hamiltonian in the projected basis.

In the PSM, one first determines a deformed basis for a calculation. We adopted the quadrupole $\varepsilon_{2}=0.22$ and hexadecapole deformations $\varepsilon_{4}=0.02$ suggested in Ref. Möller et al. (2016) for 130Ba and assumed axial symmetry. The monopole-pairing strength is taken to be $G_{M}=[20.82\mp 13.58(N-Z)/A]/A$, for neutrons and protons, respectively. The quadrupole-pairing strength $G_{Q}$ is assumed to be proportional to $G_{M}$, with the proportionality constant $0.18$ for 130Ba. For the valence single-particle space, we include three major shells, N $=3,4,5$, for both neutrons and protons. For oblate deformation, all model parameters are the same except for the quadrupole deformation which is $\varepsilon_{2}=-0.18$.

The band diagrams which display the angular-momentum-projected energies versus spins of rotational bands before configuration mixing Hara and Sun (1995) are shown in the supplemental material SM for prolate and oblate shapes.

The PSM results after configuration mixing are compared with the experimental data in Fig. 6. The theoretical results are in good agreement with the available experimental data for all rotational bands considered in the present calculation. From Fig. 6(a), we can see that band $S1$ is well reproduced by the 2-quasiparticle proton configuration that is dominated by the $\pi h^{2}[1/2,3/2]$, $K^{\pi}=1^{+}$, component. However, the calculated staggering is larger than the experimental one above spin $I=20$. The comparison of the calculated $(S2o,S2o^{\prime})$ and $S2o$-high bands with the available data is shown in Fig. 6(b). A very good agreement with the experimental data is obtained over the entire observed spin range, which strongly supports the oblate 2-quasiparticle neutron configuration assigned to bands $(S2o,S2o^{\prime})$, which is crossed by the 4-qp neutron configuration assigned to band $S2o$-high.

The energies of the prolate configurations assigned to band $D1$, $t$-band, and $K^{\pi}=8^{-}$ band shown in Fig. 6(c) increase smoothly with spin, without odd-even staggering. The 2-quasiparticle neutron configuration dominated by the $\nu h^{2}[7/2,9/2],K=8$ component is assigned to $t$-band. It reproduces well the experimental data. The 4-quasiparticle configuration dominated by the $\nu h^{-2}[7/2,9/2]+\pi h^{2}[1/2,3/2],K=9$ component is assigned to band $D1$. It is in qualitative agreement with the experimental data, with a larger deviation towards the band-head. The difference between calculated and experimental energies can be reduced by taking into account the triaxial deformation, but this is beyond the scope of the present work and will be addressed in a future publication with calculations using the triaxial projected shell model (TPSM) Sheikh and Hara (1999).

The calculated negative-parity band built on the $8^{-}$ isomer is compared with the experimental data in Fig. 6(d). The 2-quasiparticle configuration dominated by the $\nu g^{1}h^{1}[7/2,9/2],K^{\pi}=8^{-}$ and the 4-quasiparticle configuration dominated by the $\nu g^{1}h^{1}[7/2,9/2]\otimes\pi h^{2}[1/2,3/2],K^{\pi}=9^{-}$ component reproduce very well the experimental data in the spin ranges $I=8-14$ and $I=15-20$, respectively.

In the PSM calculations, the $B(E2)$, $B(M1)$ and $g$-factors are evaluated using the (final) shell model wave functions, as illustrated in the early PSM work of Ref. Hara and Sun (1995); Sun et al. (2001). We emphasize that the $g$-factor in the PSM is computed directly from the many-body wave function without a semiclassical separation of the collective and the single-particle parts.

The calculated $B(M1)/B(E2)$ and $B(E2)_{\textrm{out}}/B(E2)_{\textrm{in}}$ for $t$-band, $K^{\pi}=8^{-}$ and $D1$ bands are compared with available experimental data in Fig. 7. For $t$-band and band $D1$, the experimental $B(M1)/B(E2)$ data are well reproduced. However, the calculated results of the $K^{\pi}=8^{-}$ band are lower than the experimental data, due to the smaller calculated $B(M1)$ values, while those for the $K^{\pi}=8^{+}$ band are in agreement with the experimental values. However, we need to be cautious, because the $8^{+}$ band ($t$-band) is significantly non-yrast, and mixings due to accidental degeneracies could be involved. For the $B(E2)_{\textrm{out}}/B(E2)_{\textrm{in}}$, the PSM results are in good agreement with the data.

Summarizing, the present work reports the first observation of a two-quasiparticle $t$-band in the $A=130$ mass region, as well as the odd-spin partners of the $S$-bands built on prolate 2qp-proton and oblate 2qp-neutron configurations. Extended calculations using several theoretical models converge in a coherent interpretation of the observed bands, which represent one of the best examples of shape coexistence and exotic rotations that a $\gamma$-soft nucleus can exhibit at medium and high spins.

This research is supported by the National Natural Science Foundation of China (Grants No. U1832139, Grants No. 11505242, No. 11305220, No. U1732139, No. 11775274 and No. 11575255), by the Minister of Europe and Foreign Affairs, Partnership Hubert Curien, project Cai YuanPei 2018, n. 41458XH, by the Natural Sciences and Engineering Research Council of Canada, by the National Research, Development and Innovation Fund of Hungary (Project no. K128947), by STFC funding (grant ST/P005314/1), by the GINOP-2.3.3-15-2016-00034, National Research, Development and Innovation Office NKFIH, Contract No. PD 124717, by the Polish National Science Centre (NCN) Grant No. 2013/10/M/ST2/00427, and by the National Research Foundation of South Africa, GUN: 93531, 109134. The work of Q. B. C. is supported by Deutsche Forschungsgemeinschaft (DFG) and National Natural Science Foundation of China (NSFC) through funds provided to the Sino-German CRC 110 “Symmetries and the Emergence of Structure in QCD”.