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
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