intermediate-energyCoulombexcitation

arXiv:0708.3007v1 [nucl-th] 22 Aug 2007F.Delaunay1,2andF.M.Nunes2,3

LaboratoiredePhysiqueCorpusculaire,ENSICAEN,Universit′edeCaen,CNRS/IN2P3,14050Caen,France2

NationalSuperconductingCyclotronLaboratory,MichiganStateUniversity,EastLansing,Michigan48824,USA3

DepartmentofPhysicsandAstronomy,MichiganStateUniversity,EastLansing,Michigan48824,USA

E-mail:delaunay@lpccaen.in2p3.fr

Abstract.Coulombexcitationisastandardmethodusedtoextractquadrupoleexcitationstrengthsofeven-evennuclei.Intypicalanalysesthereactionisassumedtobeone-step,Coulombonly,andistreatedwithinasemi-classicalmodel.Inthiswork,fully-quantalcoupled-channelcalculationsareperformedforthreetestcasesinordertodeterminetheimportanceofmulti-stepe?ects,nuclearcontributions,feedingfromotherstatesandcorrectionstothesemi-classicalapproximation.Westudytheexcitationof30S,58Niand78Kron197Auat≈50AMeV.We?ndthatnucleare?ectsmaycontributemorethan10%andthatfeedingcontributionscanbelargerthan15%.Thesecorrectionsdonotaltersigni?cantlythepublishedB(E2)values,howeveranadditionaltheoreticalerrorofupto13%shouldbeaddedtotheexperimentaluncertaintyifthesemi-classicalmodelisused.Thistheoreticalerrorisreducedtolessthan7%whenperformingaquantalcoupled-channelanalysis.

1

PACSnumbers:25.70.De,24.10.Eq,23.20.-g

Submittedto:J.Phys.G:Nucl.Phys.

OnthemeasurementofB(E2)usingintermediate-energyCoulombexcitation2

Nuclearcollectivityofaneven-evennucleusiscloselyrelatedtoitsquadrupoleelectricreducedtransitionprobabilityB(E2,0+→2+Thisstrengthcanbe11).

determinedexperimentallybymeasuringeitherthelifetimeortheCoulombexcitationcrosssectionofthe2+1excitedstate[1,2].Originally,theCoulombexcitationtechnique(referredtoas“Coulex”inthefollowing)wasusedtomeasurepropertiesofthetarget(e.g.[3,4]).Thesub-Coulombenergiesatwhichthereactiontookplaceensuredanuclear-freemeasurement.Inthelastdecade,Coulexhasbeenexpandedtointermediateenergieswiththeaimofstudyingunstablenuclei[2].Inthiscasethenucleusofinterestisthebeamparticleandaheavytargetisusedtoproducethevirtualphotons.Thereactiontakesplaceathighenoughenergytoinhibitmulti-stepe?ectsanddataistakenonlyatveryforwardangles,whereoneexpectstobefreefromnuclearinterference.ThismethodhasenabledaccuratemeasurementoftheB(E2)ofalargevarietyofsystems[5,6,7,8,9,10,11,12,13,14,15,16,17].

Asystematiccomparisonbetweentheintermediate-energyCoulexmethodandthelifetimemethodshowedthatthereisconsistencybetweenthetwotechniques[1].Moreover,theaccuracyoftheB(E2)strengthsextractedthroughCoulexiscomparabletothatfromthelifetimemeasurements[1].TheworkofCooketal.[1]focusedontheexperimentalaccuracybutdidnotconsidertheuncertaintiesduetotheapproximationsintheCoulextheoryusedtoconnectcrosssectionsandelectricstrengths.Thisisexactlythefocusofthepresentstudy.

OnewaytoanalysetheseunstablebeamCoulexexperimentsisusingthesemi-classicalmodelofAlderandWinther[18].ItconvenientlyprovidesalinearrelationbetweentheCoulexcrosssectionandthereducedtransitionprobability.TheapproximationsintheAlderandWinthertheory[18]arethreefold:1)thestraight-linesemi-classicalapproximation;2)theexcitationisaone-stepprocess;3)itispurelyCoulomb.Thestraight-lineapproximationispartiallycorrectedwithinAlderandWinther[18].Thesecondpointisnotsostraight-forward.Mostofthenucleistudiedthroughthistechniqueexhibitlargecollectivityandthushaveotherexcitedstatesthatarestronglycoupledtoeitherthegroundstateorthe?rstexcited2+state.Evenifitisgenerallyassumedthatthecrosssectionstotheseotherexcitedstatesaresmallatintermediatebeamenergies,multi-stepmechanismsandinterferencescandistortthedesiredresult.Inordertosolidifythereliabilityoftheintermediate-energyCoulexmethod,itisimportanttoevaluatetheuncertaintiescomingfromtheone-stepapproximation.Finally,nuclearcontributionsneedtobeconsistentlyincludedinthecalculationssothatCoulomb-nuclearinterferenceiscorrectlyaccountedfor.Theinclusionofnucleare?ectsinrealisticquantumcoupled-channelcalculationsmayenhancemulti-stepe?ects.

Asmentionedabove,intermediate-energyCoulexreliesonrestrictingthescatteringanglestakenintoaccountforintegratingthecrosssectionstoarangecorrespondingtoimpactparameterslargerthanthesumofthetargetandprojectileradii.Adetailedstudyofthesensitivitytotheimpactparametercutwasperformedonthe46Ardata[10]andresultsvalidatetheprocedure.Incaseswherelowstatisticsforcestheinclusion

OnthemeasurementofB(E2)usingintermediate-energyCoulombexcitation3

ofawiderangularrange,nucleare?ectshavebeenestimatedwithquantumdistortedwavecalculationstobeoftheorderof6%[11,15].However,thisvalueshouldnotbetakenasde?nitive,since,asweshallsee,thenuclearcontributiondependsstronglyontheparticularanalysisconsidered.

AnotherproblemthatisconsideredwhenanalyzingCoulexdataisthepossibilityoffeeding:thereactionprocessexciteshighlyingstatesthatcouldthendecaytothe

2+enhanced2++

1state,producingan1→01signal.Inmostofthestudies,estimatesoffeedingpredictittobeunimportant(e.g.[13,15])mostlybecauseatintermediateenergiestherelativecrosssectionstohigherspinstatesaresmallandlargerexcitedstatesarehinderedcomparedtothelowertransitions.Neverthelesstherehavebeencaseswherefeedingneedstobecarefullyconsideredbeforeareliablestrengthisextracted[12,8].Inintermediate-energyCoulexexperiments,thestatisticsisoftenlowandthee?ciencyoftheγ-raydetectorsislimitedsuchthataγpeakforafeedingtransitionisrarelyseen[8].Feedingcorrectionsarebasedontheoreticalestimates[18]andsubtractedfromthe2+1crosssection,beforeextractingtheB(E2)strength.

Intermediate-energyCoulexhasbeenappliedmostlytointermediatemassnucleiboundbyafewMeVbutasbeamintensitiesimprove,itwillbeappliedtomoreexoticsystems.Thelooselyboundnatureofunstablenucleihasmodi?edmanyofthetraditionalviewsofnuclearreactions.Forexample,whentheexoticnucleushasanextendedtailinitswave-functions,one?ndsnuclearcontributionsatimpactparametersmuchlargerthanthesumofthetargetandprojectileradii[19].Inaddition,duetotheproximitytothecontinuum,multi-stepbreakupe?ectsneedtobeconsidered[20].AsystematicstudyofnuclearinterferenceintheCoulombdissociationofhalonucleihasshownlargenucleare?ects,evenintheforwardangularregionsconsideredsafeforCoulombexperiments[21].ArecentcomprehensivestudyoftheCoulexof11BeforextractionoftheB(E1)betweenthetwoboundstatesvalidatestheCoulexmethodacrossawiderangeofbeamenergies,providedallthesee?ectsaretakenintoaccountinthetheoreticalmodel[22].FortheB(E2)ofintermediatemassnuclei,itisimportanttosolidifythetheoreticalmethodsusedatpresentbeforethesenewdriplinechallengescanbefaced.

Table1.Informationontheintermediate-energyCoulexexperimentsconsideredhere.Foreachcasewegivethebeamlaboratoryenergy,themaximumcentre-of-massangleforcross-sectionintegration,thecorrespondingcrosssectionforthe2+1stateandthe

B(E2,0++

1→21)valueextractedthroughWintherandAlder’stheory[18].

Nucleus

Energy

θmaxCMσ2+

1B(E2,0++

1→21)

(AMeV)(deg.)(mb)

(e2fm4)

OnthemeasurementofB(E2)usingintermediate-energyCoulombexcitation4

Table2.Spin,parityandexcitationenergyforallthestatesincludedinthecoupled-channelcalculations.The0+assignmentforthe3.666MeVstatein30Sisbasedonacomparisonoftheexperimentalspectrumwiththespectrumofthemirrornucleusandashell-modelcalculationfortheA=30isobars.

30

S

E(MeV)0+12+1+22(0+2)

π

Jn

78

Kr

0+1+210+24+12+2

0

2.2113.4033.666

00.4551.0171.1191.148

Inthisworkweperformfully-quantumcoupled-channelcalculationsforthreetestcasesthathavebeenmeasuredbyintermediate-energyCoulex:30S,58Niand78Kr.Thesethreetestcasesspanavarietyofphysicalsituations.The?rst,30S,correspondstoaveryshortlivedisotope,twonucleonsawayfromtheprotondripline,withonlyafewexcitedstates.TheCoulexof30Swasmeasuredat35.7AMeVon197Au[9].Thesecond,58Niislessexotic,containsverystrongtransitionstohigherenergystatesandthereforehasanimportantfeedingcorrection.Ithasbeenmeasuredseveraltimesbeforeandweconsiderheretheexperimentat72.4AMeVon197Au[12].Thethird,78Kr,hasaverysmall2+1excitationenergy,andconsequentlyaverylargeB(E2).Hereweconsiderarecentmeasurementat57.4AMeVona197Autarget[14].ExperimentaldetailsfortheseexperimentsaresummarizedinTable1,whereweincludetheB(E2)extractedinthecorrespondingstudiesusingthe?rstordersemi-classicaltheory[18].

Wehaveinvestigatedthespectraofthesenucleiindetailandisolatedthestatesthatcana?ectthereactionmechanism.ThesearesummarizedinTable2.Forthetwoheaviercases,thespectraarewellknown.However,for30S,thespinandparityofthe3.666MeVstateareundetermined.Wehaveassumedtheyare0+bycomparisonwiththelevelschemeofthe30Simirrornucleus[23]andashell-modelcalculationfortheT=1statesoftheA=30isobars[24].Asitcanfeedintothe2+1state,itneedstobeincludedinthecalculations.

Stateswithunnaturalparity(e.g.1+and3+states)werenotincludedsincetheywoulddecayto2+and0+statesbymagnetictransitionswhicharenotimplementedinourcoupled-channelcalculations.Intheenergyrangeofinterest,thereisone1+statein30S,one1+statein58Niandnonein78Kr.Generally,magnetictransitionsinCoulexaremuchweakerthantheelectricones[25].

AverylargeamountoftransitionsispossiblebetweentheexcitedstateslistedinTable2.InTable3welistallthetransitionstakenintoaccountinourcoupled-channelcalculations.Wealsoprovidehal?ivesandbranchingratios[23]fromwhichwe

OnthemeasurementofB(E2)usingintermediate-energyCoulombexcitation5

Table3.Transitionsincludedinthecalculations:spinsandparitiesofthestatesinvolvedineachtransition,thecorrespondinghal?ivesT1/2,branchingratiosIγ,thereducedtransitionmatrixelementsM(Eλ)anddeformationlengthsδ=βR.Unlessotherwisenoted,thespins,parities,hal?ivesandbranchingratiosweretakenfromtheNNDCdatabase[23].

πJn,1

π

Jn,2

T1/2

Iγ

M(Eλ)

(efmλ)βR(fm)

58

Ni

2+1+414+1+222+22+22+32+32+32+32+42+42+42+40+1

+012+1+012+14+10+12+14+12+20+12+14+12+2

970fsb

970fsb0.38ps0.38ps0.38ps52fs52fs52fs52fs35fs35fs35fs35fs

1.002×10?8c

1.004.3×10?2

0.965.7×10?4

0.400.582.9×10?39.9×10?3

0.590.391.0×10?21.8×10?326.59a481.0871.561.4042.1336.759.1856.5849.6272.18d11.4140.49.3572.180.856a0.7181.0310.0450.6070.3950.2960.8150.5330.775d0.3670.5830.5301.040

(2I1+1)B(Eλ,I1→I2).

+

Forthe0+1→21transitions,weusedB(E2)directlyfromtheCoulexexperiments

OnthemeasurementofB(E2)usingintermediate-energyCoulombexcitation6

Table4.One-step(DWBA)andcoupled-channel(CC)calculationsincludingonlythegroundstateandthe2+1state:comparisonbetweencrosssectionsintegratedovertheexperimentalangularrangeforthefullprocesswiththosefromCoulombonly.Allcrosssectionsareinmb.

CC(2+1)44.740.6

58

NiCoul+nucl161.158

NiCoulomb

182.7

1052.61056.3

OnthemeasurementofB(E2)usingintermediate-energyCoulombexcitation7

Table5.Fullcoupled-channelcalculationsincludingalltransitionsspeci?edinTable

3.IJπn

→2+1

isthebranchingratiofortheJnπ

→2+1transition[23].σfeedarethecross

sectionsforfeedingintothe2+1state.Thevalueforσfeedinboldisthesumofthecrosssectionscontributingtoσ(2+1)foreachcase.

FullCCNucleus

σ

IJnπ→2+

1

σfeed

(mb)

(mb)

2+1

2+20+2

155.6188.38.61.00

8.658

Ni

1.90.961.819.10.5811.129.00.39

11.3

2+10+24+12+2

OnthemeasurementofB(E2)usingintermediate-energyCoulombexcitation8

experimentallimits.

For30Sand58Ni,nucleare?ectsarenotnegligibleandindicatethatthemaximumangleusedinintegratingtheexperimentalcrosssectionshouldbecarefullychosen.Thecontributionofthenuclearpartoftheinteractiontothecrosssectionisindeedverysensitivetothismaximumangle.For58Ni,ourtestsshowthatdecreasingthemaximumcentre-of-massanglebyonly0.5degreecutstherelativenuclearcontributionbyafactor2.

OnecanalsocompareourCoulomb-onlyDWBAcrosssectionstothosepredictedbythesemi-classicalmodel(seeTable1).Fromthiscomparisonwe?ndthatthestraight-linetrajectoryapproximationaloneintroducesanerrorinthecrosssectionof6%atthemost.

Fullcoupled-channelresultsincludingthestatesinTable2andtransitionsfromTable3arepresentedinTable5.Thecrosssectionstoindividualstates(σ)aremultipliedbythebranchingratiostothe2+1stateinordertogetthefeedingcontributions.ThesumoftheseandtheCoulex2+1crosssectiongivesthefullcrosssection(inbold)tobecomparedtotheexperiment.Feedingcontributionsareimportantin58Ni(≈17%),aswealreadyknew,butalsoin30S(≈5%).For58Ni,thefeedingcorrectionestimatedin[12]of25mbissigni?cantlysmallerthanourprediction(33mb),mainlybecauseofthecontributionfromthe4+1statewhichwasomittedintheexperimentalestimation[12].For30Sourtheoreticalcrosssectiontaking

+22

B(E2,0+1→21)=350efmisabovetheupperlimitoftheexperimentalrange.AreductionoftheB(E2)valueto303e2fm4isnecessarytogetatheoreticalcrosssectioninagreementwiththeexperimentalresult.Consideringthe7%theoreticalerror,thisvalueisstillcompatiblewiththevalueextractedin[9].For58Niand78Kr,theexpectedcrosssectioniswithin≈7%ofthemeanexperimentalvalue,i.e.withintheexperimentalrange.

Inconclusion,fullcoupled-channelcalculationsfor30S,58Niand78KrCoulexcon?rmtheagreementbetweenB(E2)extractedthroughthelifetimeandtheCoulexmethods[1].Ourstudyshowsthattheoreticalcontributionstotheerrorsinthecrosssectionscanbe≈13%ifthe?rstordersemi-classicaltheoryofAlderandWintherisused[18]butshouldbelessthan7%ifafullquantumcoupled-channelcalculationisperformedandfeedingisconsistentlytakenintoaccount.

WethankAlexandraGadeforusefuldiscussionsandcommentsonearlierversionsofthemanuscript.ThisworkwassupportedbyNSCL,MichiganStateUniversity,andtheNationalScienceFoundationthroughgrantPHY-05553.References

[1][2][3][4][5]

CookJM,GlasmacherTandGadeA2006Phys.Rev.C73024315GlasmacherT1998Annu.Rev.Nucl.Part.Sci481ShawAHandGreenbergJS1974Phys.Rev.C10263AhmadAetal.1988Phys.Rev.C371836MotobayashiTetal.1995Phys.Lett.B3469

OnthemeasurementofB(E2)usingintermediate-energyCoulombexcitation

[6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]

9

ScheitHetal.1996Phys.Rev.Lett.773967

IbbotsonRWetal.1998Phys.Rev.Lett.80002081CottlePDetal.1999Phys.Rev.C60031301(R)CottlePDetal.2002Phys.Rev.Lett.88172502GadeAetal.2003Phys.Rev.C68014302

YurkewiczKLetal.2004Phys.Rev.C70034301YurkewiczKLetal.2004Phys.Rev.C700319DincaDCetal.2005Phys.Rev.C71041302(R)GadeAetal.2005Phys.Rev.Lett.95022502ChurchJAetal.2005Phys.Rev.C720320BanuAetal.2005Phys.Rev.C72061305(R)PerruOetal.2006Phys.Rev.Lett.96232501

WintherAandAlderK1979Nucl.Phys.A319518

NunesFMandThompsonIJ1998Phys.Rev.C57R2818NunesFMandThompsonIJ1999Phys.Rev.C592652HusseinMS,Lichtenth¨alerR,NunesFMandThompsonIJ2006Phys.Lett.B091SummersNCetal.2007Phys.Lett.Binpresshttp://www.nndc.bnl.gov/

http://www.nscl.msu.edu/～brown/resources/SDE.HTMAlderKetal.1956Rev.Mod.Phys.28432

BohrAandMottelsonB1975NuclearStructurevol2NuclearDeformations(Reading:Benjamin)AlamanosNetal.1984Phys.Lett.B13737

Roussel-ChomazPetal.1988Phys.Lett.B209187ThompsonIJ1988Compt.Phys.Rep.73

BertulaniCA,StuchberyAE,MertzimekisTJandDaviesAD2003Phys.Rev.C68044609