Consistent analysis of one-nucleon spectroscopic factors involving weakly- and strongly-bound nucleons

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Consistent analysis of one-nucleon spectroscopic factors

involving weakly- and strongly-bound nucleons

J. Okolowicz, Y.H. Lam, M. Ploszajczak, A.O. Macchiavelli, N.A. Smirnova

To cite this version:

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Consistent

analysis

of

one-nucleon

spectroscopic

factors

involving

weakly- and

strongly-bound

nucleons

J. Okołowicz

a

,

Y.H. Lam

b

,

M. Płoszajczak

c

,

∗

,

A.O. Macchiavelli

d

,

N.A. Smirnova

e

a_{InstituteofNuclearPhysics,PolishAcademyofSciences,Radzikowskiego152,PL-31342Kraków,Poland}

b_{KeyLaboratoryofHighPrecisionNuclearSpectroscopy,InstituteofModernPhysics,ChineseAcademyofSciences,Lanzhou730000,China} c_{GrandAccélérateurNationald’IonsLourds(GANIL),CEA/DSM–CNRS/IN2P3,BP55027,F-14076CaenCedex,France}

d_{NuclearScienceDivision,LawrenceBerkeleyNationalLaboratory,Berkeley,CA 94720,USA}

e_{CENBG(UMR5797–UniversitéBordeaux1–CNRS/IN2P3),CheminduSolarium,LeHautVigneau,BP 120,33175GradignanCedex,France}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received29July2015

Receivedinrevisedform30March2016 Accepted31March2016

Availableonline6April2016 Editor:J.-P.Blaizot

Thereisaconsiderableinterestinunderstandingthedependenceofone-nucleonremovalcrosssections onthe asymmetryofthe neutronSn and proton Sp separation energies,followingalarge amountof

experimental data and theoretical analysesina framework ofsudden and eikonal approximations of thereactiondynamics.Thesetheoreticalcalculationsinvolveboththesingle-particlecross sectionand the shell-modeldescriptionofthe projectileinitialstate andfinal statesofthe reactionresidues.The configuration mixing in shell-model description of nuclear states depends on the proximity of one-nucleondecaythreshold butdoesit dependsensitively onSn−Sp?To answerthisquestion, weuse

theshellmodelembeddedinthecontinuumtoinvestigatethedependenceofone-nucleonspectroscopic factors ontheasymmetry ofSn and Sp formirror nuclei24Si, 24Neand 28S,28Mgand foraseriesof

neonisotopes(20≤A≤28).

©2016TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense

(http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

Single-nucleonremovalreactions atintermediateenergies pro-videabasictooltoproduceexoticnucleiwithlargecross-sections. The interpretation of thesereactions aseffective direct reactions follows from the theoretical modelling which uses an approxi-mate description of the reaction dynamics (sudden and eikonal approximations)[1]andastandardshellmodeldescriptionofthe structure of the initial state of the projectile, the final states of the

(

A

−

1

)

-nucleon reaction residues, and the relevant overlap functions. Ina series ofpapers [2,3],it was found that the ratio Rσ

=

σ

exp

/

σ

th ofthe experimental andtheoretical inclusive

one-nucleon removal cross section for a large number of projectiles showsastrikingdependenceontheasymmetryoftheneutronand proton separation energies. Is this dependence telling us some-thingimportantaboutthecorrelationsintheprojectileinitialstate and/orfinalstatesofreactionresidues,orcoulditbeanartefactof thetheoreticalmodellingused?

A possible drawback of the theoretical modelling of the one-nucleonremoval reactions is that the descriptionof reaction

dy-*

Correspondingauthor.

E-mailaddress:ploszajczak@ganil.fr(M. Płoszajczak).

namics and shell model ingredients in this description are not consistently relatedone to another. Therefore, there is no a pri-ori certainty that these reactions can be considered as effective directreactions.Acomprehensivetheoreticaldescriptionina uni-fiedframework ofthecontinuum shellmodel(CSM)[4–6] orthe coupled-channelGamow shellmodel[7]ismuchtoocomplicated to be consideredasa realistic propositionin thenearfuture. On theother hand,single-nucleon removalreactionsasexperimental tools to study exotic nuclei are too important to abandon fur-ther discussion of Rσ

(

S

)

-dependence found, where



S equals Sn

−

Sp forone-neutron removaland Sn

−

Sp forone-proton

re-movalreactions. Recently, spectroscopicfactors forproton knock-outinclosed-shell14,16,22,24,28Owerecalculatedusingthecoupled clusterformalism[8]andfoundtodependstronglyon



S, inline with the observations in [2,3]. In contrast, the experimental re-sultsinRef.[9]usingtransferreactions inOxygenisotopes show, atbest,aweakdependenceon



S. Moreover,thestudyinRef.[10]

points tosomelimitationsoftheeikonal approximation.Herewe wouldlike toaddress thequestion of whetherthededuced ratio of the experimental and theoretical one-nucleon removal cross-sectionsisrelatedinanywayto the



S-dependence of theCSM spectroscopic factors? Hence, is thisratio probing the configura-tion mixing in SM states involved inthe single-nucleon removal reactions atintermediate energies orshould it be considered as

http://dx.doi.org/10.1016/j.physletb.2016.03.086

304 J. Okołowicz et al. / Physics Letters B 757 (2016) 303–306

an empirical normalization factorof the theoretical cross-section whendeterminingspectroscopicfactorsofexoticnuclei?Ofcourse, oneshouldkeepinmindthatthespectroscopicfactorsarenot ob-servables per se and assuch are not invariant under the unitary transformationoftheHamiltonian.However,inagivenmodel,the spectroscopic factors are importanttheoretical quantities and an investigationofthe



S-dependence ofthespectroscopicfactorsin aconsistenttheoreticalframeworkprovidedbytheCSMmayshed light onour understanding ofthe one-nucleon removalreactions atintermediateenergies.

The



S-dependence ofspectroscopicfactorsisexaminedin24Si and28_{S,andtheirmirror partners}24_{Ne and}28_{Mgusingtheshell}

model embedded in the continuum (SMEC) [6] which isa mod-ern version of the CSM. The ratio Rσ , which was reported for

24_{Si and} 28_{S both for neutron and proton removalreactions [3],}

willserve asa referencein thisanalysis. Weinvestigate alsothe



S-dependence oftheratioofSMECandSMspectroscopicfactors RS F

=

SSMEC

/

SSM forachainofneonisotopesattheexperimental

separationenergies

S

nand

S

p. 2. Themodel

Inthe presentstudies,the scatteringenvironment isprovided by one-nucleondecaychannels. The Hilbertspaceis dividedinto twoorthogonalsubspaces

Q

0 and

Q

1 containing0and1particle

inthescatteringcontinuum,respectively.Anopenquantumsystem descriptionof

Q

0 spaceincludescouplingstotheenvironment of

decay channelsthrough the energy-dependenteffective Hamilto-nian[6,11]:

H

(

E

)

=

H_Q0Q0

+

W_Q0Q0

(

E

),

(1)

where

H

_Q0Q0 denotesthestandardSMHamiltoniandescribingthe

internaldynamicsandW_Q0Q0

(

E

)

:

W_Q0Q0

(

E

)

=

H_Q0Q1G(_Q+)

1

(

E

)

HQ1Q0

,

(2)

istheenergy-dependentcontinuumcouplingterm,where

G

(_Q+) 1

(

E

)

is the one-nucleon Green’s function and HQ0,Q1, HQ1Q0 are the

couplingtermsbetweenorthogonalsubspaces

Q

0 and

Q

1 [11]. E

intheabove equationsstandsforascatteringenergy.The energy scaleissettledbythelowestone-nucleonemissionthreshold.The channel stateinnucleus A is definedby thecouplingofone nu-cleon(protonorneutron)inthecontinuumtonucleus

(

A

−

1

)

ina givenSMstate.IntheSMEC calculation,weinclude18lowest de-cay channelsinthenucleus A: 9 forprotonsand9forneutrons. Thecouplingtothesechannelsgivesrisetothe mixingofall SM statesofthesametotalangularmomentumandparityinthe nu-cleus A and, hencechanges theground state (g.s.) spectroscopic factorwithrespecttoitsSMvalue.

TheSMECsolutions arefoundby solvingtheeigenproblemfor theHamiltonianEq.(1)inthebiorthogonalbasis.TheSMEC eigen-vectors



α are relatedtotheeigenstates



joftheSMHamiltonian

H_Q0Q0 by a linear orthogonal transformation:



α

=



jbαj



j.

The expectation value of any operator O can

ˆ

be calculated as:

 ˆ

O



= 

α_¯

| ˆ

O

|

α



. In case of the spectroscopic factor one has:

ˆ

O

=

a†

|

t



t

|

a, where

|

t



isthe target stateof the

(

A

−

1

)

-system. a and a† _{areannihilationandcreationoperatorsofanucleonina}

givensingle-particle(s.p.)state.

3. TheHamiltonian

Fortheisospin-symmetricpartoftheSM Hamiltonian H_Q0Q0 in the sd shell model space, we take the USD interaction [12]. The charge-dependent terms in the Hamiltonian comprise the two-body Coulomb interaction, acting between valence protons,

and isovectorsingle-particle energies [13] which account for the Coulomb effects inthe core. Both thesetermsare scaled propor-tionallyto

√

h

¯

ω

A[13],withh

¯

ω

Abeingparameterizedas

¯

hωA

=

45 A−1/3

−

25 A−2/3MeV

.

(3)

The terms HQ0,Q1, HQ1Q0 in the continuum-coupling term

Eq.(2)aregeneratedusingtheWigner–Bartlett(WB)interaction:

V12

=

V0



α

+ β

Pσ₁₂



δ (r

1

−

r2

) ,

(4)

where

α

+ β =

1 and P σ₁₂ isthespinexchangeoperator.Although theproduct

V

0

(

α

−β)

=

414 MeV fm3iskeptconstantinall

calcu-lations,themagnitudeofthecontinuumcouplingvariesdepending onspecificvaluesof

V

0,

α

,thestructureofthetarget wave

func-tion,andtheseparationenergies

S

p, Sn.

The radial s.p. wave functionsin

Q

0 andthe scatteringwave

functionsin

Q

1 aregenerated usingaWoods–Saxon(WS)

poten-tialwhichincludesspin–orbitandCoulomb parts.Theradiusand diffuseness of the WS potential are R0

=

1

.

27 A1/3fm and a

=

0

.

67 fm respectively. The spin–orbit potential is VSO

=

6

.

1 MeV,

andtheCoulombpartiscalculatedforauniformlychargedsphere with radius R0. The depth of the central part for protons

(neu-trons)is adjustedtoyield theenergyofthes.p.state involvedin the lowest one-proton (one-neutron) decaychannel equal to the one-proton (one-neutron)separationenergyintheg.s.ofthe nu-cleus A. The continuum-coupling termEq.(2)breaks the isospin conservationduetodifferentradialwavefunctionsforprotonsand neutrons,anddifferentseparationenergies

S

p, Sn.

4. Theresults

Weshalldiscussnowthedependenceofthespectroscopic fac-tors ontheasymmetry ofneutronandprotonseparationenergies and on the strength ofthe spin-exchange term which influences thestrengthofthe

T

=

0 proton–neutroncontinuumcoupling.

Fig. 1 showsthe



S dependence of theratio ofd5/2

spectro-scopic factors in SMEC andSM forthe g.s. of mirrornuclei: 24_Si

(S(pex)

=

3

.

293 MeV and S( ex)

n

=

21 MeV) [14] and 24Ne (S( ex)

p

=

16

.

55 MeV and S(nex)

=

8

.

868 MeV) [14]. The curves RS F

(

S

)

for proton (neutron) spectroscopic factors are shown with the solid (dashed) lines as a function of Sn

−

Sp (Sn

−

Sp) on a

(

Sp

,

S

n

)

-lattice. The case



S

<

0 in RS F

(

S

)

for proton

(neu-tron)spectroscopicfactorscorrespondstotheremovalofa weakly-bound proton (neutron). The values of RS F at the experimental

separation energies S(pex) and Sn(ex) are denoted by filled circles.

Alongeachcurvefortheproton(neutron)spectroscopicfactor,

S

p

(Sn)changesand

S

n(Sp)remainsconstant.

Quantitativeeffectsofthecontinuumcouplingonspectroscopic factorsdependonthedistributionofthespectroscopicstrengthin the considered SM states [15]. For

α

=

1, the rearrangement of spectroscopicfactorsissmallifthespectroscopicstrengthis con-centrated ina single SM state. This is the casein both 24_Siand 24_{Ne.Forexample,92.7%(91.9%)oftheproton(neutron)}

_d

5/2 SM

spectroscopicstrengthin24Siisintheg.s.andhence,theratio

R

S F

bothforprotonandneutrong.s.spectroscopicfactorsisalmost in-dependent of



S (see Fig. 1a). The slightbreakingofthe isospin symmetrybyboththecontinuumcouplingandtheCoulombterm in the SM interaction leads in this case to a weak dependence of RS F for the neutron spectroscopic factor (the dashed lines in

Fig. 1a)inthelimitofsmall Sn.

Fig. 1. TheratioofSMECandSMd5/2spectroscopicfactorsinmirrornuclei24Siand 24_{NeisplottedasafunctionoftheasymmetryS oftheneutronandproton}

sepa-rationenergies.Proton(neutron)spectroscopicfactorsareshownbysolid(dashed) lines.S equalsSn−Sp(Sn−Sp)fortheproton(neutron)spectroscopicfactors.

FilledcirclesdenotetheratiosRS F attheexperimentalvalueoftheasymmetry

pa-rameterS.(a)RS F(S)fortheg.s.of24Sifor

α

=1;(b)RS F(S)fortheg.s.of 24_Sifor

_α

_=0.73;(c)R

S F(S)fortheg.s.of24Nefor

α

=0.73.Formoredetails,

seethedescriptioninthetext.

showa rearrangementofthe d5/2 neutron(proton)spectroscopic

strengthin theg.s. of24_Si(24_{Ne) for}

_α

₌

₀

_.

_{73. The}

_

_S-variation

oftheratioRS F inthiscaseremainshoweverrelativelysmalland

doesnot exceed10%. One should noticealso a significant break-ingofmirrorsymmetrybycomparing

R

S F

(

S

)

curvesforneutron

spectroscopicfactorsat small Sn in 24Si(dashed linesinFig. 1b)

andprotonspectroscopicfactorsatsmall

S

p in24Ne(solidlinesin

Fig. 1c).

Theratio Rσ

=

σ

exp

/

σ

th ofcross-sectionsforoneproton

(neu-tron)removal from24Si is

∼

0

.

8

±

0

.

04 (

∼

0

.

39

±

0

.

04) [2]. The corresponding ratio of SMEC and SM proton (neutron) spectro-scopicfactorsishoweveralmostconstantandequals0.987(0.981) for

α

=

1,and0.974(0.957)for

α

=

0

.

73.

Fig. 2showsthe



S dependence oftheratioofSMECandSM g.s. spectroscopicfactors formirrornuclei: 28_{S (S}(ex)

p

=

2

.

49 MeV

and S(nex)

=

21

.

03 MeV) [14] and 28Mg (S(pex)

=

16

.

79 MeV and

S(nex)

=

8

.

503 MeV)[14].Proton(neutron)spectroscopicfactorsare

shownbysolid(dashed)lines.



S equals Sn

−

Sp (Sn

−

Sp)forthe

proton(neutron)spectroscopicfactors.

Inthesenuclei,thedistributionofSMspectroscopicstrengthis differentfromthatin24Siand24Ne.Only44% ofthe protons1/2

spectroscopicstrengthisintheg.s.of28S.Onthecontrary,97.6%of theneutron

d

5/2 spectroscopicstrengthisconcentratedintheg.s.

of28S.Consequently,evenintheabsenceofthe T

=

0 component inthecontinuumcouplinginteraction(

α

=

1),theratio

R

S F ofthe

protons1/2 spectroscopicstrengthsdependson Sp

−

Sn (seesolid

lines in Fig. 2a). This dependenceis further enhanced for

α

=

1 (see Fig. 2b). One should also noticestrong breaking of the mir-rorsymmetrybycomparingtheratioofproton

s

1/2 spectroscopic

strengthatsmall

S

p in28S(solidlinesinFig. 2b)andtheratioof

Fig. 2. TheratioofSMECandSMproton(neutron)s1/2spectroscopicfactorsand

neutron(proton)d5/2 spectroscopicfactorsinmirrornuclei28S(28Mg)isplotted

asafunctionoftheasymmetryS oftheneutronandprotonseparationenergies. (a) RS F(S)fortheg.s.of28Sfor

α

=1;(b) RS F(S)fortheg.s.of28Sifor

α

=

0.73;(c) RS F(S)fortheg.s.of28Mgfor

α

=0.73.Formoreinformation,seethe

captionofFig. 1andthedescriptioninthetext.

Fig. 3. The ratioofSMECand SM g.s.spectroscopicfactorsin thechain ofNe isotopes(20≤A≤28)isplottedasafunctionoftheasymmetryS oftheir experi-mentalneutronandprotonseparationenergiesfor

α

=0.73.S equalsS(pex)−S

(ex)

n

(S(nex)−S(pex))fortheproton(neutron)spectroscopicfactors.Theratiosofproton

(neutron)spectroscopicfactorsaredepictedwithsquares(triangles).Inagreement withtheexperimentalangularmomentumandparityofthenucleus(A−1),theg.s. protonspectroscopicfactorsared5/2forallconsideredisotopes,andneutron

spec-troscopicfactorsare(i)s1/2in20,26Ne,(ii)d3/2in22,28Ne,and(iii)d5/2in24Ne.For

moredetails,seethedescriptioninthetext.

neutron s1/2 spectroscopic strength atsmall Sn in 28Mg (dashed

linesinFig. 2c).Thiseffectisduetoanimportantdependenceof the

s-wave

continuumcouplingontheCoulombinteraction.

In28S,theratio Rσ of proton(neutron)removalcross-sections is

∼

0

.

92

±

0

.

07 (

∼

0

.

31

±

0

.

025)[2].Again,thecorrespondingratio ofSMEC andSM proton(neutron)spectroscopic factorsisalmost constant and equals 1.026 (1.0) for

α

=

1 and 0.848 (0.975) for

α

=

0

.

73.

Systematicsoftheratio RS F foreven-N Ne-isotopesisplotted

inFig. 3.Theconsideredisotopesare[14]:20_Ne(S(ex)

p

=

12

.

84 MeV and S(nex)

=

16

.

865 MeV), 22Ne (S( ex) p

=

15

.

266 MeV and S( ex) n

=

10

.

364 MeV), 24_{Ne (S}(ex)

306 J. Okołowicz et al. / Physics Letters B 757 (2016) 303–306

26_Ne(S(ex)

p

=

18

.

17 MeV and Sn(ex)

=

5

.

53 MeV),and28Ne(S(pex)

=

20

.

63 MeV and S(nex)

=

3

.

82 MeV). RS F forproton(neutron)

spec-troscopicfactors areshownwithsquares(triangles)asafunction of Sn

−

Sp (Sn

−

Sp). One can see that the ratio of SMEC and

SMspectroscopicfactorsdoesnotexhibitanysystematictendency asafunction of



S. The fluctuationsof RS F are smallexceptfor

theneutronspectroscopicfactorin22Ne.Thisnucleusishowever an exception in the Ne-chain because the g.s. SM spectroscopic factor is much smaller than the spectroscopic factor in the first excitedstateandthereforeevenasmallspin(isospin)dependence oftheproton–neutroncontinuumcouplingand/orasmall isospin-symmetry breaking term in the Hamiltonian and the continuum coupling,producesignificantchangeofthespectroscopicfactor.

In conclusion, we have consistently evaluated the fraction of single-particlespectroscopicstrengthwhichisshiftedtohigher ex-citations as a result of the coupling to the proton and neutron decay channels. We have shown that the one-nucleon spectro-scopicfactorsinSMECareweaklycorrelatedwiththeasymmetry of neutron Sn and proton Sp separation energies. Strong mirror

symmetry breaking,found in the ratioof SMEC andSM spectro-scopic factors, appears mainly for small one-nucleon separation energies suggestingthe thresholdnature of thiseffect.Whatever theprecisereasonsforastrongdependenceoftheratioof experi-mentalandtheoreticalone-nucleonremovalcrosssectionsonthe asymmetryofneutronandprotonseparationenergiesare,the ex-planation ofthisdependencedoesnotreside inthe behaviour of theone-nucleonCSM(SMEC)spectroscopicfactorsasafunctionof



S.

Acknowledgements

Thisworkwassupported inpartby theFUSTIPEN(French-U.S. Theory Institute for Physics with Exotic Nuclei) under U.S. DOE

grantNo. DE-FG02-10ER41700,bytheCOPINandCOPIGAL French-Polish scientific exchange programs, by the U.S. DOE contract No. DE-AC02-05CH11231 (LBNL), Y.H.L. andN.A.S. appreciate the financialsupportfromtheMinistryofForeignAffairsand Interna-tionalDevelopmentofFranceintheframeworkofPHCXuGuangQi 2015 under project No. 34457VA. Y.H.L. gratefully acknowledges the financial supportsfrom theNationalNatural Science Founda-tionofChina(Nos.U1232208,U1432125),MinistryofScienceand Technology ofChina(TalentedYoungScientistProgram) andfrom the ChinaPostdoctoral ScienceFoundation (2014M562481).N.A.S. acknowledgesthefundingofCFT(IN2P3/CNRS,France),APthéorie 2014.

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