Systematics of 3-qp states involved

For nuclei in the A~180 region, an abundance of high-K isomeric states has been discovered [48]. Among them, the two-quasineutron isomeric states exist systematically in even-even N = 106 isotones, which have been investigated using configuration-constrained PES calculations in Ref. [6]. The calculated excitation energies agree well with the experimental data, and strong shape polarizations were found when approaching the Z=82 shell closure. For odd-mass N = 106 isotones in this mass region, most of the observed 3-qp K isomers consistently involve a two-quasineutron configuration coupled to Kπ = 8^- states that are identified in the aforementioned even-even nuclei. For example, in nuclei such as ¹⁷⁵Tm [28, 49], ¹⁷⁷Lu [29, 50], ¹⁷⁹Ta [30, 51, 52], and ¹⁸¹Re [31, 53], 3-qp K isomers have been assigned as the two-quasineutron Kπ = 8^- configuration coupled to the energetically lowest one-quasiproton configuration. Furthermore, in ¹⁸¹Re, ¹⁷⁹Ta, and ¹⁷⁷Lu, the meta-stable Kπ=9/2^- states are found, assigned to the π9/2^- [514] configuration, leading to the presence of 3-qp states associated with the coupling of a Kπ = 8^- two-quasineutron configuration with π9/2^- [514] [29-31]. In ¹⁷⁵Tm, a K isomer that may involve the coupling of the two-quasineutron Kπ = 8^- configuration with the π 7/2^- [523] has been found [28]. The half-lives of these isomers range from a few microseconds to several days. The extent to which these high-K states are associated with shape polarization and shape isomerism remains an open question.

We performed configuration-constrained PES calculations on the 1-quasiproton and 3-qp states in N = 106 odd-mass isotones. Table 2 presents the calculated deformations and energies of the g.s., possible high-Ω 1-quasiproton, and low-lying high-K 3-qp states, compared with the available experimental data. Our calculations reproduce the experimentally assigned spin-parity of the g.s. of these nuclei, except for ¹⁷⁷Lu, in which the calculated lowest 1-quasiproton configuration is the π9/2^-[514] rather than the experimentally assigned π7/2⁺[404] [54]. However, the calculated π7/2⁺[404] configuration lies only 259 keV above the π9/2^-[514] state. Given the strong dependence of the 1-qp state energies on the ordering and spacing of the single-particle levels, the deviation in their relative positions falls within an acceptable range.

One may ask the sensitivity of these calculated results to the pairing strengths or Woods-Saxon parameters. Note that previous work has shown that the adjustment of pairing strength mainly influences the quasiparticle energy gaps and only slightly affects the deformations [37], while the ordering of single-particle levels at a certain deformation is primarily determined by the choice of Woods-Saxon potential parameters. To examine the robustness of this result, we tested several known parameter sets, including those of Blomquist and Wahlborn [55], the parameters of Chepurnov [56], the parameters given by Rost [57], and the “new” parameters [58]. We found that the deviation in the relative positions of the π9/2^-[514] and π7/2⁺[404] configurations of ¹⁷⁷Lu was preserved with different parameter sets. To accurately reproduce the ordering of the low-lying 1-qp states of ¹⁷⁷Lu, an improvement in the Woods-Saxon potential is needed.

Experimentally, the spin and parity of the g.s. of ¹⁸³Ir [59] and ¹⁸⁵Au [60-62] have been assigned to be the 5/2^- states built on the π1/2^-[541] configuration, while a strong mixing between the π1/2^-[541] and π3/2^-[532] configurations attributed to the Coriolis interactions has been proposed for the 5/2^- g.s. of ¹⁸⁵Au [63]. Our calculations show that the π1/2^-[541] configuration has the lowest energy, whereas the π3/2^-[532] state is approximately 500 keV higher. The present PES calculations show that these two low-lying 1-quasiproton states both have considerable triaxial deformations, which would reinforce substantial configuration mixing. However, the configuration mixing calculations are beyond the scope of the present study.

The present configuration-constrained PES calculations also reasonably reproduce the high-Ω 1-qp isomeric states observed in odd-mass N=106 isotones, except for the aforementioned deviation in ¹⁷⁷Lu. Notably, the configuration-constrained PES calculations show that the Kπ=9/2^- isomer of ¹⁸⁵Au and the Kπ=11/2^- state of ¹⁸⁷Tl have moderate triaxial deformations with which are remarkably polarized when compared with their ground states. This indicates the appearance of single-proton-induced shape polarization in shape-soft odd-proton nuclei when approaching the Z=82 closed shell.

Now, we turn to the investigation of energetically low-lying 3-qp states in odd-mass N=106 isotones. We mainly focus on the 3-qp states that consist of the two-quasineutron configuration coupled to different 1-quasiproton configurations, since the Kπ=8^- isomeric states are systematically identified as the configuration in even-even N=106 isotones. As shown in Table 2, the calculated energies of these 3-qp states in ¹⁷⁵Tm, ¹⁷⁹Ta, and ¹⁸¹Re agree well with the experimental data. For ¹⁷⁷Lu, the calculated configuration is overpredicted, while the state is slightly underestimated. This can be attributed to the deviation of the π7/2⁺[404] and π9/2^-[514] orbitals found in the calculations of the 1-quasiproton states.

Furthermore, the configuration-constrained PES calculations predict the candidate configurations of the 3-qp isomeric states in odd-A N=106 isotones when moving towards the Z=82 shell closure. To date, no experimental evidence has been reported for three-quasiparticle high-K isomers in ¹⁸³Ir. We proposed two possible high-K 3-qp states that are composed of two-quasineutron coupled with the proton configurations of π1/2^-[541] and 9/2^-[514], respectively. For ¹⁸⁵Au, a new isomer at an excitation energy of 1504.2(4) keV with a half-life of 630(80) ns was identified in γ-γ coincidence analysis [34]. The possible spins of this isomer are constrained to a range from 13/2 to 21/2 in comparison with the predictions from the Weisskopf estimates [34]. Our calculations suggest two possible 3-qp high-K configurations that are consistent with the systematics of 3-qp configurations in lighter odd-mass N=106 isotones. They both lie at excitation energies of approximately 1900 keV, which is slightly overpredicted. However, Ref. [34] argued that the g.s. configuration of ¹⁸⁵Au is more likely π3/2^-[532], and the calculated energy differences of these two 3-qp states with respect to the 3/2^-[532] configuration are 1444 and 1458 keV, respectively, which is in great agreement with the observation. For nucleus ¹⁸⁷Tl, two isomers with microsecond lifetimes (T_1/2 = 1.11 μs and 0.69 μs) have been reported [32]. Spin-parities Jπ = 27/2⁺, 31/2^- are tentatively assigned to the isomer at 2584 keV with a lifetime T_1/2 = 0.69 μs based on the deduced total conversion coefficient [33]. Our calculation presents a Kπ = 27/2⁺, configuration with an excitation energy of 2312 keV, which is in agreement with the prolate high-K configuration suggested in Ref. [32]. This implies that the T_1/2 = 0.69 μs isomer observed in ¹⁸⁷Tl involves the two-quasineutron configuration, which is consistent with the systematics of the 3-qp isomers observed in lighter odd-mass N=106 isotones. However, further experimental data are required to unambiguously assign the spin-parity and configuration of these observed isomers. We will discuss the other T_1/2 = 1.1 μs isomer later in Sect. 3.2.

In addition, intrinsic shape evolution is crucial for understanding the behavior of isomeric states, such as their decay properties. In a previous study [6], strong shape polarization was shown in even-even nuclei with A~180 and N=106, especially in nuclei close to the Z = 82 shell closure. For systematic comparison, we plot the variation of β₂ and γ deformations of the high-K 3-qp states and the g.s. along with the proton number Z in Fig. 1. When approaching the shell closure of Z = 82, the β₂ value of the g.s. gradually decreases, indicating that the g.s. shape evolves towards a spheroid, whereas the 3-qp states are polarized to have distinct prolate shapes. The g.s. of ¹⁸⁵Au, for example, has a remarkably γ-soft shape with a very shallow PES minimum at . Considering that the g.s. of ¹⁸⁵Au can be interpreted as the coupling of a low-Ω π 1h_9/2 proton with the ¹⁸⁴Pt core, the PES of the g.s. of ¹⁸⁵Au exhibits similar γ softness compared to ¹⁸⁴Pt. This γ-soft nature of ¹⁸⁴Pt is mainly because the prolate-to-oblate shape transition going through a transitional γ-soft shape occurs in Pt isotopes around N=110, which has been suggested by the self-consistent HFB calculation using the Gogny-D1S interaction and the interacting boson model [64], as well as the five-dimensional collective Hamiltonian (5DCH) based on covariant density-functional theory [65]. In contrast, the predicted two 3-qp high-K states of ¹⁸⁵Au both have an approximately prolate shape with and about a 60% increase in β₂ deformation. The nucleus ¹⁸⁷Tl has a spherical g.s. with a proton that singly occupies the π3s_1/2 orbital, while the calculated state is predicted to have a moderately axially-asymmetric shape with β₂≈0.23 and , exhibiting the greatest difference in quadrupole deformation between the 3-qp state and the g.s.. In fact, the calculated high-K 3-qp configurations of ¹⁸⁷Tl present an ensemble of multiple nuclear shapes, which will be analyzed in detail in Sect. 3.2.

Fig. 1Full-screen View

Calculated β₂ and γ deformations for the g.s. and 3-qp states of odd-A nuclei in the N=106 isotonic chain. For Z=69-79, the 3-qp states correspond to the coupling of the 2-qp Kπ=8^- configuration with the g.s. configurations of odd-A nuclei, which are π1/2⁺[411], π7/2⁺[404], π7/2⁺[404], π5/2⁺[402], π1/2^-[541], and π1/2^-[541], respectively. For Z=81, the 3-qp state is . These 3-qp states correspond to observed high-K isomers

Shape coexistence in high-K 3-qp states of ¹⁸⁷Tl

The shape-coexisting configurations in this mass region are mainly attributed to the large spherical and deformed shell gaps that simultaneously appear near the proton shell closure at Z=82 and the neutron mid-shell at . Moreover, the unpaired nucleon that occupies different single-particle orbitals polarizes the shape of the odd-A nucleus in different ways, leading to a more profound shape coexistence phenomenon. An abundance of experimental data has demonstrated that different-shaped configurations in these odd-A nuclei are observed not only in the low-lying 1-qp states but also in the higher-seniority isomeric states [9].

For ¹⁸⁷Tl, previous studies [66-69] have proposed the coexistence of different nuclear shapes based on an analysis of the observed low-lying collective structures. As the proton-hole neighbor of ¹⁸⁸Pb, the I=1/2⁺ g.s. of ¹⁸⁷Tl can be interpreted as the coupling of the π3s_1/2 hole with the spherical state of ¹⁸⁸Pb core. The observed Kπ=9/2^- and Kπ=13/2⁺ isomeric states can be understood as filling the π9/2^-[505] and π13/2⁺[606] intruder orbitals that are lowered along with the increase in oblate deformation, respectively [70, 71], while the rotational band built on the Iπ=11/2^- state is suggested to be the prolate deformed π 11/2^-[505] configuration [32, 33]. The calculated deformations and excitation energies for the 1-quasiproton states are listed in Table 3. The configuration-constrained PES calculations reproduce the measured excitation energies well and clearly show the coexisting shapes of these 1-qp states. Note that the π 11/2^-[505] configuration has been predicted to have a considerable axial-asymmetric shape of , which breaks down the K- and shape-hindrance and would explain why it decays rapidly to the oblate Kπ=9/2^- isomeric state [32].

In addition to the high-Ω proton orbitals mentioned above, other deformation-driving high-j high-Ω orbitals, including the high-Ω members of the proton π h_9/2 shell and the high-Ω members of the neutron νh_9/2, νi_13/2, and νf_7/2 shells, would appear close to the proton and neutron Fermi surfaces at both the oblate and prolate sides, respectively. The couplings of these high-Ω orbitals would form energetically low-lying high-K 3-qp configurations that are polarized into different shapes. We summarize the calculated deformations and excitation energies of possible high-K 3-qp configurations in Table 3. Coexisting different types of intrinsic shapes are obtained for a variety of high-K 3-qp configurations from the configuration-constrained PES calculations. Figure 2 depicts the typical PES’s corresponding to the 3-qp configurations with spherical, γ-soft prolate, oblate, and axially asymmetric shapes, respectively.

Fig. 2Full-screen View

(Color online) Calculated ¹⁸⁷Tl potential energy surfaces (PES’s) of (a) the near-spherical g.s., (b) the prolate Kπ=27/2⁺, state, (c) the oblate Kπ = 29/2⁺, state, (d) the triaxial Kπ = 25/2⁺, state. The energy difference between the neighboring contours was 100 keV. The black dots denote the minima of PES’s

Among them, the lowest prolate high-K 3-qp state given by the configuration-constrained PES is the Kπ=27/2⁺, configuration. The calculated PES for this configuration is shown in panel (b) of Fig. 2. As discussed in Sect. 3.1, this configuration is most likely to be assigned to the observed isomeric state with excitation energy Ex = 2584.6 keV and a lifetime T_1/2 = 0.69 μs. Experimentally, it was found that the 2584.6-keV isomer decays to the Iπ = 25⁺ member of the rotational band, which is assigned to the low-Ω π1/2⁺[660] configuration [32, 33]. To understand this transition, we also compute the deformation and excitation energy of the low-Ω π1/2⁺[660] state (see Table 2). The calculated energy of the π1/2⁺[660] state was 1216 keV, which was compatible with the estimation of the bandhead energy of the observed rotational band. Note that both the π1/2⁺[660] and configurations have very soft axially asymmetric deformations of . The γ-soft shape breaks down the K-conservation and allows the decay from the Kπ=27/2⁺ state to the Iπ = 25⁺ state built on the π1/2⁺[660] configuration. The configuration-constrained PES calculations predict another Kπ=27/2⁺ configuration that consists of with a smaller β₂≈0.18 and a larger . However, the large axially asymmetric deformation violates K-conservation, and hence may prohibit the formation of the K isomeric state.

Another interesting 3-qp state that we predict is the Kπ=29/2⁺, configuration. As shown in panel (c) of Fig. 2, its calculated PES shows a minimum that appears at an oblate deformation of β₂≈0.19 and . The combination of the high K value, axially symmetric shape, and calculated low energy of Ex=1839 keV supports the existence of a long-lived K isomer. Therefore, we suggest that this Kπ=29/2⁺ configuration could be assigned to the observed isomeric state with a lifetime T_1/2=1.1 μs, although the position and spin-parity of this isomer cannot be firmly determined because the γ rays linking this isomer to low-lying states are still missing [32, 33]. More recently, it was found that this isomer decays to the low-lying 13/2⁺ isomer at 1061 keV [33]. This implies that the T_1/2=1.1 μs isomeric state may be oblate deformed and be composed of the same π13/2⁺[606] configuration, which is compatible with the predicted Kπ=29/2⁺, state. Thus, we propose that two long-lived 3-qp high-K isomeric states with different shapes coexist with intermediate excitation energies in ¹⁸⁷Tl. Further measurements of observables, such as gyromagnetic ratios or electromagnetic transition properties of rotational bands built on these two long-lived states, would help us unambiguously determine the shapes and intrinsic structures of these observed isomers.

Other low-lying 3-qp high-K states are predicted by the configuration-constrained PES calculations. Among them, the Kπ = 25/2^-, configuration is of particular interest because of its very low energy and high spin value. Because it is calculated to be even energetically lower than the observed lowest Iπ=13/2⁺ state, the calculated Kπ = 25/2^- state can be expected to form a “spin trap” [1]. As Dracoulis et al. [72] have pointed out, a very low energy could result in long-lived states that would preferentially β decay and thus be missed. Therefore, it provides a challenge for both experimental and theoretical studies and, in effect, a test of the reliability of the models.