Preamble

Shape polarization and coexistence of high- three-quasiparticle states in odd-mass = 106 isotones Runyan Dong and Changfeng Jiao 1, 2, 1 School of Physics and Astronomy, Sun Yat-sen University, Zhuhai 519082, China Guangdong Provincial Key Laboratory of Quantum Metrology and Sensing, Sun Yat-sen University, Zhuhai 519082, China Three-quasiparticle -isomeric states in odd-mass = 106 isotones within the mass region are systematically investigated using configuration-constrained potential energy surface calculations. The calcula- tions successfully reproduce the excitation energies and deformations of known high- isomers in the nuclei Tm to Re. For the nuclei closer to the shell closure ( Au, and Tl), predic- tions 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 where several high- three-quasiparticle configurations with distinct prolate, oblate, and triaxial shapes are found to coexist at similar excitation energies. Notably, the oblate-deformed configuration at = 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- multi-quasiparticle states, offering valuable insights for future experimental research.

Keywords

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

INTRODUCTION

Neutron-deficient isotones around N = 106 mid-shell are 2

characterized by the existence of an abundance of low-lying,

high seniority isomeric states. In this region, the orbitals with 4

large , namely the projection of individual angular momen- tum onto the intrinsic symmetric axis, approach the neutron Fermi surface at moderate quadrupole deformations. It facil- itates the formation of broken-pair states with high values

(where K = �

) near the yrast line. According to the se- lection rules for electromagnetic transitions, the transitions of

multipolarity λ would be significantly hindered if ∆ K > λ . 11

The so-called -hindrance can lead to relatively long half- lives (on the order of nanoseconds or longer) [ ], leading to the formation of “ isomers”. One of the most well-known examples of high- isomers is found in , in which two

2-quasiparticle (qp) K π = 8 − isomers and a long-lived four- 16

quasiparticle K π = 16 + isomer with a half-life of 31 years 17

are observed [ ]. Since the occurrence of isomers is a combined effect of the unpaired nucleons occupied high- states and the nuclear deformation, the study of isomeric states is therefore pivotal for understanding the interplay be- tween the shell structure of “individual” nucleons and the col- lective behavior of a strongly correlated nucleus [ Another common feature associated with the mass region is the shape transition. In this region, the ground- state (g.s.) shape can change from a well-deformed prolate ellipsoid with Yb to a very soft spheroid ]. Moreover, the soft shape gives rise to the novel shape coexistence phenomenon, which is characterized by the emergence of low-lying states with different intrin- sic shapes in one atomic nucleus. In general, it originates Supported by the National Natural Science Foundation of China (No.12275369)

from the combining effect of approaching the Z = 82 spher- 32

ical shell closures and the deformed shell gaps around the

neutron mid-shell at N = 104 − 106 due to quadrupole- 34

quadrupole correlations. It has drawn considerable interest The most well-known example is the differently shaped triplet observed in Pb, which corresponds to the coexistence of the prolate, oblate, and spherical configu- rations [ ]. Coexisting states in even-even Pt, Hg, Pb, and Po isotopes around the neutron midshell have been extensively studied [ In addition to the shape change resulting from collective correlations such as quadrupole-quadrupole interactions, un- paired nucleons are found to strongly polarize the nuclear shape [ ]. Since isomeric states are coupled by high- unpaired nucleons, shape polarization possibly yields con- siderable differences in shape between high- states and ground states, leading to novel structures that involve both isomerism and shape isomerism. For example, the two-

quasineutron K π = 8 − ( ν { 7 / 2 − [514] ⊗ 9 / 2 + [624] } ) iso- 50

meric states which are systematically observed in the even-

even N =106 isotones between 174 Er and 188 Pb (see [ 24 , 25 ] 52

and references therein). Previous theoretical investigation has shown that the the g.s. are oblate deformed with

for 186 Hg and spherical for 188 Pb, whereas the K = 8 − iso- 55

meric states are polarized to prolate deformed with

0 . 25 . The K = 8 − isomers with shapes different from those 57

of the g.s. have later been confirmed by measuring rotational bands built on them [ ]. 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 proper- ties [ While the shape evolution and coexistence of high- states in even-even nuclei around neutron mid-shell and have been extensively studied, the structural properties such as the shape changing effects of 3-qp high- states in their odd-proton neighbors are lacking systematic investigations.

In odd- 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 op- posed to that of the high- 2-qp configuration, thereby am-

plifying or diminishing the shape difference between the 3- 74

qp states and the g.s.. Recently, the 3-qp high- isomers, originated from the coupling between the odd proton and the

aforementioned K π = 8 − configuration in even-mass cores, 77

have been observed in odd-mass N = 106 isotones from 78

175 Tm to

Tl (except the Ir) [ 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 [ ]. The ex- tent to which the observed 3-qp isomers can be considered to involve shape isomerism, in addition to isomerism, re- mains unclear. It thus has greatly stimulated our interest in pursuing theoretical studies on shape polarization and coex- istence in the 3-qp high- states within this mass region.

In this work, we investigate the 3-qp -isomeric states

of odd- A nuclei in the N = 106 isotonic chain using 89

the configuration-constrained potential energy surface (PES) method [ This method includes the axially asymmetric -degree of freedom. Furthermore, in this method, we do not introduce an adjustable parameter of the pairing strength,

and the deformation is determined self-consistently by mini- 94

mizing the corresponding PES. At prolate deformations, we mainly focus on the study of the high- 3-qp configurations composed of the coupling of the unpaired proton and the

K π = 8 − 2-qp configuration that are observed systematically 98

in even-even N = 106 nuclei. We predicted the possible con- 99

figurations of the isomers in Au and Tl, with particular attention to the shape-polarization effect from multi-qp exci- tations. Furthermore, we explored the high- states with dis- tinct shapes (oblate, prolate, and triaxial) that coexist at com- parable excitation energies in Tl, and analyzed the impact of different quasiparticle configurations on shape evolution in detail.

THE MODEL employ configuration-constrained proach [ ], based on the macroscopic–microscopic model.

The macroscopic energy contribution was computed using the standard liquid-drop model [ ] with parameters taken from Ref. [ The microscopic correction includes the Strutinsky shell correction [ ] and the pairing correction.

The single-particle levels required for the microscopic energy calculations were obtained from a non-axial deformed

Woods-Saxon potential [ 41 ] using the “universal” parameter 116

set [ 42 ]. The so-called universal parameter set (listed 117

in Table ) was optimized by simultaneously fitting the single-particle energies in Pb, particularly those corrected for nucleon-nucleus interactions beyond the independent particle model, as well as the high-spin yrast spectra of

212 Rn and

]. These parameters have been further tested for light nuclei [ ], and for heavier nuclei with ], showing satisfactory performance in describing not only the single (quasi)particle level sequences but also the nuclear equilibrium deformations.

Since these parameters were optimized for the lead region, they have been validated as the best-fit Woods-Saxon potential parameters for this mass region.

To avoid the collapse of pairing correlations in multi- quasiparticle states, we used the Lipkin-Nogami (LN) ver- an approximate particle-number projection scheme, incorpo-

rating monopole pairing. The pairing strength G was initially 134

determined via the average-gap method [ ]. Although it is often further adjusted to reproduce the odd-even mass difference using a five-point formula, we note that irregu- larities may arise near magic numbers (e.g., Au and Tl iso- approach and the artefacts from the LN correction. It is also because the odd-even mass difference equations used to ex- tract experimental pairing gaps are derived under the assump- tion that there are no non-smooth contributions to the masses apart from pairing effects, while this assumption is often not fulfilled at magic numbers [ Therefore, following the rec- ommendation of Ref. [ ], closed-shell nuclei were excluded from the pairing strength calibration. For consistency, we adopted the standard pairing strength across all isotopes un- In the PES calculations, a deformation mesh in used, with the hexadecapole deformation variation at each mesh point. For broken-pair configurations, the microscopic energy incorporates contributions from unpaired nucleons oc- cupying specific single-particle orbitals (see Ref. [ ] for de- tails).

These orbitals are continuously tracked and adiabati- cally blocked throughout the deformation plane. For axially deformed nuclei, single-particle orbitals can be spec- ified by their individual spin projection . For axially asym- metric shapes, however, is no longer conserved. In ad- dition, several single-particle orbits with approximately the same spin projection may become energetically close at given deformations. To reliably track orbits, we com- puted and compared not only the average spin projection but also the expectation values of the Nilsson numbers , and between two adjacent deformation mesh points.

We have verified that, although the Nilsson numbers , and are not strictly conserved, their expectation val- ues exhibit slow variation, allowing for a reliable configura- tion 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 origi-

nates 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 177

an equilibrium deformation for the multi-qp state that differs from that of the ground state. The configuration-constrained PES method effectively accounts for this polarization caused by the unpaired nucleon and offers a self-consistent descrip- tion of both the deformation and excitation energy of multi- qp states [ ]. The excitation energy is computed as the

energy difference between the PES minimum of the excited 184

configuration and that of the ground-state configuration, en- abling direct comparison with experimental values.

CALCULATIONS AND DISCUSSIONS Systematics of 3-qp states involved [624] [514] For nuclei in the region, an abundance of high- isomeric states has been discovered [ ]. Among them, the

two-quasineutron K π = 8 − ( ν { 7 / 2 − [514] ⊗ 9 / 2 + [624] } ) 192

isomeric states exist systematically in even-even N = 106 193

isotones, which have been investigated by means of the configuration-constrained PES calculation in Ref. [ ]. The calculated excitation energies agree well with the experi- mental data, and strong shape polarizations have been found

when approaching the Z = 82 shell closure. For odd-mass 198

N = 106 isotones in this mass region, most of the observed 199

isomers consistently involve a two-quasineutron con-

figuration coupled to K π = 8 − states that are identified in 201

the aforementioned even-even nuclei. For example, in nuclei such as ], 3-qp isomers have been assigned as

the two-quasineutron K π = 8 − configuration coupled to the 205

energetically lowest one-quasiproton configuration. Further-

more, in 181 Re, 179 Ta, and 177 Lu, the meta-stable K π = 207

states are found, assigned to the [514] config- uration, leading to the presence of 3-qp states associated with

the coupling of a K π = 8 − two-quasineutron configuration 210

[514] ]. In Tm, a isomer which may

involve coupling of the two-quasineutron K π = 8 − config- 212

uration with the [523] has been found [ ]. The half- lives of these isomers range from a few microseconds to sev- eral days. To what extent these high- states are associated with shape polarization and shape isomerism is still an open question.

We have performed the configuration-constrained PES cal-

culations on the 1-quasiproton and 3-qp states in N = 106 219

odd-mass isotones. Table presents the calculated defor- mations and energies of the g.s., the possible high- quasiproton and low-lying high- 3-qp states, compared with the available experimental data. Our calculations repro- duce the experimentally assigned spin-parity of the g.s. of these nuclei, except for Lu, in which the calculated low- est 1-quasiproton configuration is the [514] rather than the experimentally assigned [404] ]. However, the calculated [404] configuration lies only 259 keV above [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.

One may ask the sensitivity of these calculated results to the pairing strengths or the Woods–Saxon parameters. Note that previous work has shown that adjustment of pairing strength mainly influences the quasiparticle energy gaps and only slightly affects the deformations [ ], while the ordering of single-particle levels at a certain deformation is primarily determined by the choice of the Woods–Saxon potential pa- rameters. To examine the robustness of this result, we there- fore have tested with several known parameter sets, includ- ing those of Blomquist and Wahlborn [ ], the parameters of Chepurnov [ ], the parameters given by Rost [ ], and the “new” parameters [ ]. We found that the deviation in the relative positions of the [514] [404] config- urations of Lu is preserved with different parameter sets.

To accurately reproduce the ordering of the low-lying 1-qp states of Lu, an improvement of the Woods-Saxon poten- tial is needed.

Experimentally, the spin and parity of the g.s. of ] have been assigned to be the states built on the [541] configuration, while a strong mix- ing between the [541] [532] configurations attributed to the Coriolis interactions has been proposed for g.s. of ]. Our calculations show that [541] configuration has the lowest energy, while [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 configuration-constrained PES calculations also reasonably reproduce the high- 1-qp isomeric states ob-

served in odd-mass N = 106 isotones except for the afore- 265

mentioned deviation in Lu. Notably, the configuration-

constrained PES calculations show that the K π = 9 / 2 − iso- 267

mer of 185 Au and the K π = 11 / 2 − state of 187 Tl have mod- 268

erate triaxial deformations with which are remark- ably 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 = 82 272

closed shell. Now we turn to the investigation of energetically low-

lying 3-qp states in odd-mass N = 106 isotones. We 275

mainly focus on the 3-qp states that consist of the two- quasineutron [514] [624] configuration cou- pled to different 1-quasiproton configurations, since the

K π = 8 − isomeric states are systematically identified as 279

[514] [624] configuration in even-even

N = 106 isotones. As seen in Table 2 [TABLE:2] , the calculated en- 281

ergies of these 3-qp states in Ta, and Re agree well with the experimental data. For Lu, the calculated

K π = 23 / 2 − π 7 / 2 + [404] ⊗ ν { 9 / 2 + [624] ⊗ 7 / 2 − [514] } 284

configuration is overpredicted, while the K π = 25 / 2 + 285

and excitation energies for odd- nuclei in the = 106 isotopic chain. The experimental energies can be found in Refs. [ ] and references therein.

183 Ir

[541] [514] [541] [624] [514] [514] [624] [514]

185 Au

[541] [411] [532] [505] [541] [624] [514] [532] [624] [514]

187 Tl

[400] [514] [624] [514] state is slightly underestimated. This can be attributed to the deviation of the [404] [514] orbitals that we found in the cal- culations of the 1-quasiproton states.

Furthermore, the configuration-constrained PES calcula- tions predict the candidate configurations of the 3-qp isomeric

states in the odd- A N = 106 isotones when moving towards 292

the Z = 82 shell closure. To date, no experimental evidence 293

has been reported on three-quasiparticle high- isomers in 183 Ir. We proposed two possible high- 3-qp states that are [505] [505] [624] [514]

β 2 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 K π = 8 −

configuration [514] [624] with the g.s. configurations of odd- nuclei, which are [411] [404] [404] [402] [541] , and [541] , respectively. For , the 3-qp state is [505] [624] [514] These 3-qp states correspond to observed high- isomers. [514] , respectively. For Au, a new isomer at an exci- tation energy of 1504.2(4) keV with a half-life of 630(80) ns possible spins of this isomer are constrained to a range from 13/2 to 21/2 in comparison with predictions from the Weis- skopf estimates [ ]. Our calculations suggest two possible 3- qp high- configurations that are consistent with systematics

of 3-qp configurations in lighter odd-mass N = 106 isotones. 305

They both lie at excitation energies of about 1900 keV, which is a bit overpredicted. However, Ref. [ ] argued that the g.s. configuration of Au is more likely [532] , and the calculated energy differences of these two 3-qp states with re- spect to the [532] configuration are 1444 and 1458 keV, respectively, which is in great agreement with the observa- tion. For nucleus Tl, two isomers with microsecond life-

times ( T 1 / 2 = 1 . 11 µ s and 0.69 µ s) have been reported [ 32 ]. 313

Spin-parities J π = 27 / 2 + , 31 / 2 − are tentatively assigned 314

to the isomer lies at 2584 keV with lifetime T 1 / 2 = 0 . 69 315

s based on the deduced total conversion coefficient [

Our calculation presents a K π = 27 / 2 + , π 11 / 2 − [505] ⊗ 317

[624] [514] configuration with an excitation energy of 2312 keV, which is in accord with the prolate high- configuration suggested in Ref. [ ]. This implies that

the T 1 / 2 = 0 . 69 µ s isomer observed in 187 Tl involves the 321

two-quasineutron [624] [514] configuration, which is consistent with the systematics of 3-qp isomers ob-

served in lighter odd-mass N = 106 isotones. However, fur- 324

ther experimental data are required to unambiguously assign the spin-parity and configuration of these observed isomers.

We would discuss the other T 1 / 2 = 1 . 1 µ s isomer later in 327

Sect. III B In addition, intrinsic shape evolution is crucial for un- derstanding the observed behavior of isomeric states, such as their decay properties. In the previous study [ ], strong shape polarization has been shown in even-even nuclei with

A ∼ 180 and N = 106 , especially in nuclei close to the 333

Z = 82 shell closure. For systematic comparison, we plot 334

the variation of deformations of the high- qp states and the g.s. along with the proton number

When approaching the shell closure of Z = 82 , 337

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. g.s. of Au, for example, has a remarkably -soft shape

with a very shallow PES minimum at γ ≈ 24 ◦ . Consid- 342

ering that the g.s. of Au can be interpreted as the cou- pling of a low- proton with the Pt core, the PES of the g.s. of Au exhibits similar softness com- pared to Pt. This -soft nature of Pt is mainly because the prolate-to-oblate shape transition going through a transi-

tional γ -soft shape occurs in Pt isotopes around N = 110 , 348

which has been suggested by the self-consistent HFB calcu- lation using Gogny-D1S interaction and the interacting bo- son model [ ], as well as the five-dimensional collective

Hamiltonian (5DCH) based on covariant density-functional 352

theory [ In contrast, the predicted two 3-qp high- states of Au both have an approximately prolate shape and about a increase in deforma- tion.

The nucleus Tl has a spherical g.s. with a proton that singly occupies the orbital, while the calculated

K π = 27 / 2 + π 11 / 2 − [505] ⊗ ν { 9 / 2 + [624] ⊗ 7 / 2 − [514] } 358

state is predicted to have a moderately axially-asymmetric shape with , exhibiting the great- est difference in quadrupole deformation between the 3-qp state and the g.s.. In fact, the calculated high- 3-qp con- figurations of Tl present an ensemble of multiple nuclear shapes, which would be analyzed in detail in Sect.

III B Shape coexistence in high- 3-qp states of 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, 369

the unpaired nucleon that occupies different single-particle orbitals would polarize the shape of the odd- nucleus in dif- ferent ways, leading to more profound shape coexistence phe- nomenon. An abundance of experimental data has demon-

strated that differently-shaped configurations in these odd- nuclei are observed not only in the low-lying 1-qp states but

also in the higher-seniority isomeric states [ 9 ]. 376

Tl, previous studies [ ] have proposed the co- existence of different nuclear shapes by the analysis of ob- served low-lying collective structures.

As the proton-hole

neighbor of 188 Pb, the I = 1 / 2 + g.s. of 187 Tl can be in- 380

terpreted as the coupling of the hole with the spher-

ical 0 + 1 state of 188 Pb core. The observed K π = 9 / 2 − and 382

K π = 13 / 2 + isomeric states can be understood as filling 383

[505] [606] intruder orbitals that are lowered along with the increase in oblate deformation, re- spectively [ ], while the rotational band built on the

I π = 11 / 2 − state is suggested to be the prolate deformed 387

[505] configuration [ ]. The calculated defor- mations and excitation energies for these 1-quasiproton states are listed in Table . The configuration-constrained PES cal- culations well reproduce the measured excitation energies, and clearly show the coexisting shapes for these 1-qp states.

Note that the [505] configuration has been predicted to have a considerable axial-asymmetric shape of which breaks down the - and shape-hindrance and would

explain why it decays fast to the oblate K π = 9 / 2 − isomeric 396

state [ In addition to the high- proton orbitals mentioned above, other deformation-driving high- high- orbitals, including the high- members of the proton shell, the high- members of neutron , and shells, would appear close to the proton and neutron Fermi surfaces at both oblate and prolate sides, respectively. The couplings of these high- orbitals would form energetically low-lying high- 3-qp configurations that are polarized to different shapes. We summarize the calculated deformations and excitation ener- gies of possible high- 3-qp configurations in Table . Co- existing different types of intrinsic shape are obtained for a variety of high- 3-qp configurations from the configuration- constrained PES calculations. Fig. depicts the typical PES’s corresponding to the 3-qp configurations with spherical, soft prolate, oblate, and axially asymmetric shapes, respec- tively.

Among them, the lowest prolate high- 3-qp state given

by the configuration-constrained PES is the K π = 27 / 2 + , 415

[505] [624] [514] configuration. The calculated PES of this configuration is shown in the panel (b) of the Fig. . As we discussed in Sect.

III A , this con- figuration is most likely to be assigned to the observed iso-

meric state with excitation energy E x = 2584 . 6 keV and life- 420

time T 1 / 2 = 0 . 69 µ s. Experimentally, it is found that the 421

2584.6-keV isomer decays to the I π = 25 + member of the 422

rotational band, which is assigned to the low- [660] configuration [ ]. To understand this transition, we also compute the deformation and excitation energy of the low- [660] state (see them in Table ). The calculated energy of the [660] state is 1216 keV, which is com- patible with the estimation of the bandhead energy of the ob- served rotational band. Note that both the [660] [505] [624] [514] configurations have very soft axially asymmetric deformations of -soft shape breaks down the -conservation and allows

the decay from the K π = 27 / 2 + state to the I π = 25 + state 433

built on the 2 + [660] configuration. The CCPES calcu-

lations predict another K π = 27 / 2 + configuration that con- 435

sists of [505] [505] [633] with a smaller and a larger . However, the large axially asymmetric deformation violates the -conservation, and hence may prohibit the formation of the isomeric state. potential energy surfaces (PES’s) of (a) the near-spherical g.s., (b) the prolate [505] [624] [514] state, (c) the oblate [606] [624] [633] state, (d) the triaxial [505] [633] [503] state.

The energy difference between neighboring contours is 100 keV. The black dots denote the minima of PES’s.

Another interesting 3-qp state that we predict is the K π = 440

[606] [624] [633] config- uration. As seen in the panel (c) of Fig. , its calculated PES

shows a minimum that appears at oblate deformation of β 2 ≈ 443

. The combination of high value, axially

symmetric shape, and its calculated low energy of E x = 1839 445

keV, supports the existence of a long-lived isomer. There-

fore, we suggest that this K π = 29 / 2 + configuration could 447

be assigned to the observed isomeric state with a lifetime

T 1 / 2 = 1 . 1 µ s, although the position and spin-parity of this 449

isomer cannot be firmly determined because the rays link- ing this isomer to low-lying states are still missing [ More recently, it has been found that this isomer decays to the low-lying isomer that lies at 1061 keV [ ]. This

implies that the T 1 / 2 = 1 . 1 µ s isomeric state may be oblate 454

deformed and be composed of the same [606] config-

uration, which is compatible with the predicted K π = 29 / 2 + , 456

[606] [624] [633] state. We thus propose that two long-lived 3-qp high- isomeric states with

and excitation energies for the g.s., the low-lying high- 1-qp, and high- 3-qp states of Tl. The experimental data can be found in Refs. [ ] and references therein. different shapes coexist with intermediate excitation energies Tl. Further measurement of observables, such as gy- romagnetic 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- states are predicted by the configuration-constrained PES calculations.

Among them, [505] [624] [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 I π = 13 / 2 + 470

state, the calculated state can be expected to form a “spin trap” [ ]. As Dracoulis et al ] have pointed out, a very low energy could result in long-lived states that would preferentially decay and thus be missed. It there- fore provides a challenge for both experimental and theoreti- cal studies and, in effect, a test of the reliability of the models.

SUMMARY

We present a systematic theoretical study of shape polar- ization and coexistence in high- 3-qp states of odd-mass

N = 106 isotones ( 175 Tm, 177 Lu, 179 Ta, 181 Re, 183 Ir, 185 Au, 480

187 Tl) using the configuration-constrained PES method.

The investigation focuses on 3-qp states formed by cou- pling a single proton to the systematic two-quasineutron

K π = 8 − isomeric configuration known in the even-even 484

N = 106 cores. The calculations demonstrate excellent 485

agreement with the experimental data for the well-established

isomers in lighter isotones ( Z = 69 − 75 ), validating the theo- 487

retical approach. As the proton number increases towards the

Z = 82 shell closure, the ground states become progressively 489

softer and less deformed, while the high- 3-qp states ex-

hibit significant shape polarization, maintaining well-defined 491

prolate deformations. This leads to a substantial shape differ- ence between the isomers and the ground states in nuclei like

185 Au and

Furthermore, we analyze in detail the intrinsic shapes of 3-qp states in Tl. The configuration-constrained PES cal- culations identify a multitude of low-lying high- config- urations with distinctly different shapes (including prolate oblate , and triaxial ) coexisting within a narrow energy range.

Two specific long-lived isomers observed in Tl are as-

signed to configurations with different shapes: the T 1 / 2 = 501

0 . 69 µ s isomer is associated with a prolate-deformed K π = 502

27 / 2 + state, while the T 1 / 2 = 1 . 1 µ s isomer is proposed to 503

be an oblate-deformed K π = 29 / 2 + state characterized by 504

a high 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 K π = 25 / 2 − 3-qp state in 187 Tl, 507

which could act as a “spin trap” and presents a challenge for future experimental detection.

In summay, this work presents a systematic description of

high- K isomers in the N = 106 isotonic chain. The calcu- 511

lated excitation energies and deformations are in reasonable agreement with available experimental data, highlighting the crucial role of unpaired nucleons in driving shape polariza- tion and revealing the possible coexistence of distinct shapes P. Walker and G. Dracoulis, Energy traps in atomic nuclei, Na- , 35 (1999).

P. Walker and Z. Podolyák, 100 years of nuclear iso- mers—then Phys. (2020).

P. M. Walker and F. G. Kondev, K isomers in atomic nuclei, Eur. Phys. J. Spec. Top. , 983–1005 (2024).

J. Van Klinken, W. Venema, R. Janssens, and G. t. Emery, -forbidden decays in Hf; M4 decay of an yrast state, Nucl. Phys. A , 189 (1980). 9474(80)90086-X S. Mullins, G. Dracoulis, A. Byrne, T. McGoram, S. Bayer, W. Seale, and F. Kondev, Rotational band on the 31 yr isomer in Hf, Phys. Lett. B , 279 (1997).

F. Xu, P. Walker, and R. Wyss, Shape polarizations of two-

quasiparticle K π = 8 − isomeric configurations, Phys. Rev. C 538

, 731 (1999). Q.-Z. Chai, W.-J. Zhao, H.-L. Wang, and M.-L. Liu, Possible observation of shape-coexisting configurations in even–even

midshell isotones with N = 104 : a systematic total 542

Routhian surface calculation, Nucl. Sci. Tech. , 38 (2018).

K. Heyde, P. Van Isacker, M. Waroquier, J. Wood, and R. Meyer, Coexistence in odd-mass nuclei, Phys. Rep. , 291 (1983).

Heyde Wood, Shape coexistence atomic nuclei, Phys. (2011).

Garrett, Zieli´nska, Clément, experimental shape coexistence clei, Prog.

Part. Nucl. Phys. (2022). X. Guan, J. Guo, Q.-W. Sun, B. Nerlo-Pomorska, and K.

Pomorski, On shape coexistence and possible shape isomers of nuclei around Hg, Nucl. Sci. Tech. , 128 (2025).

A. Andreyev, M. Huyse, P. Van Duppen, L. Weissman, D. Ack- ermann, J. Gerl, F. Hessberger, S. Hofmann, A. Kleinböhl, G. Münzenberg, et al ., A triplet of differently shaped spin-zero states in the atomic nucleus Pb, Nature , 430 (2000).

J. Ojala, J. Pakarinen, P. Papadakis, J. Sorri, M. Sandzelius, Auranen, Badran, Davies, T. Grahn, et al ., Reassigning the shapes of the states in the Pb nucleus, Commun. Phys. 213 (2022).

R. Bengtsson and W. Nazarewicz, Shape coexistence in the neutron deficient Pb isotopes and the configuration-constrained

in neutron-deficient odd-mass nuclei near the Z = 82 shell 516

closure. We hope that the present results can provide useful guidance for future spectroscopic investigations in this mass region.

BIBLIOGRAPHY shell correction approach, Z. Phys. A , 269 (1989).

Nazarewicz, Variety shapes mercury isotopes, Phys.

Lett. (1993). R. R. Chasman, J. L. Egido, and L. M. Robledo, Persis- tence of deformed shapes in the neutron-deficient Pb region, Phys. Lett. B , 325 (2001). 2693(01)00382-3 T. Nikši´c, D. Vretenar, P. Ring, and G. A. Lalazis- Shape coexistence relativistic Hartree- Bogoliubov approach, Phys. Rev. C , 054320 (2002).

G. Dracoulis, G. Lane, A. Byrne, A. Baxter, T. Kibedi, Macchiavelli, Fallon, Clark, Isomer bands, transitions, and mixing due to shape coex- istence in Phys. Rev. C 051301 (2003).

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 Pb, EPJ direct , 1 (2000).

Egido, Robledo, Rodríguez-Guzmán, veiling origin shape coexistence isotopes, Phys.

Lett. (2004). A.-M. Oros, Heyde, Coster, Decroix, R. Wyss, B. Barrett, P. Navratil, Shape coexistence in the light Po isotopes, Nucl. Phys. A , 107 (1999).

Z. Y. Wu, C. Qi, R. Wyss, and H. L. Liu, Global calculations of microscopic energies and nuclear deformations: Isospin de- pendence of the spin-orbit coupling, Phys. Rev. C , 024306 (2015).

W. Nazarewicz, M. Riley, J. Garrett, Equilibrium defor- mations and excitation energies of single-quasiproton band heads of rare-earth nuclei, Nucl. Phys. A , 61 (1990).

P. Walker, G. Dracoulis, A. Byrne, T. Kibédi, A. Stuch-

bery, K π = 6 + and 8 − isomer decays in 172 Hf and 610

∆ K = 8 E 1 transition rates, Phys. Rev. C 49 , 1718 (1994). 611

G. Dracoulis, G. Lane, F. G. Kondev, H. Watanabe, D. Sewery-

niak, S. Zhu, M. Carpenter, C. Chiara, R. Janssens, T. Laurit- 614

sen, et al., Lifetime of the K π = 8 − isomer in the neutron-rich 615

nucleus 174 Er, and N = 106 E 1 systematics, Phys. Rev. C 79 , 616

061303 (2009). G. D. Dracoulis, A. P. Byrne, A. M. Baxter, P. M. Davidson, T. Kibédi, T. R. McGoram, R. A. Bark, S. M. Mullins, Spher- ical and deformed isomers in , Phys. Rev. C , 014303

(1999). G. D. Dracoulis, G. J. Lane, A. P. Byrne, A. M. Baxter, T. Kibédi, A. O. Macchiavelli, P. Fallon, R. M. Clark, Iso- mer bands, transitions, and mixing due to shape co- existence in , Phys. Rev. C 67, 051301 (2003).

R. Hughes, G. Lane, G. Dracoulis, A. Byrne, P. Niem- inen, Watanabe, Carpenter, Chowdhury, R. Janssens, F. G. Kondev, et al., High-spin structure, isomers, and state mixing in the neutron-rich isotopes

173 Tm and

Phys. Rev. C 054314 (2012). G. Dracoulis, F. G. Kondev, G. Lane, A. Byrne, T. Kibedi, I. Ahmad, M. Carpenter, S. Freeman, R. Janssens, N. Ham- mond, et al., Identification of yrast high- isomers in and characterisation of Lu, Phys. Lett. B , 22 (2004).

Barnéoud, André, Foin, Multi-quasiparticle yrast states in Ta, Nucl. Phys. A , 205 (1982).

C. Pearson, P. M. Walker, C. Purry, G. Dracoulis, S. Bayer, A. Byrne, T. Kibedi, F. Kondev, Multi-quasiparticle isomers and rotational bands in Re, Nucl. Phys. A , 301 (2000).

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 Tl and Eur. Phys. J. A 41 (2000).

C. Guo, W. Zhang, H. Huang, S. Zhang, Z. Li, Z. Liu, H. Hua, D. Luo, H. Wu, A. Andreyev, et al., Spectroscopy Tl: shape coexistence, Eur. Phys. J. A , 165 (2024).

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 Au, Chin. Phys. C M. Sedlák, M. Venhart, J. L. Wood, V. Matoušek, M. Balogh, A. J. Boston, T. E. Cocolios, L. J. Harkness-Brennan, R.-D.

Herzberg, D. T. Joss, et al. , Nuclear structure of Au studied /EC decay of Hg at ISOLDE, Eur. Phys. J. A , 161 (2020).

M. Venhart, J. L. Wood, M. Sedlák, M. Balogh, M. Bírová, A. J. Boston, T. E. Cocolios, L. J. Harkness-Brennan, R.-D.

Herzberg, L. Holub, et al. , New systematic features in the neutron-deficient Au isotopes, J. Phys. G: Nucl. Part. Phys. 074003 (2017).

F. Xu, P. Walker, J. Sheikh, R. Wyss, Multi-quasiparticle potential-energy surfaces, Phys. Lett. B , 257 (1998).

W. D. Myers, W. J. Swiatecki, Nuclear masses and deforma- tions, Nucl. Phys. , 1 (1966). 5582(66)90639-0 W. D. Myers, W. Swiatecki, Average nuclear properties, Ann. Phys. , 395 (1969). 4916(69)90202-4 V. Strutinsky, Shell effects in nuclear masses and de- formation energies, Nucl.

Phys. (1967). W. Nazarewicz, J. Dudek, R. Bengtsson, T. Bengtsson, I. Rag- narsson, Microscopic study of the high-spin behaviour in se- lected nuclei, Nucl. Phys. A , 397 (1985).

J. Dudek, Z. Szyma´nski, T. Werner, Woods-saxon po- tential parameters optimized spectra region, Phys. (1981).

W. Nazarewicz, M. Riley, J. Garrett and J. Dudek, Proc. Int.

Conf. on nuclear shapes, Crete, Greece, 1987, p. 67.

H. Pradhan, Y. Nogami, J. Law, Study of approximations in the nuclear pairing-force problem, Nucl. Phys. A , 357 (1973).

P. Möller, J. Nix, Nuclear pairing models, Nucl. Phys. A 20 (1992).

P. Möller, J. R. Nix, W. Myers, W. Swiatecki, Nuclear ground- state masses and deformations, At. Data Nucl. Data Tables 185 (1993).

Y. Shi, F. Xu, H. Liu, P. Walker, Multi-quasiparticle excitation:

Extending shape coexistence neutron-deficient nuclei, Phys. Rev. C , 044314 (2010).

F. G. Kondev, G. Dracoulis, T. Kibedi, Configurations and hindered decays of isomers in deformed nuclei with

A > 100 , At. Data Nucl. Data Tables 103 , 50 (2015). 704

[49] G. Løvhøiden, P. Andersen, D. Burke, E. Flynn, J. Sunier, 706

Single-proton states in Tm studied with the (t, ) reaction, Nucl. Phys. A , 64 (1979). 9474(79)90318-X F. Kondev, S. Zhu, M. Carpenter, R. Janssens, I. Ah-

mad, C. Chiara, J. Greene, T. Lauritsen, D. Seweryniak, 711

S. Lalkovski, et al., -forbidden decays of the

K π = 23 / 2 − isomer in 177 Lu, Phys. Rev. C 85 , 027304 713

(2012). F. Kondev, G. Dracoulis, A. Byrne, T. Kibédi, S. Bayer, Multi- quasiparticle states in Ta and structural changes in the yrast line of the odd tantalum isotopes, Nucl. Phys. A , 91 (1997).

F. G. Kondev, G. Dracoulis, G. Lane, I. Ahmad, A. Byrne, M. Carpenter, P. Chowdhury, S. Freeman, N. Hammond, R. Janssens, et al., -mixing and fast decay of a seven- quasiparticle isomer in Ta, Eur. Phys. J. A , 23 (2004).

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 frag- mentation of Pb at 1 GeV/nucleon, Phys. Rev. C , 064604 (2002).

U. Georg, W. Borchers, M. Keim, A. Klein, P. Lievens, R. Neu- gart, M. Neuroth, P. M. Rao, C. Schulz, the ISOLDE Collab- oration, Laser spectroscopy investigation of the nuclear mo- ments and radii of lutetium isotopes, Eur. Phys. J. A , 225 (1998).

J. Blomquist and S. Wahlborn, Shell model calculations in the lead region with a diffuse nuclear potential, Ark. Fiz. , 543 (1960).

A. Chepurnov, Average field of neutron and proton shells with

N > 126 and Z > 82 , Yad. Fiz. 6 , 955 (1967). 738

E. Rost, Proton shell-model potentials for lead and the sta- bility of superheavy nuclei, Phys. Lett. B , 184 (1968) J. Dudek, A. Majhofer, J. Skalski, T. Werner, S. ´Cwiok, W.

Nazarewicz, Parameters of the deformed Woods-Saxon poten-

tial outside A = 110 − 210 nuclei, J. Phys. G: Nucl. Phys. 5 , 744

1379 (1979). I. Ladenbauer-Bellis, H. Bakhru, P. Sen, Some decay

properties Nucl. Phys. (1975). M. Desthuilliers, C. Bourgeois, P. Kilcher, J. Letessier,

F. Beck, T. Byrski, A. Knipper, Structure of the collective 750

bands in the Au nucleus, Nucl. Phys. A , 221 (1979).

V. Berg, Z. Hu, J. Oms, C. Ekström, the Transition rates between negative-parity states in odd-mass gold nuclei, Nucl. Phys. A , 445 (1983). 9474(83)90637-1 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 (1986).

K. Wallmeroth, G. Bollen, A. Dohn, P. Egelhof, U. Krönert, M. Borge, J. Campos, A. Yunta, K. Heyde, C. De Coster, J. Wood, H.-J. Kluge, Nuclear shape transition in light gold isotopes, Nucl. Phys. A , 224 (1989).

J. E. García-Ramos, K. Heyde, L. M. Robledo, R. Rodríguez-Guzmán, Shape evolution and shape co- existence isotopes:

Comparing interacting model configuration mixing Gogny mean- field energy surfaces, Phys. Rev. C , 034314 (2014).

X. Q. Yang, L. J. Wang, J. Xiang, X. Y. Wu, Microscopic

analysis

prolate-oblate shape phase transition shape coexistence Er-Pt region, Phys. (2017).

G. Lane, G. Dracoulis, A. Byrne, P. Walker, A. Baxter, J. Sheikh, W. Nazarewicz, Shape coexistence in Tl and

187 Tl—investigation of the deformed minima, Nucl. Phys. A 780

, 316 (1995). A. Lee, G. Lane, G. Dracoulis, A. Macchiavelli, P. Fallon, R. Clark, F. Xu, D. Dong, Observation of new bands in T1, EPJ Web of Conferences, , 06002 (2012).

S. Chamoli, P. Joshi, A. Kumar, R. Kumar, R. Singh, S. Mu- ralithar, R. Bhowmik, I. Govil, Shape coexistence and life- time measurement in Tl nucleus, Phys. Rev. C , 054324 (2005).

W. Reviol, L. Riedinger, J.-Y. Zhang, C. Bingham, W. Mueller, B. Zimmerman, R. Janssens, M. Carpenter, I. Ahmad, I. Bear- den, et al., Prolate collectivity in Tl, Phys. Rev. C , R587 (1994).

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 024315 (2013).

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 Tl measured by in- source laser spectroscopy, Phys. Rev. C , 014324 (2017).

G. D. Dracoulis, G. J. Lane, A. P. Byrne, H. Watan- R. O. Hughes, N. Palalani, F. G. Kondev, Carpenter, R. V. F. Janssens, T. Lauritsen, et al., Long- lived three-quasiparticle isomers with triaxial deformation, Phys. lett. B , 59 (2012).