Shape coexistence in neutron-deficient Hg isotopes studied via lifetime measurements in 184,186Hg and two-state mixing calculations L. P. Gaffney,1,2,∗ M. Hackstein,3,† R. D. Page,1,‡ T. Grahn,1,4 M. Scheck,1,5 P. A. Butler,1 P. F. Bertone,6 N. Bree,2 R. J. Carroll,1 M. P. Carpenter,6 C. J. Chiara,6,7 A. Dewald,3 F. Filmer,1 C. Fransen,3 M. Huyse,2 R. V. F. Janssens,6 D. T. Joss,1 R. Julin,4 F. G. Kondev,8 P. Nieminen,4 J. Pakarinen,1,4,9 S. V. Rigby,1 W. Rother,3 P. Van Duppen,2 H. V. Watkins,1 K. Wrzosek-Lipska,2 and S. Zhu6 1Oliver Lodge Laboratory, University of Liverpool, Liverpool L69 7ZE, United Kingdom 2KU Leuven, Instituut voor Kern- en Stralingsfysica, B-3001 Leuven, Belgium 3Institut fu¨r Kernphysik, Universita¨t zu Ko¨ln, Zu¨lpicher Str. 77, D-50937 K¨oln, Germany 4Department of Physics, University of Jyva¨skyl¨a, P.O. Box 35, FI-40014 Jyva¨skyl¨a, Finland 5Institut fu¨r Kernphysik, TU Darmstadt, D-64289 Darmstadt, Germany 6Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA 7Department of Chemistry and Biochemistry, University of Maryland, College Park, Maryland 20742, USA 4 8Nuclear Engineering Division, Argonne National Laboratory, Argonne, Illinois 60439, USA 1 9CERN-ISOLDE, CERN, CH-1211 Geneva 23, Switzerland 0 (Dated: February 12, 2014) 2 Theneutron-deficientmercuryisotopes,184,186Hg,werestudiedwiththeRecoilDistanceDoppler- b Shift(RDDS)methodusingtheGammaspherearrayandtheKo¨lnPlungerdevice. TheDifferential e F Decay Curve Method (DDCM) was employed to determine the lifetimes of the yrast states in 184,186Hg. An improvement on previously measured values of yrast states up to 8+ is presented as 1 wellasfirstvaluesforthe9 statein184Hgand10+ statein186Hg. B(E2)valuesarecalculatedand 3 1 compared to a two-state mixing model which utilizes the variable moment of inertia (VMI) model, allowing for extraction of spin-dependent mixing strengths and amplitudes. ] x e - I. INTRODUCTION Neutron Number, N l 92 96 100 104 108 112 116 120 124 c nu beNenucolfeiinetxehriebsittiinngnduicffleeraerntstrsuhcatpuerseaetvelorwsiennceergthyehdaivse- 3.0 12+ 12+ 8+ v2 [ cdtwoiuvesee,rnya1so8sf7oHcaigalataerndgdewj1ui8tm5hHpaginodbrtashemervametidceaicnnh-saiqsnougtaeorpeiden scshhhiaafrtpgemeberaae--- y [MeV] 22..05 6+ 8+ 10+8+ 64++ 10+ 6+ 30 surements [1]. Calculations based on Strutinsky’s shell- Energ 1.5 4+ 6+ 2+ 0 correction method [2] interpreted this result as a tran- on 0+ 4+ 2+ .5 sniotuionncefdropmrolaatwe-edaekfloyr-mdeefdorsmhaedpe.obFlautrethteor aisomtoopree sphrioft- xcitati 1.0 1 E measurements reveal that the weakly-deformed oblate 0.5 0 character extends down to A = 182 in the even-mass 4 0+ Hg isotopes [3]. Calculations in these isotopes using 0.0 1 : theNilsson-Strutinskyapproach[4]predicttwodeformed 172 176 180 184 188 192 196 200 204 v minima, where the lowest-energy minimum corresponds Mass Number, A i X toanoblateshape, β (cid:39)−0.15, andthesecondtoamore deformed prolate shape with β (cid:39) 0.27. In shell-model FIG.1. (Coloronline)Levelenergysystematicsofeven-mass r a terms, these minima are associated with a proton zero- mercury isotopes. Red circles refer to the assumed intruder particle-two-hole configuration, π(0p−2h), and a two- states while blue squares refer to the assumed oblate states, proton excitation across the Z = 82 shell gap yielding a guidedbytheresultsofthecalculationsinthisworkandthat ofRef.[5]. ThefigureisanupdatedversionofthatinRef.[6] π(2p−4h) configuration, respectively. using data taken from the NNDC database [7]. Spectroscopy of the even-mass mercury isotopes re- veals a systematic trend of the intruding 0+ band head 2 (shown in Fig. 1) which minimizes in energy near the for A ≤ 186, and the 2+ levels become close enough in neutron mid-shell at N = 104. The excited states built upon these configurations become yrast above Iπ = 4+ energy to mix strongly. Large E0 components in 2+ →2+ transitions indicate 2 1 a large degree of mixing. An attempt to understand the mixing between the bands was made by measuring this ∗ Correspondingauthor(184Hg): Liam.Gaff[email protected] E0 component in-beam in 180Hg [8] and 186Hg [9]. The † Correspondingauthor(186Hg): [email protected] conversion coefficient of this transition can also be mea- ‡ Correspondingauthor(calc.): [email protected] sured following β decay, and there is an effort to provide 2 15 16 (a) (a) 15 16 (b) 356.7 1000 6+→ 4+ 2+→ 0+ 8+→ 6+ 13 409.3 646.0 381.8 13 611.0 14 680000 9-→ 7- sh us sh 11-→ 9- sh 50µm 10+s→h 8+ sh 445.2 14 11 sh 11 452.4 10 255.8 581.6 400 400.3 397.7 603.9 10 12 200 1010.7 9 12 948.9 480.8 754.9 542.0 0 329.1 7 548.8 551.3 8 1244.3 10 2000 (b) 5 273.5 1037.9 10 477.6 V] 571.8 708.8 6 488.9 ke 1500 400µm 6 489.2 8 1/ 1126.0 8 s [ 1000 4 462.6 1193.8 418.3 6 4 597.6 424.2 6 Count 500 356.8 340.1 4 719.8 4 675.1 186.4 2 0 119.4 2 402.7 2 287.0 167.6 0 2 215.6 0 2500 (c) 534.4 2000µm 366.8 405.3 2000 0 0 1500 FIG. 2. Partial level schemes of (a) 184Hg and (b) 186Hg 1000 showingstatesofinterest. DataaretakenfromRefs.[20,21]. 500 0 moreexperimentaldatainthisregion[10]. Mixingofthe 300 350 400 450 500 0+ statescanbequantifiedbycomparisonofρ(E0)2 val- Energy [keV] ues [11, 12] which have been experimentally determined in180Hg[13], 184Hg[14], and188Hg[15]. Asdescribedin FIG. 3. Gamma-ray spectra from the Gammasphere detec- tors at θ ≈ 53◦, gated on the shifted (sh) component of the a recent review on the topic [16], the microscopic shell- 4+ → 2+ transition in 184Hg at a target-to-stopper distance model approach and the theoretical mean-field approach 1 1 of(a)50µm,(b)400µm,and(c)2000µm. Transitionswhich canbothsuccessfullyreproducetheobservedmixing. An feed the 4+ state only have their shifted (sh) component in analysis of α-decay hindrance factors [13, 17] indicates a coincidenc1e, while one also observes the unshifted (us) com- smallerprolatecontributiontotheground-stateband0+ ponent of the 2+ →0+ depopulating transition. 1 1 state. To determine the magnitude and type of deformation of the two bands, their mixing strength, and to test the suppressed, high-purity Ge detectors [22] arranged into picture of shape coexistence, more precise data are re- 17 rings of constant polar angle, θ, with respect to the quired on the absolute transition strengths between the beam. For this experiment, 100 detectors split into 16 excitedstatesofthenucleiinthismassregion. Lifetimes rings were in use. The K¨oln plunger device was installed ofexcitedstatesin180,182Hghavebeenrecentlymeasured at the target position to allow for the Recoil Distance by this collaboration [18, 19]. To extend this knowledge Doppler-Shift (RDDS) lifetime measurements. The dis- andaddressthesemissingdata,lifetimemeasurementsof tance of the 11 mg/cm2-thick Au stopper foil was varied excited states in 184,186Hg have been performed. Partial withrespecttothe0.6mg/cm2 thickSmtargetwithina level schemes of the nuclei studied in this work are given range of 2–2000 µm and data were taken at 12 distances in Fig. 2. for 184Hg (10 for 186Hg). For the analysis of the data, γγ-coincidence matrices werebuiltusingGSSORT[23]andtheROOTframework[24]. II. EXPERIMENTAL DETAILS The data from different HPGe detectors of Gammas- phere were grouped by rings at similar angles. In the Excited states in 184Hg and 186Hg were popu- analysis of 184Hg, the lifetimes are obtained by taking lated in the heavy-ion induced, fusion-evaporation reac- the weighted average of the lifetimes for different rings. tions148Sm(40Ar,4n)184Hgand150Sm(40Ar,4n)186Hg,at Here,itwaspossibletoobtainthevelocityindependently beam energies of 200 MeV and 195 MeV. The average for each ring, utilizing the measured Doppler shift of the recoil velocity was v/c=1.94% and 1.90%, respectively. peaks. In the case of 186Hg, where measurement times The primary beam was provided by the ATLAS facil- wereshorter, thelifetimeswerederiveddirectlyfromthe ityattheArgonneNationalLaboratoryanddeliveredto weighted average of the intensities whilst the velocity is the target position inside the Gammasphere spectrom- obtained by weighting the velocities from the individual eter. The latter nominally consists of 110 Compton- γγ-coincidencespectraforallcombinationsofringpairs. 3 2 Typicalγγ-coincidencespectraobtainedduringtheex- ⌧(6+)=8.82(24)ps perimentarepresentedinFig.3,inwhichitispossibleto 9.6 (a) see the variation of intensity of the fully Doppler-shifted 9.2 component of the depopulating transition, I , with in- sh s] creasing distance. At the smallest distances the flight [p 8.8 times of the nuclei are very short leading to few γ rays ⌧ 8.4 emitted in-flight. Consequently, when gating in the γ-γ matrix, a low number of events are observed, an effect 8.0 present in Fig. 3 (a). Toaccountforthevaryingmeasurementtimeatdiffer- 6000 Ish ent target-to-stopper distances, the intensities had to be 5000 normalized. For184Hg,thedistanceswerenormalizedby (b) 4000 employing a gate on the total intensity of the 340-keV, 6+ → 4+ transition and averaging the intensities of the 3000 1 1 287-keV, 4+ → 2+ and 367-keV, 2+ → 0+ transitions 2000 1 1 1 1 at all angles. For 186Hg, all possible γγ coincidences for 1000 theyrasttransitionsuptothe10+ state, inspectrawith 0 1 θ < 90◦, were used for the normalization. The use of 2000 Ius thecoincidencedifferentialdecaycurvemethod(DDCM) (c) eradicates the influence of recoil de-orientation [25]. 1500 In the analysis of 184Hg, lifetimes were obtained us- 1000 ing the prescriptive coincidence DDCM [26, 27]. While here we explain only the features of the method that 500 are required for this analysis, the full details are con- tainedinRefs.[26,27],towhichthereaderisreferredfor 0 more information. The coincidence technique allows for 10 100 1000 the elimination of systematic errors usually introduced Distance[µm] in RDDS-singles measurements by the unknown feeding history from states that lie higher in energy. The decay FIG. 4. (a) Lifetime values, τ, extracted at each distance in curves,asafunctionofdistance,x,forthisanalysiswere the sensitive region, for the 340-keV, 6+1 → 4+1 transition in 184Hg,measuredinthedetectorsatθ≈53◦. Thedashedand constructed using gates on the shifted component of the dotted lines show the weighted average and the associated yrast transition directly feeding the state of interest, en- uncertainty, respectively. (b) Normalized intensity curve for suring a significant simplification whereby the lifetime is the shifted component, I , fitted (solid line) with a function determinedusingonlytheratiooftheunshifted(I )and sh us consistingofcontinuous,piecewisesecond-orderpolynomials. the time derivative of the fully-shifted (I ) components sh (c) Normalized intensity curve for the unshifted component, of the depopulating transition. I . Thecurvein(c)isproportionaltothederivative ofthat us in(1b2)00andIbusothcurvesarefittedsimultaneously. Accordingto τ(x)= Ius(x) , (1) Eq1.010,0theratioofthese⌧(g4iv+1e)s=th5e.6l(i2fe0t)impseofthestate,shown ddtIsh(x) in (a8)0.0 600 (a) Typical fits of continuously-connected second-order polynomials, performed with Napatau [28, 29], are illus- 400 III. RESULTS trated in Fig. 4. The lifetime is determined at every 200 distance and should sit at a constant value. Deviations 0 Th8.e0 final weighted averages of the mean lifetimes from this behavior indicate systematic effects, which can of all states studied are shown in Table I along with be identified easily with this method. The weighted av- 6.0 the transition strengths, B(E2), and absolute transi- erage, τav, is taken of the points inside of the sensitivity tiops]na4.l0quadrupole moments, |Q |, of the depopulating region, i.e. where the derivative of the decay curve is [ t largest. yr⌧as2t.0transitions. Transition quadrupole mome(nbt)s, Qt, are related to the B(E2) values assuming a rotating staLtiefewtiemreessimofiltahrleyydreatsetrmstianteedswoifth18t6hHegexucpepttoiotnheof1t0h+1e quad0r.0up0olede1f0o0rmed2n0u0cleus30u0singt4h0e0rota5t0io0nalm60o0del: Distance[µm] a4+1ndst2a+1te.→In0t+1histrcaanses,ittiohnesneraenrddeoruedbleittopfrtohbele4m+1at→ic2t+1o B(E2;I →I(cid:48))= 165π(cid:104)I020|I(cid:48)0(cid:105)2Q2t, (2) use the simple “gate from above” method. Instead, the method of “gating from below” [30] was used. The cor- where (cid:104)I020|I(cid:48)0(cid:105) is a Clebsch-Gordan coefficient and responding τ plot for the 4+ state is found in Fig. 5. I(cid:48) =I−2. 1 The intensities involving the 6+ and 4+ analysis were In184Hg,thesevenindependentmeasurementsofeach 1 1 corrected for a contamination from the 15− →13− tran- ofthelifetimesoftheeven-spinyraststatesarepresented 2 2 sition, using a gate on the 13− →11− transition. in Fig. 6, as a function of the “ring” angle at which they 2 2 2 ⌧(6+)=8.82(24)ps 9.6 (a) 9.2 s] [p 8.8 ⌧ 8.4 8.0 I 6000 sh 5000 (b) 4000 3000 2000 1000 0 I 2000 us (c) 1500 1000 500 0 10 100 1000 Distance[µm] 4 1200 Ius 1000 ⌧(4+1)=5.6(20)ps 10 24++ →→ 02++ 6+ → 4+ 108++ →→ 68++ 800 9 600 (a) 400 8 200 b] e 7 8.00 | [𝑄𝑡 | 6 6.0 ps]4.0 5 [ ⌧2.0 (b) 4 0.00 100 200 300 400 500 600 3 180 182 184 186 Distance[µm] Mass number, A FIG.5. (a)Thenormalizedintensityoftheunshiftedcompo- FIG. 7. (Color online) Experimental |Q | values, extracted nent of the 4+ → 2+ depopulating transition in 186Hg. The t 1 1 from measured lifetimes, for yrast transitions in mercury nu- fittedlineisasinFig.4. (b)Lifetimevaluesextractedateach clei as a function of mass number, A. Data for A=180,182 distance in the sensitive region. The solid and dashed lines aretakenfromRef.[18]andthe10+ →8+ valueforA=184 show the weighted average and the associated uncertainty. is taken from Ref [33]. Some markers are slightly offset from integer A values to maintain clarity. 45 IV. DISCUSSION 40 𝜏(2+) = 35.7(15) 35 A. Yrast states in 184Hg and 186Hg 30 s] p e, [Lifetim𝜏 122505 𝜏(4+) = 30.2(10) 𝜏𝜏𝜏𝜏((((2468++++)))) sispoiTtnorapynerssaistwtiohnsetaraleteq1su8aa0drre≤upgioAvleen≤mion1m8F6e.ingt.sT,h7|Qefto2|,r+ftoshrteatthmeeserescvhueornwy- no strong variation in |Q | with mass number, whereas t 10 𝜏(6+) = 8.7(4) the collectivity of the 4+ states reduces with increasing mass number. This can be compared to the energy level 5 𝜏(8+) = 3.19(14) systematics in Fig. 1 where the energy of the intruder 0 states reaches a minimum at A=182. 0 20 40 60 80 100 120 140 160 180 Polar angle in Gammasphere [deg] FIG. 6. (Color online) Lifetimes of yrast states in 184Hg as B. The 9 state in 184Hg 3 a function of angle. The weighted average and 1σ uncer- tainty, calculated from the weighted standard deviation, for each state are denoted by the solid and dashed lines, respec- After the even-spin, positive-parity yrast band in tively. Please note, the τ(4+) values are offset by 5◦ on the 184Hg, the most populated band is the odd-spin rota- x-axistomaintainclarityandsomeindividualerrorbarsmay tional band built upon the I = 5 state, observed at be smaller than the marker size. 1.848MeV,whichbecomesyrastataround4MeV.Anal- ogous bands have been observed in the neighboring iso- topes, specifically 178Hg [34] and 180Hg [35] and their structure discussed in terms of octupole correlations. were determined in Gammasphere. Lifetime measurements of states in this band in 182Hg have been performed and a consistency in the structure It is worth noting that the new lifetime for 4+ state of these states has been extended to 184Hg using the en- 1 in 186Hg is smaller than the previously measured value ergy displacement of states differing by 3h¯ [19]. The fromRef.[31]. Thediscrepancyinthefirstmeasurement quadrupolemomentmeasuredhereforthe9 →7 tran- 3 3 is likely due to complications that the authors encoun- sition in 184Hg, |Q | = 5.6(7) eb, is similar to that of t tered in resolving the doublet and subsequent assump- the even-spin yrast band, |Q | (cid:39) 7.7 eb, although it is t tions which were made. smaller than those measured in the lighter isotopes. 5 TABLE I. Properties of the states investigated in this study. The uncertainties presented on τ represent the 1σ statistical av error and include an additional systematic uncertainty which accounts for the choice of the fitting function and relativistic effects [27], typically ≤ 3%. Gamma-ray energies (E ) and branching fractions (b.f.) of the depopulating γ-ray transitions γ (corrected for internal conversion), as well as spin and parity (Iπ) values, are taken from Refs. [20, 21]. In cases where only one depopulating transition is observed b.f. is assumed to be equal to unity. τ values from Refs. [31–33] are shown for prev comparison. Iπ (¯h) E (keV) b.f. τ (ps) τ (ps) B(E2)↓ (W.u.) |Q | (eb) γ av prev t 184Hg 2+ 366.8 1 35.7(15) 30(7) 52(2) 4.04(8) 1 4+ 287.0 0.959(4) 30.2(10) 32.8(34) 191(6) 6.46(11) 1 6+ 340.1 1 8.7(4) 8.1(31) 308(15) 7.81(19) 1 8+ 418.3 1 3.19(14) 2.9+1.1 309(13) 7.65(17) 1 −1.6 9 329.1 0.65(16) 12.1(8) − 169(40) 5.6(7) 3 186Hg 2+ 405.3 1 24(3) 26(4) 47(6) 3.9(2) 1 4+ 402.7 0.93(2) 5.6(20) 13(4) 200(70) 6.6(12) 1 6+ 356.8 1 9.1(4) 7(3) 231(10) 6.82(15) 1 8+ 424.2 1 4.5(3) ≈4 202(14) 6.2(2) 1 10+ 488.9 1 1.9(2) − 238(25) 6.7(4) 1 C. Two-state mixing calculations calculation will be discussed. Low-lying levels in 184,186Hg have also been inter- preted assuming the mixing of spherical and deformed Inordertoshedlightonthepropertiesofthecoexisting states [39], while an alpha-plus-rotor model has been structures in these light mercury isotopes, phenomeno- usedtoextractspin-dependentinteractionstrengthsand logical two-band mixing calculations have been carried mixing amplitudes in 182,184,186Hg [5]. The latter study out using the assumption of a spin-independent interac- predicted a contribution of the more-strongly deformed tion between two rotational structures. In the calcula- structuretothe2+statein182Hgof76%,comparingvery tions, the variable moment of inertia (VMI) model [36] 1 well to the value of 71% obtained in this work. We note was used to fit known level energies of rotational bands here that this contribution drops to only 2.3% for the built upon the first two 0+ states, up to and including same state in 188Hg. The corresponding isotones in the Iπ =10+ and 4+ for yrast and non-yrast bands, respec- platinumnucleiwerealsointerpretedrecentlyusingtwo- tively. Employing the method of Lane et al. [37], one band mixing calculations and qualitatively similar con- can derive the wave-function amplitudes of the two con- clusions were drawn regarding a strong degree of mixing figurations from the mixing strengths. These are shown for the low-spin states [38, 40]. in Table II along with the mixing strength, V, for each It is possible to determine the transitional quadrupole isotope and the band-head energies of the two configura- moment of the unperturbed I →I−2 transitions in the tions. normal (n) and intruder bands (i), Q , using an aver- I Similar mixing calculations have previously been per- age of the moment of inertia of the two states, J(cid:48), such formed for 180,182,184Hg [38]. The present results for I that [36] 180,182Hg differ due to the inclusion of non-yrast states identified in recent studies [8, 10], which provide addi- (cid:113) (cid:112) tional constraints on the calculations. The calculation QI =k JI(cid:48) =k (JI +JI−2)/2, (3) for 184Hg presented in Table II is essentially identical to that in Ref. [38]. It places the I = 2 member of where J is the moment of inertia for a pure state with I the intruder band just 5 keV above the corresponding spin I, extracted from the fit. An evaluated value of the state in the normal band before mixing. This leads to constant, fitted to data in the neutron-deficient A=170 almost complete mixing between the states, producing a region,k =45(2)ebkeV−1/2 [41],wasusedinthisstudy. first-excited2+ statewhichcomprises51%ofthenormal Combining knowledge of the wave-function amplitudes configuration and 49% of the intruder configuration. An withtheintrinsicquadrupolemomentsofthepurestates, alternative calculation in which the order of these states Q , it is possible to extract the B(E2;I →I−2) values I is reversed was also performed and yielded very similar ofthemixedstates[37]. Therelativesignoftheintrinsic parameters to those presented in Table II. In this sce- quadrupole moments of the two configurations must be nario the unmixed states are nearly degenerate, so the assumed to be positive or negative and was found to be degree of mixing is even greater. The resulting B(E2) best reproduce the data when positive. This feature has values(presentedlaterinFig.8)arenotsignificantlydif- beennotedinpreviouscalculations[37,42]andisatodds ferent,soinwhatfollowsonlytheresultsfromtheformer with what is expected from the rotational model when 6 500 TABLE II. Wave-function amplitudes of the normal configu- ration, αI, at each spin, I, and spin-independent interaction 400 strengths between members of the normal (n) band and in- truder (i) band, V = |(cid:104)I |V |I (cid:105)| (equal for all values of I), 300 n I i calculated using the model described in the text. The ex- tracted unperturbed band-head energies are denoted by En 200 0 and E0i for the normal and intruder bands, respectively. 100 180Hg Nucleus I [h¯] |αI| 0 0 0.9799 500 V =82.1 keV 2 0.7722 400 4 0.1571 180Hg En =16.7 keV 0 300 6 0.0817 8 0.0546 200 E0i =403.3 keV 10 0.0408 W.u.] 100 182Hg 0 0.9606 2) [ V =89.4 keV -(cid:2) 0 2 0.5382 (cid:1) 182Hg E0n =25.9 keV 46 00..11708013 B(2;(cid:1)(cid:2)(cid:3) 450000 Ei =309.0 keV 8 0.0697 300 0 10 0.0534 200 V =84.7 keV 0 0.9725 100 184Hg 2 0.7172 4 0.2014 0 184Hg En =20.4 keV 0 6 0.1025 400 8 0.0679 Ei =353.4 keV 0 10 0.0506 300 0 0.9915 200 V =66.3 keV 2 0.9506 100 186Hg 4 0.2604 186Hg En =8.9 keV 0 6 0.0978 0 8 0.0587 2 4 6 8 10 E0i =505.4 keV (cid:2)(cid:3)[(cid:1)] 10 0.0416 0 0.9954 V =76.3 keV FIG. 8. (Color online) Experimental B(E2) values measured 2 0.9884 in this work (points) plotted as a function of spin and com- 4 0.8933 pared to those extracted from the mixing calculations (solid 188Hg En =7.8 keV 0 black line, dashed line represents the uncertainty in the con- 6 0.2467 stant, k). For reference, the intra-band B(E2) values cal- 8 0.1095 Ei =791.8 keV culated for the pure unperturbed normal (red) and intruder 0 10 0.0691 (blue) bands are also shown. Data for 180,182Hg are taken from Ref. [18] and the point at I = 10 h¯ in 184Hg is taken from Ref. [33]. two bands with the same K quantum number have an oppositesignofdeformation,i.e. anoblateandaprolate B(E2;2+ →0+)valuesaresimilarineachisotope, while 1 1 band. the B(E2;4+ →2+) varies and, in each case, is not con- 1 1 The results of the calculations for the yrast sequences sistent with a transition within either of the pure bands. arecomparedinFig.8withthoseextractedfromthelife- The low B(E2;2+ → 0+) value is interpreted as being 1 1 times measured in this work. Good agreement is found due to a transition between the weakly-deformed oblate forthemajorityofyrasttransitions,evenforcaseswhere 0+ ground state and a more strongly-deformed prolate the states are strongly mixed. The observed discrep- 2+ state[18]. However,theadmixtureofthenormaland ancies for I ≥ 6 in 186Hg may indicate a breakdown intruderconfigurationsforthe0+ and2+ statesisunique of the simple two-band picture as other structures be- ineachisotopeandasimilarB(E2)valueisnotnecessar- gin to influence the yrast states [43, 44]. For the nu- ilyindicativeofsimilarstructuresacrossthemassrange. clei presented in Fig. 8, the experimental and calculated InFig.9,calculatedB(E2)valuesareplottedasafunc- 7 -200 0 200 400 ∆𝐸2 [keV] tinct differences in deformation for the assigned normal 350 182 184 𝐼 = 2+ and intruder bands could be shown. The lifetime of the 300 180 186 𝐼 = 4+ 4+ statein186Hgwasfoundtobeshorterthantheprevi- 1 𝐼 = 6+ ously measured value. Lifetimes of the 9 state in 184Hg u.] 250 and the 10+ state in 186Hg have been m3easured for the .W -2) [𝐼 200 fivirosutstimmeea.suTrehmeeontthse,rwlhifielteimthees uanrececrotanisnisttyenctouwldithbeprree-- → 150 duced significantly. These more precise lifetime values, 𝐼 2; with those of Ref. [18], have been a vital input to the (𝐵𝐸 100 analysisofCoulombexcitationexperimentsattheREX- ISOLDE facility [45, 46]. 50 Rotational bands built upon the first two Iπ = 0+ states have been considered in terms of a two-state mix- 0 0 200 400 600 800 ing model and the mixing amplitudes of the two con- Pure-band energy difference, ∆𝐸 = 𝐸i - 𝐸n [keV] figurations extracted as a function of spin. It is ob- 0 0 0 served that, while the ground state remains composed FIG. 9. (Color online) Representative B(E2;I+ →(I−2)+) ofpredominantlyoneconfiguration,namelytheassumed 1 1 valuesasafunctionoftheenergyseparationofthepureband- weakly-deformed normal structure, in all of the even- heads, ∆E0. The curves for I =2,4,6 are calculated assum- mass mercury isotopes considered (180 ≤ A ≤ 188), the ing the mixing strength and moments of inertia are equal to first excited 2+ state changes dramatically in its com- that extracted for 184Hg. As a guide, the energy difference position. This is in contrast to a na¨ıve interpretation between the pure 2+ states, ∆E2, under this assumption is emanating from the systematics of the 2+ level energies alsoshown. Thedashedverticallinesshowtheextracted∆E 0 and B(E2;2+ → 0+) values, which are both strikingly values for each of the four isotopes. similar across the mass range. tion of the energy difference between the pure, unmixed ACKNOWLEDGMENTS 0+ band heads. The parameters used in the calculations are those of 184Hg, but since they are not too dissimilar The authors would like to thank the operators of the for all of the isotopes, the curves can be considered rep- ATLAS facility at ANL for providing the beams. This resentative of the mercury isotopes around the neutron work was supported by the United Kingdom Science mid-shell. The four isotopes compared have ∆E values 0 and Technology Facilities Council, by the German DFG, intherangeof250–550keV,markedbytheverticallines grantNo. DE1516/1-1,bytheAcademyofFinland,con- in Fig. 9. One clearly observes that above 200 keV the B(E2;2+ →0+)valuesremainconstant,eventhoughthe tract number 131665, by the German BMBF, grant No. square o1f the m1 ixing amplitude of the I = 2 states, α2, 06 KY 7153, by US Department of Energy, Office of Nu- 2 clearPhysics,underContractNo. DE-AC02-06CH11357 drops from 0.80 at 200 keV down to 0.015 at 1 MeV. In contrast,theB(E2;4+ →2+)valuesvarysignificantlyin as well as Grant No. DE-FG02-94ER40834, by FWO- 1 1 Vlaanderen (Belgium), by GOA/2010/010 (BOF KU the range of interest and are much more sensitive to α , 2 while the B(E2;6+ →4+) values become more sensitive Leuven), by the Interuniversity Attraction Poles Pro- 1 1 gramme initiated by the Belgian Science Policy Office at larger band-head energy differences. (BriX network P7/12), by the European Commission within the Seventh Framework Programme through I3- ENSAR (contract no. RII3-CT-2010-262010) and by a V. SUMMARY AND CONCLUSIONS Marie Curie Intra-European Fellowship of the European Community’s7thFrameworkProgrammeundercontract Lifetimes of excited states have been measured by em- number (PIEF-GA-2008-219175). L.P.G. acknowledges ployingtheRDDStechnique. YraststatesuptoIπ =8+ support from FWO-Vlaanderen (Belgium) as an FWO in 184Hg and Iπ =10+ in 186Hg have been studied. Dis- Pegasus Marie Curie Fellow. [1] J.Bonn,G.Huber,H.-J.Kluge,L.Kugler,andE.Otten, [5] J. D. Richards, T. Berggren, C. R. Bingham, Physics Letters B 38, 308 (1972). W. Nazarewicz, and J. Wauters, Physical Review C 56, [2] S.FrauendorfandV.Pashkkevich, PhysicsLettersB55, 1389 (1997). 365 (1975). [6] R. Julin, K. Helariutta, and M. Muikku, Journal of [3] G. Ulm et al., Zeitschrift fu¨r Physik A Atomic Nuclei PhysicsG:NuclearandParticlePhysics27,R109(2001). 325, 247 (1986). [7] J.Tuli,EvaluatedNuclearStructureDataFile(ENSDF), [4] W. Nazarewicz, Physics Letters B 305, 195 (1993). 2012. 8 [8] R.D.Pageetal., PhysicalReviewC84,034308(2011). and Nuclear Physics 67, 786 (2012). [9] M. Scheck et al., Physical Review C 83, 037303 (2011). [28] B. Saha, Napatau or Tk-Lifetime-Analysis (unpub- [10] J.Elseviersetal., PhysicalReviewC84,034307(2011). lished). [11] J. Kantele et al., Zeitschrift fu¨r Physik A Atoms and [29] A.Dewald,S.Harissopulos,andP.Brentano, Zeitschrift Nuclei 289, 157 (1979). fu¨r Physik A Atomic Nucleir Physik 334, 163 (1989). [12] K.HeydeandR.A.Meyer, PhysicalReviewC37,2170 [30] P. Petkov, A. Dewald, and P. von Brentano, Nuclear (1988). InstrumentsandMethodsinPhysicsResearchSectionA [13] J. Wauters et al., Physical Review C 50, 2768 (1994). 457, 527 (2001). [14] J. Cole et al., Physical Review Letters 37, 1185 (1976). [31] D.Proetel,R.M.Diamond,andF.S.Stephens, Physics [15] P. Joshi et al., International Journal of Modern Physics Letters B 48, 102 (1974). E 3, 757 (1994). [32] N. Rud, D. Ward, H. Andrews, R. Graham, and [16] K. Heyde and J. L. Wood, Reviews of Modern Physics J. Geiger, Physical Review Letters 31, 1421 (1973). 83, 1467 (2011). [33] W. C. Ma et al., Physics Letters B 167, 277 (1986). [17] J. Wauters et al., Zeitschrift fu¨r Physik A Hadrons and [34] F. Kondev et al., Physical Review C 61, 011303 (1999). Nuclei 345, 21 (1993). [35] F. Kondev et al., Physical Review C 62, 044305 (2000). [18] T. Grahn et al., Physical Review C 80, 014324 (2009). [36] M.A.J.Mariscotti,G.Scharff-Goldhaber,andB.Buck, [19] M. Scheck et al., Physical Review C 81, 014310 (2010). Physical Review 178, 1864 (1969). [20] J. Deng et al., Physical Review C 52, 595 (1995). [37] G. J. Lane et al., Nuclear Physics A 589, 129 (1995). [21] W. Ma et al., Physical Review C 47, R5 (1993). [38] G. D. Dracoulis, Physical Review C 49, 3324 (1994). [22] I.-Y. Lee, Nuclear Physics A 520, c641 (1990). [39] F.DickmannandK.Dietrich, Zeitschriftfu¨rPhysik271, [23] T. Lauritsen, GSSort (unpublished). 417 (1974). [24] R. Brun and F. Rademakers, Nuclear Instruments and [40] J.C.Walpeetal., PhysicalReviewC85,057302(2012). Methods in Physics Research Section A 389, 81 (1997). [41] G.D.Dracoulisetal.,NuclearPhysicsA486,414(1988). [25] P. Petkov, Nuclear Instruments and Methods in Physics [42] M. Guttormsen, Physics Letters B 105, 99 (1981). ResearchSectionA:Accelerators,Spectrometers,Detec- [43] H.Helppi,S.K.Saha,P.J.Daly,S.R.Faber,andT.L. tors and Associated Equipment 349, 289 (1994). Khoo, Physical Review C 28, 1382 (1983). [26] G. Bo¨hm, A. Dewald, P. Petkov, and P. von Brentano, [44] J. Delaroche et al., Physical Review C 50, 2332 (1994). Nuclear Instruments and Methods in Physics Research [45] A. Petts et al., Lifetime Measurements and Coulomb Section A: Accelerators, Spectrometers, Detectors and Excitation of Light Hg Nuclei, in AIP Conference Pro- Associated Equipment 329, 248 (1993). ceedings, volume 1090, pages 414–418, AIP, 2009. [27] A.Dewald,O.M¨oller,andP.Petkov,ProgressinParticle [46] N. Bree et al., Physical Review Letters (submitted) (2014).