Shape coexistence in the neutron-deficient 188Hginvestigated via lifetime measurements

M. Siciliano,1, 2, 3 I. Zanon,1, 2, 4 A. Goasduff,2, 5, 6 P.R. John,2, 5, 7 T.R. Rodŕıguez,8 S. Péru,9

I. Deloncle,9, 10 J. Libert,9 M. Zielińska,3 D. Ashad,11 D. Bazzacco,5 G. Benzoni,12 B. Birkenbach,13

A. Boso,2, 5 T. Braunroth,13 M. Cicerchia,1, 2 N. Cieplicka-Oryńczak,12, 14 G. Colucci,2, 5, 15 F. Davide,11

G. de Angelis,1 B. de Canditiis,11 A. Gadea,16 L.P. Gaffney,17 F. Galtarossa,1, 4 A. Gozzelino,1

K. Hadyńska-Klȩk,1, 15 G. Jaworski,1, 15 P. Koseoglou,7 S.M. Lenzi,2, 5 B. Melon,18 R. Menegazzo,5

D. Mengoni,2, 5 A. Nannini,18 D.R. Napoli,1 J. Pakarinen,19 D. Quero,11 P. Rath,11 F. Recchia,2, 5

M. Rocchini,18 D. Testov,2, 5, 20 J.J. Valiente-Dobón,1 A. Vogt,13 J. Wiederhold,7 and W. Witt7

1INFN, Laboratori Nazionali di Legnaro, Legnaro, Italy.2Dipartimento di Fisica e Astronomia, Università di Padova, Padova, Italy.

3Irfu/CEA, Université de Paris-Saclay, Gif-sur-Yvette, France.4Dipartimento di Fisica e Scienze della Terra, Università di Ferrara, Ferrara, Italy.

5INFN, Sezione di Padova, Padova, Italy.6Wydzia l Fizyki, Uniwersytet Warszawski, Warsaw, Poland.

7Institut für Kernphysik der Technischen Universität Darmstadt, Darmstadt, Germany.8Departamento de F́ısica Teórica and Centro de Investigación Avanzada

en F́ısica Fundamental Universidad Autónoma de Madrid, Madrid, Spain.9CEA, DAM, DIF, Arpajon, France.

10IJCLab CNRS/IN2P3, Université de Paris-Saclay, Orsay, France.11Dipartimento di Fisica and INFN Sezione di Napoli, Napoli, Italy.

12INFN, Sezione di Milano, Milano, Italy.13Institut für Kernphysik, Universität zu Köln, Cologne, Germany.

14Instytut Fizyki Ja̧drowej im. Henryka Niewodniczańskiego, Polska Akademia Nauk, Krakow, Poland.15Środowiskowe Laboratorium Ciȩżkich Jonów, Uniwersytet Warszawski, Warsaw, Poland.

16Instituto de F́ısica Corpuscular, CSIC-Universidad de Valencia, Valencia, Spain.17University of the West of Scotland, Paisley, Scotland.

18Dipartimento di Fisica and INFN Sezione di Firenze, Firenze, Italy.19Fysiikan laitos, Jyväskylän Yliopisto, Jyväskylä, Finland.

20Joint Institute for Nuclear Research, Dubna, Russia.

Background: Shape coexistence in the Z ≈ 82 region has been established in mercury, lead andpolonium isotopes. For even-even mercury isotopes with 100 ≤ N ≤ 106 multiple fingerprints ofthis phenomenon are observed, which seems to be no longer present for N ≥ 110. According to anumber of theoretical calculations, shape coexistence is predicted in the 188Hg isotope.Purpose: The aim of this work was to measure lifetimes of excited states in 188Hg to infer theircollective properties, such as the deformation. Extending the investigation to higher-spin states,which are expected to be less affected by band-mixing effects, can provide additional informationon the coexisting structures.Methods: The 188Hg nucleus was populated using two different fusion-evaporation reactions withtwo targets, 158Gd and 160Gd, and a beam of 34S provided by the Tandem-ALPI accelerator complexat the Laboratori Nazionali di Legnaro. The channels of interest were selected using the informationfrom the Neutron Wall array, while the γ rays were detected using the GALILEO γ-ray spectrom-eter. Lifetimes of excited states were determined using the Recoil Distance Doppler-Shift method,employing the dedicated GALILEO plunger device.Results: Lifetimes of the states up to spin 16 ~ were measured and the corresponding reduced tran-sition probabilities were calculated. Assuming two-band mixing and adopting, as done commonly,the rotational model, the mixing strengths and the deformation parameters of the unperturbedstructures were obtained from the experimental results. In order to shed light on the nature of theobserved configurations in the 188Hg nucleus, the extracted transition strengths were compared withthose resulting from state-of-the-art beyond-mean-field calculations using the symmetry-conservingconfiguration-mixing approach, limited to axial shapes, and the 5-dimensional collective Hamilto-nian, including the triaxial degree of freedom.Conclusions: The first lifetime measurement for states with spin ≥ 6 suggested the presenceof an almost spherical structure above the 12+1 isomer and allowed elucidating the structure ofthe intruder band. The comparison of the extracted B(E2) strengths with the two-band mixingmodel allowed the determination of the ground-state band deformation. Both beyond-mean-fieldcalculations predict coexistence of a weakly-deformed band with a strongly prolate-deformed one,characterized by elongation parameters similar to those obtained experimentally, but the calculatedrelative position of the bands and their mixing strongly differ.

arX

iv:2

001.

0903

5v2

[nu

cl-e

x] 2

2 Fe

b 20

20

2

PACS numbers: 07.85.Nc, 21.10.Tg, 21.60.-n, 25.60.Pj, 27.70.+q

I. INTRODUCTION

The regions close to Z = 50 and Z = 82 provide uniqueconditions to study the evolution of nuclear shapes andof collectivity in the vicinity of magic numbers. The leadregion, in particular, presents a wide range of phenomenarelated to the nuclear shape, as for instance shape stag-gering between odd- and even-mass nuclei [1, 2], shapeevolution with mass [3–5] and shape coexistence [6, 7].The latter is a characteristic feature of finite many-body quantum systems, such as the atomic nucleus,where structures corresponding to different shapes co-exist within a similar excitation energy.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

96 98 100 102 104 106 108 110 112 114

0+

2+

4+

6+

6+

8+

8+

10+

12+

14+

16+

0+

2+

4+

6+

8+

10+

12+

14+

16+

Ex

cit

ati

on

En

erg

y [

Me

V]

Number of neutrons N

5.35

5.40

5.45

5.50

5.55

5.60

Ch

arg

e r

ad

ius

[fm

]

Hg

Pb

Po

FIG. 1: (Color online) (top) Mean-square charge radius as afunction of the neutron number for Po (green), Pb (blue) andHg (red) isotopic chains. For the Z = 80 isotopes, the largevariation of the radius was attributed to the presence of differ-ent shapes. The values are taken from Refs. [2, 8, 9]. (bottom)Energy systematics of excited states in the neutron-deficienteven-even mercury isotopes, showing (red circles) the assumedintruder and (black squares) ground-state bands. Data aretaken from the National Nuclear Data Center database [10].

The first hint of shape coexistence in the Z≈82 re-gion came from studies of optical hyperfine structure inthe neutron-deficient mercury isotopes [8]. As shown inFigure 1 (top panel), significant staggering of the mean-

square charge radius was observed for mass 181 ≤ A ≤185, which was interpreted as resulting from the pres-ence of two structures characterized by different defor-mation. This behavious of mean-square charge radii isunique in the nuclear chart. Recent laser-spectroscopymeasurements [2] demonstrated that the shape stagger-ing is present down to A = 179 (N = 99), where thenucleus returns to sphericity in its ground state.

Another fingerprint of shape coexistence in the mer-cury isotopes is the observation of low-lying intruderbands built on second 0+ states, which are particularlyclose in energy to the ground state in the isotopes from180Hg to 188Hg (see Figure 1 (bottom panel)). Fromlevel energies within the bands, the deformations of thesenuclei in the ground-state bands were estimated to beβ2 ≈ 0.1, in contrast to β2 ≈ 0.3 obtained for the bandsbuilt on the 0+2 states [11, 12]. These structures tendto mix due to their proximity in energy. The degreeof their mixing was first estimated from the measuredα-decay hindrance factors, yielding a 3% admixture ofthe deformed configuration in the ground state of 180Hg,while mixing of 16% and 18% was obtained for 182Hg and184Hg, respectively [13]. These admixtures are consistentwith those deduced from the ρ2(E0; 0+2 → 0+1 ) value for188Hg and its upper limit for 186Hg [14], and togetherthey display a parabolic behavior as a function of neutronnumber with maximum mixing observed at the N = 104mid-shell, where the intruding structure comes the closestin energy to the ground state. The mixing of normal andintruder structures in Hg isotopes was also investigatedby applying a phenomenological two-band mixing modelto level energies in the observed rotational bands [15, 16].The most recent study [16] took into account the ener-gies of newly identified non-yrast states in 180,182Hg andyielded a lower mixing between the 0+ states than thatdeduced from the α-decay work of Ref. [13], but still witha maximum around N = 102 − 104. The conclusions ofthis analysis [16] for the 2+ states are, however, much dif-ferent, suggesting an inversion of configurations of the 2+1state between 182Hg and 184Hg, with an almost maximummixing strength (51%) for the 2+1 state in the mid-shell184Hg nucleus. As for the 0+ states, when moving awayfrom N = 104, both towards lighter and heavier nuclei,the configurations of the 2+ states become more pure.The importance of configuration mixing in the struc-ture of 2+ states in neutron-deficient Hg nuclei is fur-ther supported by enhanced ρ2(E0; 2+2 → 2+1 ) transitionstrengths observed for 180Hg [17], 182Hg [18], 184Hg [18]and 186Hg [19, 20]. Finally, the mixing of states withJ > 2 extracted from the perturbation of level energiesin rotational bands in 180−188Hg [16] decreases with spinand it no longer displays a parabolic behavior as a func-tion of neutron number, but rather a monotonic increasewith mass (e.g. for the 4+ states the admixtures increasefrom 2% in 180Hg to 20% in 188Hg). Currently, no E0

3

transition strengths are known in neutron-deficient Hgnuclei between the states of spin 4 and higher.

Further information on collective structures inneutron-deficient Hg nuclei was provided by measure-ments of γ-ray transition strengths. Lifetimes of ex-cited states were measured for several neutron-deficientspecies, populated via fusion-evaporation reactions: the178Hg [21], 180,182Hg [22, 23] and 184,186Hg [16, 24, 25]nuclei were studied using the Recoil-Distance Doppler-Shift (RDDS) method, while the Doppler-Shift Attenua-tion Method (DSAM) and the β-tagged fast-timing (FT)techniques were employed to investigate the 184Hg [26]and 186,188Hg [14, 27] isotopes, respectively. Moreover,the lifetimes of low-lying states in 190−196Hg isotopes,which could not be obtained via RDDS method due tothe presence of low-lying isomers, were recently measuredvia the FT technique [27, 28]. These studies were mostlylimited to yrast states but, as the intruder band becomesyrast at low spin (see Figure 1 (bottom panel)), theyyielded lifetimes of states in both coexisting bands in theeven-mass 180−186Hg nuclei, providing strong support forthe very different deformations of these two structures.

Finally, with the advent of radioactive ion-beam fa-cilities, the even-mass 182−188Hg isotopes were investi-gated via low-energy Coulomb excitation, yielding mag-nitudes and signs of the reduced E2 matrix elements be-tween the low-lying excited states [29, 30]. Combinedwith the mixing coefficients of Ref. [16], these results pro-vided a consistent picture of two distinct configurations(weakly-deformed oblate and strongly-deformed prolate)contributing in varying proportions to the observed low-lying states in 182−188Hg. Even though the simplest ob-servables, such as the energy of the first-excited 2+ stateand the B(E2; 2+1 → 0+1 ) value, are almost identical in182−188Hg, the structures of 2+1 states were demonstratedto be very different: the intruder configuration domi-nates in 182Hg, both configurations almost equally con-tribute in 184Hg, and the normal configuration prevailsfor 186,188Hg. Unfortunately, due to strong correlationsbetween the reduced matrix elements, the important roleof E0 transitions between the 2+ states and the lack ofsufficiently precise lifetimes, branching and mixing ratiosfor the nuclei of interest, it was not possible to determineany spectroscopic quadrupole moments from this study,except for the Qs(2

+1 ) = 0.8

+0.5−0.3 eb in

188Hg, consistentwith an oblate deformation of this state.

The first interpretation of the intruder band was givenin the work of Praharaj and Khadkikar [31], who per-formed Hartree-Fock calculations for even-mass mercuryisotopes from A = 184 to A = 204 and obtained β2 defor-mation parameters for the two coexisting structures. Intheir calculations, all these nuclei present oblate ground-state bands, with a maximum of the deformation forA = 186 (β2 = 0.117), while the excited bands areprolate-deformed. In particular, small mixing was pre-dicted between the two bands in 188Hg. More recently,Nikšić and collaborators performed relativistic Hartree-Bogoliubov (RHB) calculations [32] and predicted the

ground-state band of Hg isotopes to be weakly-deformedoblate, due to the two-proton hole in the Z = 82 shell.For the isotopes close to the neutron mid-shell, includ-ing 188Hg, this oblate ground-state band was predictedto be crossed by an intruding prolate-deformed band, re-lated to 4p − 6h proton excitations into the h9/2 andf7/2 orbitals, and the two structures were expected tobe strongly mixed. Similar conclusions could also bereached from beyond-mean-field (BMF) and interacting-boson model (IBM) calculations that were summarizedin the works of Bree et al. [29] and Wrzosek-Lipska etal. [30]. In these works, the BMF approach predictsfor N ≥ 106 a weakly-deformed ground-state band co-existing with an excited prolate band characterized by astronger deformation. For nuclei with 100 ≤ N ≤ 104the two bands cross and the ground state is expected tobe predominantly prolate, while the first excited 0+ stateis predicted to have equal contributions of the oblate andthe prolate configuration.

The intruder structure of 188Hg appears at excitationenergy slightly higher than in 182−186Hg, but it also be-comes yrast very quickly, starting from spin 6+. Thedegree of mixing between the two coexisting structuresis expected to be lower than at mid-shell, but the detailedpredictions of the models significantly differ. As 188Hgis less exotic and thus more accessible for high-precisionspectroscopy than the Hg isotopes in the nearest vicinityof N = 104, the paucity of information about its elec-tromagnetic structure, in particular that of higher-spinstates, is surprising. The transition probabilities in 188Hgwere studied via lifetime measurements using the FTtechnique [14, 27] and via Coulomb excitation [29, 30].However, the existing information is mostly limited toyrast states and the results on the 2+1 lifetime are notconsistent. In order to obtain a clear picture of shape co-existence in 188Hg and a deeper insight into configurationmixing in Hg nuclei, precise determination of the reducedtransition probabilities between the low-lying states, be-longing to both structures, is mandatory.

In the present paper, the lifetime measurement of theexcited states in both the ground-state and intruderbands in 188Hg is presented. The results are discussedand interpreted in the context of new beyond-mean-field calculations performed via the symmetry-conservingconfiguration-mixing and the five-dimensional collectiveHamiltonian methods.

II. EXPERIMENTAL DETAILS

Two experiments were performed, using differentfusion-evaporation reactions, to populate the excitedstates in 188Hg. In the first one, a 34S beam at 185-MeV energy impinged on a 600 µg/cm2 thick target of160Gd, evaporated onto a 2.5 mg/cm2 thick 181Ta foil.The second one used a 34S beam at the energy of 165MeV and a 600 µg/cm2 thick target of 158Gd evaporatedon an identical foil. In both experiments, the targets

4

0

500

1000

1500

2000

2500

3000

0 100 200 300 400 500 600 700 800 900 1000

41

+→

21

+

61

+→

41

+

81

+→

61

+

10

1+→

81

+

12

1+→

10

1+

82

+→

62

+

62

+→

41

+

14

1+→

12

1+

16

1+→

14

1+

62

+→

42

+

42

+→

22

+

12

2+→

10

1+

14

2+→

12

2+

22

+→

21

+

42

+→

21

+

61

+→

42

+

11

- → 9

-

9- →

7-

7- →

5-

10

- → 9

-

10

- → 8

-

6- →

5-

19

- → 1

7-

8- →

7-

13

- → 1

1-1

7- →

15

-

(13

- )→

(1

1- )

(15

- )→

(1

3- )

X-r

ays

5- →

41

+

7- →

62

+

7- →

61

+

Co

un

ts / 1

keV

Energy [keV]

FIG. 2: (Color online) (top) Background-subtracted γ-ray energy spectrum from the GALILEO detectors at 90◦, obtainedby gating on the 413-keV 2+1 → 0+g.s. transition and requiring the coincidence with at least one neutron. The statistics is thesum of the Exp.1 and Exp.2 datasets. The identified transitions in the 188Hg positive-parity bands are marked in red, those inand between negative-parity bands are highlighted in blue, while those between positive-parity and negative-parity bands arein green. (bottom) Partial level scheme of 188Hg, reporting the γ-ray transitions observed in the present measurement. Thearrow widths represent the efficiency-corrected transition yields.

were oriented such that the beams first passed throughthe Ta foils before striking the target material. In thefollowing, these two measurements will be referred to asExp.1 and Exp.2, respectively. As the cross sections forindividual reaction channels were different for Exp.1 andExp.2, this approach provided a better control of possible

contamination of the 188Hg data by other reaction chan-nels. The 34S beam was provided by the Tandem-ALPIaccelerator complex [33, 34] of the Laboratori Nazionalidi Legnaro (Italy). The γ rays resulting from the reac-tions were detected by the GALILEO spectrometer, anarray of 25 Compton-shielded HPGe detectors, arranged

5

into 3 rings at backward angles (152◦, 129◦, 119◦) andone ring at 90◦ [35] with respect to the beam direction.The neutrons evaporated in the reaction were detectedusing the Neutron Wall array [36], composed of 45 liq-uid scintillators placed at forward angles with respectto the beam direction. The use of Neutron Wall wasnecessary to discriminate between the events of inter-est, expected in coincidence with at least one neutron,and the Coulomb-excitation background [37]. Figure 2(top panel) shows the γ-ray energy spectrum of 188Hg,obtained in coincidence with at least one neutron andgated on the 2+1 → 0+g.s. transition in 188Hg. The par-tial level scheme, showing all γ-ray transitions observedin the present study, is presented in the bottom panel ofFigure 2. More details about the pre-sorting of the datacan be found in Ref. [38].

For the lifetime measurements, the RDDS tech-nique [39] was used by employing the GALILEOplunger [40] with a 11 mg/cm2 thick 197Au stoppermounted after the target. For each γ-ray transition twocomponents were observed, related to the radiation emit-ted before and in the 197Au foil: the γ rays emitted in-flight after the target are Doppler shifted, while thoseemitted after the implantation in the stopper are de-tected at the proper energy. From the energy differencebetween the in-flight and the stopped components of theγ-ray transitions, the average velocity of the 188Hg evapo-ration residue (ER) was determined, being β = 1.71(8)%for Exp.1 and β = 1.59(1)% in Exp.2. Considering thevelocity of the nucleus of interest, seven target-stopperdistances in the range 20-600 µm were used during Exp.1and seven in the range 7-2000 µm during Exp.2, tomeasure lifetimes between few and tens of picoseconds.Specifically, in the first experiment the distances wereoptimized to get information mostly on the 2+1 excitedstate, while for the second they were selected to extendthe measurement also to shorter-lived states.

III. LIFETIME ANALYSIS

In order to avoid the effects of unobserved feeding tran-sitions and to reduce the possibility of contaminationfrom different reaction channels, the lifetime measure-ments were performed using the γ-γ coincidence proce-dure, gating on the in-flight component of the most in-tense feeding transition. Moreover, in order to discrimi-nate against the events resulting from the Coulomb exci-tation of both 197Au stopper and 181Ta target fronting,all analyzed γ-γ matrices were constructed by requiringthe coincidence with at least one neutron identified inthe Neutron Wall array. Figure 3 shows the evolution ofthe intensities of the in-flight and stopped peaks of the2+1 → 0+g.s. transition as a function of the target-degraderdistance, after gating on the 4+1 → 2+1 in-flight compo-nent. The lifetimes of the states were extracted usingthe NAPATAU software [41], applying the DifferentialDecay Curve Method (DDCM) [39] by fitting the area of

both the in-flight (Iifi ) and the stopped (Isti ) components

with a polynomial piecewise function. These intensitieswere scaled according to an external normalization, givenby the area of the 136-keV γ-ray peak of 181Ta, comingfrom the Coulomb excitation of the target fronting. Thischoice for the normalization was due to the fact that thenumber of counts in this peak is not only proportional tothe beam intensity and duration of the run, but it alsoprovides a measure of possible degradation of the targetduring the experiment.

The lifetime τi should be the same for each i-th target-stopper distance and it is obtained as

τi =Isti − Σj (Br α Isti )j

ddtI

ifi

, (1)

where the summation is extended over the j feeding tran-sitions, each with a certain branching ratio (Br) and pa-rameter α, which includes the efficiency correction andthe angular correlation between the transition of interestand the feeding one. In the case of the γ-γ coincidenceprocedure with gating on the in-flight component of thefeeding transition, the contributions from feeding tran-sitions are eliminated and this term is null. The finalresult is given by the weighted average of the lifetimeswithin the sensitive region of the technique, i.e. wherethe derivative of the fitting function is largest.

The lifetimes of the Jπ = 2+1 , 4+1 , 6

+1 , 8

+1 , 10

+1 , 14

+1

and 16+1 excited states in188Hg were extracted via the

DDCM by gating on the in-flight component of the(J + 2)π → Jπ feeding transition. The lifetimes of thesestates were obtained using data for each of the threeGALILEO rings separately and then the weighted av-

0

40

80

120

37 µm

0

40

80

120 60 µm

0

20

40

Co

un

ts /

1.0

ke

V

150 µm

0

120

240

400 410 420

2000 µm

Energy [keV]

FIG. 3: (Color online) Background-subtracted γ-ray energyspectra of 188Hg for different target-to-stopper distances forthe Exp.2 dataset and the GALILEO detectors at 152◦. Thespectra are expanded in the vicinity of the 2+1 → 0

+1 tran-

sition and obtained by gating on the in-flight component ofthe 4+1 → 2

+1 transition and requiring the coincidence with at

least one neutron. The in-flight and stopped components areindicated by dashed-blue and solid-red lines, respectively.

6

0

60

120

180

240

300

Exp. 1

No

rma

lize

d a

rea

12

24

36

10 100 1000Lif

eti

me

[p

s]

Distance [µm]

0

40

80

120

160

200

Exp. 2

No

rma

lize

d A

rea

12

24

36

10 100 1000Lif

eti

me

[p

s]

Distance [µm]

FIG. 4: (Color online) DDCM analysis for the lifetime measurement of the 2+1 excited state, after gating on the in-flightcomponent of the 4+1 → 2

+1 transition. (top) Area of the in-flight (blue) and stopped (red) components, normalized to the area

of the 136-keV γ-ray peak of 181Ta. The dashed lines represent a fit to the in-flight-component points in the sensitive region ofthe technique. (bottom) Corresponding lifetimes obtained for individual distances. The solid line denotes the weighted averageof the lifetimes, while the filled area corresponds to 1σ uncertainty.

erages were calculated, except for the 16+ state where inorder to obtain sufficient statistics data from the threerings had to be summed. In Figure 4 the DDCM anal-ysis performed for the 2+1 state is presented for Exp.1and Exp.2, showing the results for the detectors at 129◦

and 152◦, respectively. The lifetimes obtained from thetwo experiments and data from the three HPGe ringsare in a good agreement. Thus, the weighted averagesof the results obtained for the different detector angleswere calculated, leading to τ(2+1 ) = 25(2) ps for Exp.1,τ(2+1 ) = 24(1) ps for Exp.2. The lifetime of the 4

+1 state

could be determined only from the Exp.2 dataset, whichyielded τ(4+1 ) = 1.9(8) ps. These results are in agreementwith those given in the literature [27, 29, 30].

The stopped components of the γ-ray decays from the12+2 state were observed only at the two shortest dis-tances employed in Exp.2 only, implying a very shortlifetime for the 12+2 state. Using these two data points,a lifetime of ≈ 2 ps was extracted, which would yielda transitional quadrupole moment consistent with thetrend observed for the high-spin members of the intruderband (see discussion in Sec. IV). However, due to an in-sufficient number of experimental points in the sensitiverange of the method, we adopt a more conservative upperlimit of 4 ps for this state.

The presence of the 154-ns 12+1 isomeric state pre-vented the investigation of the Jπ = 10+2 , 8

+2 , 6

+2 states

with the RDDS technique. From the Exp.1 and Exp.2measurements, an upper limit of 10 ps can be set for the6+2 excited state, since in the spectra gated on both the645-keV 8+2 → 6+2 and the 424-keV 7− → 6+2 transitionsonly the in-flight component of the 6+2 → 4+1 transitionwas observed clearly for the data obtained for the longer

plunger distances.

Due to the insufficient level of statistics for the in-flightcomponent of the 301-keV 6+1 → 4+2 transition, it wasnot possible to extract the lifetime of the 4+2 state viathe DDCM. Thus, for the Exp.2 dataset, a gate on the460-keV 8+1 → 6+1 transition was set and the lifetime wasextracted via the Decay-Curve Method (DCM), using asecond-order Bateman equation and taking into accountthe lifetime of the 6+1 state. Figure 5 shows the decaycurve for the 4+2 state, resulting in a lifetime of τ(4

+2 ) =

8(3) ps.

Table I summarizes the experimental results from thetwo datasets. The reported uncertainties represent the1σ statistical error, given by the adopted procedures and

0.0

0.2

0.4

0.6

0.8

1.0

10 100 1000

4+2

Ist /

(Iif +

Is

t )

Distance [µm]

FIG. 5: (Color online) Decay curve of the 4+2 excited state in188Hg as a function of the target-stopper distance, obtainedfor the Exp.2 dataset by gating on the in-flight component ofthe 8+1 → 6

+1 transition (see text). The red line represents

the fitted decay curve, assuming τ(6+1 ) = 5.1(5) ps.

7

TABLE I: Lifetimes (in picoseconds) of the Jπi excited statesin 188Hg. The weighted averages of the results from the threeGALILEO rings are given and compared with the literaturevalues [27, 29, 30]. The reported uncertainties are statisticalonly.

Jπi Exp.1 Exp.2 Previous

2+1 25(2) 24(1) 19(3)4+1 - 1.9(8) 2.3(2)4+2 < 10 8(3) < 586+1 5.5(6) 5.1(5) < 146+2 < 10 < 10 < 308+1 - 2.7(2) -10+1 - 2.1(2) -12+2 - < 4 -14+1 18(4) 19(3) -16+1 < 20 12(3) -

the weighted average of the results from the individualGALILEO rings. Additionally, a systematic uncertainty(typically ≤ 3%), accounting for the choice of the fittingfunction and relativistic effects [39], should be consid-ered. Because of the lack of shorter distances investi-gated in Exp.1, it was possible to study the lifetimes ofthe 2+1 , 6

+1 and 14

+1 excited states only. Similar to the

6+2 case discussed above, only an upper limit can be setfor the 4+2 and 16

+1 states.

IV. DISCUSSION

From the weighted average of lifetimes (τ̄) measured inExp.1 and Exp.2, the corresponding B(E2) values can beextracted, which can be related to the deformation pa-rameters using simple rotational approximations. In theaxially symmetric rotor model, the transition quadrupolemoment Qt is related to the E2 transition probability via

B(E2; Ji → Jf ) =5

16π〈Ji,K, 2, 0|Jf ,K〉2Q2t , (2)

where Jf = Ji − 2 and 〈Ji,K, 2, 0|Jf ,K〉 is the Clebsch-Gordan coefficient. K is the spin projection along thesymmetry axis of the nucleus, conserved in the case ofaxial symmetry (thus it is equal to 0 for the consideredstates in both coexisting structures in 188Hg). Under thisassumption, Qt does not change within a band and it isequal to the intrinsic quadrupole moment Q0, which isrelated to the β2 deformation parameter via

Q0 =3√5πZR20β2(1 + 0.36β2) , (3)

where R0 = 1.20A1/3.

In Table II the B(E2) values and the corresponding Qtand β2 values are reported for the in-band transitions.Based on the excitation energies of the states and the Qt

values, three different configurations can be identified: aslightly-deformed ground-state structure below the 12+1isomeric state, an almost spherical structure above it andfinally the deformed intruder band. For the latter, withinthe experimental uncertainties, the quadrupole momentstend to be constant for the J ≥ 4 states (Qt ≈ 5.88eb, which corresponds to β2 ≈ 0.21). Due to its lowmagnitude and slow evolution as a function of spin [16],the mixing between the ground-state and intruder bandsonly weakly affects the intra-band transition probabili-ties, leading to a rather constant trend of the quadrupolemoments. Therefore a first estimate of reduced transi-tion probabilities in the pure intruder structure can beobtained using the average Qt value for higher-spin statesin the intruder band, as shown in Figure 6.

Since there is substantial experimental evidence for themixing between the ground-state and the intruder bands(e.g. significant ρ2(E0) values, intense inter-bands tran-sitions), it is reasonable to assume that the two struc-tures are not axially symmetric. Thus, assuming thatthe J ≥ 4 intruder states result from a rotation of a non-axial nucleus characterized by two deformation parame-ters (β2, γ), one can relate the measured B(E2) values tothe intrinsic quadrupole moment Q0 using the followingextension of Equation (2), as proposed for example inRef. [42]:

B(E2; J → J − 2) = 58πQ20

J(J − 1)(2J − 1)(2J + 1)

×[cos(γ + 30◦)−

〈K2〉

(J − 1)J cos(γ − 30◦)

]2.

(4)

Applying the described model to the measured B(E2) val-ues for J ≥ 4, one obtains for the pure intruder configu-ration Q0 = 5.90(22) eb (corresponding to β2 = 0.21(1))and γ = 14(2)◦. Here, we again assumed K = 0 for allstates in the intruder band, as the departure from axialsymmetry is not large and K is likely to be approximatelya good quantum number.

In order to shed light on the nature of the coexist-ing structures, accounting for the mixing between them,the experimental reduced transition probabilities are fur-ther compared with a band-mixing model [43] and withbeyond-mean-field calculations.

A. Two-band mixing

The assumption of a purely rotational character ofthe high-spin intruder states is in line with the work ofGaffney et al. [16] where, considering a spin-independentinteraction between two rotational structures and em-ploying the method of Ref. [43], the mixing amplitude(αmix) for the excited states up to spin 10~ were ex-tracted from the excitation energies. The 6+1 , 8

+1 and

10+1 excited states were estimated to have admixture of

8

TABLE II: Comparison between the reduced transition probabilities B(E2; Jπi → Jπf ), extracted from the measured lifetimes,and the theoretical values predicted by 5DCH and SCCM for 188Hg. The transition quadrupole moment Qt and the β2deformation parameter are extracted assuming an axially symmetric rotor model. The experimental values marked with (a)

and the branching ratios (Br) are taken from Ref. [44].

Jπi → Jπf Eγ [keV] τ̄ [ps] BrB(E2) [W.u.] |Qt| [eb] β2Exp. 5DCH SCCM

2+1 → 0+1 413 24.4(9) 1.00 44(2) 82 60 3.66(10) 0.128(4)

4+1 → 2+1 592 1.9(8) 1.00 92(38) 109 94 4.5(13) 0.158(47)

6+2 → 4+1 772 < 10 0.78(1) > 4 22 113 > 0.85 > 0.03

12+1 → 10+2 62

(a)222(29) 103 1.00 1.3(2) - - 0.50(5) 0.017(2)14+1 → 12

+1 437 19(3) 1.00 40(6) - - 2.73(31) 0.095(11)

16+1 → 14+1 659 12(3) 1.00 8(2) - - 1.24(22) 0.043(8)

4+2 → 2+2 327 8(3) 0.46(2) 191(76) 215 316 6.56(184) 0.23(6)

6+1 → 4+2 301 5.3(3) 0.16(1) 156(14) 295 355 5.65(36) 0.20(2)

8+1 → 6+1 461 2.7(2) 1.00 220(16) 347 382 6.56(35) 0.23(1)

10+1 → 8+1 521 2.1(2) 1.00 155(15) 330 - 5.42(37) 0.19(1)

12+2 → 10+1 578 < 4 1.00 > 48 - - > 3.00 > 0.10

2+2 → 0+1 881

(a)203(45) 0.60(3) 0.08(2) 0.20 0.4 - -6+1 → 4

+1 504 5.3(3) 0.79(1) 57(4) 13 1.5 - -

0

50

100

150

200

250

300

0 2 4 6 8 10 12 14 16

B(E

2;

J →

J-2

) [W

.u.]

Spin J [h]

FIG. 6: (Color online) Measured B(E2) values as a func-tion of spin are compared with those calculated from the two-band mixing model [43]. The black (red) solid line denotespure normal (intruder) structures transition strengths, whilethe related shaded areas correspond to their error bars. TheB(E2) values for the pure intruder configuration, estimatedfrom the axially symmetric rotor model (see text) are pre-sented with a dashed red line. The experimental values arepresented with open squares for the ground-state band, withsolid circles for the J < 10 states of the intruder band, withan open circle for the J = 10 state of the intruder band (seetext) and with open pentagons for the almost spherical bandabove the 12+1 isomer.

the normal structure at the level of 6.1%, 1.2% and 0.5%,respectively.

By applying the two-band mixing model [43] to thetransition probabilities determined from the presentlymeasured lifetimes, information on the pure normal andintruder configurations can be obtained. Indeed, sucha two-state mixing model is a simple approach to inter-

pret the properties of physical states based on the mix-ing of different intrinsic configurations. As extensivelydiscussed in the work of Clément et al. [45], the exper-imentally observed states |Jπ1,2〉 can be written as a lin-ear combination of the intrinsic pure prolate (|Jπpr〉) andoblate states (|Jπob〉):

|Jπ1 〉 = αmix|Jπob〉+√

1− α2mix|Jπpr〉

|Jπ2 〉 = −√

1− α2mix|Jπob〉+ αmix|Jπpr〉, (5)

where the pure structures are assumed to be orthogonalbut their mixing gives rise to the observed inter-bandtransitions.

Based on the excitation energy of the states belongingto the ground-state and intruder bands (see Figure 2), itseems that only the J ≤ 8 levels should be consideredwhen applying the two-band mixing model. Indeed, inthe ground-state band the rotational-like pattern is main-tained up to the 8+2 state, while the 10

+2 state has an

energy so close to that of the 12+2 isomer that it seems tobelong to a different configuration. In the intruder band,instead, the 10+1 state is fed by a 456-keV transition fromanother 10+ level, so one cannot exclude the possibil-ity of it being mixed with this third structure. Thus,for the J ≤ 8 states one can estimate the deformationof the unperturbed configurations from the experimentalB(E2; J → J − 2) values, adopting the experimentally-deduced mixing strengths of Ref. [16]. The resultingnormal structure is characterized by |Qn0 | = 3.66(37) eb(β2 ≈ 0.13) and the intruder one by |Qi0| = 6.90(57) eb(β2 ≈ 0.25). The reduced transition probabilities, cor-responding to the unperturbed structures, are comparedwith the experimental results in Figure 6: while for the4+2 → 2+2 transition the large uncertainty does not allowto make any conclusion on the mixing, the strengths of

9

the 2+1 → 0+g.s., 4+1 → 2+1 , 6+1 → 4+2 and 8+1 → 6+1 transi-tions confirm the estimated αmix of Ref. [16]; on the otherhand, the disagreement between the B(E2; 10+1 → 8+1 )value and the prediction for the unperturbed intruderstructure supports the hypothesis previously discussed,suggesting a more complex nature of the 10+ states thatgoes beyond the two-band mixing model.

B. Beyond-mean-field calculations

In view of the measured lifetimes, many of themobtained for the first time, the neutron-deficient186,188,190Hg have been studied within a self-consistentbeyond-mean-field framework [46, 47], i.e. the five-dimensional collective Hamiltonian (5DCH) [48] and thesymmetry-conserving configuration mixing (SCCM) [49,50] methods, with the Gogny-D1M [51] and Gogny-D1S [52, 53] interactions, respectively. These calculationsare based on the mixing of a set of intrinsic states withdifferent quadrupole deformations.

In the 5DCH approach, the intrinsic states are ob-tained by constrained Hartree-Fock Bogolyubov calcu-lations (CHFB) performed for around 90 points in the(β2, γ) plane, defined by 0 ≤ β2max ≤ 0.8 and 0◦ ≤ γ ≤60◦. These calculations are performed using a triaxialharmonic oscillator basis including 13 major shells, whichis a sufficiently large model space to ensure the energyconvergence. The overlap of the intrinsic states is as-sumed to be gaussian: the Gaussian-overlap approxima-tion (GOA) allows to derive from the Hill-Wheeler equa-tion a Bohr-type Hamiltonian, called the 5DCH. ThisHamiltonian deals with quadrupole degrees of freedomonly, thus two vibrations and three rotations are takeninto account. The 5DCH potential term is determinedby the CHFB energy, obtained at the different (β2, γ)deformation points, to which the zero-point energies areadded. The kinetic terms involving three mass parame-ters and three rotational inertia parameters are also de-duced from CHFB solutions.

In the SCCM approach, the intrinsic states areHartree-Fock-Bogoliubov (HFB) like wave functions ob-tained self-consistently through the particle-number vari-ation after projection (PN-VAP) method [54]. Becausethe HFB states also break the rotational invariance ofthe system, this symmetry is restored by projecting ontogood angular momentum (particle-number and angularmomentum projection, PNAMP). The final spectrumand nuclear wave functions are obtained by mixing suchPNAMP states within the generator coordinate method(GCM). The region under study is expected to show com-peting shapes in the low-lying spectrum. Such kind ofsystem can be particularly sensitive to the convergenceof the calculations with the number of harmonic oscillatorshells, Nh.o., used to define the HFB transformation [55].

Initially the SCCM calculations were performed withNh.o. = 11 including triaxial shapes with the result thatthe prolate band was predicted to be the ground-state

configuration, contrary to what is obtained experimen-tally. Consequently, exploratory SCCM calculations lim-ited to the axial quadrupole (β2) degree of freedom wereperformed for Nh.o. = 11, 13, 15 and 17. A strong de-pendence of the excitation energies of the less-deformedband was observed as a function of Nh.o., with the inver-sion of the bands disappearing for larger bases. On theother hand, the β2 deformation of the two structures wasalmost independent on the number of harmonic oscillatorshells: β2 ≈ 0.20 for the oblate-deformed configurationand β2 ≈ 0.25 for the prolate one. Only for Nh.o. = 17convergence was obtained for the energies and wave func-tions. Such a large number of harmonic oscillator shellsmakes the SCCM with triaxial shapes extremely expen-sive from the computational point of view and, conse-quently, only axial SCCM results with Nh.o. = 17 arereported here.

A first estimation of the structure of the 186,188,190Hgisotopes can be obtained by analysing the calculatedenergy surfaces as a function of deformation param-eters. Figure 7 presents the PN-VAP and PNAMPenergies, calculated within the SCCM approach, as afunction of the deformation β2. For each of the threenuclei two clear minima are present, corresponding tooblate (β2 ≈ −0.17) and prolate superdeformed (SD)(β2 ≈ +0.65) shapes, and a double minimum struc-ture around normal-deformed (ND) prolate configura-tions with β2 ≈ +0.1,+0.3. Once the angular momentumprojection is performed, in the three cases, the absoluteminimum for J = 0 is the oblate one, then the ND config-uration and, finally, the SD one. Oblate and ND minimaare rather close in energy in 186Hg, while ND and SDminima are almost degenerate in 190Hg.

Similar conclusions can be drawn from the analysis ofthe potential-energy surfaces (PES) for 186,188,190Hg (seeFigure 8), serving as a basis for the 5DCH approach. For188Hg, the oblate minimum with β2 ≈ 0.16 is the deep-est one, followed by a slightly more shallow prolate-likedeformed minimum with a significant degree of triaxi-ality (β2, γ) = (0.28, 13

◦), and finally an axial prolate-superdeformed minimum with β2 ≈ 0.65, which is pre-dicted much higher in energy. The PES for 186Hg is verysimilar, with the only difference that the two ND minimaare almost degenerate. On the contrary, for 190Hg theprolate-like normal-deformed minimum disappears, whilethe SD prolate minimum becomes more pronounced, inline with the conclusions of Ref. [56]. One can clearlyconclude that the energy surfaces resulting from bothcalculations suggest the presence of shape coexistence inthese nuclei.

As already mentioned, the final theoretical spectra areobtained by mixing intrinsic HFB-like states having dif-ferent quadrupole deformations. In the SCCM calcula-tions, the GCM is applied to mix the PNAMP states. Inthe 5DCH approach, the intrinsic CHFB states are mixedassuming a gaussian overlap to solve the GCM-equivalentequation. Figure 9 shows the excitation energies, ob-tained with the SCCM approach, for the 186,188,190Hg

10

(a) 186Hg (b) 188Hg (c) 190Hg

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

-1470

-1465

-1460

-1455E

(MeV

)PN-VAP

J=0J=2J=4J=6J=8

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

-1485

-1480

-1475

-1470

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

-1505

-1500

-1495

-1490

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

|F(β2)|2

0+ σ

2+ σ

4+ σ

6+ σ

8+ σ

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

|F(β2)|2

0+ σ

2+ σ

4+ σ

6+ σ

8+ σ

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

|F(β2)|2

0+ σ

2+ σ

4+ σ

6+ σ

8+ σ

(a) 186Hg (b) 188Hg (c) 190Hg-0.4 -0.2 0 0.2 0.4 0.6 0.8

β2

|F(β2)|2

0+ σ

2+ σ

4+ σ

6+ σ

8+ σ

σ=1

σ=2

σ=3

FIG. 7: (Color online) PN-VAP (dashed lines) and PNAMP (continuous lines) energy curves as a function of the axialquadrupole deformation β2 for the (a)

186Hg, (b) 188Hg and (c) 190Hg isotopes. The results are obtained with the Gogny-D1Sinteraction within the SCCM approach.

10

(a) 186Hg (b) 188Hg (c) 190Hg

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

-1470

-1465

-1460

-1455

E (M

eV)

PN-VAP

J=0J=2J=4J=6J=8

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

-1485

-1480

-1475

-1470

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

-1505

-1500

-1495

-1490

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

|F(β2)|2

0+ σ

2+ σ

4+ σ

6+ σ

8+ σ

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

|F(β2)|2

0+ σ

2+ σ

4+ σ

6+ σ

8+ σ

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

|F(β2)|2

0+ σ

2+ σ

4+ σ

6+ σ

8+ σ

(a) 186Hg (b) 188Hg (c) 190Hg-0.4 -0.2 0 0.2 0.4 0.6 0.8

β2

|F(β2)|2

0+ σ

2+ σ

4+ σ

6+ σ

8+ σ

σ=1

σ=2

σ=3

FIG. 7: (Color online) PN-VAP (dashed lines) and PNAMP (continuous lines) energy curves as a function of the axial quadrupoledeformation �2 for the (a)

186Hg, (b) 188Hg and (c) 190Hg isotopes. The results are obtained with the Gogny-D1S interactionwithin the SCCM approach.

FIG. 8: (Color online) Potential energy surfaces as a function of the � and � deformation parameters for the 186,188,190Hgisotopes. The results are obtained with the Gogny-D1M interaction within the constrained Hartree-Fock-Bogoliubov approachand serve as a basis for the 5DCH calculations.

Due to the included triaxial degree of freedom, the 5DCHis able to predict a �-band (band D) appearing at low ex-citation energy. Such a structure has not been observedin the experiments presented in this work, but it wasfirst postulated in a �-decay study of Ref. [52] and thenconfirmed in Ref. [53], although the existing spin assign-ments are not firm.

Finally, while both methods predict a potential energyminimum related to a superdeformed shape, only SCCMcalculations are able to calculate the wave functions cor-responding to this minimum (band C ). To get the samespectra, the 5DCH method would require an approachdi↵erent from the one adopted in the present calcula-tions. For the moment, the existence of superdeformedstates in 188Hg has not been confirmed experimentally.

The collective wave functions (CWF), i.e. the weightsof the intrinsic quadrupole deformations in each nu-clear state, are represented in Figure 11 and Figure 12for the SCCM and 5DCH methods, respectively. TheSCCM calculations predict a rather constant deformationwithin each of the three calculated bands (oblate band Bwith �2 ⇡ �0.18, normal-deformed prolate band A with�2 ⇡ 0.22, and superdeformed band C with �2 ⇡ 0.65),

with the only exception of the 0+3 state being much lessdeformed than other members of the SD band. For theground-state and ND bands, the deformation parametersfrom the SCCM calculations are in fair agreement withthose estimated from the experimental results, assumingthe rotational model (see Table II). The overlaps betweenthe CWFs are negligible, which can be related to a verysmall mixing between the three structures and attributedto the imposition of axial symmetry in the calculations.An exploratory calculation with Nh.o = 11 including tri-axial shapes for 188Hg suggests a larger mixing betweenthe oblate-like and ND prolate-like configurations alongthe � degree of freedom.

The wave functions resulting from the 5DCH approachare much more spread on the (�, �) plane and the dis-tributions calculated for states belonging to the di↵erentstructures present considerable overlaps, see Figure 12.Due to the importance of the triaxial degree of freedom,it is impossible to refer to the two bands as “prolate” and“oblate”. The average � deformation of states belongingto the ground-state band, corresponding to the experi-mentally observed deformed band built on the 0+2 statein 188Hg, slowly increases with spin starting from �=0.2

0.0 0.2 0.6 0.80.4 β20º

15º

30º

45º

60º

!

0.0 0.2 0.6 0.80.4 β20º

15º

30º

45º

60º

!

0.0 0.2 0.6 0.80.4 β20º

15º

30º

45º

60º

!

FIG. 8: (Color online) Potential energy surfaces as a function of the β2 and γ deformation parameters for the186,188,190Hg

isotopes. The results are obtained with the Gogny-D1M interaction within the constrained Hartree-Fock-Bogoliubov approachand serve as a basis for the 5DCH calculations.

nuclei as a function of the mean value of the intrin-sic quadrupole deformation for each state provided bythe collective wave functions (CWF). Here, we clearlyobserve for the three isotopes the three different col-lective bands (with ∆J = 2) associated to the oblate(ground-state bands), ND-prolate and SD-prolate min-ima. The excitation energy of the 0+2 state in

186Hg islower than those for 188,190Hg, as observed experimen-tally. For 190Hg, the 0+2 state is found close to the band-head of the SD band.

Figure 10 presents a comparison of the partial levelscheme of 188Hg with the theoretical spectra, calculatedwithin the SCCM and 5DCH approaches. Although theabsolute minimum of the PES in Figure 8 is located at anoblate deformation, the oblate-like (band A) and prolate-like (band B) structures resulting from the 5DCH calcu-lations are inverted with respect to experimental data.Nevertheless, the energy difference between the two 0+

states is 674 keV, which is similar to what is observedexperimentally, and suggests that mixing between thetwo structures may be correctly reproduced. In contrast,

while the SCCM calculation reproduces the proper or-dering of the bands, it predicts a much larger excitationenergy of the 0+2 state, which can be related to the ab-sence of the triaxial degree of freedom and result in asmaller mixing between the excited states. At the sametime, the SCCM calculation predicts a band crossing atJπ = 6+ in 188Hg in agreement with the observations,i.e. the members of the ND-prolate band become yrastat J = 6, 8, 10. Correct band crossing is also predictedfor 186,190Hg. Regarding the level spacing of the twobands, the SCCM method reproduces better the proper-ties of the ND-prolate band (band B), while the 5DCHapproach provides a better description of the weakly de-formed structure (band A). Due to the included triaxialdegree of freedom, the 5DCH is able to predict a γ-band(band C ) appearing at low excitation energy. Such astructure has not been observed in the experiments pre-sented in this work, but it was first postulated in a β-decay study of Ref. [57] and then confirmed in Ref. [58],although the existing spin assignments are not firm.

Finally, a potential energy minimum corresponding to

11

(a) 186Hg (b) 188Hg (c) 190Hg01

02

03

01

02

03

01

02 03

21

22

23

21

22

23

21

22

23

41

42

43

41

42

43

41

42

43

61

6263 61

62

6361

62

81

83

82

8182 81++

+

+

+

+

+

+

++

+

+

+

+

+

+

+

+

+

+

+

+

++

+

++

+

+

+

+

+

+ + +

+

+

+

+

++

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

0

1

2

3

4E

(MeV

)

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

0

1

2

3

4

E (M

eV)

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

0

1

2

3

4

E (M

eV)

FIG. 9: (Color online) Energy spectra obtained with axial SCCM calculations with the Gogny-D1S interaction for the (a)186Hg, (b) 188Hg and (c) 190Hg isotopes. The energies are plotted at the mean value of the intrinsic quadrupole deformation ofthe corresponding collective wave functions. PNAMP energy curves as a function of β2 are also plotted to guide the eye.

FIG. 10: (Color online) Partial level scheme of 188Hg, reporting the low-lying excited states. The experimental results arecompared with the energy spectra predicted by the 5DCH and SCCM approaches.

a superdeformed shape is predicted in both theoreticalapproaches. In this work, the superdeformed states wereonly obtained in SCCM calculations (band D). Indeed, inthe 5DCH approach a two-center basis has to be adoptedfor diagonalising the Bohr Hamiltonian in order to ex-plore the highly deformed structure [56]. Such a struc-ture has not been observed in 188Hg, while it is known inthe neighboring 189Hg and 190Hg [59–61].

The CWF, i.e. the weights of the intrinsic quadrupoledeformations in each nuclear state, are presented in Fig-ure 11 and Figure 12 for the SCCM and 5DCH methods,

respectively. The SCCM calculations predict a ratherconstant deformation within each of the three calculatedbands (oblate band A with β2 ≈ −0.18, normal-deformedprolate band B with β2 ≈ 0.22, and superdeformed bandD with β2 ≈ 0.65), with the only exception of the 0+3state being much less deformed than other members ofthe SD band. For the ground-state and ND bands, thedeformation parameters from the SCCM calculations arein fair agreement with those estimated from the exper-imental results, assuming the rotational model (see Ta-ble II). The overlaps between the CWFs are negligible

12

(a) 186Hg (b) 188Hg (c) 190Hg

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

-1470

-1465

-1460

-1455

E (M

eV)

PN-VAP

J=0J=2J=4J=6J=8

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

-1485

-1480

-1475

-1470

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

-1505

-1500

-1495

-1490

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

|F(β2)|2

0+ σ

2+ σ

4+ σ

6+ σ

8+ σ

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

|F(β2)|2

0+ σ

2+ σ

4+ σ

6+ σ

8+ σ

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

|F(β2)|2

0+ σ

2+ σ

4+ σ

6+ σ

8+ σ

(a) 186Hg (b) 188Hg (c) 190Hg-0.4 -0.2 0 0.2 0.4 0.6 0.8

β2

|F(β2)|2

0+ σ

2+ σ

4+ σ

6+ σ

8+ σ

σ=1

σ=2

σ=3

(a) 186Hg (b) 188Hg (c) 190Hg

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

-1470

-1465

-1460

-1455

E (M

eV)

PN-VAP

J=0J=2J=4J=6J=8

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

-1485

-1480

-1475

-1470

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

-1505

-1500

-1495

-1490

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

|F(β2)|2

0+ σ

2+ σ

4+ σ

6+ σ

8+ σ

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

|F(β2)|2

0+ σ

2+ σ

4+ σ

6+ σ

8+ σ

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

|F(β2)|2

0+ σ

2+ σ

4+ σ

6+ σ

8+ σ

(a) 186Hg (b) 188Hg (c) 190Hg-0.4 -0.2 0 0.2 0.4 0.6 0.8

β2

|F(β2)|2

0+ σ

2+ σ

4+ σ

6+ σ

8+ σ

σ=1

σ=2

σ=3

(a) 186Hg (b) 188Hg (c) 190Hg

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

-1470

-1465

-1460

-1455

E (M

eV)

PN-VAP

J=0J=2J=4J=6J=8

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

-1485

-1480

-1475

-1470

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

-1505

-1500

-1495

-1490

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

|F(β2)|2

0+ σ

2+ σ

4+ σ

6+ σ

8+ σ

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

|F(β2)|2

0+ σ

2+ σ

4+ σ

6+ σ

8+ σ

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

|F(β2)|2

0+ σ

2+ σ

4+ σ

6+ σ

8+ σ

(a) 186Hg (b) 188Hg (c) 190Hg-0.4 -0.2 0 0.2 0.4 0.6 0.8

β2

|F(β2)|2

0+ σ

2+ σ

4+ σ

6+ σ

8+ σ

σ=1

σ=2

σ=3

(a) 186Hg (b) 188Hg (c) 190Hg

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

-1470

-1465

-1460

-1455

E (M

eV)

PN-VAP

J=0J=2J=4J=6J=8

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

-1485

-1480

-1475

-1470

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

-1505

-1500

-1495

-1490

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

|F(β2)|2

0+ σ

2+ σ

4+ σ

6+ σ

8+ σ

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

|F(β2)|2

0+ σ

2+ σ

4+ σ

6+ σ

8+ σ

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

|F(β2)|2

0+ σ

2+ σ

4+ σ

6+ σ

8+ σ

(a) 186Hg (b) 188Hg (c) 190Hg-0.4 -0.2 0 0.2 0.4 0.6 0.8

β2

|F(β2)|2

0+ σ

2+ σ

4+ σ

6+ σ

8+ σ

σ=1

σ=2

σ=3

(a) 186Hg (b) 188Hg (c) 190Hg

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

-1470

-1465

-1460

-1455

E (M

eV)

PN-VAP

J=0J=2J=4J=6J=8

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

-1485

-1480

-1475

-1470

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

-1505

-1500

-1495

-1490

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

|F(β2)|2

0+ σ

2+ σ

4+ σ

6+ σ

8+ σ

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

|F(β2)|2

0+ σ

2+ σ

4+ σ

6+ σ

8+ σ

-0.4 -0.2 0 0.2 0.4 0.6 0.8β2

|F(β2)|2

0+ σ

2+ σ

4+ σ

6+ σ

8+ σ

(a) 186Hg (b) 188Hg (c) 190Hg-0.4 -0.2 0 0.2 0.4 0.6 0.8

β2

|F(β2)|2

0+ σ

2+ σ

4+ σ

6+ σ

8+ σ

σ=1

σ=2

σ=3

FIG. 11: (Color online) Collective wave functions (CWF)obtained with axial SCCM calculations with the Gogny-D1Sinteraction for (a) 186Hg, (b) 188Hg and (c) 190Hg isotopes.

in most cases, which can be related to a very small mix-ing between the three structures and attributed to theimposition of axial symmetry in the calculations. Anexploratory calculation with Nh.o = 11 including triax-ial shapes for 188Hg suggests a larger mixing between theoblate-like and ND prolate-like configurations along the γdegree of freedom. In the axial SCCM calculation notice-able overlaps of CWFs can be observed only for the 0+

FIG. 12: (Color online) Collective wave functions (CWF)as a function of the β2 and γ deformation parameters for the0+A and 0

+B states, obtained with the Gogny-D1M interaction

within the 5DCH framework.

states, which suggests that the three structures (oblate,prolate and SD) mix.

The wave functions resulting from the 5DCH approachare much more spread on the (β2, γ) plane and the dis-tributions calculated for states belonging to the differentstructures present considerable overlaps, see Figure 12.Due to the importance of the triaxial degree of freedom,it is impossible to refer to the two bands as “prolate” and“oblate”. The average β2 deformation of states belong-ing to the band B, corresponding to the experimentallyobserved deformed band built on the 0+2 state in

188Hg,slowly increases with spin starting from β2 = 0.2 for the0+ state, which is in fair agreement with the β2 defor-mation estimated from the experimental data. A grad-ual evolution from an almost maximally triaxial shapewith γ = 27◦ for the 0+ state towards a more prolatedeformation higher up in the band is also predicted. No-tably, the average γ deformation parameter for the statesof spin J ≥ 4 is about 15◦, very close to the value de-rived from the experimental transition probabilities usingthe asymmetric rotor model (see Section IV). Interest-ingly, an average γ deformation parameter of about 15◦

was also obtained for the states belonging to band B inthe exploratory SCCM calculations including the triax-ial degree of freedom, but limited to Nh.o = 11. Theaverage elongation parameter of states belonging to bandA, corresponding to the experimental ground-state bandin 188Hg, is β2 ≈ 0.25 and remains constant within theband, while the triaxial parameter evolves from aboutγ ≈ 20◦ towards maximum triaxiality. The states belong-ing to this oblate-like structure present strong admixturesof the K=2 configuration, being as large as almost 50%for the 2+A state.

Both calculations yield a similar picture of two struc-tures with |β2| ≈ 0.2, coexisting at low excitation en-ergy. The 5DCH calculations suggest that at low-spinthe triaxial degree of freedom is important. Unfortu-nately, the precision of the experimentally determinedquadrupole moment of the 2+1 state in

188Hg, equal to0.8+0.5−0.3 eb [30] is insufficient to make a conclusion aboutthe degree of triaxiality in this state. This value is com-patible with both calculations: the SSCM calculations

13

give Qs(2+A) = 1.3 eb, while in the 5DCH calculations,

which attribute a triaxial character to the 2+A state, the

spectroscopic quadrupole moment is Qs(2+A) = 0.5 eb.

Table II presents a comparison of the experimental re-duced transition probabilities in 188Hg with those calcu-lated within the SCCM and 5DCH approaches. In or-der to facilitate the comparison, the 5DCH results forthe ground-state band are reported next to those for themore deformed structure (band A), while those for thestructure built on the 0+2 are listed next to those for theexperimental weakly deformed ground-state band (bandB).

The agreement for in-band transitions is reasonablygood, with the transitions within the weakly deformedband (band A) described better by SCCM, while those inthe more deformed structure (band B) are overestimatedby both models, with the 5DCH results being closer toexperimental values. The B(E2) value for the 6+1 → 4+1inter-band transition is underestimated by both calcula-tions, with the one resulting from the SCCM calculationsbeing more than one order of magnitude smaller than theexperimental value. This can be interpreted as an under-estimation of the possible mixing between the two bands,consistent with the negligible overlaps between the CWFs(Figure 11). For a more quantitative comparison, one canextract the mixing amplitudes αmix (see Equation 5) forthe 0+, 2+ and 4+ states from the calculated B(E2) tran-sition probabilities using the equations listed in SectionV.A of Ref. [45]. The values obtained using this approachfrom the experimental data and from the model calcula-tions are reported in Table III. As expected, the mixingstrengths fo r the 0+ and 2+ states obtained within the5DCH approach are much larger than for the SCCM cal-culations, and also largely exceeding what is observedexperimentally. In contrast, the large mixing strengthfor the 4+ states deduced from the experimental data isbetter reproduced by the 5DCH calculations.

Finally, the underlying shell structure of the three col-lective bands can be also analysed within the SCCMframework. The relevant spherical shells that are neededto describe such states can be qualitatively identified byNilsson-like orbitals. These orbitals are computed self-consistently for each nucleus with HFB states obtainedalong the quadrupole degree of freedom. As an example,the evolution of the orbitals as a function of the deforma-tion, β2, is plotted for the

188Hg isotope in Figure 13. TheFermi energies for protons and neutrons are also shown.The relevant regions, where the CWF are peaked (i.e.the minima of the energy surfaces), are marked by a greyband in the figure. In the oblate minimum, we see thatthe orbitals close to the Fermi energy are: 3s1/2, 0h11/2and 0h9/2 for protons and 0h9/2, 1f7/2, 0i13/2, 2p3/2 and1f5/2 for neutrons. For the ND configuration, the 1d5/2and the 1f7/2 for protons and the 0g9/2 for neutrons alsoplay a role. Finally, for the SD band, the 0i11/2 andnegative parity orbitals above N = 126 are also cross-ing the neutron Fermi energy. This picture shows thecomplexity of the single-particle structure of these heavy

0

2

4

6

8

10

Ener

gy (M

eV)

PN-VAP

-15

-10

-5

0

ε s.p

. (M

eV)

-0.8 -0.4 0 0.4 0.8β2

-15

-10

-5

0

ε s.p

. (M

eV)

(a) protons

50

82

g9/2

g7/2

d5/2

d3/2h11/2s1/2

h9/2

f7/2

82

126

h9/2 f7/2

i13/2

p3/2 f5/2p1/2

g9/2

i11/2(b) neutrons

188Hg

FIG. 13: (Color online) Single-particle energies for (a) pro-tons and (b) neutrons as a function of the deformation β2calculated for 188Hg with the Gogny D1S interaction. Dashed(continuous) lines represent negative (positive) parity statesand the thick dashed-dotted lines show the Fermi energy. Theshaded areas correspond to the position of the three minimain the energy surface shown in Figure 7.

nuclei that prevents the use of shell model calculations.We compute the occupation numbers of spherical orbitalswith each individual SCCM wave function in the threeisotopes studied here following the method developed inRef. [62]. To simplify the discussion, we define a protoncore of Z = 82 that encompasses the 0s, 0p, 1s − 0d,1p− 0f , 2s− 1d− 0g and 0h11/2 spherical orbitals, and aneutron core of N = 100 that includes the orbitals listedabove plus the 0h9/2 and 1f7/2 neutron levels. Using

such cores, the 188Hg isotope in a normal filling approx-imation should correspond to 2h-0p (protons) and 0h-8p(neutrons), respectively. However, the results obtainedfor the oblate, ND and SD bands are 4.0h-2.0p, 8.1h-6.1pand 9.0h-7.0p for protons, and 5.4h-13.4p, 7.7h-15.7p and8.3h-16.3p for neutrons, respectively. Similar configura-tions are also obtained for the other two isotopes calcu-

14

TABLE III: Average quadrupole moments (in eb) of the normal and intruder configurations as well as the mixing amplitudesαmix for the 0

+g.s., 2

+1 and 4

+1 states extracted from the theoretical B(E2) values from 5DCH and SCCM calculations using the

two-band mixing model, compared with those deduced from experimental data (see text). Due to the bands swapping, for the5DCH and SCCM (Nh.o. = 11) calculations the values are interchanged to facilitate a direct comparison with the experimentalresults.

αmix(0+g.s.) αmix(2

+1 ) αmix(4

+1 ) |Qn0 | |Qi0|

Exp. 0.995 0.988 0.893 3.66(37) 6.90(57)5DCH 0.720 0.866 0.942 4.20 7.68

SCCM (Nh.o. = 11) 0.800 0.911 0.949 3.12 9.61SCCM (Nh.o. = 17) 0.993 0.980 0.994 4.45 8.45

lated here. These results show again the complexity ofthe single-particle structure driven by the deformation ofthe system.

V. CONCLUSIONS

The nuclear structure of the neutron-deficient mercuryisotope 188Hg was investigated via lifetime measurementsat the Laboratori Nazionali di Legnaro. In order to con-trol a possible contamination of the 188Hg data by otherreaction channels, two different fusion-evaporation reac-tions were used for this study and, thanks to the powerfulcapabilities of both GALILEO and Neutron-Wall arrays,a clear identification of the channel of interest was possi-ble.

Using the RDDS technique, the lifetimes of the statesup to spin 16 ~ were measured and the results obtainedfor the 2+1 and 4

+1 states are in agreement with the val-

ues reported in literature. Thanks to the new resultsfor both low- and high-lying states, the deformation ofthe ground-state band and of the intruder structure wasestimated from the experimental data, assuming two-band mixing and two different rotational models. Thesemodels provide a similar interpretation of the structures:the ground-state band has a quadrupole deformation ofβ2 ≈ 0.13, while the intruder one has β2 ≈ 0.22. More-over, the lifetimes of the 14+1 and 16

+1 excited states

highlighted the presence of an almost spherical structureabove the 12+1 isomer.

In view of the new results, two state-of-the-art beyond-mean-field calculations were performed for the even-mass 186−190Hg nuclei using the symmetry-conservingconfiguration-mixing approach limited to axial shapesand the 5-dimensional collective Hamiltonian includingthe triaxial degree of freedom. Both calculations yield asimilar picture of two structures with |β2| ≈ 0.2, coexist-ing at low excitation energy, but predict different relativepositions of the two bands and their mixing. The un-

derlying shell structure of the collective bands was anal-ysed within the SCCM framework, identifying the rel-evant spherical shells, necessary to describe such struc-tures. For 188Hg the comparison between the theoreticalpredictions and the experimental results confirmed theproposed interpretation of both the intruder and ground-state bands.

In contrast with other beyond-mean-field calculations,the SCCM approach predicts the presence of shape co-existence also in 190Hg, with the bandhead of the SDstructure mixing with those of the ND prolate and oblatebands. Clearly, further experimental studies of 190Hg arerequired to verify these predictions. The mixing ampli-tudes for the 0+ states in 180−188Hg deduced from levelenergies and E2 transition strengths differ from those ob-tained from α-decay hindrance factors. In this context,measurements of ρ2(E0) between the coexisting struc-tures are called for, as they would provide a direct mea-sure of the degree of their mixing.

Acknowledgements

The authors would like to thank the GALILEO col-laboration. Special thanks go to the INFN-LNL technicalstaff for their help and the good quality beam and to Mas-simo Loriggiola for the extended work on the Gd plungertargets. We acknowledge the numerical calculations of S.Hilaire. This paper owes much to the collaboration withP.E. Garrett. This work was partially supported by theEspace de Structure Nucléaire Théorique (CEA/DSM-DAM). The work of T.R.R. was supported by the Span-ish MICINN under grant no. PGC2018-094583-B-I00.The work was also supported (P.R.J. and P.K.) by theGerman BMBF under contract no. 05P18RDFN9, (G.J.)by the National Science Centre Poland (NCN) under thegrant no. 2017/25/B/ST2/01569. A.G. is grateful tothe support of Fondazione Cassa di Risparmio Padova eRovigo.

[1] T. Kühl, P. Dabkiewicz, C. Duke, H. Fischer, H.J. Kluge,H. Kremmling, E.W. Otten, Phys. Rev. Lett. 39, 180(1977)

[2] B. Marsh, T. Day Goodacre, S. Sels, Y. Tsunoda, B. An-del, A.N. Andreyev, N.A. Althubiti, D. Atanasov, A.E.Barzakh, J. Billowes et al., Nature 14, 1163 (2018)

15

[3] A. Ansari, Physics Review C 33, 321 (1986)[4] P.R. John, V. Modamio, J.J. Valiente-Dobón, D. Men-

goni, S. Lunardi, T.R. Rodŕıguez, D. Bazzacco,A. Gadea, C. Wheldon, T. Alexander et al., Phys. Rev.C 90, 021301 (2014)

[5] P.R. John, J.J. Valiente-Dobón, D. Mengoni,V. Modamio, S. Lunardi, D. Bazzacco, A. Gadea,C. Wheldon, T.R. Rodŕıguez, T. Alexander et al., Phys.Rev. C 95, 064321 (2017)

[6] A.N. Andreyev, M. Huyse, P. Van Duppen, L. Weissman,D. Ackermann, J. Gerl, F.P. Hessberger, S. Hofmann,A. Kleinbhl, G. Mnzenberg et al., Nature 405, 430 (2000)

[7] P. Rahkila, D.G. Jenkins, J. Pakarinen, C. Gray-Jones,P.T. Greenlees, U. Jakobsson, P. Jones, R. Julin, S. Juu-tinen, S. Ketelhut et al., Physics Review C 82, 011303(R)(2010)

[8] J. Bonn, G. Huber, H. Kluge, L. Kluger, E. Otten,Physics Letter B 38(5), 308 (1972)

[9] I. Angeli, K. Marinova, Atomic and Nuclear Data Table99, 69 (2013)

[10] A. Sonzogni et al., National Nuclear Data Center[11] J.D. Cole, J.H. Hamilton, A.V. Ramayya, W.G. Nettles,

H. Kawakami, E.H. Spejewski, M.A. Ijaz, K.S. Toth, E.L.Robinson, K.S.R. Sastry et al., Phys. Rev. Lett. 37, 1185(1976)

[12] J.H. Hamilton, A.V. Ramayya, E.L. Bosworth,W. Lourens, J.D. Cole, B. Van Nooijen, G. Garcia-Bermudez, B. Martin, B.N.S. Rao, H. Kawakami et al.,Phys. Rev. Lett. 35, 562 (1975)

[13] J. Wauters, N. Bijnens, H. Folger, M. Huyse, H.Y.Hwang, R. Kirchner, J. von Schwarzenberg, P. Van Dup-pen, Physics Review C 50, 2768 (1994)

[14] P. Joshi, E. Zganjar, D. Rupnik, S. Robinson, P. Mantica,H. Carter, J. Kormicki, R. Gill, W. Walters, C. Binghamet al., International Journal of Modern Physics E 3(2),757 (1994)

[15] G.D. Dracoulis, Phys. Rev. C 49, 3324 (1994)[16] L.P. Gaffney, M. Hackstein, R.D. Page, T. Grahn,

M. Scheck, P.A. Butler, P.F. Bertone, N. Bree, R.J. Car-roll, M.P. Carpenter et al., Physical Review C 89, 024307(2014)

[17] R.D. Page, A.N. Andreyev, D.R. Wiseman, P.A. But-ler, T. Grahn, P.T. Greenlees, R.D. Herzberg, M. Huyse,G.D. Jones, P.M. Jones et al., Physics Review C 84,034308 (2011)

[18] E. Rapisarda, A.N. Andreyev, S. Antalic, A. Barzakh,T.E. Cocolios, I.G. Darby, R.D. Groote, H.D. Witte,J. Diriken, J. Elseviers et al., Journal of Physics G: Nu-clear and Particle Physics 44(7), 074001 (2017)

[19] J.D. Cole, A.V. Ramayya, J.H. Hamilton, H. Kawakami,B. van Nooijen, W.G. Nettles, L.L. Riedinger, F.E.Turner, C.R. Bingham, H.K. Carter et al., Phys. Rev.C 16, 2010 (1977)

[20] M. Scheck, P.A. Butler, L.P. Gaffney, N. Bree, R.J. Car-rol, D. Cox, T. Grahn, P.T. Greenlees, K. Hauschild,A. Herzan et al., Physics Review C 83, 037303 (2011)

[21] C. Müller-Gatermann, A. Dewald, C. Fransen, K. Au-ranen, H. Badran, M. Beckers, A. Blazhev, T. Braun-roth, D.M. Cullen, G. Fruet et al., Physics Review C 99,054325 (2019)

[22] T. Grahn, A. Petts, M. Scheck, P.A. Butler, A. De-wald, M.B. Gomez Hornillos, P.T. Greenlees, A. Görgen,K. Helariutta, J. Jolie et al., Physical Review C. 80,014324 (2009)

[23] M. Scheck, T. Grahn, A. Petts, P.A. Butler, A. Dewald,L.P. Gaffney, M.B. Gomez Hornillos, A. Görgen, P.T.Greenlees, K. Helariutta et al., Physical Review C. 81,014310 (2010)

[24] N. Rud, D. Ward, H.R. Andrews, R.L. Graham, J.S.Geiger, Physical Review Letter 31, 1421 (1973)

[25] D. Proetel, R. Diamond, F. Stephens, Physical Review B48, 102 (1974)

[26] W. Ma, A. Ramayya, J. Hamilton, S. Robinson, J. Cole,E. Zganjar, E. Spejewski, R. Bengtsson, W. Nazarewicz,J.Y. Zhang, Physics Letters B 167(3), 277 (1986)

[27] B. Olaizola, A.B. Garnsworthy, F.A. Ali, C. Andreoiu,G.C. Ball, N. Bernier, H. Bidaman, V. Bildstein,M. Bowry, R. Caballero-Folch et al., Physics Review C100, 024301 (2019)

[28] A. Esmaylzadeh, L.M. Gerhard, V. Karayonchev, J.M.Régis, J. Jolie, M. Bast, A. Blazhev, T. Braunroth,M. Dannhoff, F. Dunkel et al., Physical Review C 98(1),014313 (2018)

[29] N. Bree, K. Wrzosek-Lipska, A. Petts, A. Andreyev,B. Bastin, M. Bender, A. Blazhev, B. Bruyneel, P.A. But-ler, J. Butterworth et al., Physical Review Letter 112,162701 (2014)

[30] Wrzosek-Lipska, K., Rezynkina, K., Bree, N., Zielińska,M., Gaffney, L.P., Petts, A., Andreyev, A., Bastin, B.,Bender, M., Blazhev, A. et al., European Physics JournalA 55(8), 130 (2019)

[31] C. Praharaj, S. Khadkikar, J. Phys. G: Nucl. Phys. 6,241 (1980)

[32] T. Nikšić, D. Vretenar, P. Ring, G.A. Lalazissis, PhysicalReview C 65, 054320 (2002)

[33] C. Signorini, G. Bezzon, F. Cervellera, P. Spolaore,R. Ricci, Nuclear Instruments and Methods in PhysicsResearch A 220(1), 30 (1984)

[34] A. Dainelli, G. Bassato, A. Battistella, M. Bellato,A. Beltramin, L. Bertazzo, G. Bezzon, G. Bisoffi,L. Boscagli, S. Canella et al., Nuclear Instruments andMethods in Physics Research A 382, 100 (1996)

[35] J. Valiente-Dobón, D. Mengoni, F. Recchia et al., INFN-LNL Report 2014 241, 95 (2015)

[36] Ö. Skeppstedt, H. Roth, L. Lindström, R. Wadsworth,I. Hibbert, N. Kelsall, D. Jenkins, H. Grawe, M. Górska,M. Moszyński et al., Nuclear Intrumentation and Methodin Physics Research A 421, 531 (1999)

[37] I. Zanon, M. Siciliano, P. John, A. Goasduff et al., INFN-LNL Report 2017 251, 62 (2018)

[38] I. Zanon, M. Siciliano, P. John, A. Goasduff, Nuovo Ci-mento C 42, 109 (2019)

[39] A. Dewald, O. Möller, P. Petkov, Progress in Particle andNuclear Physics 67(3), 786 (2012)

[40] C. Müller-Gatermann, F. von Spee, A. Goasduff, D. Baz-zacco, M. Beckers, T. Braunroth, A. Boso, P. Cocconi,G. de Angelis, A. Dewald et al., Nuclear Instruments andMethods A 920, 95 (2019)

[41] B. Saha, Napatau or tk-lifetime-analysis (2003)[42] P. Petkov, J. Gableske, O. Vogel, A. Dewald, P. von

Brentano, R. Krcken, R. Peusquens, N. Nicolay, A. Gi-zon, J. Gizon et al., Nuclear Physics A 640(3), 293 (1998)

[43] G. Lane, G. Dracoulis, A. Byrne, S. Anderssen, P. David-son, B. Fabricius, T. Kibdi, A. Stuchbery, A. Baxter,Nuclear Physics A 589(1), 129 (1995)

[44] F. Kondev, S. Juutinen, D. Hartley, Nuclear Data Sheets150, 1 (2018)

http://www.nndc.bnl.gov/nudat2/

16

[45] E. Clément, A. Görgen, W. Korten, E. Bouchez,A. Chatillon, J.P. Delaroche, M. Girod, H. Goutte,A. Hürstel, Y. Le Coz et al., Phys. Rev. C 75, 054313(2007)

[46] J. Egido, Physica Scripta 91(7), 073003 (2016)[47] L. Robledo, T.R. Rodŕıguez, R. Rodŕıguez-Guzmán,

Journal of Physics G: Nuclear and Particle Physics 46(1),013001 (2018)

[48] J.P. Delaroche, M. Girod, J. Libert, H. Goutte, S. Hilaire,S. Péru, N. Pillet, G.F. Bertsch, Phys. Rev. C 81, 014303(2010)

[49] T.R. Rodŕıguez, J.L. Egido, Physical Review C 81,064323 (2010)

[50] T.R. Rodŕıguez, Physical Review C 90, 034306 (2014)[51] S. Goriely, S. Hilaire, M. Girod, S. Péru, Phys. Rev. Lett.

102, 242501 (2009)[52] J. Dechargé, D. Gogny, Phys. Rev. C 21, 1568 (1980)[53] J. Berger, M. Girod, D. Gogny, Computer Physics Com-

munications 63(1), 365 (1991)[54] M. Anguiano, J. Egido, L. Robledo, Nuclear Physics A

696(3), 467 (2001)[55] T.R. Rodŕıguez, A. Arzhanov, G. Mart́ınez-Pinedo,

Phys. Rev. C 91, 044315 (2015)

[56] J. Delaroche, M. Girod, J. Libert, I. Deloncle, PhysicsLetters B 232(2), 145 (1989)

[57] C. Bourgeois, M. Bouet, A. Caruette, A. Ferro,R. Foucher, J. Fournet, A. Höglund, L. Kotfila, G. Lan-dois, C. Liang et al., J. Phys. France 37, 49 (1976)

[58] J.D. Cole, J.H. Hamilton, A.V. Ramayya, W. Lourens,B. van Nooijen, H. Kawakami, L.A. Mink, E.H. Spejew-ski, H.K. Carter, R.L. Mlekodaj et al., Phys. Rev. C 30,1267 (1984)

[59] M. Drigert, M. Carpenter, R. Janssens, E. Moore, I. Ah-mad, P. Fernandez, T. Khoo, F. Wolfs, I. Bearden,P. Benet et al., Nuclear Physics A 530(2), 452 (1991)

[60] I.G. Bearden, R.V.F. Janssens, M.P. Carpenter, E.F.Moore, I. Ahmad, A.M. Baxter, P. Benet, P.J. Daly,M.W. Drigert, P.B. Fernandez et al., Zeitschrift fürPhysik A Hadrons and Nuclei 341(4), 491 (1992)

[61] I. Bearden, R. Janssens, M. Carpenter, E. Moore, I. Ah-mad, P. Daly, R. Mayer et al., Nuclear Physics A 576(3),441 (1994), ISSN 0375-9474

[62] T.R. Rodŕıguez, A. Poves, F. Nowacki, Phys. Rev. C 93,054316 (2016)

I IntroductionII Experimental detailsIII Lifetime analysisIV DiscussionA Two-band mixingB Beyond-mean-field calculations

V Conclusions Acknowledgements References