Shape Coexistence and Mixing of Low-Lying $0^+$ States in $^{96}$Sr

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Shape Coexistence and Mixing of Low-Lying 0

States in

96

_Sr

S. Cruz, P.C. Bender, R. Krucken, K. Wimmer, F. Ames, C. Andreoiu, R.A.E.

Austin, C.S. Bancroft, R. Braid, T. Bruhn, et al.

To cite this version:

S. Cruz, P.C. Bender, R. Krucken, K. Wimmer, F. Ames, et al.. Shape Coexistence and Mixing

of Low-Lying 0

+

_{States in}

96

_{Sr. Phys.Lett.B, 2018, 786, pp.94-99. �10.1016/j.physletb.2018.09.031�.}

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Shape

coexistence

and

mixing

of

low-lying

0

+

states

in

96

Sr

S. Cruz

a

,

b

,

P.C. Bender

b

,

R. Krücken

a

,

b

,

K. Wimmer

c

,

d

,

∗

,

F. Ames

b

,

C. Andreoiu

e

,

R.A.E. Austin

f

,

C.S. Bancroft

d

,

R. Braid

d

,

T. Bruhn

b

,

W.N. Catford

g

,

A. Cheeseman

b

,

A. Chester

e

,

D.S. Cross

e

,

C.Aa. Diget

h

,

T. Drake

i

,

A.B. Garnsworthy

b

,

G. Hackman

b

,

R. Kanungo

f

,

b

,

A. Knapton

g

,

W. Korten

j

,

b

,

K. Kuhn

d

,

J. Lassen

b

,

R. Laxdal

b

,

M. Marchetto

b

,

A. Matta

g

,

k

,

D. Miller

b

,

M. Moukaddam

b

,

N.A. Orr

k

,

N. Sachmpazidi

d

,

A. Sanetullaev

f

,

b

,

C.E. Svensson

l

,

N. Terpstra

d

,

C. Unsworth

b

,

P.J. Voss

e

a_{DepartmentofPhysicsandAstronomy,UniversityofBritishColumbia,Vancouver,BCV6T1Z4,Canada} b_{TRIUMF,Vancouver,BCV6T2A3,Canada}

c_{DepartmentofPhysics,TheUniversityofTokyo,7-3-1Hongo,Bunkyo-ku,Tokyo113-0033,Japan} d_{DepartmentofPhysics,CentralMichiganUniversity,MtPleasant,MI48859,USA}

e_{DepartmentofChemistry,SimonFraserUniversity,Burnaby,BCV5A1S6,Canada} f_{DepartmentofAstronomyandPhysics,SaintMary’sUniversity,Halifax,NSB3H3C2,Canada} g_{DepartmentofPhysics,UniversityofSurrey,Guildford,Surrey,GU27XH,UnitedKingdom} h_{DepartmentofPhysics,UniversityofYork,York,YO105DD,UnitedKingdom}

i_{DepartmentofPhysics,UniversityofToronto,Toronto,ONM5S1A7,Canada} j_{IRFU,CEA,UniversitéParis-Saclay,F-91191Gif-sur-Yvette,France}

k_{LPC,ENSICAEN,CNRS/IN2P3,UNICAEN,NormandieUniversité,14050Caencedex,France} l_{DepartmentofPhysics,UniversityofGuelph,Guelph,ON,N1G2W1,Canada}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received7March2018

Receivedinrevisedform12September 2018

Accepted16September2018 Availableonline27September2018 Editor:V.Metag

Keywords:

Single-particlestructure Transferreaction Shapecoexistence

Thelowenergyexcited0+_2,3statesin96_{Srareamongstthemostprominentexamplesofshapecoexistence}

acrossthenuclearlandscape.Inthiswork,theneutron[2s1/2]2contentofthe0+1,2,3statesin96Srwas

determinedby meansofthe d(95_{Sr, p) transferreactionatthe TRIUMF-ISAC2facilityusingtheSHARC}

andTIGRESSarrays.Spectroscopicfactorsof0.19(3)and0.22(3)wereextractedforthe96_Srgroundand

1229 keV 0+ states,respectively, byfitting the experimental angular distributions to DWBAreaction model calculations.Adetailedanalysisofthe

γ

-decayoftheisomeric0+₃ statewasusedtodetermine aspectroscopic factorof0.33(13). Theexperimentalresults are comparedtoshell modelcalculations, whichpredict negligiblespectroscopicstrengthfor theexcited0+ statesin96_{Sr. Thestrengths ofthe}

excited0+_2,3stateswerealsoanalyzedwithinatwo-levelmixingmodelandareconsistentwithamixing strength ofa2_{=0.40(14) and adifference inintrinsic deformations of |β|=0.31(3).These results}

suggestcoexistenceofthreedifferentconfigurationsin96Srandstrongshapemixingofthetwoexcited 0+states.

©2018TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

Describingtheshapeevolutionofatomicnucleipresentsa chal-lengetomodernnuclearstructuretheory.Theshapeofthenucleus isaresultofadelicateinterplaybetweenmacroscopic,liquid drop-like andmicroscopic shell structure effects. Nuclei witha closed shell configurationare spherical intheir ground states,butaway frommagicnumbersdeformedgroundstatesareobserved.The de-greeofdeformation resultsfromtheinteractionbetweenprotons

*

Correspondingauthor.

E-mailaddress:wimmer@phys.s.u-tokyo.ac.jp(K. Wimmer).

and neutrons depend on the exact occupation of single-particle orbitals near the Fermi surface. Therefore, small changes in the nucleonnumbercanleadtorapidchanges inboththemagnitude and type ofdeformation. One of the mostdramatic examples is the region of neutron-rich Zr( Z

=

40) and Sr ( Z

=

38) isotopes. While the properties of Zr and Sr nuclei with N

≤

58 indicate sphericalgroundstates,withtheadditionofjusttwoneutronsthe groundstatesbecome stronglydeformedfor N

=

60 andbeyond. The nucleiat thisshape transitionaround N

=

60 exhibit shape coexistence [1] withlow-lyingexcited deformed(spherical)states

https://doi.org/10.1016/j.physletb.2018.09.031

0370-2693/©2018TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

fornucleiwithN

≤

58 (N

≥

60).Thesuddenonsetofdeformation in100Zrhasbeenexplainedbythestrongresidualinteraction be-tweentheproton–neutronspin-orbitpartnerorbitals

π

[

0g9/2]and

ν

[

0g7/2].Whileinanindependentparticlepicturethe

π

[

0g9/2]

or-bitalis completely empty in Zr, addingneutrons to the

ν

[

0g7/2]

orbital enables the promotion of protons from the lower lying orbitalstothedeformation-driving

π

[

0g9/2] orbital [2,3]. The

re-sultingsuddentransitionfromsphericaltodeformedgroundstates and the emergence of shape-coexisting states in the vicinity of

N

=

60 and Z

=

40 hasbeenasubjectofconsiderableinterestfor manyyears,boththeoretically [2–21] andexperimentally[22–29]. InSr, theshape transitionisevidentthroughmeasurementsof binding energies [30], charge radii [23], excitation energies and quadrupole transition probabilities of low-lying states [24,27,28]. Low-lying0+stateswithstrongelectricmonopole(E0)transitions betweenthemindicatethecoexistenceofstateswithdifferent in-trinsicdeformationsortheoccurrenceofstronglymixed configura-tions [31].In96Sr(N

=

58)thelow-lying1229and1465 keV0+_2,3 statesareassociatedwithshape-coexistence,asevidencedby the verystrongmonopoletransitionstrength

ρ

2

₍

_E0

₎

₌

₀

_.

₁₈₅

₍

₅₀

₎

_[32]

betweenthem.However, themeasuredlifetimesandextracted E0

andE2 transitionstrengthsforthedecayofthe0+₂ and0+₃ states do not allow the conclusive determination ofthe mixing ampli-tudes betweenthe two excited 0+ states ortheir relative defor-mation.Thespectroscopicquadrupolemomentsofthe2+₁ statein

96_Srandthe2+

2 statein98Srwerefoundtobeverysimilar [27,28],

indicating that these states possess similar underlying structure. Themixing ofthe two coexistingshapes in98Sr isweak despite theirproximity [26,27].

Severaltheoreticalstudies have providedinformation on elec-tromagnetictransitionprobabilitiesin96Sr.Thelow-lying0+states in 96Sr have been studied using the complex excited VAMPIR method with a realistic effective interaction in a large model space [20].In that work,the lowestthree 0+ stateswere associ-atedwithtripleshapecoexistenceofspherical,prolate,andoblate configurations.However,thestrong E0 transitionbetweenthe 0+₂ and0+₃ statesin96Srwas notreproduced.Thebeyondmeanfield calculationswiththe Gogny D1Sinteraction of Refs. [27,28] pre-dict two excited bands in 96Sr with only moderate deformation butsubstantial triaxiality. Recently, Monte-Carlo shell model cal-culationsin a large modelspace were ableto reproduce the en-ergylevelsandB

(

E2

)

valuesoflow-lyingstatesintheZrisotopic chain [21]. When extended to the Sr isotopes [29], these calcu-lationsdescribe well the levelschemes and B

(

E2

)

valuesof 94_Sr

and98Sr butpredict alreadysubstantial deformationfor96Sr. For all these theoretical calculations the agreement withthe experi-mentalinformationismuchbetterin98Srand98Zrthan96Sr.

Inthis Letter,we report onthe first investigation ofthe low-lying 0+_1,2,3 states in 96_{Sr through the d(}95_{Sr, p) transfer reaction}

at5.5 A MeVininversekinematics,providingcrucialinsightsinto theshapemixinganddifferencesindeformationofthecoexisting shapes in 96_{Sr. In contrast to the experimental work performed}

to date, we investigate the shape transition in 96Sr from a dif-ferent perspective and study the single-particle properties. Since the95Sr 1

/

2+ groundstate isdominatedby the

[

2s1/2] spherical

single particle configuration,as we demonstrated in the present experimentalcampaign [33],the



=

0 transferinthed(95_{Sr, p)}

re-actionprobesthecomponentofthe

[

2s1

/

2

]

2 configurationinthe 0+ state wave functions. Our work allowed for the first time to establishthemixingbetweenthedifferentshapesin96_Sr.The

re-sultsindicatea strongmixingofthetwo excited 0+ stateswhile theweakpopulationofthegroundsstatehintstothecoexistence ofthreeshapesin96_Sr.

The present experiment was performed at the TRIUMF-ISAC2 facility [34] where a 95_{Sr beam was produced by impinging a}

Fig. 1. Excitation energyspectrumof96_{Srobtainedfromenergiesand anglesof}

protonsemittedatbackwardlaboratoryangles.Thetotalspectrum(filledpoints) includestheinformationfromallmeasuredprotonsintheangularrangeθlab>90◦

whilethegatedspectrum(openpoints),correctedfortheefficiencyofTIGRESS,is additionallygatedoncoincident414 keVγ-raysfromthe0+2→2+1 transition.The

fits(redsolidlines)includepeaks(bluedashedline)correspondingtothe0 keV (0+₁),1229(0+₂),1465(0+₃)and2084 keV(1+,2+)96_{Srstates,aswellasa}

con-tinuousbackgroundforthetotalspectrum.Notethatforthe 0+₃ statetheγ-ray detectionefficiencyislowerduetothe6.7(10) nshalf-lifeandthe 38% branch-ingratiotothe0+2 state [39].(Forinterpretationofthecolorsinthefigure(s),the

readerisreferredtothewebversionofthisarticle.)

480MeV protonbeamwithan intensityof10

μ

A on aUCx

tar-get.Theneutron-richSrisotopeswereproduced throughuranium fission, were laser ionized, mass separated and transported to a charge state booster [35]. The beam ( Q

=

16+) was transported to the ISAC2 facility where its kinetic energy was increased to 5.5 A MeV usingthe superconducting linearaccelerator [34].The post-accelerated 95Sr beam was delivered for approximately 2.5 dayswith an averageintensityof 1

.

5

×

106 particlesper second. The beam impinged upon a 0.44(4) mg/cm2_{, 92(1)% deuterated}

polyethylene(CD2)target,mountedinthecenter oftheSHARC

sili-condetectorarray [36].SHARC(SiliconHighly-segmentedArrayfor Reactionsand Coulex)is a compactarrangement ofdouble-sided siliconstripdetectorswhichisoptimizedforhighgeometrical effi-ciencyandexcellentspatialresolution,with

θ

lab

≥

1◦.TheSHARC

array was surrounded by the TIGRESS

γ

-ray detector array,with 12HPGeCompton-suppressedcloverdetectorsarrangedina com-pact hemispherical arrangement with approximately 50% of 4

π

geometrical coverage [37]. The beam composition was measured periodically throughout the experiment using a Bragg ionization detector [38],whichwaspositionedadjacenttotheTIGRESS exper-imental station.Thecomposition ofthe A

=

95 beamwas 95(3)%

95_Sr.

The excitation energy ofstates in 96_{Sr populated through the}

d(95Sr, p)reactionwasdeterminedbymeasuringtheprotonenergy andscattering angle.Fig.1 showstwo reconstructed 96Sr excita-tionenergy(Ex)spectraproducedusingonlydatafrombackwards

laboratory angles (

θ

lab

>

90◦) as the excitation energy resolution

is improved in this angular range. The direct population of the 0+₁ 96Sr ground state is clearly visible while the large

β

-decay background(fromaccidentallystopped95_{Sr)anddensityofstates}

made it impossible to resolve higher lying states. Excited states were thus identified usingthe de-excitation

γ

-ray in addition to an Ex gate. A

γ

-ray gate on the 414 keV 0+2

→

2+1 transition

was usedtoproducethe

γ

-rayefficiency-corrected excitation en-ergyspectrum(openpoints).Inthisspectrum,the1229 keV(0+₂), 1465 keV (0+₃) and 2084 keV (1+

,

2+) 96_{Sr states could be}

re-solved using a fit (blue dashed lines) that utilized a fixed res-olution (FWHM

=

425 keV), determined from the groundstate, andtheknownexcitation energiesofthe states.Fig.2showsthe Doppler-reconstructed

γ

-ray energy spectrum forTIGRESS detec-tors positioned at backward angles (

θ

lab

>

120◦) in coincidence

Fig. 2.γ-rayenergyspectrumcoincidentwith96_{Srstatesintherange800<E} x<

1900 keV.OnlyTIGRESSdetectorspositionedatθlab>120◦wereused.Transitions

markedwithastararefromthedecayofthe1507,1995,2084and2113 keVstates whichwerepartiallyincludedintheexcitationenergygate.Thesestateswillbe dis-cussedindetailinaforthcomingpublication.Thetransitionmarkedwithatriangle couldnotbeidentified.Apartialexperimentallevelschemeisshownontheright, includingonlystatesandtransitionsrelevantforthisworkaswellasthe235 keV

E0 transition.

withprotons. An excitation energy gate of 800

<

Ex

<

1900 keV

wasutilizedtoincludeprotonsassociatedwiththedirect popula-tionofthe1229 keV(0+₂)and1465 keV(0+₃)states.

For the Doppler reconstruction, the

γ

rays were assumed to be emittedatthe centerofthetarget. Thisis notcorrectfor the 650 keV transition from the long-lived (6.7 ns [39]) 1465 keV 0+₃ state, which decays outside the target. By using only the most backward TIGRESS detectors a reasonably narrow Doppler-correctedpeakcouldbeachievedforthistransition,sinceherethe difference betweenthereal emission angleofthe

γ

ray andthe angleassumedfortheDopplerreconstruction issufficientlysmall evenforarelativelylong-livedstate.Thisallowedforaclear iden-tification of this transition, despite the reduced

γ

-ray detection efficiency. The cross-section for the population of the 1465 keV 0+₃ statewas determined usingarelative

γ

-ray intensityanalysis between the 650 keV 0+₃

→

2+₁ line and the 414 keV 0+₂

→

2+₁ line(Fig. 2). The experimentally observedyields for these transi-tions were compared to adetailed Geant4 [40] simulation ofthe decayofthe1229and1465 keVstates,takingintoaccountthe TI-GRESSgeometry,attenuationof

γ

-raysintheSHARCchamberand beam-line material, the kinematics of the recoiling 96_{Sr nucleus}

(

β

≈

0

.

1) andtheknownhalf-lives anddecaybranching ratiosof thetwostates.Fromtheratioofthenumberofmeasured

γ

raysit wasdeducedthattherelativepopulationstrengthofthe1465 keV statecomparedtothe1229 keVstateis1.50(52).

Fig. 3 shows the experimental angular distributions for the 0+_1,2 96Sr states compared to distortedwave Born approximation (DWBA) andadiabatic distortedwaveapproximation (ADWA) cal-culationsthat were carriedoutusing FRESCO [41]. FortheDWBA calculations, the optical model (OM) parameters were optimized usingtheelasticscatteringangulardistributionsford(95Sr, d)and p(95Sr, p), the datafor which were acquired simultaneously with d(95_{Sr, p). For d(}95_{Sr, d) the OM parameters of Lohr and}

Hae-berli [42] wereused,withslightadjustmentsto betterreproduce the elastic scatteringdata. The p(95_{Sr, p) data was dominated by}

pureRutherfordscatteringandwas,withinuncertainties,not sen-sitivetodifferentOMparameters.Thesecalculationsalsoprovided the normalization of the cross section. The analysis procedure of Wilson et al. [43] was followed, and further details will be provided in a forthcoming publication [33]. For the ADWA cal-culation global nucleon-nucleus OM parameters from [44] were used. The calculations (Fig. 3) model the d(95_{Sr, p) reaction as}

Fig. 3. FitofDWBA(blue)andADWA(green)calculationstoexperimentaldatafor the96_Sr0+

1 groundstate(a)and1229 keV0+2 state(b).Forthedeterminationof

the spectroscopicfactorsonlytheforwardcenter-of-massanglesθcm<45◦ were

considered.

a single-step process where the transferred neutron populates the

ν

2s1/2 orbital via pure



=

0 angular momentum transfer.

The ADWA calculations better reproduce the data at large scat-tering angles where deuteron breakup is expected to affect the transfer cross section. The spectroscopic factors C2S are deter-mined asthe ratio of experimental to reaction model cross sec-tion, the fits were restricted to the forward angles

θ

cm

<

45◦.

For the transfer to the 0+_1,2 96Sr states the spectroscopic fac-tors amount to C2S

=

0

.

19

(

3

)

and 0.22(3), respectively, for the DWBA and 0.15(3) and 0.19(3) for the ADWA calculations. The spectroscopicfactoruncertainties includebothstatisticaland sys-tematiccontributionsrelatedtothenormalizationofthedata. Ad-ditional uncertainties for the absolute value of the spectroscopic factors arise from the choice of the reaction model and the op-tical model potential parameters especially for the DWBA. Based on the comparison of the two reaction models and various sets of optical model parameters these amount to 20%. For the rel-ative spectroscopic factors ofthe 0+ states used to extract their mixingstrengththeseuncertaintiescancel.Theinclusionof multi-step processes through coupled channels calculations leads to a slightly better description ofthe differential crosssection, at the expenseofadditionalunconstrainedparameters.Completeand de-tailed coupled-channel calculations are beyond the scope of the present work. To estimate the contribution, the inelastic excita-tion of 95Sr was taken into account through an effective defor-mation length. A value of

δ

n

=

1

.

1 fm reproduces the measured

(

d

,

d

)

cross section and is consistent with the spherical nature of 95_{Sr determined from the charge radius [23] and the}

neigh-boring 94,96_{Sr [28,29]. Changes to the spectroscopic factors are}

less than 10% and do not alter the conclusion presented in the following.

Table 1

Comparisonofexperimental(usingtheDWBAcalculations)tocalculatedspectroscopicfactors(C2_S)for0+_statesin96_{Srpopulatedviathed(}95_{Sr, p)reaction.The}

experi-mentalvaluesforthetwoexcited0+statesresultintheunmixedsphericalstate,whilefortheshellmodelcalculationsthetwo0+stateswiththehighestC2_{S arelisted}

(fordetailsseetext).

Exp. Unmixed glek a glek b glek c

Ex(keV) C2S Ex(keV) C2S Ex(keV) C2S Ex(keV) C2S Ex(keV) C2S

0 0.19(3) 0 0.19(3) 0 1.742 0 1.575 0 1.455

1229 0.22(3) 1314 0 – – – – – –

1465 0.33(13) 1380 0.55(13) 2271 0.056 1691 0.098 444 0.105

It was not possible to extract an angular distribution for the 1465 keV0+₃ state duetolow statistics.However, fromthe anal-ysisoftherelativepopulationstrengthdiscussedabove,the spec-troscopic factor for the 1465 keV state could be deduced to be 0.33(12) or 0.29(13), for DWBA and ADWA calculations, respec-tively. The relative spectroscopic factors used in the discussion beloware not affected by the choice of the reaction model and thesystematicuncertaintiesarising fromthenormalizationofthe crosssection.

Therelativestrengthsofthe0+_2,3statesin96Srwereinterpreted usingatwo-levelshapemixingmodel.The0+₁ groundstateof96Sr wasexcludedfromthemixingmodelanalysisasthemixingofthis statewiththeexcited 0+_2,3 statesisexpectedtobenegligible.This isevidencedbythefactthatnoE0 transitionsbetweentheexcited 0+_2,3 statesandthe0₁+ groundstateof 96_{Sr were reportedinthe}

workofJung [45].Thisassumptionisalsosupportedbytherecent Coulomb excitation data and beyond mean field calculations re-portedbyClémentetal. [27,28].Usingthetwo-levelmixingmodel, themonopole transitionstrength between the 0+_2,3 96Sr statesis relatedtotheirmixingstrengtha2 andintrinsicquadrupole defor-mations

β

by,

ρ

2

(

E0

)

=



3 4

π



2 Z2a2

(

1

−

a2

)



(β

2

)



2 (1)

where Z is the atomic number [31]. In the case where the un-mixedstates are a spherical configuration 0+_sph (

β

sph

=

0) and a

stronglydeformedconfiguration0+_def (with

β

def),thedifferencein

(squared)deformationbetweentheconfigurationsis

(β

2

)

= β

_def2 . Thewave functions ofthe 0+_2,3 statesin 96_{Sr wouldtherefore be}



0+₂



=

a



_

0+_sph



+

√

1

−

a2



₀+ def



and



0+₃



=

√

1

−

a2



_

₀+ sph



−

a



0+_def



, respectively,withthe mixingamplitudea.Giventhat the ground state of 95_{Sr has a nearly spherical shape [23], a substantial}

re-arrangement of the valence nucleons would be required in or-dertodirectlypopulateastronglydeformedconfigurationin96Sr throughasingle-steptransferreactionsuchas

(

d

,

p

)

.Itistherefore assumed that there was negligible direct population of the de-formedconfiguration,0+_def,inthisexperimentandsothestrength ofthe 0+_2,3 statesin 96Sr is adirect measure ofthe 0+_sph content ofthe excited 1229 and1465 keV state wave functions. The ra-tioofthespectroscopicfactorsofthe0₂+ and0+₃ statesin 96Sris thereforeequalto 1−_a2a2, givinga2

=

0

.

40

(

14

)

.This resultis

inde-pendentofthereactionmodelchoiceastheratioisdeterminedby theratioofcrosssectionswithadynamicalcorrectionaccounting forthe difference inexcitation energy. By combiningthe known value of

ρ

2

₍

_E0

₎

_{with ourexperimental constrainton the mixing}

strengtha2,equation (1) wasusedtodeterminetheabsolutevalue of

β

def

=

0

.

31

(

3

)

. In this strong mixing scenario the interaction

strength betweenthe 0+_2,3 statesin 96_{Sr is 113 keV and the}

en-ergiesoftheunmixed0+_defand0+_sph statesare1314and1380 keV, respectively.Itisinterestingtocomparetheseresultstotheshape coexistenceinneighboring98_Sr.Here,the0+

2 stateissituatedonly

215 keVabovethegroundstateandatwo-levelmixingmodel re-sultedinonlyaweakmixingbetweenthecoexistingstates [26,28]. Asaresult,the98Sr0+₁ groundstateisstronglydeformedwhereas theexcited0+₂ isnearlyspherical.Theweakmixingalsoimpliesa surprisinglysmallinteractionstrengthbetweentheunmixed con-figurationsofonly

≈

10 keV.

The strongly populated 2084 keV state (Figs.1 and2), hasa 51%branchingratioviathe855 keVtransitiontothe1229 keV0+₂ state relative to the 2084 keV ground state transition. Using the 0+_2,3 mixing strength of 0.40(14) one can calculate the expected branching ratio for the 2084 keV to 1465 keV 0+ state transi-tion ifweassume that the transitionratetothe unmixed0+_def is negligible.Basedonthis,thebranchingratioforthis619 keV tran-sition isexpectedto be 57+₋51₂₅ (30+₋27₁₃)% relative tothe 2084 keV groundstatetransitionwithin1

σ

uncertainties,assumingthatthe 2084 keVstatehasaspinandparityof1+(2+).Nomeasurement ofthistransitionhasbeenreported [39] anditwas not observed in thepresent experiment.Thus, the observationof the855 keV transitionandthenon-observationofahypothetical619 keV indi-catesthat the1229 keVstatecontains alarger componentofthe 0+_sphconfigurationthanthe1465 keVstate(a2

>

0

.

5).Atthesame time the E0 and E2 branchingratios andthevery different half-livesoftheisomeric1465 keVstate andthe1229 keVstate favor

a2

>

0

.

5 aswell.

Shell modelcalculationswere carriedout using NushellX [46], employingtheglek interaction [47].Themodelspacecomprisesof theproton f pg9/2 andneutron gds orbitals.Thetwo-bodymatrix

elementsareobtainedfromG-matrixcalculationswithsome mod-ificationstobetterdescribetheYandZrnuclei [47].Forthe calcu-lations presentedhere, the single-particle energieswere adjusted toreproducelowenergystatesforoddmassnucleiinthevicinity of Z

=

40 and N

=

58.Statesin95,96_{Srandspectroscopicfactors}

ford(95_{Sr, p)werecalculatedusingseveraldifferentvalencespaces}

toinvestigatetheinfluenceofthe variousprotondegreesof free-dom.Fortheneutronsan inertN

=

50 corewasassumedandthe valence spaceincluded the

ν

[

2s1/2],

[

1d3/2],

[

1d5/2], and

[

0g7/2]

orbitals. Calculations were carried out separately usingthree dif-ferent proton valence spaces. In valence space

a, the protons are required to be inert ina

π

[

1p3/2]4 configuration.In valence

space b

,proton excitationsinto the nearby

π

[

1p1/2] orbital are

allowed. Finally,valencespace c

furtherexpands theproton va-lence space to include the

π

[

0g9/2] orbital by allowing for two

additionalprotonexcitationsacrossthe Z

=

40 sub-shellgap.The occupancy of the

π

[

0g9/2] orbital was restricted to two protons

duetocomputationallimitations.Table1comparesthe experimen-talspectroscopicfactorsforthe0+stateswiththosepredictedby the shell model calculationsfor the three different configuration spaces.Theexperimentalvaluefortheunmixed1380 keV0+state shown in Table 1 corresponds to the total excited 0+_sph spectro-scopic factor, the sum of the experimental spectroscopic factors for the 0+_2,3 states.The shell model calculations predict a much largerneutron

ν

[

2s1/2]2 componentforthegroundstate thanthe

excited 0+ state while experimentally the opposite is observed. Includingthe

π

[

0g9/2]protonconfigurationintheshellmodel

cal-culationsonly slightlyincreasesthepopulation ofthe excited 0+ state,whiledramaticallyloweringits energy(seeTable1). Exper-imentally, about half of the overall predicted

ν

[

2s1/2]2 strength

is observed in the low-lying 0+ states. The present shell model interactionsandsingle-particleenergiesalsodescribewellthe ex-perimental spectroscopic factors for the low-lying states in 95Sr populated via the d(94_{Sr, p) reaction, carried out as part of the}

same experimental campaign. For the 1

/

2+ ground state of 95_Sr

a C2S

=

0

.

45 (forvalencespace

b which described bestthe ex-citation energies) is predicted while the measurement using the sameanalysisaspresentedhereresultsin C2_S

₌

₀

_.

₄₁

₍

₉

₎

_.Further

results on the d(94,95_{Sr, p) reactions will be reported in a}

forth-comingpublication [33].

Theseresultsshow,thatwhilethe94,95_{Srnucleiaswellasthe}

strongly deformed98_{Srnucleus can be described rather well}

us-ingshellmodel [29,33] andbeyondmeanfieldcalculations [27,28],

96_SratN

₌

_{58,justbeforetheshapetransition,hasamuchmore}

complicatedstructure.Thepresenttransferreactionstudyenables to selectivelypopulate thespherical component ofthe 0+ states in96Sr. Thespherical componentisfound mainly inthestrongly mixedexcited 0+states.Themixingratiowasdeterminedforthe firsttimeinthepresentstudy.Thegroundstateontheotherhand is only weakly populated in the d(95Sr, p) reaction suggesting a triple shape coexistence in 96Sr witha (weakly) oblate or triax-ialgroundstate.

In summary, we have measured the population of low-lying statesin96Srvia thed(95Sr, p)reactionat5.5 A MeV.The results showasurprisinglystrongpopulationofanexcitedspherical con-figurationin96_{Sr,whichitself isstronglymixedwitha deformed}

(

β

=

0

.

31

(

3

)

) configuration,givingrise to two 0+ statesat 1229 and1465 keV.Owingtothealmostpure

ν

[

s1/2]groundstate

con-figuration ofthe 1

/

2+ ground state in95_{Sr, the}

₍

_d

_,

_p

₎

_transferis

mostlysensitivetothe

ν

[

s1/2]2 configurationinthefinal0+state.

Contrarytotheexperimentaldata,shellmodelcalculationsusinga constrainedmodelspacepredictapredominant

ν

[

s1/2]2

configura-tionforthe96_{Srgroundstate.Thissuggeststheoccurrenceofthree}

distinct shapesin 96Sr. Clearly,more extensivetheoreticalstudies are required to gain better insights into thesingle-particle wave functionsintheSrisotopesinthiscontext.Extensionsofthework carriedoutfortheZrisotopes withlargescaleshellmodel calcu-lations [15] andMonteCarloShellModelcalculations [21],aswell asfurtherdevelopmentsofthebeyondmeanfieldcalculations [14,

16,27,28],willbeofinterest.

TheeffortsoftheTRIUMFoperationsteaminsupplyingthe95Sr beam are highly appreciated. We acknowledge support fromthe ScienceandTechnologiesFacilityCouncil(UK,grantsEP/D060575/1 and ST/L005727/1), the National Science Foundation (US, grant PHY-1306297), the Natural Sciences and Engineering Research CouncilofCanada, theCanadaFoundation forInnovationandthe British Columbia Knowledge and DevelopmentFund. TRIUMF re-ceives funding via a contribution through the National Research CouncilCanada.

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