Multiple chiral doublets in four-j shells particle rotor model : five possible chiral doublets in ¹³⁶₆₀Nd₇₆

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Multiple

chiral

doublets

in

four- j shells

particle

rotor

model:

Five

possible

chiral

doublets

in

136

₆₀

Nd

₇₆

Q.B. Chen

a

,

B.F. Lv

b

,

C.M. Petrache

b

,

J. Meng

c,d,e,

∗

a_{Physik-Department,TechnischeUniversitätMünchen,D-85747Garching,Germany}

b_{CentredeSciencesNucléairesetSciencesdelaMatière,CNRS/IN2P3,UniversitéParis-Saclay,Bât.104-108,91405Orsay,France} c_{StateKeyLaboratoryofNuclearPhysicsandTechnology,SchoolofPhysics,PekingUniversity,Beijing100871,China}

d_{YukawaInstituteforTheoreticalPhysics,KyotoUniversity,Kyoto606-8502,Japan} e_{DepartmentofPhysics,UniversityofStellenbosch,Stellenbosch,SouthAfrica}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received18May2018

Receivedinrevisedform31May2018 Accepted12June2018

Availableonline15June2018 Editor:W.Haxton

Aparticlerotormodel,whichcouplesnucleonsinfoursingle- j shellstoatriaxialrotorcore,isdeveloped to investigate the five pairs of nearly degenerate doublet bandsrecently reported in the even-even nucleus 136Nd. The experimental energy spectra and available B(M1)/B(E2) values are successfully reproduced.Theangularmomentumgeometriesofthevalencenucleonsandthecoresupportthechiral rotationinterpretationsnot onlyforthepreviouslyreportedchiraldoublet,butalsofortheotherfour candidates.Hence,136Ndisthefirsteven-evencandidatenucleusinwhichthemultiplechiraldoublets exist.Fivepairsofchiraldoubletbandsinasinglenucleusisalsoanewrecordinthestudyofnuclear chirality.

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

Chiralrotation isan exotic rotationalmode in a nucleuswith triaxial ellipsoidal shape. The rotations about an axis out of the threeprincipalplanesofthetriaxialnucleuscangiverisetoapair ofneardegenerate



I

=

1 bandswiththesameparity,i.e.,chiral doubletbands [1].Chiralrotationwaswellestablishedinthe A

∼

80,100, 130,and190massregions inodd–odd nuclei [2–7] and odd- A nuclei [8–10].Fordetails,seerecentreviews [11–17] ordata tables [18].

However, chiral doublet bands were rarely observed in even-even nuclei. The general opinion for this is that the multi-quasiparticleconfigurationsbecome morecomplexandinvolveat leasttwo valenceprotonsandtwo valenceneutrons.InRef. [19], two doublet bands were observed in the even-even isotopes 110,112_{Ruandinterpretedassoftchiralvibrations.}

Very recently, five pairs of nearly degenerate doublet bands were reported in the even-even nucleus136_{Nd, which were} dis-coveredin ahigh-statisticsexperimentperformedwiththe high-efficiencyJurogamIIarray [20].Itwasdemonstratedthatthechiral partners of strongly populated bands in the triaxial nucleus are presentclosetoyrast,asinthecaseoftheodd–oddandodd–even nuclei, but are far weaker than the yrast partners andtherefore noteasy to observe.Theobserved fivepairsofnearly degenerate

*

Correspondingauthor.

E-mailaddress:mengj@pku.edu.cn(J. Meng).

bandswere investigatedbythe constrainedandtilted axis crank-ing covariant density functional theory (TAC-CDFT) [21–25]: one ofthemisrevealedtobe achiraldoublet,andtheother fourare chiral candidates [20]. These observations shednew light on the investigations of chiral doublets in even-even nuclei. If the four chiralcandidatesarefinallyconfirmed,thenthey willconstitutea multiplechiraldoublet(M

χ

D),aphenomenonpredictedby covari-antdensityfunctionaltheory(CDFT) [21,26–29] andparticlerotor model (PRM) [30–33], andobserved experimentally [34–36]. The futureidentificationofsuch bandsin136_{Ndwillhopefullyopen a} campaignofmeasurements forother even-eventriaxialnuclei, in whichthechiralityortheM

χ

Dphenomenoncouldexist.

Theoretically, various approaches have been developed ex-tensively to investigate the chiral doublet bands. For example, the PRM [1,37–41], the TAC approach [24,42–44], the TAC plus random-phase approximation (RPA) [45], the collective Hamilto-nian method [46,47], the interacting boson model [48,49], and theangularmomentumprojection(AMP) [50–53].Inaddition,the generalizedcoherentstate model(GCSM)was alsoapplied to tri-axialeven-evennucleiandadifferentformalism ofchiralrotation wasproposed [54–57].

In Ref. [20], asmentioned above, the observed doublet bands in 136_{Nd were investigated in the framework of the TAC-CDFT} [22–25],whichisafullymicroscopicapproach,butcannotdescribe theenergysplittingandthequantumtunneling betweenthe chi-https://doi.org/10.1016/j.physletb.2018.06.030

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

tal model couplingthe collective rotation and the single-particle motions.IncontrasttotheTACapproach,itdescribesasystemin the laboratory frame. The total Hamiltonianis diagonalized with totalangularmomentumasagoodquantumnumber,andthe en-ergysplittingandquantumtunneling betweenthedoubletbands canbeobtaineddirectly.Moreover,thebasicmicroscopicinputsfor PRMcanbe obtainedfromtheconstrainedCDFT [10,21,25,34–36,

58].

Various versions of PRM have been developed to investigate thechiraldoubletbandswithdifferentkindsofconfigurations.For example, the PRM with one-particle-one-hole configuration was used to describe the chirality in odd–odd nuclei [1,33,37,38,59]. Tosimulatetheeffectsofmanyvalencenucleons,pairing correla-tionswereintroducedandPRMwithtwoquasiparticles configura-tionwasdeveloped [39,60–65].Todescribe theodd- A nuclei,the many-particle-many-hole versions of PRM with nucleons in two single- j shells [40,66] orthree single- j shells [34,35,58,67],have beendeveloped.It is notedthat the unpaired nucleon configura-tionsofthedoubletbandsintheeven-evennucleus136_Ndinvolve fourdifferentsingle- j shells.SuchPRMisstillunavailable.

In this letter, a PRM that couples nucleons in four single- j shells to a triaxial rotor core is developed and applied to study theenergyspectra,theelectromagnetictransitionprobabilities,as wellastheangularmomentumgeometriesfortheobserved dou-bletbandsin136Nd.

TheformalismofthePRM inthepresentworkisan extension ofthatinRef. [40],wherethemany-particle-many-holeversionof PRMwithtwosingle- j shellswasdeveloped.ThetotalHamiltonian ofPRMisexpressedas

ˆ

HPRM

= ˆ

Hcoll

+ ˆ

Hintr

,

(1) withthecollectiverotorHamiltonian

ˆ

Hcoll

=

3



k=1

ˆ

R2_k 2Jk

=

3



k=1

(ˆ

Ik

− ˆ

Jk

)

2 2Jk

,

(2)

where the indexes k

=

1, 2, and 3 refer to the three principal axes ofthe body-fixed frame. The R

ˆ

k and

ˆ

Ik denotethe angular

momentumoperators ofthecoreandofthetotalnucleus, respec-tively,andthe

ˆ

Jk isthetotalangularmomentum operatorofthe

valencenucleons. Themoments ofinertia oftheirrotational flow type [68] are adopted,i.e.,

J

k

=

J

0sin2

(

γ

−

2k

π

/

3

)

,with

γ

the triaxialdeformation parameter.Inaddition,the intrinsic Hamilto-nianiswrittenas Hintr

=

4



i=1



ν

ε

i,νa†i,νai,ν

,

(3)

where

ε

i,ν is thesingle particle energy inthe i-th single- j shell

providedby

Here, the plus or minus sign refers to particle or hole, andthe coefficient C is proportional to the quadrupole deformation

β

as inRef. [69].

The single particle state and its time reversal state are ex-pressedas a†ν

|

0

 =



α cν_α

|

α

,

j



,

(5) a†_ν¯

|

0

 =



α

(

−

1

)

j−cν_α

|

α

,

j

− ,

(6)

where



isthe projection ofthe single-particle angular momen-tum j along the 3-axis of the intrinsic frame and restricted to

. . .

,

−

3

/

2,1/2,5/2,

. . .

duetothetime-reversaldegeneracy,and

α

denotes theother quantum numbers.Fora systemwith



4_i=1Ni

valencenucleons (Ni denotesthe numberof thenucleons inthe i-thsingle- j shell),theintrinsicwavefunctionisgivenas

|

ϕ

 =

4



i=1



ni l=1 a†_i,ν l

n_i



l=1 a†_i,μ¯ l

|

0

,

(7) withni

+

ni

=

Ni and0

≤

ni

≤

Ni.

Thetotal wave functioncanbe expandedintothe strong cou-plingbasis

|

I M

 =



Kϕ cKϕ

|

I M K

ϕ

,

(8) with

|

I M K

ϕ

 =



1 2

(

1

+ δ

K 0

δ

ϕ,ϕ¯

)



|

I M K

|

ϕ

 + (−

1

)

I−K

|

I M

−

K

| ¯

ϕ





,

(9) where

|

I M K



istheWignerfunction



2I+1

8π2 DIM K.The basisstates

are symmetrized under the point group D2, which leads to K

−

1

2



4

i=1

(

ni

−

ni

)

beinganeveninteger.

It is noted that due to the inclusion of many-particle-many-holeconfigurationswithfoursingle- j shells, thesizeofthebasis spaceisratherlarge.Forexample,forthecalculationsofbandD1 in 136_{Nd(see its configurationin Table 1), the dimension of the} basis space is 864

(

2I

+

1

)

.For I

=

10h,

_¯

this value is 18144, and for I

=

20h,

_¯

itis 35424. Itis quitetime-consuming inthe diago-nalizationofthe PRM Hamiltonianmatrix.Tosolvethisproblem, similarintheshell-model-likeapproach(SLAP) [70,71],weadopta properlytruncatedbasisspacebyintroducingacutoffforthe con-figuration energy



_i,ν

ε

i,ν . In such a way,the dimension of the

PRM matrixis reducedto

∼

5000–10000 with the energy uncer-taintywithin0.1%.

Fig. 1. (Color online.) TheenergyspectraofbandsD1–D6andtheirpartners calcu-latedbyPRMincomparisonwithcorrespondingdata.Theexcitationenergiesare relativetoarigid-rotorreference.

AfterobtainingthewavefunctionsofPRM,thereduced transi-tionprobabilitiesB

(

M1

)

andB

(

E2

)

,andexpectationvaluesofthe angularmomentumofthesystemcanbecalculated.

Therearefivepairsofdoubletbandsin136_{Nd(labeledasbands} D1–D5),inwhichthreeofthem(bandsD1,D2,andD5)have pos-itiveparity. Besides, thereis adipole band (labeledas band D6), which has no partner band. In the PRM calculations for these bands, the unpaired nucleon configurations are consistent with thosein Ref. [20] and the corresponding quadrupoledeformation parameters

(β,

γ

)

areobtainedfromtriaxialconstrainedCDFT cal-culations [21].Themomentsofinertia

J

0andCoriolisattenuation factors

ξ

areadjustedto reproducethetrendoftheenergy spec-tra.ThecorrespondingdetailsarelistedinTable1.Inaddition,for theelectromagnetictransitions,theempiricalintrinsicquadrupole moment Q0

= (

3

/

√

5

π

)

R2₀Z

β

, and gyromagnetic ratios for rotor

gR

=

Z

/

A andfornucleons gp(n)

=

gl

+ (

gs

−

gl

)/(

2l

+

1

)

(gl

=

1

(

0

)

forprotons(neutrons)andgs

=

0

.

6gs

(

free

)

) [68] areadopted.

ThecalculatedenergyspectraforthebandsD1–D6in136_Ndare presentedinFig.1,together withthecorrespondingdata.The ex-perimentalenergyspectraarereproduced excellentlyby thePRM calculations.Beingaquantummodel,PRMisabletoreproducethe energysplittingforthewholeobservedspinregion.Itisseenthat exceptingband D2,the trendandamplitudefortheenergy split-tingbetweenpartnerbandsarereproducedwell.

For D2 and D5, the energy differences between the doublet bands decrease gradually with the spin. In detail, for D2, the energy splitting is

∼

360 keV at I

=

21

_¯

h, and finally goes to

∼

110 keV at I

=

25

_¯

h. The PRM calculations underestimate this energyseparation.ForbandsD5andD5-C,whichareidentifiedas chiraldoublets inRef. [20],their energysplittingis

∼

410 keV at

I

=

18h,

¯

andreducesto

∼

150 keV at I

=

27h.

¯

Fortheotherthreepairsofdoubletbands(D1,D1-C),(D3,D3-C), and (D4,D4-C), the energy separations are about 70, 40, and 120 keV, respectively, and do not change much with the spin. For band D4, the PRM calculations can not reproduce the data below I

=

18h,

_¯

indicating that theused configurationisnot suit-ableforthelower spinpart.Nevertheless,therathersmallenergy

Fig. 2. (Color online.) ThestaggeringparametersofbandsD1–D6calculatedbyPRM incomparisonwithcorrespondingdata.

Fig. 3. (Color online.) TheB(M1)/B(E2)ofbandsD1–D6andtheirpartners calcu-latedbyPRMincomparisonwithcorrespondingdata.

differencesbetweenthesedoubletssupportthattheirchiral inter-pretation.

From the energy spectra, the staggering parameters S

(

I

)

=

[

E

(

I

)

−

E

(

I

−

1

)

]/

2I areextractedanddisplayedinFig.2.The stan-dard fingerprintsforchiralbandsoutlinedinRef. [3] require that

S

(

I

)

isindependentofspin.Overall,thePRM calculationscan re-producethebehaviorsofexperimental S

(

I

)

.Moreover,the S

(

I

)

of allbandsvarysmoothlyanddonotchangemuchwithspin.These phenomenafurtherprovidethesupportthatthebandsD1–D5are chiralpartners.

In Fig.3, the B

(

M1

)/

B

(

E2

)

valuesof bandsD1–D6 calculated by PRM incomparisonwiththe correspondingexperimental data are shown. One observes that thePRM calculationsshow an im-pressive agreement with the data. Moreover, excepting band D2

Fig. 4. (Color online.) Therootmeansquarecomponentsalongtheintermediate(i-,squares),short(s-,circles)andlong(l-,triangles)axesoftherotor,valenceprotons,and valenceneutronsangularmomentacalculatedasfunctionsofspinbyPRMforthedoubletbandsD2andD2-Cin136_Nd.

Fig. 5. (Color online.) Same as Fig.4, but for D5.

thecalculatedB

(

M1

)/

B

(

E2

)

valuesofthedoubletbandsarerather similar.

ForbandsD1 andD6, the B

(

M1

)/

B

(

E2

)

valuesdecrease with spin. For D2, an abrupt increase of B

(

M1

)/

B

(

E2

)

is observed at

I

=

21h.

_¯

ThelargecalculatedB

(

M1

)/

B

(

E2

)

valueatI

=

21h results

_¯

from the small B

(

E2

)

value. After analyzing the corresponding PRM wave function, we find that, at I

<

20h,

¯

the largest com-ponent ofthe state is Is

∼

I (Is the angular momentum

compo-nent along the short axis), while for I

≥

20h,

¯

the largest one is

Is

∼

I

−

2.ThisleadstothesmallB

(

M1

)

valueatI

=

20h and

¯

small B

(

E2

)

valuesatI

=

20 and21

¯

h,andhencealarge B

(

M1

)/

B

(

E2

)

value at I

=

21h.

_¯

For D5 and D5-C, their B

(

M1

)/

B

(

E2

)

values are quite similar and fulfill the characteristics of chiral doublet bands [38,59]. Therefore, they were identified as chiral doublet bandsinRef. [20].

For bands D3 and D4, their B

(

M1

)/

B

(

E2

)

values are similar. Theyexhibit a trendthat firstincreases andthen decreaseswith increasing spin. Moreover, their quasi-particle alignments show pronouncedsimilarityoverawideintervalofrotationalfrequency, showninRef. [20].Itseems thattheyare M

χ

Dbuiltonidentical configurationasin103_{Rh [35].However,astheirspectraare} inter-weavedeachotheratseveralspins, thispossibilityisexcluded.In thecalculations,weuseaconfigurationwiththreesingle- j shells to describe D3 and a configuration with four single- j shells to describe D4,shown inTable 1.Admittedly, the presentPRM cal-culationsdonotagreeverywell withthedataofD3.ForD4,the

calculated results reproduce very well the experimental data for

I

≥

19

_¯

h.

Note that onlythe B

(

M1

)/

B

(

E2

)

ratios forthe doublet bands D5 andD5-Chavebeen measured.Hence,the otherfourpairs of doublet bands are considered only aschiral candidates. As men-tioned before, the calculated B

(

M1

)/

B

(

E2

)

values are similar in the doublet bands. This suggests that the other four candidates mightalso be chiraldoublets. Definitely,furtherexperimental ef-forts are highly demanded toobtain solid evidenceforthechiral doubletsinterpretation.

Thesuccessinreproducingtheenergyspectraand electromag-netic transitionprobabilities for thedoublet bands in136_Nd mo-tivate us to examine the angular momentum geometries of the observed bands. For this purpose, we calculate the expectation values ofthe squaredangular momentum componentsalong the intermediate (i-), short(s-), andlong (l-) axes for the rotor, va-lenceprotons,andvalenceneutrons.Here,theobtainedresultsof bandsD2,D5,andD4areshowninFigs. 4,

5

,and

6

asexamples, respectively.

As shown in Fig. 4, for both bands D2 and D2-C, the collec-tive core angular momentum mainly aligns along the i-axis at

I

≥

25h,

_¯

because it has the largest moment of inertia. It should bementionedthatthe s-componentofthecollectivecoreangular momentumislargeandcannotbeneglected.Moreover,itexhibits adiscontinuous behaviorbetween I

=

19 and20h in

¯

the D2,and between I

=

17 and18h and

¯

I

=

22 and23

¯

h inD2-C.Thisis un-derstoodasthereasonofabruptincreasesofB

(

M1

)/

B

(

E2

)

values,

Fig. 6. (Color online.) Same as Fig.4, but for D4.

asdiscussedpreviously.Theangularmomentumofthethreeh11/2 valenceprotonparticlesmainlyaligns alongthe s-axis,andthose ofvalenceprotonandneutronholesmainlyalongthel-axis.Such orientations form the chiral geometry of aplanar rotation. But it shouldbenotedthatduetothelarges-componentoftherotorand proton,thetotalangularmomentumliesclosetothes–i plane.

Forthechiraldoublet bandsD5andD5-C, asshowninFig. 5, theangularmomentahavesimilarorientationsatI

≥

21h,

¯

asthose inD2.Namely, theangularmomentumoftherotor mainlyaligns along the i-axis,the two h11/2 valence proton andone f7/2 va-lenceneutronparticlesmainlyalignalongthes-axis,andtwo g7/2 valenceprotonandoneneutronh11/2 valenceholesmainlyalong thel-axis.IncomparisonwiththoseinD2,thes-axiscomponents oftheangularmomentaoftherotorandh11/2valenceproton par-ticlesinD5are about2h smaller.

_¯

Suchorientationsformabetter chiralgeometryofaplanarrotationthanthatofD2.AtI

≤

21h,

_¯

the

l-axiscomponentsofangularmomentaoftwo g7/2 valenceproton holes are differentin bands D5 andD5-C. For D5, the two pro-tonholesarealignedandcontribute

∼

5h.

_¯

However,forD5-C,the alignment happenswhenthe spin increasesfrom 17h to

_¯

21

_¯

h.At

I

=

17h,

_¯

the two proton holes contribute with

∼

2h.

_¯

At I

=

21h,

_¯

thetwo protonholescontribute

∼

5

_¯

h.Such differencecausesthe energydifference betweenthe doublet bands inthis spin region

∼

400keV asshowninFig.1.

Forthe bands D4 andD4-C, asshownin Fig. 6,similar apla-narorientationoftheangularmomentaoftherotor, theparticles, andtheholescanbeobserved.ThissupportsthatD4andD4-Care chiraldoublets.Asdiscussedpreviously,thereisaband-crossingat

I

=

19h,

_¯

andtheadoptedconfigurationisonlysuitablefor describ-ingthedataaboveband-crossing.OneobservesthatatI

≥

19

_¯

h,the angularmomentaofthetwoh11/2valenceprotonparticlestendto alignalong i-axis.Thisleads tothe increaseof B

(

E2

)

, andhence tothedecreaseofB

(

M1

)/

B

(

E2

)

withthespinasshowninFig.3. In summary, a PRM couplingnucleons in four single- j shells to a triaxial rotor core is developed to investigate the five pairs of nearly degenerate doublet bands recently reported in the even-even nucleus 136Nd. The configurations and corresponding quadrupoledeformation parameters(

β

,

γ

)are obtainedfromthe constrained CDFT calculations. The experimental energy spectra and available B

(

M1

)/

B

(

E2

)

values are successfully reproduced. The angular momentum geometries of the valencenucleons and the core support the chiral rotation interpretations not only for thepreviouslyreportedchiraldoublet,butalsotheotherfour can-didates.Therefore,136_{Ndis thefirsteven-even candidatenucleus} inwhich theM

χ

Dexists. Fivepairs ofchiral doubletbands ina singlenucleusisalsoanewrecordinthestudyofnuclear

chiral-ity. Further experimental efforts are highly encouraged to obtain solidevidenceforM

χ

Dinterpretations.

This work was partly supported by the National Key R&D Program of China (Contract No. 2018YFA0404400), the Deutsche Forschungsgemeinschaft(DFG)andNationalNaturalScience Foun-dationofChina(NSFC)throughfundsprovidedtotheSino-German CRC110“SymmetriesandtheEmergenceofStructureinQCD”,and theNSFCunderGrantsNo. 11335002andNo. 11621131001.

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