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

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.


INTRODUCTION
The nuclei of the A = 130 mass region around N = 76 often have properties in agreement with a γ-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 [1], and highly-deformed or superdeformed bands (see e. g. [2,3]). 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-Ω 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-Ω 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 γ-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 bandheads have often long lifetimes and can become isomeric. This situation has been recently discussed in 130 Ba [1], in which a dipole band built on the long-lived K π = 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 ≈ 160−180, like the Os and W nuclei [4]. The 130 Ba nucleus also exhibits a γ-band with strong energy staggering between the odd and even spins, a fingerprint of the γ-softness or O(6) symmetry (see [5][6][7]). This softness facilitates the coexistence of different shapes, which can change depending on the specific configuration, rang-ing from prolate to oblate. In addition to the RAL Sbands 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 Fermialigned (FAL) by Frauendorf [8,9]. One example is 180 W, in which a rotational band built on the 8 + , 2132-keV state has been interpreted as a νi 2 13/2 FAL t-band [10], and another is 182 Os, in which two bands built on 8 + and 9 + states are also interpreted as a νi 2 13/2 FAL t-band [11]. No two-quasiparticle t-bands were reported until now in the A ≈ 130 mass region, even though the Fermi level for N ≈ 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 ≈ 130 mass region the observation in 130 Ba 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) [2], tilted axis cranking (TAC) [8], particle rotor model (PRM) [12], and projected shell model (PSM) [13,14].

EXPERIMENTAL RESULTS
The 130 Ba nucleus has been populated via the 122 Sn( 13 C,5n) reaction at a beam energy of 65 MeV. The target consisted of a stack of two self-supporting 122 Sn foils with a thickness of 0.5 mg/cm 2 each. The 13 C beam of 5 pnA was provided by the XTU Tandem accelerator of the Laboratori Nazionali di Legnaro. The γ-rays were detected by the GALILEO spectrometer [15,16], 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 [17] and the Neutron Wall array [18,19], respectively.Data were recorded by the GALILEO data acquisition system which was designed for the GALILEO-EUCLIDES-NWALL Experiment [20]. More details of the experimental setup and data analysis can be found in Ref. [1] and a forthcoming paper [21].
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 [22]. The 11 + and 12 + levels of the tband were previously established by means of the decay-ing 441-, 540-and 981-keV transitions [23], 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 [21], to the t-band, γ-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 δ from the analysis of the angular distributions, and deduced the B(M 1)/B(E2) and B(E2) out /B(E2) in ratios of reduced transition probabilities (see the supplemental material [22]).
The bands S1 and S2o previously reported in Ref. [23] 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 and S2o , as well as their connecting transitions to the even-spin partners are completely new. Detailed experimental information on the transitions of bands S1, S1 , S2o and S2o will be published in a forthcoming paper [21].
We also identified the high-spin part of the groundstate 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 [1].

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 130 Ba constitute one of the most complete sets of bands built on different shapes rotating around axes with different orienta- tions, offering thus a fertile field of investigation of multiquasiparticle collective excitations. In fact, we already had indications that 130 Ba, a nucleus with a pronounced γ-softness at low spin, can acquire a stable axial shape in two-quasiparticle configurations, as in the νh −1 11/2 g −1 7/2 configuration assigned to the 8 − isomer [1].
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 − ω curves, which were subsequently used to determine the single-particle aligned angular momenta i x following the procedure of Ref. [24,25] (see the supplemental material [22]).
The first important result of the present work is the observation of the odd-spin bands S1 and S2o , 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 [26][27][28][29][30][31][32]. Their interpretation is often handwaving, invoking the γ-band built on the S-bands, or is completely missing. These weakly populated oddspin 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 and S2o 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 , S2o, S2o 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 νh 2 11/2 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 132 Ce, which has a similar level scheme [33]. This structure can, however, be com- pared with the K π = 23/2 + band in 129 Ba [34], as well as with the recently reported high-K band in 127 Xe [35], both being built on isomeric states that support their high-K assignments. In 130 Ba the band-head of the tband is not isomeric, but has a significant decay branch to the K π = 8 − isomer, indicating its high-K character. Such t-bands, built on νi 2 13/2 configurations, have also been identified in the A ≈180 region [36]. A specific comparison can be made with 180 W [10], in which a band built on a 8 + state interpreted as having a νi 2 13/2 configuration has a prompt decay branch to a K π = 8 − isomer, like in the case of 130 Ba. We therefore suggest that the newly identified t-band of 130 Ba 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 ∆I = 1 transitions, which strongly suggests the presence in its configuration of two high-Ω neutron orbitals, which can be the same as in the t-band (νh 2 11/2 FAL) and two low-Ω πh 11/2 protons. We therefore assign the πh 2 11/2 ⊗ νh 2 11/2 FAL configuration to band D1, which is obtained by coupling the prolate configurations πh 2 11/2 assigned to band S1 and νh 2 11/2 assigned to the t-band.

TRS calculations
The bands S1 and S2o have been reported previously in Ref. [37], 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 [38][39][40][41][42][43].

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 [9] for the two axial deformations ε 2 = 0.20 and -0.19, and ε 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 νh 2 11/2 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 π = 8 − band have similar behavior. They are much smaller than that of band D1 (see Fig.  4), because the low-Ω quasiproton pair drives J toward 90 • . The energies and B(M 1)/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 π = 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 180 W and 182 Os [10,11] 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 α = −1/2, 1/2, −1/2, respectively. The band S2o is assigned to ν[ef ] and S2o' to ν[eg]. We have discarded the possible assignment of S2p to ν[f g] because there are no connecting transitions to S2o like the ones observed from S2o to S2o. At prolate shape the analogous structure appears for the RAL quasiprotons. We have discarded the possible assign-ment π[f g] because there are no connecting transitions observed from S2p to S1 . Ruling out the two possibilities leaves the combination of the two FAL quasineutrons to S2p as alternative.
For the νh 2 11/2 configuration the TRS calculations show a minimum at γ ≈ −60 • and a kind of plateau around γ = 0 • . 130 Ba is very γ-soft (close to the O(6) limit [6]), and large-scale collective motion in the γ-degree of freedom is expected. The collective wave functions for a Bohr Hamiltonian with a qualitatively similar potential are shown in Ref. [7] (Fig. 8, case χ = 50, change γ → 60 • − γ. The presence of such a state at 1179 keV was demonstrated in Ref. [6].) The collective ground state (S2o) is centered near oblate shape. The first exited spin-zero state (γγ 0 ) has a node and two maxima, one on the oblate side and one (with larger probability) near γ = 30 • . The calculation assumes a constant mass parameter. It is possible that a γ-dependent mass parameter gives more weight to the prolate side. With increasing γ 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.

PRM calculations
The active nucleon configurations that can be assigned to the bands observed in 130 Ba have been calculated using the configuration-fixed constrained triaxial covariant density functional theory (CDFT) framework [44] and those calculated using PC-PK1 effective interaction [45] are given in Table I. With the obtained deformation parameters, quantal PRM calculations [12,46,47] 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 J 0 and Coriolis attenuation factors ξ in the PRM calculations are also listed in Table I. The obtained calculated results in comparison with the experimental data are shown in Fig. 5. It is seen that the experimental energy spectra, B(M 1)/B(E2), as well as B(E2) out /B(E2) in of bands K π = 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(M 1)/B(E2) of the K π = 8 − band and the t-band are similar. The B(M 1)/B(E2) values of the t-band are overall a bit larger that those of the K π = 8 − band, because the calculated B(M 1) values of the t-band are larger that those of the K π = 8 − band. This can be attributed to the different effective g-factors (g ν − g R ) (with g R = Z/A) of the νh 11/2 and νg 7/2 orbitals. For the νh 11/2 orbital, the (g ν − g R ) = −0.64, while for the νg 7/2 orbital (g ν − g R ) = −0. 18  the square of the magnetic moment, are larger than those of the K π = 8 − band. Moreover, the B(E2) values are determined by the Clebsch-Gordan coefficients I i K20|I f K for a band with good K [48]. 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) out is larger than the B(E2) in . This feature is seen in Fig. 5 for the t-band and the K π = 8 − band. Furthermore, due to the same nominal K = 8 value in the two bands, their B(E2) out /B(E2) 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 θ = J 2 3 /(I + 1/2). The results are in good agreement with the TAC calculations.

PSM calculations
PSM [13,49] has been successfully applied for studying the structure of high-spin states, such as tilted rotation in the A≈ 180 mass region [50], multi-quasiparticle configurations [51,52] and multiple dipole bands [53,54]. 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 ε 2 = 0.22 and hexadecapole deformations ε 4 = 0.02 suggested in Ref. [55] for 130 Ba and assumed axial symmetry. The monopole-pairing strength is taken to be G M = [20.82 ∓ 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 130 Ba. 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 ε 2 = −0.18.
The band diagrams which display the angular- momentum-projected energies versus spins of rotational bands before configuration mixing [13] are shown in the supplemental material [22] 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 πh 2 [1/2, 3/2], K π = 1 + , component. However, the calculated staggering is larger than the experimental one above spin I = 20. The comparison of the calculated (S2o, S2o ) 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 ), which is crossed by the 4-qp neutron configuration assigned to band S2o-high.
In the PSM calculations, the B(E2), B(M 1) and gfactors are evaluated using the (final) shell model wave functions, as illustrated in the early PSM work of Ref. [13,14]. 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(M 1)/B(E2) and B(E2) out /B(E2) in for t-band, K π = 8 − and D1 bands are compared with available experimental data in Fig. 7. K π = 8 − band are lower than the experimental data, due to the smaller calculated B(M 1) values, while those for the K π = 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) out /B(E2) 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 γ-soft nucleus can exhibit at medium and high spins.