Pairing-quadrupole interplay in the neutron-deficient tin nuclei: First lifetime measurements of low-lying states in $^{106,108}$ Sn

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Pairing-quadrupole interplay in the neutron-deficient tin

nuclei: First lifetime measurements of low-lying states in

106,108

_Sn

M. Siciliano, J.J. Valiente-Dobón, A. Goasduff, F. Nowacki, A.P. Zuker, D.

Bazzacco, A. Lopez-Martens, E. Clément, G. Benzoni, T. Braunroth, et al.

To cite this version:

M. Siciliano, J.J. Valiente-Dobón, A. Goasduff, F. Nowacki, A.P. Zuker, et al.. Pairing-quadrupole

in-terplay in the neutron-deficient tin nuclei: First lifetime measurements of low-lying states in

106,108

_Sn.

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Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Pairing-quadrupole

interplay

in

the

neutron-deﬁcient

tin

nuclei:

First

lifetime

measurements

of

low-lying

states

in

106 ,

108 Sn

M. Siciliano

a

,

b

,

c

,

∗

,

J.J. Valiente-Dobón

a

,

A. Goasduff

a

,

b

,

d

,

F. Nowacki

e

,

A.P. Zuker

e

,

D. Bazzacco

d

,

A. Lopez-Martens

f

,

E. Clément

g

,

G. Benzoni

h

,

T. Braunroth

i

,

F.C.L. Crespi

h

,

j

,

N. Cieplicka-Ory ´nczak

h

,

1

,

M. Doncel

k

,

S. Ertürk

l

,

G. de France

g

,

C. Fransen

i

,

A. Gadea

m

,

G. Georgiev

f

,

A. Goldkuhle

i

,

U. Jakobsson

n

,

G. Jaworski

a

,

2

,

P.R. John

b

,

d

,

3

,

I. Kuti

o

,

A. Lemasson

g

,

T. Marchi

a

,

D. Mengoni

b

,

d

,

C. Michelagnoli

g

,

4

,

T. Mijatovi ´c

r

,

C. Müller-Gatermann

i

,

D.R. Napoli

a

,

J. Nyberg

p

,

M. Palacz

q

,

R.M. Pérez-Vidal

m

,

B. Say˘gi

a

,

5

,

D. Sohler

o

,

S. Szilner

r

,

D. Testov

b

,

d

,

M. Zieli ´nska

c

,

D. Barrientos

s

,

B. Birkenbach

i

,

H.C. Boston

t

,

A.J. Boston

t

,

B. Cederwall

n

,

J. Collado

u

,

D.M. Cullen

v

,

P. Désesquelles

f

,

C. Domingo-Pardo

m

,

J. Dudouet

f

,

6

,

J. Eberth

i

,

F.J. Egea-Canet

a

,

V. González

u

,

L.J. Harkness-Brennan

t

,

H. Hess

i

,

D.S. Judson

t

,

A. Jungclaus

w

,

W. Korten

c

,

M. Labiche

x

,

A. Lefevre

g

,

S. Leoni

h

,

j

,

H. Li

n

,

A. Maj

y

,

R. Menegazzo

d

,

B. Million

h

,

A. Pullia

h

,

j

,

F. Recchia

b

,

d

,

P. Reiter

i

,

M.D. Salsac

c

,

E. Sanchis

u

,

O. Stezowski

z

,

Ch. Theisen

c

a_INFN,_Laboratori_Nazionali_di_Legnaro,_Legnaro_(PD),_Italy

b_Dipartimento_di_Fisica_e_Astronomia,_Università_di_Padova,_Padua_(PD),_Italy c_{CEA/Irfu/DPhN,}_Université_de_{Paris-Saclay,}_{Gif-sur-Yvette,}_France d_INFN,_Sezione_di_Padova,_Padua,_Italy

e_IPHC,_CNRS/IN2P3_Université_de_Strasbourg,_Strasbourg,_France f_CSNSM,_CNRS/IN2P3,_Université_de_{Paris-Saclay,}_Orsay,_France

g_Grand_{Accélérateur}_National_d’Ions_Lourds,_CEA/Irfu/DRF_and_CNRS/IN2P3,_Caen,_France h_INFN,_Sezione_di_Milano,_Milan,_Italy

i_Institut_für_Kernphysik,_Universität_zu_Köln,_Cologne,_Germany j_Dipartimento_di_Fisica,_Università_di_Milano,_Milan,_Italy k_Universidad_de_Salamanca,_Salamanca,_Spain l_Ömer_Halisdemir_{Üniversitesi,}_Ni˘gde,_Turkey

m_Instituto_de_Física_Corpuscular,_{CSIC-Universidad}_de_Valencia,_Valencia,_Spain n_Department_of_Physics,_Royal_Institute_of_Technology,_Stockholm,_Sweden o_INR,_Hungarian_Academy_of_Sciences,_Debrecen,_Hungary

p_Department_of_Physics_and_Astronomy,_Uppsala_University,_Uppsala,_Sweden q_´_Srodowiskowe_Laboratorium_{Cie¸ ˙zkich}_Jonów,_Uniwersytet_Warszawski,_Warsaw,_Poland r_Ruder_Boškovi´c_Institute_and_University_of_Zagreb,_Zagreb,_Croatia

s_CERN,_Geneva,_Switzerland

t_Oliver_Lodge_Laboratory,_University_of_Liverpool,_Liverpool,_UK

u_Departamento_de_Ingeniería_{Electrónica,}_Universitad_de_Valencia,_Valencia,_Spain v_Schuster_Laboratory,_University_of_Manchester,_Manchester,_UK

w_Instituto_de_Estructura_de_la_Materia,_CSIC,_Madrid,_Spain x_STFC_Daresbury_Laboratory,_Warrington,_UK

y_Instytut_Fizyki_Ja¸drowej_im._Henryka_{Niewodnicza´}_nskiego,_Polska_Akademia_Nauk,_Krakow,_Poland z_IPN-Lyon,_CNRS/IN2P3,_Université_de_Lyon,_{Villeurbanne,}_France

*

Correspondingauthorat:CEA/Irfu/DPhN,UniversitédeParis-Saclay,Gif-sur-Yvette,France.

E-mailaddress:marco.siciliano@lnl.infn.it(M. Siciliano).

1 _Present_address:_Instytut_Fizyki_Ja¸drowej_im._Henryka_{Niewodnicza´}_nskiego,_Polska_Akademia_Nauk,_Krakow,_Poland. 2 _Present_address:_Srodowiskowe_´ _Laboratorium_{Cie¸ ˙zkich}_Jonów,_Uniwersytet_Warszawski,_Warsaw,_Poland. 3 _Present_address:_Institut_für_Kernphysik,_Technische_Universität_Darmstadt,_Darmstadt,_Germany. 4 _Present_address:_Institut_{Laue-Langevin,}_Grenoble,_France.

5 _Present_address:_Ege_{Üniversitesi,}_˙Izmir,_Turkey.

6 _Present_address:_IPN-Lyon,_CNRS/IN2P3,_Université_de_Lyon,_{Villeurbanne,}_France.

https://doi.org/10.1016/j.physletb.2020.135474

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2 M. Siciliano et al. / Physics Letters B 806 (2020) 135474

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Articlehistory:

Received24December2019

Receivedinrevisedform27March2020 Accepted4May2020

Availableonline11May2020 Editor: B.Blank Keywords: Lifetime Nuclearstructure Multi-nucleontransfer LightSn Trackingarray

The lifetimesofthe low-lyingexcitedstates2+ and 4+ have beendirectlymeasuredinthe neutron-deﬁcient 106,108_Sn _isotopes. _The _nuclei _were_populated _via _a_{deep-inelastic} _reaction_and _the _lifetime measurementwasperformedemployingadifferentialplungerdevice.Theemitted

γ

raysweredetected by the AGATAarray,whilethe reactionproductswereuniquely identiﬁedbythe VAMOS++ magnetic spectrometer. Large-Scale Shell-Modelcalculations withrealistic forcesindicatethat,independentlyof thepairingcontentoftheinteraction,thequadrupoleforceisdominantintheB(E2;2+₁→0+g.s.)values

anditdescribeswelltheexperimentalpatternfor104−114Sn;theB(E2;4+1→2+1)values,measuredhere fortheﬁrsttime,dependcriticallyonadelicatepairing-quadrupolebalance,disclosedbytheveryprecise resultsin108_Sn.

1. Introduction

A little over a decade ago, the Sn isotopes were considered theparadigmsofpairingdominance: low-lyingstatesofgood se-niority,nearly constant J π

=

2+₁ excitation energiesandparabolic B

(

E2

;

2+₁

→

0+_g_._s_.

)

behavior. Thelatter was observedfor A

≥

116. Forthe lighter species, experimental results on transition proba-bilities were scarce as the presence of low-lying isomeric states hindered direct measurements of lifetimes below them. From Coulomb-excitationmeasurementswithradioactiveionbeamsonly the reduced transition probability between the ﬁrst excited 2+ state and the ground state could be determined [1–8]. Within experimental uncertainties,they suggest arather-constant behav-ior for 106

≤

A

≤

110, instead of the parabolic trend expected when isovector T

=

1 pairing dominates.This “plateau” includes alsothe stable112,114Sn nuclei, forwhich measurements ofboth B

(

E2

;

2+₁

→

0+_g_._s_.

)

andB

(

E2

;

4+₁

→

2+₁

)

valuesexist [9].Beforethis work, the B

(

E2

;

4+₁

→

2+₁

)

values were completely absent inthe neutron-deﬁcientSnisotopes.

The experiment described in this Letter was devoted to de-termine the strength of 2+₁

→

0+_g_._s_. and 4+₁

→

2+₁ transitions in

106,108_Sn_by_measuring_the_lifetime_of₂+

1 and4+1 states.Although

severaltheoretical interpretations havebeen proposed [10,1,4,11,

12], the evolution of the B

(

E2

;

2₁+

→

0+_g_._s_.

)

values in the light Snnucleiremains puzzling.The experimental and theoretical re-sultsthatwillbepresentedinthisLetterprovideafurtherinsight, which reveals the underlying structure of the light tin isotopes, namelythecounterbalanceofquadrupoleandpairingforcesinthe Snisotopicchain.

2. Experiment

A multi-nucleon transfer reaction, that is commonly used to investigate neutron-rich nuclei [13–15], was unconventionally adoptedtopopulatetheSnisotopesclosetotheprotondripline, so the isotopes of interest were populated in the collision of a

106_Cd_beam_and_a92_Mo_target._The_beam-target_combination_and

beam energy were selected as a compromise between two re-quirements. On one hand the reaction fragments energy had to be suﬃcientlyhightoallow their identiﬁcationby the spectrom-eter.Ontheother,inorderto performthelifetimemeasurement, thepopulationofthestatesabovethe6+isomershadtobe min-imized; this condition imposed an upper limit on the excitation energyandconsequentlyonbeamenergy,evenattheexpenseof thecrosssectiontopopulatemoreexoticspecies.The106_Cd_beam,

providedbytheseparated-sectorcyclotronoftheGANIL facilityat anenergyof770MeV,impingedonto a0.8mg/cm2 92Motarget. ThelifetimemeasurementwasperformedwiththeRecoilDistance Doppler-Shift(RDDS)method[16–18].Thetargetwasmountedon

thedifferentialCologneplungerwitha1.6mg/cm2_thicknat_Mg

de-graderdownstream.Inordertomeasurethelifetimesofinterest, 8differenttarget-degraderdistancesintherange31-521 μmwere used.Thecomplete(A,Z)identiﬁcation,together withthe velocity vector forthe projectile-like products was obtainedonan event-by-event basis using the VAMOS++ spectrometer [19–21], placed at the grazing angle

θ

lab=25◦. In coincidence with the magnetic

spectrometer, the

γ

rays weredetected bythe

γ

-raytracking de-tector arrayAGATA[22,23],consistingof8triple-clusterdetectors placed at backward angles in a compact conﬁguration (18.5 cm fromthetarget).Thecombinationofthepulse-shapeanalysis[24] and the Orsay Forward-Tracking (OFT) algorithm [25] allowed to reconstructthetrajectoryofthe

γ

raysemittedby thefragments. More details about theion identiﬁcationand the analysis proce-durecanbefoundinRefs. [26,27].

3. Results

Thanks to the precise determination of the ion velocity vec-tor andthe identiﬁcationof theﬁrst interaction point ofeach

γ

ray inside AGATA, Doppler correction was applied on an event-by-event basis.The magnetic spectrometer directlymeasured the fragments velocity after the degrader (

β

af ter

≈

9%). However, for

each

γ

-raytransitiontwopeakswereobserved,relatedtoits emis-sion before and after the Mg foil: the

γ

rays emitted after the degrader areproperlyDoppler corrected,whilethoseemitted be-foreareshiftedtolowerenergiesbecauseofthedifferentvelocity ofthereactionfragment(

β

be f ore

≈

10%).Therelativeintensitiesof

thepeaksasafunctionofthetarget-degraderdistancearerelated tothelifetimeofthestateofinterest.Theareasareobtainedviaa

χ

2_{-minimization}_ﬁt,_performed_by_considering_a_gaussian_shape_of

thepeaks,whosecentroidandFWHMwereconstrainedfromthe closest and longesttarget-degrader distancespectra, anda linear background.

The completeidentiﬁcationoftheVAMOS++spectrometerand theemploymentofamulti-nucleontransfer(MNT)mechanism al-lowedustoreconstructtheTotalKinetic-EnergyLoss(TKEL)ofthe reaction. Thisquantity was crucialforthelifetimemeasurements in108Sn,sinceitallows ustocontrolthedirectpopulationofthe excited states [18,28], in order to reduce possible contamination fromthehigh-lyingstatesabovethe6+₁ isomer.Forthisnucleus,in fact,theenergiesofthe2+₁

→

0g.s.(1206keV)and8+1

→

6+1 (1196

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Fig. 1. Doppler-correctedγ-rayenergyspectraof108_Sn_before_(black)_and_after_(green)_the_gate_on_the_Total_{Kinetic-Energy}_Loss_(TKEL),_obtained_by_summing_up_the

statisticsofallthetarget-degraderdistances.The2+1 →0+g.s.(red),4+1→2+1 (blue)and8+1→6+1 (orange)transitionsaremarked,indicatingtheunshifted (u)andshifted (s)

centroidswithasolidandadashedline,respectively.Intheinset,theTKELdistributionandthesetgate(green)arepresented.

abovethe6+₁ isomer:forTKEL

<

21 MeV,the8+₁

→

6+₁ transition peaksbecamenegligibleandthemeasuredlifetimeofboth4+₁ and 2+₁ statesremainedconstant,evenformorerestrictiveconditions. ThedeterminationofsuchaTKELcondition anditseffectsonthe lifetimemeasurement arediscussed indetail inRef. [29].In that work, the ratio of the 4+₁

→

2+₁ transition components remains constant also for larger values of the TKEL, which is consistent withthe unalteredline-shapeoftheFig.1spectra at

≈

900 keV; thisis duetothefact thatthe 6+₁ isomer“blocks” the depopula-tionfromhigher-lyingstates,sotheirpresencedoes notaffectthe lifetimemeasurementofthe4+₁ state.The sameisobviouslytrue forthe2+₁ statebut,inthisspeciﬁccase,ablindintegrationofthe spectrum will includethe unshifted componentof the 8+

→

6+ transition. Being the 8+ a long lived state with a lifetime more thantwo ordersofmagnitudelarger thanthe 2+₁,thiswould re-sultinanartiﬁciallyshorterlifetime.Forthisreason,theTKELcut allowstogetridofthecontaminationofourtransitionofinterest fromthe8+deexcitation.Fig.2(left)showstheDoppler-corrected

γ

-ray energyspectraof 108Snfor severaldistances,requiringthe TKEL

<

21 MeVcondition.Thisnontraditionalprocedureallowedus totakeintoaccountjustthe6+₁,4+₁ and2+₁ statesinthe measure-mentofthelifetimesviaDecay-Curve Method(DCM).Duetothe directpopulation ofthe excited states,the presence ofthe long-lived6+₁ isomersimpliﬁesthelifetimemeasurementby“blocking” the depopulation from the higher-lying states and, as shown in Fig.2(right),itcontributesjustasanoffsettothedecaycurvesof the4+₁ and2+₁ states.

For106Snthedescribed TKEL-gateprocedurewas notrequired and, becauseof thepresence of thelong-lived isomer, thedecay cascadeofthe6+₁,4+₁ and2+₁ stateswastakenintoaccountwhile measuringthelifetimeviaDCM.Alsointhiscase,thepresenceof the6+₁ isomericstateaffectsthelifetimeanalysisbyintroducingan offset,whichisrelatedtothedirectpopulationofthestates [17], in the decaycurves of the 2₁+ and 4+₁ excited states. The direct populationof the excited stateswas extracted fromthe single-

γ

spectra.Thisinformationwasusedtoconstraintheparametersof thedecaycurves.Thepossiblepresence ofadditionalfeeders was investigatedin the single-

γ

spectra andalso in the

γ

-

γ

coinci-dences:exceptforthoseconsideredinthemeasurement,noother transitionsfeedingthe4+₁ and2+₁ excitedstateswereobservedfor both106,108_Sn._In_Fig.₂_(right_pad)_the_decay_curves_are_presented

Table 1

Measured lifetime of the excited states Iπ in 106,108_Sn _and _{corresponding} B(E2;Iπ_→_Iπ₋₂₎_values._The_last_column_shows_the_theoretical_predictions_from theextensionofthecalculationsofRef. [1] (seetext).

Iπ Eγ [keV] τ[ps] B(E2) [e2fm4] Exp. Theo. 108_Sn ₂+ 1 1206.1 0.76 (8) 422 (44) 425 4+₁ 905.1 3.7 (2) 364 (20) 349 106_Sn ₂+ 1 1207.7 1.3 (7) 245 (132) 339 4+₁ 811.9 5 (4) 446 (334) 379

for the 4+₁ and 2+₁ states: the extractedlifetime ofthe 2+₁ state is inperfect agreementwith theliterature, supporting the valid-ityoftheexperimentalmethod.Therefore,thankstothepowerful setupandtheunconventionalexperimentaltechnique,thelifetime ofthe2+₁ and4+₁ stateshavebeenmeasured,fortheﬁrsttime,in

106,108_Sn.

Table1summarizestheexperimentalresults,showingthe life-times and the derived reduced transition probabilities B

(

E2

)

for

108_Sn_and106_Sn_isotopes,_as_well_as_the_theoretical_values_from_the

extensionofthecalculationsofRef. [1],obtainedbyemployingthe sameinteractioninthefull gds valencespaceforbothprotonsand neutrons,usingtheeffectivecharges

(

eπ

,

eν

)

= (

1

.

35

,

0

.

65

)

, allow-ingupto4p

−

4h excitationsandwithoutanysenioritytruncation. The extracted B

(

E2

)

values forthe 106,108_Sn_isotopes _are _shown

in Fig. 3 together withall previous experimental results for the whole isotopic chain. The B

(

E2

;

2₁+

→

0+_g_._s_.

)

strengths previously measuredarecompatiblewiththeresultsobtainedinthis experi-ment,whilefortheB

(

E2

;

4+₁

→

2+₁

)

valuesnodataexistedinthis region. Unfortunately, the exoticity of 106_Sn _and_the _necessity _to

avoid the population of the states above the long-lived 6+ iso-merresultinaratherlargestatisticalerrorontheB

(

E2

;

4+₁

→

2+₁

)

valueforthisisotope. 4. Discussion

(5)

Fig. 2. (left)Doppler-correctedγ-rayenergyspectraof108_Sn_for_different_{target-to-degrader}_distances,_gated_on_the_Total_{Kinetic-Energy}_Loss_(TKEL)._For_each_transition

theunshifted (u)andshifted (s)centroidsareindicatedbyasolidandadashedline,respectively.(right)Ratioofthetransitioncomponentsintensityasafunctionofthe distance,obtainedbygatingontheTKEL.Thedashedlinesrepresenttheﬁtteddecaycurvesforthetwoexcitedstates.

newly measured B

(

E2

;

4+₁

→

2+₁

)

values.Given a proper interac-tion, it is very simple to describe the “anomalous” B

(

E2

;

2+₁

→

0+_g_._s_.

)

patternoftheSnisotopes,whilethe –hithertounexplored– B

(

E2

;

4+₁

→

2+₁

)

behaviordemandsmorecare.

The context of these theoretical results is provided by the Pseudo-SU(3) symmetry, which acts in the space of gds orbits above(andexcept)g9/2.ThebasicideaisinspiredbyElliott’sSU(3)

scheme [31,32] andconsistsinbuildingintrinsicstatesthat max-imizethe quadrupole operator [33–35]. As shown in Figure 2 of Ref. [30] (top rightpad), for the light Snnuclei only the ﬁrst 6 neutronsplayaroleinthevalueofthequadrupoleoperator,while the contribution of the following 6 neutrons is null, leading to a “plateau” inthe B

(

E2

;

2+₁

→

0+_g_._s_.

)

reducedtransition probabil-ities. For a strong enough quadrupole force, the system exhibits rotationalfeatures,leadingto B4/2

≡

B

(

E2

;

4+1

→

2+1

)/

B

(

E2

;

2+1

→

0+_g_._s_.

)

≈

1

.

43 (Alaga rule). This holds in the Cd case, while for Snthe quadrupole strength ismuch reduceddue to theabsence of g9/2 proton holes. It is still strong enough to produce a

sta-ble B

(

E2

;

2+₁

→

0+_g_._s_.

)

pattern analogous to the Cd one, but the B

(

E2

;

4+₁

→

2+₁

)

behaviorbecomessensitivetopairingand single-particlebehavior.Thus,itappearsthatthequadrupoledominance inCd gives wayto a formofpairing-quadrupole interplayin Sn. Forthe sake ofcompleteness, inwhat follows we brieﬂy explain thestepsinvolvedintheshellmodelcalculations.

The interaction must be extracted from a realistic potential, properly renormalized and monopole corrected. All realistic po-tentials give very similar results [34], N3LO was chosen [46] and Vlow−k-regularized [47]. The CDB [48] or AV18 [49]

poten-tials would yield the same results. The indispensable renormal-izations amount to a –rigorously established– 30% boost of the quadrupoleforceandaphenomenological40%increaseofthe pair-ing force [50]. The replacementofthemonopoletermis impera-tive, since the realistic interactions have bad monopole behavior [34, Sec. II.B.3]. Thus, thiscorrection was done by replacing the monopolepart of the interaction with the Hamiltonianprovided by the GEMO (GEneral MOnopole) code [51], which is based on the Duﬂo-Zuker mass formula [52,53], adding the single particle spectrumof101Sn(inparenthesestheenergies inMeV):

g9/2

(

−

6

.

0

),

d5/2

(

0

.

0

),

g7/2

(

0

.

5

),

s1/2

(

0

.

8

),

d3/2

(

1

.

6

).

TheresultinginteractioniscalledI.3.4.Possibleuncertainties con-cernthepairingcontent,sothepairingstrengthhasbeentreated asa free parameter and different values have been investigated, yieldingtotheI.3.0andI.3.2interactions(withnoand20% correc-tions,respectively).Furthermore,accordingtothestudyofRef. [54, Fig.3.2.1],analternativetotheGEMOspectrum(DZinthatFigure)

Fig. 3. ReducedtransitionprobabilityB(E2)forthe(a)2+1→0+g.s.and(b)4+1 →2+1

transitionsalongtheSnisotopicchain.Theresultsofthepresentwork(redsquares) arecomparedwiththosefrompreviousexperiments[9,36,1–3,37,4,38,5,39–41,6,42,

7,43,44,8,45].ThepredictionsfromLarge-ScaleShell-Modelcalculations fromthe I.3.4interactionarealsoshown(openblackpentagons).

isanextrapolatedestimate(EX)equivalenttopushingupthes1/2

orbitalto1.6MeV,whichresultsintheI.3.4s1.6interaction. The calculations were performed with the ANTOINE pro-gram [34] inut M spaces in the gds shellofup to ug9/2-proton

holes and t g9/2-neutron holes, for a total of M holes. Here we

presentut M

=

202 casesofm-schemedimensionsofupto6

·

107_,

butitwascheckedthattheyreproducewellthe1010-dimensional ut M

=

444 cases.

Fig.4(a)establishesthat theamountofpairingmakesno dif-ference inthe B

(

E2

;

2+₁

→

0+_g_._s_.

)

transitionprobabilities.Theshift inthe positionofthes1/2 orbithasan inﬂuence,albeitminor.In

the Fig.4 (b)both the pairingandthe single particle shiftmake an enormous difference in the B

(

E2

;

4+₁

→

2+₁

)

behavior. In the caseof strongquadrupole dominance, the B

(

E2

;

2+₁

→

0+_g_._s_.

)

pat-ternwouldbeasinFig.4(a)andtheB

(

E2

;

4+₁

→

2+₁

)

onewould bethesamemultipliedby1.43.Thisisnotfarfromthe1.25forthe I.3.0case,asshowninFig.4(c).The B4/2ratioisfurtherreduced

(6)

Fig. 4. Experimentalandcalculated(a)B(E2;2+1→0+g.s.)and(b)B(E2;4+1→2+1)

valuesand(c)B4/2ratio.Effectivechargesare(eπ,eν)= (1.40,0.72)throughout. ForA=112 and114 theneutroneffectivechargeeν shouldbeincreasedto0.75 toaccountfortheomissionofthe h11/2 shell,thatplaysasmallbutsigniﬁcant

role.ExperimentalB(E2;2+1→0+g.s.)valuesaretheweightedaveragesoftheFig.3 results,whiletheB(E2;4+1→2+1)datacomesfromRef. [9] andthepresentwork.

I.3.0s1.6hasthes1/2singleparticleenergymovedupby800keVwithrespectto

GEMO.

lostforthetwoI.3.4cases,butbothareclosetotheobserved val-uesin112−114Sn [9]. Moreover,ournewmeasurein 108Snbreaks the ambiguity in favor of the I.3.4 chosen standard with GEMO spectrum,providingapotentiallyinteresting suggestionaboutthe spectrumof101_Sn.

Traditionally all emphasis has been put in explaining the B

(

E2

;

2+₁

→

0+_g_._s_.

)

patterns,while the B

(

E2

;

4+₁

→

2+₁

)

oneshave been ignored. To understand the present situation, we do well to remember that nuclear structure is a compromise between monopole, quadrupole and pairing. Under a favorable monopole structure (andgoodmonopolebehavioroftheinteraction), quadru-poledominanceobtains andleads torotational features.Energies (momentsofinertia)aresensitivetopairing,buttheB

(

E2

;

2+₁

→

0+_g_._s_.

)

patternsareimmune.Thisiswhatseemstobehappeningin thecalculationsofTogashiet al.[12],whoseB

(

E2

;

2+₁

→

0+_g_._s_.

)

pat-ternis identicalto ours, while thewave-functions exhibit strong spin and mass dependence in the g9/2 proton-hole occupancy,

which are nearly constant in our case. We attribute the coinci-denceinthepatternstogoodmonopolebehavior.

Inadditionto the B

(

E2

)

strengths, thetheoretical estimations arein goodagreement withthe excitation energyofthe 2+₁ and

4+₁ states: besides the I.3.0 that represents the no-pairing limit, the variousinteractions presented inthepaper arewithin

≈

200 keVaccuracy.

5. Conclusions

The unconventional use of multi-nucleon transfer reactions withaplungerdevice hasallowedustomeasurelifetimesinthe neutron-deﬁcient106,108_Sn_isotopes. _Since_{deep-inelastic}_reactions

directlypopulatethelow-lyingexcitedstatesofthechannelsof in-terest,theexperimentallimitationscausedbythepresenceofthe low-lyingisomers were overcome. Moreover, theTKEL-gate tech-nique allowedusto avoidthe possiblecontamination from high-lyingstatesabovethe6+₁ isomers.ThisledtotheﬁrstB

(

E2

;

4+₁

→

2+₁

)

measurementinthismassregion.Thetheoreticalresultsshow thattheexperimentaltrendofthe B

(

E2

;

2₁+

→

0+_g_._s_.

)

valuesinthe mass region 104

≤

A

≤

114 can be reproduced within the gds model space. Traditionally, pairing was thought to be dominant in the Sn isotopes but the calculations indicate that the rather constant pattern ofthe B

(

E2

;

2+₁

→

0+_g_._s_.

)

values is associatedto quadrupoledominance,independentofthepairingstrength.Inthe caseof B

(

E2

;

4+₁

→

2+₁

)

values,instead,pairing becomescrucial: itisclearthatquadrupoledominanceholdsforthe J π

=

0+_g_._s_.

,

2+₁ states,butitisstronglychallengedforthe4+₁ statethrough mix-ing with a pairing dominated intruder. On the other hand, the lackofprecise informationaboutthe4+₁ stateshindersthe char-acterization of such an intruder state. What is beyond doubt is theimportanceoftheveryprecisemeasurementsin108Sn, which haveshowntoopennewperspectivesintheunderstandingofthe quadrupole-pairinginterplay.

These results representa step forward to our current knowl-edge of the region. Indeed, the precise measurement of the B

(

E2

;

4+₁

→

2+₁

)

value in 108_Sn _has _led _to _a _unique _theoretical

work which,inaddition todiscussin detailthebalance between pairing-quadrupole correlations in the nuclear interaction, ques-tions the “goodness” ofprevious theoretical works.In fact, while many different theoretical works have claimed to reproduce the trend ofthereduced transitionprobabilities B

(

E2

;

2+₁

→

0+_g_._s_.

)

in the N

=

Z

=

50 region,ourpaperhighlights thatthe B

(

E2

;

2+₁

→

0+_g_._s_.

)

values are not the only ones to be of key importance for understanding the nuclear structure close to 100Sn; actually, the B

(

E2

;

4+₁

→

2+₁

)

values,whose theoreticaldiscussion is faced for theveryﬁrsttimeinthisarticle,resulttobeofcrucialimportance forunderstandingthenuclearstructureoftheregion.

Furtherexperimentalandtheoreticalstudiesshouldfocustheir attentionsontheelectromagneticpropertiesnotonlyofthe2+₁

→

0+_g_._s_.transitions,butalsoofthe 4+₁

→

2+₁ ones.Thiswillallowus toshedlightonthepeculiarstructureoftheSnisotopicchain,of the Z

≈

50 regionandofother regionswheresimilar behaviorof the B

(

E2

)

strengths havebeenobserved,suchasthe N

=

50 iso-tonicchain.Inaddition,theproton- andneutron-transfer spectro-scopicfactorswillprovidefurtherinformationonthemicroscopic natureofthelow-lyingstatesinthe Z

≈

50 region.

Declarationofcompetinginterest

Theauthorsdeclarethattheyhavenoknowncompeting ﬁnan-cialinterestsorpersonalrelationshipsthatcouldhaveappearedto inﬂuencetheworkreportedinthispaper.

Acknowledgement

(7)

and to A. Poves for his fruitful comments. This work was par-tiallysupported bytheEuropeanUnion’sSeventhFramework Pro-gramme for Research and Technological Development (grant no. 262010).A.G. acknowledges the supportof theFondazione Cassa diRisparmio Padova e Rovigounder theproject CONPHYT, start-ing grant in 2017. The work was also supported (B.C, H.L., J.N. andU.J.)bySwedishResearchCouncilunderthegrantagreements nos. 822-2005-3332, 821-2010-6024, 821-2013-2304, 621-2014-5558and2017-0065,andbytheKnutandAliceWallenberg Foun-dationgrantno.2005.0184,(B.S.)bytheScientiﬁcand Technolog-ical Council of Turkey (TUBITAK) underthe project no. 114F473, (A.G.,A.J.andR.P.)bytheMinisteriodeCienciaeInnovacíonunder the contracts nos. SEV-2014-0398, FPA2017-84756-C4 and EEBB-I-15-09671,by the Generalitat Valenciana underthe grant agree-ment no.PROMETEO/2019/005 andby the EU-FEDER funds,(T.M andS.S.)bytheCroatianScienceFoundationundertheprojectno. 7194,(I.KandD.S.)bytheHungarianNationalResearchand Inno-vation Oﬃce (NKFIH) underthe project nos. K128947,PD124717 andGINOP-2.3.3-15-2016-00034,(M.P.)bythePolishNational Sci-enceCentrewiththegrantsnos.2014-14-M-ST2-00738, 2016-22-M-ST2-00269 and 2017-25-B-ST2-01569 under the COPIN-IN2P3, COPIGALandPOLITAprojects.

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