Chiral geometry and rotational structure for ¹³⁰Cs in the projected shell model

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Chiral

geometry

and

rotational

structure

for

130

Cs

in

the

projected

shell

model

F.Q. Chen (

)

a

,

J. Meng (

)

a

,

b

,

c

,

∗

,

S.Q. Zhang (

)

a

a_{StateKeyLaboratoryofNuclearPhysicsandTechnology,SchoolofPhysics,PekingUniversity,Beijing100871,People’sRepublicofChina} b_{YukawaInstituteforTheoreticalPhysics,KyotoUniversity,Kyoto606-8502,Japan}

c_{DepartmentofPhysics,UniversityofStellenbosch,Stellenbosch,SouthAfrica}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received18June2018

Receivedinrevisedform30July2018 Accepted22August2018

Availableonline24August2018 Editor: W.Haxton

Keywords:

Nuclearchirality Projectedshellmodel Chiralgeometry

K plot Azimuthal plot

The projectedshellmodel withconfiguration mixingfornuclearchiralityisdeveloped andapplied to theobservedrotationalbandsinthechiralnucleus130_{Cs.Forthechiralbands,theenergyspectraand}

electromagnetictransitionprobabilitiesarewellreproduced.Thechiralgeometryillustratedinthe

K plot

andthe

azimuthal plot is

confirmedtoberobustagainsttheconfigurationmixing.Theotherrotational bandsarealsodescribedinthesameframework.

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

Spontaneouschiralsymmetrybreakinginatomicnucleihas at-tractedintensivetheoreticalandexperimentalstudiessinceitsfirst predictionbyFrauendorfandMeng[1] in1997.Ittakesplacewhen the angular momentum of a triaxial nucleus has non-vanishing components on all of the three intrinsic principal axes. These three components, contributed by the valenceproton(s), the va-lence neutron(s), and the rotating core, respectively, can form a left- oraright-handedconfiguration.

The picture of chiral geometry is described in the intrinsic frame, and it is manifested in the laboratory system by the ob-servation of the chiral doublet bands, which are a pair of near-degenerate



I

=

1 bands with the same parity. Intensive efforts havebeendevotedtothesearchofchiraldoubletbandsinvarious massregions.Sofararound50pairsofchiraldoubletshavebeen discoveredinA

∼

80 [2,3],100 [4–9],130 [10–16] and190 [17,18] mass regions, see Refs. [19–21] for reviews. The chiral doublet bandsobservedexperimentallyuptonowhasbeencompiledvery recentlyinRef. [22].Theoreticalstudiesofthechiraldoubletbands havebeencarriedoutbytheparticlerotormodel(PRM)[1,23–26], thetilted axiscrankingmodel(TAC) [1,27–31], theTAC plus

ran-*

Correspondingauthorat:SchoolofPhysics,PekingUniversity,Beijing100871, People’sRepublicofChina.

E-mailaddresses:mengj@pku.edu.cn(J. Meng),sqzhang@pku.edu.cn

(S.Q. Zhang).

domphaseapproximation[32],thecollectiveHamiltonian[33,34], the interactingboson–fermion–fermion model(IBFFM) [12,35,36], andthegeneralizedcoherentstatemodel[37].

Theangularmomentum projection(AMP)approach,which re-storesthebrokenrotationalsymmetryinthemeanfield wavefunc-tions, is a promising tool for the microscopic description to the nuclearsystem. TheAMPapproach couldbe basedonmeanfield descriptions fromNilsson+BCS [38] tovarious densityfunctionals [39–43],andhasbeenappliedtovariousproblemsassummarized inarecentreview[44].AsoneoftheimplementationsoftheAMP approach,theprojectedshellmodel[38] wasused inattemptsto understand the chiral doublet bands [45,46] in Cs isotopic chain by Bhat et al. The observed energy spectra and electromagnetic transitionsforthedoubletbandsarewellreproducedinthese cal-culations.However,itwasfounddifficulttogiveanillustrationfor theunderlying chiralgeometry.The difficultyliesinthe factthat theangularmomentumgeometryisdefinedintheintrinsicframe, whiletheangularmomentumprojectedwavefunctionsarewritten inthelaboratoryframe.Thisdifficultyisovercomerecentlybythe introductionoftheK plot and theazimuthal plot inRef. [47].The

K plot isdefinedastheprobabilitydistributionsofthecomponents oftheangularmomentumonthethreeaxesoftheintrinsicframe, whiletheazimuthal plot isdefinedastheprobabilitydistributions ofthe polarandtheazimuthal anglesof theangularmomentum inthe intrinsicframe. The K plot andthe azimuthal plot provide

probabilitydistributionsoftheorientationrelatedquantitiesinthe https://doi.org/10.1016/j.physletb.2018.08.039

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

intrinsic frame, as demonstrated by the chiral doublet bands in 128_{Cs [47]. With the help of the K plot and the azimuthal plot,} theintrinsicorientationofthe angularmomentumcanbe shown explicitly. Consequently, the existence of aplanar orientation for chiralbandscouldbeconfirmedwithoutambiguity.Intheangular momentumprojectedframework,thechiralgeometrycanalsobe illustratedbytheroot-mean-squarevaluesoftheangular momen-tumcomponentsintheintrinsicframe[48].

Itisnoted, however,that theprojectedcalculationinRef. [47] islimitedtotheconfigurationofthelowest

π

h11/2 andthefourth

ν

h11/2orbitalsonly,andtheconfigurationmixingisneglected.For a betterdescription tothe nuclearsystem, the theoretical frame-workinRef. [47] needstobe generalizedtoincludevarious con-figurations, i.e., to the projected shell model. In order to have a betterunderstandingofthechiraldoubletbands,itisnecessaryto confirm whetherthe chiralgeometryis robust against configura-tionmixing. Another advantageis that theprojected shellmodel canprovidea simultaneousdescriptionforall rotationalbandsin onenucleusonthesamefoot.

Inthisworktheprojectedshellmodelwithconfiguration mix-ingfornuclearchiralityisdevelopedandappliedto130Cs[10].The chiralgeometryforthechiraldoubletbandsisillustratedinterms ofthe K plot andtheazimuthal plot,inasimilarwayasRef. [47]. Theeffectoftheconfigurationmixingonthechiralgeometryand thedescriptionoftheotherrotationalbandsarediscussed.

The framework of the projected shell model is based on the standardpairingplusquadrupoleHamiltonian[49],

ˆ

H

= ˆ

H0

−

χ

2



μ

ˆ

Q_μ+Q

ˆ

μ

−

GMP

ˆ

+P

ˆ

−

GQ



μ

ˆ

P_μ+P

ˆ

μ

,

(1)

which includes a sphericalsingle-particle partand the separable twobodyinteractions,i.e.,thequadrupole–quadrupoleinteraction, the monopole pairing, and the quadrupole pairing. The intrinsic ground state of an odd–odd nucleus can be denoted as

|

ν0π0



,

where

ν

0 and

π

0 are single-particle orbitals blocked. The state

|

ν0π0



is determined by the variation principle with the

con-straintsonthequadrupolemoments andtheaverageparticle num-bers:

δ



ν0π0

| ˆ

H

− λ

q0Q

ˆ

0

− λ

q2Q

ˆ

2

− λ

NN

ˆ

− λ

ZZ

ˆ

|

ν0π0

 =

0

,

(2)

andcanbewrittenasatwo-quasiparticlestate ontopofa quasi-particlevacuumoftheeven-evencore

|

0



:

|

ν0π0

 = β

ν+0

β

+

π0

|

0

.

(3)

By the variation in Eq. (2), the state

|

ν0π0



and the vacuum

|

0



areobtained,aswellascorrespondingquasiparticleoperators

{β

+

ν

,

β

+π

}

,with

ν

and

π

thesingle-particle orbitals. Various

two-quasiparticle states can be constructed as

{|

κ

}

= {β

ν+

β

π+

|

0

}

, in which

κ

specifies different two-quasiparticle configurations. The effect of the configuration mixing, which was neglected in Ref. [47],canbethentakenintoaccount.

Thetwo-quasiparticle states

|

N,Z,κ



withgoodparticle

num-berN and Z canbeprojectedfrom

|

κ



:

|

N,Z,κ

 ≡ ˆ

PNP

ˆ

Z

|

κ

.

(4)

The symmetry restored basis is constructed by the angular mo-mentumprojection:

{ ˆ

PI_{M K}

|

N,Z,κ

}.

(5)

The diagonalization ofthe Hamiltonian in the projected basis (5) leads to theHill-Wheeler equation (Notethat theseprojected basisvectorsarenotorthogonal[38]):



Kκ

(



N,Z,κ

| ˆ

HP

ˆ

K KI 

|

N,Z,κ



−

EIσ



N,Z,κ

| ˆ

PK KI 

|

N,Z,κ

)

fKIσκ

=

0

,

(6)

inwhich

σ

specifiesdifferenteigenstatesofthesamespin I.By solvingEq. (6) theeigenenergiesEI_{σ andthewavefunctions}

|

σ

I M

 =



Kκ

f_KIσ_κP

ˆ

_{M K}I

|

N,Z,κ



(7)

are obtained. With the wave functions (7) the electromagnetic transitionsandotherphysicalquantitiescanbecalculated.

In order to examine the configuration mixing, it is necessary to knowthe weightsof differenttwo-quasiparticle configurations

|

κ



inthewavefunction(7).Theseweightscouldbeobtainedby

resorting the concept ofcollective wave functions inthe genera-tor coordinate method(GCM), andregarding

{

K

,

κ

}

inEq. (7) as generatorcoordinates.Thecorresponding generatingfunctionsare theprojectedbasisinEq. (5),andthecorrespondingnormmatrix elementswrite

N

I

(K,

κ

;

K

,

κ



)

≡ 

N,Z,κ

| ˆ

PIK K

|

N,Z,κ

.

(8)

The collective wave functions gIσ

₍

_K

_,

_κ

₎

_{can be obtained by the}

squarerootofthenormmatrix(8) andthefunction f_KIσ_{κ :}

gIσ

(

K,

κ

)

=



K,κ

N

1/2

I

(K

,

κ

;

K

,

κ



)

fKIσκ

,

(9)

and they are proved to satisfy the normalizationandorthogonal relation



Kκ

gIσ1∗

₍

_K,

_κ

_)g

Iσ2

_(K,

_κ

₎

= δ

σ1σ2

.

(10)

IntheGCM,thecollectivewavefunctionsareunderstoodas prob-ability amplitudes of the generator coordinates. Therefore, the weightoftheconfiguration

|

κ



inthestate

|

σI M



couldbe

writ-tenas:

Wκ

=



K

|

gIσ

(K

,

κ

)

|

2

.

(11)

Thecollectivewave functions(9) arealsousedinthecalculations ofthe K plot andtheazimuthal plot,whichhavebeenintroduced inRef. [47].

In the following, the five rotational bands observed in 130Cs [50–52],includingthechiraldoubletbands,areinvestigatedbythe presentapproach.TheparametersintheHamiltonian(1) aretaken fromRef. [53].The quadrupoledeformationparameters(

β,

γ

) are constrainedtobe(0

.

20

,

30◦). Thischoice of

(β,

γ

)

isthesameas thatadoptedinthecalculationfor128CsinRef. [47].Itagrees rea-sonably with thedeformation parameters (0

.

19

,

39◦) usedin the projectedshellmodelcalculationinRef. [53].

Different from the calculation in Ref. [47], where the lowest

π

h11/2 andthe fourth

ν

h11/2 orbitalsare blocked,herethe two-quasiparticle configurations

β

_ν+

β

_π+

|

0



are constructed using or-bitals

ν

and

π

fromthe N

=

4 andN

=

5 majorshells.The quasi-particle energycutoff Eν

+

Eπ

≤

3

.

5MeV isadoptedforthe two-quasiparticleconfigurations,inwhichEν and Eπ arethe quasipar-ticle energies ofthe orbitals

ν

and

π

,respectively. There are 60 two-quasiparticle configurations withpositiveparity and68with negative paritytakenintoaccountintheconfigurationspace.Half of themcan be projected ontoeven K valuesandthe other half canbeprojectedontooddK values[54].Thereforethedimension oftheprojected basis(5) forspin I is30

(

2I

+

1

)

forthe positive paritysubspaceand34

(

2I

+

1

)

forthenegativeparitysubspace.

Fig. 1. (Coloronline)(Upperpanels)Experimentalandcalculatedrotationalbands in130_{Cs.Panel(a)showsbandswithpositiveparitywhilepanel(b)showsthose}

withnegativeparity.ExperimentaldataaretakenfromRefs. [50–52].Thecalculated energylevelsareshiftedbytakingthelevel9+ inbandAasareference.(Lower panels)ExperimentalandcalculatedvaluesofS(I) = [E(I) −E(I−1)]/2I forbands A,B(c)andbandsC,D(d).

Thecalculatedenergyspectraoftherotationalbands(denoted asbandsA–E)in130_{CsareshowninFigs.1(a)and(b),in} compar-isonwiththeavailabledata[50–52].

As shown in Fig. 1 (a), the calculated yrast and yrare bands withpositive parityagreewell withtheexperimental chiral dou-blet bands, band A and band B, including the near degeneracy between the partner bands. Note that the configuration mixing, which is absent in the calculation in Ref. [47], is takeninto ac-countin thepresentcalculation. Thepresentresultsdemonstrate thattheneardegeneracybetweenthechiraldoubletbandsis per-sistentevenwiththeconfigurationmixing.

Thecalculatedyrastandyrarebandswithnegativeparityagree reasonably with the experimental bands D and C, as shown in Fig.1(b). Thelevels with I

≥

18h in

_¯

band Dare overestimated, whichmightbeduetoacrossingwithfour-quasiparticle configu-rationsandisbeyondthe presenttwo-quasiparticleconfiguration space.

Thecalculatedlowest band withthedominatingconfiguration

ν

g7/2

π

h11/2 suggestedin Ref. [51] lies around 1 MeV above the experimentalband E.Thedominatingconfiguration(s) willbe ex-plainedinthe following.Onemaynoticethat themomentof in-ertia for band E is larger than those for bands C and D, which suggestsa largerdeformation forband E.Inthissense,band Eis beyondthe presentscope. Thereforeweexclude bandEfromthe followingdiscussions.

Thecompositionofeachstate canbecalculatedfromEq. (11), andthen thedominating configuration(s)can be recognized. The dominating configurations for bands A–D at the band head are given in Table 1, together with those suggested in the previous studies.Thedominatingconfigurationsfoundbythepresent calcu-lationcoincidewiththepreviousassignments,exceptbandC.The dominatingconfigurationfoundforbandC is

ν

h₁₁[5th]_/2

π

d[2nd]_5/2 while thepreviously assignedoneis

ν

h11/2

π

g7/2.Thisdifferenceisdue to the strong mixing between the g7/2 and the d5/2 orbitals as showninFig.2.

InFig.2,thecompositionsofconfigurationsforbandsA–D are shownasfunctionsof spin.ForbandsA andB, theconfiguration

ν

h[₁₁5th_/2]

π

h₁₁[1st_/2] isdominantuntilI

=

16h,

_¯

afterwhichstrong config-urationmixingoccurs.ForbandC,theconfigurations

ν

h[₁₁5th_/2]

π

d[₅2nd_/2]

Table 1

ThedominatingconfigurationsforbandsA–Datthebandheadinthepresent cal-culation,incomparisonwiththosesuggestedinthepreviousstudies.Theordinal numbersinthesecondcolumndenotespecificsingle-particleorbitals.For exam-ple,thenotationπh₁₁[1st_/]₂representsthefirst(lowest)single-particleorbitalinthe

πh11/2subshell.

Bands Present configuration Previous configuration References A, B νh[₁₁5th_/2]πh[₁₁1st_/2] νh11/2πh11/2 [10,52,51]

C νh[₁₁5th_/2]πd₅[2nd_/2] νh11/2πg7/2 [50,51]

D νh[115th/2]πd

[2nd]

5/2 νh11/2πd5/2 [50,51]

Fig. 2. (Coloronline)CompositionofconfigurationforbandsA–D.Foreachband, contributionsofthedominatingconfigurationsasfunctionsofspinarepresented bycoloredlineswiththeircorrespondingconfigurationlabeledbythesamecolor. Contributionsofotherconfigurations(iflargerthan1%)arepresentedbydark yel-lowlines.

and

ν

h[₁₁5th_/2]

π

g₇[2nd_/2] stronglycompete witheachother.The configu-ration

ν

h[₁₁5th_/2]

π

d₅[2nd_/2] winsatthebandhead,buttheconfiguration

ν

h[₁₁5th_/2]

π

g₇[2nd_/2] takesoverat I

=

8h.

¯

Forband D,theconfiguration

ν

h[₁₁5th_/2]

π

d₅[2nd_/2] isdominantuntil I

=

13h,

_¯

afterwhichstrong con-figurationmixingoccurs.

The amplitudes of signature splitting of bands A–D, reflected bythequantity S

(

I

)

= [

E

(

I

)

−

E

(

I

−

1

)

]/

2I,areshowninFigs.1(c) and(d).ForbandsAandB,theangularmomentumofthevalence neutronorientatesalongthelongaxis(l-axis),andthatofthe va-lenceprotonorientatesalongtheshortaxis(s-axis).Bothofthem areperpendicular approximatelytotheangularmomentumofthe collective rotation.Thevaluesof S

(

I

)

stay almostindependent of spin,asisseeninFig.1(c).Infact,thespinindependenceofS

(

I

)

has beenrecognized as one ofthe criteriafor the chiraldoublet bands[4,55]. ForbandsCandD, the strongconfigurationmixing leadstotheincreasingsignaturesplittingwithspin[56] asshown inFig.1(d).Moreover,it isnotedinFig.1that theamplitudesof

S

(

I

)

increase withspin, whichisduetotheincreasing configura-tionmixing.

In thefollowing we focus ourdiscussion on theproperties of thechiraldoubletbandsAandB.

Thecalculated intrabandB

(

E2

)

andB

(

M1

)

,andtheinterband

B

(

M1

)

forbandsAandBareshowninFig.3,incomparisonwith thedata[57].Figure3(a)showsthattheintrabandB

(

E2

)

are simi-larforbandsAandB,inbothexperimentalandtheoreticalresults, as expectedfor a pair ofchiral doublets [24,55]. The B

(

E2

)

val-uesaresomewhatoverestimatedbythecalculation.Thesimilarity oftheintrabandB

(

M1

)

betweenthechiraldoublets,togetherwith

Fig. 3. (Coloronline)(a)TheintrabandB(E2),(b)theintrabandB(M1),and(c)the interbandB(M1)forthechiraldoubletbandsin130_{Cscalculatedbytheprojected}

shellmodel,comparedwiththeexperimentaldatatakenfromRef. [57].

the staggering behavior of both intra- and interband B

(

M1

)

, are found in Figs. 3(b) and(c), in both experimental andtheoretical results.Thesefeaturesarealsoexpectedassignaturesofthechiral modesaccordingtoRefs. [24,55].Thecharacteristicfeaturesofthe electromagnetictransitions expectedforthe chiral doublet bands showninFig.3,whichhavebeenreproducedwithasingle config-uration[47],arepersistentevenwhentheconfigurationmixingis takenintoaccount.

TheangularmomentumgeometryofthechiraldoubletsAand B can be illustrated by the K plot and the azimuthal plot, as in Ref. [47].TheK plot,i.e.,theK distributionspIσ

(

|

K

|)

forthe angu-larmomentumonthethreeprincipalaxes,areshowninFig.4at thespins I

=

10

,

11

,

14

,

15

,

18

,

19h.

_¯

Theseresults couldbe com-paredwiththecorresponding resultsobtainedbythePRM,which havebeendiscussedinRefs. [26,58].

AsseenintheK plot inFig.4,theevolutionofthechiralmodes canbeexhibited.

Forspins I

=

10 and11h,

¯

the K plots shownin Fig.4are in accordancewiththe expectationforchiral vibrationwithrespect to the s–l plane,which takesplace nearthe band headwith in-sufficientcollectiverotation.TheprobabilityatKi

=

0 issignificant

forband A, which indicates a wave function symmetric with

re-specttoKi

=

0 andcorrespondstoa0-phononstate.Ontheother

hand,the vanishingprobability at Ki

=

0 forband B indicatesan

antisymmetricwavefunctioncorrespondingtoa1-phononstate. Forspins I

=

14 and15

_¯

h,themostprobablevalue forKi

ap-pears atKi

∼

13h for

¯

bandA, whichsuggeststhatthe collective

rotation aroundthei-axisdevelopsandtheangularmomentafor band Adeviate fromthes–l plane withtheincrease ofspin. The peaks of the Kl- and Ks-distributions also correspond to K

val-uessubstantiallyawayfromzero,sothemostprobableorientation oftheangularmomentumisaplanar,indicatingtheoccurrenceof staticchirality.TheK plots forbandBaresimilartothoseforband A, and the occurrence of static chirality is thus supported. The tunneling betweenthe left-handedand the right-handed config-urations isresponsiblefortheenergyseparationbetweenthetwo bands,whichexplainswhythedoublets areclosestto eachother inenergyatI

=

14 and15h.

¯

For spins I

=

18 and 19 h,

_¯

the Ks- and Kl-distributions

be-come broad,andthe peaksofthe Ki-distributions become sharp

at Ki

∼

I. Both features suggest that the angular momentamove

closetothei-axis.Thereforethestaticchiralitydisappearsandthe aplanarrotationisreplacedbytheprincipalaxisrotation.

InFig.5,theprofilesfortheorientationoftheangular momen-tumonthe

(θ,

φ)

plane,theazimuthal plot,areshownforthesame spinsasinFig.4.Thedefinitionsoftheangles

(θ,

φ)

canbefound inRef. [47].

ForspinsI

=

10 and11h,

_¯

theprofilesfortheorientationofthe angularmomentumforbandAhaveasinglepeakat

(θ

∼

60◦

,

φ

=

90◦

)

, which suggests that the angular momentum stays within the s–l plane,inaccordancewiththeexpectationfora 0-phonon state. On the other hand, the profiles for band B show a node at

(θ

∼

60◦

,

φ

=

90◦

)

, withtwo peaks at

(θ

∼

65◦

,

φ

∼

45◦

)

and

(θ

∼

65◦

,

φ

∼

135◦

)

, respectively. The existence of the node and thetwopeakssupportstheinterpretationofa1-phononvibration. Thereforetheinterpretationofchiralvibrationisdemonstrated.

ForspinsI

=

14 and 15h,

¯

theazimuthal plots forbandsAand Baresimilar.Twopeakscorrespondingtoaplanarorientationsare found,i.e.

(θ

∼

75◦

,

φ

∼

35◦

)

and

(θ

∼

75◦

,

φ

∼

145◦

)

forband A, while

(θ

∼

70◦

,

φ

∼

40◦

)

and

(θ

∼

70◦

,

φ

∼

140◦

)

forbandB.These featurescouldbeunderstoodasarealizationofstaticchirality.The non-vanishing distribution for

θ

=

90◦ and

φ

=

90◦ reflects the tunnelingbetweentheleft- andright-handedconfigurations.

For spins I

=

18 and 19 h,

_¯

the peaks for the azimuthal plot

for band A move toward (

θ

∼

80◦

,

φ

∼

20◦) and (

θ

∼

80◦

,

φ

∼

160◦), namely closeto the i-axis.This isin accordance withthe

Fig. 4. (Color online) The K plot,i.e., the K distributions for the angular momentum on the short (s),intermediate (i) and long (l)axes, calculatedat spins I=

Fig. 5. (Coloronline)Theazimuthal plot, i.e.,profilefor the orientationofthe angularmomentum onthe (θ, φ)plane,calculatedat spins I=10, 11, 14, 15, 18, 19¯h,

respectively,forthechiraldoubletbandsin130_Cs.

dominance of rotation around the i-axis reflected in the K plot

discussed above, suggesting the disappearance of chiral geome-try and the onset of principal axis rotation. The peaks for the

azimuthal plot for band B locate at

(θ

∼

75◦

,

φ

∼

35◦

)

and

(θ

∼

75◦

,

φ

∼

145◦

)

,whicharesimilartothose atI

=

14

,

15h but

¯

ap-proaching the i-axis. Thus the chiral geometry is weakened. In general,theevolutionoftheangularmomentumgeometryinthe

azimuthal plot isconsistentwiththoseinthe K plot.

Byblockingthe lowest

π

h11/2 and thefourth

ν

h11/2 orbitals, the chiral geometry in K plot and azimuthal plot has been dis-cussedinRef. [47].Figs.4and

5

confirmthatthechiralgeometry in K plot and azimuthal plot canbe robust against configuration mixing.

Insummary,theprojectedshellmodelwithconfiguration mix-ing for nuclear chirality is developed and applied to the ob-servedrotationalbandsinthechiralnucleus130_{Cs.Boththechiral} andachiral bandsare described on the samefoot. Forthe chiral bands, the energy spectra, S

(

I

)

, B

(

M1

)

and B

(

E2

)

are well re-produced.Thechiralgeometryisdemonstratedinthe K plot and

theazimuthal plot. The robustness ofthe chiral geometryagainst theconfiguration mixingisconfirmed. As another advantage,the otherrotationalbandsaredescribedsimultaneouslywiththesame Hamiltonian.

Acknowledgements

Thiswork waspartlysupported bytheNationalKeyR&D Pro-gramofChina No.2018YFA0404400, theNationalNaturalScience FoundationofChina(GrantsNo.11335002andNo.11621131001), and the China Postdoctoral Science Foundation under Grant No. 2017M610688.

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