Search for the p_1tn1_1tn/_1tn2_1th- resonance in _1hn7He with the _1hn7Li(d,_1hn2He) reaction and measurement of the deuteron electrodisintegration under 180 ̊at the S-DALINAC [Elektronische Ressource] / angefertigt von Natalya Ryezayeva
135 pages
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Search for the p_1tn1_1tn/_1tn2_1th- resonance in _1hn7He with the _1hn7Li(d,_1hn2He) reaction and measurement of the deuteron electrodisintegration under 180 ̊at the S-DALINAC [Elektronische Ressource] / angefertigt von Natalya Ryezayeva

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135 pages
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Description

7Search for the p ¡ Resonance in He1=27 2with the Li(d, He) ReactionandMeasurementof the Deuteron Electrodisintegration–under 180 at the S-DALINACVom Fachbereich Physikder Technischen Universitat Darmstadt˜zur Erlangung des Gradeseines Doktors der Naturwissenschaften(Dr. rer. nat.)genehmigteD i s s e r t a t i o nangefertigt vonNatalya Ryezayevaaus Polyarny (Russland)Juli 2006DarmstadtD 17Referent: Professor Dr. rer. nat. Dr. h.c. mult. A. RichterKorreferent: Dr. rer. nat. J. WambachTag der Einreichung: 18.07.2006Tag der Prufung:˜ 06.11.2006ZusammenfassungDie vorliegende Arbeit enthalt zwei Teile. Beide beschaftigen sich mit der Unter-˜ ˜suchung von leichten Kernen.7Im ersten Teil wird die Struktur des exotischen Kerns He studiert. Jungste Ex-˜perimente geben widersprechende Aussagen uber die mogliche Existenz des p -˜ ˜ 1=27Spin-Bahnpartnersdes HeGrundzustandsmiteinemdominantenp ¡ Einteilchen-3=27 2 7Charakter. Zur Kl˜arung dieser Frage wurde die Reaktion Li(d, He) He bei ei-ner Einschussenergie von 171 MeV am Kernfysisch Versneller Insituut (KVI) inGroningen untersucht. Das Experiment fand im April 2003 statt. Mit dem expe-rimentellen Aufbau am KVI wurde eine Energieau ˜osung von ¢E… 150 keV inden gemessenen Spektren erreicht, die geringer ist als die naturlic˜ he Linienbreite7 2des He-Grundzustands. Das ungebundene He-System wurde identiflziert durchdie Koinzidenzmessungen der beiden Protonen mit kleiner relativer Energie.

Sujets

Informations

Publié par
Publié le 01 janvier 2006
Nombre de lectures 10
Langue Deutsch
Poids de l'ouvrage 2 Mo

Exrait

Searchforthep1/2−Resonancein7He
withthe7Li(d,2He)Reaction
andtMeasuremenoftheDeuteronElectrodisintegration
under180◦attheS-DALINAC

of

VomFachbereichPhysik
derTechnischenUniversit¨atDarmstadt

GradesdesErlangungzurhaftenNaturwissenscderDoktorseinesrer.(Dr.nat.)

genehmigte

Dissertation

onvangefertigt

NatalyaRyezayeva
ausPolyarny(Russland)

2006Juli

Darmstadt17D

t:Referen

t:Korreferen

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agT

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ung:hEinreic

ufung:¨Pr

Professor

Professor

Dr.

Dr.

18.07.2006

06.11.2006

rer.

rer.

nat.

nat.

Dr.

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terhRic

Zusammenfassung

DievorliegendeArbeitenth¨altzweiTeile.Beidebesch¨aftigensichmitderUnter-
suchungvonleichtenKernen.
ImerstenTeilwirddieStrukturdesexotischenKerns7Hestudiert.J¨ungsteEx-
perimentegebenwidersprechendeAussagen¨uberdiem¨oglicheExistenzdesp1/2-
Spin-Bahnpartnersdes7HeGrundzustandsmiteinemdominantenp3/2−Einteilchen-
Charakter.ZurKl¨arungdieserFragewurdedieReaktion7Li(d,2He)7Hebeiei-
nerEinschussenergievon171MeVamKernfysischVersnellerInsituut(KVI)in
Groningenuntersucht.DasExperimentfandimApril2003statt.Mitdemexpe-
rimentellenAufbauamKVIwurdeeineEnergieaufl¨osungvonΔE≈150keVin
dengemessenenSpektrenerreicht,diegeringeristalsdienat¨urlicheLinienbreite
des7He-Grundzustands.Dasungebundene2He-Systemwurdeidentifiziertdurch
dieKoinzidenzmessungenderbeidenProtonenmitkleinerrelativerEnergie.Die
Messungenerstreckensich¨ubereinenWinkelbereichvonΘcm=0◦-11.3◦.F¨urdie
EntfaltungdesSpektrumswurdenbekannteResonanzen,dasAufbruchverhalten
von7HeundquasifreieLadungsaustauschbeitr¨ageverwendet,unterBer¨ucksichti-
gungderCluster-Strukturvon7Li.DabeiweisendieexperimentellenErgebnisse
aufeinem¨oglicheniedrigligendeResonanzbeiEx=(1.45−+00..57)MeVmiteiner
BreiteΓ=(2.0−+11..10)MeVhin.DieGamow-TellerSt¨arkef¨urdie¨Uberg¨angezu
denniedrigstenZust¨andenin7HestimmenmittheoretischenVorhersageneines
abinitioQuantumMonte-CarloModells¨uberein.Weiterhinwurdederspektro-
skopischeFaktorSn=0.64±0.09des7HeGrundzustands(7He=6He⊗n)mit
extrahiert.Analyse-matrixReinerIndemzweitenTeilderArbeitwirdderDeuteronaufbruchinder2H(e,e)Re-
aktionuntereinemStreuwinkelΘ=180◦untersucht.DieMessungenwurden
amsupraleitendemDarmst¨adterElektronenlinearbeschleunigerS-DALINACbei
EinschussenergienE0von27.8MeVund74MeVimM¨arzundApril2006durch-
gef¨uhrt.BeiniedrigenImpuls¨ubertr¨agen(q=0.28fm−1bzw.0.73fm−1)do-
minierenmagnetische¨Uberg¨angedenAufbruchwirkungsquerschnitt.Daherkann
manausdengemessenenWirkungsquerschnittenanderAufbruchschwelledie
Wirkungsquerschnittef¨urdenEinfangprozessnp−→dγextrahierenunterVer-
wendungdesPrinzipsdesdetailliertenGleichgewichtes.DiegenaueInformation
¨uberdiesenProzessistvongroßemInteressef¨urdieBig-BangNukleosynthese

(BBN).

Die

experimentellen

Daten

sind

in

schenVorhersagenunterVerwendungeines

und

terun

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¨ungereinstimmUb

mit

theoreti-

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h-StrMesonaustausc

ome¨

und

onfigurationen.Isobark

Summary

Thepresentworkcontainstwoparts,bothdevotedtotheinvestigationoflight
uclei.n

Inthefirstpartofthethesisthestructureoftheexotic7Henucleusisstu-
died.Thedisappearanceoftheusualmagicnumbersinextremelyneutron-rich
nucleiimpliesaconsiderablemodificationinthespin-orbitinteraction.Recent
experimentsyieldcontradictoryresultsaboutapossibleexistenceofthep1/2−
spin-orbitpartnerofthe7Hegroundstatewithadominantp3/2−single-particle
character.Inordertoclarifythisquestionastudyofthe7Li(d,2He)7Hereaction
hasbeenperformedusinga171MeVdeuteronbeamprovidedbythecyclotron
atKernfysischVersnellerInsituut(KVI)inGroningen.Theexperimentwascar-
riedoutinApril2003.ThesetupatKVIoffersaresolutionΔE≈150keV
(FWHM)inthemeasuredspectra,betterthanthelinewidthoftheground
stateof7He.Theunbound2Hesystemwasidentifiedbydetectingcoincidences
betweentwoprotonswithsmallrelativeenergy.Thedataweretakenoverthe
angularrangeΘcm=0◦-11.3◦.Apossibleresonanceatanexcitationenergy
Ex=(1.45−+00..57)MeVwithawidthΓ=(2.0−+11..10)MeVissuggestedbyadecompo-
sitionofthespectrumusingknownresonances,thebreakupbehaviourof7Heand
quasifreecharge-exchangecontributions,takingintoaccounttheclusterstructure
of7Li.Gamow-Tellerstrengthsfortransitionstotheloweststatesin7Hearein
remarkableagreementwithresultsfromabinitioQuantumMonteCarlocalcula-
tions.TheneutronspectroscopicfactorSn=0.64±0.09ofthe7Hegroundstate
(7He=6He⊗n)isextractedbyanR-matrixanalysis.
Inthesecondpartofthethesisthedeuteronbreakuphasbeenstudiedinthe
2H(e,e)reactionatΘ=180◦.Thepresentmeasurementswereperformedin
MarchandApril2006atthesuperconductingDarmstadtelectronlinearaccele-
ratorS-DALINACatanincidentelectronenergyE0=27.8MeVand74MeV.
Atlowmomentumtransfer(q=0.28fm−1and0.73fm−1,respectively)magne-
tictransitionsareexpectedtogivethedominantcontributionstothebreakup
crosssections.Thus,themeasureddeuteron-electrodisintegrationcrosssectionat
thresholdcanbeusedtoobtainthecrosssectionforthenp−→dγreactionapply-
ingtheprincipleofdetailedbalance.Theaccurateinformationaboutthisprocess
isofgreatinterestinnuclearastrophysics,specificallywithregardtoBig-Bang

ucleosynnthesis.

The

talerimenexp

data

are

calcalculationsbasedonanucleon-nucleon

inclusion

of

meson-exchange

and

isobar

in

texcellen

tialotenp

effects.

agreement

mo

del

with

(Bonn-B),

theoreti-

under

the

tstenCon

Searchforthep1/2−Resonancein7He
withthe7Li(d,2He)Reaction

ductiontroIn1

kgroundbacTheoretical22.1Scatteringformalism.........................
2.2DistortedwaveBornapproximationandopticalpotential.....
2.3Effectiveinteraction..........................
2.4Shellmodelandone-bodytransitiondensities...........
2.5CrosssectionsandGamow-Tellerstrength.............
2.6AbinitioQuantumMonteCarlomethod..............
2.7R-matrixtheory............................

setuptalerimenExp33.1AGORcyclotronatKVI.......................
3.2Big-bitespectrometer.........................
3.2.1Raytracing..........................
3.2.2Recoilcorrections.......................
3.3EUROSUPERNOVAdetectionsystem...............
3.3.1Focal-planedetectionsystem.................
3.3.2Focal-planepolarimeter...................
3.3.3Scintillators..........................
3.4Electronics...............................
3.4.1Triggerlogic..........................
i

1

1

5571012151719

2323232527272829292930

4

5

6

7

8

3.4.2DSP-basedonlinedataprocessingsystem..........
3.5Dataacquisitionandonlineanalysis.................
23.6Heeventreconstruction.......................
3.7Experiment..............................

analysisData4.1Reconstructionofthescatteringvariables..............
4.2Acceptancecorrection........................
4.3Instrumentalbackgroundsubtraction................
4.4Angularbinsandexperimentalcrosssections............
4.5Quasifreecontinuumbackground..................
4.6Fitofthespectra...........................
4.6.1Decompositionofthespectrum...............
−π4.6.2Possiblelow-lyingJ=1/2spin-orbitpartneroftheground
state..............................

ExtractionofGamow-Tellerstrength
5.1CrosssectionandB(GT).......................
5.2DWBAcalculationsoftheangulardistributions..........
5.3ComparisonwithQMCcalculations.................

Spectroscopicfactorofthegroundstate
6.1Deconvolution.............................
6.2R-matrixanalysis...........................

Continuumstructureathigherenergies

outloandSummaryok

ii

31333434

363637414345505053

56565759

626265

67

70

MeasurementoftheDeuteronElectrodisintegration
under180◦attheS-DALINAC

9ductiontroIn

10Experimentalprocedure

10.1S-DALINAC..............................
◦10.2QCLAMspectrometerand180facility...............

10.3Experiment..............................

discussionandResults11

11.1Analysisofthespectrum.......................

11.2Determinationofthecrosssections.................

11.3Decompositionofthespectrum...................

11.4Discussion...............................

11.5Extractionoftheastrophysicalnp−→dγcrosssection......

okoutloandSummary12

ADouble-differentialcrosssectionsforthe7Li(d,2He)reaction

iii

72

72

77

7777

80

82

82

87

90

91

94

96

97

iv

I:TARPSearchforthep1/2−Resonancein7He
withthe7Li(d,2He)Reaction

ductiontroIn1

Inrecentyearsthestudyofthestructureandthereactionmechanismsinvolving
lightexoticnucleihasleadtothedevelopmentofnewtheoreticalmodelsbasedon
abinitiomethods[1,2]andalsoonrefinedoperatormethodslikeunitarycorrela-
tionoperatormethodUCOM[3].Duetonovelexperimentaltechniquesandthe
rapidprogressintheproductionofradioactivebeamsvariousunexpectedpheno-
menahavebeenobservedatthedriplineswhichcannotbeexplainedwithinthe
traditionalmodelsusedforstablenuclei.Asanexample,nuclearhalostateshave
beenfoundinanumberoflightnucleiclosetothenucleondriplines.Insimple
words,nuclearhaloesarecloudsofnucleonsextendingwellbeyondthesurfaceof
aboundcorewithprotonsandneutrons.Butthereisnoprecisedefinitionwhen
anuclearstateshouldbecalledahalostate.Forexample,thematterradiusfor
suchsystemsshouldbesignificantlylargerthanthestandardnuclearradius,and
ahalosystemshoulddivideintoclustersthathavealargerdistancefromeach
otherthaninusualnuclei.Thepairingisveryimportantinlooselyboundsy-
stems.Thisismostremarkablyseenintheheliumisotopes.Addingoneneutron
to4Heyields5Hewhichisunbound.However,addingtwoneutronstothe4He
6Hewillbeformedwhichisabound,twoneutronhalonucleus.Continuationto-
wardstheneutrondripline,thestoryrepeatsitselfwith7Heand8He.Continuing
evenfurther9Heisreachedandthedoublymagicnucleus10He.Thus,thorough
investigationsoftheneutron-richHeisotopeshavedemonstratedtheirdominant
α-clusterstructure[4],leadingtoatwo-neutronhaloin6Heandapeculiar4n+
αstructurein8Hewhiletheodd-massisotopes5,7Heareparticle-unbound.Still
manyopenquestionsregardingthestructureoftheunstableprotonandneutron
richnucleihavetobeanswered[5].Theinvestigations,thatarepresentedinthis

1

Fig.1.1:Simplifiedviewofthe7Hehalonucleusconsistingoftwoprotons(dark)
andfiveneutrons(lightgrey).

work,focusontheexperimentalstudyoftheunbound7Hesystem.Asimplified
viewofthe7HenucleuscanbeseeninFig.1.1.

Themainreasonforinvestigating7Hehasbeenarecentveryinterestingobserva-
tionwhichwasreportedbyMeisteretal[6].InthisworkaJπ=1/2−resonance
atexcitationenergyEx=0.56(10)MeVwithawidthΓ=0.75(8)MeVwasobser-
ved.Thisstate,assumedtobethespin-orbitpartnerofthe7Hegroundstate,was
populatedina8Hebreakupreactionona12Ctargetat227MeV/nucleon.Sucha
lowvalueforthespin-orbitsplittingisatvariancewithtypicalvaluesof5-6MeV
observedinheaviernucleilike15O,17Oor39Ca.Evenincaseof5He,thespin-orbit
splittingisbelievedtobesignificantlylargerataround4MeV.Thestrengthof
thespin-orbitforceismeasuredbytheenergysplittingofspin-orbitpartners.For
halo-nucleiingenerallittleisknownaboutthespin-orbitsplitting,exceptthat
onecouldarguethatthelargeradialextentoftheexcessnucleonsmightsignifi-
cantlyreducethespin-orbitinteraction(Vs.o.∼r1∂V∂r(r)).Theobservationofthe
spin-orbitpartnerin7Hecouldthusshedlightuponthespin-orbitinteractionin
neutron-halonucleiwithimportantconsequencesfortheunderstandingoftheir
structure.

The7Henucleushasbeenstudiedinvariousexperiments.Thegroundstate(g.s.)
isfairlywellknownandwasforthefirsttimeobservedbyStokesandYoungin
1967ina(t,3He)reaction[7].Thisisacomparativelylong-livedresonancewith
atotalwidthΓ≈160keVwhichissituatedatabout440keVabovethe(6He
+n)threshold.Itwasmorethan30yearslaterwhenanexcitedstatewasfound
inthisexoticnucleus.Thestatewasobservedinthep(8He,d)7Hereactionby

2

Korsheninnikovetal.[8]atEx=2.9MeVwithawidthofΓ=1.9MeV.Itwas
alsofoundthatitpredominantlydecaystothefirstexcitedstateof6He(Jπ=
2+).ThisledtoaspinassignmentofJπ=5/2−forthenewresonance.Another
broadstateatEx=2.9MeVandevidenceforafurtherresonanceat5.8MeV
excitationenergywithalargewidthbetween3and5MeVwerefoundin[9].The
workofMeisteretal.[6]promptedanumberofnewmeasurements.Studiesofthe
7Hestructurethroughtheneutron-transferreactiond(6He,p)7He[10]contradict
thefindingofRef.[6].Noresonancesexceptthegroundstatehavebeenseenin
theexcitationenergyregionupto8MeV.Moreover,inrecentexperiment[11,
12]investigatingisobaricanalogstatesofthe7Hein7Lithroughthep(6He,n)
compoundnucleusreactionnoevidenceforalow-lying1/2−resonancewasfound.
Theauthorssuggestabroad1/2−stateaboveEx≈2.3MeVin7He,thehighest
energyaccessibleintheirexperiment.However,basedoncontinuumshell-model
calculationsithasbeenpointedout[13]thattheresonancewiththeparameters
ofRef.[6]wouldnotbedetectableintheparticularkinematicschoseninRef.[12].
Ontheotherhand,inaratherrecentwork[14]theresultsofthep(8He,d)7He
experimentalsoindicatedthelow-lyingexcitedstatewithparametersEx=0.9
±0.5MeV,Γ=1.0±0.9MeV.Finally,therecentd(6He,p)7Heexperiment
performedatArgonneNationalLaboratory[15]isnotconsistentwiththeresult
ofMeisteretal.[6]sinceitsuggestsabroadresonancestructurebetween2and
3MeVexcitationenergywithΓ=2MeV,whichcouldcorrespondtoa1/2−
state.

Thepresentworkprovidesanalternativeaccesstothisimportantquestionbyuti-
lizingthe7Li(d,2He)7Hecharge-exchangereactionatzerodegrees,whereGamow-
Teller(GT)transitionsareselectivelyexcited.Thisreactionatintermediateener-
gieshasbeendevelopedrecentlyasahigh-resolutionspectroscopictoolforthe
studyofGTstrengthdistributions[16].Likethe7Hegroundstate,theJπ=3/2−
g.s.ofthe7Liisalsointerpretedas1p3/2single-particlestate.Gamow-Tellertran-
sitionspopulatingspin-orbitpartners-likethe1p3/2and1p1/2state-should
havesimilarstrengths.ThisisalsopredictedbyGreen’sFunctionMonteCarlo
(GFMC)calculations.Sincetheg.s.issufficientlypopulatedinacharge-exchange
reaction[7],alow-lyingresonancein7HewithparametersfromRef.[6]should
giveaclearsignal.Theunbounddiprotonsystemisreferredtoas2He,ifthetwo
protonscoupletoan1S0,T=1state.Thus,becauseofthequantumnumbers
3

oftheinitialandfinalparticles(dand2He,respectively),the(d,2He)reaction
inducesexclusivelypurespin-isospinexcitations.Experimentally,the1S0stateis
selectedbylimitingtherelativeenergyofthetwo-protonsystemto1MeVin
detection.theirThemeasurementofthe(d,2He)reactionrequiresthecoincidentdetectionofthe
outgoingprotonswithsmallopeningangle,closetothedirectionoftheincident
deuteron.Inthiscasetheuseofamagneticspectrometerwithalargemomentum-
acceptanceandwithadetectorsystemcapableofsimultaneouslydetectingboth
protonsisneeded.Withthistypeofsetuparatherhighresolutioncanbeachieved.
However,thedifficultyofsuchmeasurementsresidesinthecapabilityofthe
detectionsystemtosuppressthetremendousrandombackgroundduetothe
breakupofthedeuterononthetargetanditsabilitytofilteroutthevalid2He
events.The(d,2He)probeisofpurespin-flipnatureandevenmoreselectivethan
(n,p)and(t,3He)reactions,wherenon-spin-fliptransitionscancompete.

ThePartIofthisthesisisorganizedinthefollowingway.InChapter2someof
thetheoreticaltoolsusedfortheinterpretationofthepresentdataareintroduced.
Chapter3isdevotedtothedescriptionoftheexperimentalsetupandprocedu-
re.Chapter4dealswiththeanalysismethodstoextractobservablesfromthe
measuredquantities.TheextractionoftheGamow-Tellerstrengthforlow-lying
statesin7HeisdiscussedinChapter5.Chapter6givestheresultsofanR-matrix
analysisthatyieldsthespectroscopicfactorofthe7Hegroundstate.Chapter7
presentsashortdiscussionaboutthehigher-lyingstructureinthe7Henucleus.
AsummaryandoutlookgiveninChapter8concludethefirstpart.Mainresults
ofthepresentpartarepublishedin[17,18].

4

kgroundbacTheoretical2

Thefollowingsectionspresentthebasicconceptsforthetheoreticalcalculations
performedinthisthesis.Afterageneralapproachtowardsthescatteringforma-
lism,thedistortedwavedescriptionofnucleon-nucleusscatteringispresented,
whichisthenusedforthecalculationsoftheangulardistributions.Amorecom-
pleteaccountofscatteringtheoryandoftheDistortedWaveBornApproximation
(DWBA)canbefoundin[19–21].Section2.4dealswithashortintroductionto
theshellmodel.Theone-bodytransitiondensitiescalculatedwithintheshellmo-
delcodecanbeusedfortheDWBAcalculations.Chapter2.6givesanoverview
oftheQuantumMonteCarlo(QMC)model.TheGTstrengthsobtainedfrom
theexperimentaldatacanbecomparedwithabinitioQMCcalculations.The
descriptionoftheQMCmodelwasextractedfromtheoriginalpapers[22–26].

Scattering2.1formalism

ThescatteringofaparticlefromapotentialV(r)isdescribedbythetime-
independentSchr¨odingerequation

(H−E)ψ(k,r)=0,
wheretheHamiltonianHcanbedividedintotwoparts

(2.1)

H=H0+V.(2.2)
ThefirstpartH0,describestheunperturbedmotionofthesystemandthese-
cond,V,theinteraction,whichdisappearsforsufficientlylargeseparationofthe
interactingparticlesofthesystem.Theenergyinthecenter-of-masssystemis
denotedbyE.Theparticleisrepresentedbyawavefunctionψthathastobe
asolutionofequation(2.1).Thesolutionhastobefoundwiththeboundary
conditionthatthetotalwavefunctionmusthavetheasymptoticbehaviorofan
incomingplanewaveandanoutgoingsphericalwave
5

ψ(k,r)−→φ+f(Θ)eikr,(2.3)
rwhereristherelativedistancebetweentheparticlesintheentrancechannel,k
isthewavevector,φisaplanewaveandf(Θ)isthescatteringamplitudewhich
isconnectedtothedifferentialcrosssectionsby
ddσΩ(Θ)=|f(Θ)|2.(2.4)
TheSchr¨odingerequation(2.1)canalsobeexpressedinintegralform.Inthis
casetheformalsolutionisusuallycalledtheLippmann-Schwingerequationand
canberepresentedas

notationDiracinor

Ψ(±)=Φ+G0±VΨ±,

(2.5)

|ψ±=|φ+G0±V|ψ±,(2.6)
wheretheG0istheGreen’sfunctionoftheunperturbedequation

1G0±=E−H0±iε.(2.7)
Thus,rewriting(2.5)usingtherelation(2.7)fortheGreenfunctiononegets

1Ψ(±)=Φ+E−H0±iεVΨ±.(2.8)
IngeneraltheLippmann-Schwingerequationhastwosolutions.Thefirstconsists
ofaplanewaveandanoutgoingscatteredwaveandisusuallyindicatedwitha’+’.
Thesecondsolutionisaplanewaveplusaningoingscatteredwavedenotedwith
a’−’.TheGreen’sfunctionoperatorG0canbeusedtodefinetheTtransition
aseratorop

T±=V+VG0±T±,
withthetransitionmatrixelement
Tfi=φ|T+|φ=φ|V|ψ+,
6

(2.9)

(2.10)

whereiandfareusedtolabeltheinitialandfinalscatteringwavefunctions.
Thestatevector|ψ±in(2.6)andtheTmatrixoperatorin(2.9)canbeseenas
aseriesexpansioninV.Theiriteratedformsare

|ψ±=|φ+G0±V|φ+G0±VG0±V|φ+...(2.11)
andT±=V+VG0±T±+VG0±VG0±V+...,(2.12)

respectively.Thetwoseriescanbeapproximatedbytakingonlythefirstnterms
intoaccount,whichiscalledBornapproximationtonthorder.Inthesecondseries
G0±propagatestheparticlefromscatteringtoscatteringpoint.Takingonlythe
firstntermsintoaccount,onelimitsthenuclearreactiontoan-stepscattering
process.ThematrixelementTfidefinedin(2.10)canberelatedtothescattering
amplitudeµf(Θ)=−2πh¯2Tfi(2.13)
withµbeingthereducedmassofthesystem.

2.2DistortedwaveBornapproximationandop-
ticaltialotenp

Forthetreatmentofthecharge-exchangereactiontheDistortedWaveBornAp-
proximationisused.Thistakesintoaccounttheelasticscatteringoftheincident
particlesbeforethereaction,andoftheemergentparticlesafterwards[27,28].To
treatthescatteringprocessinDWBA,thepotentialVisusuallysplitintotwo
parts

V=U+W.(2.14)
TheopticalpotentialUdescribestheelasticscatteringandisresponsiblefor
thedistortionoftheincomingandoutgoingwaves.ThepotentialWcontainsthe
residualinteractionandaccountsfortheinelasticity.Thescatteringproblemwith
thepotentialUalonehasaknownsolutionχwhichfulfilstheequation
7

(H0+U)χ±=Eχ±.(2.15)
Thus,replacingtheplanewaveinequation(2.6)bythedistortedwaveonecan
expressthewavestatevector|ψ±intermsofthewavefunctions|χ±

|ψD±W=|χ±+GU±W|ψD±W,(2.16)
whereGU±belongstotheHamiltonianHU=H0+U.Applyingequations(2.11)
and(2.12)thetransitionmatrixelementcanbethanwrittenas

Tfi=φ|U|χ++χ−|W|ψ+,(2.17)
whereiandfareusedtolabeltheinitialandfinalscatteringwavefunctions,
respectively.IntheDWBAthestatevector|ψ+isreplacedbythedistortedwave
statevector|ψD+Wdefinedin(2.16)andisapproximatedtothefirstorderinthe
seriesexpansion.ThisallowstocalculatethetransitionmatrixinDWBAas

TDfiWBA(post)=χ−|W|ψD+WBA.(2.18)
Theformofthetransitionmatrixderivedin(2.18)iscalledpostrepresentation.
Thepriorrepresentationofthetransitionmatrix,whichisderivedfromthetime-
reversedreaction,canbecalculatedtobe

TfDiWBA(prior)=ψD−WBA|W|χ+.(2.19)
Itcanbeshownthatthepostandthepriorformareidentical

TDfiWBA=χ−|W|ψD+WBA=ψD−WBA|W|χ+.(2.20)
Thedistortedwavesχ±areobtainedfromsolvingtheSchr¨odingerequation(2.15)
withtheopticalpotentialUfromtherelation(2.14).Theconventionalformof
thispotentialcontainsrealandimaginarypartsofcentralandspin-orbittype.In
analogywithopticstherealpartofthepotentialdescribesthescatteringinthe
dispersivemedium,whiletheimaginarypartcorrespondstotheabsorption.The
standardformoftheopticalpotentialisparameterizedas
8

dV(r)=−VRf(r,r0,a0)+i4aIWDdrf(r,rI,aI)−iWSf(r,rI,aI)(2.21)
h¯21d
+VLSmπc(L∙S)rdrf(r,rLS,aLS)+VCoul.
UsuallyforthedescriptionoftherealpartofthepotentialtheWoods-Saxon
shapeforthecentraltermVRandtheThomasformforthespin-orbittermVLS
ischosen.Theimaginarypartissplitintotwopartsnamedvolumeandsurface,
wherethevolumetermWShastheWoods-SaxonformandthesurfacetermWD
isdescribedbythederivativeoftheWoods-Saxonform.Theradialformfactors
ofWoods-Saxontypeinequation(2.21)canbewrittenas
1f(r,rx,ax)=1+exp((r−rxA1/3)/ax)(2.22)
withradiusr,diffusenessaandx=0,I,LS.TheCoulombpotentialVCoulisassu-
medtobethepotentialgeneratedbyauniformlychargedsphereofradiusrcA1/3,
whereAisthemassnumberofthenucleus.
OrdinaryDWBAcalculationsmaygiverisetoambiguitiesintheinterpretation
oftheexperimentaldifferentialcrosssectionsfortheA(d,2He)Breaction,where
AisthetargetandBistheresidualnucleus.The2Hesystemisinrealityapairof
protonscoupledtothe1S0state.Thedisagreementofthetheoreticalcalculations
withexperimentaldatacanbeexplainedbythefactthattheopticalpotentialof
twofragmentsinathree-bodychanneldoesnotresembleelasticopticalpotentials
obtainedinatwo-bodychannel.Furthermore,nophenomenologicalinformation
isavailableonthe2He-nucleusinteraction.
Insteadoftreatingboth,theincidentandtheexitchannel,inDWBA,inthe
presentwork,theadiabaticapproximationmethod[29–31]isappliedthatusesa
three-bodywavefunctionintheexitchannelratherthanusinganartificialfinal
statedistortinginteractionbetween2Heandthefinalnucleus.Thetransition
amplitudefor(d,2He)reactionlookslike

TDfiWBA=χ2−He(r)ψ2He(x1−2)Ψf|V|Ψiφdχd+(r),(2.23)
wheretheprojectileandejectilewavefunctionsoftheincidentandexitchannels
aredenotedbyφdandψ2He,thetargetwavefunctionsbyΨiandΨfandthe

9

distortedwavesbyχd+andχ2−He,respectively.Therelativecoordinatebetween
projectileandtargetisrepresentedbyrandtherelativedistancebetweentwo
protonsbyx1−2.Theincidentchannelwavefunctionisgeneratedusingthecon-
ventionalopticalpotentialwhereastheexitchannelwavefunctionistreatedin
theadiabaticapproximation.Thethree-bodywavefunction|ψ+=|χ2−Heψ2He
intheexitchannelsatisfiestheSchr¨odingerequation
2−2µh¯r2+ViB(r,x1−2)+H12−Eψ+(kB−12,k1−2;r,x1−2)=0,
B−12i=1,2
(2.24)withthereducedmassµB−12inthefinalstate.IftheconditionEppisfulfilled,
whereppistherelativeenergybetweentwoprotons,theexcitationwithinthe
1-2system2Hecanbeneglected,i.e.thesub-HamiltonianH12isreplacedby
itseigenvaluepp.Thisisknownasadiabaticorsuddenapproximation.Similar
tocommonDWBAtheinteractionViBofprotonsi=1,2withtheresidual
nucleusBisapproximatedbyanopticalpotentialevaluatedat21(E−pp).The
DWBAtechniqueadaptedtothe(d,2He)reactionisusedtocalculateangular
distributionsofdifferentialcrosssections.Thecalculationsrequirethedefinition
ofthetransition,aneffectiveinteraction,adeuteronopticalpotentialforthe
entrancechannelandaprotonopticalpotentialfortheexitchannel.

teractionineEffectiv2.3

InordertocalculatethetransitionmatrixelementTfi,aneffectiveinteraction
betweenthenucleonsoftheprojectileandthetargetisneeded.Thiseffective
interactionis,ingeneral,verycomplicated.Itdependsontheincidentenergy
andonspecificpropertiesofprojectileandtarget.Thesimplestmethodforthe
descriptionofthecrosssectionisbasedontheimpulseapproximation(IA)with
planeordistortedwaves(PWIA/DWIA,respectively).Thisapproachisvalidat
incidentenergiesabove∼100MeV/nucleonandassumesthattheinteraction
takesplacebetweenonlyonenucleonoftheprojectileandoneofthetargetina
singlecollision.Furthermore,theeffectiveinteractionisassumedtobethefree
nucleon-nucleoninteraction.TheeffectiveinteractionusedinthisworkistheT-
10

matrixinteractionparameterizedbyLoveandFraney[32,33].
Theeffectivenucleon-nucleon(NN)interactionisasumofcentral(C),spin-orbit
(LS)andtensor(T)termsandusuallyhasaform

Vip=VipC(r)+VipLS(r)L∙S+VipT(r)Sip.(2.25)
Thethreepartscanbesplitupagainintotheirspinandisospindependence

CCCCCVip(r)=V0(r)+Vσ(r)σi∙σp+Vτ(r)τiτp+Vστ(r)σi∙σpτi∙τp,(2.26)

VipLS=VLS(r)+VτLS(r)τi∙τp,

(2.27)

VipT=VT(r)+VτT(r)τi∙τp,(2.28)
wherepisreferredtoaprojectilenucleonanditoatargetnucleon,r=ripisthe
distancebetweenparticipatingnucleons,andSipisthetensoroperator.
ForsmallmomentumtransferqandlargeEcmvaluesthesimplestapproachto
constructaneffectiveNN-interactioncanbeused,i.e.usingthefreetransition
matrixelementt0NNaszero-rangeeffectiveinteraction

Vip=t0NN(E,q)δ(rip).(2.29)
TheNNscatteringamplitudeisusuallyexpressedinaform

M(Ecm,Θ)=A+Bσi∙nσp∙n+C(σi∙n+σp∙n)+Eσi∙qσp∙q+Fσi∙Qσp∙Q,(2.30)
wherethecoefficientsA,B,C,E,andFarefunctionsofthecenter-of-massenergy
Ecm,ofthescatteringangleΘ,andofthetwo-bodyisospin,e.g.A=A0+A1τiτp.
Theunitvectorsq,Q,ncanbecalculatedfrom

n=k×k,q=k−k,Q=k+k.(2.31)
Anapproximationoftheeffectiveinteractioncanbeobtainedbyassuminga
localfinite-rangeformofeffectivenucleon-nucleoninteraction.Inthiscasethe
11

parametersofViphavetobeadjusteduntiltheNNtransitionmatrixelementis
reproducedinmomentumspace[32]

22t0NN=−4πE(¯hc)M=e−ikrVip(1−Pip)eikrdr,(2.32)
cmwheretheoperatorPipistheexchangeoperatorwhichgeneratesantisymmetri-
zation.Sinceonlythespin-isospincomponentsareinvolvedinthe(d,2He)reaction,the
interactionfrom(2.25)isgivenby

Vip=[VσCτ(rip)(σi∙σp)+VσTτ(rip)Sip]τi∙τp.(2.33)
AnexplicitrepresentationfortheinteractioncoefficientsasasumofYukawa
termsforthecentralpartandasasumofr2×Yukawatermsforthetensorpart
[32].inengivis

2.4Shellmodelandone-bodytransitiondensi-
ties

Thenuclearshellmodelwasdiscoveredinthelate40’sandhasprovidedasuc-
cessfulapproachtothemicroscopicdescriptionofnuclearstructure.Forinstance,
one-bodytransitiondensitiescanbecalculatedfrommodelwavefunctionsofthe
nuclearstatesusingshellmodelcodeslikeOXBASH[34].Thebasicassumption
ofthenuclearshellmodelisthat,tofirstorder,eachnucleonmovesindepen-
dentlyinanaveragefield.Thus,thenuclearinteractioncanbeapproximatedby
anaveragecentralpotential.Becauseofthespecialcharacteroftheinteraction,
namelybeingashort-rangeattractiveforce,andbecauseofthePauliprinciple,
theshell-modelpotentialisessentiallyconstantatthecenterofthenucleusand
goesquicklytozeroatthenuclearsurface.Animportantrepresentationofthis
behaviourisgivenbytheWoods-Saxonpotential

V(r)=−V0f(r,r0,a0)
12

(2.34)

wheref(r,r0,a0)isdefinedinEq.(2.22),thewelldepthV0isoftheorderof
50MeV.TothenuclearpotentialV(r)onemustaddtheCoulombpotential
VCoul(r)andaspin-orbitpartVLS(r)L∙S,whichisrequiredtogiveacorrect
descriptionoftheobservedshellclosuresandremovethedegeneracyofstates
withthesameorbitalangularmomentumL.IncaseofaWoods-Saxonpotential
theSchr¨odingerequationhastobesolvednumerically.Analyticformsofthe
nuclearwavefunctionscanbeobtainedbyusinganeasilytractableharmonic-
tialotenposcillatorV(r)=−V0+mω2r2/2(2.35)
andneglectingthespin-orbitforce.Here,V0isthewelldepth,misthemassof
anucleonandωistheoscillatorfrequencyofthesimpleharmonicmotionofthe
particle.Asrincreases,thepotentialtendstoinfinity.Theenergylevelsofthe
harmonic-oscillatorpotentialaregivenby

ENL=[2(N−1)+L+3/2]¯hω,(2.36)
whereNistheradialquantumnumberandListheorbitalangularmomentum.
Thenuclearpotentialcanbewrittenasasumofthetwo-bodyinteractionsVij
whichtakeplacebetweenthenucleonsinsidethenucleus
AAAV=Vij=Vi+Wij,(2.37)
i>jii>jwhereViistheaveragemean-fieldpotentialtheithnucleonfeelsasaresultof
thecombinedforceoftherestofthenucleonsandWijrepresentstheresidu-
alinteractionswhichcannotbeincorporatedintotheaveragepotential.The
mean-fieldpotentialVicausescompletelyindependentmotionofthenucleons
andleadstopuresingle-particlewavefunctions.Theresidualinteractionsper-
turbthispictureandcausethenuclearwavefunctiontobecomeasuperposition
ofthesingle-particlestates.Thissituationisknownasconfigurationmixing.
ConsideringEq.(2.37)thenuclearHamiltonianbecomes
AAA22H=h¯pi+Vi(ri)+Wij(|ri−rj|),(2.38)
m2i>jiiwherethefirsttermdenotesthekineticenergyoftheindividualnucleonsand
thenexttwotermsrepresentthepotential.Aschematicviewofthe7Henucleus

13

Fig.2.1:Schematicrepresentationofthe7Henucleusintheshellmodel.

withintheshellmodelcanbeseeninFig.2.1.Whenapplyingtheapproximation
describedinprevioussections,thescatteredparticleplaystheroleofaone-body
operatoractingonthetargetnucleons.Thus,anyinelasticnucleartransitioncan
bewrittenasasuperpositionofsingle-particletransitionstakingtargetnucleons
fromanorbitjtoanorbitjwithamplitudesSJjj.
Theseparationofnuclear-structureinteractioncontributionstothescattering
amplitudeisachievedbyperformingamultipoleexpansionoftheeffectiveinter-
actioninmomentumspace[35],giving

ATfi=2(−1)Mvβα(q,Q)χf|TˆLS−MJ(β)|χiΦJMff|TˆLSM(Jα)(j)|ΦJMiiq2dq
παβJMj=1
(2.39)withparametersQandqdefinedinequation(2.31),αandβrepresentthe
quantumnumbersoftheprojectileandtarget,respectively,vβα(q,Q)areBessel
transformationsofthevariouspartsoftheeffectiveinteractionandTˆLSM(Jα,β)are
sphericaltensoroperators.UsingtheWigner-Eckarttheoremthenuclearmatrix
elementcanbewrittenas

AΦJMff|TˆLSMJ(j)|ΦJMii
=1j

14

(2.40)

A=(−1)Jf−MfJfJJiΦJf||TˆLSJ(j)||ΦJi.
−MfMMiJ=1
Thereducedmatrixelementinequation(2.40)canbeexpressedasasumof
single-particlematrixelementsj||TˆLSJ||jweightedwithso-calledone-body
transitiondensitiesSJjj(Jf,Ji)
AΦJf||TˆLSJ(j)||ΦJi=SJjj(Jf,Ji)j||TˆLSJ||j,(2.41)
jj=1JasdefinedarehwhicSJjj(Jf,Ji)=−√1ΦJf||[βj†β˜j]||ΦJi.(2.42)
1+J2Here,βj†isacreationoperatorproducingaparticlewithangularmomentumj
andβ˜janannihilationoperatorcreatingaholestatewithangularmomentumj.

2.5CrosssectionsandGamow-Tellerstrength

ThedifferentialcrosssectionofascatteringreactionA(a,b)Bisconnectedtothe
transitionmatrixelementviathefollowingrelation[28]
dσ=µiµ2fkf1JˆA−2|T(MA,MB,ma,mb)|2.(2.43)
dΩ(2πh¯)2ki2MA,MB,ma,mb
√Hereµi,µfarethereducedmassesoftherespectiveparticles,Jˆ≡2J+1is
aspinfactorandthesumisoverthespinprojectionsintheinitialandfinal
statesrespectively.AccordingtotheDWBA,thetransitionmatrixelementfrom
Eq.(2.20)canbeexpressedas
ATfDiW=χf−∗(r,k)Φf|tjp|Φiχi+(r,k)d3r,(2.44)
jwherethetransitionoperatortjpinthemomentumrepresentationisidenticalto
theNNt-matrix(Eq.2.32)andcanbewritteninthefollowingway
tip=VST(rjp)(1−Pjp)Oˆj(ST)Oˆp(ST),(2.45)
15

wherePipinterchangestargetnucleoniandprojectile,andoperatorsOˆaredefi-
asned

Oˆ(ST)=στS=1,T=1(GT)(2.46)
τS=0,T=1(F).
Thetransitionmatrixismostusefulinitsmomentumrepresentation
TfDiW=t(q)ρST(q)D(q,k,k)d3q.(2.47)
Heret(q)istheFouriertransformoftheinteraction,ρST(q)isthetransition
densityandDisthedistortionfactor
t(q)=4πV(r)[j0(qr)+(−1)ljl(kr)]r2dr=JST(q),(2.48)
AρST(q)=ΦB|Oˆj(ST)eiqr|ΦA∙b|Oˆa(ST)|a,(2.49)
jD(q,k,k)=13χf−∗(r,k)χ+(r,k)e−iqrd3r,(2.50)
)π(2whereJSTisthevolumeintegraloftherelevantinteraction,listheangular
momentumtransferandjisthemodifiedBesselfunction.
ThetransitiondensityρSTreducestoasingletermundertheassumptionthat
onlyangularmomentumtransferΔl=0isimportantforsmall-anglescattering.
assumptionthisUnder√√ρST(q)=I0(q)MST2JA+12S+18(2.51)
×JAJBSJaJbS(−1)JB−Ma−Ms,
MsMA−MBMsma−mbMs
where,againundertheassumptionofthesmallmomentumtransfer,
1I0(q)=|R(r)|2j0(qr)r2drexp(−6q2r2ρ).(2.52)
Herer2ρisthemean-squareradiusofthetransitiondensity.Thereducedmatrix
elementsMSTcorrespondtoGamow-TellerandFermistrengths,respectively:
TS|M|2=2JA1+1|JB||στ±||JA|2=B(GT)S=1(2.53)
2JA1+1|JB||τ±||JA|2=B(F)S=0
16

Usingtheeikonalapproximationforthedistortedwavesanapproximationfor
thedistortionfactorDcanbeobtained
D(q,k,k)exp1[−xA1/3+p(ω)]δ(|k−k|−q)eiφ(2.54)
2=NDδ(|k−k|−q).
Herep(ω)isasecond-orderpolynomialandω=Ex−Qistheenergyloss,φand
xcomefromtherealandimaginarypartoftheopticalpotential,respectively.
InsertingEqs.(2.48)-(2.54)intotheEq.(2.47)andperformingtheintegration,
forq=0onecanobtainasimpleformofthetransitionmatrixelement[36]

22|Tfi|(q=0)=NDB(ST)|JST|(2JA+1)(2S+1)(2.55)2
×JAJBSJaJbS(−1)JB−Ma−Ms.
MsMA−MBMsma−mbMs
TherelationbetweenthecrosssectionandtheGamow-Tellerstrengthcanbeseen
intheexpression(2.43).Summationoverprojectionsofspin(ja=jb=1/2),for
a(n,p)-or(p,n)-reaction,providestheresult[37]
2dσ(q=0)=µ2kfNDJσ2τB(GT),(2.56)
dΩπh¯ki
whichisalsoagoodapproximationforthe(d,2He)reaction.

2.6AbinitioQuantumMonteCarlomethod

Sincethestandardshellmodelfailstodescribemanyoftheessentialfeatures
oflightexoticnucleus7He,theexperimentalresultsdiscussedinthisthesisare
comparedwiththosefromtheabinitioQuantumMonteCarlomodel.Inthelast
decade,theQMCmethodhasbeendevelopedintoapowerfultoolfordescription
oflightnuclei(uptoA=12)usingrealistictwo-nucleon(NN)andthree-nucleon
(NNN)potentials.Itgivesprecisespectroscopicinformationlikeradii,excitati-
onenergies,moments,transitionstrengths,bindingenergies,etc.Thequantum
MonteCarlomethodsconsistofvariationalMonteCarlo(VMC)andGreen’s
17

functionMonteCarlo(GFMC)methods.TheVMCisanapproximatemethod
thatisusedasastartingpointforthemoreaccurateGFMCcalculations.The
GFMCmethodstartsfromthetrialfunction,whichisobtainedbytheVMC,
andmakesaEuclideanpropagationthatconvergestothelowestenergyforthe
investigatednuclearstatewithgivenquantumnumbersJπandT.
TheHamiltonianusedinthecalculationsincludesanonrelativisticone-bodyki-
neticenergyKi,theArgonnev18two-nucleonpotential[38]andtheUrbanaIX
three-nucleonpotential[39],
H=Ki+vij+Vijk.(2.57)
<ki<ji<jiThetwo-nucleonpotentialvijcanbewrittenasasumofelectromagneticand
one-pionexchangetermsandashorter-rangephenomenologicalpart[1].The
UrbanapotentialVijkispresentedasasumoftwo-pion-exchangeandshorter-
terms.phenomenologicalrangeThevariationalmethod(VMC)canbeusedtoobtainapproximatesolutionsto
themany-bodySchr¨odingerequationfordifferentnuclearsystems.Themethod
findsanupperbound,ET,toaneigenenergyofHbyevaluatingtheexpectation
valueofHinatrialwavefunctionΨT.TheparametersinΨTarevariedto
minimizeET,andthelowestvalueistakenastheapproximateenergy.Agood
variationaltrialfunctionisdefinedas
|ΨT=1+UˆijTkNIS(1+Uˆij)|ΨJ.(2.58)

i<j<ki<jHereUˆijandˆUijTkNIarenon-commutingtwo-andthree-nucleoncorrelationope-
rators,SisthesymmetrizationoperatorandtheΨJisthetotallyantisymmetric
Jastrowfunctionwhichdeterminesthequantumnumbersofthenuclearstate
beinginvestigated.ThecorrelationoperatorUˆijincludesspin,isospin,andtensor
termsinducedbythetwo-nucleonpotential,whileUˆijTkNIreflectsthestructureof
thedominantpartsofthethree-nucleoninteraction.
ByusingtheGFMCmethod[40],oneisabletodeterminebindingenergiesand
energiesoflow-lyingexcitedstates.Inotherwords,theaimoftheGFMCisto
projectouttheexactlowestenergystate,|Ψ0=limτ→∞exp[−(H−E0)τ]|ΨT,
whereτistheimaginarytime.The|Ψ0isassociatedwithachosensetofquan-
tumnumbersfromtheapproximation|ΨTgivenbyEq.(2.58).Theeigenvalue
18

E0iscalculatedexactlywhileotherexpectationvaluesaregenerallyestimated
neglectingtermsoforder||Ψ0−|ΨT|2andhigher.Incontrast,theerrorin
thevariationalenergyisoforder||Ψ0−|ΨT|2,andotherexpectationvalues
calculatedwith|ΨThaveerrorsoforder||Ψ0−|ΨT|.

R2.7theory-matrix

R-matrixtheory[41]isoneoftheessentialtoolsforreliableinterpretationof
nuclearreactionandscatteringdata.Itisparticularlyimportantfortheextrac-
tionoftheresonanceparametersofaninvestigatedstate.Forinstance,onecan
experimentallyobtaintheobservedreducedwidthwhichisthedecisivequantity
forthedeterminationofthespectroscopicfactor.Spectroscopicfactorsareba-
sicquantitiescharacterizingthesingle-particlenatureofnuclearexcitationsand
thereforeserveasanimportanttestofwavefunctionscalculatedwithrecently
developedabinitiomethods.Itcanalsobecomparedtotheclassicalnuclear
shell-modelcalculations.Usingthe(d,2He)reactionwithgoodenergyresolution
forastudyofthestructureof7Heonegetsnewinformationabouttheg.s.reso-
nanceparametersandthenintheframeofanR-matrixanalysisonecanobtain
theneutronspectroscopicfactor,assumingthattheJπ=3/2−groundstateof
7Hemightbedescribedasa6He⊗ν1p3/2configuration.Theneutronspectroscopic
factorSnisdefinedasγ2
Sn=γ2obs(2.59)
spwithγ2obsandγ2spbeingtheobservedandsingle-particlereducedwidths,respec-
tively,wheretheformerisprovidedbytheresonancefitwhilethelattercanbe
computedusingaWoods-Saxonsingle-particlepotential.
Fordescriptionoftheunstablep3/2single-particlegroundstatein7HeaWoods-
Saxonsingle-particlepotentialwithspin-orbitinteractionisused.Theparameters
arechosentogiveanoptimalfittotheexperimentallyknownnucleardatain
the6He-7Hemassregion,suchasmatterandchargeradiiof6He,energyofthe
decaying7Hegroundstate.ThechoiceofaWoods-Saxonshapewasinspiredby
thepotentialusedby[42]forthereproductionoftheexperimentalp-resonance
19

(2.60)

statesin5He.ThepotentialhasbeenalreadydefinedinEq.(2.34).Theexplicit
is:hereusedformparttralcenU(r)=−Vc/exp(x(r)+1)(2.60)
partspin-orbit1dUls=Vlsdrexp(x(r)+1)[j(j+1)−l(l+1)−3/4]/2(2.61)
parttrifugalcentheand

2Ucen(r)=2mh¯redl(l+1)/r2(2.62)
withparametersfor7He:Vc=28.13MeV;Vls=12.22MeVfm;x(r)=(r−Rc)/a
whereRc=2.79fmanda=0.25fm;mredc2=804.786MeV.Thegroundstate
energyresultingwiththesedataisE(p3/2)=0.446MeVwhichcoincideswith
theexperimentalvalueof0.445MeV.
Forthe6HeboundstatesEs=-26.52MeVandE(p3/2)=-1.87MeV.Thedepth
ofthecentralpartisadoptedtoreproducethesevalues.Thisresultsinthematter
rmsradiusrm=2.64fmandthechargeradiusrch=2.01fm(theexperimental
valueis2.05fm[43]).Withthepotentialchosenthedecayinggroundstateof7He
isobtainedassolutionofthecentralpartoftheSchr¨odingerequationwiththe
boundaryconditionofapurelyoutgoingwaveataradiusRwhichiswelloutside
therangeofthepotentialU(r).Aslongasthisconditionisfulfilledtheresult
doesnotdependonaspecificchoiceofR.Fromthesolutionfollowsthecomplex
alueveigeniEgs=Ereal−2Γgs(2.63)
withEreal=0.446MeVandΓgs=0.258MeV.
Thereducedsingle-particlewidthγ2spdependsonthechoiceofamatchingradi-
usrmatch.AccordingtoR-matrixtheorythisisthechannelradiusatwhichthe
motionfrominsidethepotentialturnsoverintothefreerelativemotionofthe
decayingparticles.Anindicationforthisisthematchingofthelogarithmicderi-
vativeofthepotentialmotionwiththatofafreeoutgoingwave.InFig.2.2the
realpartofthelogarithmicderivativesRe[fl(r)]forbothpotentialmotionanda
freeoutgoingwaveispresentedleadingtoanoptimumvaluermatch=4.0fm.
20

Fig.2.2:Logarithmicderivativesofthepotentialmotionofthe7Heg.s.andthat
ofthefreeoutgoingwave.

Todeterminetheobservedreducedwidthγ2obsin(2.59),R-matrixtheoryinone-
levelapproximationhasbeenused[41].Fortheneutrondecaychannelof7Hethe
decaycrosssectionisgivenby

ΓcΓc(E)
σcc(E)=[E−Ec+Δ(E)]2+41Γc(E)2.(2.64)
Here,c,carechannelnumbers,Γcisthestrengthofthereactionchannel
7Li(d,2He)7He,andΓc(E)denotestheenergydependenttotalwidthofthedecay
ofthe7He1p3/2groundstateinto6He+n.ThequantityΔ(E)istheenergyshift
engivfunction,yb

Δ(E)=−Sc(E,rmatch)γ2obs(2.65)
withtheshiftfactorSc.Therelationbetweentheenergydependenttotalwidth
Γc(E)andγ2obsisgivenby

Γc(E)=2Pc(E,rmatch)γ2obs.(2.66)

ThequantitiesScandPcarefunctionsofthechannelwavefunctions

FcdFc+GcdGc
Sc(E,rmatch)=rmatchdr22dr,
G+Fcc21

(2.67)

rkmatchPc(E,rmatch)=Fc2+Gc2
whereFcandGcaretheregularandirregularsolutionsof

freewaveequation.FromtheEq.(2.64)followstheenergy

as

the

solution

of

Eres=Ec+Δ(Eres).

22

(2.68)

theofpartradialthe

ellevresonancetheof

(2.69)

talerimenExp3setup

3.1AGORcyclotronatKVI

ThemeasurementswerecarriedoutattheKernfysischVersnellerInstituut(KVI)
Groningen.TheheartofthefacilitiesforexperimentalnuclearphysicsatKVI
istheAcceleratorGroningenOrsay(AGOR)cyclotron[44]whichwasbuiltin
collaborationbetweentheInstitutdePhysiqueNucleared’OrsayandKVI.This
compactmachineiscapableforacceleratingbothlightandheavyions.Thereare
threedifferentionsources:ECRIS-3produceshighlychargedheavyions,POLIS
polarizedprotonsanddeuteronsandCUSPunpolarizedprotonsandlightions.
Themaximumenergylimitationoftheprotonbeams,190MeV,isimposedby
thefocusingpropertiesofAGOR.Themaximumenergyforheavyionsdepends
ontheircharge-to-massratioQ/A,withamaximumenergyof95MeV/nucleon
forQ/A=0.5.AnoverviewoftheexperimentalfacilitiesatKVIisgivenin
3.1.Fig.

ectrometerspBig-bite3.2

Bothprotonsfrom2HedecayweremomentumanalyzedwiththemagneticBig-
BiteSpectrometer,so-calledBBS[45].ItconsistsoftwoquadrupolesQ1and
Q2forfocusingtheparticleswhichenterthespectrometer,andadipoleDfor
selectingtheparticlesaccordingtotheirmomentum.AlayoutoftheBBSandof
therelateddetectorsystemusedforthisexperimentisshowninFig.3.2.

Therearethreeconfigurationswhichdifferinthepositionsofthequadrupole
doubletwithrespecttothescatteringchamber,theyarecalledmodeA,BandC.
InmodeAthequadrupoledoubletismovedtowardsthetargetandthesolid-angle
acceptancereaches13msr.Atthesametimethemomentum-biteacceptanceis
13%.Particleswithamagneticrigiditydeviatinglessthan6.5%fromthenominal
oneareacceptedbythespectrometer.InmodeC,wherethedoubletismoved
awayfromthetargettowardsthedipolemagnet,thesituationisreversed.The

23

Fig.

:3.1

Cyclotron

and

exp

24

talerimen

halls

at

KVI.

Fig.3.2:SchematicdrawingoftheBig-Bitespectrometerandtheattachedde-
systems.tector

solidangledecreasesto6.7msr,andthemomentumacceptanceincreasesto25%.
ModeBwasusedinthepresentexperiment,whichisanintermediatesettingto
modesAandC.However,duringthemeasurementsthemagneticfieldofthe
firstquadrupolewasincreasedby5%toimprovetheimagingpropertiesofthe
spectrometerintheverticaldirection.Thisleadstoareductionofthevertical
openingangleoftheBBSfrom140mradtoabout100mradandresultsina
solidangleof6.6msr(modeB∗).TheparametersoftheBBSinthemodeB∗
aregiveninTable3.1.Toachieveoptimalmomentumresolutionthebeamline
tothespectrometerwasadjustedfordispersion-matchedbeamtransport.

tracingyRa3.2.1

Bothprotonsfromthe(d,2He)reactionaremeasuredatthefocalplaneofthe
BBS.Inordertorelatethemeasuredquantitiestoparametersofthereactionat
thetarget,theparticleshavetobetransportedthroughthespectrometer.The
particlecoordinatesatthetarget(subscriptt)areconnectedtothoseatthefocal

25

Tab.3.1:DesignparametersfortheBBSinmodeB∗.

MeV430limitBendingMomentumbiteΔpp19%
msr6.6ΔΩangleSolidHorizontalopeningangleΔθ66mrad
VerticalopeningangleΔφ100mrad

plane(subscriptd)byafive-foldTaylorexpansion

αt=(α|xµθνyλφη)xdµθdνydλφdη.(3.1)
,λ,ηµ,νThetargetvariablesaredenotedbyαandcanbeθ,φorδ.Thecoordinatesx
andθdescribepositionsandanglesinthehorizontalplaneandyandφsimilarly
intheverticalplane.ThedeviationoftherigidityBρoftheparticlewithrespect
toaparticletravellingalongthecentralrayofthespectrometerisdefinedasδ
δ=Bρ−Bρ0,(3.2)
ρB0whereBisthemagneticfieldandρ0thebendingradiusofthecentraltrajectory.
Therigidityislinearlyrelatedtothemomentump=mvvia
Bρ=qmv,(3.3)
withqbeingthechargeandmtherelativisticmassoftheparticle.Thecoefficients
oftheTaylorexpansionusedtoreconstructthescatteringvariablesatthetarget
positionweredeterminedina11B(p,p)experiment[46]atEp=150MeVwhich
isalsosuitableforthepresentexperimentwithlowerenergyprotons(inthiscase
Ep≤85MeV[47]).Foraprecisedeterminationofthescatteringanglesnear0◦,
bothhorizontalandverticalscattering-anglecomponentshavetobemeasured
withgoodaccuracy.Byadjustingthefieldstrengthofthequadrupolemagnets,
onecanchangethefocusingpropertiesofthespectrometer.Attheirnominal
settings,thequadrupolesproduceacrossoverintheverticalfocal-planecoordi-
natewhengoingfromthelow-tothehigh-momentumside.Becausenotonly

26

theverticalpositionydbutalsotheangleφdcontainpracticallynoinformation
whichwouldallowtoreconstructtheverticaltargetangleφt,thefieldstrengthof
thefirstquadrupolewasadjustedtoperformnearpoint-to-parallelfocusingover
theregionofthefocalplaneusedduringtheexperiments.Thiswasachievedby
increasingthemagneticfieldofthefirstmagnetby5%,asdescribedabove.

correctionsRecoil3.2.2

Duetothereactionkinematicsparticlestransferdifferentrecoilenergiestothe
targetnucleusatdifferentscatteringanglesandthusarriveatdifferentpositions
atthefocalplane.Thisleadstothesocalledkinematiclinebroadening.This
dependenceisdescribedinfirstorderbyaconstantdefinedby

K=p1ddpΘ

(3.4)

in[1/mrad]andcanbecalculatedusingatwo-bodykinematicscode[48]like
KINEMA.Thiseffectcanbecompensatedbysoftwarecorrectionsandhasbeen
successfullyappliedin[46]tothefocal-planedetectionsystemusedinthepresent
t.erimenexp

3.3EUROSUPERNOVAdetectionsystem

Forthedetectionoftheoutgoingprotonsfrom(d,2He)reactionthefocal-plane
detectorsystem[49]builtbytheEuroSuperNovacollaborationhasbeenused.
Itconsistsoftwosubsystems,theFocal-PlaneDetectionSystem(FPDS),and
theFocal-PlanePolarimeter(FPP).AtopviewisshowninFig.3.3.TheFPDS
consistsofapairofgas-filledverticaldriftchambers(VDC[50])formomentum
reconstructionoftheprotons.TheFPPcomprisesmulti-wireproportionalcham-
bers(MWPCs)andapairofscintillatorpaddlesS1andS2fortime-of-flightand
ts.measuremenenergy-loss

27

Fig.3.3:Topviewofthefocal-planedetectionsystemandthefocal-planepola-
rimeter;dimensionsaregiveninmm.

3.3.1Focal-planedetectionsystem

Whenaparticletraversesthedriftchambers,itionizesthegasmoleculescontai-
nedinthechambers.Theelectronsthendrifttothewiresclosesttothepointof
traversal.Bymeasuringthearrivaltimesofthepulseswithrespecttoacommon
referencetime,onegetsasetofdrifttimes.Fromtheobtaineddrifttimesone
cancalculatetheintersectionpointbetweenthetrackandthewireplane.

ThetwopairsofVerticalDriftChambers(VDCs)arepositionedalongthefocal
planeatanangleof39◦withrespecttothebeamdirection.Eachchamberhas
anX-planeandanU-planewiththewiresoftheU-planetitledby32.86◦against
theverticaldirection.Bothdriftchambersarefilledwithagas-mixtureconsisting
of50%argonand50%iso-butane.TheactivedetectionareaoftheVDCsis
1030×367mm2.Thewireplaneshavealternatingsenseandguardwires,which
actasanodes,whereastheVDCfoilsserveascathodes.Thenumberofsense
wiresinbothchambersis240.Thewirepitchfortheanodewiresis4mm.The
designedresolutionofeachVDCis45µminthedispersiveand80µminthe
non-dispdirection.eersiv

28

3.3.2Focal-planepolarimeter

TheFocal-PlanePolarimeter(FPP)consistsoffourMWPCs(Multi-WirePro-
potionalChambers)D1-D4,whichallowapreciseandeasytrackreconstruction,
andoftwosegmentedscintillatorplanesS1andS2fortime-of-flightandenergy
measurements.TheFPPisplacedperpendicularlytothecentralray.Asmallveto
scintillatorcanbemountednearthefocalplaneoftheBBSallowingtosuppress
eventsfromelasticscattering.EachMWPCconsistsofanXandYplanewith
alltogether2896wires.Thewiredistance2.5mmdefinesthespatialresolution.
LiketheVDCs,theMWPCsarefilledwithamixtureof50%argonand50%
iso-butaneandanegativehighvoltageisappliedtothecathodefoils.Thewires
arekeptatzeropotential.Ifthedetectorisoperatedinthepolarimetermode,
thepolarizationofscatteredparticlesisdeducedbymeasuringtheasymmetryof
secondaryscatteringinthegraphiteanalyzerCwhichwasremovedduringthe
(d,2He)experiments.

Scin3.3.3tillators

AscanbeseeninFig.3.3,theS1andS2scintillatorplanesarelocatedinfrontof
theanalyzerandbehindthelastMWPC,respectively.Thecoincidencebetween
signalsofthetwoplanesdefinestheeventtriggerwhichstartstheread-outof
thefront-endelectronicsandsubsequentonlineeventprocessing.Bothscintillator
planesconsistoffiveoverlappingscintillatorpaddles,whicharereadoutfromthe
topandbottombyphoto-multipliers.ThescintillatingmaterialusedisNE102A.
Thethicknessofthepaddlesofthesecondplaneis6mmwhereasitisonly2mm
inthefirstplanetoreducemultipleCoulombscattering.

Electronics3.4

Figure3.4showsanoverviewoftheread-outsystems[46]ofthefocal-planedetec-
tionsystemandthefocal-planepolarimeterconsistingofpreamplifiers,levelcon-
verters,CAMAC-basedread-outsystemsandVME-basedreal-timeprocessing.

29

ThetiminginformationoftheVDCsisdigitizedbytwoFERA(FastEncoding
ReadoutADC)systemsconsistingofCAMACbasedLeCroy3377driftchamber
time-to-digitalconverters(TDC).EachTDCmodulecontains32channelsandis
abletostoreupto16signalsperchannelwithadigitizingresolutionof500ps.
Themulti-hitcapabilityoftheseTDCsisespeciallyimportantinthecaseofthe
(d,2He)measurementbecausetwoprotontrackspereventhavetobereconstruc-
tedinthedetectionsystem.ThesignalsfromtheS1andS2scintillatorplanes
arealsoreadoutbyLeCroy3377TDCs.TheMWPCsarereadoutusingLeCroy
PCOS(proportionalchamberoperatingsystem)systemwhichhasbeenspecially
developedforwirechamberread-out.Insteadofproducingtiminginformation
PCOSonlyreportswhichwireshavefiredinanevent,makingitfasterthan
TDCsystems.Asanexample,thetotalread-outtimein(p,p)measurements
amountst=6.6µsandallowsthesystemtohandleread-outratesof100kHzat
adead-timeof67%.MoredetaileddescriptionscanbefoundinRefs.[46,51].

logicriggerT3.4.1

Theeventtriggerisbasedonacoincidenceofsignalsfromthetwoscintillator
planesintheabsenceofasignalfromthevetoscintillator.ItinitiatesCAMAC
read-outbydistributingcommonstopandgatesignalstotheTDCandPCOS
systemsandallowstoinhibitthegenerationofnewtriggersignalsincaseoneof
thesystemsis’busy’.Thesignalsofbothphotomultipliersfromeachscintillator
arecombinedbyameantimer,whichrequiresbothsignalstobepresentwithina
timeintervalof30ns.AlogicalORofthesesignalsisusedtocreatetheoutput
signalforeachscintillatorarrayseparately.Themaintriggersignalisgeneratedas
alogicalANDofthesignalsofscintillatorarraysS1andS2.Fromthemaintrigger
signalthegatesforthedifferentreadoutsystemsaregeneratedviaanotherlogical
AND.Itsuppressesthetriggersignal,ifthereadoutofthelasteventisnotyet
complete,i.e.someofelectronicmodulesprovidea’busy’signal.Theseparation
ofthemaintriggersignalfromtheincorporationofthe’busy’signalshasthe
advantagethatthelifetimeofthereadoutsystemcanbedetermineddirectlyas
thequotientofthenumberofcoincidencesfromthetwologicalANDs.

30

3.4.2

Fig.3.4:Schematicviewoftheread-outsystem.

DSP-basedonlinedataprocessingsystem

ThecompletedetectoristhenreadoutbyaDSP(digitalsignalprocessor)sy-

stem[52].TheDSPsSTR8090actasaneventfilter,astheymakethedecision

whetheraneventisrecordedtotapeordiscarded.Ineithercasethedetector

dead-timecanbekeptbelow8µs,therebyallowinghightriggerrates.TheDSP

softwarechecksifthedatafromtheMWPCsareconsistentwithtwocoincident

protontracks.Onlyinthiscase,thedatafromtheVDCsatthefocalplane,

togetherwithallotherdetectordata,arerecordedontape.Thesystemshownin

Fig.3.5consistsoffivefirst-in-first-out(FIFO)modulesSTR7090tobufferevent

31

Fig.3.5:VME-basedreal-timeprocessingelectronics.

data,anarbitermoduleSTR8090TRANtodistributethedataandanumber

ofDSPswhichperformthenecessarytests.Thedataflowproceedsasfollows:

afterreceiptofatriggersignalandsubsequentread-outofthedetectorsbythe

CAMACsystems,alldataaretransferredtoVMEFIFOmodules.Additional

triggersignalsareinhibitedbyreturninga’busy’signaltothetriggerlogicuntil

thedatatransferhasbeencompletedoratime-outsignaloccurs.Onceacom-

pleteeventhasbeenread,the’busy’signalisremovedtoallowthenextevent

tobereadoutbythefront-endelectronics.Completeeventsaretransferredvia

anarbitermoduletooneofseveralDSPmoduleswhichperformthenecessary

calculations.TheFIFOmoduleisconnectedtothearbiterandtheDSPmodules

bytheVME-independentlocalreadoutbus(LRB).

TheLRBprovidesafastdatatransferbetweentheFIFOmodulesandtheDSP

section,whichiscontrolledbythearbitermoduleusingadedicatedtokenpro-

tocol.ThisallowstheuseofmultipleparallelworkingDSPmodulesconnected

tothesameLRB.Besides,thecomputingpowerofthesystemcanalwaysbe

increasedbyaddingfurtherDSPmodules.TheDSPsoftwarerunningduringthe
(d,2He)experimentscanbesubdividedintothreemainsteps:datareadoutand

dataconsistencychecks,softwaretriggerdecision,anddataoutput.Tosummari-

32

zetheappliedDSPeventfilter,atleastonehithastoberecordedineachMWPC
planeandtheremustbetwowirechamberswithatleast3hits.

Theoveralldead-timeofthecompletesystemiskeptbelow10µswhichcorre-
spondstoatheoreticallimitfortheeventthroughputof100kHz.Moreinforma-
tionontheonlineprocessingsystemcanbefoundinRefs.[52,53].

3.5Dataacquisitionandonlineanalysis

Theheartofthedataacquisition[46]isanAlphaAXP-VME4/288Workstation
operatedwiththesystemVxWorks.Itconnectselectronicsandnetworkandruns
thedataacquisitionsoftwareD2HEDAQandtheslow-control.TheDAQsoftware
onecanuseforbothonlineandoff-lineanalysis.ItisbasedontheCDAQlibrary
whichwasdevelopedbyZwartsatKVI[54].Therearethreeprocesseswhich
runduringtheexperimentondifferentcomputers:processeddatafromtheDSP
systemarereadoutbythefirstDAQprocessontheVMECPU.Thedataare
thentransferredtoaLinuxPCinthecontrolroomviaEthernetwhereitis
pickedupbyasecondprocess.ThesecondprocessstoresalldataonDLTtape
andsendsaload-dependentpercentageofthedataviatheKVInetworktoathird
processonanotherLinuxPCwhichisusedforonlineanalysisandvisualization.
AllthesemodulesarecontrolledviaTCP/IPconnectionsbyagraphicaluser
FPPGUI.terfacein

Tobeabletocontinuouslymonitorandcontrolthemostimportantexperimen-
talparametersofthedetectorsystemsaslow-controlsoftwarehasbeendevelo-
ped[47].Theslow-controlconsistsoftwocomponents:slowserverwhichrunson
theVMEworkstationandslowclientononeoftheLinuxPCs.Viathegraphi-
caluserinterfaceitispossibletodisplayscalerdata,setandmonitorthewire
chamberhighvoltagesandchangepreamplifierthresholds.

Datavisualizationintheonlineandfurthertreatmentofthedataintheoff-line
caseisrealizedusingthePAWpackage[55]fromtheCERNlibrary.Duringthe
experimentstheresultsoftheonlineanalysisarebookedintohistogramsina
shared-memoryregion.UsingPAWonecanaccessanddisplaythesehistograms.

33

Foroff-lineanalysistheresultingvariablesarewrittenonevent-per-eventbasisto
afileinNTUPLEformat.ThisformatisunderstooddirectlybyPAWandallows
aneffectivefurtheranalysisofthedata.

3.62Heeventreconstruction

Theanalysisisdividedintofewdifferentsteps.First,usingtheD2HEDAQanaly-
sistherawdataareunpackedandthensenttotheinterpretationroutine,which
extractstherelevantinformationincludingsignalsfromscintillatordetectorsand
wirechambers.Themostcomplexpartoftheinterpretationresidesinthedeter-
minationoftheVDCintercepts,sincetwocorrelatedtracks(protonsfrom2He
decay)havetobereconstructedoutofseveraldrifttimes.Thesetwotracksare
almostsimultaneous(lessthan10nsoftimedifference)andcanonlyberesolved
becauseofthemulti-hitcapabilityoftheTDCsused,whichregistermorethan
onetimeperchannelbeforeastopsignalissetbythetrigger.TheVESNAn(VDS
EuroSuperNovaAnalysisfornparticles)softwaredevelopedbySchmidt[56]was
appliedinthepresentexperimentforthetrackreconstruction.Havingtheinter-
ceptsfromtheVDCsdata,thefocal-planevariableslikehorizontalandvertical
positionsandanglescanbedetermined.Thenthekinematicrecoilcorrectionhas
tobedone(seeSection3.2.2).Finally,usingtheray-tracinginformation(see
Section3.2.1),thetargetvariablesforeachprotonarecalculated.Thus,allthe
informationcharacterizingthe2Heparticlelikekineticenergy,scatteringangle
andinternalenergycanbeextracted.Thedetailsoftheanalysiscanbefoundin
4.Chapter

terimenExp3.7

ThepresentexperimentwasperformedinApril2003atKVIinGroningen.Deu-
teronsfromtheCUSPionsourcewereacceleratedtoakineticenergyof171MeV.
Thedeuteronbeamwasincidentona7Litargetisotopicallyenrichedto99.9%and

34

havingathicknessof9mg/cm2.Measurementsweremadeatfourdifferentspec-

trometeranglesettingscorrespondingtocenter-of-massanglesΘcm=0◦−11.3◦.
Thetwoprotonsfrom2HeweremomentumanalyzedbytheBBSanddetec-

tedbytheESN-detectionsystem.Thebeamcurrentvariedbetween0.3nAand

1.5nA,dependentlyonthespectrometerangle.Asatest,foreachangle,mea-
surementson12Cwereperformedasthisnucleushasbeenstudiedextensively
withthe(d,2He)reaction[47,51].Measurementsona9.4mg/cm2thicknatural

carbontargetservedforthedeterminationoftheexperimentalenergyresolution

ΔE150keV(FWHM).Thetestmeasurementsprovidealsoapossibilityto
verifythecorrectoperationofthedetectorsystemandoftheanalysisprocedure

ork.wthisinapplied

35

analysisData4

Inthischapter,thedifferentstepsperformedduringthedataanalysiswillbe
described.First,thedeterminationofthescatteringvariablesneededtoobtain
theexcitationenergyspectrawillbeoutlined.Theacceptanceandbackground
correctionprocedureswillbegiveninSections4.2and4.3.Thentheextraction
oftheexperimentalcrosssectionswillbedescribed.Finally,thequasifreeconti-
nuumbackgroundwillbedeterminedandresultingfitofthe7Hespectrawillbe
shown.ThelasttwoSectionsdiscussthedecompositionofthespectra,aimingat
apossibleidentificationofalow-lyingresonanceinthe7Henucleus.

4.1Reconstructionofthescatteringvariables

ThereconstructedVDCinterceptsfromthedrift-timeinformationusingaso-
phisticatedalgorithm(shortlydescribedinSection3.6)arerelatedtothefocal
planecoordinatesbysimplegeometry.Thehorizontalpositionatthefocalpla-
neisdirectlygivenbythecrossingpointoftheX-planeofthefirstVDC.The
determinationoftheremainingcoordinatesliketheverticalposition,thehori-
zontalandverticaldetectoranglesisthenstraightforwardusingtheinformation
fromthethreeotherVDCplanes.AsexplainedinSection3.2.1,thetargetvaria-
bleswereobtainedusingaray-tracingprocedure.Thehorizontalscatteringangle
θt,1(2),theverticalscatteringangleφt,1(2)andthemomentump1(2)aredetermined
forbothprotons,denotedbysubscripts1and2.Thekineticenergyiscalculated
firstforeachprotonseparately,the2Hetotalenergyfollowsfromthesumofthese
twoprotonenergies.Themomentumoftheprotonscanbecalculatedfromthe
BBSdipolefieldBandrelativemagneticrigidity(givenin%)
p=qBρ0(1+δ−)3(4.1)
103356.3×whereBρ0isthemagneticrigidityofthecentralrayinTm,qisthechargeofthe
particleandδisdefinedintheSection3.2.1.Thekineticenergyofeachproton
ascalculatedisEkin,1(2)=m2cp4+p21(2)−mcp2,(4.2)
36

wherempistherestmassoftheproton.Thedirectionvectorsrˆ1(2)oftheprotons
canbeobtainedfromthetargetcoordinatesaccordingto

sin(ΘBBS+θ1(2),zx,t)∙cos(φ1(2),zy,t)
rˆ1(2)=sin(φ1(2),zy,t)(4.3)
cos(ΘBBS+θ1(2),zx,t)∙cos(φ1(2),zy,t).
Therelativeanglebetweentwoprotons(openingangle)inthelaboratorysystem,
thetotalmomentumandthescatteringangleford→2Heinthelaboratory
systemcanbeexpressedinthefollowingway

Θpp=arccos(rˆ1,rˆ2),

(4.4)

p=rˆ1∙p1+rˆ2∙p2,(4.5)
Θ=arccospz.(4.6)
pUsingthekinematicslibraryKINEMA[48]onecanconvertscatteringvariables
intothecenter-of-masssystemandcalculatetheexcitationenergyExfromthe
differencebetweenbeamenergyEd,kineticenergyE2He,recoilenergyERand
theQ-valueofthereaction.Finally,therelativeenergyofthetwoprotonscan
beobtainedfromkineticenergiesoftheprotonsandfromtheopeningangle
=E1+E2−E1E2cosΘpp.(4.7)
2

Acceptance4.2correction

DuetothelimitedmomentumandangularacceptanceoftheBBSspectrometer
andthediparticlenatureof2He,theexperimentalspectraneedtobefolded
withanacceptancefunction[57].Thiscorrectionfunctiontakesthesettingsof
allBBSmagnetsandthereactionkinematicsintoaccountandisobtainedby
aMonte-Carlosimulation[51].Theeasiestwaytodotheacceptancecorrection
istocalculatethedetectionprobabilityoftwocorrelatedprotonsasafunction
ofthesolidangleandtheexcitationenergyintheresidualnucleusforagiven
37

Fig.4.1:Calculatedinternalenergyspectrumofthetwoprotonsin2Hefrom
Watson-MigdalFSItheory[58,59].

rangeofinternalenergies.Inthepresentworkthelimitsforwerechosento
be0<<1MeV,becausethemaximumrelativeprotonenergywhichcanbe
observedwiththeBBS+ESNsystemisabout1MeV.

Todeterminetheacceptancefunction,the(d,2He)reactionissimplifiedtotwo
independenttwo-bodyreactions.Inthefirststepofthescatteringreactionthe
2Heparticleistreatedlikeaboundparticle

d+A→2He+B.

(4.8)

The2Hesystemdecaysintotwoprotonsinthesecondstep.Thisistheso-called
(FSI)teractioninfinal-state

2He→p+p.(4.9)
Thefirststepisgovernedbytwo-bodykinematicsandonlyneedstherandom
generationofanexcitationenergyoftheresidualnucleus,aninternalenergy
for2He,polarandazimuthalscatteringangles.Theinternalenergy,computed
usinginputdatafromtheFSItheoryofWatson[58]andMigdal[59]isshownin
Fig.4.1.Inthesecondstepitisassumed,thatthe2Henucleusdecaysisotropically
intotwoprotonsinthecenter-of-masssystem.ALorentztransformationbrings
thecenter-of-masscoordinatesofthetwoprotonstothelaboratoryframe.

38

Afterthesimulationofthe2Hedecay,thefullsetofkinematicparametersis
availableforbothprotonsatthereactionpoint(target),suchashorizontalΘzy,i(i
=1,2)andverticalcoordinatesΦzy,i(i=1,2),andthesingle-protonmomentum
Δp/p0,i(i=1,2)coordinates.Thisallowstocalculatethetransportofthe
twoprotonsfromthetargetthroughthespectrometertothefocalplaneusing
theray-tracingtechniquedescribedinSection3.2.1,yieldingthecoordinatesof
eachprotonatthedetectorsurface.Thesamegatesasusedintheanalysisof
measureddataareappliedonthedetectorcoordinatesinasimulationprocedure.
ToincluderesolutioneffectsthedetectorpositionsareconvolutedwithGaussian
peakswithappropriatewidths.Asthenextstep,newtargetcoordinatesofboth
protonsarecalculatedbytransportingthenbackthroughthespectrometerfrom
itsfocalplanetothetarget,usingagaintheray-tracingtechnique.Finally,one
obtainsasetofvariablesforeachinitialprotonandasetofvariablesafterits
transportfromtargettodetectorandback,withoutanyconstraintsonsolid-angle
acceptance,detectoracceptanceandgates.Fromthesetofvariablesbelongingto
theprotonsthathavebeentransportedthroughthespectrometer,onecalculates
ectrum.spenergyexcitationthe

Thesametwo-dimensionalcutsthatusedintheexperiment,areappliedtoobtain
theacceptance-correctionfunction.Asanexampletheexperimentalgatesforall
acceptance-relevantvariablesasmomentum,y-coordinateandscatteringangles
Θzy,i(i=1,2)andΦzy,i(i=1,2)aredisplayedinFig.4.2.Onthel.h.s.of
theFig.4.2thecutsindicatedassolidlinesforΘBBS=0◦withoutanaperture
attheentranceoftheBBSandonther.h.s.forΘBBS=7.8◦withanaperture
areshown.TheoffsetforYdcanbedeterminedwhenusingtheaperture.Asthe
openingangleofthedi-protonsystemvarieseventbyeventina(d,2He)reac-
tion,thesingle-protoncoordinatesinthefocal-planearebroaddistributionsas
showninFig.4.2.Thesimulatedacceptance-correctionfunctionF(ΔΩ,ΔE)for
acertainsolidangleisdeterminedbycalculatingtheratiobetweenhypotheti-
callynon-decaying2Heparticlesanddi-protonsfromthe2Hedecayreachingthe
detector.AtypicalexampleisshowninFig.4.3forthe7Li(d,2He)reactionat
c.m.anglesbetween0◦and1◦.Theshapeistriangularwithanindicationofaflat
topandcanbeparameterizedbytwoGaussians.Sincetheacceptancefunction
dependsonthecentralmomentumofthemagneticspectrometeritneedstobe
reevaluatedeachtimethefieldischanged.The2Hedetectionprobabilityistaken

39

Fig.

4.2:

Focal-plane

ΘBBS=0◦

spectra

panel)(left

of

panel)

protons

Θandpanel)

from

Θ

the

7.8=

7Li(d,2He)7He

measurement

at

(leftpanel)andΘBBS=7.8◦(rightpanel).Thedifferent

formsintheleftandrightpanelsresultfromtheuseofanapertureat

theentranceoftheBBSforthelatter.Experimentalcutsusedforthe

determinationoftheacceptance-correctionfunctionareshownasfull

lines.Seetextforfurtherexplanations.

40

Fig.4.3:Simulatedacceptance-correctionfunctionforthe7Li(d,2He)reactionat
Θcm=0◦-1◦andtheresultofthefitbytwoGaussians.

intoaccountbydefininganeffectivesolidangleΔΩeff=F(ΔE)ΔΩwhichis
thenused(seeSection4.4)tocalculatetheexperimentaldouble-differentialcross
section.Systematicerrorsduetotheuncertaintiesinthedeterminationofthe
effectivesolidangleareoftheorderof15%.

4.3Instrumentalbackgroundsubtraction

Thebackgroundinthemeasuredrawspectraisduetouncorrelatedprotonsde-
tectedwithinthecoincidencewindowintheFPDSandcomesmainlyfromthe
deuteronbreakupreactiononthetargetnucleus.Thefirststepinthebackground
subtractionprocedureistheexperimentalseparationofprotonsoriginatingfrom
differentbeambunchesofthecyclotron.Theintervalbetweentwobeambunches
wasassmallas23ns(repetitionrateofthecyclotron).Atypicalexampleofthe
timedifferencebetweentwoprotonsmeasuredatthefocalplaneofthespectro-
meterisshowninFig.4.4.Ascanbeseen,thepromptpeakat0ns,stemming
fromtwoprotonsofthesamebeambunch,iswellseparatedfromrandompeaks
(coincidencesbetweentwoprotonsfromdifferentbeambunches).Smallcontri-
butionofrandomprotonswithinthepromptpeakcanbesubtractedbyapplying

41

Fig.4.4:Spectrumofthetimedifferencebetweentwoprotonsatthefocalplaneof
theBBS.Apromptandfourrandompeaksarevisible.Thetimeinterval
betweentwopeaksisequivalenttotheinverseofthebeamrepetition
rateof43MHz(i.e.23ns).

gatesontherandomandthepromptpeak,respectively.Figure4.5presentsthe
backgroundspectrumobtainedaftergatingontherandompeak.Tominimizethe
statisticalvariationoftherandomdistributionandtosmoothit,therandomexci-
tationenergyspectrumwasfittedwithafourth-orderpolynomialfunctionbefore
subtractionfromthepromptone.Thebackgroundsubtractionprocedureisshown
inFig.4.6.Thereisasmallcontaminationfromthe12C(d,2He)12Bobservedin
thespectrum.Besidesthe12Bgroundstate,thecontributiontothespectrumis
verysmall.Inordertoremovethiscontribution,themeasured12C(d,2He)spec-
trumhasbeenscaledbyanormalizationfactorwhichiscalculatedfromtheratio
betweenthe12Bg.s.peakareainthe7Hespectraandthoseofthemeasurements
usinga12Ctarget.AscanbeseenbycomparisonofFig.4.7andFigs.4.8,4.9,
afterthisproceduretheboronpeakscanbesubtractedfromthe7Hespectrum.

42

Fig.4.5:Excitationenergyspectrumproducedbygatingontherandompeakof
thetimedifferencedistributionbetweentwoprotons.

4.4Angularbinsandexperimentalcrosssecti-
ons

Data,measuredatdifferentBBSsettings,havetobedividedintoangularbins
inordertoobtainangulardistributions.Eachsettingofthespectrometerangle
wassplitintotwoorthreebins,dependingonthestatistics,withawidthof
ΔΘcm=1◦asisshownintheTab.4.1.AMonte-Carlosimulationhasbeen
performedforeveryexperimentalsettingofthespectrometerandfordifferent
angle.scatteringinbins

Theexperimentaldouble-differentialcrosssectionintheexcitationenergybin
ΔExandthesolid-anglebinΔΩiscalculatedaccordingto
d2σA1NΔEx,ΔΩmb
dΩdEx=0.266TkQ(1−τ)α2ΔExΔΩeffsrMeV(4.10)

with

43

Fig.4.6:Spectrumof7Heaftergatingonthepromptandtherandompeaksof
thetimedifferencedistributionbetweentwoprotons.Thedashedline
istheresultofthefourth-orderpolynomialfitappliedtotherandom
spectrumshowninFig.4.5.

Fig.4.7:Subtractionof12Bfromthe7Hespectrumafterbackgroundandaccep-
correction.tance

44

Tab.4.1:Angularbinscreatedinthedataanalysis.

◦Θlab(BBS)ΔΘcm=1
◦◦◦◦◦◦◦
00-11-22-3
3◦3.7◦-4.7◦4.7◦-5.7◦
5◦5.9◦-6.9◦6.9◦-7.9◦
7.8◦9.3◦-10.3◦10.3◦-11.3◦

A:targetmass[g/mol],
T:targetthickness[mg/cm2],
k:isotopicenrichment,
Q:integratedchargein[nC],
dead:τtime,,efficiencydetector:αNΔEx,ΔΩ:numberofcountsinthebinΔΩΔEx,
ΔΩeff:effectivesolidangle[sr](seeSection4.2),
ΔEx:energybin[MeV].

Theacceptanceandbackground-correcteddouble-differentialcrosssectionsat
excitationenergiesupto25MeVareshowninFigs.4.8and4.9andlistedinthe
tableinAppendix.ThedatacanalsobefoundinWorldWideWeb[60].

4.5Quasifreecontinuumbackground

Quasifreecharge-exchangeisamajorsourceofnonresonantcontinuumback-
ground.Thequasifreescattering(QFS)referstoreactionsinwhichtheneutron
scattersfromasingleprotoninthetarget,essentiallyasiftherestofthenucleus
wasnotpresent.InthiscontextthetermQFSmeansthesingle-stepquasifree
nucleonknockoutreaction.Thestruckprotonisemittedfromthetargetnucleus
andisdetectedinthespectrometer.Inparticular,the(d,2He)charge-exchange

45

Fig.

4.8:

Double-differential

at

various

cross

scattering

sections

angles

Θc.m.

46

for

=

0

the

7

Li(d,

5.7

.

2

He)

reaction

measured

Fig.

4.9:

reaction

Double-differential

cross

sections

atvariousscatteringanglesΘc.m.

for

the

27He)Li(d,

◦◦=5.9−11.3.

reaction

measured

onaboundprotonleadstotheejectionofthecharge-exchangedneu-

tronfromtheresidualnucleusinitsgroundstateorexcitedstate.Todealwith

theQFSintheanalysis,thetechniquedevelopedbyErelletal.[61]inastudy

ofpioncharge-exchangereactionisused.Thissemi-phenomenologicalparame-

trizationhasbeenappliedsuccessfullytointermediate-energycharge-exchange

47

Fig.4.10:Spectrumofthe6Li(d,2He)reactionatEd=171MeVandΘcm=
20◦−21◦(fromananalysisofRef.[67]).Thelong-dashedlineisafitof
asemi-phenomenologicalmodel[61]forthequasifreescatteringcross
section.

reactionspectra[62–66].Theshapeofthebackgroundisdescribedby

(4.11)

21−expE−E0
σdTdΩdE=NE−EQF2,(4.11)
+1WLwhereE,EQFandE0denotetheoutgoing2Heenergy,themaximumofthe
quasifreepeakapproximatedbyaLorentzfunctionandacutoffenergydueto
Pauliblocking,respectively.ThecutoffenergyE0representsthethresholdfor
breakupdyothree-bE0=E2He(7Heg.s.)−Sn.(4.12)
Thequasifreepeakenergyisdeterminedfromthecomparisonofthequasifree
(d,2He)reactiononthetargetwiththeanalogouselementaryreactiononthe
protonEQF=E2He(1H)−Sn.(4.13)
Here,E2Hedenotesthekineticenergyofthe2Heparticlesforthe1H(d,2He)
reactionandSn=-0.445MeVtheneutronseparationenergyin7He.Inother
words,thecentroidenergyEQFofthequasifreeprocessisshiftedrelativeto
thatofthecharge-exchangereactiononafreeprotonbytheneutronbinding

48

energyinresidualnucleus.Unlikewith(p,n)-typereactionsnoshiftduetoa
Coulombbarrierneedstobeincluded[63].TheLorentzianwidthWLdependson
qtransfertummomenthe

2WL=WL01+αkq.(4.14)
FThescalingparameterT=4.0MeVandtheparametersWL0=16.26MeV,
α/kF2=0.363fm−2fromEq.(4.14)weredeterminedbyameasurementofthe
6Li(d,2He)6Hereaction[67]underthesamekinematicalconditionsasthepresent
experimentatlargescatteringangles,wherethequasifreecrosssectionshould
dominate.Itmaybenoted,thattheresultsofRef.[67]obtainedindependently
atdifferentmomentumtransfersindicate,thatthenormalizationfactorNis
q-independent.ThisisalsoconsistentwithfindingsofWangetal.[68]forthe
quasifreecrosssectionsinthe(p,n)reactiononp-shellnuclei.Asdemonstratedin
Fig.4.10fortheexampleoftheangularbinΘcm=20◦−21◦,theapproachof[61]
providesagooddescription6Li(d,2He)6Hedata.Thebackgroundparametersfor
the7Li(d,2He)reactionasafunctionofangle,arelistedinTab.4.2

Tab.4.2:Angle-dependentparametersforthequasifreebackgroundparametriza-
(4.11).Eq.oftion

Θcmqcm(fm−1)WL(MeV)EQF(MeV)E0(MeV)
0◦−1◦0.03916.27168.744158.719
1◦−2◦0.08716.30168.618158.702
2◦−3◦0.14016.38168.367158.667
3.7◦−4.7◦0.23316.58167.651158.568
4.7◦−5.7◦0.28816.75167.060158.486
5.9◦−6.9◦0.35417.00166.187158.365
6.9◦−7.9◦0.40917.25165.313158.244
9.3◦−10.3◦0.54117.99162.721157.886
9.3◦−10.3◦0.59718.36161.430157.707

49

4.6Fitofthespectra

Thenextstepoftheanalysisafterbackgroundsubtraction,acceptancecorrecti-
on,andextractionoftheexperimentalcrosssectionwasadecompositionofthe
resultingspectra.ThecomputerprogramFIT[69]wasusedinthisprocedure.

4.6.1Decompositionofthespectrum

Figure4.11displaysexamplesoftheresultingdouble-differentialcrosssections
asafunctionoftheexcitationenergyin7Heforthreeangularbins.Theg.s.
transitionisresolvedinallspectra.Thelowthresholdenergy(besidesthealready
open6He(g.s.)+nchannel)forα+3ndecayatEth=0.53MeVleadstoabroad
distributionofstrengthevenatlowexcitationenergies.Twopreviouslyobserved
resonancesin7HeatEx=2.9(1)and5.8(3)MeVwithwidthsΓ=1.99(11)and
4(1)MeV,respectively,foundinreactions[8,9]wheretheyprovideaclearsignal,
arenotexcitedselectivelyinthepresentexperiment.Theprominentstructure
aroundEx≈20MeVwasalsoobservedinthe7Li(n,p)reaction[70]andmay
resultfromanexcitationoftheisovectorgiantdipoleresonanceoftheαcluster
corein7Hesimilartoobservationsin7Li[71,72].The7Helevelschemewith
knownresonancesandwithparticlethresholdsrelevanttothepresentanalysisis
showninFig.4.12.Notethat7Heg.s.isunboundby0.445MeV.Thus,resonance
andexcitationenergiesareshiftedbythisvaluerelativetoeachother.

InspectingFig.4.11,theidentificationofapossibleadditionallow-lyingreso-
nanceisclearlyadifficulttask.Thefirststepoftheanalysisisadecomposition
intoBreit-Wignerresonanceswithanenergy-dependentpenetrabilityshownin
Fig.4.12plusanadditionalresonanceatEx≈20MeVfrom[70].Contributi-
onsduetothequasifreenucleonknockoutreactionsarealsoexpectednotonly
from7Li,butalsoduetothecharge-exchangereactionsontand4Hebecause
ofthepronouncedclusterstructureofthe7Ligroundstatewitha(4He⊗t)con-
figuration.Forthequasifreescatteringon7Liasawholethereexistsnotonly
adistributionforthe6Hegroundstate,butalsoforthefirstexcited2+state.
Theexcitationenergydependenceofbothprocesses(6He(g.s.)+nand6He(2+)+n
channels)isdescribedbythesemi-phenomenologicalparametrizationofErellet

50

Fig.4.11:Selectedspectraofthe7Li(d,2He)reactionatEd=171MeVfordiffe-
rentangularbinsandtheirdecomposition.Solidlines:experimentally
establishedresonancesandresultingfit.Long-dashedlines:background
fromquasifreescatteringon7Liasawhole(6He(g.s.)+nand6He(2+)
+nchannels)usingthemodelofRef.[61],onthe4Heclusterin7Li
(t+t+nchannel)andonthetritoncluster(4He+3nchannel)usingthe
datafrom[74,75].(a):relativemagnitudesdeterminedbyafittothe
data.(b):quasifreescatteringon7Li,assuming6He(g.s.)+nchannelor
(c):assuming6He(2+)+nchannel,respectively,fixedbyameasurement
ofthe6Li(d,2He)reaction.

al.[61],whichhasbeenpresentedinthepreviousSection.Todeterminetheener-
gydependenceofthecharge-exchangereactionsontheclustercomponentsthe
dataonthe3,4He(p,n)reactions[74,75]atmomentumtransferscomparableto
thepresentcasewereused.ThecorrespondingthresholdsareEx=0.53MeVand
Ex=11.87MeVforthe4He+3nandt+t+nchannels,respectively.Inorderto
applythe(p,n)results[74,75]forthe(n,p)reactionsonthe7Lig.s.clusters,one
canemploychargesymmetry.Furthermore,3He(p,n)3prepresentsthemirrorre-

51

Fig.4.12:Levelschemeofthe7Henucleus.

actiontotherequiredt(n,p)3nchannel.Inordertofitthe7Hespectrausingthese
twoclustercomponents,astandardfunctioninthelibraryprogrammeFITwas
chosenwhichdescribestheformofthecorrespondingexperimentalspectra[69]
usingaGaussfunctionandapolynomialoffourthorder.

Returningtothe7Li(d,2He)7Hedata,threedifferentanalysesofthespectraare
presentedinthefollowing.Thefitstakeintoaccountallknownresonances(g.s.,
2.9and5.8MeV)atlowexcitationenergies.Theircentroidsandwidthsare
allowedtovarywithintheexperimentaluncertainties[73].Additionally,thepro-
minentbumpat(Ex≈20MeV)isdescribedasasingleresonancewiththe

52

parametersdeducedby[70].Furthermore,thethreequasifreebackgroundchan-
nelsdiscussedaboveareincluded.Agooddescriptionofalldatacanbeachieved
(seee.g.solidlinesinFig.4.11).Independentofdetailedassumptionsaboutthe
centroidenergiesandwidthsofpossibleresonancesathigherexcitationenergies,
thedecompositionsshowninFig.4.11demonstratethattheydonotcontribu-
tesignificantlytothecrosssectionsinthelow-energyregion.Thesameistrue
forbackgroundprocesseslikethe4He+3nandt+t+nchannels,whichare
structurelessintheregionofinterestandslowlyandsmoothlyincreasinghaving
maximaatmuchhigherenergies.
Ontheotherhand,themagnitudeofthe6He+ncontributionisthemostcritical
aspectintheanalysisofthe7Hespectra.IntheFig.4.11(a)thedecompositionof
thespectraisshown,wheretheoverallnormalizationNfromEq.(4.11)forboth
the6He(g.s.)+nand6He(2+)+nchannelsistreatedasafreeparameterduring
thefit.However,the6He(2+)+npartwithathresholdenergyEx=1.35MeV
ispredictedtobezerointhefreefit.Moreover,theresultingangulardistributi-
onofthe6He(g.s.)+nchannelshowsconsiderablescatteringand,inparticular,
astrongdecreaseatlargermomentumtransfersincompatiblewiththephysical
interpretationofthequasifreeknockoutprocess.ThiscanbeseeninFig.4.13,
wheretheangulardistributionofthe6He(g.s.)+nquaisifreechannelfromthe
freefit[Fig.4.11(a)]iscomparedwiththatfromthedecompositionconstrained
bythe6Li(d,2He)6Hemeasurement[Fig.4.11(b)]ispresented.Therefore,inan
alternativeanalysis[Fig.4.11(b,c)]itisassumedthatthemagnitudeofthesin-
glenucleonknockoutquasifreecrosssectionisnotchangingsignificantlywhen
goingfrom6Lito7LiandthereforetheoverallnormalizationNinEq.(4.11)
thuswastakenfromthecorresponding6Li(d,2He)6Hedata[67].Twoextreme
casesareconsidered:thetotal6He+ncontributionisdescribedexclusivelybythe
6He(g.s.)+nchannel[Fig.4.11(b)]orbythe6He(2+)+nchannel[Fig.4.11(c)].

4.6.2Possiblelow-lyingJπ=1/2−spin-orbitpartnerofthe
stateground

WhilethedecompositionwiththreeanalysesdescribedinpreviousSectionshows
comparableresultsoverthewiderangeoftheexcitationenergy,differencesare
observedinthelow-energyregionofthespectrum.Anextendedviewofthe
53

Fig.4.13:Angulardistributionofthetotal6He(g.s.)+nquasifreecontribution
derivedfromafreefitshowninFig.4.11(a)(opencircles)andfixedby
the6Li(d,2He)6HemeasurementshowninFig.4.11(b)(solidcircles).

low-energypartoftheΘcm=0◦−1◦spectrumisplottedinFig.4.14(a)-(c)in
comparisonforthethreedifferentdecompositions.Theg.s.resonanceandthe
regionaboveEx3MeVarewelldescribedinallcases.However,inbetween
thedataovershootthefit,indicatingthepresenceofapossiblefurtherresonance.
Indeed,thisisnotonlyobservedat0◦butalsointheotherspectra,exceptfor
thelargestscatteringanglesmeasured.Ontheotherhand,inclusionofanad-
ditionalresonancewithEx1.45MeVandΓ2MeVprovidesanexcellent
descriptionofthedata,seeFigs.4.14(b),(c).Consideringonlythe6He(2+)+n
channeltheresonancebecomesevenmorepronounced.Forbothcasesthecor-
respondingχ2/d.o.f.improvesfrom2.3to1.7.Theestimateduncertaintiesfor
thecentroidenergyandtheresultingwidthofapossibleadditionalresonanceat
lowExareratherlarge,inparticular,duetothelargeexperimentalerrorofthe
5.8MeVresonancewidth.ArangeofacceptablevaluesEx=(1.45−+00..57)MeV,
Γ=(2.0−+11..10)MeVwasdeterminedbytheuncertaintyofthetheoreticalχ2distri-
bution.Thesystematicuncertaintiesoftheextractedresonanceparametersdue
toabsolutenormalizationofthedataandacceptancecorrections[57]areofthe
15%.oforder

AssumingalternativelyanadditionalresonancewiththeparametersofRef.[6]

54

Fig.4.14:Low-energyregionofthe7HespectrumobtainedforΘcm=0◦−1◦(top
rowofFig.4.11).Solidanddashedlinesarethesameasindecompo-
sitions(a)-(c)ofFig.4.11.Hatchedareain(b)and(c):additional
low-energyresonancenecessarytodescribethedata.(d):additional
resonanceassumingtheparametersof[6].

andestimatingthecrosssectionat0◦fromthepredictionsoftheabinitiocal-
culationsdiscussedbelowleadstothepoorfitshowninFig.4.14(d).Evidently,
sucharesonanceshouldbeclearlyvisibleinthedata.

AsdiscussedinthenextSection,wheretheB(GT)strengthsextractedfrom
themeasuredcrosssectionarecomparedwithmodelcalculations,thislow-lying
resonancewouldbeacandidatefortheJπ=1/2−spin-orbitpartnerofthe7He
state.ground

55

5ExtractionofGamow-Tellerstrength

Afurthertestofthepossibleevidenceforalow-lyingresonancein7Hewith
thepropertiesextractedfromthedataisprovidedbyacomparisonofGamow-
Tellerstrengthsextractedfromthemeasuredcharge-exchangecrosssectionswith
GFMCcalculations[1].TheproceduretoextracttheB(GT)strengthisbased
onitsproportionalitytotheΔL=0partofthecharge-exchangecrosssections
atmomentumtransferq=0asdiscussedby[37]whichcanbeextractedfroman
extrapolationofthemeasuredangulardistributions.

5.1CrosssectionandB(GT)

Inthecaseofvanishingmomentumtransfer,the(d,2He)reactionproceedsthrough
theστpartoftheeffectiveinteraction.Asdescribedaboveinthesection2.5,the
measuredcrosssectionisdirectlyproportionaltotheB(GT)strength,whichin
the(n,p)caseis[36,37]

2dσ(q=0)=Cµ2kfNDJσ2τB(GT+).(5.1)
dΩπh¯ki
ThescalingfactorCisinsertedbecausethe(d,2He)responseadditionallyscales
withthedistributionofthed→2Hetransitionstrength,whosedetectionislimited
bytheexperimentalsetup.Thescalingfactorcanbedeterminedbycomparing
the(d,2He)crosssectionwithknownGTstrengthfromβdecay,whereavailable,
orbyusingthe(d,2He)reactiononself-conjugatenuclei,wheretheGTstrengthis
expectedtobethesameinbothisospindirections(B(GT−)=B(GT+)),sothat
inthiscase,B(GT−)datafrom(p,n)experimentscanbetakenasareference.
InourcaseanempiricalnormalizationfactorC=0.320±0.027derivedfrom
dataonp-andsd-shellnuclei[76]isusedforthedeterminationoftheB(GT)
strengths.Thevolumeintegralofthespin-dependentisovectorcentralpartof
theeffectivenucleon-nucleoninteractionatq=0isgivenin[32]andamounts
to|Jστ|=165MeVfm3atE/A=85MeV.ThedistortionfactorNDisusually
estimatedbycalculatingtheratioofthedistorted-wave(DW)andplane-wave

56

sections,cross(PW)ND=σDW(q=0).(5.2)
σPW(q=0)
ThePWcrosssectionsareobtainedbysettingallpotentialstrengthstozero.
Thecrosssectionsareextrapolatedtozeromomentumtransfer(q=0)usingthe
calculation,WBAD

dσ(q=0)=σcalc(q=0)dσexp(Θ,q).(5.3)
dΩσcalc(Θ,q)dΩ
Thisisareliableprocedureifmeasurementsarebeingperformedinaregionclose
◦to.0

5.2DWBAcalculationsoftheangulardistribu-
tions

Theoreticalpredictionsoftheangulardistributionsfordifferenttransitionsare
obtainedfromdistortedwaveBornapproximationcalculationsemployingtheco-
deACCBA[29],whichisspecializedforthe(d,2He)reactionandappliesthe
techniquesdiscussedinSection2.2.Thisisasemi-microscopiccodethatuses
aneffectivenucleon-nucleon(NN)interactiondescribedinSection2.3andshell-
modelwavefunctions(seeSection2.4).TheeffectiveNNinteractionwastaken
from[32,33],forwhichaparametersetat85MeVpernucleonhasbeenextra-
polatedearlier[77].Thespin-orbitpotentialisnotincluded.Theoptical-model
parametersusedtocreatethedistortedwavesintheincidentchannelwerecal-
culatedfromaglobalfittodeuteronelasticscatteringupto90MeV[78].They
wereextrapolatedtothepresentenergyof171MeV.However,theextrapolated
parametersareidenticalwiththoseofextractedfromtheexperimental(d,d)da-
tameasuredatthesameincidentenergyatKVI[79].Fortheexitchannel,the
optical-modelparametershavebeentakenfromGuptaetal.[80].Nuclearwa-
vefunctionsandone-bodytransitiondensities(OBTDs)weregeneratedbythe
shell-modelcodeOXBASH[34]asabinitiowavefunctionsarestillnotavailable.
Theangulardistributionsofthe7Li(d,2He)reactionpopulatinglow-lyingstatesin
7HeincomparisonwiththeoreticalDWBAcalculationsaredisplayedinFig.5.1.

57

Fig.5.1:Experimentalangulardistributionsofthetransitionstothelevelsat
Ex=0.0,1.45and2.9MeVin7He(fullcircles)andDWBAcalculati-

ons(solidlines)usingshell-modelwavefunctionsandtheLove-Franey

effectiveprojectile-targetinteraction[32,33].Thedashed-dotted,das-

hedanddottedlinesshowthedecompositionintoΔL=0,2,and4

contributions,respectively.

Thedataexhibitaquiteunexpectedbehavior:whileangulardistributionsofpro-

minentGTtransitionsincharge-exchangereactionsarenormallystronglypeaked

atΘcm=0◦,onlyaweakangledependenceisvisibleinFig.5.1.Inparticular,

theg.s.crosssectionangulardistributionisalmostconstant.Thecrosssections

58

atanglescloseto0◦arecomparativelysmall,about70timesweakerthanthe
well-knownp3/2→p1/2GTtransitionpopulatedinthe12C(d,2He)12Breaction
atacomparableincidentenergy[77].Thisstrongreductioncouldbeattributed
tothedominantclusterstructureoftheinvolvednuclei.Inordertoseparatethe
ΔL=0andΔL>0piecesofthe(d,2He)crosssectionsweperformedasyste-
maticstudytestingavarietyofp-shellresidualinteractionsusingtheOXBASH
code.Threedifferentinteractionswereemployed:CKPOTandCKIbyCohen
andKurath[81]andCKIHE,whichisalsobasedontheCohenandKurathinter-
actionandadjustedbyStevensonandBrown[82]todifferentHeisotopes.The
predictedGTtransitionstrengthsandthusthecorrespondingcharge-exchange
crosssectionsdifferwidely,butforagiventransitiontheshapesofthepartial
ΔL=0,2,4DWBAangulardistributionsareratherinsensitivetotheparticular
choiceoftheinteraction.Thus,thedecompositionofthecrosssectionsisde-
terminedbyafitallowingseparatevariationofaveragedΔL=0,2,4angular
distributions.ThentheexperimentaldatacanbedescribedwellandtheΔL=0
fractionatΘcm=0◦amountsto62%,68%and85%forthelevelsatEx=0.0,
1.45and2.9MeV,respectively.Resultsobtainedusinganyoftheinteractions
individuallyagreewithin5%.Theimpactofareducedisovectortensorforceas
suggestedin[77]hasalsobeeninvestigated.Again,theresultingΔL=0cross
sectionsvarylessthan5%.

calculationsQMCwithComparison5.3

Theabinitiocalculationsprovidearemarkablysuccessfuldescriptionofthepro-
pertiesoflightnucleiincludingthetransitionfromstablenucleitotheproton
andneutrondriplines.Theyalsoreproducethesingle-particlespectroscopicfac-
torofthe7Heg.s.deducedfromanR-matrixanalysisofthepresentdata(see
Section6and[18]).Calculationsfor7Li→7HeGTtransitionsareavailable[83]
usingavariationalMonteCarloapproachwhichpreciselyreproducesweakdecay
propertiesinA=6,7nuclei[84].BasicequationsoftheabinitioQMCcalcu-
lationscanbefoundinSection2.6.Thepredictionsareshownonthel.h.s.of
Tab.5.1.ThededucedB(GT)valuesaresummarizedinther.h.sofTab.5.1.

59

Tab.5.1:ComparisonofVMCmodelpredictionsandexperimentalexcitation
energiesandGTtransitionstrengthspopulatingthelowestresonances
7inHe.

VMCmodelExperiment
JπEx(MeV)B(GT)Ex(MeV)B(GT)
3/2−0.00.0039(1)0.00.0044(14)
1/2−2.9(3)0.0055(1)1.45−+00..570.0076(23)a
5/2−3.4(1)0.0110(2)2.9(1)0.0252(78)

a6Asp+ectroscopicfactorratioof1:3[85]isassumedforthepopulationofthe6He(g.s.)+n
andHe(2)+nchannels.

Theexperimentaluncertaintiesincludestatisticalandsystematicerrorsfromthe
unitcrosssectionnormalizationandthemodeldependenceoftheDWBAana-
lysis.TheexperimentalB(GT)valuefortheJπ=1/2−statecorrespondstoa
spectroscopicfactorratioof1:3forthequasifree6He(g.s.)+nand6He(2+)
+nchannelstakenfromashell-modelprediction[85].Goingfromoneextreme
(only6He(g.s.)+n)totheother(onlythe6He(2+)+n)inthespectrumdecom-
positiondescribedinSection4.6,theB(GT)strengthchangesfrom0.0056(17)
to0.0084(26).TheweaknessoftheGTtransitionsmayraisesomedoubtsabout
theapplicabilityoftheproportionalityassumptionbetweenβdecaymatrixele-
mentsand0◦charge-exchangecrosssections[86].However,thecomparisonwith
theVMCpredictionsinTab.5.1demonstratesaremarkableagreementbetween
experimentandtheory,notonlyfortheratioofthepossiblespin-orbitpartners
butalsofortheabsolutevalues.
Finally,acomparisonofexperimentalandtheoreticallypredictedexcitationener-
giesforthelow-energyresonancesin7Hecanbedone.Theexcitationenergyof
the1/2−statedependssensitivelyontheinclusionofathree-bodyinteraction.
TheVMCcalculationgivesEx=2.0MeV.Resultsforvariouscombinationsof
two-andthree-bodyinteractionsarepresentedinTableXIIofRef.[23]allowi-
ngforarangeofExvaluesbetween0.4and3.2MeV.Thecombinationofthe
Argonnev18nucleon-nucleonandIllinois-2three-nucleoninteractiongenerally
givesthebestoverallagreementforlightnuclei[22]andthecorrespondingvalues
60

Fig.5.2:Experimentalexcitationenergiesin7He(left)incomparisonwiththe

QMCcalculatedvalues(right).Hatchedarea:widthofthestates.

areincludedinTab.5.1.Acomparisonoftheexperimentallyestablishedand

theoreticallycalculatedexcitationenergiesforthelow-lyingstatesin7Hecanbe

seeninFig.5.2.Thepredictionforthe1/2−stateisabout1.5MeVhigherthan

theexperimentalfinding.Ofcourse,ifthe1/2−statehadanexcitationenergy

closetotheresonanceatEx=2.9MeVthesecouldnotbeseparatedinthe

presentexperiment,buttheexcessofcrosssectionatlowenergieswouldremain

unexplained.

61

6Spectroscopicfactorofthegroundstate

Spectroscopicfactorsarebasicquantitiescharacterizingthesingle-particlenature
ofnuclearexcitationsandthereforeserveasanimportanttestofwavefunctions
calculatedwithrecentlydevelopedabinitiomethods.Inparticular,predictions
fromGreen’sfunctionMonteCarlocalculations[23]areavailable[83]fortheneu-
tronspectroscopicfactorsSnofthelowestJπ=3/2−and1/2−statesin7He,
respectively:Sn(3/2−)=0.527(4)andSn(1/2−)=0.873(6).Itcanalsobecom-
paredtotheclassicalnuclearshell-modelcalculations.Thecorrespondingvalues
withintheCohenandKurathmodel[81]are0.591and0.685,respectively.The-
rearealsothefermionicmoleculardynamicsFMD[87]calculationspredicting
Sn(3/2−)=0.53.Usingthe(d,2He)reactionwithgoodenergyresolutiononecan
providenewinformationabouttheg.s.resonanceparametersandtheninthe
frameofanR-matrixanalysisobtaintheneutronspectroscopicfactor,assuming
thattheJπ=3/2−groundstateof7Hecanbedescribedas6He⊗ν1p3/2configu-
ration.Theanalysisconsistsoftwosteps:subtractionofbackgroundunderthe
resonanceanddeconvolutionoftheexperimentaldatabecauseofthefiniteenergy
resolution,andasingle-levelR-matrixanalysisofthedeconvolutedresonance.

olutionvDecon6.1

Asalreadymentionedabove,thegroundstateof7HeisunboundbyEr=0.445MeV
withrespecttothe6He+nthreshold[73].Thelow-energyregionuptoresonance
energyEr=2.5MeVisshowninFig.6.1.Thespectrumissummedoverthean-
gularregionΘcm=0◦−4◦inordertoobtaingoodstatistics.TheJπ=3/2−g.s.
resonanceclearlystandsout.However,theexperimentaldatashowabackground
contributionbelowtheresonancepeak.Thisneedstobesubtractedbeforeapp-
lyinganone-levelR-matrixanalysistoobtaintheobservedreducedwidthfrom
whichonecandeterminethespectroscopicfactor.

ThedecompositionofthespectrumhasbeenalreadydiscussedinSection4.6whe-
rethreedifferentanalyseshavebeenpresented[seeFigs.4.11(a)andFigs.4.14(b,c)].

62

Fig.6.1:Low-energypartofthe7Li(d,2He)spectrum.Dashedline:background
contribution.Solidline:groundstatefitfunctionandtotalfit.

ThedashedlineinFig.6.1isasumoftheresonancesandquasifreechannelscon-
tributionsexceptthegroundstate.Here,asanexample,onlyacasepresented
inFig.4.14(b)isshown.Itisnotimportantwhichanalysisisusedforthede-
terminationoftheratio–groundstateversusthe’rest’.Moreover,theresults
oftheanalysesdifferlessthan2%fromeachother.Itisclearthatthereisa
systematicalerrorinthefitting,inparticular,duetothemodellingofthe6He
+nquasifreescatteringandexperimentaluncertaintiesoftheknownresonances
(2.9MeVand5.8MeV).Investigatingtheexperimentallimitsoftheparameters
forbothresonancesthesystematicerrorisestimatedtobeoftheorderof10%.

TheexperimentalenergyresolutionofΔE150keV(FWHM)requiresade-
convolutionoftheg.s.resonancelineshape.Usingthespectrumasadiscrete
representationofthemeasuredlineshape,theproblemreducestosolvingasy-
equationslinearofstem

(6.1)

2m/=ku(Ei)=w(Ei−Ek)r(Ek)δE.(6.1)
2m/−=kHere,w(E)standsforthetruespectralshapeandr(E)fortheinstrumental
resolutionfunction.Theindexirunsfrom1tom+1,andδE=Em+1−Em.
Fornmeasureddatapointsm=n−1.Sincetheu(Ei)arenoisybecauseofthe
63

Fig.6.2:DeconvolutedexperimentalcrosssectionandfitwiththeR-matrixone-
levelresonance(solidline).Thedashedlineindicateszerocrosssection.
Widthofinstrumentalresolutionfunction(FWHM)is150keV.

experimentalerror,theresultingsystemoflinearequationsisunstable,leadingto
unpredictableerrorsinthesolutionsw(Em).Stabilizationmightbeachievedby
usingTikhonov’sregularizationmethod[88,89].Alternatively,Eq.(6.1)canbe
solvedbycalculatingtheFouriertransformofthebackgroundsubtractedfitted
spectraandapplyingtheFourierconvolutiontheorem.Sincebothmethodsare
linearprocedures,theresultsshouldbeidentical,andthiswasindeedobserved.
TheTikhonovregularizationmethodwaspreferredbecausetheinputdataare
thenoisymeasureddata,andtheoutputisasetofdeconvolutednoisydatawhich
enterdirectlytheR-matrixanalysiswithoutuseofinterpolatingfunctions.The
resultofthedeconvolutioncanbeseeninFig.6.2.Thefitwasperformedwith
thepackage’NonlinearFit’[90]ofMathematica5.1.

Fortheinstrumentalfunctionr(E)anormalizedGaussianwithawidthσde-
terminedbytheexperimenthasbeenused.Theexperimentaluncertaintiesare
dominatedbyvariationsoftheexperimentalresolutionΔEduringtheexperi-
ment,whichweremeasuredtobe±10keV.Forthispurposethe12C(d,2He)12B
measurementswereperformed.Theenergyresolutionwasdeterminedfromthe
12Bgroundstate.Theresultingtruewidthoftheg.s.resonanceof7Hecanbe

64

determinedfromthedeconvolutedspectrumwiththeresultΓ=183(22)keV
(FWHM).ThisvalueisconsiderablylargercomparingwithΓ=150(20)keV
[73].literaturethefrom

analysis-matrixR6.2

TheR-matrixanalysisprocedureisdescribedindetailsinSection2.7.Table
6.1summarizesresultsforEres,γ2sp,γ2obsandtheextractedspectroscopicfactor
Snatrepresentativevaluesofthechannelradiusrmatch.Resultsarepresented
fortheestimatedrangeoftheinstrumentalenergyresolutionFWHM=140-
160keV,andforthreedifferentmatchingradiirmatchinthevicinityoftheoptimal
one,rmatch=4.0fm.Thermatchvariedbetween3.5fmand4.5fminorderto
estimatethedependenceoftheresultsonthechoiceoftheWoods-Saxonpotential
parameters.Usingtheoptimalmatchingradiusandaveragingtheresultsforthe
differentestimatedexperimentalresolutionswithequalweight,oneobtainsas
theresultoftheanalysisforthe7HegroundstatespectroscopicfactorSn=
0.64±0.09whichisslightlylargerthantheGFMCandFMDpredictionsbut
agreeswithinuncertaintieswithashell-modelcalculationbasedontheCohen

Tab.6.1:ResultsoftheR-matrixanalysisofthedeconvolutedexperimentalcross
sectioninthereaction7Li(d,2He)7He:resonanceenergy(Eres),single-
particle(γ2sp)andobserved(γ2obs)reducedwidths,andneutronspectros-
copicfactor(Sn)forrepresentativevaluesofthechannelradiusrmatch.
Thequotederrorsresultfromtheexperimentaluncertainties.

rmatchEresγ2spγ2obsSn
(MeV)(MeV)(MeV)(fm)0.60(9)1.767(20)2.9510.4423.50.64(9)1.145(13)1.7480.4424.00.67(9)0.802(9)1.1870.4424.5

65

andteraction.inKurath

Thus,theresultsuggestsalarges.p.spectroscopic

Asaconsequence,the

single-particle

neutron

ground

state.

state

of

7

He

66

can

be

factorofthe

considered

7

as

Hegroundstate.

a

predominantly

7Continuumstructureathigherener-
gies

Clusteringinlightnucleihasanimportantinfluenceontheirstructure.Asa
possiblebuildingblockanα-particleisconsideredbecauseofitsstableand
inertbehaviourduetothestrongbindingoftwoprotonsandtwoneutrons.Ty-
picalexamplesare8Be,whichhasanα−αdinuclearmolecularstructureinthe
groundstateandlow-lyingexcitedstates,and12Cwhichhasbeenidentifiedasa
many-cluster3-αstructure.Therearealsosystemswhichcannotentirelydecom-
posedintoα-particlesubunits,forexample6,7Lipossesα+dandα+tcluster
structures,respectively.Theexperimentalsignaturesofclusterstateshavebeen
traditionallyinvestigatedbyselectiveexcitationinα-transferreactions.

Forthe7Li(d,2He)7Hereactionstudiedhere,itmeansthatthereareexcitations
possiblenotonlyduetothecharge-exchangereactionin7Liasawhole,but
alsoduetotheintrinsicexcitationsoftheclustersitself,inthepresentcasean
α-particleandtritium.Twocontributionsfromclustersinthe7Heexcitation
spectraarethusexpectedfromcharge-exchangequasifreereactions,occurring
eitheronaprotonofthe4Heorontheprotonofthetritium.Thiscorresponds
totheα+3nandt+t+nchannelsinFig.4.11,withthethresholdenergies
Ex=0.53MeVandEx=11.87MeV,respectively.UsingdatafromRefs.[74,75]
onecansuccessfullydescribetheexperimentalspectra(seeSection4.6).

Anotherpossibleexcitationoftheα-particlein7Liistheresonancestructureat
Ex=20.2−+01..29MeVexcitationenergywithawidthΓ=7.3−+00..36MeV(Fig.4.11)
whichwasalreadyobservedinothercharge-exchangereactions[70–72].Yamagata
etal.[71]searchedforanexcitationofanα-cluster,namely,theisovectorgiant
dipoleresonance(GDR)of4Hein7Libyusingthe7Li(p,p)reactionat300MeV
andtheanalogoftheGDRoftheα-clusterin7Hebythe7Li(7Li,7Be)reaction
at455MeV[72].Thedata,asmentionedabove,suggestthattheresonanceat
Ex≈20MeVisacandidatefortheGDRinanα-cluster.Theexcitationenergy
andthewidthoftheresonanceobservedinnuclearreactionslike(7Li,7Be),(n,p),
(d,2He)isverysimilartothosefortheGDRin4Heobservedinthe4He(γ,n)
data[91].Allreactionsmentionedaboveareselectiveforisovectorexcitations
withspin-transferΔS=0andΔS=1,whereasthe(d,2He)reactionhasapure

67

Fig.7.1:ExperimentalangulardistributionofthetransitiontothelevelatEx=
20.0MeVin7Hefromthepresentexperiment(fullcircles,errorbarsare
statisticalonly)incomparisonwithdata(opencircles)fromRef.[72].

spin-flip(ΔS=1)character.TheGDRcanbeexcitedviabothspin-flipandnon-
spin-fliptransitionsalthoughnon-spin-flippartsusuallydominate.Intherecent
workofRef.[72]itwasreportedthattheinvestigatedhigher-lyingresonancein
7Heisassignedtothedipoleresonancewithspin-transfercomponentsΔS=0
andΔS=1.Thecontributionofeachcomponentisfoundtobeapproxima-
telythesame.Thismeansthatinthe7Li(d,2He)7HeonlytheΔS=1partof
theGDRisobserved,becauseoftheselectivityofthereaction.InFig.7.1re-
markableagreementisobservedcomparingthedifferentialcrosssectionsofthe
Ex≈20MeVresonancestructurefrompresent(d,2He)datawiththeΔS=1
crosssectionsmeasuredinthe(7Li,7Be)reaction[72].Theangulardistributions
oftheprominentstructureatEx≈20MeV(fullcirclesinFig.7.1)isflatand
almostconstantoverthewholeangularrangeavailableinthepresentexperiment.
Noconclusionaboutthespinispossible,becausesuchanbehaviourcanbedes-
cribedwithanycombinationofthepartialΔL=1andΔL=3DWBAangular
distributionsforJπ=1/2+,3/2+,5/2+(assumingaGDRexcitation).

Tosummarize,thebroadstructureathighexcitationenergiesin7Heconsistsof
twoparts,bothattributedtoexcitationsintheα-cluster.Asalreadymentioned,
thefirstoneresultsfromthequasifreeknockoutreactionin4Hewhichleads

68

to

the

t

+

t

+

n

resonancesecond

resonance

[70–72]

hannelc

at

in

outab

the

α

75][74,

with

isMeV20

-

cluster.

a

troidcen

terpretedin

69

as

at

an

Ex

=

analog

16.4

of

MeV

the

and

tgian

the

oledip

okoutloandSummary8

Asearchforthep1/2spin-orbitpartnerofthe7Hegroundstatehasbeenperformed
utilizingthepropertiesofGTtransitionsselectivelyexcitedinthe7Li(d,2He)7He
reactionatzerodegrees.Thedatadonotsupportanarrow1/2−resonanceat
Ex=0.56(10)MeVasclaimedbyMeisteretal.[6],inagreementwithconclu-
sionsofRefs.[11,12].However,contraryto[11]thepresentresultssuggesta
resonancewithparametersEx=(1.45−+00..57)MeV,Γ=(2.0−+11..10)MeVpartially
overlappingwiththerangeofpossibleparametersdeducedin[12]andaswell
asthosein[14].Adecompositionofthespectrumisperformedtakingintoac-
countknownresonancesandquasifreecharge-exchangereactionson7Liaswell
asontritonand4Heclustersinthe7Ligroundstate.Asdiscussedindetailin
Chapter4,thisfindingdependssensitivelyonthemodellingofthe6He+nqua-
sifreescatteringcontributiontothespectra.Thechoiceoftheparametrization
describedin[61]isjustifiedbythegooddescriptionofananalogousmeasurement
ofthe6Li(d,2He)6Hereactioninakinematicalregimewherethequasifreecross
dominate.sectionsTheB(GT)strengthstotheloweststatesin7He,extractedfromthe0◦cross
sectionsafteradecompositionofthespectraincludingthisadditionalresonance,
areinexcellentagreementwithQuantumMonteCarlocalculations.Furthertests
oftheseresultsmaybeprovidedbystudiesemployingthe(d,p)reactionwitha
radioactive6Hebeam[10,15].Also,alternativetheoreticalapproacheslikethe
Gamowshellmodel[92–94]orfermionicmoleculardynamics[87]mayhelpto
clarifythequestionofthep-shellspin-orbitsplittingin7He.
Usingthepresentmeasurementofthe7Li(d,2He)7Hereactionwithgoodenergy
resolution,theneutronsingle-particlespectroscopicfactorSnofthe7Heground
statewasextractedbyanR-matrixanalysis.Thewidththatresultsfromthe
deconvolutionofthespectrumisΓ=183(22)keV(FWHM).Thespectroscopic
factorobtainedfromtheexperimentaldataisinagreementwithpredictionsfrom
CohenandKurathbutslightlylargerthanrecentabinitioGreen’sfunctionMonte
CarlocalculationsofPieperandWiringaaswellasthosefromfermionicmolecular
del.modynamics

Physicsofexoticnucleiisaveryintriguingbranchofmodernphysics.Therecent
70

developmentofradioactivebeamfacilitiesandtheoreticalmethodshasopened

new

The

and

perspectivesintheinvestigationof

ofexploration

hucm

remains

regionsextremethe

to

eb

ered.vdisco

eculiarpthe

of

71

the

mass

propertiesofdrip-linenuclei.

table

is

just

at

the

eginningb

PARTMeasuremenII:t
ofunderthe180◦DeuteronattheS-DElectroALINAdisinCtegration

troIn9duction

Primordialnucleosynthesis(Big-BangNucleosynthesis,BBN)providesatestof
cosmologicalmodelsandconstrainscosmologicalkeyparameters,suchasaverage
densityofmatterintheuniverse,baryondensity,numberoflightneutrinospecies
etc.[95].TheStandardBig-Bangmodelofcosmologyisthesimplest,basedon
theobservedlarge-scaleisotropyandhomogeneityoftheuniverse.Cosmological
modelscanbetestedbycomparingpredictedlightelementabundanceswith
observedabundances.Itisknown,thatprimordialnucleosynthesiscreatedonly
thefirstthreelightelements:hydrogen,heliumandlithium.Noelementsheavier
thanberylliumwereproducedbecausethereisnostablenucleuswith8nucleons,
sotherewasabottleneckinthenucleosynthesis,thatstoppedtheprocessthere.
Thesynthesisofthelightelements(D,3He,4Heand7Li)isdeterminedbyevents
occurringinthetimesfromt≈1tot≈1000sinthehistoryoftheuniverse
whentemperatureswentdownfromT≈1000KorhighertoT≈100K.A
smallpartofthereactionnetworkoftheprimordialnucleosynthesisisshown
inFig.9.1.Asisshown,thenucleosynthesischainbeginswiththeformation
ofdeuteriuminthep(n,γ)dreactionwhichisknowntocreatealldeuteriumin
theearlyuniverse.Thecrosssectionofthisprocessisnecessarytoestimatethe
productionyieldsofprimordiallightelements.Deuterium(D)abundanceprovides
directinformationonthebaryondensityintheearlyuniverse,whatmakesDa
perfect’baryometer’[96–98].However,deuteriumisfragileandisdestroyedin
starsevenbeforetheyreachthemainsequence.Thus,localmeasurements,where
probablyabout50%ofthematerialhasbeenthroughstars,donotdirectlyreflect
itsprimevalabundances.Theimportantsynthesis-reactionp(n,γ)dtakesplaceat
T<0.3MeV,whenthephotodestructionrateislowerthantheproductionrate.
Knowingaccuratelythen-pcapturecrosssectionandusingtheexperimental
valuefortheprimevaldeuteriumnumberdensitywouldallowforanaccurate
72

Fig.9.1:PartofthereactionchaininBig-BangNucleosynthesisfrom[95].

determinationofthebaryondensityΩBh2(histheHubbleconstantinunitsof
100kms−1Mpc−1).

Withthebaryondensityathandonecanpredicttheabundancesoftheother
threelightelements.Asanexample,Fig.9.2displaysthecontributionofvarious
primordialprocessestothetotaluncertaintyoftheprimordial7Liabundance.
Thelargestuncertaintyisassociatedwiththep(n,γ)dreaction.Then-pcapture
crosssectionshavebeenmeasuredatthermalenergies[99,100],at270keVin
thecenterofmass(c.m.)andabove,whereasintheBig-Bangenergyregion(20-
200keVn-pc.m.equivalentenergies)measurementshavenotbeenperformed,
becauseneutronbeamswithlowenergyspreadarenotavailable.

Then-pcapturecrosssectioncanalsobeinferredviadetailedbalancefromdeu-
teronphotodisintegrationexperimentsbyusingtheγ-dcrosssection.Theγ-d
crosssectioniseasiertomeasurewithhighaccuracythanthen-pprocessitself.
Thus,theexperimentalinformationobtainedformtheγ-dreactioninthethres-
holdenergyregionprovidesparametersusedinevaluationsofnucleosynthesis
intheearlyuniverse.Experimentally,theγ-dprocesshasbeenrecentlystudied
usingquasimonoenergeticgamma-raysproducedvialaser-Comptonbackscatte-
ring[101,102]forphotonanalyzingpowermeasurementsintheenergyrange
between2.39and4.05MeV.Theγd→npcrosssectionsmeasuredattheener-
giesrelevanttotheBBNinseveralexperimentalworks[103–107]aresummarized
inFig.9.3togetherwithrecenttheoreticalcalculations[108]withintheframe-
workofeffectivefieldtheory(EFT).Here,theresultofthetotaltheoreticalcross

73

Fig.9.2:Uncertaintiesinthepredicted7Liabundancefromtheindividualreac-
[96].Ref.fromtions;

sectionisshownassolidline,andalsoseparatelythecontributionsfromtheM1
(dashed)andE1(dotted)transitionamplitudes.Therealsoexiststheoreticalcal-
culationswithinthenucleon-nucleonpotentialmodel[109].Asonecanseefrom
theFig.9.3,therearenodataavailableattheenergiesbelow2.3MeV.Itis
alsoimportanttoemphasizethatthecontributionofthemagneticdipole(M1)
dominatesovertheelectricdipole(E1)inthethresholdenergyregion.

VeryrecentworkofNakayama[110]reportsaboutameasurementofthecharge-
exchange2H(7Li,7Be)reactionat0◦withanenergyresolutionof800keV
(FWHM)todeducethedistributionoftheB(M1)reducedtransitionstrength
forthephotodisintegrationofthedeuteron.However,itisnecessarytomeasure
theM1γ-dcrosssectioningreaterdetailwithgoodenergyresolutiontoimprove
theuncertaintiesintheBBNmodelpredictions.

Thefirstexcitedstate(Ex=2.2MeV)ofthedeuteronisunboundbyafew
hundredkeVonlyandisexcitedfromthegroundstatepredominantlythrough
anisovectorspin-flipM1transition.Previousphotodisintegrationexperiments
focusedontheangulardistributionsasamethodofextractingtherelativeE1
andM1contributions.However,thesedistributionsaremostsensitivetotheM1

74

Fig.9.3:Totalcrosssectionfortheγd→npprocess.Theexperimentaldataare
from[103]:opensquares,[104]:opencircles,[105]:filledsquares,[106]:fil-
ledcirclesand[107]:opentriangles.Dashedanddottedcurvesinthe
theoreticalEFTcalculations[108]representtheM1andE1contributi-
onstothetotalcrosssection(solidline),respectively.

contributionnear0◦and180◦,wherethemeasurementsaredifficult.Someearlier
electronscatteringworks[111,112]at180◦havebeenperformedtostudydeute-
ronmagneticdipoledesintegrationatlow-momentumtransfer.Theseresultsare
summarizedinFig.9.4.Thetrianglesandcirclesshowthe2Hbreakupspectrum
measuredwithanelectroninitialenergyof41.5MeV[111]and56.4MeV[112],
respectively.However,asonecanseefromthefigure,theresolutionwasrelatively
low(notbetterthan200keV).

Thesystemfor180◦electronscatteringatthesuperconductingDarmstadtelec-
tronlinearacceleratorS-DALINAC[113]isanidealplacetomeasuretheM1
breakupcrosssectionwithgoodenergyresolution.At180◦thetransversalcon-
tributiontotheelectronscatteringcrosssectionremainsfinite,whereasthelon-
gitudinalpartvanishes,greatlysuppressingthebackgroundandincreasingthe
sensitivityfortransversetransitions.Thus,electronscatteringat180◦servesas
a’spinfilter’sothatmagnetictransitions,whichareofpurelytransversenature,
arestronglyenhancedwhiletheelasticradiativetailislargelysuppressed[114].

75

Fig.9.4:Deuteronelectrodisintegrationspectraat180◦forE0=41.5MeV[111]

(triangles)andforE0=56.4MeV[112](circles).

Thesecondpartofthepresentworkisstructuredasfollows.Informationabout

experimentaltechniquesisprovidedinChapter10.Chapter11givesanoverview

ofthedataanalysis,presentstheexperimentalfindingsandacomparisonofthe

experimentalcrosssectionswiththeoreticalcalculations.InChapter12ashort

summaryandoutlookaregiven.

76

10Experimentalprocedure

CALINAS-D10.1

ThesuperconductingDarmstadtelectronlinearacceleratorS-DALINACisin
operationattheInstituteforNuclearPhysicsofDarmstadtUniversiyofTechno-
logysince1991[113].Itisusedforexperimentsonnuclearandradiationphysics
withenergiesbetween2.5and120MeV.Theelectronsareemittedfromacathode
andthenacceleratedelectrostaticallytoanenergyof250keV.Therequiredtime
structureoftheelectronbeamforradio-frequencyaccelerationina3GHzfieldis
preparedbyachopper/prebunchersystemoperatingatroomtemperature.The
superconductinginjectorlinacconsistsofone2-cell,one5-cell,andtwostandard
20-cellNiobiumstructures,cooledtoatemperatureof2Kbyliquidhelium.When
leavingtheinjector,thebeamhasanenergyupto10MeVandcanbeusedfor
nuclearresonancefluorescenceexperiments[115].Alternativelyitcanbebentby
180◦,andinjectedintothemainacceleratorsection.Thissuperconductinglinac
haseight20-cellcavitieswhichprovideanenergygainofupto40MeV.After
passingthroughthemainlinactheelectronbeammaybeextractedtowardsthe
experimentalhalloritcanbereinjectedtwiceintothemainlinacusingtwosepa-
ratedrecirculatingbeamtransportsystems.Afteraccelerationtheelectronbeam
isdeliveredtoseveralexperimentalfacilities,schematicallyshowninFig.10.1.
Awiderangeofelectronscatteringexperimentsiscarriedoutusingalargesolid
angleandmomentumacceptancemagneticspectrometerofQCLAMtype[116]
orwithahigh-resolutionmagneticspectrometerofenergy-losstype[117,118].

10.2QCLAMspectrometerand180◦facility

The180◦systemwasbroughtintooperationin1994[119,120].Comparedto
previous180-degreesystemsthepresentdeviceshowsanumberofexceptional
features:averylargemomentumacceptance;theabilitytoreconstructbothho-
rizontalandverticalcomponentsofthescatteringangleforeachevent,which

77

Fig.10.1:ThesuperconductingDarmstadtelectronlinearacceleratorS-DALI-
NACwithexperimentalfacilities.

allowsboththedefinitionofsolidangleandtheabilitytochecktheangularali-
gnment;alargesolidangleacceptancewhichmaybereducedbysoftwarecuts
inthedataanalysis.TheQCLAMspectrometer[116]consistsoftwoelements:
aclamshell-typedipolemagnetwithadeflectionangleof120◦andaquadrupole
magnetwhichprovidesadditionaltransversefocusingtoincreasethesolidan-
gleacceptance.Themagneticspectrometeriscoupledtoascatteringchamber
throughaslidingseal.Thisallowsonetovarytheanglebetween25◦and155◦
withoutbreakingthevacuum.Thus,theangulardistributionsoftheelectrons
measured.ebcan

Thepropertiesofthe180◦systemcanbesummarizedasfollows

ummaximtralcentummomen•momentumacceptance±10%

95MeV/c•horizontalopeningangle±60mrad

78

Fig.10.2:Schematicviewofthe180◦facility

•verticalopeningangle±40mrad
msr9.6anglesolid••intrinsicmomentumresolution2×10−4.

TheDarmstadt180◦systemisillustratedinFig.10.2.Thefirstpartisachicane
consistingofthreemagnetswithcircularpoles.Eachmagnetbendstheincident
beamby25◦.Themostimportantpartofthebeamtransportin180◦modeis
a’separationmagnet’,locatedinthecenterofthescatteringchamberbetween
targetandspectrometer.The’separationmagnet’deflectsthebeambacktoits
originaldirectionandontothetarget.Thetargetisshifted16.5cmdownstream
fromthecenterofthescatteringchamber.Backscatteredelectronsaredeflected
bythe’separationmagnet’toafiniteanglewherethespectrometerisplaced.After
traversingthetarget,thebeamtravelstoaFaradaycup,wheretheelectronbeam
currentisintegratedtodeterminethenumberofincidentelectrons.Inorderto
refocusthebeamaftermultiplescatteringinthetarget,aquadrupoledoublet
islocatedimmediatelydownstreamofthescatteringchamber.Thequadrupole
magnetsaremountedonrailssothattheirpositioncanbeeasilyadjusted.

79

Thefocal-planedetectionsystem[121]consistsofthreemulti-wiredriftchambers,
athinplasticscintillationcounterandaplexiglassCherenkovdetector.Twodrift
chambers,X1andX2,measurethepositioninthedispersivedirection(xdi-
rection)andintersectionangles.Thewiresinthethirdchamber(Ychamber)
arerotatedby26.6◦withrespecttothewiresinX1andX2sothatitisalso
possibletodetermineintersectionpointsinnondispersivedirection.Thesedata
allowacompletereconstructionofscatteringangleandexcitationenergy.The
scintillatorisservingasatriggerdetectorandaCherenkovdetectorisusedfor
suppression.kgroundbacthe

terimenExp10.3

Fig.10.3:ExamplesofrawCD2(e,e)spectrameasuredat180◦atE0=27.8MeV
(left)andE0=74MeV(right).

Forthemeasurementspolyethylenfoilsenrichedto98%D2withdifferentthick-
nessesbetween4.8to9.6mg/cm2wereused.TheelasticlinesofH,Dand12C
arewellseparatedinthespectrumbytheirrespectiverecoilenergies.Examp-
lesofrawspectraareshowninFig.10.3.Dataweretakenforelectronenergies
E0=27.8(1)MeVand74.0(1)MeVcorrespondingtomomentumtransfersq=
0.28fm−1and0.73fm−1.Thespectrometersettingcoveredanexcitationenergy

80

rangeEx≈0-5.0MeVatE0=27.8(1)MeVandEx≈0-12.5MeVat74MeV.
TheenergyresolutionwasΔE≈45keV(FWHM)atlowerelectronbeamenergy
and≈140keVat74MeVincidentelectronenergy.Itwaslimitedbythetarget
thicknessandenergyspreadofthebeam.Typicalbeamcurrentswere80-250nA.

Athighercurrents(200-250nA)aspecialtargetholder[122]wasused.Thiswas

necessarybecausethedeuteriumfoilsmayevaporateandbedestroyedrapidlyif

theygettoohotintheareaofthebeamspot.Thetargetmovedwithavelocityof

fewhundredrotationsperminutebothhorizontallyandverticallyreproducinga

Lissajousfigureonanareaof0.8×0.8cm2.Alsoa12Ctargetandanemptyframe
wereputintothebeamregularly.The12Cmeasurementservedfortheenergy

calibration.

trolled.con

With

the

y-frameempt

tmeasuremen

81

talinstrumen

kgroundbac

asw

discussionandResults11

11.1Analysisofthespectrum

Thefollowingdataanalysisdealsonlywithmeasurementsperformedwithin-
cidentelectronshavinganenergyof27.8MeV.Dataobtainedatanincident
electronenergyof73.5MeVwillbeanalyzedinfuture.

Anessentialpointforthesuccessofthepresentexperimentisthesuppression
oftheinstrumentalbackground.Forthispurposea20MHzpulsedbeamwas
employedtodistinguishbetweentheelectronsscatteredoffthetargetfromthose
ofbackgroundsources(e.g.theFaradaycuporslitsystems)bytime-of-flight
(TOF)measurements.Amacrostructurewithawidthofabout2nsandrepetition
rateof20MHzisimprintedontheusual3GHztimestructure.

Withthistechniquethesignal-to-backgroundratiointhemeasuredspectracould
beincreasedbyuptoanorderofmagnitude[123,124].Figure11.1presentsa

Fig.11.1:TimeofflightspectrumoftheCD2(e,e)measuredatE0=27.8MeV
◦180at.

samplespectrumofcountsversustimeofflight.Thepeakfromthetargetis

82

Fig.11.2:Two-dimensionaltime-of-flightspectrumandasoftwarecutappliedto
line).(soliddatathe

visiblearound40nsandcanbeidentifiedbyputtinganemptytargetintothe
beam.Anadditionalbroadpeakatabout30nsresultsfromdifferentbackground
sources.Individualsourcescannotberesolvedwiththepresenttimestructure
andresolution.Accordingtopreviousinvestigations[123,124],however,themain
contributionscomefromtheFaradaycup,thechicane,refocusingquadrupole
magnetsandtheslitsystem.Withanelectronicwindowonthetarget-related
events,thebackgroundisalreadyreducedconsiderably.

Furtherimprovementofthetimeresolutionisachievedbycorrectingforthepath-
lengthdifferencesthroughthespectrometer.Thistechniqueallowsonetomake
moreaccuratetimegateswithoutcontributionsfromnearesttothetargetback-
groundsources.Theyarestemmingfromelectronsscatteredoffthepolesofthe
separatingmagnetandfromthebackwallofthescatteringchambersurroun-
dingthebeamlineexit.Figure11.2showsatypicalhistogramoftimeofflight
versusexcitationenergy.Toproducethehistogram,gatesontheverticalandho-

83

rizontalanglesaretakingintoaccount,thecorrespondingprocedureisdescribed
furtherbelow.Makinghereatwo-dimensionalsoftwarecutonthetrueeventsthe
backgroundcouldbereducedbyafactorof5inthepresentexperiment.

Figure11.3depictsasanexampleaspectrumwithout(upper)andwith(lower)
softwarecutsonthecorrecttimeofflight.Onlyafractionofthebackground
remains,partlyduetotheradiativetailsoftheelasticlines;thisillustratesthe
advantageof180◦electronscattering.

Fig.11.3:SpectrumoftheCD2(e,e)reactionat180◦without(top)andwith
corrections.ttime-of-flighottom)(b

Theintersectionpointsandtheintersectionangleinthefocalplaneofthespectro-
meteraredefinedusingamethoddevelopedin[121,125].Therelativemomentum
deviationΔp/pcent,thehorizontal(ΘH)andtheverticalscatteringangle(ΘV)
canbedeterminedfromtheseparametersandareusedtoobtaintheexcitation
relationtheybenergy

pΔEx=E0−c∙pcentpcent−Trec−ΔETarget,(11.1)
whereE0isthebeamenergy,TrecistherecoilenergyandΔETargetistheenergy
lossinthetarget.Themomentumofelectronsonthereferencetrajectorythrough

84

thespectrometerpcentisdefinedbythemagneticfieldofthespectrometerdipole
magnet.DuringtheexperimentthefieldisreadoutwithaHallprobewitha
relativeaccuracyof1×10−4andiscontrolledbymeasuringthecurrentinthe
dipolemagnet.ThebeamenergyE0canbedeterminedfromtheaccelerator
settingswithanaccuracyofonly±500keV.Thus,E0isreconstructedfromthe
positionoftheelasticlineorprominentnuclearexcitationswithknownexcitation
ectrum.sptheinenergy

Inthe180◦modetheverticalscatteringangleisbasicallylimitedbythegap
oftheseparationmagnet.Thus,withinΔp/p=10%theverticalacceptance
isalmostindependentoftherelativemomentumdeviation.Intherightpart
ofFig.11.4countsarepresentedasafunctionoftheverticalangle.Thecross

Fig.11.4:Distributionofthereconstructedhorizontal(left)andvertical(right)
scatteringangleswithanexcitationenergylessthan100keVforthe
12CelasticlineinCD2.

section,whichisdominatedbytheelasticscatteringon12C,isminimalatan
angle180◦.Anexcitationenergycutappliedhereforthe12Celasticlinewasless
than100keV.Thedecreaseforverticalanglesmallerthan177◦andlargerthan
183◦showstheacceptancelimitsduetotheabovementionedfiniteapertureof
theseparationmagnet.Onlycountsinsidetheangularregion177◦-183◦were
usedintheanalysis.Theasymmetryappearsbecauseofacouplingofthevertical
angleacceptancetothehorizontalscatteringangleduetodispersiveproperties

85

Fig.11.5:SpectrumoftheCD2(e,e)reactionat180◦andE0=27.8MeV.The
spectrumisscaledbyafactorof1/4for12Celasticline.

ofthe’separatingmagnet’inthehorizontalplaneandduetoacircularshape
ofthespectrometeraperture.Aminorcontributionmightalsoariseifthebeam
passesnotexactlyperpendiculartothetargetand/oritspositionisnotproperly
adjusted.ThehorizontalangleΘHislimitedbytheapertureoftheQCLAM
spectrometerandbythesizeofthedetectorsystem.Thus,acceptancedepends
onthemomentumdeviationΔp/pcent.Therearealsoelectrictransitionsinthe
observedexcitationenergyregion,whosecrosssectionisverysensitivetothe
changesinthescatteringangle.Thus,asoftwarecutindependentontheΔp/pcent
wasappliedforΘHinordertoobtainaconstantsolid-angleacceptanceinthe
wholespectrumandafixedeffectivescatteringangle.Thehorizontalanglelike
theverticalanglewaslimitedtotheregion177◦-183◦resultinginasolidangle
acceptanceof9.47msr.Figure11.5displaystheCD2excitationenergyspectrum
includingallcutsandcorrections.Here,the12Celasticline,thedeuteronand
hydrogenelasticlines,andthebreakupofthedeuteronareobserved.

86

11.2Determinationofthecrosssections

Differentialcrosssectionsherecalculatedforthedeuterononly.Itcanbedeter-
minedabsolutelybythefollowingexpression
d2σAeNΔEx,ΔΩµb
dΩdEx=TNAQeffΔExΔΩsrMeV.(11.2)

Here

•N:numberofeventsintheexcitationenergybinΔExandthesolidangle
ΔΩacceptance•A:targetmass[g/mol]
•T:targetthickness[mg/cm2]
•NA:Avogadronumber[1/mol]
•e:electroncharge[C]
•Qeff:accumulatedchargecorrectedfordeadtime[C]
•ΔEx:energybin[MeV]
•ΔΩ:solidangleacceptance[sr].
UnderΘ=180◦elasticscatteringisstronglysuppressedfortargetswithground
stateJπ=0+.SincetheeffectivescatteringangleΘeffwasabout177.5◦intheex-
periment,the12Celasticlineisstillvisibleinthespectrum.Itmightthusbeused
forrelativenormalizationofthedeuteroninelasticscatteringcrosssection.Since
Θeffcan,however,bedeterminedonlywiththeuncertaintyofapproximately
0.1◦[123],thisleadstoasystematicerrorofmorethan10%intheextractedela-
sticcrosssection(e.g.in12C).However,thedeuterongroundstatehasJπ=1+.
Thismeansthattheelasticcrosssectionismainlyoftransversecharacterat
Θeff=177.5◦.Thus,aratherpreciserelativenormalizationoftheinelasticcross
sectionispossiblebyusingthedeuteronelasticline.Theelasticcrosssectionis
usuallydetermined[126]bytwostructurefunctionsA(q2)andB(q2)
ddσΩ=ddσΩMottA(q2)+B(q2)tan22Θ,(11.3)
87

Fig.11.6:ElasticdeuteronstructurefunctionsA(q2)(left)andB(q2)(right)as
functionsofthemomentumtransfer.Thesolidlinesshowafittothe
experimentaldata(circles)takenfrom[127].Thestarsrepresentthe
valuesextractedforthecalculationofthecrosssectioninthepresent
t.erimenexp

wheredσ/dΩMottistheMottcrosssectionandΘisthescatteringangle.The
structurefunctionsAandBarewellmeasured[127]forthemomentumtrans-
ferq2>0.25fm−2.Theextrapolationofthesedatatothepresentcase(q2
=0.078fm−2)isshowninFig.11.6.Thedifferencebetweentheabsolutelycal-
culatedcrosssectionsfromEq.(11.2)andthatdeterminedthroughtherelative
normalizationfromEq.(11.3)is6%only,wellwithintherespectiveerrorbars.

Becauseofthesmallmassoftheelectron,aseriousdisadvantageoftheelectron
scatteringisthegeneratedradiation.Itresultsinabroadeningofthelinesand
radiationtailswhichappearinanyspectralpeak.Thereisabigadvantageof180◦
scatteringthatgenerallytheintensitiesofthetailbackgroundsarecomparable
forelasticandinelasticpeaks.Thefirstoftheradiativecorrections,theSchwinger
correction,accountsforthelossofthepeakareaduetothoseelectronsdegraded
becauseoftheemissionofrealsoftphotonsaswellasemissionandabsorption
ofvirtualphotonsofanyenergy.Bremsstrahlungcorrectionstakeintoaccount

88

effectswhichcauseanasymmetricdistortionofthepeakduetosmallangle
scatteringfromelectronsandnucleiotherthanthescatteringnucleus.Thethird
effect,Landaustraggling,describesthebroadeningofthepeakduetothelosses
ofenergyfromatomicexcitationandionisation.Thus,thecrosssectionshould
bemultipliedbycorrectionfactors

dσdσexp1
dΩ=dΩexp(δS+δB)1−δI.(11.4)
ThevalueδSstandsfortheSchwingercorrection,δBforbremsstrahlungandδI
forionisationcorrections.Adetaileddescriptionofthesecorrectionscanbefound
in[126,128,129].At180◦theanomalousmagneticmomentoftheelectroncon-
tributestotheradiativecorrectiontoscatteringbyacharge.Thiseffectslightly
increasestheSchwingercorrection.AnexplicitformulafortheSchwingercorrec-
tiontobothchargeandmagneticscatteringwasgivenbyBorie[130]andused
forthepresentcalculation.Thetotalcorrectionfactorisaslargeas36%,where
thelargestcontributioncomesfromtheSchwingercorrection.

Themeasuredcrosssectionscontainnotonlystatisticalbutalsothefollowing
systematicerrorswhichcontributetothetotaluncertaintyinthedetermination
sections:crosstheof

•UncertaintyinthedeterminationoftheaccumulatedchargeintheFaraday
(5%)cup

•Errorinthedead-timecorrection(2%)

argetTinhomogenitInaccuracyinthey(5%)solidangle(7%).Theerrorsalthoughbeingsystematiconceweretreatedasindependentofeach
otherandthereforecanbeaddedquadraticallygivingthetotalsystematicerror
10%.ximatelyapproof

89

11.3Decompositionofthespectrum

AdecompositionoftheCD2spectrumintoindividualpeaksandbackground
(Fig.11.7)hasbeenperformedusingtheprogramFIT[69].Thelineshapeofthe

Fig.11.7:SpectrumoftheCD2(e,e)reactionat180◦andE0=27.8MeV.Solid
lines:adjustedmodelfunctionsforeachindividualpeakandresulting
fit.Dashedline:backgroundfunctiondeterminedbytheemptytarget
ts.measuremenframe

elasticlineshasbeendescribedbyaspecialfunction,whichtakesintoaccountthe
radiativetailaswellasaGaussianlineshapeduetothedetectorresponse[123].
exp[−C(x−x0)2/σ12]x<x0
y=y0∙exp[−C(x−x0)2/σ22]x0<x≤x0+ησ2(11.5)
A/(B+x−x0)γx>x0+ησ2.
Itconsistsofthreesmoothlyconnectedparts:aGaussianrisingflankofwidth
σ1,aGaussiandroppingflankofwidthσ2andahyperbolicfunctionsimulating
theradiativetail.Thesymbolx0denotestheenergyofthepeakmaximum,y0
isthevalueatthepeakmaximum,ηisthestartingpointoftheradiativetail
inunitsσ2andγistheexponentofthehyperbolicfunction.ThefactorsA,B
90

andCresultfromtheconditionofthesmoothlydifferentiableconnectionofthe
individualfunctionsattheinterconnectionpointsx0andx0+ησ2.Thesame
lineshape,butwithdifferentparameters,hasbeenappliedtofitthebreakup
ofthedeuteron.Theshapeandthemagnitudeoftheinstrumentalbackground
wasdeterminedbytheemptyframemeasurementsduringtheexperimentand
approximatedinthespectrumwithapolynomial.Alllinesandthebackground
havebeenfittedsimultaneously.

Discussion11.4

Inthenextstepoftheanalysistheelasticradiativetailsandtheinstrumental
backgroundhavebeensubtractedfromtheinelasticspectrum.Inspiteofgood
energyresolutionthespectrumofthedeuteronbreakupstillshouldbedeconvolu-
tedinordertoobtainthecorrectlineshape.Theinstrumentalresponsefunction
hasbeenexpectedtobeaGaussianfunctionwithawidthof45keV(FWHM).
ThedeconvolutionprocedurehasbeendescribedinChapter6.1.Alsointhiscase
itwasobserved,thatboth,FouriertransformandTikhonov’sregularizationme-
thodsfurnishidenticalresults.Thespectrumbeforeandafterthedeconvolution
isshowninFig.11.8.

Thecrosssectionswerethenmultipliedbytheradiativecorrectionfactorineach
energybinoftheinelasticcontinuumandcomparedwiththeoreticalcalculations
fortheelectrodisintegrationofthedeuteronbasedonaphenomenologicalnucleon-
nucleonpotential(Bonn-B)providedbyArenh¨ovel[131,132].Mesonexchange
currentsandisobarconfigurationsareincludedinthecalculations.Anexcellent
agreementoftheexperimentalmagneticdipoledisintegrationcrosssectionswith
theoreticalpredictionsisobserved,ascanbeseeninFig.11.9.

Therearealsoeffectivefieldtheory[134]calculationsofd(e,ep)ncoincidence
crosssectionsat180◦.Theresultsdescribedin[134]showaperfectagreement
withthepotentialmodelcalculationsinthesamekinematicalregime.Thus,one
canexpect,thatfuturecalculations[133,135]oftheinclusived(e,e)crosssections
mightalsogivegoodagreementwiththepresentexperimentaldata.

91

Fig.11.8:Solidline:deuteronelectrodisintegrationspectrumat180◦forE0
=27.8MeVwithanenergyresolutionΔE=45keVFWHM.Das-
hedline:deconvolutedspectrum.

Fig.11.9:Solidline:deconvoluteddeuteronspectrumat180◦forE0=27.8MeV
withexperimentaluncertainties(hatchedarea).Dashedline:theoreti-
calcalculationsbasedonphenomenologicalnucleon-nucleonpotential
[131].(Bonn-B)

92

Fig.11.10:Dashedline:deuteronelectrodisintegrationspectrumat180◦forE0
=27.8MeV.Solidline:excitationenergyspectrumofthe2H(d,2He)
reactionatΘ=0◦(seeRef.[136]forfurtherdetails).

Thecharge-exchange(d,2He)reactionusedinthefirstpartofthepresentthesis
tostudytheexotic7Henucleushasalsobeenutilized[136]toinvestigatethe2H
system.Takenfromthatworktheexcitationenergyspectrumofthe2H(d,2He)2n
reactionmeasuredatzero-degreeisshowninFig.11.10(solidline)togetherwith
the180◦deuteronelectrodisintegrationspectrumfromthepresentexperiment
(dashedline)normalizedontherespectivemaximaofthecrosssections.Excellent
agreementbetweentheirshapesisobserved.Thisisaquiteremarkablefinding
sincethesetworeactionsarebasedontotallydifferenttypesofinteraction.Itis
knownthatthe(d,2He)reactioninducesexclusivelyisovectorspin-flipexcitations.
Fromtheabovementionedagreementitbecomesevidentthatalsointhe2H(e,e)
reactionat180◦atlow-momentumtransfertheM1contributiondominatesinthe
section.crossbreakupmeasured

93

11.5Extractionoftheastrophysicalnp−→dγ
crosssection

Underthesimplifyingassumptionthatthedeuteronelectrodisintegrationcross
sectionmeasuredat180◦isexclusivelydescribedbyamagneticdipole(M1)
transition,onecanextracttheformfactor

22◦dσd(180Ω)=2eE2EE3Mq2[GM(q2)]2,(11.6)
1d1whereE1andE3arerespectivelytheenergiesoftheincidentandscatteredelec-
tron,GMisthedeuteronmagneticformfactor,Mddenotesthedeuteronmass,
andqisthefourmomentumtransfer.Fromthedefinitionofmagneticcrosssec-
tion[137]forscatteringatanangleΘandforamomentumtransferq

dσdσ2dσq2q2Θ
dΩ(Θ,q)=dΩMottVTFT(q)(11.7)
=dΩMott6Md2[GM(q2)]21+21+4Md2tan22.
Here,dσ/dΩMottistheMottcrosssectionandVTisthetransversalkinematic
factor.Fromthisexpressiononecanextracttheformfactor

2FT2(q)=q2[GM(q2)]2.(11.8)
M3dInordertoextrapolatetheformfactorobtainedintheelectronscatteringtothe
photonpoint,oneneedsalsothedatameasuredathigherenergy(E0=74MeV).
Usingthetwodatapointsfromthepresentexperimenttogetherwithdatafrom
theliterature[127],anextrapolationproceduresimilartothatdescribedinSec-
tion11.2canbeinvoked.TheB(M1)strengthiscalculatedfromthefollowing
equationB(M1,q)=92FT2(q).(11.9)
q2Sincetheabsorptioncrosssectionsareproportionaltotheradiativewidths,one
fromwidthstheobtaines

Γγ(↑)=0.01157Eγ3B(M1,↑),
94

(11.10)

with(2j0+1)
Γγ(↓)=(2jx+1)Γγ(↑).(11.11)
Forthedeuteroncasethespinofthegroundstatej0=1,andthespinofthe
investigatedexcitedstateatEx=2.2MeVisassumedtobejx=0.Thus,one
writecan

Γγ(↓)=3Γγ(↑).(11.12)
Fortheradiativecaptureofneutronstheabsorptioncrosssectioniswrittenas

(2jR+1)1
σ=πλ¯(2ja+1)(2jb+1)(E−ER)2+41Γ2ΓnΓγ,(11.13)
whereja=jb=1/2arethespinsofthenucleonsandjR=0isthespinofthe
resonance,andλ¯isdefinedas
Ma+Mb¯h
λ¯=Mb√2MaEL.(11.14)
InthepresentcasethevaluesMa,MbarethenucleonmassesandtheEListhe
kineticenergyinthelaboratoryframe.Atthecontinuumintheexcitationenergy
spectrumonecansetE=ERandΓisequaltothebinwidth.Onecanalsoset
Γ=Γn,becausethepredominantdecaymodeistheelasticchannel.Thecross
sectionthenlookslike

σ=πλ¯2(2j4a∙+(2j1)(2Rj+b1)+1)ΓγΓ(↓).(11.15)

Duringtheperiodofinterest(seeSection9)thenucleonsarenonrelativistic.
AssumingthatbothprotonandneutronaredistributedaccordingtotheMaxwell-
BoltzmanndistributionfortemperatureT,thenp→dγreactionrateisexpressed
intermsofthecrosssectionby

σv=8σ(E)exp(−E/kT)dE,
1/2
µπwhereµisthereducedmass.

95

(11.16)

outloandSummary12ok

Electronscatteringat180◦providesnumerousadvantagesforthestudyofnuclear
structure,inparticularmagneticexcitationsinnuclei.TheM1crosssectionofthe
deuteronbreakuphasbeenpreciselymeasuredclosetothresholdwithgoodenergy
resolutionofΔE=45keV(FWHM)usingthed(e,e)reactionat180◦forE0=
27.8MeV.DatawerealsotakenathigherelectronenergyE0=74MeVandwillbe
analyzedinthenearfuture.Fromthiselectrodisintegrationcrosssectiononecan
inferinafurtheranalysistheastrophysicallyrelevantcrosssectionfornp−→dγ
processwhichinturnprovidesinformationaboutabundancesoflightelements
inBig-Bangnucleosynthesis.Dataareinexcellentagreementwiththeoretical
calculationsbasedonaphenomenologicalnucleon-nucleonpotential(Bonn-B),
andinclusionofmeson-exchangecurrentsandisobarconfigurations.Itwouldbe
alsoofinteresttocomparethedatawitheffectivefieldtheorycalculationswhich
progress.inwnoare

Thed(e,e)dataobtainedclosetothethresholdcanmakeanimportantcontri-
butiontotheGerasimov-Drell-Hearn(GDH)sumrule.GDHconnectsanenergy
weightedintegralofspin-polarizedphotoabsorptioncrosssectionwiththean-
omalousmagneticmomentofthetarget[138].Sincethemagneticmomentofthe
deuteronisverysmall,theabsolutevalueoftheintegralisalsosmall.Athigher
energiesitcanbeestimatedbytheincoherentsumofquasifreeproductionfrom
theindividualnucleonsinthedeuteronthatmeansthatthiscontributionislarge
andpositive.Thiscanbecancelledbylargenegativecontributionatlowerener-
giesdominatedbythebreakuppeak.ItisexpectedthattheGDHsumruleis
dominatedbytheM1componentofthecrosssectionnearthebreakupthreshold.
Thismeansthatthemeasuredcrosssectionscanbedirectlyrelatedtothesum
[139].rule

96

A

the7Li(d,Double-differen2He)tialreactioncrosssectionsfor

Thedouble-differentialcrosssectionsforthe7Li(d,2He)reactionforvariousscat-

tering

table.

angleshavebeencalculatedinSection4.4andare

The

aluesv

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given

in

bµ[

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(sr

)].MeV

97

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theforsectionscrosstialDouble-differenA.1:ab.T7Li(d,
He)2

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±±◦4.70.412.68.919.494.245.623.624.717.721.517.624.817.525.0
◦6703700703030
.03.3..6.7.4.4.4.2.4.3.4.3.4
–4.7±11±±±±±±±±±±±
±±◦25.719.724.521.326.210.528.424.339.894.260.30.80

3.7◦333.44.270003000000.077.2.4.2.2.1.2.2.2.2.2.2
..–3.0±00±±±±±±±±±±±
±±◦2.00.464.14.834.275.728.422.613.121.319.718.520.723.029.2
◦2773033030000
.07.7..2.3.2.1.1.2.1.2.2.2.2
–2.0±0±0±±±±±±±±±±±±
◦1.00.383.56.048.776.425.217.317.518.217.322.422.326.227.3
33◦.00.0.3.7.3.7.7.7.7.7.7.3.
±7.245232222223
–1.00±±±±±±±±±±±±±
◦0.0-2.160.412.656.692.825.926.717.616.416.524.520.523.331.0
MeV,-0.5-0.3-0.20.00.20.30.50.60.80.91.11.21.4
-0.60xE

98

7angles.scatteringariousvforreactionHe
theforsectionscrosstialDouble-differenA.1:ab.T7Li(d,
He)2

◦–11.3◦10.3◦–10.3◦9.3◦–7.9◦6.9◦–6.9◦5.9◦–5.7◦4.7

◦–4.7

◦3.7◦–3.0◦2.0◦–2.0◦1.0

◦–1.0◦0.0MeV,xE

7.7.7.7.7.7.7.3.3.3.3.3.3.3.
00000001111111
±±±±±±±±±±±±±±
14.915.816.016.817.916.418.822.222.923.725.824.725.726.5
3.3.3.3.3.3.3.3.3.3.3.3.3.3.
11111111111111
±±±±±±±±±±±±±±
13.111.721.921.921.823.623.224.527.428.830.128.030.331.6
3.3.3.0.0.3.3.0.0.0.0.0..00.
11122112222222
±±±±±±±±±±±±±±
20.828.529.137.725.331.735.240.039.844.644.049.447.546.0
3.3.0.0.0.0.0..00.0.0.0.0.0.
11222222222222
±±±±±±±±±±±±±±
24.128.333.742.824.533.340.842.744.947.951.649.656.957.7
3.0..77.3.7.7.3.7.3.3.3.7.3.
34445445455545
±±±±±±±±±±±±±±
25.236.340.047.435.042.546.255.949.955.658.359.552.063.3

0.0.0.3.0.3.3.0.3.3.0.3.0.0.
44456556556566
±±±±±±±±±±±±±±
33.848.648.365.153.960.766.933.834.043.760.557.769.876.6

07773337333333.2.2.2.2.3.3.3.2.3.3.3.3.3.3
±±±±±±±±±±±±±±
28.934.732.939.915.952.354.950.459.160.561.859.066.463.3
0.20.27.27.20.47.27.27.27.27.27.27.27.23.3
±±±±±±±±±±±±±±
27.237.338.431.637.547.758.351.961.164.461.863.162.566.1

3.3.3.3.3.0.0.0.0.7.0.0.7.7.
33335444444444
±±±±±±±±±±±±±±
38.130.542.733.147.843.445.361.755.365.858.261.972.269.6

1.51.71.82.02.12.32.42.62.72.93.03.23.33.5

99

7ariousvforreactionHeangles.scattering
theforsectionscrosstialDouble-differenA.1:ab.T7Li(d,
He)2

◦–11.3◦10.3◦–10.3◦9.3◦–7.9◦6.9◦–6.9◦5.9◦–5.7◦4.7

◦–4.7

◦3.7◦–3.0◦2.0◦–2.0◦1.0

◦–1.0◦0.0MeV,xE

3.3.3.3.3.3.3.3.3.3.3.3.3.3.
11111111111111
±±±±±±±±±±±±±±
26.926.329.229.328.027.627.728.732.632.031.131.133.734.2

3.3.3.3.3.3.3.3.3.3.3.3.3.3.
11111111111111
±±±±±±±±±±±±±±
30.231.635.332.733.434.135.437.136.939.538.638.638.639.2
0.0.0.0.0.0.0.0.0.0.0.0.0.0.
22222222222222
±±±±±±±±±±±±±±
47.148.853.252.756.753.050.759.853.454.757.654.358.058.2
0.0.0.0.0.0.0.0.0.0..00.0.0.
22222222222222
±±±±±±±±±±±±±±
53.558.453.259.062.363.458.861.862.159.561.362.661.460.0
3.3.3.0.0..30.3.3.0.3.3.3.3.
55566565565555
±±±±±±±±±±±±±±
68.455.065.080.670.764.974.161.763.179.568.365.267.661.9

.30.0.0.0.0.7.7.7.0.7.0.0.7.
56666666666666
±±±±±±±±±±±±±±
62.066.475.884.483.886.475.973.276.878.191.279.576.787.5

33333333333333.3.3.3.3.3.3.3.3.3.3.3.3.3.3
±±±±±±±±±±±±±±
70.367.570.667.266.274.474.673.375.269.079.969.778.979.0
3.33.33.33.33.33.33.33.33.33.33.33.33.33.3
±±±±±±±±±±±±±±
73.668.973.969.069.969.874.879.772.280.777.974.683.780.1

7.7.7.7.7.7.7.7.7.7.7.7.7.7.
44444444444444
±±±±±±±±±±±±±±
74.971.374.368.380.779.674.172.178.272.282.975.073.587.0

3.63.83.94.14.24.44.54.74.85.05.15.35.45.6

100

7Heangles.scatteringariousvforreaction
TtheforsectionscrosstialDouble-differenA.1:ab.7Li(d,
He)2

◦–11.3◦10.3◦–10.3◦9.3◦–7.9◦6.9◦–6.9◦5.9◦–5.7◦4.7◦–4.7

◦3.7◦–3.0◦2.0◦–2.0◦1.0

◦–1.0◦0.0MeV,xE

3.3.3.3.3.3.3.3.3.3.3.3.3.3.
11111111111111
±±±±±±±±±±±±±±
33.334.234.436.940.936.835.437.938.439.436.037.839.738.9
3.3.3.3.0.3.3.3.3.3.0.0.3.0.
11112111112212
±±±±±±±±±±±±±±
40.436.440.341.844.543.144.744.145.341.647.446.544.245.8
0.0.0.0.0.0.0.0.0.0.0.0.0.0.
22222222222222
±±±±±±±±±±±±±±
55.257.661.161.854.462.361.862.761.361.661.062.860.262.9
0.0.0.0.7.7.7.7.0.0.0.0.0..7
22222222222222
±±±±±±±±±±±±±±
63.264.669.164.967.665.567.670.764.767.266.867.965.769.2
3.0.3.3.0.0.3.0..03.3.0.0.0.
56556656655666
±±±±±±±±±±±±±±
70.577.264.562.376.178.264.072.976.769.367.482.282.276.9
3.0.0.0..00.7.7.0.7.70.7.7.0.
666666666±6666
±±±±±±±±±±±±±
80.077.381.875.380.081.687.074.882.1102.075.082.482.778.9

33333303030030.3.3.3.3.3.3.4.3.4.3.4.4.3.4
±±±±±±±±±±±±±±
74.375.076.576.174.275.477.671.380.667.979.487.171.682.9
3.33.33.33.33.33.33.33.33.33.33.33.33.33.3
±±±±±±±±±±±±±±
74.277.971.972.978.278.082.580.079.682.185.677.685.275.3

7.7.7.7.7.7.7.7.7.7.7.7.7.7.
44444444444444
±±±±±±±±±±±±±±
79.080.370.882.382.384.775.778.486.279.379.279.171.772.0

5.75.96.06.26.36.56.66.86.97.17.27.47.57.7

101

7angles.scatteringariousvforreactionHe
crosstialDouble-differenA.1:ab.Ttheforsections7Li(d,
He)2

◦–11.3◦10.3◦–10.3◦9.3◦–7.9◦6.9◦–6.9◦5.9◦–5.7◦4.7

◦–4.7

◦3.7◦–3.0◦2.0◦–2.0◦1.0

◦–1.0◦0.0MeV,xE

3.3.3.3.3.3.3.3.3.3.3.3.3.3.
11111111111111
±±±±±±±±±±±±±±
41.743.140.040.441.840.740.941.038.041.441.540.442.439.0

0.0.0.0.3.3.3.3.3.3.3.0.0.0.
22221111111222
±±±±±±±±±±±±±±
46.945.948.747.843.844.344.042.342.143.041.843.344.743.7
0.20.20.20.20.20.20.20.20.20.20.20.20.20.2
±±±±±±±±±±±±±±
60.160.161.759.063.961.160.560.560.657.560.659.060.657.9
7.20.20.20.27.27.20.20.27.27.27.27.27.27.2
±±±±±±±±±±±±±±
68.766.367.366.768.271.462.863.866.566.564.863.963.263.4
3.0.3.0.0.0.0.0.0.0.0..07.0.
56566666666666
±±±±±±±±±±±±±±
62.273.367.173.674.277.275.576.481.378.478.367.882.874.3

7.7.0.0.0.0..70.7.0.0.7.7.7.
66666666666666
±±±±±±±±±±±±±±
76.077.791.769.169.984.985.562.378.985.375.276.974.469.5

330303033330003..3.4.3.4.3.4.3.3.3.3.4.4.4
±±±±±±±±±±±±±±
69.875.587.472.878.973.174.970.372.468.768.777.170.568.9
3.33.33.33.33.33.33.33.33.33.33.33.33.33.3
±±±±±±±±±±±±±±
80.380.379.675.380.673.681.475.081.579.179.077.177.574.2

7.7.7.7.3.7.7.7.7.7.7.7.7.7.
44445444444444
±±±±±±±±±±±±±±
76.677.277.877.686.380.078.778.580.278.871.970.377.278.3

7.88.08.18.38.48.68.78.99.09.29.39.59.69.8

102

7vforreactionHeangles.scatteringarious
theforsectionscrosstialDouble-differenA.1:ab.T7Li(d,
He)2

◦–11.3◦10.3◦–10.3◦9.3◦–7.9◦6.9◦–6.9◦5.9◦–5.7◦4.7

◦–4.7

◦3.7◦–3.0◦2.0◦–2.0◦1.0

◦–1.0◦0.0MeV,xE

3.3.3.3.3.3.3.3.3.3.3.3.3.3.
11111111111111
±±±±±±±±±±±±±±
39.442.236.037.840.640.940.639.839.540.537.639.537.440.1
0.0.0.0.0.3.0.0.0.0.0.0.0.0.
22222122222222
±±±±±±±±±±±±±±
47.545.645.746.345.739.945.244.544.045.542.046.345.239.5
0.0.0.0.0.0.0.0.0.0.0.0.0.0.
22222222222222
±±±±±±±±±±±±±±
56.960.162.656.961.062.456.257.761.058.255.057.356.557.0
7.7.7.7.7.7.0.0.7.7.7.7.7.7.
22222222222222
±±±±±±±±±±±±±±
62.262.964.167.062.660.958.758.259.960.264.664.360.464.9
0.0.7.0.0.0.0.0.7.0.0.3.0.0.
66666666666566
±±±±±±±±±±±±±±
74.465.087.277.664.163.177.362.680.170.470.153.170.966.8

7.0.0.0.7.0.7.0.7..77.7.0.3.
66666666666667
±±±±±±±±±±±±±±
76.071.264.969.772.167.069.172.374.683.685.667.488.574.7

00000000000000.4.4.44..4.4.4.4.4.4.4.4.4.4
±±±±±±±±±±±±±±
68.067.666.571.776.867.281.578.776.568.764.971.975.968.8
3.33.33.33.33.33.33.33.33.33.30.40.43.33.3
±±±±±±±±±±±±±±
75.672.678.576.869.272.676.578.576.873.880.178.765.075.0

7.3.3.3.3.7.7.3.7.7.3.7.3.3.
45555445445455
±±±±±±±±±±±±±±
70.080.784.077.278.270.472.378.272.766.572.668.773.072.7

9.910.110.210.410.510.710.811.011.111.311.411.611.711.9

103

7angles.scatteringariousvforreactionHe
theforsectionscrosstialDouble-differenA.1:ab.T7Li(d,
He)2

◦–11.3◦10.3◦–10.3◦9.3◦–7.9◦6.9◦–6.9◦5.9◦–5.7◦4.7

◦–4.7

◦3.7◦–3.0◦2.0◦–2.0◦1.0

◦–1.0◦0.0MeV,xE

3.3.3.3.3.3.3.3.3.3.3.3.3.3.
11111111111111
±±±±±±±±±±±±±±
38.537.440.140.038.138.138.237.434.834.335.739.440.037.9

3.0.0.0.0.0.0.0.0.0.0.0.0.0.
12222222222222
±±±±±±±±±±±±±±
38.540.144.240.842.841.342.639.337.339.838.836.938.336.5
0.20.20.20.20.20.20.20.27.20.20.20.27.27.2
±±±±±±±±±±±±±±
58.055.657.556.558.955.755.952.859.755.251.654.357.054.9
7.27.27.27.27.27.27.27.27.27.27.27.27.227.
±±±±±±±±±±±±±±
55.060.260.855.057.956.259.256.059.856.956.051.958.356.4
0.0.0.0.0.7.0.0.7.0.3.7.7.0.
66666666665666
±±±±±±±±±±±±±±
68.467.966.665.066.577.754.659.769.952.847.366.867.761.3

7.0.0.7.7.7.7.7.0.0.7.3..73.
66666666666767
±±±±±±±±±±±±±±
72.068.053.573.375.668.365.478.774.661.168.061.161.972.9

00000000000007.4.4.4.4.4.44..4.4.4.4.4.4.4
±±±±±±±±±±±±±±
75.864.064.066.163.357.871.062.066.564.764.361.062.877.5
3.33.33.30.43.33.33.33.33.30.40.40.43.33.3
±±±±±±±±±±±±±±
73.568.265.272.769.062.162.666.564.067.570.171.862.862.2

7.3.7.3.3.7.3.3.3.3.3.3.3.3.
45455455555555
±±±±±±±±±±±±±±
69.279.863.369.371.061.975.168.566.769.471.569.267.771.8

12.012.212.312.512.612.812.913.113.213.413.513.713.814.0

104

7angles.scatteringariousvforreactionHe
theforsectionscrosstialDouble-differenA.1:ab.T7Li(d,
He)2

◦–11.3◦10.3◦–10.3◦9.3◦–7.9◦6.9◦–6.9◦5.9◦–5.7◦4.7

◦–4.7

◦3.7◦–3.0◦2.0◦–2.0◦1.0

◦–1.0◦0.0MeV,xE

3.3.3.3.3.3.3.0.0.0.0.0.0.0.
11111112222222
±±±±±±±±±±±±±±
36.637.835.641.837.239.941.243.647.242.247.048.446.649.4
0.0.0.0.0.0.0.0.0.0.0.0.0.0.
22222222222222
±±±±±±±±±±±±±±
41.039.441.841.641.741.850.249.948.951.554.048.652.948.7
7.7.7.7.7.7.7.7.7.7.7.7.7.3.
22222222222223
±±±±±±±±±±±±±±
52.453.354.261.060.261.059.762.768.369.165.365.565.877.6
7.7.7.7.7.7.7.7.3.3.3.3.3.3.
22222222333333
±±±±±±±±±±±±±±
51.453.156.455.661.258.761.463.771.269.670.973.376.071.6
7.7.7.0.0.7.7.3.3.0.3.0.0.7.
66666667787888
±±±±±±±±±±±±±±
71.163.563.555.451.263.466.476.872.090.472.087.882.590.8

7.7.0.7.3.3.3.7.0.0.3.0.3.0.
66667778887878
±±±±±±±±±±±±±±
51.169.666.386.465.665.663.562.766.590.077.171.775.562.1

07777377733033.4.4.4.4.4.5.4.4.45..5.6.5.5
±±±±±±±±±±±±±±
63.866.074.166.565.480.671.271.073.079.885.195.387.875.7
0.40.40.40.40.47.40.47.47.47.47.47.47.47.4
±±±±±±±±±±±±±±
70.374.070.868.475.981.177.581.377.385.485.680.589.991.4

.33.3.0.3.0.0.7.0.0.0.7.7.7.
55565666666666
±±±±±±±±±±±±±±
70.761.167.777.868.372.474.487.380.080.779.184.987.990.5

14.114.314.414.614.714.915.015.215.315.515.615.815.916.1

105

7forreactionHeangles.scatteringariousv
A.1:ab.TtheforsectionscrosstialDouble-differen7Li(d,
He)2

◦0.0.0.0.0.0.0.0.0.0.0.0.0.0.
◦–11.3±±±±±±±±±±±±±±
22222222222222
10.351.647.450.252.351.556.451.861.055.357.557.256.856.855.2
◦7.7.7.7.7.7.7.7.7.7.7.7.7.7.
22222222222222
–10.3±±±±±±±±±±±±±±
◦9.353.857.255.459.157.852.159.558.761.063.356.158.858.057.4
◦3.3.3.3.3.3.3.3.3.3.3.3.3.3.
33333333333333
–7.9±±±±±±±±±±±±±±
◦6.970.772.668.677.675.573.572.071.478.673.069.868.273.472.6
◦3.3.3.3.3.3.3.3.0.0.0.0.0.0.
33333333444444
–6.9±±±±±±±±±±±±±±
◦5.969.068.574.277.575.276.071.776.488.884.485.176.171.373.7
0◦3.3.7.0.0.7.7.0.3.7.7.7.3..
778888889888910
–5.7±±±±±±±±±±±±±±
◦4.771.570.784.780.875.381.382.072.787.982.879.171.589.696.4
000◦7.0.3.3.7.7.7.7.3.7....3.
88798888981010109
–4.7±±±±±±±±±±±±±±
77.086.872.174.384.282.191.161.677.885.478.194.894.493.7

◦3.7◦30000000703.7773.7.5.6.6.6.6.6.6.6.6.6±.66.±
–3.0±±±±±±±±±±±±
◦2.078.386.481.990.592.289.189.990.296.582.5107.690.388.8113.6
◦7.43.53.53.53.53.53.53.53.50.63.53.53.50.6
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◦1.086.789.487.789.799.782.597.093.791.299.793.190.886.596.7

0.◦7.7.7..783.7.3.3.3.0.0.0.7.
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◦0.084.384.976.279.4105.696.081.487.794.689.696.698.790.469.5
MeV,x16.216.416.516.716.817.017.117.317.417.617.717.918.018.2
E

106

7angles.scatteringariousvforreactionHe
forsectionscrosstialDouble-differenA.1:ab.Tthe7Li(d,
He)2

◦◦–11.3±±±±±±±±±±±±±±
7.20.20.27.27.27.27.27.27.27.27.27.27.27.2
10.357.455.254.354.952.861.253.458.655.956.358.060.452.058.5
◦7.7.7.7.3.3.3.3.3.3.3.3.3.3.
–10.32±2±2±2±3±3±3±3±3±3±3±3±3±3±
◦9.360.161.557.060.462.160.266.260.363.661.259.561.067.265.2
◦3.3.3.0.0.0.0.0.0.0.0.0.7.0.
3±3±3±4±4±4±4±4±4±4±4±4±4±4±
–7.9◦6.969.167.871.573.373.478.973.878.465.471.774.975.381.773.8
◦0.0.0.0.0.0.0.0.7.0.7.7.7.7.
44444444444444
–6.9±±±±±±±±±±±±±±
◦5.980.478.474.869.070.372.378.676.078.873.577.482.571.380.7
◦7.3.0.3.0.7.0.7.0.7.3.3.0.7.
898988881010991010
–5.7±±±±±±±±±±±±±±
◦4.773.986.656.876.058.969.253.963.382.886.571.265.474.781.6
0.◦0.0.7.0.70.3.123.7.7.33.3
10101010.81011±111010.911.9
–4.7±±±±±±±±±±±±±
◦3.786.488.094.581.762.481.899.8106.598.080.884.360.889.863.7

◦333733330.803007.7.7.7.6.7.7.7.7±.8.7.8.8.8
–3.0±±±±±±±±±±±±±
◦2.093.591.290.185.089.592.489.591.3101.494.578.786.190.795.5
70..◦6±0.60.66±0.60.67.67.60.67.67.60.67.67.6
–2.0±±±±±±±±±±±±
◦1.0103.198.388.1105.296.580.796.392.686.689.490.179.691.790.9

7.3.3.3.
◦0.3.0.890..097.7.7.97.3.
878±±88±888±89
–1.0±±±±±±±±±±
◦0.089.682.286.6102.0108.889.583.5110.999.190.783.2101.787.495.4
MeV,x18.318.518.618.818.919.119.219.419.519.719.820.020.120.3
E

107

7scatteringariousvforreactionHeangles.
theforsectionscrosstialDouble-differenA.1:ab.T7Li(d,
He)2

◦◦–11.3±±±±±±±±±±±±±±
7.27.27.27.27.27.23.37.27.27.23.333.3.33.3
10.359.658.352.760.061.351.461.356.056.051.761.654.058.554.7
◦3.3.0.3.0.3.3.0.0.0.0.0.0.0.
–10.33±3±4±3±4±3±3±4±4±4±4±4±4±4±
◦9.357.967.066.261.064.950.656.558.857.159.959.755.251.652.5
◦0.0.0.0.7.0.7.7.7.7.7.7.7.7.
4±4±4±4±4±4±4±4±4±4±4±4±4±4±
–7.9◦6.971.569.567.661.573.167.371.470.069.377.962.165.165.361.0
◦7.7.7.7.7.7.7.3.7.3.7.3.3.3.
44444445454555
–6.9±±±±±±±±±±±±±±
◦5.972.973.783.068.678.469.173.887.268.369.363.274.472.874.1
◦0.7.3.0.37.7.3.7.70.0.3.7.
10101110.910101110.810121110
–5.7±±±±±±±±±±±±±±
◦4.765.375.187.769.857.765.867.374.366.542.251.274.467.359.9
◦7.70.0.3.3.7.7.7.0.0.0.0.7.
10.8101211111010121014121212
–4.7±±±±±±±±±±±±±±
◦3.780.251.765.191.676.275.466.961.888.151.799.471.267.876.3

◦7000073770.770.0.8.8.8.8.8.8.9.8.810.8.810.8
–3.0±±±±±±±±±±±±±±
◦2.089.081.175.077.677.076.291.179.673.990.873.270.685.056.3
◦7.7.3.7.7.3.7.3.3.3.3.3.0.3.
66766767777787
–2.0±±±±±±±±±±±±±±
◦1.087.890.693.985.675.686.175.681.785.781.173.771.785.175.2
7.◦7730.3377.30.0.0.0.
.8.8.910±10.9.9.810.910101010
–1.0±±±±±±±±±±±±±
◦0.080.479.790.5111.697.677.577.364.197.572.780.982.077.979.1
MeV,x20.420.620.720.921.021.221.321.521.621.821.922.122.222.4
E

108

7reactionHeangles.scatteringariousvfor
theforsectionscrosstialDouble-differenA.1:ab.T7Li(d,
He)2

◦◦–11.3±±±±±±±±±±±±±±
3.33.33.33.33.33.33.33.333.3.33.33.33.30.4
10.362.251.853.851.949.250.554.843.354.047.945.542.048.849.8
◦.00.0.0.0.0.7.0.7.0.0.7.7.7.
–10.34±4±4±4±4±4±4±4±4±4±4±4±4±4±
◦9.364.254.545.659.655.350.757.643.451.545.941.750.849.643.7
◦7.7.7.7.3.3.3.3.3.3.3.3.7.0.
4±4±4±4±5±5±5±5±5±5±5±5±4±6±
–7.9◦6.965.260.165.560.467.274.065.959.856.956.557.456.447.561.9
◦3.0.3.0.3.3.3.0.0.3.3.0.3.3.
56565556655655
–6.9±±±±±±±±±±±±±±
◦5.972.974.668.879.061.451.456.868.263.656.455.054.550.344.5
◦0.3.3.0.0.0.7.3.7.3.7.30.7.
1011111212121013101112.91212
–5.7±±±±±±±±±±±±±±
◦4.746.060.362.662.762.760.448.374.844.548.858.733.148.656.2
◦0.7.3.7.7.7.7.3.0.3.0.0.3.3.
1210131412121411121314121113
–4.7±±±±±±±±±±±±±±
◦3.766.251.075.293.966.263.685.852.551.767.169.351.444.157.8

◦0.3770.07.0.033.7.0.7.......
10±9±8±8±10±8±10±10±8±9±11±12±12±10±
–3.0◦2.079.365.459.251.068.745.074.066.341.451.868.886.477.154.8
◦0.87.87.80.80.83.70.83.73.77.87.80.87.80.8
–2.0±±±±±±±±±±±±±±
◦1.077.190.087.270.673.263.670.355.858.877.169.561.061.955.1

◦0.3.7.70.0.0.0.3.30.0..73.
..10±11±10±8±10±10±10±10±11±9±±1210±10±11±
–1.0◦0.070.990.780.950.266.865.963.160.372.352.478.952.261.765.8
MeV,x22.522.722.823.023.123.323.423.623.723.924.024.224.324.5
E

109

7angles.scatteringariousvforreactionHe
theforsectionscrosstialDouble-differenA.1:ab.T7Li(d,
He)2

◦–11.3◦10.3

◦–10.3◦9.3

◦–7.9◦6.9

◦–6.9◦5.9

◦–5.7◦4.7

◦–4.7

◦3.7◦–3.0◦2.0

◦–2.0◦1.0

◦–1.0◦0.0

MeV,xE

0.4±48.3

0.4±40.7

3.5±52.4

0.6±50.0

3.13±58.1

7.10±

34.40.12±67.2

3.9±67.9

7.8±36.4

24.6

0.4±46.7

7.4±41.7

3.5±54.1

0.6±52.4

7.10±35.0

7.16±84.3

3.11±59.7

7.8±58.9

0.12±68.5

24.8

3.3±41.8

0.4±32.1

3.5±44.7

0.6±53.4

7.12±49.0

7.12±49.9

0.10±46.5

7.8±55.0

3.11±58.1

24.9

110

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Danksagung

Alljenen,diezumEntstehenundGelingendieserArbeitbeigetragenhaben,
m¨ochteichandieserStellemeinenDankaussprechen.

Zuallererstm¨ochteichmeinemDoktorvaterHerrnProfessorDr.Dr.h.c.mult.
AchimRichterdankenf¨urdieM¨oglichkeit,dieseinteressanteArbeitinseiner
ArbeitsgruppeanderTechnischenUniversit¨atDarmstadtdurchf¨uhrenzuk¨onnen,
insbesonderef¨urdieUnterst¨utzungw¨ahrendalldieserJahre.Erhatmirzwei
sehrfaszinierendeundanspruchsvolleAufgabengestellt.Durchseinewertvolle
BeratungundMotivationhatermichzurerfolgreicheL¨osungdieserAufgaben
st¨andiggef¨uhrt.

HerrnProfesseorDr.JochenWambachdankeichf¨urdiefreundliche¨Ubernahme
Korreferates.des

MeinbesondererDankgiltHerrnProfessorDr.PetervonNeumann-Cosel,der
michbetreuthatunddenFortgangdieserArbeitbeijederGelegenheitun-
terst¨utzt.Auchf¨urseingroßesInteresseundvielefruchtbareDiskussionenund
Anregungendankeichihmsehrherzlich.

HerrnProfessorDr.FrederBeckm¨ochteichf¨urdieR-MatrixAnalyseder7Li(d,2He)
Datendanken.ErhatdurchseineausgezeichneteZusammenarbeitundDiskus-
sionsbereitschaftsehrzumGelingendieserArbeitbeigetragen.

MeinDankgiltHerrnProfessorDr.HartmuthArenh¨ovelf¨urdieZurverf¨ugung-
stellungtheoretischerRechnungen.

BeiHerrnProfessorDr.JoachimEndersm¨ochteichmichf¨urdasgr¨undliche
KorrekturlesenunddiezahlreichenRatschl¨agebedanken.

ManythanksgotoProfessorDr.CharyRangacharyuluforfruitfuldiscussions
andhispermanentinterestintheprogressofthed(e,e)dataanalysis.

IwouldliketoexpressmygratitudetoDr.LeonidChulkovforhispermanent
helpintheinterpretationofthe7Li(d,2He)7Hedataaswellasforsharinghis
tremendousknowledgeandabilityinphysicsofexoticnuclei.Itwasagreat
pleasureformetoworktogetherwithhimtheselasttwoyears.

HerrnDr.HaraldGenzdankeichbesonderesf¨urseinegroßartigeHilfeundUn-
terst¨utzunginallenFragenundProblemen.
BeiallenMitarbeiternderEUROSUPERNOVA-Kollaborationm¨ochteichmich
f¨urdieHilfebeiderDurchf¨uhrungdesExperimentesamKVIundf¨urdiegute
Zusammenarbeitbedanken.F¨urdieHilfebeiderDatenanalysedankeichinsbe-
sondereHerrnDr.SvenRakersundFrauDipl.-Phys.PetraHaefner.

BeimeinenKollegenundFreundenausderQCLAM-Gruppebedankeichmichf¨ur
dieTeilnahmeandenMessschichten.DerFrauIrynaPoltoratskaunddenHerren
AnatoliyByelikovundMaksymChernykhm¨ochteichmeinenDankaussprechen
f¨urihrenEinsatzbeiderVorbereitungdesExperimentsamS-DALINAC.Mein
besondererDankf¨urdiefreundlicheUnterst¨utzungundenormeHilfew¨ahrend
derVorbereitungundDurchf¨uchrungderbeidenExperimentegehtandieHerren
Dr.YaroslavKalmykovundDr.ArtemShevchenko.

AllenMietgliedernderBeschleunigergruppeamS-DALINACdankeichf¨urdie
BereitstellungeinesexzellentenStrahlsundwertvolleHilfew¨ahrenddesExperi-
ts.men

MeineFamiliem¨ochteichherzlichdanken,diemichimmerunterst¨uzthat,wenn
auchmeistensausgr¨oßergeographischeDistanz!
DievorliegendeArbeitwurdegef¨ordertdurchMittelausdemDFGGraduierten-
kolleg410”PhysikundTechnikvonBeschleunigern”unddemSonderforschungs-
bereichSFB634”Kernstruktur,nukleareAstrophysikundfundamentaleExperi-
mentebeikleinenImpuls¨ubertr¨agenamsupraleitendenDarmst¨adterElektronen-
beschleunigerS-DALINAC”.

123

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