Preamble

Shape polarization and coexistence of high-K three-quasiparticle states in odd-mass N = 106 isotones∗

Runyan Dong¹,² and Changfeng Jiao¹,²,†

¹School of Physics and Astronomy, Sun Yat-sen University, Zhuhai 519082, China
²Guangdong Provincial Key Laboratory of Quantum Metrology and Sensing, Sun Yat-sen University, Zhuhai 519082, China

Three-quasiparticle K-isomeric states in odd-mass N = 106 isotones within the A ∼ 180 mass region are systematically investigated using configuration-constrained potential energy surface calculations. The calculations successfully reproduce the excitation energies and deformations of known high-K isomers in the nuclei from ¹⁷⁵Tm to ¹⁸¹Re. For the nuclei closer to the Z = 82 shell closure (¹⁸³Ir, ¹⁸⁵Au, and ¹⁸⁷Tl), predictions for the configurations of observed and yet-to-be-observed isomers are provided. The results reveal strong shape polarization, where the three-quasiparticle states are driven to larger deformations compared to the often shape-soft or spherical ground states. A particularly rich spectrum of shape coexistence is predicted in ¹⁸⁷Tl, where several high-K three-quasiparticle configurations with distinct prolate, oblate, and triaxial shapes are found to coexist at similar excitation energies. Notably, the oblate-deformed Kπ = 29/2⁺ configuration at Ex = 1839 keV is proposed to be responsible for a long-lived isomer. This study provides a comprehensive picture of shape evolution and coexistence in high-K multi-quasiparticle states, offering valuable insights for future experimental research.

Keywords: Shape polarization, shape coexistence, high-K isomeric state, configuration-constrained potential energy surface.

Introduction

Atomic nuclei in the A ∼ 180 mass region whose neutron numbers are close to the midshell are characterized by the existence of an abundance of low-lying, high seniority isomeric states. In this region, the orbitals with large Ω, namely the projection of individual angular momentum onto the intrinsic symmetric axis, approach the neutron Fermi surface at moderate quadrupole deformations. This facilitates the formation of broken-pair states with high K values (where K = Σi Ωi) near the yrast line. According to the selection rules for electromagnetic transitions, the transitions of multipolarity λ would be significantly hindered if ΔK > λ. The so-called K-hindrance can lead to relatively long half-lives (on the order of nanoseconds or longer) [1, 2], leading to the formation of "K isomers." One of the most well-known examples of high-K isomers is found in ¹⁷⁸Hf, in which two 2-quasiparticle (qp) Kπ = 8⁻ isomers and a long-lived four-quasiparticle Kπ = 16⁺ isomer with a half-life of 31 years are observed [3, 4]. Since the occurrence of K isomers is a combined effect of the unpaired nucleons occupying high-Ω states and the nuclear deformation, the study of K isomeric states is therefore pivotal for understanding the interplay between the shell structure of "individual" nucleons and the collective behavior of a strongly correlated nucleus [1].

Another common feature associated with the A ∼ 180 mass region is the shape transition. In this region, the ground-state (g.s.) shape can change from a well-deformed prolate ellipsoid with β₂ ≥ 0.28 for ¹⁷⁶Yb to a very soft spheroid for ¹⁸⁸Pb [5]. Moreover, the soft shape gives rise to the novel shape coexistence phenomenon, which is characterized by the emergence of low-lying states with different intrinsic shapes in one atomic nucleus. In general, it originates from the combining effect of approaching the Z = 82 spherical shell closures and the deformed shell gaps around the neutron midshell at N = 104–106 due to quadrupole-quadrupole correlations. It has drawn considerable interest [6–8]. The most well-known example is the differently shaped 0⁺ triplet observed in ¹⁸⁶Pb, which corresponds to the coexistence of the prolate, oblate, and spherical configurations [9, 10]. Coexisting 0⁺ states in even-even Pt, Hg, Pb, and Po isotopes around the neutron midshell have been extensively studied [11–14].

In addition to the shape change resulting from collective correlations such as quadrupole-quadrupole interactions, unpaired nucleons are found to strongly polarize the nuclear shape [15]. Since K isomeric states are coupled by high-Ω unpaired nucleons, shape polarization possibly yields considerable differences in shape between high-K states and ground states, leading to novel structures that involve both K isomerism and shape isomerism. For example, the two-quasineutron Kπ = 8⁻(ν{7/2⁻[514] ⊗ 9/2⁺[624]}) isomeric states are systematically observed in the even-even N = 106 isotones between ¹⁷⁴Er and ¹⁸⁸Pb (see [16, 17] and references therein). Previous theoretical investigation has shown that the g.s. are oblate deformed with |β₂| ≈ 0.13 for ¹⁸⁶Hg and spherical for ¹⁸⁸Pb, whereas the K = 8⁻ isomeric states are polarized to prolate deformed with |β₂| ≈ 0.25. The K = 8⁻ isomers with shapes different from those of the g.s. have later been confirmed by measuring rotational bands built on them [18, 19]. Furthermore, it is found that for shape-soft nuclei, the shape changes, particularly in the triaxial deformations, can be important for understanding the observed behaviors of isomeric states, such as decay properties [5].

While the shape evolution and coexistence of high-K states in even-even nuclei around neutron mid-shell and A ∼ 180 have been extensively studied, the structural properties such as the shape changing effects of 3-qp high-K states in their odd-proton neighbors are lacking systematic investigations. In odd-A nuclei, although the unpaired nucleon introduces additional complexity, it also serves as a sensitive probe of the underlying shell structure. The shape polarization effect induced by the single nucleon can be either parallel to or opposed to that of the high-K 2-qp configuration, thereby amplifying or diminishing the shape difference between the 3-qp states and the g.s.. Recently, the 3-qp high-K isomers, originated from the coupling between the odd proton and the aforementioned Kπ = 8⁻ configuration in even-mass cores, have been observed in odd-mass N = 106 isotones from ¹⁷⁵Tm to ¹⁸⁷Tl (except the ¹⁸³Ir) [20–26]. In addition, a substantial amount of experimental data also suggest that the low-lying 1-qp states of neutron-deficient odd-mass Au and Tl isotopes exhibit shape coexistence [6–8]. The extent to which the observed 3-qp isomers can be considered to involve shape isomerism, in addition to K isomerism, remains unclear. It thus has greatly stimulated our interest in pursuing theoretical studies on shape polarization and coexistence in the 3-qp high-K states within this mass region.

In this work, we investigate the 3-qp K-isomeric states of odd-A nuclei in the N = 106 isotonic chain using the configuration-constrained potential energy surface (CCPES) method [27]. This method is a deformation-pairing self-consistent PES calculation that includes the axially asymmetric γ-degree of freedom. At prolate deformations, we mainly focus on the study of the high-K 3-qp configurations composed of the coupling of the unpaired proton and the Kπ = 8⁻ 2-qp configuration that are observed systematically in even-even N = 106 nuclei. We predicted the possible configurations of the isomers in ¹⁸⁵Au and ¹⁸⁷Tl, with particular attention to the shape-polarization effect from multi-qp excitations. Furthermore, we explored the high-K states with distinct shapes (oblate, prolate, and triaxial) that coexist at comparable excitation energies in ¹⁸⁷Tl, and analyzed the impact of different quasiparticle configurations on shape evolution in detail.

II. The Model

We employ the CCPES approach [27], based on the macroscopic–microscopic model. The macroscopic energy contribution was computed using the standard liquid-drop model [28] with parameters taken from Ref. [29]. The microscopic correction includes the Strutinsky shell correction [30] and the pairing correction. The single-particle levels required for the microscopic energy calculations were obtained from a non-axial deformed Woods-Saxon potential [31] using a universal parameter set [32]. To avoid the collapse of pairing correlations in multi-quasiparticle states, we used the Lipkin-Nogami (LN) method [33] as an approximate particle-number projection incorporating monopole pairing. The pairing strength G was initially determined via the average-gap method [34, 35]. Although it is often further adjusted to reproduce the odd-even mass difference using a five-point formula, we note that irregularities may arise near magic numbers (e.g., Au and Tl isotopes) [34]. Therefore, following the recommendation of Ref. [34], closed-shell nuclei were excluded from the pairing strength calibration. For consistency, we adopted the standard pairing strength across all isotopes under investigation.

In the PES calculations, a deformation mesh in (β₂, γ) is used, with the hexadecapole deformation β₄ variation at each mesh point. The intrinsic PES is assumed to be reflection-symmetric with respect to γ = 0°; that is, the shape with γ = -60° is the same as the one with γ = 60° for non-collective excitations. For broken-pair configurations, the microscopic energy incorporates contributions from unpaired nucleons occupying specific single-particle orbitals (see Ref. [27] for details). These orbitals are continuously tracked and adiabatically blocked throughout the deformation plane throughout the calculation. Although Nilsson quantum numbers are not strictly conserved, their expectation values ⟨N⟩, ⟨n_z⟩, ⟨Λ⟩, and ⟨|Ω|⟩ exhibit slow variation, allowing a reliable configuration assignment. Therefore, each configuration is identified by computing the average Nilsson quantum numbers of the blocked orbitals. The total energy of a multi-qp state with unpaired nucleons can be decomposed into the deformation energy and the configuration energy, where the latter originates from qp excitations due to pair breaking and excitations of particles that define the specific configuration.

Quasiparticle excitations, particularly in deformation-soft nuclei, can induce significant shape polarization, resulting in an equilibrium deformation for the multi-qp state that differs from that of the ground state. The CCPES method effectively accounts for this polarization caused by the unpaired nucleon and offers a self-consistent description of both the deformation and excitation energy of multi-qp states [5, 36]. The excitation energy is computed as the energy difference between the PES minimum of the excited configuration and that of the ground-state configuration, enabling direct comparison with experimental values.

III. Calculations and Discussions

A. Systematics of 3-qp states involving ν{9/2⁺[624] ⊗ 7/2⁻[514]}

For nuclei in the A ∼ 180 region, an abundance of high-K isomeric states has been discovered [37]. Among them, the two-quasineutron Kπ = 8⁻(ν{7/2⁻[514] ⊗ 9/2⁺[624]}) isomeric states exist systematically in even-even N = 106 isotones, which have been investigated by means of the configuration-constrained PES calculation in Ref. [5]. The calculated excitation energies agree well with the experimental data, and strong shape polarizations have been 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 [20, 38], ¹⁷⁷Lu [21, 39], ¹⁷⁹Ta [22, 40, 41], and ¹⁸¹Re [23, 42], 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] [21–23]. In ¹⁷⁵Tm, a K isomer which may involve coupling of the two-quasineutron Kπ = 8⁻ configuration with the π7/2⁻[523] has been found [20]. The half-lives of these isomers range from a few microseconds to several days. To what extent these high-K states are associated with shape polarization and shape isomerism is still an open question.

We have performed the CCPES calculations on the 1-quasiproton and 3-qp states in N = 106 odd-mass isotones. Table 1 presents the calculated deformations and energies of the g.s., the 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] [43]. However, the calculated π7/2⁺[404] configuration lies only 259 keV above the π9/2⁻[514] state. Given the strong dependence of 1-qp state energies on the ordering and spacing of single-particle levels, the deviation in their relative positions falls within an acceptable range.

Experimentally, the spin and parity of the g.s. of ¹⁸³Ir [44] and ¹⁸⁵Au [45–47] 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 π5/2⁻[532] configurations attributed to the Coriolis interactions has been proposed for the 5/2⁻ g.s. of ¹⁸⁵Au [48]. Our calculations show that the π1/2⁻[541] configuration has the lowest energy, while the π5/2⁻[532] state is about 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, configuration mixing calculations are beyond the scope of the present work.

The present CCPES 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 CCPES calculations show that the Kπ = 9/2⁻ isomer of ¹⁸⁵Au and the Kπ = 11/2⁻ state of ¹⁸⁷Tl have moderate triaxial deformations with β₂ ∼ 0.2 which are remarkably polarized when compared with their g.s.. It 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 ν{7/2⁻[514] ⊗ 9/2⁺[624]} configuration coupled to different 1-quasiproton configurations, since the Kπ = 8⁻ isomeric states are systematically identified as the ν{7/2⁻[514] ⊗ 9/2⁺[624]} configuration in even-even N = 106 isotones. As seen in Table 1, the calculated energies of these 3-qp states in ¹⁷⁵Tm, ¹⁷⁹Ta, and ¹⁸¹Re agree well with the experimental data. For ¹⁷⁷Lu, the calculated Kπ = 23/2⁻ π7/2⁺[404] ⊗ ν{9/2⁺[624] ⊗ 7/2⁻[514]} configuration is overpredicted, while the Kπ = 25/2⁺ π9/2⁻[514] ⊗ ν{9/2⁺[624] ⊗ 7/2⁻[514]} state is slightly underestimated. This can be attributed to the deviation of the π7/2⁺[404] and π9/2⁻[514] orbitals that we found in the calculations of the 1-quasiproton states.

Furthermore, the CCPES calculations predict the candidate configurations of the 3-qp isomeric states in the odd-A N = 106 isotones when moving towards the Z = 82 shell closure. To date, no experimental evidence has been reported on three-quasiparticle high-K isomers in ¹⁸³Ir. We proposed two possible high-K 3-qp states that are composed of two-quasineutron ν{7/2⁻[514] ⊗ 9/2⁺[624]} 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 recently [26]. The possible spins of this isomer are constrained to a range from 13/2 to 21/2 in comparison with predictions from the Weisskopf estimates [26]. Our calculations suggest two possible 3-qp high-K configurations that are consistent with systematics of 3-qp configurations in lighter odd-mass N = 106 isotones. They both lie at excitation energies of about 1900 keV, which is a bit overpredicted. However, Ref. [26] 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 (T₁/₂ = 1.11 µs and 0.69 µs) have been reported [24]. Spin-parities Jπ = 27/2⁺, 31/2⁻ are tentatively assigned to the isomer lies at 2584 keV with lifetime T₁/₂ = 0.69 µs based on the deduced total conversion coefficient [25].

Our calculation presents a Kπ = 27/2⁺, π11/2⁻[505] ⊗ ν{9/2⁺[624] ⊗ 7/2⁻[514]} configuration with an excitation energy of 2312 keV, which is in accord with the prolate high-K configuration suggested in Ref. [24]. This implies that the T₁/₂ = 0.69 µs isomer observed in ¹⁸⁷Tl involves the two-quasineutron ν{9/2⁺[624] ⊗ 7/2⁻[514]} configuration, which is consistent with the systematics of 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 would discuss the other T₁/₂ = 1.1 µs isomer later in Sect. III B.

In addition, intrinsic shape evolution is crucial for understanding the observed behavior of isomeric states, such as their decay properties. In the previous study [5], strong shape polarization has been 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 very γ-soft shape with an energy minimum at γ ≈ 24°, while the predicted two 3-qp high-K states both have an approximately prolate shape with γ ≈ 1° and about a 60% increase in β₂ deformation. The nucleus ¹⁸⁷Tl has a spherical g.s. with a proton that singly occupies the π3s₁/₂ orbital, while the calculated Kπ = 27/2⁺ π11/2⁻[505] ⊗ ν{9/2⁺[624] ⊗ 7/2⁻[514]} state is predicted to have a moderately axially-asymmetric shape with β₂ ≈ 0.23 and |γ| ≈ 12°, 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 would be analyzed in detail in Sect. III B.

B. 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 N ≈ 106. Moreover, the unpaired nucleon that occupies different single-particle orbitals would polarize the shape of the odd-A nucleus in different ways, leading to more profound shape coexistence phenomenon. An abundance of experimental data has demonstrated that differently-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 [7].

For ¹⁸⁷Tl, previous studies [49–52] have proposed the coexistence of different nuclear shapes by the analysis of 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 π3s₁/₂ hole with the spherical 0⁺₁ 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 [53, 54], while the rotational band built on the Iπ = 11/2⁻ state is suggested to be the prolate deformed π11/2⁻[505] configuration [24, 25]. The calculated deformations and excitation energies for these 1-quasiproton states are listed in Table 2. The CCPES calculations well reproduce the measured excitation energies, and clearly show the coexisting shapes for these 1-qp states. Note that the π11/2⁻[505] configuration has been predicted to have a considerable axial-asymmetric shape of γ ≈ 18°, which breaks down the K- and shape-hindrance and would explain why it decays fast to the oblate Kπ = 9/2⁻ isomeric state [24].

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 πh₉/₂ shell, the high-Ω members of neutron νh₉/₂, νi₁₃/₂, and νf₇/₂ shells, would appear close to the proton and neutron Fermi surfaces at both 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 to different shapes. We summarize the calculated deformations and excitation energies of possible high-K 3-qp configurations in Table 2. Coexisting different types of intrinsic shape are obtained for a variety of high-K 3-qp configurations from the CCPES calculations. Fig. 2 depicts the typical PES's corresponding to the 3-qp configurations with spherical, γ-soft prolate, oblate, and axially asymmetric shapes, respectively.

Among them, the lowest prolate high-K 3-qp state given by the CCPES is the Kπ = 27/2⁺, π11/2⁻[505] ⊗ ν{9/2⁺[624] ⊗ 7/2⁻[514]} configuration. The calculated PES of this configuration is shown in the panel (b) of Fig. 2. As we discussed in Sect. III A, this configuration is most likely to be assigned to the observed isomeric state with excitation energy Ex = 2584.6 keV and lifetime T₁/₂ = 0.69 µs. Experimentally, it is 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 [24, 25]. To understand this transition, we also compute the deformation and excitation energy of the low-Ω π1/2⁺[660] state (see them in Table 2). The calculated energy of the π1/2⁺[660] state is 1216 keV, which is compatible with the estimation of the bandhead energy of the observed rotational band. Note that both the π1/2⁺[660] and π11/2⁻[505] ⊗ ν{9/2⁺[624] ⊗ 7/2⁻[514]} configurations have very soft axially asymmetric deformations of γ ∼ 15°. 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 CCPES calculations predict another Kπ = 27/2⁺ configuration that consists of π11/2⁻[505] ⊗ ν{9/2⁻[505] ⊗ 9/2⁺[633]} with a smaller β₂ ≈ 0.18 and a larger |γ| ≈ 30°. However, the large axially asymmetric deformation violates the 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⁺, π13/2⁺[606] ⊗ ν{9/2⁺[624] ⊗ 7/2⁺[633]} configuration. As seen in the panel (c) of Fig. 2, its calculated PES shows a minimum that appears at oblate deformation of β₂ ≈ 0.19 and |γ| ≈ 60°. The combination of high K value, axially symmetric shape, and its 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 T₁/₂ = 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 [24, 25]. More recently, it has been found that this isomer decays to the low-lying 13/2⁺ isomer that lies at 1061 keV [25]. This implies that the T₁/₂ = 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⁺, π13/2⁺[606] ⊗ ν{9/2⁺[624] ⊗ 7/2⁺[633]} state. We thus propose that two long-lived 3-qp high-K isomeric states with different shapes coexist with intermediate excitation energies in ¹⁸⁷Tl. Further measurement of observables, such as gyromagnetic ratios or electromagnetic transition properties of rotational bands built on these two long-lived states, would help us to unambiguously pin down the shapes and intrinsic structures of these observed isomers.

Other low-lying 3-qp high-K states are predicted by the CCPES calculations. Among them, the Kπ = 25/2⁻, π9/2⁻[505] ⊗ ν{9/2⁺[624] ⊗ 7/2⁺[633]} configuration is of particular interest because of its very low energy and high spin value. Since 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. [55] have pointed out, a very low energy could result in long-lived states that would preferentially β decay and thus be missed. It therefore provides a challenge for both experimental and theoretical studies and, in effect, a test of the reliability of the models.

IV. Summary

We present a systematic theoretical study of shape polarization and coexistence in high-K 3-qp states of odd-mass N = 106 isotones (¹⁷⁵Tm, ¹⁷⁷Lu, ¹⁷⁹Ta, ¹⁸¹Re, ¹⁸³Ir, ¹⁸⁵Au, ¹⁸⁷Tl) using the configuration-constrained potential energy surface (CCPES) method.

The investigation focuses on 3-qp states formed by coupling a single proton to the systematic two-quasineutron Kπ = 8⁻ isomeric configuration known in the even-even N = 106 cores. The calculations demonstrate excellent agreement with the experimental data for the well-established isomers in lighter isotones (Z = 69–75), validating the theoretical approach. As the proton number increases towards the Z = 82 shell closure, the ground states become progressively softer and less deformed, while the high-K 3-qp states exhibit significant shape polarization, maintaining well-defined prolate deformations. This leads to a substantial shape difference between the isomers and the ground states in nuclei like ¹⁸⁵Au and ¹⁸⁷Tl.

Furthermore, we analyze in detail the intrinsic shapes of 3-qp states in ¹⁸⁷Tl. The CCPES calculations identify a multitude of low-lying high-K configurations with distinctly different shapes (including prolate, oblate, and triaxial) coexisting within a narrow energy range. Two specific long-lived isomers observed in ¹⁸⁷Tl are assigned to configurations with different shapes: the T₁/₂ = 0.69 µs isomer is associated with a prolate-deformed Kπ = 27/2⁺ state, while the T₁/₂ = 1.1 µs isomer is proposed to be an oblate-deformed Kπ = 29/2⁺ state characterized by a high K value, an axial shape, and a low excitation energy of 1839 keV, which favors a long lifetime. The study also predicts a very low-lying Kπ = 25/2⁻ 3-qp state in ¹⁸⁷Tl, which could act as a "spin trap" and presents a challenge for future experimental detection.

In conclusion, this research provides a consistent and systematic description of high-K isomers in the N = 106 chain, highlighting the crucial role of unpaired nucleons in driving shape polarization and revealing a complex landscape of shape coexistence in neutron-deficient odd-mass nuclei near shell closures. The predictions offered herein serve as a strong motivation and guide for future spectroscopic studies.

V. Bibliography

[1] P. Walker, G. Dracoulis, Energy traps in atomic nuclei, Nature 399, 35 (1999). https://doi.org/10.1038/19911
[2] P. Walker, Z. Podolyák, 100 years of nuclear isomers—then and now, Phys. Scr. 95, 044004 (2020). https://doi.org/10.1088/1402-4896/ab635d
[3] J. Van Klinken, W. Venema, R. Janssens, G. t. Emery, K-forbidden decays in ¹⁷⁸Hf; M4 decay of an yrast state, Nucl. Phys. A 339, 189 (1980). https://doi.org/10.1016/0375-9474(80)90086-X
[4] S. Mullins, G. Dracoulis, A. Byrne, T. McGoram, S. Bayer, W. Seale, F. Kondev, Rotational band on the 31 yr 16⁺ isomer in ¹⁷⁸Hf, Phys. Lett. B 393, 279 (1997). https://doi.org/10.1016/S0370-2693(96)01660-7
[5] F. Xu, P. Walker, R. Wyss, Shape polarizations of two-quasiparticle Kπ = 8⁻ isomeric configurations, Phys. Rev. C 59, 731 (1999). https://doi.org/10.1103/PhysRevC.59.731
[6] K. Heyde, P. Van Isacker, M. Waroquier, J. Wood, R. Meyer, Coexistence in odd-mass nuclei, Phys. Rep. 102, 291 (1983). https://doi.org/10.1016/0370-1573(83)90085-6
[7] K. Heyde, L. Wood, Shape coexistence in atomic nuclei, Rev. Mod. Phys. 83, 1467 (2011). https://doi.org/10.1103/RevModPhys.83.1467
[8] P. E. Garrett, M. Zielińska, E. Clément, An experimental view on shape coexistence in nuclei, Prog. Part. Nucl. Phys. 124, 103931 (2022). https://doi.org/10.1016/j.ppnp.2021.103931
[9] A. Andreyev, M. Huyse, P. Van Duppen, L. Weissman, D. Ackermann, J. Gerl, F. Hessberger, S. Hofmann, A. Kleinböhl, G. Münzenberg, et al., A triplet of differently shaped spin-zero states in the atomic nucleus ¹⁸⁶Pb, Nature 405, 430 (2000). https://doi.org/10.1038/35013012
[10] J. Ojala, J. Pakarinen, P. Papadakis, J. Sorri, M. Sandzelius, D. M. Cox, K. Auranen, H. Badran, P. J. Davies, T. Grahn, et al., Reassigning the shapes of the 0⁺ states in the ¹⁸⁶Pb nucleus, Commun. Phys. 5, 213 (2022). https://doi.org/10.1038/s42005-022-00990-4
[11] G. Dracoulis, G. Lane, A. Byrne, A. Baxter, T. Kibedi, A. Macchiavelli, P. Fallon, R. Clark, Isomer bands, E0 transitions, and mixing due to shape coexistence in ¹⁸⁸₈₂Pb₁₀₆, Phys. Rev. C 67, 051301 (2003). https://doi.org/10.1103/PhysRevC.67.051301
[12] Y. L. Coz, F. Becker, H. Kankaanpää, W. Korten, E. Mergel, P. Butler, J. Cocks, O. Dorvaux, D. Hawcroft, K. Helariutta, et al., Evidence of multiple shape-coexistence in ¹⁸⁸Pb, EPJ direct 1, 1 (2000). https://doi.org/10.1007/s1010599a0003
[13] J. Egido, L. Robledo, R. Rodríguez-Guzmán, Unveiling the origin of shape coexistence in Pb isotopes, Phys. Rev. Lett. 93, 082502 (2004). https://doi.org/10.1103/PhysRevLett.93.082502
[14] A.-M. Oros, K. Heyde, C. De Coster, B. Decroix, R. Wyss, B. Barrett, P. Navratil, Shape coexistence in the light Po isotopes, Nucl. Phys. A 645, 107 (1999). https://doi.org/10.1016/S0375-9474(98)00602-2
[15] W. Nazarewicz, M. Riley, J. Garrett, Equilibrium deformations and excitation energies of single-quasiproton band heads of rare-earth nuclei, Nucl. Phys. A 512, 61 (1990). https://doi.org/10.1016/0375-9474(90)90004-6
[16] P. Walker, G. Dracoulis, A. Byrne, T. Kibédi, A. Stuchbery, Kπ = 6⁺ and 8⁻ isomer decays in ¹⁷²Hf and ΔK = 8 E1 transition rates, Phys. Rev. C 49, 1718 (1994). https://doi.org/10.1103/PhysRevC.49.1718
[17] G. Dracoulis, G. Lane, F. G. Kondev, H. Watanabe, D. Seweryniak, S. Zhu, M. Carpenter, C. Chiara, R. Janssens, T. Lauritsen, et al., Lifetime of the Kπ = 8⁻ isomer in the neutron-rich nucleus ¹⁷⁴Er, and N = 106 E1 systematics, Phys. Rev. C 79, 061303 (2009). https://doi.org/10.1103/PhysRevC.79.061303
[18] G. D. Dracoulis, A. P. Byrne, A. M. Baxter, P. M. Davidson, T. Kibédi, T. R. McGoram, R. A. Bark, S. M. Mullins, Spherical and deformed isomers in ¹⁸⁸Pb, Phys. Rev. C 60, 014303 (1999). https://doi.org/10.1103/PhysRevC.60.014303
[19] G. D. Dracoulis, G. J. Lane, A. P. Byrne, A. M. Baxter, T. Kibédi, A. O. Macchiavelli, P. Fallon, R. M. Clark, Isomer bands, E0 transitions, and mixing due to shape coexistence in ¹⁸⁸₈₂Pb₁₀₆, Phys. Rev. C 67, 051301 (2003). https://doi.org/10.1103/PhysRevC.67.051301
[20] R. Hughes, G. Lane, G. Dracoulis, A. Byrne, P. Nieminen, H. Watanabe, M. Carpenter, P. Chowdhury, R. Janssens, F. G. Kondev, et al., High-spin structure, K isomers, and state mixing in the neutron-rich isotopes ¹⁷³Tm and ¹⁷⁵Tm, Phys. Rev. C 86, 054314 (2012). https://doi.org/10.1103/PhysRevC.86.054314
[21] G. Dracoulis, F. G. Kondev, G. Lane, A. Byrne, T. Kibedi, I. Ahmad, M. Carpenter, S. Freeman, R. Janssens, N. Hammond, et al., Identification of yrast high-K isomers in ¹⁷⁷Lu and characterisation of ¹⁷⁷mLu, Phys. Lett. B 584, 22 (2004). https://doi.org/10.1016/j.physletb.2004.01.056
[22] D. Barnéoud, S. André, C. Foin, Multi-quasiparticle yrast states in ¹⁷⁹Ta, Nucl. Phys. A 379, 205 (1982). https://doi.org/10.1016/0375-9474(82)90564-4
[23] C. Pearson, P. M. Walker, C. Purry, G. Dracoulis, S. Bayer, A. Byrne, T. Kibedi, F. Kondev, Multi-quasiparticle isomers and rotational bands in ¹⁸¹Re, Nucl. Phys. A 674, 301 (2000). https://doi.org/10.1016/S0375-9474(00)00143-3
[24] A. P. Byrne, A. M. Baxter, G. D. Dracoulis, S. M. Mullins, T. Kibédi, T. R. McGoram, K. Helariutta, J. F. C. Cocks, P. Jones, R. Julin, et al., Microsecond isomers in ¹⁸⁷Tl and ¹⁸⁸Pb, Eur. Phys. J. A 7, 41 (2000). https://doi.org/10.1007/s100500050008
[25] C. Guo, W. Zhang, H. Huang, S. Zhang, Z. Li, Z. Liu, H. Hua, D. Luo, H. Wu, A. Andreyev, et al., Spectroscopy of ¹⁸⁷Tl: shape coexistence, Eur. Phys. J. A 60, 165 (2024). https://doi.org/10.1140/epja/s10050-024-01395-3
[26] X. Y. Fu, Z. Liu, A. N. Andreyev, C. F. Jiao, R. Y. Dong, W. Q. Zhang, H. Huang, W. Sun, Q. B. Chen, S. Q. Zhang, et al., Observation of a new isomer in ¹⁸⁵Au, Chin. Phys. C 49, 011001 (2025). https://doi.org/10.1088/1674-1137/add70c
[27] F. Xu, P. Walker, J. Sheikh, R. Wyss, Multi-quasiparticle potential-energy surfaces, Phys. Lett. B 435, 257 (1998). https://doi.org/10.1016/S0370-2693(98)00857-0
[28] W. D. Myers, W. J. Swiatecki, Nuclear masses and deformations, Nucl. Phys. 81, 1 (1966). https://doi.org/10.1016/0029-5582(66)90639-0
[29] W. D. Myers, W. Swiatecki, Average nuclear properties, Ann. Phys. 55, 395 (1969). https://doi.org/10.1016/0003-4916(69)90202-4
[30] V. Strutinsky, Shell effects in nuclear masses and deformation energies, Nucl. Phys. A 95, 420 (1967). https://doi.org/10.1016/0375-9474(67)90510-6
[31] W. Nazarewicz, J. Dudek, R. Bengtsson, T. Bengtsson, I. Ragnarsson, Microscopic study of the high-spin behaviour in selected A ≈ 80 nuclei, Nucl. Phys. A 435, 397 (1985). doi:https://doi.org/10.1016/0375-9474(85)90471-3.
[32] J. Dudek, Z. Szymański, T. Werner, Woods-saxon potential parameters optimized to the high spin spectra in the lead region, Phys. Rev. C 23, 920 (1981). https://doi.org/10.1103/PhysRevC.23.920
[33] H. Pradhan, Y. Nogami, J. Law, Study of approximations in the nuclear pairing-force problem, Nucl. Phys. A 201, 357 (1973). https://doi.org/10.1016/0375-9474(73)90071-7
[34] P. Möller, J. Nix, Nuclear pairing models, Nucl. Phys. A 536, 20 (1992). https://doi.org/10.1016/0375-9474(92)90244-E
[35] P. Möller, J. R. Nix, W. Myers, W. Swiatecki, Nuclear ground-state masses and deformations, At. Data Nucl. Data Tables 59, 185 (1993). https://doi.org/10.1006/adnd.1995.1002
[36] Y. Shi, F. Xu, H. Liu, P. Walker, Extending shape coexistence in A ∼190 nuclei: multi-quasiparticle excitation, Phys. Rev. C 82, 044314 (2010). https://doi.org/10.1103/PhysRevC.82.044314
[37] F. G. Kondev, G. Dracoulis, T. Kibedi, Configurations and hindered decays of K isomers in deformed nuclei with A > 100, At. Data Nucl. Data Tables 103, 50 (2015). https://doi.org/10.1016/j.adt.2015.01.001
[38] G. Løvhøiden, P. Andersen, D. Burke, E. Flynn, J. Sunier, Single-proton states in ¹⁷⁵Tm studied with the (t, α) reaction, Nucl. Phys. A 327, 64 (1979). https://doi.org/10.1016/0375-9474(79)90318-X
[39] F. Kondev, S. Zhu, M. Carpenter, R. Janssens, I. Ahmad, C. Chiara, J. Greene, T. Lauritsen, D. Seweryniak, S. Lalkovski, et al., M3 and E4 K-forbidden decays of the Kπ = 23/2⁻ isomer in ¹⁷⁷Lu, Phys. Rev. C 85, 027304 (2012). https://doi.org/10.1103/PhysRevC.85.027304
[40] F. Kondev, G. Dracoulis, A. Byrne, T. Kibédi, S. Bayer, Multi-quasiparticle states in ¹⁷⁹Ta and structural changes in the yrast line of the odd tantalum isotopes, Nucl. Phys. A 617, 91 (1997). https://doi.org/10.1016/S0375-9474(97)00007-9
[41] F. G. Kondev, G. Dracoulis, G. Lane, I. Ahmad, A. Byrne, M. Carpenter, P. Chowdhury, S. Freeman, N. Hammond, R. Janssens, et al., K-mixing and fast decay of a seven-quasiparticle isomer in ¹⁷⁹Ta, Eur. Phys. J. A 22, 23 (2004). https://doi.org/10.1140/epja/i2004-10025-9
[42] M. Pfützner, P. Regan, P. Walker, M. Caamano, J. Gerl, M. Hellström, P. Mayet, K.-H. Schmidt, Z. Podolyak, M. Mineva, et al., Angular momentum population in the fragmentation of ²⁰⁸Pb at 1 GeV/nucleon, Phys. Rev. C 65, 064604 (2002). https://doi.org/10.1103/PhysRevC.65.064604
[43] U. Georg, W. Borchers, M. Keim, A. Klein, P. Lievens, R. Neugart, M. Neuroth, P. M. Rao, C. Schulz, the ISOLDE Collaboration, Laser spectroscopy investigation of the nuclear moments and radii of lutetium isotopes, Eur. Phys. J. A 3, 225 (1998). https://doi.org/10.1007/s100500050172
[44] I. Ladenbauer-Bellis, H. Bakhru, P. Sen, Some decay properties of ¹⁸³Ir, Nucl. Phys. A 252, 524 (1975). https://doi.org/10.1016/0375-9474(75)90517-5
[45] M. Desthuilliers, C. Bourgeois, P. Kilcher, J. Letessier, F. Beck, T. Byrski, A. Knipper, Structure of the collective bands in the ¹⁸⁵Au nucleus, Nucl. Phys. A 313, 221 (1979). https://doi.org/10.1016/0375-9474(79)90577-3
[46] V. Berg, Z. Hu, J. Oms, C. Ekström, Transition rates between negative-parity states in odd-mass gold nuclei, Nucl. Phys. A 410, 445 (1983). https://doi.org/10.1016/0375-9474(83)90637-1
[47] A. Larabee, M. Carpenter, L. Riedinger, L. Courtney, J. Waddington, V. Janzen, W. Nazarewicz, J.-Y. Zhang, R. Bengtsson, G. Leander, Shape coexistence and alignment processes in the light Pt and Au region, Phys. Lett. B 169, 21 (1986). https://doi.org/10.1016/0370-2693(86)90678-7
[48] K. Wallmeroth, G. Bollen, A. Dohn, P. Egelhof, U. Krönert, J. Campos, A. Yunta, K. Heyde, C. De Coster, M. Borge, J. Wood, H.-J. Kluge, Nuclear shape transition in light gold isotopes, Nucl. Phys. A 493, 224 (1989). https://doi.org/10.1016/0375-9474(89)90396-5
[49] G. Lane, G. Dracoulis, A. Byrne, P. Walker, A. Baxter, J. Sheikh, W. Nazarewicz, Shape coexistence in ¹⁸⁵Tl and ¹⁸⁷Tl—investigation of the deformed minima, Nucl. Phys. A 586, 316 (1995). https://doi.org/10.1016/0375-9474(94)00515-
[50] A. Lee, G. Lane, G. Dracoulis, A. Macchiavelli, P. Fallon, R. Clark, F. Xu, D. Dong, Observation of new h₉/₂ and h₁₁/₂ bands in ¹⁸⁷Tl, EPJ Web of Conferences, 35, 06002 (2012). https://doi.org/10.1051/epjconf/20123506002
[51] S. Chamoli, P. Joshi, A. Kumar, R. Kumar, R. Singh, S. Muralithar, R. Bhowmik, I. Govil, Shape coexistence and lifetime measurement in ¹⁸⁷Tl nucleus, Phys. Rev. C 71, 054324 (2005). https://doi.org/10.1103/PhysRevC.71.054324
[52] W. Reviol, L. Riedinger, J.-Y. Zhang, C. Bingham, W. Mueller, B. Zimmerman, R. Janssens, M. Carpenter, I. Ahmad, I. Bearden, et al., Prolate collectivity in ¹⁸⁷Tl, Phys. Rev. C 49, R587 (1994). https://doi.org/10.1103/PhysRevC.49.R587
[53] A. Barzakh, L. K. Batist, D. Fedorov, V. Ivanov, K. Mezilev, P. Molkanov, F. Moroz, S. Y. Orlov, V. Panteleev, Y. M. Volkov, Changes in the mean-square charge radii and magnetic moments of neutron-deficient Tl isotopes, Phys. Rev. C 88, 024315 (2013). https://doi.org/10.1103/PhysRevC.88.024315
[54] A. Barzakh, A. Andreyev, T. E. Cocolios, R. De Groote, D. Fedorov, V. Fedosseev, R. Ferrer, D. Fink, L. Ghys, M. Huyse, et al., Changes in mean-squared charge radii and magnetic moments of ¹⁷⁹⁻¹⁸⁴Tl measured by in-source laser spectroscopy, Phys. Rev. C 95, 014324 (2017). https://doi.org/10.1103/PhysRevC.95.014324
[55] G. D. Dracoulis, G. J. Lane, A. P. Byrne, H. Watanabe, R. O. Hughes, N. Palalani, F. G. Kondev, M. Carpenter, R. V. F. Janssens, T. Lauritsen, et al., Long-lived three-quasiparticle isomers in ¹⁹¹Ir and ¹⁹³Ir with triaxial deformation, Phys. Lett. B 709, 59 (2012). https://doi.org/10.1016/j.physletb.2012.02.008