EFFECTSOFNONLOCALITYONTRANSFERREACTIONS By LukeTitus ADISSERTATION Submittedto MichiganStateUniversity inpartialentoftherequirements forthedegreeof Physics{DoctorofPhilosophy 2016 ABSTRACT EFFECTSOFNONLOCALITYONTRANSFERREACTIONS By LukeTitus Nuclearreactionsplayakeyroleinthestudyofnucleiawayfromstability.Single- nucleontransferreactionsinvolvingdeuteronsprovideanexceptionaltooltostudythesingle- particlestructureofnuclei.Theoretically,thesereactionsareattractiveastheycanbe castintoathree-bodyproblemcomposedofaneutron,proton,andthetargetnucleus. Opticalpotentialsareacommoningredientinreactionsstudies.Traditionally,nucleon- nucleusopticalpotentialsaremadelocalforconvenience.Theofnonlocalpotentials havehistoricallybeenincludedapproximatelybyapplyingacorrectionfactortothesolution ofthecorrespondingequationforthelocalequivalentinteraction.Thisisusuallyreferred toasthePereycorrectionfactor.Inthisthesis,wehavesystematicallyinvestigatedthe ofnonlocalityon( p;d )and( d;p )transferreactions,andthevalidityofthePerey correctionfactor.Weimplementedamethodtosolvethesinglechannelnonlocalequation forbothboundandscatteringstates.Wealsodevelopedanimprovedformalismfornonlocal interactionsthatincludesdeuteronbreakupintransferreactions.Thisnewformalism,the nonlocaladiabaticdistortedwaveapproximation,wasusedtostudytheofincluding nonlocalityconsistentlyin( d;p )transferreactions. Forthe( p;d )transferreactions,wesolvedthenonlocalscatteringandboundstateequa- tionsusingthePerey-Bucktypeinteraction,andcomparedtolocalequivalentcalculations. UsingthedistortedwaveBornapproximationweconstructtheT-matrixfor( p;d )transfer on 17 O, 41 Ca, 49 Ca, 127 Sn, 133 Sn,and 209 Pbat20and50MeV.Additionallywestudied ( p;d )reactionson 40 Causingthethenonlocaldispersiveopticalmodel.Wehavealsoin- cludednonlocalityconsistentlyintotheadiabaticdistortedwaveapproximationandhave investigatedtheofnonlocalityonon( d;p )transferreactionsfordeuteronsimpinged on 16 O, 40 Ca, 48 Ca, 126 Sn, 132 Sn, 208 Pbat10,20,and50MeV. WefoundthatforboundstatesthePerrycorrectedwavefunctionsresultingfromthe localequationagreedwellwiththatfromthenonlocalequationintheinteriorregion,but discrepancieswerefoundinthesurfaceandperipheralregions.Overall,thePereycorrec- tionfactorwasadequateforscatteringstates,withtheexceptionforafewpartialwaves. Nonlocalityintheprotonscatteringstatereducedtheamplitudeofthewavefunctioninthe nuclearinterior.Thesamewasseenfornonlocalityinthedeuteronscatteringstate,butthe wavefunctionwasalsoshiftedoutward.IndistortedwaveBornapproximationstudiesof ( p;d )reactionsusingthePerey-Buckpotential,wefoundthattransferdistributionsatthe peakby15 35%ascomparedtothedistributionresultingfromlocalpoten- tials.Whenusingthedispersiveopticalmodel,thisdiscrepanciesgrewto ˇ 30 50%.When nonlocalitywasincludedconsistentlywithintheadiabaticdistortedwaveapproximation,the disagreementwasfoundtobe ˘ 40%. Ifonlylocalopticalpotentialsareusedintheanalysisofexperimental( p;d )or( d;p ) transfercrosssections,theextractedspectroscopicfactorsmaybeincorrectbyupto50%in somecasesduetothelocalapproximation.Thishighlightsthenecessitytopursuereaction formalismsthatincludenonlocalityexactly. ACKNOWLEDGMENTS IwouldliketothankmyadvisorProf.FilomenaNuneswhosesupportandguidancehas helpedmebothprofessionallyandpersonally.Iamgratefulforhercontinuedsupport throughoutmyjourneyinnuclearphysics,andherencouragementinmyexplorationof otherandcareers.Herconsistentpositiveattitudeisamodelformetofollow,and hasmadethetaskofcompletingmydegreeapleasure.UndoubtedlyIwouldnot bewhereIaminlifewithouther.Iamveryfortunatetohaveherasanadvisor. Iwouldliketoexpressmythankstomyguidancecommittee,Prof.MarkVoit,Prof. PhilDuxbury,Prof.MortenHjorth-Jensen,andProf.RemcoZegersfortheirinvaluable adviceandsuggestionsalongtheway.I'dliketothankagainProf.MarkVoitfortakingme underhiswinginastronomyforasemester,andforthecountlesshoursofdiscussionswe hadaboutgalaxyclustersandtheuniverse.IwouldalsoliketothankProf.IanThompson, Prof.RonJohnson,Prof.PierreCapel,Prof.Tostevin,andDr.GregoryPotel,without whom,thisworkwouldnothavebeenpossible. IoweadebtofgratitudetotheDepartmentofPhysicsandAstronomyatMichiganState University,theNationalSuperconductingCyclotronLaboratory,andthetheorygroupfor providingacademic,andtechnicalsupport.IwouldalsolikethanktheNational ScienceFoundationandtheDepartmentofEnergyfortheirsupport. IthankmycolleaguesandfriendsatMichiganStateUniversity,andespeciallymycurrent andpastgroupmembers:BichNguyen,NeelamUpadhyay,MuslemaPervin,AmyLovell, AlainaRoss,TerriPoxon-Pearson,GregoryPotel,JimmyRotureau,andIvanBrida. Lastbutnotleast,Ithankmyparents,CarrieGossenandJoeTitus,mybrotherBlake Titus,andtherestofmyfamilyforloveandsupportthroughoutmyentirelife,andtheir iv constantencouragementasIdevelopedmyinterestinscience.Withoutthem,Iwouldnever havemadeitthisfar. v TABLEOFCONTENTS LISTOFTABLES .................................... viii LISTOFFIGURES ................................... x Chapter1Introduction ............................... 1 1.1NuclearInteractions...............................7 1.2Nonlocality....................................9 1.2.1MicroscopicOpticalPotentials......................10 1.2.2PhenomenologicalNonlocalOpticalPotentials.............13 1.2.3SolvingNonlocalEquations.......................14 1.3Motivationforpresentwork...........................15 1.4Outline.......................................18 Chapter2ReactionTheoryfortheTransferofNucleons .......... 20 2.1ElasticScattering.................................21 2.2Two-BodyT-Matrix...............................24 2.2.1BornSeries................................27 2.3Three-BodyT-Matrix...............................28 2.4Three-BodyModels................................36 2.5AdiabaticDistortedWaveApproximation...................37 2.6NonlocalAdiabaticDistortedWaveApproximation..............41 2.7SpectroscopicFactors...............................46 Chapter3OpticalPotentials ............................ 48 3.1GlobalOpticalPotentials.............................48 3.2MotivatingNonlocalPotentials.........................51 3.3Perey-BuckType.................................53 3.3.1CorrectionFactor.............................58 3.4Giannini-RiccoNonlocalPotential........................60 3.5NonlocalDispersiveOpticalModelPotential..................61 3.6LocalEquivalentPotentials...........................63 Chapter4Results ................................... 66 4.1NumericalDetails.................................68 4.1.1ofNeglectingRemnant......................70 4.2DistortedWaveBornApproximationwiththePerey-BuckPotential.....70 4.2.1ProtonScatteringState.........................71 4.2.2NeutronBoundState...........................73 4.2.3( p;d )TransferCrossSections-DistortedWaveBornApproximation.74 4.2.4Summary.................................80 vi 4.3DistortedWaveBornApproximationwiththeDispersiveOpticalModelPo- tential.......................................80 4.3.1ProtonScatteringState.........................81 4.3.2NeutronBoundState...........................83 4.3.3( p;d )TransferCrossSections-DistortedWaveBornApproximation.84 4.3.4Summary.................................87 4.4NonlocalAdiabaticDistortedWaveApproximationwiththePerey-BuckPo- tential.......................................88 4.4.1TheSourceTerm.............................88 4.4.2DeuteronScatteringState........................90 4.4.3( d;p )TransferCrossSections......................93 4.4.4ComparingDistortedWaveBornApproximationandtheAdiabatic DistortedWaveApproximation.....................99 4.4.5EnergyShiftMethod...........................101 4.4.6Summary.................................103 Chapter5ConclusionsandOutlook ....................... 105 5.1Conclusions....................................105 5.2Outlook......................................107 APPENDICES ...................................... 110 AppendixASolvingtheNonlocalEquation.....................111 AppendixBDerivingthePereyCorrectionFactor.................116 AppendixCNonlocalAdiabaticPotential......................123 AppendixDDerivingtheT-Matrix.........................141 AppendixEChecksoftheCodeNLAT.......................176 AppendixFMirrorSymmetryofANCs.......................186 AppendixGListofAcronyms............................196 BIBLIOGRAPHY .................................... 197 vii LISTOFTABLES Table3.1:PotentialparametersforthePerey-Buck[1]andTPM[2]nonlocal potentials.................................56 Table4.1:Percentofthe( d;p )transfercrosssectionatthepeak foracalculationincludingtheremnanttermrelativetoacalculation withouttheremnantterm........................70 Table4.2:Percentofthe( p;d )transfercrosssectionsatthepeak whenusingthePCF(2ndcolumn),oranonlocalpotential(3rdcol- umn),relativetothelocalcalculationwiththeLPEpotential,fora numberofreactionsoccurringat20MeV................79 Table4.3:Percentofthe( p;d )transfercrosssectionsatthepeak whenusingthePCF(2ndcolumn),oranonlocalpotential(3rdcol- umn),relativetothelocalcalculationwiththeLPEpotential,fora numberofreactionsoccurringat50MeV................79 Table4.4:Percentofthe( p;d )transfercrosssectionsatthe peakatthelistedbeamenergiesusingtheDOMpotentialrelative tothecalculationswiththephase-equivalentpotential.Resultsare listedseparatelyfortheofnonlocalityontheboundstate,the scatteringstate,andthetotal......................85 Table4.5:SameasTable4.4,butnowforthePerey-Buckpotential.......85 Table4.6:Percentofthe( d;p )transfercrosssectionsatthepeak whenusingnonlocalpotentialsinentranceandexitchannels(1st column),nonlocalpotentialsinentrancechannelonly(2ndcolumn), andnonlocalpotentialsinexitchannelonly(3rdcolumn),relative tothelocalcalculationwiththeLPEpotentials,foranumberof reactionsoccurringat10MeV......................94 Table4.7:Percentofthe( d;p )transfercrosssectionsatthepeak whenusingnonlocalpotentialsinentranceandexitchannels(1st column),nonlocalpotentialsinentrancechannelonly(2ndcolumn), andnonlocalpotentialsinexitchannelonly(3rdcolumn),relative tothelocalcalculationwiththeLPEpotentials,foranumberof reactionsoccurringat20MeV......................97 viii Table4.8:Percentofthe( d;p )transfercrosssectionsatthepeak whenusingnonlocalpotentialsinentranceandexitchannels(1stcol- umn),nonlocalpotentialsinentrancechannelonly(2ndcolumn),and nonlocalpotentialsinexitchannelonly(3rdcolumn),relativetothe localcalculationwiththeLPEpotentials,foranumberofreactions occurringat50MeV.Figurereprintedfrom[3]withpermission...98 TableE.1:Thenonlocaladiabaticintegral, rhs ofEq.(2.49),calculatedwith MathematicaandNLATusinganalyticexpressionsforthewavefunc- tionsandpotentials............................183 TableF.1:RatioofprotontoneutronANCsforthedominantcomponent:Com- parisonofthiswork R withtheresultsoftheanalyticformula R 0 Eq.(F.10)andtheresultsofthemicroscopictwo-clustercalculations R MCM [4,5]includingtheMinnesotainteraction.Theuncertainty in R accountforthesensitivitytotheparametersof V Bx ......191 TableG.1:Listofacronymsusedinthiswork...................196 ix LISTOFFIGURES Figure1.1:Thechartofthenuclides.Theprotondriplineisindicatedbytheline abovethestablenuclei,andtheneutrondriplineisindicatedbelow thestablenuclei.Theproton(neutron)driplineindicateswherethe additionofasingleproton(neutron)willmaketheresultingnucleus unbound.Figurereprintedfrom[6]withpermission..........2 Figure1.2:TypicalNuclearShellStructure.....................4 Figure1.3:Dependenceofthetransferangulardistributiononthetransferred angularmomentumfor 58 Ni( d;p ) 59 Niat8MeV,withdatafrom[7]. Reprintedfrom[8]withpermission...................5 Figure1.4:Angulardistributionsforelasticscatteringofnucleons 208 Pbat 25MeV.(a) n + 208 Pb(b) p + 208 Pbwithtialcrosssection normalizedtoRutherford........................8 Figure1.5:Anexampleofachannelcouplingnonlocality.Inthiscase,the deuteronisimpingedonsometarget.Thechannelcouplingnon- localityresultsfromthedeuteronbreakingupasitapproachesthe nucleus,propagatingthroughspaceinitsbrokenupstate,andthen recombiningtoformthedeuteronagain.................10 Figure2.1:First,second,andall-ordercouplingswithinasetof0 + ,2 + ,and4 + nuclearlevels,startingfromthegroundstate.............28 Figure2.2:Thecoordinatesusedinaoneparticletransferreaction.......29 Figure2.3:ThecoordinatesusedtocalculatetheT-matrixfor( d;p )transfer..34 Figure2.4:ThefourWeinbergStateswhenusingacentralGaussianwhich reproducesthebindingenergyandradiusofthedeuteronground state.Theinsetshowstheasymptoticpropertiesofeachstate....40 Figure2.5:Thecoordinatesusedforconstructingtheneutronnonlocalpotential. Theopendashedcirclerepresentstheneutroninatpointin spacetoaccountfornonlocality.....................42 Figure2.6:Thecoordinatesusedforconstructingthesystemwavefunctionfor the d + A deuteronscatteringstate...................44 x Figure3.1:tialelasticscatteringrelativetoRutherfordasafunctionof scatteringangle.(a) 48 Ca( p;p ) 48 Caat15 : 63MeVwithdatafrom[9] (b) 208 Pb( p;p ) 208 Pbat61 : 4MeVwithdatafrom[10].........57 Figure3.2:Calculatedandexperimentalelasticscatteringangulardistributions usingthenonlocalDOMpotential.Dataforeachenergyarefor claritywiththelowestenergyatthebottomandhighestatthetop. Datareferencesin[11].Figurereprintedfrom[11]withpermission..62 Figure3.3: 49 Ca( p;p ) 49 Caat50.0MeV:Thesolidlineisobtainedfromusing thePerey-Bucknonlocalpotential,theopencirclesareatothe nonlocalsolution,andthedottedlineisobtainedbytransformingthe depthsofthevolumeandsurfacepotentialsaccordingtoEq.(B.14). Figurereprintedfrom[12]withpermission...............65 Figure4.1:Realandimaginarypartsofthe J ˇ =1 = 2 partialwaveofthescat- teringwavefunctionforthereaction 49 Ca( p;p ) 49 Caat50.0MeV: NL (solidline), PCF (crosses),and loc (dashedline).Top(bot- tom)panel:absolutevalueofthereal(imaginary)partofthescat- teringwavefunction.Figurereprintedfrom[12]withpermission...72 Figure4.2:Realandimaginarypartsofthe J ˇ =11 = 2 + partialwaveofthe scatteringwavefunctionforthereaction 49 Ca( p;p ) 49 Caat50.0MeV. SeecaptionofFig.4.1.Figurereprintedfrom[12]withpermission..72 Figure4.3:Groundstate,2 p 3 = 2 ,boundwavefunctionfor n + 48 Ca. ˚ NL (solid line), ˚ PCF (crosses),and ˚ loc (dashedline).Theinsetshowsthe ˚ NL ˚ PCF .Figurereprintedfromfrom[12]withper- mission...................................74 Figure4.4:Angulardistributionsfor 49 Ca( p;d ) 48 Caat50MeV:Inclusionofnon- localityinboththeprotonscatteringstateandtheneutronbound state(solid),usingLPEpotentials,thenapplyingthecorrectionfac- tortoboththescatteringandboundstates(crosses),usingtheLPE potentialswithoutapplyinganycorrections(dashedline),including nonlocalityonlyintheprotonscatteringstate(dottedline)andin- cludingnonlocalityonlyintheneutronboundstate(dot-dashedline). Figurereprintedfrom[12]withpermission...............75 Figure4.5:SameasinFig.4.4butfor 49 Ca( p;d ) 48 Caat E p =20MeV.Figure reprintedfrom[12]withpermission...................77 Figure4.6:SameasinFig.4.4butfor 133 Sn( p;d ) 132 Snat E p =20MeV.Figure reprintedfrom[12]withpermission...................77 xi Figure4.7:SameasinFig.4.4butfor 209 Pb( p;d ) 208 Pbat E p =20MeV.Figure reprintedfrom[12]withpermission...................78 Figure4.8:AngulardistributionsforelasticscatteringnormalizedtoRutherford forprotonson 40 Caat E p =20MeV.Theelasticscatteringwith theDOMpotential(solidline),theDOMLPEpotential(opencir- cles),thePerey-Buckinteraction(dashedline),andthePerey-Buck LPEpotential(opensquares).Thedata(closeddiamonds)from[13]. Figurereprintedfrom[14]withpermission...............81 Figure4.9:SameasinFig.4.8butfor E p =50MeV.Datafrom[15].Figure reprintedfrom[14]withpermission...................82 Figure4.10:Therealandimaginarypartsofthe J ˇ =1 = 2 + partialwaveofthe scatteringwavefunctionforthereaction 40 Ca( p;p ) 40 Caat E p =50 MeV.ThisshowsthewavefunctionresultingfromtheDOMpoten- tial(solidline)anditsLPEpotential(dottedline),thePerey-Buck potential(dashedline)anditsLPEpotential(dot-dashedline).The top(bottom)panelshowstheabsolutevalueofthereal(imaginary) partofthescatteringwavefunction.Figurereprintedfrom[14]with permission.................................83 Figure4.11:Theneutrongroundstate1 d 3 = 2 boundwavefunctionfor n + 39 Ca. ShownisthewavefunctionobtainedusingtheDOMpotential(solid line),thePerey-Buckpotential(crosses)andthelocalinteraction (dashedline).Theinsetshowstheasymptoticpropertiesofeach wavefunction.Figurereprintedfrom[14]withpermission......84 Figure4.12:Angulardistributionsforthe 40 Ca( p;d ) 39 Careactionat(a) E p =20 MeV,(b) E p =35MeV,and(c) E p =50MeV.Inthisisthe transferdistributionresultingfromusingthenonlocalDOM(solid line)anditsLPEpotential(dottedline),thePerey-Buckpotential (dashedline)andthePerey-BuckLPEpotential(dot-dashedline). Figurereprintedfrom[14]withpermission...............86 Figure4.13:Absolutesvalueofthe d + A sourcetermwhennonlocalandlocal potentialsareused.(a) d + 48 Caat E d =50MeV.(b) d + 208 Pbat E d =50MeV.Bothareforthe L =1and J =0partialwave.Figure reprintedfrom[3]withpermission....................89 Figure4.14:Absolutevalueofthe d + A sourcetermwhennonlocalandlocal potentialsareused.(a) d + 48 Caat E d =50MeV.(b) d + 208 Pbat E d =50MeV.Bothareforthe L =6and J =5partialwave.Figure reprintedfrom[3]withpermission....................90 xii Figure4.15:Absolutevalueofthe d + A scatteringwavefunctionusingtheADWA theorywhennonlocalandlocalpotentialsareused.(a) d + 48 Caand (b) d + 208 Pb.Bothforthe L =1and J =0partialwaveat E d =50 MeVinthelaboratoryframe.Figurereprintedfrom[3]withpermis- sion....................................91 Figure4.16:Absolutevalueofthe d + A scatteringwavefunctionusingtheADWA theorywhennonlocalandlocalpotentialsareused.(a) d + 48 Caand (b) d + 208 Pb.Bothforthe L =6and J =5partialwaveat E d =50 MeVinthelaboratoryframe.Figurereprintedfrom[3]withpermis- sion....................................92 Figure4.17:Angulardistributionsfor( d;p )transfercrosssections.Theinsets arethetheoreticaldistributionsnormalizedtothepeakofthedata distribution.(a) 48 Ca( d;p ) 49 Caat E d =10MeVwithdata[16]at E d =10MeVinarbitraryunits.(b) 132 Sn( d;p ) 133 Snat E d =10 MeVwithdata[17]at E d =9 : 4MeV.(c) 208 Pb( d;p ) 209 Pbat20 MeVwithdata[18](Circles)and[19](Squares)at E d =22MeV. Figurereprintedfrom[3]withpermission...............95 Figure4.18:Angulardistributionsfor( d;p )transfercrosssections.Theinsetis thetheoreticaldistributionsnormalizedtothepeakofthedatadistri- bution.(a) 48 Ca( d;p ) 49 Caat E d =50MeVwithdata[20]at E d =56 MeV.(b) 132 Sn( d;p ) 133 Snat E d =50MeV.(c) 208 Pb( d;p ) 209 Pbat 50MeV.Figurereprintedfrom[3]withpermission..........96 Figure4.19:Comparisonof( d;p )transfercrosssectionswhenusingtheDWBAas comparedtotheADWA.(a) 16 O( d;p ) 17 O,(b) 48 Ca( d;p ) 49 Cawith datafrom[20].(c) 132 Sn( d;p ) 133 Sn.Alldistributionsat E d =50 MeV.Figurereprintedfrom[3]withpermission............100 Figure4.20:Comparisonof( d;p )angulardistributionswhenusingtheenergy shiftmethodof[21,22].(a) 16 O( d;p ) 17 Oat E d =10MeV(b) 40 Ca( d;p ) 41 Caat E d =10MeV(c) 208 Pb( d;p ) 209 Pbat E d =20 MeV.Thesolidlineiswhenfullnonlocalitywasincludedintheen- trancechannel,dashedlineiswhentheLPEpotentialwasused,dot- dashedlinewhentheCH89potential[23]wasusedwiththeadditional energyshiftquanin[21],andthedottedlinewhentheCH89po- tentialwasusedatthestandard E d = 2value.Figurereprintedfrom [3]withpermission............................102 FigureE.1:erentialelasticscatteringrelativetoRutherforedasafunctionof scatteringangle. 209 Pb( p;p ) 209 Pbat E p =50 : 0MeV:Thesolidline isobtainedfromNLAT,thedottedlineisobtainedfromNLATand setting =0 : 05fm,andthedashedlineisfrom FRESCO .....177 xiii FigureE.2:erentialelasticscatteringasafunctionofscatteringangle. 208 Pb( n;n ) 208 Pb at E p =14 : 5MeV:Thesolidlineisobtainedanonlocalcalculation usingNLAT,andthedashedlineisthenonlocalcalculationpublished byPereyandBuck[1]..........................178 FigureE.3: n + 48 Caboundwavefunction,andthedeuteronboundwavefunction. n + 48 Ca:ThesolidlineisobtainedfromNLAT,thedottedlineis obtainedfromNLATandsetting =0 : 05fm,andthedashedlineis from FRESCO .Deuteron:Dot-dashedlineisdeuteronboundwave functionobtainedfromNLAT,andtheopencirclesareobtainedwith FRESCO ................................179 FigureE.4:Thelocaladiabaticpotentialfor d + 48 Caat E d =20MeVcalculated withNLATandwith TWOFNR [24].(a)Realpart,(b)Imaginary part....................................180 FigureE.5: 48 Ca( d;d ) 48 Caat E d =20MeV.Thesolidlineiswhenusingthe localadiabaticpotential,thedottedlineiswhendoinganonlocal calculationwith =0 : 1fminthenucleonopticalpotentials,andthe dashedlineisacalculationdonein FRESCO .............181 FigureE.6: 132 Sn( d;p ) 133 Snat E d =50MeV.SolidlineisalocalDWBAcalcula- tionwithNLAT,thedashedlineisacalculationdonewith FRESCO .181 FigureE.7:Angulardistributionsfor 208 Pb( d;p ) 209 Pbat E d =50MeVobtained byusingtstepsizestocalculatethe rhs ofEq.(2.49).The solidlineusesastepsizeof0 : 01fm,thedashedlineastepsizeof 0 : 03fm,andthedottedlineastepsizeof0 : 05fm...........184 FigureE.8:Angulardistributionsfor 208 Pb( d;p ) 209 Pbat E d =50MeVobtained byusingtstepsizesandvaluesofacutparameter(CutL)to calculatethe rhs ofEq.(2.49).Thesolidlineusesastepsizeof0 : 01 fmwithCutL=2,thedashedlineastepsizeof0 : 01fmwithCutL=3, andthedottedlineastepsizeof0 : 05fmwithCutL=2........185 FigureF.1:Neutronandprotonspectroscopicfactorsfor 17 Oand 17 F,respec- tively,consideringthe 16 Ocoreinits0 + groundstateand2 + excitedstate:(a)5 = 2 + groundstateand(b)1 = 2 + excitedstate. Figurereprintedfrom[25]withpermission..............193 FigureF.2:RatioofprotonandneutronANCsfor 17 Oand 17 F,respectively,in- cluding 16 O(0 + ; 2 + ):(a)5 = 2 + groundstateand(b)1 = 2 + excited state.Figurereprintedfrom[25]withpermission...........193 xiv Chapter1 Introduction Sincethedawnofnuclearphysics,reactionstudieshavebeenperformedtoinvestigatethe propertiesofthenucleus.Oneofthemanyreasonsthesestudieshavebeencarriedout istoaddresstheoverarchinggoalofnuclearphysics.Thisistounderstandwhereallthe matterintheuniversecamefromandhowitwasformed.Tosolvethisproblem,wenotonly needtounderstandtheenvironmentsinwhichnuclearreactionsoccur,butwealsoneedto understandthenatureofthenucleiundergoingthereactions.Thisisadauntingtaskwith hundredsofstablenuclei,andthousandsofunstablenucleiknowntoexist[6]. InFig.1.1thechartofthenuclidesisshownwiththecorrespondingprotonandneutron driplines.Thedriplineisthepointthatseparatesboundfromunboundnuclei.Theneutron dripline,forexample,thepointwheretheadditionofasingleneutronwillmakethe resultingnucleusunbound.Whileanextraordinaryamountofprogresshasbeenmadein experimentallymeasuringunstablenuclei,itisremarkablehowfartheneutrondriplineis expectedtoextend,andhowmanynucleiareyettobediscovered. Formanydecades,intenseexperimentalandtheoreticalhasbeenputintostudying stablenuclei.Whileexperimentsaimedatstudyingstableisotopesarestillperformed, thefocusinmoderntimeshasshiftedtowardsthestudyofexoticnuclei.Inthecontext ofunderstandingtheoriginofthematterintheuniverse,exoticnucleiplayacrucialrole. Whileexoticnucleiliveforaveryshortperiodoftime,reactionsonexoticnucleiareessential tocreatingheavyelements[8].Incertainastrophysicalenvironments,nucleirapidlycapture 1 protonsorneutrons,pushingthemtowardsthedripline.Theseunstablenucleithen decay backtothevalleyofstability.Tofullyunderstandthepaththenucleosynthesistakes,and theelementsthatareproduced,wemustunderstandthepropertiesoftheexoticnucleivery farfromstability,andthereactionmechanismsofneutrons,protons,orheavierelementson thoseexoticnuclei. Figure1.1:Thechartofthenuclides.Theprotondriplineisindicatedbythelineabove thestablenuclei,andtheneutrondriplineisindicatedbelowthestablenuclei.Theproton (neutron)driplineindicateswheretheadditionofasingleproton(neutron)willmakethe resultingnucleusunbound.Figurereprintedfrom[6]withpermission. Formanynuclearreactionexperiments,agoodreactiontheoryisrequiredtoextract reliableinformation.Thesamecanbesaidaboutthepotentialsweputintoourtheories.In fact,thetwoworkhandinhand.Anexcellentmodelcanbeheldbackbytheuseofpoor interactions,whilethebestinteractionavailablewillprovidelittleinsightwhenusedina poormodel. Animportantpartofunderstandingthepropertiesofnucleiisknowingthespinandparity assignmentsofthevariousenergylevels.Singlenucleontransferreactionsareanexcellent 2 toolforunderstandingtheseproperties.Theprotonsandneutronsinsideanucleusarrange themselvesinanorganizedway,roughlyfollowingthewaylevelsorganizethemselvesina harmonicoscillatorpotentialwithaspin-orbitinteraction.Fillingashellprovidesadditional stability.IndicatedinFig.1.2(right)arethemagicnumberscorrespondingtothenumber ofneutronsorprotonsneededtoinashell.TheorderingshowninFig.1.2providesa guidetoassigningenergylevels.Asonemovesawayfromstabilitythereisshellreordering andtmagicnumbersemerge. Theuseofsinglenucleontransferreactionssuchas( d;p )or( p;d )asaprobetostudy nuclearstructurebeganintheearly1950s.Butlerrealizedthatthespinsandparitiesof nuclearenergylevelscanbeobtainedfromangulardistributions,withouttheneedtoknow propertiesofexcitedstates[26].ThisfactwasreiteratedbyHuby[27,28],andlaterfollowed upwiththeoreticalcalculationsbyBhatiaandcollaborators[29].Whiletheseearlystudies reliedontheverysimpleplanewaveBornapproximation,itdrewconsiderableattention to( d;p )reactionsasameanstostudynuclearstructurethroughtheanalysisofangular distributionsoftransferreactions. Sincethesepioneeringstudies,theshellstructureofnucleihasbeenstudiedwiththeaid ofsinglenucleontransferreactions.Ofparticularinterestforthisworkarethestripping ( d;p )orpickup( p;d )reactions.Thesetypesofreactionsareanexcellenttoolformeasuring theenergylevelsofnuclei,aswellasthespinandparityassignmentsofthecorresponding energylevels.Itistransferreactionssuchasthesewhichprovidedmuchofthestructure informationofstableisotopesintheearlydaysofnuclearphysics[30,31,32,33,34,35]. InFig.1.3weshowthedependenceofthetransferangulardistributiononthetransferred angularmomentumfor 58 Ni( d;p ) 59 Niat10MeV.Thetransferredangularmomentumhas aneontheshapeofthetransferdistribution,aswellasthelocationofthepeakof 3 Figure1.2:TypicalNuclearShellStructure. thetransferdistribution.Itisseenthatforthe s 1 = 2 statethepeakoccursat0 ,andfor increasingangularmomentumtransferthepeakgetsshiftedtoincreasingangles.The oscillationsofthetransferdistributioncanbeunderstoodintermsofaractionpattern, analogoustothatofasingleslitpattern.Withincreasingenergytheaction patternisfoundtohavemoreoscillations.Also,asthebeamenergyincreases,thetransfer distributiongetsshiftedtomoreforwardangles.Themagnitudeofthecrosssectionisrelated tothe Q -value,orenergymismatch,ofthereaction.Themagnitudeofthecrosssectionis largestwhen Q =0,anddecreasesasenergymismatchincreases. Modernreactiontheorieshaveprogressedgreatlysincethe1950s,allowingformorereli- ablenuclearstructureinformationtobeextractedfromexperimentaldata.Thetheoretical 4 Figure1.3:Dependenceofthetransferangulardistributiononthetransferredangularmo- mentumfor 58 Ni( d;p ) 59 Niat8MeV,withdatafrom[7].Reprintedfrom[8]withpermission. advancesofreactiontheorycoupledwithadvancesinexperimentaltechniqueshavemadethe useoftransferreactionstostudyexoticnucleifeasible.Intheearlydaysofnuclearphysics, transferreactionswereperformedbymakingatargetusingstablenuclei,andimpinging protons,deuterons, 3 He,orothernucleionthetargettoinitiatethetransferprocess.When studyingunstablenuclei,thesereactionsaredoneininversekinematics[17,36,37,38,39,40]. Sinceexoticnucleiaretooshortlivedtomakeatarget,adeuteratedtarget,forexample, issometimesused,andabeamofexoticnucleiisimpingedonthetargettoperformthe experiment. As( d;p )or( d;n )transferreactionsareausefultoolforstudyingtheoverlapfunctionof thealnucleus,thesereactionsarealsoapreferredmethodtoextractthenormalization 5 ofthetailoftheoverlapfunction.Thisquantityisknownastheasymptoticnormalization cot(ANC),andisinEq.(F.8).Atverylowenergies,thetransfercrosssection isdominatedbytheamplitudeoftheoverlapfunctionintheasymptoticregion.Thus, a( d;n )transferreactioncanprovideinformationontheprotonboundstateofthe nucleus.TheANCcanbeusedtodetermineastrophysicallyimportant( p; )reactionrates atenergiesunobtainableexperimentallyviatheANCmethod[41]. MakinguseoftheANCmethod,transferreactionshavealsobecomeacommontool toextractinformationrelevantintheunderstandingofastrophysicallyimportantprocesses [42,43].Sometimes,theANCforthesystemofinterestisnotaccessibledirectly,while themirrorsystemis.Whenthisisthecase,chargesymmetryofthenuclearforcecanbe exploitedtoderiveamodelindependentquantityrelatingtheratioofANCsofthetwo systems[44].ThishasbeenshowntobeareliablemethodtoindirectlyextractanANC [4,25],andhasbeenusedinpractice[45,46]. WhereasANCscalculatedtheoreticallycanbeverytdependingonthemodel thatisused,theideabehindthemethodproposedin[44]suggeststhattheratioofANCs ofmirrorpairsismodelindependent.ThismethodisveryusefultoextracttheANCofthe protonstate,usefulin( p; )reactionsimportantforastrophysics,bymeasuringthemirror partner.Intheearlystageofmygraduatework,weperformedastudytotestthemodel independenceoftheratioofANCsofmirrorpairs,andthevalidityoftheanalyticformula derivedin[44].ThisprojectisdiscussedinAppendixF. 6 1.1NuclearInteractions Theelasticscatteringofanucleonofanucleusisacomplicatedquantummany-body problem.Tosolvetheproblemexactlywouldrequirethefullyanti-symmetrizedmany-body wavefunctionthatincludesthecouplingsoftheelasticchanneltoalltheothernon-elastic channelsavailable(transfer,inelasticscattering,chargeexchange,fusion,etc.).This isaveryultproblemtosolve,andinpractice,thescatteringprocessisnotsolvedinthis manner.However,theelasticscatteringofaparticlefromsomearbitrarypotential, U ( R ), iswellunderstood[8,47].Assumingthatthecomplicatedinteractionbetweensomeparticle andthenucleuscanberepresentedbyacomplexisthebasisoftheopticalmodel. InFig.1.4weshowtheangulardistributionsforelasticscatteringofnucleons 208 Pbat 25MeV.Inpanel(a)is n + 208 Pb,andinpanel(b)is p + 208 Pb.DuetotheCoulombpotential, protonelasticscatteringisusuallynormalizedtoRutherford,whichisthepoint-Coulomb crosssection,andalwaysgoestounityat0 .Whenthisisdone,theangulardistributions forprotonelasticscatteringareunitless.Theoscillationsresultfromapattern whichcanbeunderstoodqualitativelyinasimilarwayassingleslitForalarger targetorlowerenergy,therewillbefeweroscillationsbetween0 and180 ,andtherewill bemoreoscillationsforasmallertargetorahigherenergy. Intheopticalmodel,elasticscatteringdataarebyvaryingpotentialparametersinan assumedformfor U ( R ).Thiscomplexinteraction,referredtoastheopticalpotential,isused todescribetheelasticscatteringprocessoftheparticlethenucleus,withtheimaginary parttakingintoaccountlossofuxtonon-elasticchannels.Oncetheopticalpotentialis itcanthenbeusedasaninputtoamodelthatdescribessomeotherprocesswiththe goalofobtaininganobservableotherthanelasticscattering,suchastransfercrosssections. 7 Figure1.4:Angulardistributionsforelasticscatteringofnucleons 208 Pbat25MeV.(a) n + 208 Pb(b) p + 208 PbwithtialcrosssectionnormalizedtoRutherford. Elasticscatteringdataforthedesiredtargetandenergyareoftentimesnotavailable.To remedythisproblem,opticalpotentialsareconstructedthroughsimultaneoustolarge datasetsofelasticscattering.Thesearereferredtoasglobalopticalpotentials.Theenergy, target,andprojectiledependentparametersarevariedtoproduceabesttotheentire dataset.Thepurposeofusingaglobalpotentialisthatonecaneasilyinterpolateinorderto obtainapotentialforanucleusinwhichthereisnoexperimentaldataavailable.Obtaininga potential,andthereforepredictionsonobservables,ofun-measurednucleiisaveryattractive featureofusingaglobalpotentialandisacredittotheirsuccessoverthedecades.Itisfor thisreasonthatconsiderablehasbeenputintocreatingmanytglobaloptical potentialsovertheyearswhichhavereceivedwidespreaduse[48,23,49]. Globalpotentialsareaveryusefultoolforstudyingnuclearreactionsandpredicting observables.However,thewaytheyareconstructedleavesoutaconsiderableamountof physics.Elasticscatteringonlyconstrainsthenormalizationofthescatteringwavefunction outsidetherangeoftheinteraction.Itisnotsensitivetotheshort-rangepropertiesof thewavefunction.Therefore,theshort-rangephysicsisnotconstrainedatallbyelastic 8 scattering.Also,muchoftheelasticscatteringdatathatexistsisforstablenuclei.With theincreasinginterestofthestudyofrareisotopes,theextrapolationstoexoticnucleimay notbereliable.Itisforthisreasonthatamorephysicallymotivatedformfortheoptical potentialshouldbepursued. Allwidelyusedglobalopticalpotentialsarelocal.However,whenderivedfromamany- bodytheory,theresultingopticalpotentialisnonlocal.Thestrongenergydependenceof globalpotentialsisassumedtoaccountforthenonlocalitythatisneglected.Withincreasing interestinmicroscopicallyderivedopticalpotentials,itisbecomingnecessarytoinvestigate thevalidityofthelocalassumption,anddevelopmethodstoincorporatenonlocalpotentials intomodernreactiontheories. 1.2Nonlocality Ithaslongbeenknownthattheopticalpotentialisnonlocal[50].IntheHartree-Fockthe- ory,theexistenceofanexchangetermintroducesanexplicitnonlocalpotential[51].For scattering,thecomplicatedcouplingoftheelasticchanneltoallothernon-elasticchannels accountsforanothertsourceofnonlocality[52,53].Thesetwosourcesofnonlo- cality,anti-symmetrizationandchannelcouplings,havebeenknownandstudiedfordecades (e.g.[54]). Asaphysicalexample,consideradeuteronimpingingonatarget,andlet R and R ' locatethecenterofthedeuteronrelativetothecenterofthetarget.Let'ssaythatthe deuteronbreaksupat R 0 asitapproachesthetarget.Thedeuteroncanthenpropagate throughspaceinitsbrokenupstate,thenrecombinetoformthedeuteronagainat R . ThisprocessisdepictedinFig.1.5.Suchaprocesswouldconstituteachannelcoupling 9 nonlocality.Thiswillresultinapotentialoftheform V ( R ; R 0 )sincetheinteractionata givenpointisdependentonthevalueofthepotentialandthescatteringwavefunctionat allotherpointsinspace. Figure1.5:Anexampleofachannelcouplingnonlocality.Inthiscase,thedeuteronisim- pingedonsometarget.Thechannelcouplingnonlocalityresultsfromthedeuteronbreaking upasitapproachesthenucleus,propagatingthroughspaceinitsbrokenupstate,andthen recombiningtoformthedeuteronagain. Asanotherexample,considerasinglenucleonscatteringanucleus.Sincethesystem wavefunctionisafullyanti-symmetricmany-bodywavefunction,itisnotguaranteedthat theprojectileintheincidentchannelisthesameparticleastheoneintheexit.ThePauli principlealsoplaysarolewhentheprojectileispropagatingthroughthenuclearmedium, andmostnotablyhastheofreducingtheamplitudeofthewavefunctioninthenuclear interior.Allofthesewillmanifestinapotentialoftheform V ( R ; R 0 ). 1.2.1MicroscopicOpticalPotentials Thisworkisnotconcernedwithconstructingamicroscopicopticalpotential,butrather withusingcurrentphenomenologicalnonlocalopticalpotentials,andstudyingtheof nonlocalityontransferobservables.However,itisimportanttounderstandtheconsiderable 10 amountofthathasbeenputforthinrecentdecadestoconstructopticalpotentials frommicroscopictheories.Inthisthesiswewilldemonstratethatnonlocalityisanimportant featureofthenuclearpotentialthatmustbeconsideredexplicitly.Movingforward,the developmentofab-initiomany-bodytheoriesthepromiseofrealisticmicroscopicoptical potentials.Themethodsoutlinedherewillbethetoolsforfuturestudies. Severalstudieshavebeenmadetoconstructamicroscopicallybasedopticalpotential.In thepioneeringworkofWatson[55,56],andlaterbyKerman,McManus,andThaler (KMT)[57],thetheoryofmultiplescatteringwasdeveloped,wheretheopticalpotentialto describeelasticscatteringisconstructedintermsoftheamplitudesforthescatteringofthe incidentparticlebytheindividualneutronsandprotonsinthetargetnucleus.Thistheory forconstructingtheopticalpotentialislimitedtorelativelyhighenergies( > 100MeV). DerivingthemultiplescatteringexpansionoftheKMTopticalpotentialisacomplicated task,buthasbeendonesuccessfully,suchasfor 16 O[58]. Theopticalpotentialcanalsobeidenwiththeself-energy,asindicatedbyBell andSquires[50].HufnerandMahauxstudiedtheopticalpotentialingreatdetailthrough useofasystematicexpansionoftheself-energywithintheGreensfunctionapproachtothe many-bodyproblem[59].ThisapproachisanalogoustotheBethe-Bruecknerexpansion forthecalculationofthebindingenergy[60].Thisformulationoftheopticalpotentialin termsoftheself-energyisattractiveasitissuitableforbothintermediateandhighenergy scattering,andreducestotheexpressionsofmultiplescatteringtheoryathighenergies. Itisthroughtheconnectiontotheself-energythatJeukenne,Lejeune,Mahaux(JLM) formulatedtheiropticalmodelpotentialfornuclearmatter[61].Inenuclear matter,theconceptofaprojectileandtargetlosetheirmeaning.Instead,apotentialenergy andalifetimeforaquasiparticlestateobtainedbycreatingaparticleorholewithmomentum 11 k abovethecorrelatedgroundstateisLater,theJLMapproachwasextendedto nucleiusingalocaldensityapproximation[62]. Thelinkbetweentheself-energyandtheopticalpotentialwasfurtherexploredbyMa- hauxandSartor[63].Thisimplementationisknownasthedispersiveopticalmodel(DOM). Theadvantageofthismethodisthatitprovidesalinkbetweennuclearreactionsandnu- clearstructurethroughadispersionrelation.Inrecentyears,alocalversionoftheDOM wasintroducedforCalciumisotopes[64],andanonlocalDOMwassubsequentlydeveloped for 40 Ca[65,11].TransferreactionstudieshaveshownthatthelocalDOMisabletode- scribetransfercrosssectionsaswellasorbetterthanglobalpotentials[66],andthatthe nonlocalDOMcantlymodifytheshelloccupancy,orspectroscopicfactor,ofthe statespopulatedintransferreactions[14]. Variousothertechniquesexistwhichconstructanopticalpotentialthroughtheself- energyusingmodernadvancesinnucleartheory.Makinguseoftheprogressthathasbeen achieved,Holtandcollaboratorsconstructedamicroscopicopticalpotentialfromtheself- energyfornucleonsinamediumofisospin-symmetricnuclearmatterwithinthe frameworkofchiraletheory[67]. Thetwosourcesofnonlocality,channelcouplingandanti-symmetrization,havebeen studiedovertheyearsbynumerousauthors[54,68,69]tonameonlyafew.Manyof thesestudiesderivethenonlocalpotentialusingsomemicroscopictheory,thencompare thepotentialobtainedtocommonlyusedphenomenologicalnonlocalpotentials.Suchwas donein[54]wherethemultichannelalgebraicscattering(MCAS)method[70]wasused toobtainthenonlocalpotentialresultingfromchannelcoupling.Theresultingnonlocal potentialwasfoundtobeverytfromthesimpleGaussiannonlocalitiesassumed inphenomenologicalpotentials.However,theMCASmethodisonlysuitableforverylow 12 energyprojectiles,wherejustafewexcitedstatesarerelevanttothecoupling,andthus,can beexplicitlycoupledtogethertogeneratethechannelcouplingnonlocalpotential. 1.2.2PhenomenologicalNonlocalOpticalPotentials Theformalismtodevelopamicroscopicopticalpotentialiscomplicated,andrequirescon- siderablecomputationtimetoimplement.However,constructinganonlocalpotentialphe- nomenologicallyprovidesapracticalalternativetoconstructanonlocalpotentialapplicable forwidespreaduse.TheseminalworkofPereyandBuck,[1],wastheattempttocon- straintheparametersofanonlocalpotentialthroughtoelasticscatteringdata.This workwasdoneinthesixties,butitisstillthemostcommonlyreferencednonlocaloptical potential.Inthelateseventies,GianniniandRiccoconstructedaphenomenologicalnonlo- calopticalpotential,[71,72].Inthatwork,thepotentialparameterswereconstrainedwith toalocalform,thenatransformationformulawasusedtoobtainthenonlocalpoten- tial.Veryrecently,Tian,Pang,andMa(TPM)introducedathirdnonlocalglobaloptical potential,[2].Thesethreeworksaretoourknowledgetheonlyattemptstoconstructa phenomenologialnonlocalglobalopticalpotential. Acommonfeatureofusinganonlocalpotentialisthattheamplitudeofthewavefunction isreducedinthenuclearinteriorascomparedtothewavefunctionresultingfromusinga localpotential.Numerousstudieshavebeenperformedtoinvestigatethisandto waystocorrectforit[73,74,75].Thesestudieswerefocusedonpotentialsofthe formofthephenomenologicalPerey-Bucknonlocalpotential.Alocalequivalentpotential tothenonlocalpotentialshouldformallyexist.Attemptshavebeenmadetothislocal equivalentpotential[76,77].Innearlyallthesecases,thePerey-Buckformforthenonlocal potentialwasassumed. 13 1.2.3SolvingNonlocalEquations Whilethetheoreticalfoundationforconstructingnonlocalpotentialshasbeenaroundfor manydecades,thebroadapplicationofnonlocalpotentialsintheofnuclearreactions hasnevercometofruition.Withanonlocalpotential,theScodingerequationtransforms fromatialequationtoanintetialequation.Therefore,themoststraight- forwardwaytosolvetheequationisthroughiterativemethods,whichdramaticallyincreases thecomputationalcost. Sincetheknowledgeofnonlocalitydatesbacktothe1950swhencomputerpowerwas muchmorelimitedthantoday,thepreferredmethodwastoincludenonlocalityapproxi- matelythroughacorrectionfactor[73,74,75].However,severalmethodsnowexistthat improvetheofthebasiciterationscheme.KimandUdagawahavepresenteda rapidmethodusingtheLanczostechnique[78,79].AmethodbyRawitscheruseseither ChebyshevorSturmianfunctionsasabasistoexpandthescatteringwavefunction[80]. Also,animprovediterativemethodhasbeenproposedbyMichel[81]. Computationtimeisnolongeranissue.Inthiswork,weusedaniterativemethodout- linedinAppendixAtosolvetheintialequation.Thisis,byfar,theeasiest,but notthemosttwaytosolvetheequation.Sincetheincreaseincomputation timeisminimal,pursuingafasterwaywasnotapriorityandwillbepursuedatalatertime. Ifonedesiredtoconstructtheirownglobalnonlocalpotentialbyglargeamountsof elasticscatteringdata,itwouldbeadvantageoustofurtheroptimizeourtechnique. 14 1.3Motivationforpresentwork Inthiswork,wewouldliketodescribesinglenucleontransferreactionsinvolvingdeuterons whileusingnonlocalopticalpotentials.Eversincetheearlydaysofnuclearphysics,right uptothemodernday,thedistortedwaveBornapproximation(DWBA)hasbeenacommon theoryusedtoanalyzedatafromtransferreactionexperiments[82,83].IntheDWBA,the transferprocessisassumedtooccurinaone-stepprocess,andanopticalpotentialto deuteronelasticscatteringisusedtodescribethedeuteronscatteringstate.Theshortcoming oftheDWBAisthatthedeuteronislooselybound,soitislikelythatthedeuteronwill breakupasitapproachesthenucleus.Nottakingdeuteronbreakupintoaccountexplicitly canhaveatontransfercrosssections.Inallknownimplementationsofthe DWBAtodescribetransfercrosssections,localdeuteronopticalpotentialshavebeenused. Thesedeuteronopticalpotentialswereobtainedeitherbyasingleelasticscattering angulardistribution,orusingaglobalparameterizationsuchasthatfromDaehnick[84]. Inordertoincludedeuteronbreakupexplicitly,itisnecessarytoincludethe n p degrees offreedom.Thisthenrequiressolvingthe n + p + A three-bodyproblem.Athree-body approachwasintroducedinthezerorangeapproximationbyJohnsonandSoper[85],and laterextendedtoincluderangebyJohnsonandTandy[86].Thisisknownas theadiabaticdistortedwaveapproximation(ADWA).Arecentsystematicstudyof( d;p ) reactionswithintheformalismof[86]showstheimportanceofrange[87].In thesetheoriesthedeuteronscatteringstateistreatedasathree-bodyproblem,composed of n + p + A .Thebreakupofthedeuteronisincludedexplicitly,andtheinputpotentials areneutronandprotonopticalpotentials,whicharemuchbetterconstrainedthandeuteron opticalpotentials.Inthissense,ADWAisamoreadvancedtheorythantheDWBAwith 15 theaddedadvantagethatnucleonopticalpotentialsexistinanonlocalform.Therefore,in thisworktheexplicitinclusionofnonlocalityinsinglenucleontransferreactionswithinthe ADWAwillbepursued. Asmentionedbefore,nonlocalityin( d;p )transferreactionshastraditionallybeenin- cludedapproximatelythroughacorrectionfactor.Thisisthemethodexploitedincom- monlyusedtransferreactioncodessuchasTWOFNR[24].Theboundandscatteringwave functionsarecalculatedusingasuitablelocalpotential,normallyaglobalpotentialforelas- ticscatteringandameanreproducingtheexperimentalbindingenergyforthebound state.ThecorrectionfactorusedimpliesthatthenonlocalityassumedisofthePerey-Buck form.Frommicroscopiccalculations,itisknownthatasingleGaussianisnottto takeintoaccountthecomplexnatureofnonlocality[54].Therefore,notonlyisthismethod ofincludingnonlocalitynotaccurate,butitislimitedtoaformforthenonlocalitythatmay notadequatelyrepresentthetruenonlocalityinthenuclearpotential. Recently,someattemptshavebeenmadetoincludenonlocalitywithintheadiabatic modelbyintroducinganenergyshifttotheopticalpotentialsusedtocalculatethescattering wavefunctions[22,21].Thismethodisveryattractiveasalllocalcodeswhichcalculate ( d;p )transfercanstillbeusedwithoutmoHowever,theadequacyofthisenergy shifttotakenonlocalityintoaccountmustbequanAnotherlimitationofthismethod isthatitreliesonenergyindependentnonlocalnucleonopticalpotentialsassumedtohave thePerey-Buckform. Whiletheexistenceofnonlocalityintheopticalmodelhasbeenknownformanydecades, notmanycalculationsoftransferreactionswiththeexplicitinclusionofnonlocalityhaveever beenperformed.Whiletheapproximatewaystocorrectfornonlocalityarecommon,itis notknowniftheseapproximatemethodsaret.Themethodofconstructinglocal 16 opticalpotentialsthroughtoelasticscatteringdatahasbeenpracticalanduseful,but sinceelasticscatteringdoesnotconstraintheshortrangenonlocalitiespresentinthenuclear potential,itisunlikelythisapproachtoconstructingtheopticalpotentialwillbereliable whenmovingtowardsexoticnuclei.Also,itmustbeunderstoodhowotherobservablesare duetothewayinwhichtheopticalpotentialsareconstructed. Thegoalofthisthesisistostudytheexplicitinclusionofnonlocalityonsinglenucleon transferreactionsinvolvingdeuterons.Sincenonlocalityhaseitherbeenignoredorincluded approximatelyinnearlyallreactioncalculationsforoverhalfacentury,theeofneglect- ingnonlocalyonreactionobservablesmustbequanAlso,thequalityofthecommonly usedapproximatetechniquesneedtobeassessed.Forthispurposeweextendtheformalism oftheADWAtoincludenonlocality.Finally,withrenewedinterestinmicroscopicoptical potentials,theformalismmustbekeptgeneralsothatnonlocalpotentialsofanyformcan beused. Inthisthesis,wewilltesttheconceptofthecorrectionfactorusingthePerey-Buck potential.ThiswillbedonebyperformingDWBAcalculationsof( p;d )reactionsonawide rangeofnucleiandenergies.Thecorrectionfactorwillbeappliedtotheprotonscattering state,andtheneutronboundstateintheentrancechannel.Wewillthenincludenonlocality explicitlyintheentrancechannelinordertoquantifytheadequacyofthecorrectionfactor toaccountfornonlocality.Forthispartofthestudy,alocaldeuteronopticalpotentialwill beusedtodescribethedeuteronscatteringstatewithintheDWBA. Sinceitiswellknownthatdeuteronbreakupplaysanimportantroleindescribingthe reactiondynamics,itiscrucialtoincorporatenonlocalityintoareactiontheorythatex- plicitlyincludesdeuteronbreakup.Thus,wechosetoextendtheformalismoftheADWA toincludenonlocalpotentials.Also,sincethePerey-Buckformforthenonlocalityisnot 17 consistentwithmicroscopiccalculations,theformalismwaskeptgeneralsothatitcanbe usedwithanonlocalpotentialofanyformthatmayresultfromamicroscopiccalculation. Finally,throughasystematicstudy,theofignoringnonlocalityintheoptical potentialontransferobservablescanbequanWewillchoosearangeofnucleiand energies,andperformcalculationsof( d;p )transferreactionsusingnonlocalpotentialsin boththeentranceandexitchannels.Theresultingcrosssectionswillbecomparedtocross sectionsgeneratedfromlocalphaseequivalentpotentialsinordertoquantifytheof neglectingnonlocalityintheopticalpotential. 1.4Outline Thisthesisisorganizedinthefollowingway.Inchapter2wewillpresentthenecessary theory.Wewillbeginwithadiscussionofelasticscattering,andthetwo-bodyT-matrix. Wewillextendthetwo-bodyT-matrixtothree-bodies.Thenwewillintroducetheadiabatic distortedwaveapproximation,andextendthistheorytoincludenonlocalpotentials. Inchapter3wewilldiscussopticalpotentials.Firstwewillintroducetheconceptofaglobal opticalpotential,thenturnourattentiontononlocalpotentials.WewillintroducePerey- Bucktypepotentials,andthecorrespondingcorrectionfactor.Wewillthendescribethe Giannini-RiccopotentialandtheDOMnonlocalpotential.Lasttherewillbeadiscussion oflocalequivalentpotentials.InChapter4wewillpresentourresultsbeginninginSec.4.2 withadiscussionof( p;d )reactionsusingthePerey-Buckpotentialintheentrancechannel withintheDWBA.InSec.4.3wecomparetheofincludingtheDOMpotentialand thePerey-Buckpotentialintheentrancechannelof( p;d )reactionsusingtheDWBA.Lastly, inSec.4.4westudy( d;p )transferreactionswithintheADWAwhileincludingnonlocality 18 consistently.Finally,inChapter5wewilldrawourconclusionsanddiscusstheoutlookfor futurework. Someoftheworkdevelopedduringthisthesis,whilecritical,istootechnicaltopresent inthemainbody.Wehavethuscollectedthatinformationinthefollowingappendices. InAppendixAwediscussthemethodbywhichwesolvethescatteringandboundstate nonlocalequations.InAppendixBwederivethecorrectionfactorthatisappliedtowave functionsresultingfromalocalpotentialinordertoaccountfortheneglectofnonlocality. InAppendixCwederivethenonlocaladiabaticpotential,andinAppendixDwederivethe partialwavedecompositionoftheT-matrixusedtocalculatetransferreactioncrosssections. InAppendixEwegooversomecheckstoensuretheaccuracyofthecodeIdevelopedto computetransfercrosssections,NLAT(NonLocalAdiabaticTransfer).InAppendixF, wediscussamethodtoextractastrophysicallyrelevantANCsusingtheconceptofmirror symmetry.WhileAppendixFisaresearchprojectofrelevancetothethatstandson itsown[25],itdoesnotthethemeofthethesis.Therefore,wechosetoincludeitasa separateappendix. 19 Chapter2 ReactionTheoryfortheTransferof Nucleons Elasticscatteringistheanchorofmanyreactiontheoriessinceelasticscatteringwavefunc- tionsareoftentimesinputstothesetheories,andareusedtocalculatequantitiessuchas transfercrosssections.Elasticscatteringisalsotheprimarymeansbywhichweconstruct thenuclearpotential.Therefore,forreactiontheorytomakeusefulpredictions,wemust haveagoodunderstandingofelasticscattering. Thetheoreticalstudyoftransferreactionscommonlyusesthedistorted-waveBornap- proximation(DWBA).Inthistheory,thetransferprocessisassumedtobeasinglestep,and thebreakupofthedeuteronisincludedimplicitlythroughthedeuteronopticalpotential. Thedeuteronislooselybound,andislikelytobreakupduringthecourseofthereaction. Therefore,notincludingthebreakupofthedeuteronexplicitlyisknowntobeinaccurate [88].Despitebreakupnotbeingincludedexplicitly,theDWBAtheoryisstillcommonly usedtodescribetransferreactionsduetoitssimplicityandthelegacyofcodesavailable. Modernreactiontheoriesthatincorporatebreakupbeginwiththethree-bodypicture oftheprocess.Thethreebodiesaretheneutronandtheprotonmakinguptheincident deuteron,andthetargetnucleus.Apracticalmethodforincludingdeuteronbreakupwas introducedbyJohnsonandTandy[86].Thismethodisusuallyreferredtoastheadiabatic 20 distorted-waveapproximation(ADWA).TheADWAhasbeenbenchmarkedwithmoread- vancedtechniques[89,90],andshowntobecompetitive.In[89],( d;p )angulardistributions fortheADWAandtheexactFaddeevmethodarecompared.Itwasfoundthattheresults fromtheADWAarewithin10%ofthefullsolutionatforwardangles,demonstratingthat theADWAisareliableandpracticalmethodforcalculatingangulardistributionsoftransfer reactions.TheADWAtheorywillbethefocusofthiswork. AnattractivefeatureoftheADWAisthatitincludesbreakupexplicitly,andalsorelies onnucleonopticalpotentials,whicharemuchbetterconstrainedthanthedeuteronoptical potentialsusedintheDWBA.InallknownusesoftheADWA,localnucleonopticalpo- tentialswereused.However,recentstudieshaveshownthatthenonlocalityofthenuclear potentialcanhaveatimpactontransfercrosssections[91,12,14].Thus,ithas becomenecessarytoextendtheADWAformalismtoincludenonlocalpotentials[3]. 2.1ElasticScattering Todescribeelasticscatteringdistributions,webeginbysolvingthepartialwavedecomposed Scodingerequation ~ 2 2 @ 2 @R 2 L ( L +1) R 2 + U N ( R )+ V C ( R ) E ( R )=0 ; (2.1) with U N ( R )beingsomeshort-rangenuclearpotential, V C theCoulombpotential, the reducedmassoftheprojectiletargetsystem,and E theprojectilekineticenergyinthecenter ofmassframe.Here, = f LI p J p I t g isasetofquantumnumbersthateachpartial 21 wave,where L istheorbitalangularmomentumbetweentheprojectileandthetarget, I p and I t arethespinoftheprojectileandtargetrespectively,and J p istheangularmomentum resultingfromcouplingtheorbitalangularmomentumwiththespinoftheprojectile.In theasymptoticlimitwherethenuclearpotentialgoestozero,thescatteringwavefunction takestheform ( R )= i 2 H L ( L ;kR ) S H + L ( L ;kR ) ; (2.2) where = Z 1 Z 2 e 2 ~ 2 k istheSommerfeldparameter,kisthewavenumber, S isthe scatteringmatrixelement(S-matrix),and H and H + aretheincomingandoutgoingHankel functions[92],respectively.Forneutrons, =0.Thetheoreticalscatteringamplitudefor elasticscatteringisrelatedtotheS-Matrixby f p t p i t i ( )= p p i t t i f c ( )+ 2 ˇi k i X L i LJ p i J p M p i M p M i J T C J p i M p i L i M i I p i p i C J tot M tot J p i M p i I t i t i C J p M p LMI p p C J tot M tot J p M p I t t Y LM ( ^ k ) Y L i M i ( ^ k i ) (1 S ) e i ˙ L ( )+ ˙ L i ( i ) (2.3) with p i and t i beingtheprojectionsofthespinoftheprojectileandtarget,respectively, beforethescatteringprocess,while p and t arethespinprojectionsafterthescattering process.Inthisequation, f c isthepointCoulombscatteringamplitude: 22 f c ( )= 2 k sin 2 ( = 2) exp h ln(sin 2 ( = 2))+2 i˙ 0 ( ) i ; (2.4) withtheCoulombphasegivenby ˙ L ( )=ar+ L + ). The S aredeterminedbymatchinganumericalsolutionofEq.(2.1)totheknownasymp- toticform(2.2).Thisisdonebyconstructingthe R -Matrix,whichissimplyaninverse logarithmicderivative. R = 1 R match H L S H + L H L 0 S H + L 0 (2.5) withtheprimesindicatingderivativeswithrespectto R .The R -Matrixisevaluatedat somematchingpointoutsidetherangeofthenuclearinteraction,denotedby R match .The R -matrixuniquelydeterminesthe S -matrixby S = H L R match R H L 0 H + L R match R H + L 0 : (2.6) Oncethe S -matrixforeachpartialwaveiscalculated,thetheoreticaltialcross section,whichisthequantitythatiscomparedwithexperiment,isobtainedbysumming thesquaredmagnitudeofthescatteringamplitudeoverthe m -states,andaveraging overtheinitialstates: 23 d˙ d = 1 ^ I p i ^ I t i X p t p i t i f p t p i t i ( ) 2 (2.7) 2.2Two-BodyT-Matrix Wewouldliketothetransitionamplitude(T-matrix)fora( d;p )transferreaction.Before wegettotransferreactions,letusconsidertheT-matrixfortwo-bodyscattering,such aselasticscattering.ThediscussionofSec.2.1formulatedelasticscatteringintermsofan S-matrix.Thisisthewaymostcodessolveelasticscattering.Anotherwayofformulating elasticscatteringisintermsoftheT-matrix,andleadstoanaturalgeneralizationtothree- bodyscattering,whichisthecasefor d + A reactions. Webeginwithapartialwavedecomposedtwo-bodycoupledchannelequation[8] ~ 2 2 d 2 dR 2 L ( L +1) R 2 + V c ( R ) E ( R )= X 0 h j V j 0 i 0 ( R 0 ) : (2.8) TheT-matrixisanimportantquantityasitgivestheamplitudeoftheoutgoingwaveafter scattering.InEq.(2.2)wewrotetheasymptoticformofthescatteringwavefunctionin termsoftheS-matrix.Wecanwriteanequivalentexpressionfortheasymptoticformofthe wavefunctionintermsoftheT-matrix i ( R ) ! i F L i ( L ;kR )+ T i H + L ( L ;kR ) ; (2.9) 24 where F ( R )istheregularCoulombfunction,andagain, H + istheoutgoingHankel function.If U =0then T =0.ThegoalisthustoanexpressionfortheT-matrix. UsingGreen'sfunctiontechniques,theT-matrixfortwo-bodyscatteringisgivenby[8] T i = 2 ~ 2 k h ˚ ( ) j V j i ; (2.10) where ˚ isthehomogeneoussolutionwhennocouplingpotentialsarepresent, isthe reducedmassofthetwo-bodysystem,and k isthewavenumber.The ( ) superscript indicatesthat ˚ ( ) hasincomingsphericalwavesastheboundarycondition. ˚ ( ) isthus thetimereverseof ˚ .Thecomplexconjugationimpliedinthebra-ketnotationcancelsthe complexconjugationimpliedinthe ( ) . Oftentimes,wecandecompose V intotwopartssothat V = U 1 + U 2 .Wewouldlike tocalculatetheT-matrixforthetransitionwhentwopotentialsarepresent,andderive the two-potentialformula .WebeginbywritingtheT-matrixsubstitutingintheseparated expressionfor V ~ 2 k 2 T 1+2 = Z ˚ ( U 1 + U 2 ) dR: (2.11) Usingthesetwopotentials,wecanvariousfunctions. ˚ isthefreesolution, ˜ isthesolutiondistortedby U 1 only,and isthefullsolution.Thesearerelatedtoeach otherthroughtherelations 25 [ E T ] ˚ =0 ˜ = ˚ + ^ G 0 U 1 ˜ = ˚ + ^ G 0 ( U 1 + U 2 ) = ˜ + ^ G 1 U 2 ; (2.12) withthetwoGreen'sfunctionsgivenby ^ G 0 =[ E T ] 1 ^ G 1 =[ E T U 1 ] 1 : (2.13) Usingtheserelations,wecanrewritetheT-matrixas ~ 2 k 2 T 1+2 i = Z h ˜ ( U 1 + U 2 ) ( ^ G 0 U 1 ˜ )( U 1 + U 2 ) i dR = Z [ ˚U 1 ˜ + ˜U 2 ] dR = h ˚ ( ) j U 1 j ˜ i + h ˜ ( ) j U 2 j i : (2.14) Considertheelasticscatteringofprotonsasanillustrativeexample.Inthiscase, U 1 could betheCoulombpotential,and U 2 couldbethenuclearpotential.Thetermwouldbethe Coulombscatteringamplitude, f c ( ),andthesecondtermwouldbetheCoulomb-distorted 26 nuclearamplitude f n ( ).Thus,thenuclearscatteringamplitudewhenCoulombispresent isnotsimplytheamplitudeduetotheshort-rangednuclearforcesalone,butfromthe ofCoulombontopofnuclear.Fromthesescatteringamplitudesweobtainthetial elasticcrosssectionbycalculating j f c ( )+ f n ( ) j 2 .Thisisusedinourstudiesforcomputing elasticscatteringofchargedparticles. 2.2.1BornSeries UsingEq.(2.14),andtheimplicitformfor inEq.(2.12),wecan,byiteration,formwhatis knownastheBornseries: T (1+2) i = T (1) + T 2(1) = T (1) 2 ~ 2 k h h ˜ ( ) j U 2 j ˜ i + h ˜ ( ) j U 2 ^ G 1 U 2 j ˜ i + ::: i : (2.15) Truncatingtheseriesafterthetermisknownasthedistorted-waveBorn approximation(DWBA). TheDWBAisparticularlyusefulwhenwearedescribingsomekindoftransition.If U 1 isacentralopticalpotentialforallnon-elasticchannels,itcannotcausethetransitionsince centralpotentialsarenotabletochangethequantumnumbersofthescatteredparticle, orchangetheirenergy.Whenthisisthecase, T (1) =0,andwegetanexpressionforthe T-matrixtodescribethetransitionfromanincomingchannel i toanexitchannel 6 = i . T DWBA i = 2 ~ 2 k h ˜ ( ) j U 2 j i i (2.16) 27 Figure2.1:First,second,andall-ordercouplingswithinasetof0 + ,2 + ,and4 + nuclear levels,startingfromthegroundstate. Letusconsider,asanexample,inelasticexcitationofarotationalbandinanucleus.Fig. 2.1illustratessecond,andallordercouplingsbetweenthe0 + groundstate,andthe2 + , and4 + excitedstates.TheDWBAcanbethoughtofasaonestepprocess,where thegroundstatecouplestoeitherthe2 + stateorthe4 + state.Similarly,thesecond-order DWBAisatwo-stepprocesswhere,forexample,thegroundstatecancoupletothe2 + state,andthenthe2 + statecaneithercoupletothe4 + stateorthe0 + .Forapartofthe transferreactionstudiesinthisthesis,thederDWBAwasused.Fromhereonout, theDWBAwillsimplybereferredtoastheDWBA. 2.3Three-BodyT-Matrix Wecangeneralizetheabovediscussiontoathree-bodysystem.Considerthecollectionof thethreebodies n + p + A ,withthecoordinatesappropriatefor A ( d;p ) B giveninFig.2.2. Thecoordinates r np and R dA refertothebeforethetransferoccurs,andthe coordinates r nA and R pB areforimmediatelyafterthetransfer.TheHamiltonianforthe 28 threebodiesisgivenby H = T r np + T R dA + V p ( r np )+ V t ( r nA )+ U pA ( R pA ) ; (2.17) where U pA ( R pA )isthecore-coreopticalpotential.Wecanequivalentlyexpressthetwo kineticenergytermsas T r np + T R dA = T r nA + T R pB .Thisallowsustowritetwot internalHamiltoniansfortheboundstates, H d = T r np + V p ( r np )and H B = T r nA + V t ( r nA ). Thus,wecanwritetheHamiltonianintwoways,calledthepostandthepriorform Figure2.2:Thecoordinatesusedinaoneparticletransferreaction. H = H prior = T R dA + U i ( R dA )+ H p ( r np )+ V i = H post = T R pB + U f ( R pB )+ H t ( r nA )+ V f ; (2.18) where U i;f aretheentranceandexitchannelopticalpotentials,respectively,andthe V i;f interactiontermsaregivenby 29 V i = V t ( r nA )+ U pA ( R pA ) U i ( R dA ) V f = V p ( r np )+ U pA ( R pA ) U f ( R pB ) : (2.19) For( d;p )reactionsitisadvantageoustoworkinthepostform.Insuchacase,wesee that U pA ( R pA ) U pB ( R pB ) ˇ 0.Thistermiscalledtheremnanttermandapproximately cancelsforallbutlighttargets.Thisisbecausetheopticalpotentialsbetween p + A and p +( A +1)arenotlikelytobetlyt.Wedemonstratethattheremnantcan beneglectedinSec.4.1.1. Justlikeinthecaseofelasticscattering,thetialcrosssectionforan A ( d;p ) B reactionisfoundbysummingthesquaredmagnitudeofthescatteringamplitudeoverthe m-states,andaveragingoverinitialstates.TheT-matrixisrelatedtothescattering amplitudeby f A M d p M B ( k f ; k i )= f 2 ˇ ~ 2 r v f v i T A M d p M B ( k f ; k i ) ; (2.20) wherethesubscript i ( f )representstheinitialstate, f isthereducedmass,and v isthevelocityoftheprojectile.Here, A , M d , p ,and M B aretheprojectionofthespin ofthetargetintheentrancechannel,thedeuteron,theproton,andthetargetintheexit channel,respectively. WewouldliketotheT-matrixfor A ( d;p ) B reactions.Usingthepostrepresentation fortheHamiltonian,Eq.(2.18),andthetwo-potentialformula,Eq.(2.14),wecanidentify 30 U 1 = U f ( R pB )+ V t ( r nA )and U 2 = V f .Since U f ( R pB )+ V t ( r nA )producestheelastic scatteringstateof p + B ,itcannotcausethetransfertransition.Therefore, T (1) =0.As aresult, T 2(1) istheonlynon-zeroterm.Inournotation,theexactT-matrixforagiven projectionofangularmomentuminthepostformis: T post A M d p M B ( k f ; k i )= h p M B k f j V np + j A M d k i i : (2.21) Theremnantterm= U pA U pB isnegligibleforallbutlighttargets.TheketinEq. (2.21)fortheT-matrixisthefullthree-bodywavefunctionfor n + p + A ,whilethebra istheproductofaprotondistortedwaveandthe n + A boundstatewavefunction.As aapproximation,wecanapproximatetheketasaproductofadeuteronboundstate andadeuterondistortedwave.ThisisthewellknowndistortedwaveBornapproximation (DWBA).Inthiscase,theketisgivenby j M d A k i i = I A A ( ˘ A ) ˚ j i ( r np ) ˜ (+) i ( k i ; r np ; R dA ;˘ p ;˘ n ) ; (2.22) where I A A ( ˘ A )isthespinfunctionforthetarget,withspin I A andprojection A . ˚ j i ( r np ) istheradialwavefunctionfortheboundstate,whichinthiscaseisthedeuteron,and j i is theangularmomentumresultingfromcouplingthespinofthefragmentintheboundstate (theneutron)totheorbitalangularmomentumbetweenthefragmentandthecore.The distortedwave, ˜ (+) i ,isgivenby 31 ˜ (+) i ( k i ; r np ; R dA ;˘ p ;˘ n )= 4 ˇ k i X L i J P i i L i e i˙ L i ^ J P i ^ J d ˜ L i J p i ( R dA ) R dA (2.23) 8 > < > : ~ Y L i ( ^ k i ) ( ˆ I p ( ˘ p ) n ~ Y ` i (^ r np ) I n ( ˘ n ) o j i ˙ J d ~ Y L i ( ^ R dA ) ) J P i 9 > = > ; J d M d ; where I p ( ˘ p )and I n ( ˘ n )arethespinfunctionsfortheprotonandneutronrespectively, withspin I p = I n = 1 2 .Thespinofthedeuteronisgivenby J d =1,andthespinofthe deuteroncoupledtotheorbitalangularmomentumbetweenthedeuteronandthetarget, L i , givesthetotalprojectileangularmomentum, J P i .Thesphericalharmonics, ~ Y L ,are withthephaseconventionthathasabuiltinfactorof i L .Therefore, ~ Y L = i L Y L with Y L onp.133ofthebook[93].Thehattedquantitiesaregivenby ^ J = p 2 J +1.The function ˜ L i J P i ( R dA )theequation " ~ 2 2 i @ 2 @R 2 dA L i ( L i +1) R 2 dA ! + U dA + V SO 1 L i J P i + V C ( R dA ) E d # ˜ L i J P i ( R dA )=0 ; (2.24) where V SO isthespin-orbitpotential,and V C istheCoulombpotential. U dA isadeuteron opticalpotential.Forthebrawehave h p M B k f j = ( I A ( ˘ A ) n ~ Y ` f (^ r nA ) I n ( ˘ n ) o j f ) J B M B ˚ j f ( r nA ) ˜ ( ) f ( k f ; R pB ) (2.25) 32 where ˚ j f ( r )isthe n + A boundstateradialwavefunction, j f istheangularmomentum oftheboundstateresultingfromcouplingthespinoftheneutrontotheorbitalangular momentumoftheboundstate, ` f ,while J B isthetotalangularmomentumofthe nucleus, B .Theexitchanneldistortedwaveisgivenby ˜ ( ) f ( k f ; R pB ;˘ p )= 4 ˇ k f ^ I p X L f J P f i L f e i˙ L f ^ J P f ˜ L f J P f ( R pB ) R pB 8 < : ~ Y L f ( ^ k f ) n I p ( ˘ p ) ~ Y L f ( ^ R pB ) o J P f 9 = ; I p p (2.26) andthefunction ˜ L f J P f ( R pB ) " ~ 2 2 f @ 2 @R 2 pB L f ( L f +1) R 2 pB ! + U pB + V SO I p L f J P f + V C ( R pB ) E p # ˜ L f J P f ( R pB )=0 (2.27) with U pB beingtheprotonopticalpotentialintheexitchannel. WeneedtodoapartialwavedecompositionoftheT-matrix,Eq.(2.21),sothatwecan calculatethescatteringamplitude,Eq.(2.20),andhence,thecrosssection,inanumerically tway.WeshowinAppendixDthatforageneral ` i and ` f relativeorbitalangular momentumintheinitialandboundstates,thepartialwavedecompositionoftheT- matrixisgivenby 33 Figure2.3:ThecoordinatesusedtocalculatetheT-matrixfor( d;p )transfer. T QM Q m f = C X K X L i J P i X L f J P f A K;L i J P i L f J P f QM Q m f ( ^ k f ) I K;L i J P i L f J P f ; (2.28) wherephaseandstatisticalfactorsarecollectedin C = 32 ˇ 3 ^ I n ^ I p k i k f ( ) 3 I p + j i + J d +2 j f ^ j i ^ j f ; (2.29) angularmomentumcouplingsaremostlyputin, 34 A K;L i J P i L f J P f QM Q m f ( ^ k f )= ( ) K ^ K h ` f ` i ( K ) I n I n (0) KM j ` f I n ( j f ) ` i I n ( j i ) KM i X L i J P i X L f J P f i 3 L i + L f + ` f + ` i e i ( ˙ L i + ˙ L f ) ^ L i ^ L f ^ J P i ^ J P f I p J d ( j i ) L f L i ( K ) j f m f j I p L f ( J P f ) J d L i ( J P i ) j f m f i X g h L f L i ( g ) J P f J P i ( j f ) QM Q j L f J P f ( I p ) L i J P i ( J d ) QM Q i X m g C QM Q gm g j f m f C gm g L f m g L i 0 Y L f m g ( ^ k f ) ; (2.30) andtheradialintegralsarecontainedin I K;L i J P i L f J P f = X M K ( ) M K C K; M K L f 0 L i ; M K X ~ m f ~ m i C KM K ` f ~ m f ` i ~ m i Z ˚ j f ( r nA ) ˜ L f J P f ( R pB ) V ( r np ) ˚ j i ( r np ) ˜ L i J p i ( R dA ) R pB r 2 nA R dA Y L i ; M K ( ^ R dA ) Y ` f ~ m f (^ r nA ) Y ` i ~ m i (^ r np )sin dR pB dr nA : (2.31) The9jsymbol, h j 1 j 2 ( j 12 ) j 3 j 4 ( j 34 ) jm j j 1 j 3 ( j 13 ) j 2 j 4 ( j 24 ) j 0 m 0 i ,isgivenonp.334of[93],while the C j 3 m 3 j 1 m 1 j 2 m 2 aretheClebsh-Gordancots.Thecoordinatesusedtocalculatethe integralintheequationabovearegiveninFig.2.3. Withthispartialwavedecomposition,thetialcrosssectionisgivenby 35 d˙ d = k f k i i f 4 ˇ 2 ~ 4 ^ J 2 B ^ J 2 d ^ J 2 A ^ j 2 f X m f QM Q T QM Q m f T QM Q m f : (2.32) IntroducingEq.(2.28)intoEq.(2.32)weobtaintheformforthetransfercrosssectionused inthiswork. 2.4Three-BodyModels Weareinterestedindescribingthereaction A ( d;p ) B wherethenucleus B = A + n is aboundstate.Inprinciple,thescatteringstateforthedeuteroncanbemodeledasa d + A two-bodyproblem.Thisisoftendonewherethe d + A opticalpotentialistakenfrom todeuteronelasticscattering.However,duetothelooselyboundnatureofthedeuteron, itisimportanttoconsiderdeuteronbreakupexplicitly.Thus,webeginwithathree-body Hamiltonianforthe n + p + A system, H 3 B = T R + T r + U nA + U pA + V np : (2.33) Here T R and T r arethekineticenergyoperatorsforthecenterofmassmotionandthe n p relativemotion,respectively. V np istheneutron-protoninteraction,while U pA and U nA aretheproton-targetandneutron-targetinteractions.Thewavefunction r ; R )describes adeuteronincidentonanucleus A andisasolutiontotheequation H 3 B = E Avarietyofmethodsexisttosolvethethree-bodyproblem.TheFaddeevapproach anexactmethodtosolvethethree-bodyproblemforaparticularHamiltonian[94],such 36 astheHamiltoniangiveninEq.(2.33).Faddeevmethodsarecomputationallyexpensive, andsofarcurrentimplementationshavewithhandlingheavysystemsdueto theCoulombpotential.TheContinuumDiscretizedCoupledChannel(CDCC)method anothermeansofsolvingthethree-bodyproblem[95].However,thismethodtoois computationallyexpensive.TheADWAcanprovideareliabledescriptionoftransfercross sectionswhilerequiringminimalcomputationcosts.Studieshavebenchmarkedthesethree methodsandhaveshownthattheADWAcanreliablyreproducetransfercrosssectionswhen comparedtotheothertwomoreadvancedmethodsintheenergyrangesrelevantforthis study[89,90]. 2.5AdiabaticDistortedWaveApproximation Considerthethree-bodywavefunctiondescribingthedeuteronscatteringstate.Aformal expansionofthiswavefunctionisgivenby r ; R )= d ( r ) X d ( R )+ Z d k k ( r ) X K ( R ) ; (2.34) where d ( r )isthedeuteronboundstatewavefunction,and X d ( R )istheelasticdeuteron centerofmassscatteringwavefunction. k ( r )describestherelativemotionofan n p pair,andthecontinuumcomponents X K ( R )describethemotionofthecenterofmassof this n p pairscatteredwithrelativeenergy K . IntheDWBA, r ; R )= d ( r ) X d ( R ),sobreakupisnotincludedsincethesecondterm inEq.(2.34)isneglected,whichcontainsallthebreakupcomponents.Whileitisknownthat 37 breakupisimportanttothedynamicsofdeuteroninducedtransferreactions,calculatingthe secondterminEq.(2.34)toallordersaccuratelyis InformulatingtheADWA,JohnsonandTandy[86]realizedthattocalculatetransfer crosssections,weneedtoknowthethree-bodywavefunctiononlyinthecombination V np j i , asisseeninEq.(2.21)withtheremnanttermneglected.Therefore,analternativeexpansion shouldbesoughtthataccuratelyrepresentsthethree-bodywavefunctionwithintherange of V np .TheessenceoftheADWAmethod[86]istoexpandthethree-bodywavefunction inadiscretesetofWeinbergstates, r ; R )= 1 X i =0 i ( r ) X i ( R ) : (2.35) TheWeinbergstatesareacompletesetofstateswithintherangeofthe V np interaction, andaregivenby T r + i V np ( r )+ d i ( r )=0 ; (2.36) where d isthedeuteronbindingenergy,andachstateisorthogonalbytherelation h i j V np j j i = ij : (2.37) TheWeinbergcomponent 0 ( r )occurswhen 0 =1.Therefore,thecomponent issimplythedeuterongroundstatewavefunctionwithatnormalizationcondition. 38 EachsuccessiveWeinbergcomponentwillcontainanadditionalnode.SinceeachWeinberg statehasthesamebindingenergy,theasymptoticpropertiesofeachWeinbergstatewillbe identical.InFig.2.4weshowthefourWeinbergstateswhenusingacentralGaussian whichreproducesthebindingenergyandradiusofthedeuterongroundstate,asin[96]. Forthefourstates, i = f 1 ; 5 : 2 ; 12 : 7 ; 23 : 4 g .Theinsetshowstheasymptoticproperties ofeachstate.SinceeachWeinbergstatehasthesamebindingenergy,theydecaywiththe samerateoutsidetherangeoftheinteraction. Sinceweareonlyinterestedindescribingtheshort-rangedpropertiesofthethree-body wavefunction,havingthewrongasymptoticsisnotaconcern.Foraneexpansion, onlyanumberoftermsshouldbenecessaryforanadequatedescriptionofthewave functionwiththeinclusionofbreakup.KeepingallthetermsintheexpansionofEq.(2.35) resultsinacomplicatedcoupledchannelsetofequationstodescribethescatteringprocess. Toeliminatethiscomplication,thetypicalprocedureistokeeponlythetermofthe expansion.Thishasbeenshowntobeanexcellentapproximation[97]. Toderivetheadiabaticpotential,weinserttheexpansionofthethree-bodywavefunc- tion,Eq.(2.35),intotheScodingerequationusingourparticularthree-bodyHamiltonian, Eq.(2.33).Sincewearekeepingonlytheterm,wewillwritethewavefunctionas r ; R ) ˇ 0 ( r ) X AD ( R ).Thisgivesus T R + T r + U nA ( R n )+ U pA ( R p )+ V np E 0 ( r ) X AD ( R )=0(2.38) Here, E isthetotalsystemenergygivenby E = E d d ,where E d istheincidentdeuteron kineticenergyinthecenterofmassframe,and d isthedeuteronbindingenergy.Since 39 Figure2.4:ThefourWeinbergStateswhenusingacentralGaussianwhichreproduces thebindingenergyandradiusofthedeuterongroundstate.Theinsetshowstheasymptotic propertiesofeachstate. ( T r + V np ( r 0 ( r )= d 0 ( r ),wecanmakethisreplacementgivingus T R + U nA ( R n )+ U pA ( R p ) E d 0 ( r ) X AD ( R )=0(2.39) Wenowmultiplyby h 0 j V np andusetheorthogonalitypropertiesoftheWeinbergstatesto obtain h T R + U Loc AD ( R ) E d i X AD ( R )=0 ; (2.40) wherethelocaladiabaticpotential, U Loc AD ( R )isgivenby 40 U Loc AD ( R )= 0 j V np ( U nA ( R n )+ U pA ( R p )) j 0 i : (2.41) Itisimportanttonotethat X AD ( R )isnotthesameastheelasticscatteringwave function X d ( R )intheplanewavebasisofEq.(2.34). X d ( R )describeselasticscattering, andthepotentialusedtogenerate X d ( R )wouldbeadeuteronopticalpotentialobtainedby elasticscatteringdata.Ontheotherhand, U AD doesnotdescribedeuteronelastic scattering.Infact,theadiabaticpotentialisonlyofusetodescribetransferreactions. However,theinputopticalpotentials, U nA and U pA ,dodescribeelasticscattering,andare obtainbytonucleondata.ThisisanadvantageousfeatureoftheADWAasnucleon opticalpotentialsaremuchbetterconstrainedthandeuteronopticalpotentials. 2.6NonlocalAdiabaticDistortedWaveApproxima- tion WewouldliketoconsidertheadiabaticpotentialinEq.(2.41)whenweareusingnonlocal nucleonopticalpotentials.AdetailedderivationispresentedinAppendixC.Herewe willgiveanoverviewofthederivation.Asanexample,considertheneutronnonlocal operatoractingonthethree-bodywavefunction: ^ U nA r ; R )= Z U nA ( R n ; R 0 n R 0 n ; R 0 p ) ( R 0 p R p ) d R 0 n d R 0 p =8 Z U nA R r 2 ; 2 R 0 R r 2 r 2( R 0 R ) ; R 0 ) d R 0 : (2.42) 41 ThecoordinatesusedforcalculatingtheneutronnonlocalpotentialareshowninFig.2.5, wheretheopendashedcirclerepresentstheneutroninatpointinspacetoaccount fornonlocality.Sincewearecalculatingtheopticalpotentialfortheneutroninteracting withthetarget,theprotonremainsstationarywhenintegratingtheneutroncoordinateover allspace.Hence,thereasonforthedeltafunctioninEq.(2.42). Figure2.5:Thecoordinatesusedforconstructingtheneutronnonlocalpotential.Theopen dashedcirclerepresentstheneutroninatpointinspacetoaccountfornonlocality. InEq.(2.42),theJacobianforthecoordinatetransformationisunity,andweintegrated over d r 0 toeliminatethedeltafunction.Weusedthevectortions R p;n = R r 2 ,where R p usesthe\+"signand R n usesthe\ "sign.Asimilarexpressionisfoundfortheproton nonlocaloperator. SinceweareusingonlytheWeinbergstate,wewilldropthe\0"subscriptonthewave functions,andwritetheexpansionofthethree-bodywavefunctionas r ; R ) ˇ r ) X ( R ). Thus,thegeneralnucleonnonlocaloperatoris 42 ^ U NA r ) X ( R )=8 Z U NA R r 2 ; 2 R 0 R r 2 r 2( R 0 R )) X ( R 0 ) d R 0 : (2.43) Addingandsubtracting R inthesecondargumentof U NA andmakingtheon s = R 0 R ,wecanrewritethenucleonnonlocaloperatoras ^ U NA r ) X ( R )=8 Z U NA R p;n ; R p;n +2 s r 2 s ) X ( R + s ) d s : (2.44) InEq.(2.23)wegavethedeuterondistortedwaveforeachprojectionofangularmomen- tumofthedeuteronandtarget.Nowweneedthedeuteronwavefunctionforrelativemotion between d and A foreachvalueandprojectionoftotalangularmomentum, J T M T .Thisis givenby r ; R ) ˇ r ) X ( R )= X `LJ p ˚ ` ( r ) ˜ J T M T LJ p ( R ) R (2.45) ˆ nnn 1 = 2 ( ˘ n ) 1 = 2 ( ˘ p ) o 1 ~ Y ` (^ r ) o 1 ~ Y L ( ^ R ) o J p I t ( ˘ t ) ˙ J T M T : ThedescriptionofeachtermisgivenafterEq.(2.23).Thecoordinatesforconstructingthe systemwavefunctionforthedeuteronscatteringstatearegiveninFig.2.6. Wewouldliketothepartialwavedecompositionof 43 Figure2.6:Thecoordinatesusedforconstructingthesystemwavefunctionforthe d + A deuteronscatteringstate. h ^ T R + V C ( R )+ U so ( R ) E d i r ) X ( R )= ^ U nA + ^ U pA r ) X ( R ) ; (2.46) where U so ( R )isthesumoftheneutronandprotonspin-orbitpotentials.Tobeginthe partialwavedecomposition,multiplyEq.(2.46)by X ` 0 ˚ ` 0 ( r ) V np ( r ) ( nnn 1 = 2 ( ˘ n ) 1 = 2 ( ˘ p ) o 1 ~ Y ` 0 (^ r ) o 1 ~ Y L 0 ( ^ R ) o J 0 p I t ( ˘ t ) ) J T M T andintegrateover d r , d R , d˘ n , d˘ p and d˘ t .The lhs oftheequationbecomes 1 R ~ 2 2 @ 2 @R 2 L ( L +1) R 2 + E d ˜ J T M T LJ p ( R ) : (2.47) 44 Asweonlyconsidered ` =0deuteronsinourcalculations,letusmakethisassumptionright atthebeginningofourpartialwavedecompositionofthe rhs .Therefore,thetwo ~ Y ` (^ r ) termsgive1 = 4 ˇ ,andthepartialwavedecompositionofthe rhs ofEq.(2.46)is 8 4 ˇ X L 0 J 0 p Z ˚ 0 ( r ) V np ( r ) U NA R p;n ; R p;n +2 s ˚ 0 ( j r 2 s j ) ˜ J T M T L 0 J 0 p ( j R + s j ) j R + s j (2.48) ˆ n 1 ( ˘ np ) ~ Y L ( ^ R ) o J p I t ( ˘ t ) ˙ J T M T ˆ n 1 ( ˘ np ) ~ Y L 0 ( \ R + s ) o J 0 p I t ( ˘ t ) ˙ J T M T d s d r d R d˘ t d˘ n d˘ p : Ourgoalistocoupletheintegranduptozeroangularmomentum.Thiswillbespherically symmetricsowecanusesymmetrytoreducethedimensionalityoftheintegral.Afterseveral additionalstepsofalgebrawearriveat: ~ 2 2 @ 2 @R 2 L ( L +1) R 2 V C ( R ) U so ( R )+ E d ˜ J T M T LJ p ( R )(2.49) = 8 R p ˇ ^ L Z ˚ 0 ( r ) V np ( r ) ˜ J T M T LJ p ( j R + s j ) j R + s j Y L 0 ( \ R + s ) U nA ( R n ; R n +2 s ) ˚ 0 ( j r 2 s j )+ U pA R p ; R p +2 s ˚ 0 ( j r +2 s j ) r 2 sin r dr r d s : Thisisultimatelythenonlocalequationwesolvetoobtaintheadiabaticwavethatrepresents d + A initialscatteringtobeintroducedintotheT-matrix,Eq.(2.21).Moredetailsofthe derivationaregiveninAppendixC. 45 2.7SpectroscopicFactors Transferreactionsareperformednotonlytoextractspinandparityassignmentsofenergy levels,butalsotoextractspectroscopicfactors.Asanexampletounderstandtheconceptof aspectroscopicfactor,letusconsiderthe 17 Onucleus,whichcanbemodeledasa 16 Ocore plusavalenceneutron.Letusassumethat 16 Ocontainsonlya0 + groundstateanda2 + excitedstate.Thegroundstateof 17 Oisa5 = 2 + state.Duetothepossibleexcitedstatesof thecore,thegroundstateof 17 Ocanoccurinvariousurations.Hereweconsideronly twoforsimplicity: j 17 O g:s: i = 1 16 O(0 + ) n 1 d 5 = 2 5 = 2 + + 2 h 16 O(2 + ) n 2 s 1 = 2 i 5 = 2 + (2.50) Thesetwofor 17 Oare:thegroundstateofthe 16 Ocorecoupledtothevalence nucleonina1 d 5 = 2 orbital,andthe 16 Ocoreinitsexcited2 + statecoupledtothevalence neutronina2 s 1 = 2 orbital.Bothurationsmustcorrespondtothegroundstateenergy, whichmeansthattheavailableenergyfortheneutroninthe1 d 5 = 2 orbitalistthan thatfortheneutroninthe2 s 1 = 2 orbitalduetocoreexcitation.Thespectroscopicfactor tellsushowprobableitistothevalenceneutronin 17 O g:s: ina1 d 5 = 2 with 16 O g:s: ,andisgivenby: S 1 d 5 = 2 = jh 16 O(0 + ) j 17 O g:s: ij 2 = 2 1 (2.51) 46 Thespectroscopicfactorforthewith 16 Ointhegroundstatecanoftenbe cleanlyextractedfromthe 16 O( d;p ) 17 O g:s: .Thereasonbeingthefastradialfallforthe othercoduetotheadditionalbindingoftheneutroncausedbycoreexcitation. InasimpletheoreticalDWBAanalysis,onlytheofEq.(2.50)isincludedin thecalculation.Thepeakofthetransferdistributioncorrespondstoimpactparametersfor thedeuterongrazingthesurface.Therefore,oneexpectsthetransferprocesstoadequately bedescribedasaone-stepprocess.Sinceweleftoutalloftheotherour theoryassumedthat Theory 1 =1,somostoftenitwillover-predictthetransfercross sectionatthepeak.Bynormalizingthetheoreticaltransferdistributionatthepeakto theexperimentaldistributionatthepeak,wecanextractthephysical j 1 j 2 value.It isforthisreasonthatweareinterestedinthemagnitudeofthetransfercrosssectionatthe peakthroughoutthisthesis. 47 Chapter3 OpticalPotentials epotentialsdescribingthescatteringprocessareneededwhendoingcalculationsof reactions,andanaccuratetheoreticaldescriptionofthesereactionsisrequiredforthereliable extractionofdesiredquantities.Opticalpotentialshavebeenobtainedphenomenologically, primarilyfromelasticscatteringdata,butsometimesfromabsorptioncrosssectionsand polarizationobservables[98,23,49,99,48].Inallopticalpotentials,thenuclearpotentialis assumedtobecomplex,wheretheimaginaryparttakesintoaccountlossoftonon-elastic channels. Inallcommonlyusedglobalopticalpotentials,theinteractionisassumedtobelocal. Asaconsequence,thesepotentialsallhaveastrongenergydependence.Inherentinthe localassumptionofthepotentialisafactoringoutofthemany-bodydegreesoffreedom. Therefore,theanti-symmetrizationofthemany-bodywavefunction,andthecouplingto allthenon-elasticchannels,isnotexplicitlytakenintoaccount,andmustbeintroduced elyintothelocalpotentialthroughanenergydependenceoftheparameters. 3.1GlobalOpticalPotentials Globalopticalpotentialsareoftenusedintheanalysisofnuclearreactions.Globalpotentials areveryconvenientastheycaneasilybeextrapolatedtoregionsofthenuclearchartwhere dataisnotavailable,ortheycanbeusedatenergieswheredatahasnotbeentaken.However, 48 suchextrapolationsshouldalwaysbedonecarefully.Aglobalopticalpotentialconstructed fromtostablenucleimaynotgivesensibleresultswhenextrapolatedtoexoticnuclei. Nonetheless,usingaglobalpotentialissometimestheonlyoptionavailablewhenmaking theoreticalpredictionsofexperimentsonexoticnuclei. Globalopticalpotentialsattempttodescribethenuclearpotentialacrosssomerange ofmassandenergy.Todothis,somekindofformforthecomplexmeanmustbe assumed.Inmostconstructionsofglobalopticalpotentials,therealandimaginaryparts containcombinationsofVolume(v),Surface(d),andSpin-Orbit(so)termsgivenby U v ( R )= V v f ( R;r v ;a v ) U d ( R )=4 a d V d d dR f ( R;r d ;a d ) U so ( R )= ~ m ˇ c 2 V so 1 R d dR f ( R;r so ;a so )2 L s ; (3.1) where f ( R;r;a )= " 1+exp R rA 1 = 3 a !# 1 : (3.2) TheCoulombpotentialistakentobethatofahomogeneoussphereofcharge V C ( R )= 8 > > > < > > > : Z 1 Z 2 e 2 2 3 R 2 R 2 C if R 50MeV)areconsideredinthe Onceafunctionalformischosen,thefreeparametersarevariedtoobtainabest toalargeamountofelasticscatteringdata.Inthesethedepth,andsometimesthe radiusandofthevarioustermscanbeenergyandmassdependent.When 50 usingthepotential,thetargetmass,charge,andtheprojectileenergymustbesp Then,thevalueforthedepth,radius,andusenessofeachtermiscalculated.Thereare severalglobalopticalpotentialsonthemarket,someofthecommononesarediscussedhere [49,23,48]. 3.2MotivatingNonlocalPotentials WealreadydiscussedinSec.1.2thesourcesofnonlocalityfortheeNAinteraction. HereweprovideadditionalperspectivebasedonFeshbach'swork[52,53].Whenderived fromthemany-bodyproblem,thesingleparticleScodingerequationdescribingthemotion ofnucleonsinnucleiisnonlocal.IntheprojectionoperatortheoryofFeshbach,aformal equationforthesingleparticlemotioncanbederived.TheformalismofFeshbachusesthe projectionoperator P toprojectthemany-bodywavefunctionontothechannelsthatare consideredexplicitly,andtheprojectionoperator Q projectsontoallchannelsleftoutfrom themodelspace.Considerthecasewhen P projectsthemany-bodywavefunction,onto theelasticchannel.Whenthisisthecase,theoperatorsareby P = j gs ih gs j ; Q =1 P ; Qj gs i =0 ; (3.4) where j gs i givestheelasticscatteringchannelwherethetargetremainsinitsgroundstate, j gs i andtheprojectileundergoeselasticscattering, j X el i : j gs i = j X el ij gs i : (3.5) 51 Usingthisprojectionoperatorformalism,Feshbachshowedthataformalexpressionforthe Scodingerequationtodescribeelasticscatteringisgivenby E T R h gs j V j gs ih gs j VQ 1 E QHQ QV j gs i X el ( R )=0 ; (3.6) with V beingthebareprojectile-targetinteraction.Fromthisequation,wecanidentifythe opticalpotentialas U = V + VQ 1 E QHQ QV: (3.7) Thetermappearslocalwhilethesecondisinherentlynonlocal.Ifweallowforanti- symmetriztionbetweentheprojectileandallthenucleonsofthetarget,wherebytheincident nucleonmaynotthebesameastheexitingnucleon,eventhetermbecomesnonlocal. IntheHartree-Focktheory,usedforboundstatecalculations,thenaturallyarisingexchange termisadirectresultofanti-symmetrization. WhileEq.(3.7)isaformalequation,itgivessomephysicalinsightintothenatureof nonlocality.Thenucleonbeginsinthespaceofelasticscattering, P -space.Thesystemthen couplestosomenon-elasticchannelandpropagatesthroughthatspace, Q -space,before returningbacktotheelasticchannelatsomelaterlocationinspace.Thisalsogivesa physicalfortheneedtohaveapotentialwithtwoarguments, U ( R ; R 0 ).Flux leaves P -spaceandgoesinto Q -spaceat R 0 .Thepropagatesthrough Q -space,before beingdepositedbackinto P -spaceat R . 52 Withanonlocalpotential,theScodingerequation,Eq.(2.1),getstransformedintoan intialequation ~ 2 2 r 2 R )+ E R )= U o ( R R )+ Z U NL ( R ; R 0 R 0 ) d R 0 : (3.8) Todescribethephysicsofleaving P -spaceat R 0 ,propagatingthrough Q -space,and returningto P -spaceat R ,itbecomesnaturaltodescribethepotentialatthepoint R tobe dependentontheoverlapofthewavefunctionandthepotentialatallotherpointsinspace, hence,theneedfortheintegral. 3.3Perey-BuckType Duetothesuccessoflocalglobalopticalpotentials,itwouldbenaturaltoassumethat similarglobalparameterizationshavebeenmadetononlocalpotentials.Unfortunately,this isnotthecase.Toourknowledge,thereareonlythreeglobalnonlocalopticalpotentials thatareconstructedphenomenologicallyfromelasticscattering.TheseminalpaperofPerey andBuckin1962wastheattempttomakeaparametersetforanonlocalmodel[1]. Inthelate70sGianniniandRiccoconstructedtheirpotentialbyalargeamountof datatoalocalform.Theythenusedanapproximatetransformationformulatoobtainthe nonlocalparameters[71,72].Finally,in2015,Tian,Pang,andMa(TPM)constructedtheir potentialthroughtoelasticscatteringandanalyzingpowers[2]. Whiletheexistenceofnonlocalityinthenuclearpotentialhaslongbeenknown,there hashistoricallybeengreatyinspecifyingtheexactformforthenonlocalnuclear 53 potential.AsimpleformwasproposedbyFrahnandLemmer[100],andlaterdeveloped andimplementedbyPereyandBuck[1].ThePerey-Buckpotentialisthemostcommonly referredtophenomenologicalnonlocalopticalpotentialduetoitssimplicity.ThePerey-Buck potentialisgivenby U NL PB ( R ; R 0 )= U R + R 0 2 H R R 0 ; ; (3.9) wherethefunction U R + R 0 2 isofaWoods-Saxonform,andthefunction H ( R R 0 ; ) ischosentobeanormalizedGaussianfunction, H R R 0 ; = exp R R 0 2 ˇ 3 2 3 : (3.10) Makingtheion p = R + R 0 2 , U ( p )hasaformsimilartothoseinlocalopticalmodel calculations.ForthePerey-Bucknonlocalpotential, U ( p )consistsofanonlocalrealvolume, nonlocalimaginarysurface,andalocalrealspin-orbitpotential. Theparameterthattherangeofthenonlocalityis .Asaphysicalexample tounderstandthisparameter,consideranti-symmetrization,whichaswealreadydiscussed isasourceofnonlocality.Sincethetruemany-bodywavefunctionisanti-symmetric,it ispossiblefortheincidentnucleontonotbethesameasthescatterednucleon.Forthe incidentnucleonto\switchplaces"withoneofthenucleonswithinthetarget,itisreasonable toassumethatthetwonucleonsmustberelativelyclosetoeachotherforthistooccur. Typically,nonlocalityrangesareoftheorderofthesizeofthenucleon.ForthePerey-Buck potential, isat0 : 85fm.Forothernonlocalpotentials,suchastheTPM, isan 54 additionalparameterintheirTheresultingvaluefor inneutronandprotonversions oftheTPMareverysimilartothatofthePerey-Buckpotential. AsthePerey-Buckpotentialisphenomenological,theparametersinvolvedareobtained byelasticscattering.Twodatasetswereused: n + 208 Pbat7 : 0and14 : 5MeV.Perey andBuckassumedthattheparameterswereenergyandmassindependent.Therefore,a singleparametersetcompletelythenonlocalpotentialofPereyandBuck.The parametersetforthePerey-BuckpotentialisgiveninTable3.1. TheworkofTian,Pang,andMa(TPM)wasthemodernattempttoaparameter setforanonlocalpotential[2].Intheirt,amultitudeofdatawasconsidered,spanning energyandmass.ThisisagreatimprovementoverthetwodatasetsPereyandBuckused intheirAseparatepotentialforprotonsandneutronswasfoundfortheTPMpotential, unlikePereyandBuckwherenoprotonswereusedintheAswithPereyandBuck,the parametersintheTPMpotentialareassumedenergyandmassindependent.Forhigher energyreactions,theTPMpotentialwasfoundtoprovideabetter ˜ 2 thanthePerey-Buck potential,whileatlowerenergiesthetwopotentialsarecomparable.Theparametersetfor theTPMpotentialisgiveninTable3.1. TheTPMpotentialwaspublishedaftermuchoftheworkforthisstudywascompleted. Hence,forthisreason,andduetothepopularityandwidespreaduseofthePerey-Buck potential,thepotentialweusedtoassesstheofnonlocalitywasthatofPereyand Buck.Inthisstudy,weareinterestedinbetweennonlocalandlocalequivalent calculations,andnotsomuchonthequalityofthenonlocalcalculationsthemselves.When itbecomesnecessarytousenonlocalpotentialstoextractinformationfromexperiments,the improvedTPMpotentialisthebetterchoice. Withlocalopticalpotentials,hundredsofelasticscatteringdatasetsusingbothprotons 55 Perey-Buck TPM TPM Neutrons Protons V v 71 : 00 70 : 00 70 : 95 r v 1 : 22 1 : 25 1 : 29 a v 0 : 65 0 : 61 0 : 58 W v | 1 : 39 9 : 03 r w v | 1 : 17 1 : 24 a w v | 0 : 55 0 : 50 W d 15 : 00 21 : 11 15 : 74 r w d 1 : 22 1 : 15 1 : 20 a w d 0 : 47 0 : 46 0 : 45 V so 7 : 18 9 : 00 8 : 13 r so 1 : 22 1 : 10 1 : 02 a so 0 : 65 0 : 59 0 : 59 r c 1 : 22 | 1 : 34 0 : 85 0 : 90 0 : 88 Table3.1:PotentialparametersforthePerey-Buck[1]andTPM[2]nonlocalpotentials. andneutronsscatteringofarangeofnucleiatarangeofenergiesareusedtoconstrainthe parametersofthepotential.Therefore,itwouldbeexpectedthatthetwodatasetsPerey andBuckusedtoconstraintheirparameterswouldnotbettoreproduceelastic scatteringoverawiderangeofnucleiandenergies.Nonetheless,reasonableagreementwith dataisseendespitethesimplisticwayinwhichthepotentialparametersareconstrained,as isseeninFig.3.1. Inordertosolvethenonlocalequation,werstneedtodoapartialwaveexpansionof thenonlocalpotential, U NL PB ( R ; R 0 )= X LM g L ( R;R 0 ) RR 0 Y LM ( ^ R ) Y LM ( ^ R 0 ) = X L 2 L +1 4 ˇ g L ( R;R 0 ) RR 0 P L (cos ) ; (3.11) 56 Figure3.1:tialelasticscatteringrelativetoRutherfordasafunctionofscattering angle.(a) 48 Ca( p;p ) 48 Caat15 : 63MeVwithdatafrom[9](b) 208 Pb( p;p ) 208 Pbat61 : 4MeV withdatafrom[10]. wherewe astheanglebetween R and R 0 .Multiplyingbothsidesby P L (cos ), integratingoverallangles,usingtheorthogonalityoftheLegendrepolynomials,andsolving for g L ( R;R 0 ),wethat g L ( R;R 0 )=2 ˇRR 0 Z 1 1 U NL PB ( R ; R 0 ) P L (cos ) d (cos ) : (3.12) Now,insertingthePerey-Buckformforthenonlocalpotential,replacing 1 2 R + R 0 with 1 2 ( R + R 0 ),anddoingafewlinesofalgebraoutlinedinAppendixA,wearriveat g L ( R;R 0 )= 2 i L z ˇ 1 2 j L ( iz )exp R 2 + R 0 2 2 U 1 2 ( R + R 0 ) (3.13) withz= 2 RR 0 2 ,and j L beingsphericalBesselfunctions.Wenowhaveapartialwaveequation intermsof g L ( R;R 0 )foreachfunction ˜ L ( r ): ~ 2 2 d 2 dr 2 L ( L +1) R 2 ˜ L ( R )+ E˜ L ( R )= Z g L ( R;R 0 ) ˜ L ( R 0 ) dR 0 : (3.14) 57 3.3.1CorrectionFactor EversincePereyandBuckintroducedtheirpotentialin1962,nearlyallanalyticwork involvingapproximationstononlocalpotentials,orcorrectionstowavefunctionsduetoa nonlocalpotential,assumedthePerey-Buckformforthenonlocality.Thatis,thereisonly onenonlocalityparameter, ,andthenonlocalpartofthepotentialtakestheformEq.(3.9). ThisisnottruefortheDOMortheGiannini-Riccononlocalpotential,discussedlater,where thereareseveraltermswithatnonlocalityparameter. Accountingforthenonlocalitythroughtheenergydependenceofalocalopticalpotential isknowntobet.Onekeyfeatureofanonlocalpotentialisthatitreducesthe amplitudeofthewavefunctioninthenuclearinteriorcomparedtothewavefunctionfroman equivalentlocalpotential.Thisistheso-calledPerey[101].Physically,thereductionof thewavefunctioncanbeunderstoodtoresultfromtherepulsionduetothePauliprinciple. Sinceitwasn'tpracticaltosolvetheintialequationwithanonlocalpoten- tialinthe1960s,therewasgreatinterestinawaytoaccountforthisreductionof amplitudewhilestillkeepingthesimplicityofsolvingalocalequation.Thiswasaccom- plishedbyAustern,whostudiedthewavefunctionsofnonlocalpotentialsanddemonstrated thePereyinonedimension[73].Later,Fiedeldeydidasimilarstudyforthethree dimensionalcase[75].Usingatmethod,Austernpresentedawaytorelatewave functionsobtainedfromanonlocalandalocalpotentialinthethree-dimensionalcase[74]. Sincethen,nonlocalcalculationshavebeenavoidedusingthePereycorrectionfactor(PCF). ThePereycorrectionfactorisderivedindetailinAppendixB.Herewesimplyoutline thederivation.ToderivethePCF,webeginwiththethreedimensionalScodingerequation 58 ~ 2 2 r 2 NL ( R )+ E NL ( R )= U o ( R NL ( R )+ Z U NL ( R ; R 0 NL ( R 0 ) d R 0 ; (3.15) where U o ( R )isthelocalpartofthepotential,andtypicallycontainsspin-orbitandCoulomb terms.Letusafunction, F ( R )thatconnectsthelocalwavefunction loc ( R )resulting fromthepotential U LE ( R )withthewavefunctionresultingfromanonlocalpotential, NL ( R ): NL ( R ) F ( R loc ( R ) : (3.16) Thepotential U LE ( R )issuchthatitreproducestheexactsameelasticscattering asthenonlocalpotential.Sincethelocalandnonlocalequationsdescribethesameelastic scattering,thewavefunctionsshouldbeidenticaloutsidethenuclearinterior.Thus, F ( R ) ! 1as R !1 .Thelocalequationthat loc is: ~ 2 2 r 2 loc ( R )+ E loc ( R )= U LE ( R loc ( R ) : (3.17) CombiningEq.(3.15)andEq.(3.17)withtheassumptionofEq.(3.16)weobtain: F ( r )= 1 2 2 ~ 2 h U LE ( R ) U o ( R ) i 1 = 2 : (3.18) 59 ItshouldbenotedthatthePCFisonlyvalidfornonlocalpotentialsofthePerey-Buck form.However,thereisnoreasontoexpectthatthefullnonlocalityintheopticalpotential willlookanythinglikethePerey-Buckform.Onphysicalgrounds,theopticalpotential mustbeenergydependentduetononlocalitiesarisingfromchannelcouplings.Whilethe spformchosenforthePerey-Buckpotentialisconvenientfornumericalcalculations,a singleGaussiantermmockingupallenergy-independentnonlocalislikelytobean oversim 3.4Giannini-RiccoNonlocalPotential ThePerey-Buckpotentialremainedtheonlywidelyknownandusednonlocalpotential availableforthefollowing15yearsafteritsdevelopment.AnattemptbyGianniniand Riccowasmadetoconstructanonlocalpotentialofasimilarformbutwithmoredata constrainingtheparameters[71,72].Theirworkfocusedon N = Z sphericalnuclei, whiletheirsecondworkmadeanextensionto N 6 = Z nuclei.Unfortunately,indoingthe nononlocalcalculationswereperformed.Instead,theweredoneusingapurelylocal opticalpotential,andatransformationformulawasusedwhichrelatednonlocalandlocal formfactors.ThistransformationformulaisderivedinAppendixB. Toconstructtheirpotential,GianniniandRiccoderivedageneralexpressionofthe nonlocalpotentialintheframeworkofWatsonmultiple-scatteringtheory[55].Theformof thederivednonlocalpotentialisaguidefortheparametrizationofthephenomenological opticalpotential,whoseparametervaluesaretobothelasticscatteringandboundstate properties.Todothealocalformfortheopticalpotentialwaschosen.Theparameters werevariedtoobtainabestoftheavailabledata,andthenthetransformationformulas 60 wereusedtogetthenonlocalGiannini-Riccopotentialfor N = Z nuclei(GR76)[71],and laterfor N 6 = Z nuclei(GR80)[72].Wechosenottousethispotentialsincethetodata weredoneusinglocalpotentials,andanunreliabletransformationformulawasusedtoget thenonlocalpotential. 3.5NonlocalDispersiveOpticalModelPotential Analternativemethodforobtainingtheopticalpotentialisthroughtheself-energy,which canbecalculatedmicroscopicallyusingmoderndaystructuretheory.Thisisthemethod bywhichtheDispersiveOpticalModel(DOM)isconstructed.TheDOMmakesuseofthe Kramers-Kronigdispersionrelationthatlinkstheimaginaryandrealpartsofthenucleon self-energy[63,65].Theopticalpotentialisconstrainedbythisdispersionrelation.This methodwasintroducedbyMahauxandSartor[102].Thenuclearmeanisafunction ofenergy,wherefor E< 0itistheshell-modelpotentialthatdescribessingle-particlestates, whilefor E> 0itistheopticalmodelpotentialthatdescribesscatteringcrosssections. Whilethenuclearmeanisacontinuousfunctionofenergy,itsbehaviorastheenergy changessignisnotsimpleduetothecouplingbetweenelasticandinelasticchannels.Itis thiscouplingthatgivesrisetotheenergydependenceandtheimaginarycomponentinthe opticalpotential.Throughthedispersiverelation,thescatteringandboundstatepartsof thenuclearmeancanbelinked.Thescatteringparameterscanbeconstrainedbyuseof elasticscattering,andtheboundstateparameterscanbeconstrainedbycomparisons tosingleparticleenergiesand( e;e 0 p )observables.AsseeninFig.3.2,thenonlocalDOM potentialreproducesexperimentalelasticscatteringdataacrossawiderangeofenergies. Usingthisdispersiverelation,alocalversionoftheDOMhasbeendeveloped[103].The 61 Figure3.2:Calculatedandexperimentalelasticscatteringangulardistributionsusingthe nonlocalDOMpotential.Dataforeachenergyareforclaritywiththelowestenergy atthebottomandhighestatthetop.Datareferencesin[11].Figurereprintedfrom[11] withpermission. localDOMwassubsequentlyusedtointheanalysisof( d;p )transferreactionsonclosed shellnucleiandshowntodescribetransferangulardistributionswithsimilaradequacyas someofthelocalglobalopticalpotentialsonthemarket[66]. Recently,thedispersiveopticalmodelformalismhasbeenextendedtoexplicitlyinclude nonlocality,spfor 40 Ca[11].AscomparedtothePerey-Buckpotential,thenonlocal DOMhasverydtrangesforthenonlocality,andatvalueforthenonlocal rangeinthevolumeandsurfaceabsorptionterms.Thetrangesofnonlocalityfor eachterminthepotentialmakestheapplicationofacorrectionfactorHowever,as theadequacyofthecorrectionfactortotakeintoaccountnonlocalectshasbeenputinto question[12],acorrectionfactorfortheDOMpotentialshouldnotbesought. 62 3.6LocalEquivalentPotentials Toassesstheofnonlocalpotentials,alocalphaseequivalent(LPE)potentialneeds tobefound.Alocalpotentialisconsidered'phaseequivalent'toanonlocalpotentialifit reproducesthesameelasticscattering.Thisischosensinceopticalpotentialsare constructedthroughtoelasticscatteringdata.Therefore,iftwopotentialsareableto generatethesameelasticscatteringdistribution,thenthetwopotentialsareindistinguishable atthelevelofelasticscattering,regardlessoftheirform. Thedownsideofthisisthattheshort-rangednonlocalarenotcon- strainedthroughelasticscattering.ToaLPEpotential,weassumesomeform fortheLPEpotential.Thisformisnormallychosentomimictheshapeofthenonlocal potential.Asanexample,thePerey-Bucknonlocalpotentialhasrealvolume,realspin- orbit,andimaginarysurfaceterms.Therefore,theLPEpotentialwaschosentohavethe sameterms.Wecalculatetheelasticscatteringdistributiongeneratedfromthenonlocal potential,thenvarytheparametersofourLPEpotentialtoobtainabesttotheelastic scatteringdistribution.Thiswasdonewiththecode SFRESCO [104]whichperformsa ˜ 2 minimization. AnothermethodtoobtainaLPEpotentialisthroughS-matrixinversion[105,106]. Thishastheadvantageoverelasticscatteringsincetheresultingpotentialwillexactly reproducetheS-matrixelementsyoustartedwith[107].However,itisimportanttonote thattheS-matrixisnotanobservable,soonecannotextractanS-matrixforeachpartial wavefromelasticscatteringdata. ExactlyreproducingtheS-matrixelementsfromthenonlocalcalculationwhendoingthe localwasoneyweencounteredinthisstudy.Whilethevisuallylookedvery 63 good,thereweresomeveryminorbetweentheS-matrixelementsgeneratedwith thenonlocalpotentialandtheLPEpotential.Therewasalwaysparticulardiy forsurfacepartialwaves.Nonetheless,whentherewererences,thewere small,andnotnoticeableinelasticangulardistribution. FindingaLPEpotentialisalsoanattractivewaytomakeverysophisticatedcalculations oftheopticalpotentialpractical,andtoassesstheirvalidity.Suchhasbeendonewith thenonlocalopticalpotentialgeneratedfrommultiplescattering[108]usingtheS-matrix inversiontechnique.Here,thePCFwascalculatedbytakingtheratioofthewavefunction generatedfromthenonlocalpotentialwiththatfromthelocalpotential.Asimilarprocedure wasdoneusingtheg-foldingmodelfor p + 12 Cscatteringatvariousenergies[109].Inthat study,theyinvestigatedtheenergydependenceoftheequivalentlocalpotential,showing thatthisenergydependencedoesnottakeintoaccountthefullnonlocality,andthatthe nonlocalityitselfmustbeenergydependent. WhiletheS-matrixinversiontechniquewasusefultoaLPEpotentialinthesestudies, itmaynotalwaysbethemostattractivewaytoobtainaLPEpotentialinpractice.A toelasticscatteringisamuchmorepracticalandnaturalwaytoobtainalocalpotential, sinceitisbasedonanobservableforwhichonemayhavedata.Inpractice,whenanoptical potentialisdesired,theremaysometimesbeelasticscatteringdataavailableonthenucleus ofinterestatthecorrectenergy.Whenthisisthecase,acommonproceduremaybeto theelasticscatteringdatadirectly,ratherthanrelyontheextrapolationsofsomeglobal potential.Itisthisphilosophywewantedtofollowwhenobtaininglocalpotentialsthatare phaseequivalenttoagivennonlocalpotential.However,ratherthantheorytodata, wetheorytotheory. AnexampleofonesuchisshowninFig.3.3.Weshowthetialcrosssection 64 overtheRutherfordcrosssectionasafunctionofthescatteringangle.Thesolidlineisthe elasticscatteringdistributiongeneratedusingthePerey-Bucknonlocalpotential.Theopen circlesareatothenonlocalsolution,andthedottedlineisobtainedbytransformingthe depthsofthevolumeandsurfacepotentials[12].Noticethatthelocalisessentiallyexact allthewayoutto180 .ThetransformationformulasreliedonbyGianniniandRiccoto constructtheirpotentialrepresentsthedottedline.Theinadequacyofthetransformation formulatoreproducethesolutionwiththenonlocalpotentialiswhytheGiannini-Ricco potentialwasnotfavoredinthisstudy. Figure3.3: 49 Ca( p;p ) 49 Caat50.0MeV:ThesolidlineisobtainedfromusingthePerey-Buck nonlocalpotential,theopencirclesareatothenonlocalsolution,andthedottedlineis obtainedbytransformingthedepthsofthevolumeandsurfacepotentialsaccordingtoEq. (B.14).Figurereprintedfrom[12]withpermission. 65 Chapter4 Results FormanyyearsnonlocalityhasbeenelyincludedincalculationsbyuseofthePerey correctionfactor(PCF)[73,75],asdiscussedinSec.3.3.1.However,thePCFisonly suitableforusewithpotentialsofthePerey-Buckform,andthusnotofusefortheDOM potentialoramicroscopicallyderivedopticalpotential.Inaddition,thequalityofthePCF hasneverbeenrigorouslytested.Therefore,thestpartofthisstudywastoinvestigate ifthecorrectionfactorwasadequatelyabletoaccountfortheimpactofnonlocalityonthe wavefunctions. Thereductionofthewavefunctioncanbeunderstoodphysicallyintermsoftherepulsion betweenfermionsduetothePauliprinciple.Sinceonemajorsourceofnonlocalityisdue toanti-symmetrization,thisrepulsionwillnaturallyhavetheofpushingsomeofthe wavefunctionoutoftheinteriorascomparedtoaninteractionthatdoesn'ttakeanti- symmetrizationintoaccount. Thedeuteronscatteringstateisalsobynonlocality.WhenusingtheDWBA,it ispossibletoapplyacorrectionfactortothedeuteronscatteringwavefunction.However,a nonlocalglobaldeuteronopticalpotentialdoesnotexistforthepurposeofcomparison.Also, aswehavediscussed,theDWBAdoesnottakedeuteronbreakupintoaccountexplicitly. Therefore,wewouldliketousethemoreadvancedADWAwhichdoesconsiderbreakup,and reliesonbetterconstrainednucleonopticalpotentials,ofwhichnonlocalglobalpotentials exist(i.e.Perey-Buck). 66 ThenumericaldetailsofthecalculationsperformedinthisthesisarepresentedinSec. 4.1.Theresultsofthisthesiswillbepresentedinthreeparts.ThepartinSec. 4.2investigates( p;d )transferreactionson 17 O, 41 Ca, 49 Ca, 127 Sn, 133 Sn,and 209 Pbat protonenergiesof E p =20and50MeV.Thetransfercrosssectionswerecalculatedwithin theDWBA,andnonlocalityinthedeuteronchannelisnotincluded.Inthisstudy,we investigatedtheofnonlocalityontheprotonscatteringwavefunctionandtheneutron boundstatewavefunction.WealsoexaminedthevalidityofthecommonlyusedPCFto elyincludenonlocality. Next,inSec.4.3,westudied( p;d )transferreactionson 40 Caatprotonenergiesof E p =20,35,and50MeVusingthenonlocalDOMpotential,aswellasthePerey-Buck potential.Onceagain,thetransfercrosssectionwascalculatedwithintheDWBA,and nonlocalityinthedeuteronchannelisnotincluded.Herewestudiedholestatesrather thansingleparticlestates,asinthepreviousstudy.Hence,thegoalofthisstudywasto understandiftheofnonlocalityseeninthepreviousstudycouldbegeneralizedto holestates,andtoseeifthesameconclusionscanbedrawnwhenusingatformfor thenonlocalpotential. Finally,inSec.4.4,westudied( d;p )reactionson 16 O, 40 Ca, 48 Ca, 126 Sn, 132 Sn,and 208 Pbatdeuteronenergiesof E d =10,20and50MeV.Forthesecases,nonlocalitywas includedexplicitlyinthedeuteronscatteringstatewithintheADWA,aswellasinthe protonchannel.Inallwavefunctions,thePerey-Bucknonlocalpotentialwasused.This studysoughttoquantifytheofnonlocalitywhenincludedconsistentlyincalculations ofsinglenucleontransferreactionsincludingdeuteronbreakup. Itisimportanttonotethatthepurposeofthisworkisnottodescribethedata.Wedonot expectthatthePerey-Buckpotential,developedinthesixtiesfor n + 208 Pbatintermediate 67 energiesusingtwodatasets,willdowellforawiderangeoftargetsandenergies.The focusshouldbeonthebetweenthenonlocalandthelocalcalculationsunder theconstraintofthesamephysicalinput,namely,thatboththenonlocalandlocaloptical potentialsintroducedreproducetheexactsameelasticscattering. 4.1NumericalDetails Tocomparetheresultsofthenonlocalcalculations,wemustcompareourresultstocalcu- lationsusinglocalpotentialswiththesameconstraints.Therefore,toconstrainthelocal nucleon-targetopticalpotentials,werequirethattheyreproducethesameelasticscattering obtainedwhenusingthePerey-BuckortheDOMpotentialattherelevantenergies.Forthe protonscatteringstates,wecalculate( p;p )elasticscatteringattherelevantenergyusingthe Perey-BuckorDOMpotential,thentheresultingdistributiontoalocalform.The oftheselocalphaseequivalent(LPE)potentialswasperformedusingthecode SFRESCO [104]. Forthedeuteronscatteringstates,theprocedureissomewhatt.InSecs.4.2 and4.3weusethelocalglobaldeuteronopticalpotentialofDaehnick[84]evaluatedatthe relevantenergy.InSec.4.4,wecalculate( n;n )and( p;p )elasticscatteringathalfthe deuteronenergyusingthePerey-Buckpotential,andagainfoundLPEpotentialsforthe elasticscatteringdistributions.Thelocaladiabaticpotentialisthencalculatedwiththe protonandneutronLPEpotentials. Fortheneutronboundstates,wecalculatedthenonlocalequationusingarealWoods- Saxonformwithanonlocalityrangeof =0 : 85fm.Wealsousedalocalspin-orbit interactionwithadepthat6MeV.Foreachtermweusedaradiusof r =1 : 25fm 68 and a =0 : 65fm.ThedepthofthenonlocalrealWoods-Saxonformwasthenadjustedto reproducethephysicalbindingenergy.Also,inSec.4.3weusedthenonlocalDOMpotential tocalculatetheneutronboundstate.Thecorrespondingboundstateresultingfromlocal potentialswasobtainedbysetting =0andadjustingthelocalrealWoods-Saxondepthto reproducethebindingenergy. InSecs.4.2and4.3,thecalculatedwavefunctionswerereadintothecode FRESCO to calculatethe( p;d )transfercrosssections.WeusedtheReidsoftcoreinteraction[110]in the( p;d )T-matrixandtocalculatethedeuteronboundstate.InSec.4.4,theboundand scatteringstatesthatarecalculatedareinsertedintothe( d;p )T-matrixEq.(2.21).This wasimplementedinthecodeNLAT(NonLocalAdiabaticTransfer).TheNNinteraction inthiscasewasacentralGaussianwhichreproducesthebindingenergyandradiusofthe deuterongroundstate,asin[96]. InSecs.4.2and4.3thescatteringwavefunctionsweresolvedbyusinga0 : 05fmradial stepsizewithamatchingradiusof40fm.Fortheboundstatessolutions,weuseda radialstepsizeof0 : 02fm.Thematchingradiuswashalftheradiusofthenucleusunder consideration,andthemaximumradiuswas30fm.Thecrosssectionscontaincontributions ofpartialwavesupto J =30. InSec.4.4thescatteringwavefunctionswerecalculatedinstepsof0 : 01fmwitha matchingradiusof30fm.Thenonlocaladiabaticpotentialwasobtainedonaradialgridof step0 : 05fm.Weusedlinearinterpolationtocalculatethenonlocaladiabaticpotentialin stepsof0 : 01fminordertocalculatetheadiabaticdeuteronwavefunctionwiththesame stepsize.Theboundstatewavefunctionswerealsocalculatedinstepsof0 : 01fmwitha maximumradiusof30fmandamatchingradiusofhalftheradiusofthenucleusunder consideration.Again,convergedcrosssectionscontainpartialwavesupto J =30. 69 16 O( d;p ) 40 Ca( d;p ) 48 Ca( d;p ) 126 Sn( d;p ) 132 Sn( d;p ) 208 Pb( d;p ) 10MeV 1 : 92% 2 : 69% 0 : 39% 0 : 32% 0 : 73% 1 : 48% 20MeV 1 : 87% 2 : 26% 0 : 34% 0 : 65% 0 : 17% 0 : 11% 50MeV 5 : 57% 0 : 07% 2 : 59% 0 : 41% 0 : 08% 0 : 38% Table4.1:Percentdofthe( d;p )transfercrosssectionatthepeakforacalcu- lationincludingtheremnanttermrelativetoacalculationwithouttheremnantterm. 4.1.1ofNeglectingRemnant Togetanideaoftheoftheremnantterm,weshowinTable4.1thepercent atthepeakofthe( d;p )transfercrosssectionforacalculationwiththe remnanttermrelativetoacalculationwithouttheremnanttermforawiderangeoftargets. TheseDWBAcalculationsusedthedeuteronglobalopticalpotentialofDaehnick[84]to describethedeuteronscatteringstate,andtheLPEpotentialstothePerey-Buckpotential fortheprotonscatteringstate,acentralGaussianforthedeuteronboundstate,andareal Woods-Saxonformfortheneutronboundstatethatreproducestheexperimentalbinding energy. 4.2DistortedWaveBornApproximationwiththePerey- BuckPotential ThepartofthisstudywastoinvestigatetheofthePerey-Buckpotentialonthe entrancechannelof( p;d )transferreactions.Forthisstudy,nonlocalitywasincludedexplic- itlyintheprotonscatteringstateandtheneutronboundstate.Usingthewavefunctions generatedwiththenonlocalpotentials,( p;d )transferreactionswerecalculated.Thesecross sectionswerecomparedtothosegeneratedwithLPEpotentials,discussedinSec.3.6.Also, 70 wavefunctionsweremodwiththePCF,andthecorrespondingtransfercrosssections werecalculated.Thegoalinthisstudywastoassesstheofnonlocalityontransfer crosssectionswhencomparedtocrosssectionsgeneratedwithLPEpotentials,aswellasto determinethequalityofthePCFanditsabilitytoreproducetheofnonlocality. 4.2.1ProtonScatteringState Whendoingcalculationsof( p;d )or( d;p )reactionsusingtheT-matrixformalismofEq.(2.21), itisrequiredtocalculateaprotonelasticscatteringwavefunctionineithertheentrance orexitchannel.Becausesomecodes,suchas TWOFNR [24],allowfornonlocalitytobe includedthroughthePCF,thisapproachhasbecomecommonpractice.However,untilre- cently,theaccuracyofthisapproachwasnotunderstood.UsingthePerey-Buckpotential, thismethodologytoincludenonlocalityhasbeentested.Todothischeck,aLPEpotential neededtobefound.TheopencirclesinFig.3.3areonesuchexampleofaLPEpotential. TheLPEpotentialfoundfromtheisthe U LE terminEq.(3.18). Asanexample,wewillusetheLPEpotentialfromFig.3.3andconsiderthescattering wavefunctionforthereaction 49 Ca( p;p ) 49 Caat50MeV.The J ˇ =1 = 2 partialwaveis showninFig.4.1.Asisseeninthethereductionofthewavefunctionresulting fromthenonlocalpotential(solidline)relativetothewavefunctionfromtheLPEpotential (dashedline)isapparent.Alsoseenisthatthewavefunctionsfromthenonlocalpotential andthelocalpotentialwiththePCFapplied(crosses)areingoodagreement.Thiswas ageneralresultformostpartialwaves.However,inallcasesthatwerestudied,problems aroseforpartialwavescorrespondingtoimpactparametersaroundthesurfaceregion,shown inFig.4.2.Sincetransfercrosssectionstendtobemostsensitivetothesurfaceregion, thefortheseangularmomentaareparticularlyrelevant.Wewillseehowthe 71 Figure4.1:Realandimaginarypartsofthe J ˇ =1 = 2 partialwaveofthescatteringwave functionforthereaction 49 Ca( p;p ) 49 Caat50.0MeV: NL (solidline), PCF (crosses),and loc (dashedline).Top(bottom)panel:absolutevalueofthereal(imaginary)partofthe scatteringwavefunction.Figurereprintedfrom[12]withpermission. inadequacyofthePCFforsurfacepartialwavetheresultingtransfercrosssections inSec.4.2.3. Figure4.2:Realandimaginarypartsofthe J ˇ =11 = 2 + partialwaveofthescattering wavefunctionforthereaction 49 Ca( p;p ) 49 Caat50.0MeV.SeecaptionofFig.4.1.Figure reprintedfrom[12]withpermission. 72 ThisinabilityofthePCFtocorrectsurfacepartialwavesispartlyduetothewayinwhich itwasderived.WhenderivingthePCF,termsrelatedto r 2 F wereneglected,suchasthe oneinEq.(B.21).Thistermonlycontributesaroundthenuclearsurface.Inaddition,when performingthelocalweoccasionallyfoundslightintheS-matrixelementsfor aparticularpartialwave.Sincethescatteringwavefunctionsarenormalizedaccordingto Eq.(2.2),thesesmallchangesintheS-matrixwillresultintamplitudesforthereal andimaginarypartsofthescatteringwavefunctionintheasymptoticregion. 4.2.2NeutronBoundState Wenowturnourattentiontotheneutronboundstatethatexistsintheentrancechannel of( p;d )reactions.Inordertoinvestigatetheofnonlocalityontheboundstatewave functions,andtheadequacyofthePCFtocorrectfornonlocality,thePCFwasappliedto thelocalboundstatewavefunction,andtheresultingwavefunctionwasrenormalizedto unity. Toillustrate,the2 p 3 = 2 groundstatewavefunctionfor n + 48 CaisshowninFig.4.3. Visually,thecorrectionfactordoesanexcellentjobcorrectingfornonlocalityinthebound state.However,itisimportanttonoticethatinthesurfaceregion(2 5fm),thePCF doesverylittletobringthewavefunctionresultingfromthelocalequivalentpotentialinto agreementwiththewavefunctionresultingfromthenonlocalpotential.Theinset,which showsthebetween ˚ NL and ˚ PCF ,emphasizesthisfact. AsstatedinSec.4.2.1,thereasonfortheinadequacyofthePCFinthesurfaceregion goesbacktothewayinwhichthePCFwasderived.Inthiscase,theboundwavefunction hasalargeslopearoundthesurfaceresultinginlargebetweenthewavefunction generatedusingnonlocalinteractionsandtheonegeneratedfromlocalinteractions. 73 Figure4.3:Groundstate,2 p 3 = 2 ,boundwavefunctionfor n + 48 Ca. ˚ NL (solidline), ˚ PCF (crosses),and ˚ loc (dashedline).Theinsetshowsthe ˚ NL ˚ PCF .Figure reprintedfromfrom[12]withpermission. Anotherimportantpointtonoteisthatnonlocalityhastheofincreasingthe normalizationoftheasymptoticpropertiesofthewavefunction(theANC).Sincenonlocality reducestheamplitudeofthewavefunctioninthenuclearinterior,andthewavefunctionis alwaysnormalizedtounity,theANCmustincrease.Therefore,theANCoftheboundwave functionresultingfromnonlocalpotentialswasfoundtoalwaysbelargerthantheANCfrom localpotentials,andtheANCofthecorrectedwavefunctionwassomewhereinbetween. 4.2.3 ( p;d ) TransferCrossSections-DistortedWaveBornAp- proximation Nowthatwehavestudiedtheofnonlocalityonthescatteringandboundstatewave functions,wecaninvestigatethenonlocalityhason( p;d )transferreactionswhen 74 nonlocalityisincludedexplicitlyintheentrancechannel. Asaexample,considerthetransferreactioncorrespondingtothewavefunctionswe havebeenstudyinginSecs.4.2.1and4.2.2, 49 Ca( p;d ) 48 Caataprotonenergyof E p =50 MeVinthelaboratoryframe.Theseparateandcombinedofnonlocalityinthebound andscatteringstatesareshowninFig.4.4.Thesolidlinecorrespondstowhennonlocality isincludedinboththeprotonscatteringstateandtheneutronboundstate,thedashed linecorrespondstothedistributionobtainedwhenonlylocalequivalentpotentialsareused, thecrossescorrespondtothecrosssectionobtainedwhentheprotonscatteringstateand neutronboundstatewavefunctionsarebothcorrectedwiththePCF.Alsoshownwiththe dottedlineisthecrosssectionwhennonlocalitywasonlyaddedtothescatteringstate,and thedot-dashedlinewhennonlocalityisonlyaddedtotheneutronboundstate. Figure4.4:Angulardistributionsfor 49 Ca( p;d ) 48 Caat50MeV:Inclusionofnonlocalityin boththeprotonscatteringstateandtheneutronboundstate(solid),usingLPEpotentials, thenapplyingthecorrectionfactortoboththescatteringandboundstates(crosses),using theLPEpotentialswithoutapplyinganycorrections(dashedline),includingnonlocalityonly intheprotonscatteringstate(dottedline)andincludingnonlocalityonlyintheneutron boundstate(dot-dashedline).Figurereprintedfrom[12]withpermission. 75 TheresultsofFig.4.4areuniqueforthecasesweconsideredinthattheshapeofthe distributionwasantlychanged.Thereasonforthetchangesaroundzero degreescanbeseenfromananalysisofthescatteringandboundwavefunctions.Thebound wavefunctionhasanodewhichoccursataradiuscorrespondingtothesurfaceregionfor 49 Ca.Sincetheboundwavefunctionhasalargeslopeinthisregion,thepercent betweenthenonlocalandlocalwavefunctionscanbequitelarge.Forthiscase,thenonlocal boundwavefunctionissmallerthanthelocalwavefunctionsinthisregion,reducingthe crosssectionatthepeakaround20degrees.Ontheotherhand,themagnitudeofthebound wavefunctionislargeintheasymptoticregion,whichincreasesthecrosssectionatzero degrees. Forthescatteringwavefunction,themosttwereforpartialwaves correspondingtothesurfaceregion.Also,theasymptoticsofthescatteringwavefunctions weretduetosmallintheS-matrix,againmostlyforsurfacepartialwaves. Thereisaninterplaybetweentherealandimaginarypartsofthescatteringwavefunction whichthecrosssectionatforwardangles.Then,thecomplexcombinationofall theseproducestheinterestingbehaviorofthetransfercrosssectionatforwardangles. Nowconsiderthesamereactionbutatalowerenergy.InFig.4.5weshow 49 Ca( p;d ) 48 Ca ataprotonenergyof E p =20MeVinthelaboratoryframe.Thiscaseismorerepresentative ofthegeneralfeatureswesawinthissystematicstudy.Thenonlocalityinthescatteringstate hadtheofreducingthetransfercrosssectionduetothereductionofthescattering wavefunction,whilethenonlocalityintheboundstatehadtheofenhancingthecross sectionduetotheincreaseofthewavefunctionintheasymptoticregion.Atthislower energy,theoverallwasanenhancementofthetransfercrosssectionatthepeak. Inaddition,itisseenthatthePCFmovesthetransferdistributiongeneratedwithlocal 76 Figure4.5:SameasinFig.4.4butfor 49 Ca( p;d ) 48 Caat E p =20MeV.Figurereprinted from[12]withpermission. Figure4.6:SameasinFig.4.4butfor 133 Sn( p;d ) 132 Snat E p =20MeV.Figurereprinted from[12]withpermission. potentialsinthedirectionofthatgeneratedwithnonlocalpotentials.However,thePCF wasnotabletofullytaketheofnonlocalityintoaccount. Nextweconsiderheaviertargets,suchas 133 Snand 209 Pb,at E p =20MeV.Inbothof thesecasestheinclusionofnonlocalityinthescatteringstatedecreasedthecrosssectionby 77 Figure4.7:SameasinFig.4.4butfor 209 Pb( p;d ) 208 Pbat E p =20MeV.Figurereprinted from[12]withpermission. asmalleramountthanwasseenbefore.Thisisduetothelowenergyoftheprotonandthe highchargeofthetarget.Nonlocalityintheprotonscatteringstatereducedthemagnitude ofthewavefunctioninthenuclearinterior,buttheenergyoftheprotonwasnothighenough topenetratedeeplyduetothelargeCoulombbarrier.Ontheotherhand,nonlocalityinthe boundstateisveryt.Sincetheprojectileenergywaslow,andthechargeofthe targetwashigh,thereactionisdominatedbytheasymptotitcsoftheboundwavefunction, whichisenhancedinthenonlocalcase. For 133 Sn( p;d ) 132 Snat E p =20MeV,Fig.4.6,thePCFdoesareasonablejobtak- ingnonlocalityintoaccount,buttherearestilldiscrepanciesbetweenthefullnonlocaland correctedlocalsolutions.For 209 Pb( p;d ) 208 Pbat E p =20MeV,Fig.4.7,therearediscrep- anciesatforwardangles,butthedistributionsresultingfromnonlocalandlocalpotentials coincidentallyagreeatthepeak.Thisagreementisaccidental,andcomesfromthenonlocal intheboundstatecancelingthatinthescatteringstate. 78 Corrected Nonlocal E lab =20MeV RelativetoLocal RelativetoLocal 17 O(1 d 5 = 2 )( p;d ) 7 : 1% 18 : 8% 17 O(2 s 1 = 2 )( p;d ) 20 : 1% 26 : 5% 41 Ca( p;d ) 11 : 4% 21 : 9% 49 Ca( p;d ) 10 : 4% 17 : 3% 127 Sn( p;d ) 17 : 5% 17 : 3% 133 Sn( p;d ) 18 : 2% 24 : 4% 209 Pb( p;d ) 19 : 4% 20 : 8% Table4.2:Percentofthe( p;d )transfercrosssectionsatthepeakwhenusing thePCF(2ndcolumn),oranonlocalpotential(3rdcolumn),relativetothelocalcalculation withtheLPEpotential,foranumberofreactionsoccurringat20MeV. Corrected Nonlocal E lab =50MeV RelativetoLocal RelativetoLocal 17 O(1 d 5 = 2 )( p;d ) 17 : 0% 35 : 4% 17 O(2 s 1 = 2 )( p;d ) 0 : 2% 12 : 7% 41 Ca( p;d ) 2 : 9% 5 : 8% 49 Ca( p;d ) 16 : 0% 17 : 1% 127 Sn( p;d ) 10 : 1% 4 : 5% 133 Sn( p;d ) 6 : 7% 16 : 9% 209 Pb( p;d ) 8 : 6% 8 : 6% Table4.3:Percentofthe( p;d )transfercrosssectionsatthepeakwhenusing thePCF(2ndcolumn),oranonlocalpotential(3rdcolumn),relativetothelocalcalculation withtheLPEpotential,foranumberofreactionsoccurringat50MeV. Thepercentatthepeakofthetransferdistributionsforallthecasesthat werestudiedaresummarizedinTables4.7and4.8forthe( p;d )reactionsat20and50MeV. Forthelowerenergycases,itisseenthattheinclusionofnonlocalityprovidedageneral enhancementtothetransfercrosssectionatthepeak.Thisisduetotheenhancement oftheboundstatewavefunctionintheasymptoticregionplayingamoretrolein themagnitudeofthetransfercrosssectionatlowenergies.Athigherenergies,thereisa competitionbetweentheenhancementduetotheboundstate,andthereductionduetothe scatteringstate.Inmostcases,thereisstillanenhancementofthecrosssection,butthe overallislesstthanforthelowerenergycases. 79 4.2.4Summary Inthisstudy,thelongestablishedPereycorrectionfactor(PCF)andtheofnonlocal- ityontheentrancechannelof( p;d )reactionswerestudied.Theintegro-ditialequation containingthePerey-Bucknonlocalpotentialwassolvednumericallyforsinglechannelscat- teringandboundstates.Alocalphaseequivalent(LPE)potentialwasobtainedby theelasticdistributiongeneratedbythePerey-Buckpotential.ThePCFwasappliedtothe wavefunctionsgeneratedwiththeLPEpotentialsorthelocalequivalentbindingpotentials, andthescatteringandboundstatewavefunctionswerethenusedinaDWBA calculationinordertoobtain( p;d )transfercrosssections. Wefoundthattheexplicitinclusionofnonlocalityintheentrancechannelincreasedthe transferdistributionatthepeakby15 35%.Inmostcases,thetransferdistribution fromusinganonlocalpotentialincreasedrelativetothedistributionfromthelocalpotential. Inallcases,thePCFmovedthetransferdistributioninthedirectionofthedistributionwhich includednonlocalityexplicitly.However,nonlocalitywasneverfullytakenintoaccountwith thePCF.ThePCFcanbeimprovedbyincludingthesurfacetermsthatwereneglected,and notassumingthatthelocalmomentumapproximationisvalid.Suchadditionalcorrections werenotpursuedsincethefullnonlocalsolutioncanbecalculated. 4.3DistortedWaveBornApproximationwiththeDis- persiveOpticalModelPotential TheresultsofthepreviousworkbyTitusandNunes[12],coveredinSec.4.2,demonstrated thatnonlocalityistinthestudyoftransferreactions,andthatthePCFisnot 80 abletofullyreproducethecomplexofnonlocality.However,thepreviousstudyonly consideredthePerey-Bucknonlocalpotential.Recently,thenonlocalDOMpotentialfor 40 Cawasdeveloped[11].Wewantedtoseeifnonlocalityremainsanimportantingredientto transferreactionswhenatnonlocalpotentialisused.SincetheDOMwasconstructed onlyfor 40 Ca,wewereonlyabletoconsider 40 Ca( p;d ) 39 Careactions.Forthisstudy,we investigatedlaboratoryprotonenergiesof E p =20,35,and50MeV. Figure4.8:AngulardistributionsforelasticscatteringnormalizedtoRutherfordforprotons on 40 Caat E p =20MeV.TheelasticscatteringwiththeDOMpotential(solidline),the DOMLPEpotential(opencircles),thePerey-Buckinteraction(dashedline),andthePerey- BuckLPEpotential(opensquares).Thedata(closeddiamonds)from[13].Figurereprinted from[14]withpermission. 4.3.1ProtonScatteringState IninvestigatingtheofnonlocalitywhenusingtheDOMpotential,wenolongercon- sideredthePCFsincethereisnoeasygeneralizationofthePCFtotheDOMpotential. ThisisbecauseeachtermoftheDOMpotentialhasatvalueforthenonlocality parameter, .WhileitwouldbepossibletoconstructaPCFfortheDOMpotential,the 81 Figure4.9:SameasinFig.4.8butfor E p =50MeV.Datafrom[15].Figurereprintedfrom [14]withpermission. resultsofthepreviousworkbyTitusandNunes[12]madepursuingaPCFfortheDOM potentialirrelevant. InFigs.4.8and4.9weshowtheelasticdistributionsgeneratedfromtheDOMandthe Perey-BucknonlocalpotentialsalongwiththecorrespondingLPEpotentials.Wealsoshow thecorrespondingelasticscatteringdataattheclosestrelevantenergy.Itisseenfromthe distributionsthattheDOMpotentialissuperiorwhenitcomestodescribingthedata.This shouldnotbeasurpriseastheDOMpotentialwasconstructedfromtonucleonelastic scatteringdataon 40 CawhilethePerey-Buckpotentialwasconstructedfromneutronelastic scatteringon 208 Pbatlowenergy.Nonetheless,thePerey-Buckpotentialdoesareasonably goodjobatdescribingtheelasticscatteringdatafortheenergyandangularrangethatthe dataisavailable. Toinvestigatethescatteringwavefunctions,weconsiderthe J ˇ =1 = 2 + partialwave forscatteringat E p =50MeVinFig.4.10.ForboththeDOMandthePerey-Buck 82 Figure4.10:Therealandimaginarypartsofthe J ˇ =1 = 2 + partialwaveofthescattering wavefunctionforthereaction 40 Ca( p;p ) 40 Caat E p =50MeV.Thisshowsthewavefunction resultingfromtheDOMpotential(solidline)anditsLPEpotential(dottedline),thePerey- Buckpotential(dashedline)anditsLPEpotential(dot-dashedline).Thetop(bottom) panelshowstheabsolutevalueofthereal(imaginary)partofthescatteringwavefunction. Figurereprintedfrom[14]withpermission. nonlocalpotential,weseethereductionofthescatteringwavefunctionrelativetothewave functionfromtheLPEpotential,whichisconsistentwithearlierstudies[12,73,75].Since thetwononlocalpotentialsdescribetelasticscatteringdistributions,theywillhave tS-matrixelementsforeachpartialwave,andhence,thetnormalizationsin theasymptoticregionisexpected. 4.3.2NeutronBoundState Theneutron1 d 3 = 2 boundstatewavefunctionsusingthevariouspotentialsareshowninFig. 4.11.TheDOMboundwavefunctionwasfoundusingthepotentialdin[11].The samegeneralfeaturesoftheboundwavefunctionsin[12]areseenhere.Nonlocalityreduces theamplitudeoftheboundwavefunctionandthuspushesthewavefunctionoutward. 83 Figure4.11:Theneutrongroundstate1 d 3 = 2 boundwavefunctionfor n + 39 Ca.Shownis thewavefunctionobtainedusingtheDOMpotential(solidline),thePerey-Buckpotential (crosses)andthelocalinteraction(dashedline).Theinsetshowstheasymptoticproperties ofeachwavefunction.Figurereprintedfrom[14]withpermission. 4.3.3 ( p;d ) TransferCrossSections-DistortedWaveBornAp- proximation InFig.4.12weshowthetransferdistributionsforthethreeenergiescalculated.Shownis thetransferdistributionresultingfromtheDOMnonlocalpotentialanditsLPEpotential, aswellasthePerey-BucknonlocalpotentialanditsLPEpotential.Ingeneralweseethat nonlocalityforbothpotentialsprovidesanenhancementofthecrosssectionatthe peak.Thisisconsistentwiththeconclusionsof[12].However,athigherenergies,thereis notasmuchcancellationbetweenthescatteringandboundstatessothatthefullnonlocal calculationstillresultedinafairlytincreaseinthecrosssection.Thekeyerence isthattheneutronwasboundby15 : 6MeVinthisstudywhereastheneutronwasalways boundlessthan10MeVinallcasesinthestudyofSec.4.2. 84 E p (MeV) boundstate scatteringstate fullnonlocal 20 27% -14% 15% 35 31% 10% 52% 50 31% -3% 29% Table4.4:Percentofthe( p;d )transfercrosssectionsatthepeakatthe listedbeamenergiesusingtheDOMpotentialrelativetothecalculationswiththephase- equivalentpotential.Resultsarelistedseparatelyfortheofnonlocalityonthebound state,thescatteringstate,andthetotal. E p (MeV) boundstate scatteringstate fullnonlocal 20 42% -15% 27% 35 55% -8% 52% 50 42% -11% 29% Table4.5:SameasTable4.4,butnowforthePerey-Buckpotential. ShowninTable4.4and4.5arethepercentofthe( p;d )transfercrosssections atthepeakwhenusingtheDOMandPerey-Buckpotentials,respectively.Inthetables weshowtheseparateoftheneutronboundstateandtheprotonscatteringstate,as wellasthepercentforthefullnonlocalcalculation. Inmostcasesnonlocalityinthescatteringstatehadthetofreducingthetransfer crosssection.Oneexceptionwasfor E p =35MeVwhenusingtheDOMpotential.The increasewasduetotheshapeofthescatteringwavefunctionnearthesurfaceregion.Inthis particularcase,obtainingandexacttothenonlocaldistributionwasmuchmore thanintheothercases.Allothercasesreducedthecrosssectionbyasimilaramount.Since theCoulombbarrierisnotlargefor 40 Ca,therewasnosuppressionofthenonlocal inthescatteringstateasweseeninthepreviousstudyforheaviersystems,suchas 133 Sn or 209 Pb. Theofnonlocalityintheboundstateatallprotonenergieswastoincreasethe crosssection.Thisisbecausenonlocalityshiftstheboundwavefunctiontowardsthesurface regionwherethesetransferreactionsaremoresensitive.Wealsonotethatthenonlocal 85 Figure4.12:Angulardistributionsforthe 40 Ca( p;d ) 39 Careactionat(a) E p =20MeV,(b) E p =35MeV,and(c) E p =50MeV.Inthisgureisthetransferdistributionresulting fromusingthenonlocalDOM(solidline)anditsLPEpotential(dottedline),thePerey- Buckpotential(dashedline)andthePerey-BuckLPEpotential(dot-dashedline).Figure reprintedfrom[14]withpermission. 86 forthePerey-Buckinteractionaregenerallylargerthanforthepreviousstudy[12]. Thisisbecauseinthepreviousstudywewerestudyingsingleparticlestatesinclosedshell nuclei,whileherewefocusonholestatesin 40 Ca. 4.3.4Summary Inthisworkwestudiedtheofaddingnon-localityintheentrancechanneloftransfer reactionsusinganonlocalpotentialobtainedfromthedispersiveopticalmodel(DOM)and comparingittotheresultsfromtheolderPerey-Buckinteraction.Ourstudiesfocusonthe 40 Ca( p;d ) 39 Careactionat E p =20,35and50MeV.Weconsiderthenonlocalityinthe protonchannel,andsolvetheintialequationtoobtaintheprotonscattering andneutronboundstatesolutionsforbothnonlocalpotentials.Wethencomputedthe transfermatrixelementintheDWBA,ignoringnonlocalityinthedeuteronchannel. Ourresultsshowthat,irrespectiveofthedetailsofthepotential,nonlocalityreduces thestrengthofthewavefunctioninthenuclearinterior,anmostnoticeableinthe boundstates,butalsotinscatteringstates.Duetothenormalizationcondition, nonlocalityintheboundstatealsoshiftsthewavefunctiontotheperipheryregion,causing anincreaseinthetransfercrosssections.Typically,nonlocalityinthescatteringstateacts intheoppositedirection,reducingtheoverallWhennonlocalityisincludedinboth theboundandscatteringstates,thetransfercrosssectionsareincreasedby ˇ 15 50%for theDOMpotential,incontrastwith ˇ 30 50%obtainedwiththePerey-Buckinteraction. 87 4.4NonlocalAdiabaticDistortedWaveApproxima- tionwiththePerey-BuckPotential Theprevioustwostudies(TitusandNunes[12]discussedinSec.4.2,andRoss,Titus, Nunes, et.al. [14]discussedinSec.4.3)havedemonstratedthattheexplicitinclusionof nonlocality,atleastintheentrancechannelof( p;d )reactions,isveryimportanttotake intoaccountexplicitly.Theyhavealsoshownthatcommonlyusedcorrectionfactorsarenot ttoectivelyincludenonlocality,andthatnonlocalityisimportantregardlessof theformchosenforthenonlocalpotential. Inbothofthesestudies,alocaldeuteronopticalpotentialwasusedtodescribethe deuteronscatteringstate.Wewillnowturnourattentiontostudyingtransferreactions withintheADWA,discussedinSec.2.5,whichincludesdeuteronbreakupexplicitly.As theADWAisbasedonathree-bodyHamiltonian,weincludednonlocalityconsistentlyin allnucleon-targetinteractions.Forthisstudy,wewillfocuson( d;p )transferreactions. Itshouldbenotedthatduetotimereversalinvariance,thecrosssectionsfor( d;p )and ( p;d )transferreactionsronlybyastatisticalconstant,assumingthattheinitialand statesarethesame.Thestatisticalconstantcanbedeterminedbydetailedbalance [8].Therefore,eventhoughweareconsideringatreaction,wearebuildingonthe learningfromthepreviousstudies. 4.4.1TheSourceTerm Inordertocomparetheofnonlocalityontheadiabaticpotential,wethe rhs ofEq.(2.49)tobe S NL AD ( R ).Itistocomparethenonlocaladiabaticpotential 88 directlytothelocaladiabaticpotentialsince S NL AD ( R )hasthescatteringwavefunction builtintoit.However,wecancompare S NL AD ( R )withthelocalcorrespondingquantity, S Loc AD ( R )= U Loc AD ( R ) X Loc AD ( R ).Afterapartialwavedecomposition,thesourcetermsbecome functionsdependentonlyonthescalar R foreach LJ combinationofthedeuteronscattering state. Figure4.13:Absolutesvalueofthe d + A sourcetermwhennonlocalandlocalpotentialsare used.(a) d + 48 Caat E d =50MeV.(b) d + 208 Pbat E d =50MeV.Bothareforthe L =1 and J =0partialwave.Figurereprintedfrom[3]withpermission. Togetanideaoftheofnonlocalityontheadiabaticpotential,wemakeacomparison inFig.4.13ofthisradialsourcetermfortheangularmomentumvaluesof L =1and J =0 ofthe d + 48 Caand d + 208 Pbwavefunction,bothatabeamenergyof E d =50MeV.In Fig.4.14wemakethesamecomparisonbutforthe L =6and J =5partialwave.In boththesolidlinecorrespondstothenonlocalsourceterm,whilethedashedline isitslocalequivalent.Themagnitudeofthenonlocalsourcetermisreducedcomparedto thelocalsourceterm.Itisalsoseenthatthesourceterminthenonlocalcasegetsshifted outwardrelativetothelocalcase.Boththeseimprintthemselvesontheadiabatic deuteronwavefunction,aswewillseeinSec.4.4.2. 89 Figure4.14:Absolutevalueofthe d + A sourcetermwhennonlocalandlocalpotentialsare used.(a) d + 48 Caat E d =50MeV.(b) d + 208 Pbat E d =50MeV.Bothareforthe L =6 and J =5partialwave.Figurereprintedfrom[3]withpermission. 4.4.2DeuteronScatteringState ThenecessaryformalismforthelocalimplementationoftheADWAandthenonlocalexten- sionoftheADWAhasbeenaddressedinChapter2.Theradialequationthatmustbesolved foreachpartialwaveisgivenbyEq.(2.49).The rhs ofthisequationactsasasourceterm, andthebetweenthenonlocalsourcetermandthecorrespondinglocalsource termwascomparedinSec.4.4.1. Turningourfocustothedeuteronscatteringwavefunction,Figs.4.15and4.16showthe absolutevaluesofthe d + A scatteringwavefunctionwhenusingtheADWAwithnonlocal andlocalpotentials.Thesolidlinecorrespondstothescatteringwavefunctionresultingfrom usingthenonlocalPerey-BuckpotentialinEq.(2.49),whilethedashedlineisthescattering wavefunctionthatresultsfromusingthelocaladiabaticpotential,Eq.(2.41),wherethe necessaryareusedforthenucleonopticalpotentials.Panel(a)isfor d + 48 Cawhilepanel (b)isfor d + 208 Pb. 90 Figure4.15:Absolutevalueofthe d + A scatteringwavefunctionusingtheADWAtheory whennonlocalandlocalpotentialsareused.(a) d + 48 Caand(b) d + 208 Pb.Bothforthe L =1and J =0partialwaveat E d =50MeVinthelaboratoryframe.Figurereprinted from[3]withpermission. Here,itisimportanttonotethattheindividual n + A and p + A localopticalpotentialsare phaseequivalenttothenonlocalPerey-Buck,butthenonlocalandlocaladiabaticpotentials arenotphaseequivalent.Theadiabaticpotentialisonlyusefulforcalculatingthedeuteron scatteringwavefunctionwithintherangeofthe V np interaction,andisnotapplicablefor calculatingdeuteronelasticscattering.Itisforthisreasonthatwechosefortheinputoptical potentialstobephaseequivalent,andnotthefulladiabaticpotential. WhencomparedtothesourcetermthatdrivesthiswavefunctioninFig.4.13,weseethat boththewavefunctionandthesourcetermarereducedrelativetothelocalcounterpart.This isthesamefeaturethatwesawwhenstudyingtheprotonscatteringstatein( p;d )reactions. Thereductionofthewavefunctionintheinteriorisacommonfeatureofusingnonlocal potentials,andcanbeunderstoodphysicallyintermsofthePauliexclusionprinciple. Whenstudyingprotonscatteringstates,nonlocalityonlyhadtheofreducingthe amplitudeofthescatteringwavefunction.However,eringfromtheprotonscattering 91 state,the d + A scatteringwavefunctionisalsoshiftedoutwardrelativetothewavefunction resultingfromlocalpotentials(seeFig.4.15).Thisisanalogoustotheboundstatecase wherethewavefunctionwasbothreducedandshiftedoutwardduetononlocality.This shiftingoutwardofthe d + A scatteringwavefunctionchangestheamplitudeofthewave functionatthenuclearsurface.Sincethesurfaceregioniswhere( d;p )crosssectionsare mostsensitive,theshiftingoutwardcanhaveatonthecrosssection.In fact,aswewillseeshortly,nonlocalityinthedeuteronscatteringstateincreasesthetransfer crosssectioninmostcasesthatwerestudied. Figure4.16:Absolutevalueofthe d + A scatteringwavefunctionusingtheADWAtheory whennonlocalandlocalpotentialsareused.(a) d + 48 Caand(b) d + 208 Pb.Bothforthe L =6and J =5partialwaveat E d =50MeVinthelaboratoryframe.Figurereprinted from[3]withpermission. Theabsolutevaluesofthe d + A scatteringwavefunctionsforthe L =6and J =5 partialwaveareshowninFig.4.16for d + 48 Caand d + 208 Pbat E d =50MeV.Wesee similarfeaturesaswedidforthe L =1and J =0case:thewavefunctionisbothreduced andpushedoutwardduetononlocality.Forthe d + 208 Pbcaseinparticular,thereisdramatic shiftinthewavefunctionaroundthenuclearsurface( ˘ 7 : 5 8 : 5fm),whichwewillis 92 veryimportantwhencalculatingtransfercrosssections. 4.4.3 ( d;p ) TransferCrossSections Forthecalculationof( d;p )transfercrosssections,thenonlocalPerey-Buckpotentialwas usedfortheneutronandprotonopticalpotentialsintheentranceandexitchannels.The separateofnonlocalityintheprotonscatteringstateandtheneutronboundstate havealreadybeenstudied.Theresultsofsuchastudyhasbeenpublishedinourprevious papers[12,14],andisnotdiscussedhere.Inaddition,sincewehavealreadydetermined thatthePCFisttotakenonlocalityintoaccount,wedidnotinvestigatethePCF inthisstudyandfocusedinsteadontheofexplicitlyincludingnonlocalityinthe entranceandexitchannelsin( d;p )reactions. Inouranalysis,wecomputedangulardistributionsforawidevarietyofcasesfrom 16 O to 208 Pb.SomeillustrativeexamplesareshowninFig.4.17and4.18.Extensivetablesfor allcasesareshowninTables4.6,4.7,and4.8.Inthetablesweshowthepercent betweencrosssectionsproducedbynonlocalandlocalinteractions,atthepeakoftheangular distribution,relativetoapurelylocalcalculation.Inthecolumnweincludenonlocality inallnucleon-targetinteractions.Inthesecond(third)columnweincludenonlocalityinthe entrance(exit)channelonly. InFig.4.17and4.18weincludetheresultswhennonlocalityisincludedconsistently (solidline),onlyinthedeuteronchannel(dashedline),onlyintheprotonchannel(dot- dashedline),andwhereonlyaLPEpotentialisused(dottedline).InFig.4.17wepresent ( d;p )calculationsfordeuteronsimpingingon:(a) 48 Caat E d =10MeV,(b) 132 Snat E d =10MeV,and(c) 208 Pbat E d =20MeV.ThesamecasesarepresentedinFig.4.18, butfor E d =50MeV.Whenavailable,wealsopresentdatapoints.ThedatainFig.4.17a 93 ispublishedinarbitraryunits.Therefore,thisdatasetwasnormalizedtothepeakofthe theoreticaldistributionthatisgeneratedwhennonlocalityisfullyincluded. Nonlocal Nonlocal Final Nonlocal Entrance Exit Angle E lab Bound Relative Relative Relative of 10MeV State toLocal toLocal toLocal Peak 16 O( d;p ) 1 d 5 = 2 27 : 2% 3 : 0% 32 : 7% 26 16 O( d;p ) 2 s 1 = 2 15 : 5% 0 : 2% 13 : 5% 0 40 Ca( d;p ) 1 f 7 = 2 48 : 5% 11 : 4% 46 : 5% 39 48 Ca( d;p ) 2 p 3 = 2 19 : 4% 6 : 8% 27 : 8% 15 126 Sn( d;p ) 1 h 11 = 2 36 : 9% 8 : 7% 26 : 9 72 132 Sn( d;p ) 2 f 7 = 2 25 : 7% 0 : 2% 30 : 1% 55 208 Pb( d;p ) 2 g 9 = 2 52 : 5% 2 : 0% 47 : 3% 180 Table4.6:Percentofthe( d;p )transfercrosssectionsatthepeakwhen usingnonlocalpotentialsinentranceandexitchannels(1stcolumn),nonlocalpotentials inentrancechannelonly(2ndcolumn),andnonlocalpotentialsinexitchannelonly(3rd column),relativetothelocalcalculationwiththeLPEpotentials,foranumberofreactions occurringat10MeV. Atlowenergies,nonlocalityintheexitchannelprovidesatenhancementofthe crosssectionforallcases,whichisduetotheneutronboundstate.Asmentionedbefore, theANCoftheboundstateresultingfromnonlocalpotentialsislargerthanthatfromlocal potentials.Sincelowenergytransferreactionsareprimarilysensitivetotheasymptotic propertiesofthewavefunctions,thisresultsinanincreaseofthecrosssection. Thenonlocalityintheprotonscatteringstateisnotfelttlyatlowenergies, sothereductionofthecrosssectionduetothereducedamplitudeoftheprotonscattering stateissmall.Thisisconsistentwiththeresultspublishedinourpreviouspapers,[12,14]. Athigherenergies,thenonlocalityoftheprotonscatteringstatebecomesmoresignt, andthereisacompetitionbetweentheofnonlocalityintheneutronboundstateto enhancethecrosssection,andtheofnonlocalityintheprotonscatteringstateto reducethecrosssection.Nonlocalityintheprotonscatteringstatehadalargerfor 94 Figure4.17:Angulardistributionsfor( d;p )transfercrosssections.Theinsetsarethe theoreticaldistributionsnormalizedtothepeakofthedatadistribution.(a) 48 Ca( d;p ) 49 Ca at E d =10MeVwithdata[16]at E d =10MeVinarbitraryunits.(b) 132 Sn( d;p ) 133 Snat E d =10MeVwithdata[17]at E d =9 : 4MeV.(c) 208 Pb( d;p ) 209 Pbat20MeVwithdata [18](Circles)and[19](Squares)at E d =22MeV.Figurereprintedfrom[3]withpermission. 95 Figure4.18:Angulardistributionsfor( d;p )transfercrosssections.Theinsetisthetheo- reticaldistributionsnormalizedtothepeakofthedatadistribution.(a) 48 Ca( d;p ) 49 Caat E d =50MeVwithdata[20]at E d =56MeV.(b) 132 Sn( d;p ) 133 Snat E d =50MeV.(c) 208 Pb( d;p ) 209 Pbat50MeV.Figurereprintedfrom[3]withpermission. 96 Nonlocal Nonlocal Final Nonlocal Entrance Exit Angle E lab Bound Relative Relative Relative of 20MeV State toLocal toLocal toLocal Peak 16 O( d;p ) 1 d 5 = 2 24 : 9% 2 : 6% 25 : 7% 0 16 O( d;p ) 2 s 1 = 2 7 : 1% 0 : 7% 6 : 0% 0 40 Ca( d;p ) 1 f 7 = 2 43 : 3% 11 : 0% 34 : 1% 26 48 Ca( d;p ) 2 p 3 = 2 14 : 9% 7 : 1% 12 : 2% 8 126 Sn( d;p ) 1 h 11 = 2 33 : 6% 7 : 7% 26 : 4 35 132 Sn( d;p ) 2 f 7 = 2 3 : 2% 2 : 5% 4 : 2% 16 208 Pb( d;p ) 2 g 9 = 2 35 : 0% 12 : 6% 20 : 5% 32 Table4.7:Percentofthe( d;p )transfercrosssectionsatthepeakwhen usingnonlocalpotentialsinentranceandexitchannels(1stcolumn),nonlocalpotentials inentrancechannelonly(2ndcolumn),andnonlocalpotentialsinexitchannelonly(3rd column),relativetothelocalcalculationwiththeLPEpotentials,foranumberofreactions occurringat20MeV. theheaviernucleiduetoalargersurfaceregionbeingprobed.ThisisseeninFig.4.18. Comparingthedot-dashedlinedwiththedottedline,weseethatthereisanenhancementof thecrosssectionfor 48 Ca,butareductionfor 132 Snand 208 Pb.Thenetofnonlocality intheexitchanneldependsonacomplexinterplayofthepropertiesoftheboundstate(i.e. numberofnodes,bindingenergy,andorbitalangularmomentum),aswellasthemagnitude oftherealandimaginarypartsofthescatteringwavefunctionnearthenuclearsurface. Dependingonthecase,theshiftingoutwardofthedeuteronwavefunctionsseeninFigs. 4.15and4.16didnotalwayshavethesameonthetransfercrosssections.Comparing thedashedanddottedlinesinFigs.4.17and4.18,weseethatfor 48 Ca,nonlocalityin thedeuteronscatteringstatehasasimilarasfortheprotonscatteringstateinthat itreducesthecrosssection.Asthesizeofthetargetincreases,theoutwardshiftofthe wavefunctionbecomesmoreimportant.Thisisseeninthecomparisonofthe d + 208 Pband the d + 48 CawavefunctionsinFigs.4.15and4.16.The d + 208 Pbwavefunctionisshifted outwardmorethanthe d + 48 Cawavefunction,whichchangestheamplitudeatthenuclear 97 Nonlocal Nonlocal Final Nonlocal Entrance Exit Angle E lab Bound Relative Relative Relative of 50MeV State toLocal toLocal toLocal Peak 16 O( d;p ) 1 d 5 = 2 22 : 3% 3 : 1% 15 : 8% 10 16 O( d;p ) 2 s 1 = 2 20 : 7% 0 : 4% 21 : 2% 0 40 Ca( d;p ) 1 f 7 = 2 4 : 8% 4 : 4% 0 : 2% 0 48 Ca( d;p ) 2 p 3 = 2 41 : 9% 8 : 1% 39 : 9% 0 126 Sn( d;p ) 1 h 11 = 2 6 : 9% 6 : 7% 2 : 5 13 132 Sn( d;p ) 2 f 7 = 2 10 : 9% 20 : 4% 26 : 2% 0 208 Pb( d;p ) 2 g 9 = 2 64 : 8% 86 : 5% 1 : 7% 0 Table4.8:Percentofthe( d;p )transfercrosssectionsatthepeakwhen usingnonlocalpotentialsinentranceandexitchannels(1stcolumn),nonlocalpotentials inentrancechannelonly(2ndcolumn),andnonlocalpotentialsinexitchannelonly(3rd column),relativetothelocalcalculationwiththeLPEpotentials,foranumberofreactions occurringat50MeV.Figurereprintedfrom[3]withpermission. surface,andhasatimpactonthetransfercrosssection.AsseeninTable4.8, nonlocalityinthedeuteronscatteringstatefor 208 Pb( d;p ) 209 Pbhasthemostt ofallthecasesstudies. TheinsetsinFig.4.17showthatwhenthetheoreticalcrosssectionsarenormalizedto thedataatthepeakofthedistribution,thelowenergydatacannotdistinguishbetween thevariousmodelssincetheshapeofthetheoreticaldistributionsaresimilar.However,for 48 Ca( d;p ) 49 Caat E d =50MeV,nonlocalitytlyimprovesthedescriptionofthe data.Inallcases,ifoneweretoextractaspectroscopicfactorfromthedata,theresults includingnonlocalitywouldconsiderablyfromthosewhenonlylocalinteractionsare used. 98 4.4.4ComparingDistortedWaveBornApproximationandthe AdiabaticDistortedWaveApproximation TheDWBAisstillthework-horseusedintheanalysisofmosttransfercrosssections.The DWBAisbasedonaseriesexpansiondescribedinSec.2.2.1.Thisexpansionisusually truncatedtost-ordersothatdeuteronbreakupisonlyincludedimplicitlythroughthe imaginarypartofthedeuteronopticalpotential.ThisisunliketheADWAwhichisbased onathree-bodymodel,includesbreakupexplicitly,andreliesonnucleonopticalpotentials. HereweshowthattheintheDWBAandADWAformalismcanleadtovery tpredictionsforthe( d;p )crosssections. InFig.4.19theangulardistributionsforthreet( d;p )reactionsobtainedusing theDWBAarecomparedtothoseobtainedwiththeADWA.Therewasnoobviousway tocomparetheofnonlocalityintheentrancechannelsinceDWBAandADWAtreat thedeuteronchannelverytly,andanonlocalglobaldeuteronopticalpotentialdoes notexist.Bothlocalandnonlocalpotentialswereusedintheexitchannel.Forauseful comparison,thesamenonlocalandlocalpotentialsareusedintheexitchannel.Forthe ADWA,theLPEpotentialsobtainedfromtothedistributiongeneratedwiththePerey- Buckpotentialareused,whilefortheDWBAweusedthedeuteronopticalpotentialof Daehnick[84]. Wefocusonthelocalresults,andcompareinFig.4.19theDWBA(dottedline)with theADWA(dashedline).Theshapesaretlyt,aswellasthemagnitudeof thecrosssectionatthepeak.Includingnonlocalityintheexitchanneldoesnotresolve thisdiscrepancy.Weseethatintroducingnonlocalityintheexitchannelhasthesimilar ofincreasingthecrosssectionforboththeDWBAandADWAcalculations.Wealso 99 Figure4.19:Comparisonof( d;p )transfercrosssectionswhenusingtheDWBAascompared totheADWA.(a) 16 O( d;p ) 17 O,(b) 48 Ca( d;p ) 49 Cawithdatafrom[20].(c) 132 Sn( d;p ) 133 Sn. Alldistributionsat E d =50MeV.Figurereprintedfrom[3]withpermission. 100 showtheinFig.4.19b.Itisclearthatforthiscase,theDWBAisnotabletodescribethe angulardistributionfromexperiment.Thisisoneexamplethatdemonstratestheneedto explicitlyincludedeuteronbreakupintothecalculation. 4.4.5EnergyShiftMethod Sincemanyreactionproblemsaresolvedincoordinatebasedtheories,localinteractionshave beenpreferredduetothesimplicityofsolvingtheequations.Forthisreason,Timofeyuk andJohnson,[21,22],developedamethodtoelyincludenonlocalityinthedeuteron scatteringstatewithintheformalismoftheADWA.Theirmethodreliesonlocalpotentials sothatthenonlocalequationdoesnotneedtobesolved.AssumingthePerey-Buckform forthenonlocalpotential,andthroughexpansions,theythatbyshiftingtheenergyat whichthelocalpotentialsareevaluatedby ˘ 40MeVfromthestandard E d = 2value,one cancapturetheofnonlocality.Sincewearenowabletoincludenonlocalityexplicitly, thismethodcanbetested. Asweareonlyconcernedwithnonlocalityinthedeuteronchannel,wethepotentials intheprotonchannelsowecanmakeameaningfulcomparison.Intheexitchannel,theLPE potentialsfoundfromtoPerey-Buckprotonelasticscatteringdistributionswereused, alongwiththelocalbindingpotentialusedtoreproducetheexperimentalbindingenergy. Intheentrancechannel,weusedthenonlocalPerey-Buckpotential,andthecorresponding LPEpotentials.Tousethemethodof[21,22],weneededanenergydependentlocaloptical potential.ForthisweusedtheCH89potential[23]evaluatedatthestandard E d = 2value, andwiththeadditionalenergyshiftthatwasquanin[21]. TheresultsofthisstudyareshowninFig.4.20.Weshowangulardistributionsfor( d;p ) reactionson(a) 16 Oat E d =10MeV,(b) 40 Caat E d =10MeV,and(c) 208 Pbat E d =20 101 Figure4.20:Comparisonof( d;p )angulardistributionswhenusingtheenergyshiftmethod of[21,22].(a) 16 O( d;p ) 17 Oat E d =10MeV(b) 40 Ca( d;p ) 41 Caat E d =10MeV(c) 208 Pb( d;p ) 209 Pbat E d =20MeV.Thesolidlineiswhenfullnonlocalitywasincludedin theentrancechannel,dashedlineiswhentheLPEpotentialwasused,dot-dashedlinewhen theCH89potential[23]wasusedwiththeadditionalenergyshiftquanin[21],andthe dottedlinewhentheCH89potentialwasusedatthestandard E d = 2value.Figurereprinted from[3]withpermission. 102 MeV.Thesolidlineisthedistributionformnonlocalityexplicitlyincludedinthedeuteron channel,thedashedlinearetheresultsofthelocalcalculationswiththeLPEpotentials forthedeuteronscatteringstate,thedot-dashedlineusedthemethodof[21,22],andthe dottedlineiswhenthelocalpotentialsareevaluatedatthestandard E d = 2value. Ourresultsshowthattheenergyshiftmethodalwaysincreasesthecrosssection.How- ever,theexplicitinclusionofnonlocalityinthedeuteronscatteringstatecansometimes decreasethecrosssection,asisseeninFig.4.20a.Oftentimes,theenergyshiftmovesthe transferdistributiontowardslargerangles,analsoseeninthefullnonlocalcalcula- tions.Insomecases,theenergyshiftmethodovershootsthefullnonlocalcalculation,as inFig.4.20b.InFig.4.20c,weseeanexamplewheretheenergyshiftdoesaverygood jobatreproducingthenonlocalects.Ingeneral,wefoundthatformostcases,theenergy shiftcapturedthequalitativeofnonlocality,butwasunabletoprovideanaccurate accountofthenonlocal 4.4.6Summary Inthisworkwestudiedtheofnonlocalityon( d;p )transferreactions.Anextensionof theADWAtheorywasdevelopedtoincludenonlocalityinthedeuteronscatteringstateusing thePerey-Bucknonlocalnucleonopticalpotential[1].IntheexitchannelthePerey-Buck potentialwasusedtodescribetheprotonscatteringstate,anditsrealpartwasadjusted fortheneutronboundstate.Forthescattering,alocalphaseequivalent(LPE)potential wasobtainedbytheelasticscatteringgeneratedfromthecorrespondingnonlocal potential.Boththelocalandnonlocalboundstatesreproducedthesameexperimental bindingenergies. Forthe( d;p )reactionsstudied,wefoundthattheinclusionofnonlocalityinboththe 103 entranceandexitchannelsincreasedthetransfercrosssectionby ˘ 40%.Inmostcases, nonlocalityinthedeuteronscatteringstatecausedamoderateincreaseinthetransfercross section.However,forheavytargetsathighenergies,thisincreasewaslarge.Nonlocality intheexitchannelcaused,almostexclusively,anincreaseinthetransfercrosssection, exceptforheavytargetsathighenergiesforwhichthecrosssectionswerereduced.Wealso comparedourADWAresultwiththosefromDWBAandfoundtheofnonlocalityin thestatetobeconsistentinbothformulations,evenifquantitativelyt.Wealso comparedourADWAresultswiththeenergyshiftmethodintroducedbyTimofeyukand Johnson[21,22]andfoundthatmethodtobequalitativelyconsistentwithourresults. Theconclusionofthepresentstudythoseof[12,14].Thereareimportant inthetransfercrosssectionswhenincludingnonlocalityexplicitlyascomparedto whenusingLPEpotentials.Thishighlightsthenecessityofexplicitlyincludingnonlocality todescribetransferreactions.Sincetheinclusionofnonlocalitynormallyincreasesthe crosssection,are-analysisoftransferreactiondatawilllikelyreducecurrenltyaccepted spectroscopicfactors,suchasthosereportedin[111]. 104 Chapter5 ConclusionsandOutlook 5.1Conclusions Inthisthesiswestudiedtheofnonlocalityoftheopticalpotentialintransferreactions. Forthispurposewedevelopedamethodforsolvingtheinrentialequationsand extendedtheadiabaticdistortedwaveapproximationfortransfer( d;p )reactionstoinclude nonlocalinteractionsofgeneralform.Weperformedseveralsystematicstudies,includinga rangeofenergies,targets,andinteractions. Forthe( p;d )reactionstudyusingthePerey-BucknonlocalpotentialofSec.4.2,we consideredarangeofnuclei,andprotonenergiesof E p =20and50MeV[12].Wecalculated thetransfermatrixelementinthedistortedwaveBornapproximation(DWBA),anduseda localopticalpotentialtodescribethedeuteronscatteringstate.Wefoundthattheexplicit inclusionofnonlocalityincreasedthetransfercrosssectionatthepeakby15 35%, relativetowhenlocalpotentialswereuse.Wefoundthatinallcases,thePereycorrection factortraditionallyuseddoesnotprovideaquantitativedescription.Ourresultssuggest thatsuchacorrectionfactortoaccountfornonlocalityshouldnotbeused. InSec.4.3wecomparedtheDOMpotentialandthePerey-Buckpotentialtostudy( p;d ) reactionson 40 Caat E p =20,35,and50MeV[14].Weincludednonlocalityintheentrance channel,thencomputedtransfercrosssectionsintheDWBA,ignoringnonlocalityinthe deuteronchannel.Wegeneratedtwolocalphaseequivalent(LPE)potentials,oneforthe 105 DOMandoneforthePerey-Buckpotential.BoththeDOMandthePerey-Buckpotential producedverylargeincreasesinthemagnitudeofthetransfercrosssection, ˇ 15 50%for theDOMpotential,and ˇ 30 50%forthePerey-Buckpotential.Likeinthestudy of( p;d )reactions,whennonlocalitywasincludedonlyintheboundstate,largeincreasesin themagnitudeofthetransfercrosssectionwereseen.Typically,nonlocalityintheproton scatteringstateactsintheoppositedirection,reducingthetransfercrosssection. InthelaststudyofSec.4.4involving( d;p )reactions,nonlocalitywasincludedinthe deuteronchannelwithintheadiabaticdistortedwaveapproximation(ADWA),whichunlike theDWBA,includesdeuteronbreakupexplicitly[3].TheformalismforthelocalADWA theoryhadtobeextendedtoincludenonlocalpotentials,andwasdoneinSec.2.6.We foundthattheinclusionofnonlocalityincreasedthetransfercrosssectionby ˘ 40%.Inmost cases,nonlocalityinthedeuteronscatteringstatecausedamodestincreaseinthetransfer crosssection.However,forheavytargetsathighenergy,theincreaseduetononlocalityin thedeuteronchannelwasverylarge.Thisisincontrasttowhennonlocalityisaddedtothe protonscatteringstate,whichoftentimesreducedthetransfercrosssection.Thereason fortherenceisthatnonlocalityinthedeuteronscatteringstatehadboththeof reducingthemagnitudeofthescatteringwavefunctionwithinthenuclearinterior,butalso shiftingthewavefunctionoutwardstowardstheperiphery. Allthesethreestudies[12,14,3]demonstratetheimportantinthetransfer crosssectionswhenincludingnonlocalityexplicitly,ascomparedtowhenusinglocalphase equivalentpotentials.Thisemphasizesthenecessityofincludingnonlocalitytodescribe transferreactionsifaccuratestructureinformationistobeextracted. Mostoften,transferreactionsareperformedtoextractaspectroscopicfactor.Theanaly- sisisdoneusinglocalpotentials.Asexplicitinclusionofnonlocalityincreasedthepredicted 106 crosssections,onewouldexpectlowerspectroscopicfactorstoresultfromtheanalysisif nonlocalpotentialsareused.Thismaywellcontributetosolvingthediscrepancybetween spectroscopicfactorsextractedfromknockoutandtransfer[112],butthatstudyhasyetto beperformed. 5.2Outlook Goingforward,nonlocalitymustbecarefullytakenintoaccountinanyadvancedreaction theory.Itwillbecomeincreasinglyimportanttoconstructamodernnonlocalglobaloptical potential.WorkalongtheselineshasalreadybeendonebyTian,Pang,andMa(TPM) withtherecentpublicationoftheirnonlocalpotential,[2].However,thispotentialisstill energyindependent,andisbasedonthesimplePerey-Buckform. Onphysicalgrounds,theopticalpotentialmustbeenergydependentduetochannel couplingPreliminaryindicationsfromanunpublishedstudybyBacq,Lovell,Titus, andNunes[113]showthatthereisindeedanenergydependenceinnonlocalpotentialsofthe Perey-Buckform.Workonspecifyingthepreciseenergydependenceisongoing.Itwould beadvantageoustoconstructanenergydependentnonlocalglobalopticalpotentialthrough ˜ 2 minimizationofalargequantityofelasticscatteringangulardistributions,andperhaps polarizationobservablesandotherdataaswell. ThePerey-Buckformforthepotentialcomprisesasinglenonlocalityparameter, .This simpleformhasbeenusefulformanydecadessinceitallowedforsimpleimplementation. Itisunlikelythatasinglenonlocalityparameterisabletorepresentthecomplexnature ofnonlocalityintherealisticmany-bodyproblem.TheDOMpotential,forexample,has at foreachtermofthepotential.Amoresophisticatedformforthenonlocal 107 potentialshouldbeconsidered. Thereareseveralmethodsavailabletoconstructamicroscopicallybasedopticalpoten- tial.SomeofthesemethodswerediscussedinChapter1,andshouldbepursuednowthatit isknownthatnonlocalityisveryimportantandmustbeincludedexplicitly.Sinceprevious microscopiccalculationsofnonlocalpotentialshaveshownthattheirformdoesnotresem- blethesimpleGaussiannonlocalityofthePerey-Buckpotential,itisimportanttobetter understanditsanalyticproperties. Withanimprovednonlocalglobalopticalpotential,existingtransferdatacanbere- analyzed.Thelargediscrepanciesbetweenspectroscopicfactorsextractedfromthenonlocal andlocalcalculationsinthisstudydemonstratesthatthestructureinformationofmost nucleiarelikelytobealteredwhenthedataisanalyzedwithnonlocalpotentials. Whilethisthesisfocusedon( d;p )and( p;d )reactions,theroleofnonlocalityshouldbe investigatedinotherreactionsaswell.Wearecurrentlyinvestigatingtheroleofnonlocality in( d;n )reactions[114].Surprisingly,theofnonlocalityin( d;n )reactionsareeven moretthanin( d;p )reactions. Alongwithtransfer,therearemanyotherreactionsthatareperformedtoextractsin- gleparticlestructureofnuclei.Nuclearknockoutreactions( A ( a; ) X )areanalternative methodtoextractaspectroscopicfactor.Suchareactionalsorequiresanopticalpotential betweenthecollidingnuclei.Understandingtheofnonlocalityinthiscaseisalsoim- portant.Inelasticscatteringprovidesthetransitionstrengthbetweenthegroundstateanda boundexcitedstateinanucleus.Sincetheseareobtainedbycomparingexperimentaldata totheoreticaldistributionsresultingfromaDWBAorcoupledchannelanalysis,onemay againexpectthatinclusionofnonlocalityinthedescriptionoftheprocesswillimprovethe reliabilityoftheextractedtransitionstrength.Three-bodymodelsexisttocalculatetrans- 108 fer,suchasthecontinuumdiscretizedcoupledchannelmethod[95].Inthiscase,coupled channelinerentialequationswouldneedtobesolved,andnewnumericalmethods maybeneededtoaccomplishthistask.Finally,charge-exchangereactions,suchas( p;n ), probethespinandisospinpropertiesofnuclei.Onceagain,informationfromexperiments aresometimesextractedfromaDWBAanalysis,andincludingnonlocalityintheoptical potentialwillimprovethereliabilityoftheoreticalpredictions,consequentlythereliability oftheextractedGamow-Tellerstrengths. Oneaspectthatwasnotaddressedinthisthesisconcernserrorquanion.Lovell andNunes[115]arecurrentlyaddressingerrorquanindirectreactiontheorieswhen usinglocalpotentials.Anextensionofthatworkwillbenecessaryastheuseofnonlocal nonlocalpotentialsbecomesmorewidespread.Muchworkhashighlightedtheimportance ofnonlocalityinthereactiondynamics.Howthencanweconstrainnonlocalityfromdata? ThePerey-Buckpotential,forexample,hasaGaussianformwithanonlocalityrangeof =0 : 85fm.Elasticscatteringisnotsensitivetoshort-rangeproperties,soconstraining inthiswaymaynotbethebestmethod.Reactionssensitivetoshortrangecorrelationsmay abetteravenue. Theresultsofthisthesis,alongwiththeadventofmicroscopictheoriestoconstruct nonlocalpotentials,improvedphenomenologicalnonlocalglobalopticalpotentials,andever increasingcomputerpower,hasthepotentialtoelevatereactiontheorytoanewlevel. Whilewehavefocusedon( p;d )and( d;p )reactions,withourincreasedunderstandingof nonlocality,ithasbecomenecessarytoupdatetheotherformalismsandcodescommonly usedinnuclearphysicstoproperlyincludenonlocalityinreactiontheory. 109 APPENDICES 110 AppendixA SolvingtheNonlocalEquation InordertoassessthevalidityofthelocalapproximationweneedtosolveEq.(3.14)exactly. Severalmethodsexistforsolvingthescatteringstate,suchas[78,80].Ourapproachfollows PereyandBuck[1]whereEq.(3.14)issolvedbyiteration.Forsimplicity,wewilldropthe localpartofthenonlocalpotential, U o ( r ),inourdiscussion,althoughitisincludedinour calculations. TosolvethepartialwaveequationofEq.(3.14)numerically,weneedtothekernel function g L ( R;R 0 ).Inordertodothis,weneedtodoapartialwaveexpansionofthe nonlocalpotential, U NL PB ( R ; R 0 )= X L 2 L +1 4 ˇ g L ( R;R 0 ) RR 0 P L (cos ) ; (A.1) wherewe astheanglebetween R and R 0 .InsertingthePerey-Buckformforthe nonlocalpotential,multiplyingbothsidesby P L (cos ),integratingoverallangles,usingthe orthogonalityoftheLegendrepolynomials,andsolvingfor g L ( R;R 0 ),wethat g L ( R;R 0 )=2 ˇRR 0 U 1 2 ( R + R 0 ) Z 1 1 exp R R 0 2 ˇ 3 2 3 P L (cos ) d (cos ) : (A.2) 111 Foramoment,consideronlytheintegralontheright: Z 1 1 exp R R 0 2 ˇ 3 2 3 P L (cos ) d (cos )= exp R 2 + R 0 2 2 ˇ 3 2 3 Z 1 1 e i ( iz ) P L (cos ) d (cos ) : (A.3) UsingtheintegralrepresentationforthesphericalBesselfunctions, j L ( x )= 1 2 i L Z 1 1 e ixu P L ( u ) du; (A.4) with u =cos ,wendthat g L ( R;R 0 )= 2 i L z ˇ 1 2 j L ( iz )exp R 2 + R 0 2 2 U 1 2 ( R + R 0 ) = h L ( R;R 0 ) U 1 2 ( R + R 0 ) : (A.5) Calculating h L ( R;R 0 )numericallyisfor z ˛ 1duetolargecancellationsbetween theterms,soweneedtoapproximatethisfunctionwhendoingnumericalcalculationsfor largevaluesoftheargument.MakinguseoftheasymptoticexpressionforthesphericalBessel function,andneglectingtermsproportionaltoexp R + R 0 2 ,wethat h L ( R;R 0 )for z ˛ 1canbeapproximatedas 112 h L ( R;R 0 ) ˇ 1 ˇ 1 2 e R R 0 2 for j z j˛ 1 : (A.6) Scatteringsolutionsareconsideredwherethesubscript n denotesthe n thorder approximationofthecorrectsolution.Theiterationschemestartswithaninitialization, ~ 2 2 d 2 dr 2 L ( L +1) R 2 ˜ n =0 ( R )+[ E U init ( R )] ˜ n =0 ( R )=0 ; (A.7) where U init ( R )issomesuitablelocalpotentialusedtogettheiterationprocessstarted. Knowing ˜ o ( R )onethenproceedswithsolving: ~ 2 2 d 2 dr 2 L ( L +1) R 2 ˜ n ( R )+[ E U init ( R )] ˜ n ( R ) = Z g L ( R;R 0 ) ˜ n 1 ( R 0 ) dR 0 U init ( R ) ˜ n 1 ( R ) ; (A.8) withasmanyiterationsnecessaryforconvergence.Thenumberofiterationsdependsmostly onthepartialwavebeingsolvedfor(lowerpartialwavesrequiremoreiterations)andthe qualityof U init ( R ).Itwasrareforanypartialwavetorequiremorethan20iterationsto converge,evenwithaverypoorchoicefor U init ( R ).IftheLPEpotentialisusedas U init ( R ), thenanypartialwaveconvergeswithlessthan10iterations. Fortheboundstateproblem,themethodissomewhatt.Avarietyofmethods existintheliterature,somedevelopedspallytohandlenon-analyticforms(e.g.[81]). Ourapproachmaynotbethemostt,butitisstraightforward,general,andeasyto implement.Tosolvetheboundstateproblemwithanonlocalpotentialwebeginbysolving 113 Eq.(A.7).Sinceweareusingthewavefunctionfromthepreviousiterationtocalculatethe nonlocalintegral,weneedtokeeptrackofthetnormalizationsoftheinwardand outwardwavefunctionsthatresultsfromthechoicefortheinitialconditionsforeachwave function.Thus,theequationsweiterateare: ~ 2 2 d 2 dr 2 ` ( ` +1) r 2 ˚ In n ( r )+[ E U init ( r )] ˚ In n ( r ) = Z R Max 0 g ` ( r;r 0 ) ˚ In n 1 ( r 0 ) dr 0 U init ( r ) ˚ In n 1 ( r ) ; (A.9) and ~ 2 2 d 2 dr 2 ` ( ` +1) r 2 ˚ Out n ( r )+[ E U init ( r )] ˚ Out n ( r ) = Z R Max 0 g ` ( r;r 0 ) ˚ Out n 1 ( r 0 ) dr 0 U init ( r ) ˚ Out n 1 ( r ) ; (A.10) where R Max issomemaximumradiuschosengreaterthantherangeofthenuclearinterac- tion.Notethat ˚ In ( r )isthewavefunctionforintegratingfromtheedgeoftheboxinward andhasanormalizationsetbytheWhittakerfunctionastheinitialcondition,while ˚ Out ( r ) isthewavefunctionforintegratingfromtheoriginoutwardandhasthenormalizationset usingthestandard r L +1 initialconditionneartheorigin. Eventhough ˚ Out and ˚ In byonlyaconstant,thesetwoequations(Eq.(A.9)and Eq.(A.10))arenecessarybecausethevalueofthenormalizationconstantisonlyknownafter convergence.Foragiveniteration, ˚ Out and ˚ In convergewhentheirlogarithmicderivatives agreeatthematchingpoint.Tokeepthepropernormalizationthroughouttheentirerange 114 [0 ;R Max ],weneedtoretaintwoversionsoftheconvergedwavefunctionforeachiteration: ˚ In n ( r )= 8 > < > : C Out n ˚ Out n ( r )for0 r < > : ˚ Out n ( r )for0 r > > > > < > > > > > : I d LJ p I d L 0 J p 000 9 > > > > > = > > > > > ; nn I d ( ˘ np ) I d ( ˘ np ) o 0 n ~ Y L ( ^ R ) ~ Y L 0 ( ^ R ) o 0 o 00 = ^ J p LL 0 ^ I d ^ L nn I d ( ˘ np ) I d ( ˘ np ) o 0 n ~ Y L ( ^ R ) ~ Y L 0 ( ^ R ) o 0 o 00 : (C.25) ReplacingEq.(C.25)intoEq.(C.24),andsummingover L 0 ,weobtain: 131 1 R ~ 2 2 @ 2 @R 2 L 0 ( L 0 +1) R 2 + E d ˜ J T M T L 0 J p ( R ) Z ˚ ` ( r ) V np ( r ) ˚ ` 0 ( r ) r 2 dr ( ) 2 J p 1 ^ I d Z n I d ( ˘ np ) I d ( ˘ np ) o 00 d˘ np 1 ^ L Z n ~ Y L ( ^ R ) ~ Y L ( ^ R ) o 00 d R (C.26) TheintegraloverthetwoWeinbergstatesmultipliedby V np gives 1bythenormalization conditionofEq.(2.37).Also,since I d =1, J p isaninteger,so( ) 2 J p =1.Thus,wehave 1 R ~ 2 2 @ 2 @R 2 L 0 ( L 0 +1) R 2 + E d ˜ J T M T L 0 J p ( R )(C.27) 1 ^ I d Z n I d ( ˘ np ) I d ( ˘ np ) o 00 d˘ np 1 ^ L Z n ~ Y L ( ^ R ) ~ Y L ( ^ R ) o 00 d R : TheintegraloverthespinfunctionscanbeworkedoutwithEq.(11),p.70,andEq.(4),p.358 of[93]: Z n I d ( ˘ np ) I d ( ˘ np ) o 00 d˘ np = Z nn 1 = 2 ( ˘ n ) 1 = 2 ( ˘ p ) o 1 n 1 = 2 ( ˘ n ) 1 = 2 ( ˘ p ) o 1 o 00 d˘ n d˘ p = Z ^ 1 2 8 > > > > > < > > > > > : 1 2 1 2 1 1 2 1 2 1 000 9 > > > > > = > > > > > ; nn 1 = 2 ( ˘ n ) 1 = 2 ( ˘ n ) o 0 n 1 = 2 ( ˘ p ) 1 = 2 ( ˘ p ) o 0 o 00 d˘ n d˘ p = Z ^ 1 2 ^ 1 2 2 ^ 1 Z n 1 = 2 ( ˘ n ) 1 = 2 ( ˘ n ) o 00 d˘ n Z n 1 = 2 ( ˘ p ) 1 = 2 ( ˘ p ) o 00 d˘ p : (C.28) 132 DoingtheintegralovertheneutronspinfunctionsbyusingEq.(13),p.132,andEq.(1),p.248, of[93]wearrriveat: Z n 1 = 2 ( ˘ n ) 1 = 2 ( ˘ n ) o 00 d˘ n = X n Z C 00 1 = 2 ; n ; 1 = 2 n 1 = 2 ; n ( ˘ n 1 = 2 n ( ˘ n ) d˘ n = X n C 00 1 = 2 ; n ; 1 = 2 n ( ) 1 = 2 n Z 1 = 2 n ( ˘ n 1 = 2 n ( ˘ n ) d˘ n = X n ( ) 1 = 2+ n d 1 2 ( ) 1 = 2 n Z 1 = 2 n ( ˘ n 1 = 2 n ( ˘ n ) d˘ n = p 2 ; (C.29) Asimilarprocedureisfollowedfortheintegralovertheprotonspinfunctions.Thus,with I d =1 1 ^ I d Z n I d ( ˘ np ) I d ( ˘ np ) o 00 d˘ np =1 : (C.30) TheintegralinEq.(C.27)canbeworkedoutusingEq.(6),p.62,andEq.(1),p.248,of [93] 133 1 ^ L Z n ~ Y L ( ^ R ) ~ Y L ( ^ R ) o 00 d R = 1 ^ L X M C 00 L; MLM Z ~ Y L; M ( ^ R ) ~ Y LM ( ^ R ) d R = 1 ^ L X M C 00 L; MLM ( ) L + M Z ~ Y LM ( ^ R ) ~ Y LM ( ^ R ) d R = 1 ^ L X M ( ) L + M ^ L ( ) L + M =1 : (C.31) Therefore,introducingEq.(C.31)intoEq.(C.27)andjoiningthe rhs ofEq.(C.16)weobtain: 1 R ~ 2 2 @ 2 @R 2 L ( L +1) R 2 + E d ˜ J T M T LJ p ( R ) = X ` 0 L 0 J 0 p X ` 8 Z ( ˆ n I d ( ˘ np ) ~ Y ` (^ r ) o j p ~ Y L ( ^ R ) ˙ J p I t ( ˘ t ) ) J T M T U NA R p;n ; R p;n +2 s ( ˆ n I d ( ˘ np ) ~ Y ` 0 ( \ r 2 s ) o j p ~ Y L 0 ( \ R + s ) ˙ J 0 p I t ( ˘ t ) ) J T M T ˚ ` ( r ) V np ( r ) ˚ ` 0 ( j r 2 s j ) ˜ J T M T L 0 J 0 p ( j R + s j ) j R + s j d s d r d R d˘ t d˘ np (C.32) Wenowconcentrateonthetensorcouplingsinthe rhs ofEq.(C.32).First,weintroduce ` =0forthedeuteron,andintegrateover d˘ t : 134 ( ˆ n I d ( ˘ np ) ~ Y 0 (^ r ) o j p ~ Y L ( ^ R ) ˙ J p I t ( ˘ t ) ) J T M T ( ˆ n I d ( ˘ np ) ~ Y 0 ( \ r 2 s ) o j p ~ Y L 0 ( \ R + s ) ˙ J 0 p I t ( ˘ t ) ) J T M T = 1 4 ˇ ˆ n I d ( ˘ np ) ~ Y L ( ^ R ) o J p I t ( ˘ t ) ˙ J T M T ˆ n I d ( ˘ np ) ~ Y L 0 ( \ R + s ) o J 0 p I t ( ˘ t ) ˙ J T M T = 1 4 ˇ X M p M 0 p X t 0 t C J T M T J p M p I t t C J T M T J 0 p M 0 p I t 0 t n I d ( ˘ np ) ~ Y L ( ^ R ) o J p M p n I d ( ˘ np ) ~ Y L 0 ( \ R + s ) o J 0 p M 0 p Z I t t ( ˘ t I t 0 t ( ˘ t ) d˘ t = 1 4 ˇ X M p M 0 p X t C J T M T J p M p I t t C J T M T J 0 p M 0 p I t t n I d ( ˘ np ) ~ Y L ( ^ R ) o J p M p n I d ( ˘ np ) ~ Y L 0 ( \ R + s ) o J 0 p M 0 p : (C.33) InsertingEq.(C.33)intoEq.(C.32)wearriveat: 1 R ~ 2 2 @ 2 @R 2 L ( L +1) R 2 + E d ˜ J T M T LJ p ( R ) = 1 4 ˇ X L 0 J 0 p 8 Z ˚ 0 ( r ) V np ( r ) U NA R p;n ; R p;n +2 s X M p M 0 p X t C J T M T J p M p I t t C J T M T J 0 p M 0 p I t t n I d ( ˘ np ) ~ Y L ( ^ R ) o J p M p (C.34) n I d ( ˘ np ) ~ Y L 0 ( \ R + s ) o J 0 p M 0 p ˚ 0 ( j r 2 s j ) ˜ J T M T L 0 J 0 p ( j R + s j ) j R + s j d s d r d R d˘ np : 135 Nextwecouplethetwotensorstogetheruptozeroangularmomentumusing,Eq.(6),p.62, andEq.(1),p.248,of[93]: n I d ( ˘ np ) ~ Y L ( ^ R ) o J p M p n I d ( ˘ np ) ~ Y L 0 ( \ R + s ) o J 0 p M 0 p =( ) J p M p n I d ( ˘ np ) ~ Y L ( ^ R ) o J p ; M p n I d ( ˘ np ) ~ Y L 0 ( \ R + s ) o J 0 p M 0 p =( ) J p M p X SM S C SM S J p ; M p J 0 p M p ˆ n I d ( ˘ np ) ~ Y L ( ^ R ) o J p n I d ( ˘ np ) ~ Y L 0 ( \ R + s ) o J 0 p ˙ 00 ! ( ) J p M p C 00 J p ; M p J 0 p M p ˆ n I d ( ˘ np ) ~ Y L ( ^ R ) o J p n I d ( ˘ np ) ~ Y L 0 ( \ R + s ) o J 0 p ˙ 00 =( ) J p M p ( ) J p + M p J p J 0 p M p M 0 p ^ J p ˆ n I d ( ˘ np ) ~ Y L ( ^ R ) o J p n I d ( ˘ np ) ~ Y L 0 ( \ R + s ) o J 0 p ˙ 00 (C.35) =( ) 2 J p J p J 0 p M p M 0 p ^ J p ˆ n I d ( ˘ np ) ~ Y L ( ^ R ) o J p n I d ( ˘ np ) ~ Y L 0 ( \ R + s ) o J 0 p ˙ 00 : WereplaceEq.(C.35)inEq.(C.34)andsumover J 0 p and M 0 p toobtain: 136 1 R ~ 2 2 @ 2 @R 2 L ( L +1) R 2 + E d ˜ J T M T LJ p ( R ) = 1 4 ˇ X L 0 8 Z ˚ 0 ( r ) V np ( r ) U NA R p;n ; R p;n +2 s X M p X t C J T M T J p M p I t t C J T M T J p M p I t t ( ) 2 J p 1 ^ J p ˆ n I d ( ˘ np ) ~ Y L ( ^ R ) o J p n I d ( ˘ np ) ~ Y L 0 ( \ R + s ) o J p ˙ 00 ˚ 0 ( j r 2 s j ) ˜ J T M T L 0 J p ( j R + s j ) j R + s j d s d r d R d˘ np : (C.36) Nextwesumover M p and t using,Eq.(8),p.236,of[93], 1 R ~ 2 2 @ 2 @R 2 L ( L +1) R 2 + E d ˜ J T M T LJ p ( R ) = 1 4 ˇ X L 0 8 Z ˚ 0 ( r ) V np ( r ) U NA R p;n ; R p;n +2 s ( ) 2 J p 1 ^ J p ˆ n I d ( ˘ np ) ~ Y L ( ^ R ) o J p n I d ( ˘ np ) ~ Y L 0 ( \ R + s ) o J p ˙ 00 ˚ 0 ( j r 2 s j ) ˜ J T M T L 0 J p ( j R + s j ) j R + s j d s d r d R d˘ np : (C.37) WenowuseEq.(11),p.70,andEq.(4),p.358,of[93]tocouplethespinfunctionsandthe sphericalharmonicstozeroangularmomentum, 137 ˆ n I d ( ˘ np ) ~ Y L ( ^ R ) o J p n I d ( ˘ np ) ~ Y L 0 ( \ R + s ) o J p ˙ 00 = ^ J 2 p 8 > > > > > < > > > > > : I d LJ p I d L 0 J p 000 9 > > > > > = > > > > > ; nn I d ( ˘ np ) I d ( ˘ np ) o 0 n ~ Y L ( ^ R ) ~ Y L 0 ( \ R + s ) o 0 o 00 = LL 0 ^ J p ^ I d ^ L nn I d ( ˘ np ) I d ( ˘ np ) o 0 n ~ Y L ( ^ R ) ~ Y L 0 ( \ R + s ) o 0 o 00 : (C.38) Using( ) 2 J p =1andsummingover L 0 , 1 R ~ 2 2 @ 2 @R 2 L ( L +1) R 2 + E d ˜ J T M T LJ p ( R ) = 1 4 ˇ X L 0 8 Z ˚ 0 ( r ) V np ( r ) U NA R p;n ; R p;n +2 s ( ) 2 J p 1 ^ I d ^ L n I d ( ˘ np ) I d ( ˘ np ) o 00 n ~ Y L ( ^ R ) ~ Y L ( \ R + s ) o 00 ˚ 0 ( j r 2 s j ) ˜ J T M T LJ p ( j R + s j ) j R + s j d s d r d R d˘ np = 1 4 ˇ X L 0 8 Z ˚ 0 ( r ) V np ( r ) U NA R p;n ; R p;n +2 s (C.39) 1 ^ L n ~ Y L ( ^ R ) ~ Y L ( \ R + s ) o 00 ˚ 0 ( j r 2 s j ) ˜ J T M T LJ p ( j R + s j ) j R + s j d s d r d R : Bringingthe1 =R termfromthe lhs overtothe rhs givesus: 138 ~ 2 2 @ 2 @R 2 L ( L +1) R 2 + E d ˜ J T M T LJ p ( R ) = R ^ L 8 4 ˇ Z ˚ 0 ( r ) V np ( r ) U NA R p;n ; R p;n +2 s n ~ Y L ( ^ R ) ~ Y L ( \ R + s ) o 00 ˚ 0 ( j r 2 s j ) ˜ J T M T LJ p ( j R + s j ) j R + s j d s d r d R = R ^ L 8 4 ˇ X M C 00 L; MLM Z ˚ 0 ( r ) V np ( r ) U NA R p;n ; R p;n +2 s ~ Y L; M ( ^ R ) ~ Y LM ( \ R + s ) ˚ 0 ( j r 2 s j ) ˜ J T M T LJ p ( j R + s j ) j R + s j d s d r d R = R ^ L 8 4 ˇ X M ( ) L M ^ L Z ˚ 0 ( r ) V np ( r ) U NA R p;n ; R p;n +2 s ~ Y L; M ( ^ R ) ~ Y LM ( \ R + s ) ˚ 0 ( j r 2 s j ) ˜ J T M T LJ p ( j R + s j ) j R + s j d s d r d R : (C.40) Sincetheintegrandiscoupledtozeroangularmomentum,itissphericallysymmetric,which meansthatitisinvariantunderrotationsofthethreevectors R , r ,and s .Thus,wecan evaluateitinanywewant.Byplacingthe R inthe^ z -direction, M =0, and ~ Y L 0 (^ z )= i L ^ L p 4 ˇ .Wewillplace r inthe xz -planesothatthe ˚ r -dependenceisremoved. Integrationover d R yieldsafactorof4 ˇ forallotherchoicesforthedirectionof R .Since weare r tobeinthe xz -plane,wegetafactorof2 ˇ fromeachvectortotakecareof rotationsaroundthez-axis.Thus,weneedtomultiplytheintegralby(4 ˇ ) (2 ˇ )=8 ˇ 2 . Thereisnoadditionalsymmetryto s .Finally,introducingthesesymmetriesinthe integralofEq.(C.40)andthephase i L ofthesphericalharmonics,wearriveatthe expressionwehaveusedinourimplementation: 139 ~ 2 2 @ 2 @R 2 L ( L +1) R 2 + E d ˜ J T M T LJ p ( R ) = R ^ L 8 4 ˇ ^ L p 4 ˇ ( ) L i 2 L 1 ^ L 8 ˇ 2 Z ˚ 0 ( r ) V np ( r ) U NA R p;n ; R p;n +2 s Y L 0 ( \ R + s ) ˚ 0 ( j r 2 s j ) ˜ J T M T LJ p ( j R + s j ) j R + s j d s r 2 dr sin r r = 8 R p ˇ ^ L Z ˚ 0 ( r ) V np ( r ) U NA R p;n ; R p;n +2 s Y L 0 ( \ R + s ) ˚ 0 ( j r 2 s j ) ˜ J T M T LJ p ( j R + s j ) j R + s j d s r 2 dr sin r r (C.41) where R p;n , R ,and s aretobeevaluatedinthedescribedbefore. U NA is thenucleonopticalpotentialforeithertheprotonorneutron.Makingthereplacement U NA ! U nA + U pA givesusthenonlocaladiabaticpotential,andtheresultingpartialwave equationforthedeuteronscatteringstatewhenusingnonlocalpotentialswithintheADWA. 140 AppendixD DerivingtheT-Matrix HerewewillderivetheexplicitformfortheT-matrixfor( d;p )transferreactions.This equationwasgiveninthepostforminEq.(2.21),andisrepeatedherewithouttheremnant term T A M d p M B ( k f ; k i )= h p M B k f j V np j A M d k i i : (D.1) WeneedtotheexplicitpartialwaveforallwavefunctionsinEq.(D.1).Webeginby thewavefunctionforrelativemotionbetween d + A ,whichisgivenby: ` i j i = X L i X J P i M P i ˆ n I p ( ˘ p ) j i ( r np ;˘ n ) o J d ~ Y L i ( ^ R dA ) ˙ J P i M P i I A A ( ˘ A ) ˜ L i J P i ( R dA ) R dA = X L i X J P i M P i ( ˆ I p ( ˘ p ) n ~ Y ` i (^ r np ) I n ( ˘ n ) o j i ˙ J d ~ Y L i ( ^ R dA ) ) J P i M P i I A A ( ˘ A ) ˚ j i ( r np ) ˜ L i J p i ( R dA ) R dA : (D.2) AsinAppendixC, I p ( ˘ p ), I n ( ˘ n )and I A ( ˘ A )arethespinfunctionsfortheproton, 141 neutron,andtarget,respectively,eachwithprojections p , n ,and A . ~ Y ` i isthespherical harmonicsfortherelativemotionbetweentheneutronandprotoninthedeuteron,and ~ Y L i isthesphericalharmonicfortherelativemotionbetweenthedeuteronandthetarget.As inAppendixC,wearegourtensorswiththebuiltinfactorof i L sothat ~ Y ` i = i ` i Y ` i with Y ` i onp.133,Eq.(1),of[93].Asareminder, ˚ j i ( r np )istheradialwavefunction fortheboundstate,and j i resultsfromcouplingtheorbitalmotionofthedeuteronbound statewiththespinoftheneutron. ˜ L i J p i ( R dA )istheradialwavefunctionforthedeuteron scatteringstate,and J p i resultsfromcouplingthespinofthedeuteron, J d =1totheorbital motionbetweenthedeuteronandthetarget. Theincomingdistoredwaveshoulddependonlyontheprojectionsoftheprojectileand target,andreducetoaplanewaveinthelimitofzeropotential.Therefore,wemultiply Eq.(D.2)bytheincomingcot 4 ˇ k i i L i e i˙ L i P M 0 i ~ Y L i M 0 i ( ^ k i ) C J P i M P i J d M d L i M 0 i givingus: M d A ` i j i = 4 ˇ k i X L i J P i i L i e i˙ L i I A A ( ˘ A ) ˚ j i ( r np ) ˜ L i J p i ( R dA ) R dA X M 0 i M P i ~ Y L i M 0 i ( ^ k i ) C J P i M P i J d M d L i M 0 i ( ˆ I p ( ˘ p ) n ~ Y ` i (^ r np ) I n ( ˘ n ) o j i ˙ J d ~ Y L i ( ^ R dA ) ) J P i M P i : (D.3) Now,wemakethefollowingsubstitutionsusingEq.(13),p.132,andEq.(10),p.245,of[93] ~ Y L i M 0 i ( ^ k i )=( ) L i + M 0 i ~ Y L i ; M 0 i ( ^ k i ) C J P i M P i J d M d L i M 0 i =( ) L i + M 0 i ^ J P i ^ J d C J d M d L i ; M 0 i J P i M P i ; (D.4) 142 andinsert( ) 2( L i + M 0 i ) =1,Eq.(D.3)becomes: j M d A i i = 4 ˇ k i X L i J P i i L i e i˙ L i I A A ( ˘ A ) ˚ j i ( r np ) ˜ L i J p i ( R dA ) R dA ^ J P i ^ J d X M 0 i M P i C J d M d L i ; M 0 i J P i M P i ~ Y L i ; M 0 i ( ^ k i ) ( ˆ I p ( ˘ p ) n ~ Y ` i (^ r np ) I n ( ˘ n ) o j i ˙ J d ~ Y L i ( ^ R dA ) ) J P i M P i = 4 ˇ k i X L i J P i i L i e i˙ L i I A A ( ˘ A ) ˚ j i ( r np ) ˜ L i J p i ( R dA ) R dA ^ J P i ^ J d ( ~ Y L i ( ^ k i ) ( ˆ I p ( ˘ p ) n ~ Y ` i (^ r np ) I n ( ˘ n ) o j i ˙ J d ~ Y L i ( ^ R dA ) o J P i ) J d M d = I A A ( ˘ A ) ˚ j i ( r np ) ˜ (+) i ( k i ; r np ; R dA ;˘ p ;˘ n ) : (D.5) Thepartialwavefortheincomingdistortedwaveiswrittenas: ˜ (+) i ( k i ; r np ; R dA ;˘ p ;˘ n )= 4 ˇ k i X L i J P i i L i e i˙ L i ^ J P i ^ J d ˜ L i J p i ( R dA ) R dA (D.6) 8 > < > : ~ Y L i ( ^ k i ) ( ˆ I p ( ˘ p ) n ~ Y ` i (^ r np ) I n ( ˘ n ) o j i ˙ J d ~ Y L i ( ^ R dA ) ) J P i 9 > = > ; J d M d : Thewavefunctionforrelativemotionbetween p + B isgivenby 143 ` f j f = n I A ( ˘ A ) j f ( r nA ;˘ n ) o J B M B X L f X J P f M P f n I p ( ˘ p ) ~ Y L f ( ^ R pB ) o J P f M P f ˜ L f J P f ( R pB ) R pB = ( I A ( ˘ A ) n ~ Y ` f (^ r nA ) I n ( ˘ n ) o j f ) J B M B ˚ j f ( r nA ) X L f X J P f M P f n I p ( ˘ p ) ~ Y L f ( ^ R pB ) o J P f M P f ˜ L f J P f ( R pB ) R pB : (D.7) Inthisequation ` f istheorbitalangularmomentumbetweenthetargetandthebound neutron, j f isthequantumnumberresultingfromcoupling ` f tothespinofthetarget, I A . Thetotalangularmomentumofthetargetisgivenby J B andresultsfromcoupling j f to I A .Theorbitalangularmomentumbetweentheprotonandthetargetisgivenby L f ,and thetotalangularmomentumoftheprojectile, J p f resultsfromcoupling L f tothespinof theproton, I p . Aswedidfortheentrancechannel,weneedtomultiplytheexitchannelwavefunction bytheoutgoingcot: 4 ˇ k f i L f e i˙ L f P M 0 f ~ Y L f M 0 f ( ^ k f ) C J P f M P f I p p L f M 0 f sothattheremaining quantumnumbersarefortheprojectionsoftheprojectileandtargetintheexitchannel: 144 j p M B ` f j f i = 4 ˇ k f ( I A ( ˘ A ) n ~ Y ` f (^ r nA ) I n ( ˘ n ) o j f ) J B M B ˚ j f ( r nA ) X L f J P f i L f e i˙ L f ˜ L f J P f ( R pB ) R pB X M 0 f M P f Y L f M 0 f ( ^ k f ) C J P f M P f I p p L f M 0 f n I p ( ˘ p ) ~ Y L f ( ^ R pB ) o J P f M P f : (D.8) Wefollowthesamestepsasbefore,usingEq.(13),p.132,andEq.(10),p.245,of[93] Y L f M 0 f ( ^ k f )=( ) L f + M 0 f ~ Y L f ; M 0 f ( ^ k f ) C J P f M P f I p p L f M 0 f =( ) L f + M 0 f ^ J P f ^ I p C I p p L f ; M 0 f J P f M P f ; (D.9) anduse( ) 2( L f + M 0 f ) =1.Thisresultsin: 145 j p M B f i = 4 ˇ k f ( I A ( ˘ A ) n ~ Y ` f (^ r nA ) I n ( ˘ n ) o j f ) J B M B ˚ j f ( r nA ) X L f J P f i L f e i˙ L f ˜ L f J P f ( R pB ) R pB ^ J P f ^ I p X M 0 f M P f C I p p L f ; M 0 f J P f M P f ~ Y L f ; M 0 f ( ^ k f ) n I p ( ˘ p ) ~ Y L f ( ^ R pB ) o J P f M P f = 4 ˇ k f ( I A ( ˘ A ) n ~ Y ` f (^ r nA ) I n ( ˘ n ) o j f ) J B M B ˚ j f ( r nA ) X L f J P f i L f e i˙ L f ˜ L f J P f ( R pB ) R pB ^ J P f ^ I p 8 < : ~ Y L f ( ^ k f ) n I p ( ˘ p ) ~ Y L f ( ^ R pB ) o J P f 9 = ; I p p = ( I A ( ˘ A ) n ~ Y ` f (^ r nA ) I n ( ˘ n ) o j f ) J B M B ˚ j f ( r nA ) ˜ (+) f ( k f ; R pB ;˘ p ) : (D.10) IntheT-Matrix,Eq.(D.1)theexitchannelappearsasabra: h p M B f j = ( I A ( ˘ A ) n ~ Y ` f (^ r nA ) I n ( ˘ n ) o j f ) J B M B ˚ j f ( r nA ) ˜ ( ) f ( k f ; R pB ) ; (D.11) wheretheoutgoingdistortedwave ˜ ( ) ( k ; R )isthetimereverseof ˜ (+) ,sothat ˜ ( ) ( k ; R )= ˜ (+) ( k ; R ) .Therefore,tomakethismoreexplicitweuseEq.(2),p.141,of[93] 146 h ˜ ( ) ( k ; R ) j = ˜ ( ) ( k ; R ) = ˜ (+) ( k ; R ) = ˜ (+) ( k ; R ) =( ) L ˜ (+) ( k ; R ) ; (D.12) where k ! k givesafactorof( ) L fromthesphericalharmonics,asseeninEq.(2),p.141, of[93],andthetwocomplexconjugationscancel. Theincomingandoutgoingdistortedwavesaregivenby ˜ (+) i ( k i ; r np ; R dA ;˘ p ;˘ n )= 4 ˇ k i ^ J d X L i J P i i L i e i˙ L i ^ J P i ˜ L i J p i ( R dA ) R dA (D.13) 8 > < > : ~ Y L i ( ^ k i ) ( ˆ I p ( ˘ p ) n ~ Y ` i (^ r np ) I n ( ˘ n ) o j i ˙ J d ~ Y L i ( ^ R dA ) ) J P i 9 > = > ; J d M d and, ˜ ( ) f ( k f ; R pB ;˘ p )= 4 ˇ k f ^ I p X L f J P f i L f e i˙ L f ^ J P f ˜ L f J P f ( R pB ) R pB 8 < : ~ Y L f ( ^ k f ) n I p ( ˘ p ) ~ Y L f ( ^ R pB ) o J P f 9 = ; I p p : (D.14) Intheadiabatictheory, ˜ L i J P i ( R dA )theequation 147 " ~ 2 2 i @ 2 @R 2 dA L i ( L i +1) R 2 dA ! + U ad + V SO 1 L i J P i + V C ( R dA ) E # ˜ L i J P i ( R dA )=0 (D.15) where U ad istheadiabaticpotential,and V SO 1 L i J P i isthespin-orbitpotential.Thefunction ˜ L f J P f ( R pB )asinglechannelopticalmodelequation: " ~ 2 2 f @ 2 @R 2 pB L f ( L f +1) R 2 pB ! + U pB + V SO I p L f J P f + V C ( R pB ) E # ˜ L f J P f ( R pB )=0 (D.16) with U pB beinganucleonopticalpotential.Intheseequations i and f arethereduced massintheinitialandalstates,nottobeconfusedwithspinprojections A , p ,and n . Also, U ad and U pB canbeeitherlocalornonlocal. AsmentionedinSec.2.3,thescatteringamplitudeisrelatedtotheT-matrixby f A M d p M B ( k f ; k i )= f 2 ˇ ~ 2 ~ T A M d p M B ( k f ; k i ) = f 2 ˇ ~ 2 r v f v i T A M d p M B ( k f ; k i ) = f 2 ˇ ~ 2 v u u u u t ~ k f f ~ k i i h p M B f j V np j A M d i i : (D.17) Thetialcrosssectionisobtained,byaveragingthemodofthescatteringamplitude 148 squaredoverinitialstates,andsummingover m -states: d˙ d = 1 ^ J 2 d ^ J 2 A X A M d p M B f A M d p M B ( k f ; k i ) 2 == k f k i i f 4 ˇ 2 ~ 4 1 ^ J 2 d ^ J 2 A X A M d M B p h p M B f j V np j A M d i i 2 : (D.18) WenowputEq.(D.5)andEq.(D.11)into h p M B f j V np j A M d i i : h p M B f j V np j A M d i i = Z ( I A ( ˘ A ) n ~ Y ` f (^ r nA ) I n ( ˘ n ) o j f ) J B M B ˚ j f ( r nA ) 4 ˇ k f ^ I p X L f J P f i L f e i˙ L f ^ J P f ˜ L f J P f ( R pB ) R pB 8 < : ~ Y L f ( ^ k f ) n I p ( ˘ p ) ~ Y L f ( ^ R pB ) o J P f 9 = ; I p p (D.19) V ( r np I A A ( ˘ A ) ˚ j i ( r np ) 4 ˇ k i ^ J d X L i J P i i L i e i˙ L i ^ J P i ˜ L i J p i ( R dA ) R dA 8 > < > : ~ Y L i ( ^ k i ) ( ˆ I p ( ˘ p ) n ~ Y ` i (^ r np ) I n ( ˘ n ) o j i ˙ J d ~ Y L i ( ^ R dA ) ) J P i 9 > = > ; J d M d d R pB d r nA d˘ n d˘ p d˘ A : Breakingthecouplingbetweenthetargetandtheboundstate,andgroupingthe twospinfunctionsforthetargettogether,weobtain: 149 h p M B f j V np j A M d i i = 0 B @ X 0 A Z I A 0 A ( ˘ A I A A ( ˘ A ) d˘ A 1 C A Z X m f C J B M B I A 0 A j f m f n ~ Y ` f (^ r nA ) I n ( ˘ n ) o j f m f ˚ j f ( r nA ) 4 ˇ k f ^ I p X L f J P f i L f e i˙ L f ^ J P f ˜ L f J P f ( R pB ) R pB 8 < : ~ Y L f ( ^ k f ) n I p ( ˘ p ) ~ Y L f ( ^ R pB ) o J P f 9 = ; I p p (D.20) V ( r np ) ˚ j i ( r np ) 4 ˇ k i ^ J d X L i J P i i L i e i˙ L i ^ J P i ˜ L i J p i ( R dA ) R dA 8 > < > : ~ Y L i ( ^ k i ) ( ˆ I p ( ˘ p ) n ~ Y ` i (^ r np ) I n ( ˘ n ) o j i ˙ J d ~ Y L i ( ^ R dA ) ) J P i 9 > = > ; J d M d d R pB d r nA d˘ n d˘ p : Theintegralinthelinegives 0 A A .Performingthesumover 0 A provides: 150 h p M B f j V np j A M d i i = Z X m f C J B M B I A A j f m f n ~ Y ` f (^ r nA ) I n ( ˘ n ) o j f m f ˚ j f ( r nA ) 4 ˇ k f ^ I p X L f J P f i L f e i˙ L f ^ J P f ˜ L f J P f ( R pB ) R pB 8 < : ~ Y L f ( ^ k f ) n I p ( ˘ p ) ~ Y L f ( ^ R pB ) o J P f 9 = ; I p p (D.21) V ( r np ) ˚ j i ( r np ) 4 ˇ k i ^ J d X L i J P i i L i e i˙ L i ^ J P i ˜ L i J p i ( R dA ) R dA 8 > < > : ~ Y L i ( ^ k i ) ( ˆ I p ( ˘ p ) n ~ Y ` i (^ r np ) I n ( ˘ n ) o j i ˙ J d ~ Y L i ( ^ R dA ) ) J P i 9 > = > ; J d M d d R pB d r nA d˘ n d˘ p : Wenowcouplethefollowingtensorstogether: 8 < : ~ Y L f ( ^ k f ) n I p ( ˘ p ) ~ Y L f ( ^ R pB ) o J P f 9 = ; I p p 8 > < > : ~ Y L i ( ^ k i ) ( ˆ I p ( ˘ p ) n ~ Y ` i (^ r np ) I n ( ˘ n ) o j i ˙ J d ~ Y L i ( ^ R dA ) ) J P i 9 > = > ; J d M d = X QM Q C QM Q I p p J d M d 8 > < > : 8 < : ~ Y L f ( ^ k f ) n I p ( ˘ p ) ~ Y L f ( ^ R pB ) o J P f 9 = ; I p (D.22) 8 > < > : ~ Y L i ( ^ k i ) ( ˆ I p ( ˘ p ) n ~ Y ` i (^ r np ) I n ( ˘ n ) o j i ˙ J d ~ Y L i ( ^ R dA ) ) J P i 9 > = > ; J d 9 > > = > > ; QM Q ; 151 toobtain: h p M B f j V np j A M d i i = (4 ˇ ) 2 k i k f ^ J d ^ I p Z X m f X QM Q C J B M B I A A j f m f C QM Q I p p J d M d n ~ Y ` f (^ r nA ) I n ( ˘ n ) o j f m f ˚ j f ( r nA ) X L i J P i X L f J P f i L f e i˙ L f ^ J P f ˜ L f J P f ( R pB ) R pB V ( r np ) ˚ j i ( r np ) i L i e i˙ L i ^ J P i ˜ L i J p i ( R dA ) R dA 8 > < > : 8 < : ~ Y L f ( ^ k f ) n I p ( ˘ p ) ~ Y L f ( ^ R pB ) o J P f 9 = ; I p 8 > < > : ~ Y L i ( ^ k i ) ( ˆ I p ( ˘ p ) n ~ Y ` i (^ r np ) I n ( ˘ n ) o j i ˙ J d ~ Y L i ( ^ R dA ) ) J P i 9 > = > ; J d 9 > > = > > ; QM Q d R pB d r nA d˘ n d˘ p : (D.23) WecanrewriteEq.(D.23)inamorecompactform: h p M B f j V np j A M d i i = X m f X QM Q C J B M B I A A j f m f C QM Q I p p J d M d T QM Q m f ; (D.24) sothat 152 X A M d M B p jh p M B f j V np j A M d i ij 2 = X A M d M B p X m f QM Q X m 0 f Q 0 M 0 Q C J B M B I A A j f m f C J B M B I A A j f m 0 f C QM Q I p p J d M d C Q 0 M 0 Q I p p J d M d T QM Q m f T Q 0 M 0 Q m 0 f : (D.25) NowweconsiderthepairofClebsch-GordansanduseEq.(10),p.245,of[93], C J B M B I A A j f m f C J B M B I A A j f m 0 f = ( ) I A A ^ J B ^ j f ! 2 C j f ; m f I A A J B ; M B C j f ; m 0 f I A A J B ; M B : (D.26) ThistogetherwithEq.(8),p.236,of[93]allowsustosimplifyEq.(D.25)to X A M d M B p jh p M B f j V np j A M d i ij 2 = ^ J 2 B ^ j 2 f X m f QM Q X m 0 f Q 0 M 0 Q 0 @ X A M B C j f ; m f I A A J B ; M B C j f ; m 0 f I A A J B ; M B 1 A 0 @ X p M d C QM Q I p p J d M d C Q 0 M 0 Q I p p J d M d 1 A T QM Q m f T Q 0 M 0 Q m 0 f = ^ J 2 B ^ j 2 f X m f QM Q X m 0 f Q 0 M 0 Q m f m 0 f QQ 0 M Q M 0 Q T QM Q m f T Q 0 M 0 Q m 0 f = ^ J 2 B ^ j 2 f X m f QM Q T QM Q m f T QM Q m f ; (D.27) 153 andthenthetialcrosssectionis d˙ d = k f k i i f 4 ˇ 2 ~ 4 1 ^ J 2 d ^ J 2 A X A M d M B p h p M B f j V np j A M d i i 2 = k f k i i f 4 ˇ 2 ~ 4 ^ J 2 B ^ J 2 d ^ J 2 A ^ j 2 f X m f QM Q T QM Q m f T QM Q m f : (D.28) Inessence,ourtaskistoworkout,explicitly, T QM Q m f T QM Q m f = (4 ˇ ) 2 k i k f ^ J d ^ I p X L i J P i X L f J P f i L i L f e i ( ˙ L i + ˙ L f ) ^ J P i ^ J P f Z ˚ j f ( r nA ) ˜ L f J P f ( R pB ) V ( r np ) ˚ j i ( r np ) ˜ L i J p i ( R dA ) R pB R dA 8 > < > : 8 < : ~ Y L f ( ^ k f ) n I p ( ˘ p ) ~ Y L f ( ^ R pB ) o J P f 9 = ; I p 8 > < > : ~ Y L i ( ^ k i ) ( ˆ I p ( ˘ p ) n ~ Y ` i (^ r np ) I n ( ˘ n ) o j i ˙ J d ~ Y L i ( ^ R dA ) ) J P i 9 > = > ; J d 9 > > = > > ; QM Q n ~ Y ` f (^ r nA ) I n n ( ˘ n ) o j f m f d R pB d r nA d˘ n d˘ p : (D.29) Ourstrategyistocouplethesphericalharmonicswiththeargument ^ k togethersowecanpull themoutoftheintegral.Wewanttocouplethesphericalharmonicswiththearguments ^ r and ^ R togetheruptozeroangularmomentumsowecanusesymmetrytoreducethe dimensionalityoftheangularintegral.Also,wewanttocouplethespinorswithcommon 154 argumentsuptozeroangularmomentumsowecanintegratethemout.Thisisdetailedin thenextfewpages. Wecangroupthe ^ k sphericalharmonicstogetherrightaway.Letusintroducethe A J P f = n I p ( ˘ p ) ~ Y L f ( ^ R pB ) o J P f B J P i = ( ˆ I p ( ˘ p ) n ~ Y ` i (^ r np ) I n ( ˘ n ) o j i ˙ J d ~ Y L i ( ^ R dA ) ) J P i (D.30) sothetensorEq.(D.29)is ( ˆ ~ Y L f ( ^ k f ) A J P f ˙ I p ˆ ~ Y L i ( ^ k i ) B J P i ˙ J d ) QM Q = j L f J P f ( I p ) L i J P i ( J d ) QM Q i = X gh j L f L i ( g ) J P f J P i ( h ) QM Q ih L f L i ( g ) J P f J P i ( h ) QM Q j L f J P f ( I p ) L i J P i ( J d ) QM Q i = X gh h L f L i ( g ) J P f J P i ( h ) QM Q j L f J P f ( I p ) L i J P i ( J d ) QM Q i ˆ n ~ Y L f ( ^ k f ) ~ Y L i ( ^ k i ) o g ˆ A J P f B J P i ˙ h ˙ QM Q ; (D.31) whereweusedtheofthe9jinEq.(5),p.334,of[93]: 155 h j 1 j 2 ( j 12 ) j 3 j 4 ( j 34 ) jm j j 1 j 3 ( j 13 ) j 2 j 4 ( j 24 ) j 0 m 0 i = jj 0 mm 0 ^ j 12 ^ j 13 ^ j 24 ^ j 34 8 > > > > > < > > > > > : j 1 j 2 j 12 j 3 j 4 j 34 j 13 j 24 j 9 > > > > > = > > > > > ; : (D.32) InsertingEq.(D.31)intoEq.(D.29): T QM Q m f = (4 ˇ ) 2 k i k f ^ J d ^ I p X L i J P i X L f J P f i L i L f e i ( ˙ L i + ˙ L f ) ^ J P i ^ J P f Z ˚ j f ( r nA ) ˜ L f J P f ( R pB ) V ( r np ) ˚ j i ( r np ) ˜ L i J p i ( R dA ) R pB R dA X gh h L f L i ( g ) J P f J P i ( h ) QM Q j L f J P f ( I p ) L i J P i ( J d ) QM Q i ˆ n ~ Y L f ( ^ k f ) ~ Y L i ( ^ k i ) o g ˆ A J P f B J P i ˙ h ˙ QM Q n ~ Y ` f (^ r nA ) I n n ( ˘ n ) o j f m f d R pB d r nA d˘ n d˘ p : (D.33) Nowweconsidertheproduct fABg : 156 ˆ A J P f B J P i ˙ h = 8 < : n I p ( ˘ p ) ~ Y L f ( ^ R pB ) o J P f ( ˆ I p ( ˘ p ) n ~ Y ` i (^ r np ) I n ( ˘ n ) o j i ˙ J d ~ Y L i ( ^ R dA ) ) J P i 9 > = > ; h = j I p L f ( J P f ) J d L i ( J P i ) hm h i = X g 0 h 0 j I p J d ( g 0 ) L f L i ( h 0 ) hm h ih I p J d ( g 0 ) L f L i ( h 0 ) hm h j I p L f ( J P f ) J d L i ( J P i ) hm h i = X g 0 h 0 h I p J d ( g 0 ) L f L i ( h 0 ) hm h j I p L f ( J P f ) J d L i ( J P i ) hm h i 8 < : ( I p ( ˘ p ) ˆ I p ( ˘ p ) n ~ Y ` i (^ r np ) I n ( ˘ n ) o j i ˙ J d ) g 0 n ~ Y L f ( ^ R pB ) ~ Y L i ( ^ R dA ) o h 0 o h : (D.34) InsertingEq.(D.34)intoEq.(D.33)wearriveat: 157 T QM Q m f = (4 ˇ ) 2 k i k f ^ J d ^ I p X L i J P i X L f J P f i L i L f e i ( ˙ L i + ˙ L f ) ^ J P i ^ J P f Z ˚ j f ( r nA ) ˜ L f J P f ( R pB ) V ( r np ) ˚ j i ( r np ) ˜ L i J p i ( R dA ) R pB R dA X gh h L f L i ( g ) J P f J P i ( h ) QM Q j L f J P f ( I p ) L i J P i ( J d ) QM Q i X g 0 h 0 h I p J d ( g 0 ) L f L i ( h 0 ) hm h j I p L f ( J P f ) J d L i ( J P i ) hm h i 8 < : n ~ Y L f ( ^ k f ) ~ Y L i ( ^ k i ) o g 8 < : ( I p ( ˘ p ) ˆ I p ( ˘ p ) n ~ Y ` i (^ r np ) I n ( ˘ n ) o j i ˙ J d ) g 0 n ~ Y L f ( ^ R pB ) ~ Y L i ( ^ R dA ) o h 0 o h ˙ QM Q n ~ Y ` f (^ r nA ) I n n ( ˘ n ) o j f m f d R pB d r nA d˘ n d˘ p : (D.35) ThefollowingtensorinEq.(D.35)canbeusing,Eq.(27),p.64,andEq.(8),p.70, of[93] ( I p ( ˘ p ) ˆ I p ( ˘ p ) n ~ Y ` i (^ r np ) I n ( ˘ n ) o j i ˙ J d ) g 0 =( ) I p + J d g 0 ( ˆ I p ( ˘ p ) n ~ Y ` i (^ r np ) I n ( ˘ n ) o j i ˙ J d I p ( ˘ p ) ) g 0 =( ) I p + J d g 0 ( ) J d + I p + g 0 X q ^ J d ^ q 8 > < > : I p j i J d g 0 I p q 9 > = > ; ˆ n ~ Y ` i (^ r np ) I n ( ˘ n ) o j i n I p ( ˘ p ) I p ( ˘ p ) o q ˙ g 0 : (D.36) Sincethespinfunctionsmustbecoupledtozeroangularmomentum,thisimpliesthat q =0 158 and g 0 = j i .Therefore,Eq.(D.36)to,with( ) 2 J d =1,andusingEq.(1),p.299,of [93] ( ) 2 I p ^ J d 8 > < > : I p j i J d j i I p 0 9 > = > ; ˆ n ~ Y ` i (^ r np ) I n ( ˘ n ) o j i n I p ( ˘ p ) I p ( ˘ p ) o 0 ˙ j i =( ) 2 I p ( ) I p + j i + J d ^ J d ^ I p ^ j i ˆ n ~ Y ` i (^ r np ) I n ( ˘ n ) o j i n I p ( ˘ p ) I p ( ˘ p ) o 0 ˙ j i : (D.37) Rememberingthat g 0 = j i ,insertingEq.(D.37)intoEq.(D.35)weobtain: T QM Q m f = (4 ˇ ) 2 ^ I 2 p k i k f ( ) 3 I p + j i + J d ^ j i X L i J P i X L f J P f i L i L f e i ( ˙ L i + ˙ L f ) ^ J P i ^ J P f Z ˚ j f ( r nA ) ˜ L f J P f ( R pB ) V ( r np ) ˚ j i ( r np ) ˜ L i J p i ( R dA ) R pB R dA X gh h L f L i ( g ) J P f J P i ( h ) QM Q j L f J P f ( I p ) L i J P i ( J d ) QM Q i X h 0 h I p J d ( j i ) L f L i ( h 0 ) hm h j I p L f ( J P f ) J d L i ( J P i ) hm h i (D.38) ( n ~ Y L f ( ^ k f ) ~ Y L i ( ^ k i ) o g ( ˆ n ~ Y ` i (^ r np ) I n ( ˘ n ) o j i I n ( ˘ p ) I n ( ˘ p ) 0 ˙ j i n ~ Y L f ( ^ R pB ) ~ Y L i ( ^ R dA ) o h 0 o h ˙ QM Q n ~ Y ` f (^ r nA ) I n n ( ˘ n ) o j f m f d R pB d r nA d˘ n d˘ p : WenowtakethelasttwolinesofEq.(D.38),breakallthecouplingsbetweenthepairs,and introducethenecessaryClebsch-Gordancots: 159 ( n ~ Y L f ( ^ k f ) ~ Y L i ( ^ k i ) o g ( ˆ n ~ Y ` i (^ r np ) I n ( ˘ n ) o j i n I p ( ˘ p ) I p ( ˘ p ) o 0 ˙ j i n ~ Y L f ( ^ R pB ) ~ Y L i ( ^ R dA ) o h 0 o h ˙ QM Q = X m g m h C QM Q gm g hm h n ~ Y L f ( ^ k f ) ~ Y L i ( ^ k i ) o gm g X m i m h 0 C hm h j i m i h 0 m h 0 n ~ Y ` i (^ r np ) I n ( ˘ n ) o j i m i n I p ( ˘ p ) I p ( ˘ p ) o 00 n ~ Y L f ( ^ R pB ) ~ Y L i ( ^ R dA ) o h 0 m h 0 : (D.39) InsertingEq.(D.39)backintoEq.(D.38)weobtain: T QM Q m f = (4 ˇ ) 2 ^ I 2 p k i k f ( ) 3 I p + j i + J d ^ j i X L i J P i X L f J P f i L i L f e i ( ˙ L i + ˙ L f ) ^ J P i ^ J P f Z ˚ j f ( r nA ) ˜ L f J P f ( R pB ) V ( r np ) ˚ j i ( r np ) ˜ L i J p i ( R dA ) R pB R dA X gh h L f L i ( g ) J P f J P i ( h ) QM Q j L f J P f ( I p ) L i J P i ( J d ) QM Q i X h 0 h I p J d ( j i ) L f L i ( h 0 ) hm h j I p L f ( J P f ) J d L i ( J P i ) hm h i X m g m h C QM Q gm g hm h X m i m h 0 C hm h j i m i h 0 m h 0 n ~ Y L f ( ^ k f ) ~ Y L i ( ^ k i ) o gm g n ~ Y ` i (^ r np ) I n ( ˘ n ) o j i m i n I p ( ˘ p ) I p ( ˘ p ) o 00 n ~ Y L f ( ^ R pB ) ~ Y L i ( ^ R dA ) o h 0 m h 0 n ~ Y ` f (^ r nA ) I n n ( ˘ n ) o j f m f d R pB d r nA d˘ n d˘ p : (D.40) Wenowconsider n ~ Y L f ( ^ R pB ) ~ Y L i ( ^ R dA ) o h 0 m h 0 n ~ Y ` f (^ r nA ) I n n ( ˘ n ) o j f m f anduseEq.(23), p.64,of[93]toobtain: 160 ( ) j f m f n ~ Y ` f (^ r nA ) I n ( ˘ n ) o j f ; m f n ~ Y ` i (^ r np ) I n ( ˘ n ) o j i m i =( ) j f m f X KM C KM j f ; m f j i m i ( n ~ Y ` f (^ r nA ) I n ( ˘ n ) o j f n ~ Y ` i (^ r np ) I n ( ˘ n ) o j i ) KM =( ) j f m f X KM C KM j f ; m f j i m i j ` f I n ( j f ) ` i I n ( j i ) KM i =( ) j f m f X KM C KM j f ; m f j i m i X g 00 h 00 h ` f ` i ( g 00 ) I n I n ( h 00 ) KM j ` f I n ( j f ) ` i I n ( j i ) KM i j ` f ` i ( g 00 ) I n I n ( h 00 ) KM i =( ) j f m f X KM C KM j f ; m f j i m i X g 00 h 00 h ` f ` i ( g 00 ) I n I n ( h 00 ) KM j ` f I n ( j f ) ` i I n ( j i ) KM i ˆ n ~ Y ` f (^ r nA ) ~ Y ` i (^ r np ) o g 00 I n ( ˘ n ) I n ( ˘ n ) h 00 ˙ KM : (D.41) Sincethespinsmustbecoupleduptozero,weseethat h 00 =0and g 00 = K .Imposingthis conditioninEq.(D.41): 161 T QM Q m f = (4 ˇ ) 2 ^ I 2 p k i k f ( ) 3 I p + j i + J d + j f m f ^ j i X L i J P i X L f J P f i L i L f e i ( ˙ L i + ˙ L f ) ^ J P i ^ J P f Z ˚ j f ( r nA ) ˜ L f J P f ( R pB ) V ( r np ) ˚ j i ( r np ) ˜ L i J p i ( R dA ) R pB R dA X gh h L f L i ( g ) J P f J P i ( h ) QM Q j L f J P f ( I p ) L i J P i ( J d ) QM Q i X h 0 h I p J d ( j i ) L f L i ( h 0 ) hm h j I p L f ( J P f ) J d L i ( J P i ) hm h i X m g m h C QM Q gm g hm h X m i m h 0 C hm h j i m i h 0 m h 0 X KM C KM j f ; m f j i m i h ` f ` i ( K ) I n I n (0) KM j ` f I n ( j f ) ` i I n ( j i ) KM i n ~ Y ` f (^ r nA ) ~ Y ` i (^ r np ) o KM I n ( ˘ n ) I n ( ˘ n ) 00 n ~ Y L f ( ^ k f ) ~ Y L i ( ^ k i ) o gm g n I p ( ˘ p ) I p ( ˘ p ) o 00 n ~ Y L f ( ^ R pB ) ~ Y L i ( ^ R dA ) o h 0 m h 0 d R pB d r nA d˘ n d˘ p : (D.42) Nowwecouplethe^ r and ^ R sphericalharmonicsuptozerousingEq.(1),p.248,of[93] n ~ Y L f ( ^ R pB ) ~ Y L i ( ^ R dA ) o h 0 m h 0 n ~ Y ` f (^ r nA ) ~ Y ` i (^ r np ) o KM = X SM S C SM S h 0 m h 0 KM nn ~ Y L f ( ^ R pB ) ~ Y L i ( ^ R dA ) o h 0 n ~ Y ` f (^ r nA ) ~ Y ` i (^ r np ) o K o SM S ! C 00 h 0 m h 0 KM nn ~ Y L f ( ^ R pB ) ~ Y L i ( ^ R dA ) o h 0 n ~ Y ` f (^ r nA ) ~ Y ` i (^ r np ) o K o 00 =( ) h 0 m h 0 h 0 K m h 0 ; M ^ h 0 nn ~ Y L f ( ^ R pB ) ~ Y L i ( ^ R dA ) o h 0 n ~ Y ` f (^ r nA ) ~ Y ` i (^ r np ) o K o 00 =( ) h 0 m h 0 h 0 K m h 0 ; M ^ h 0 X M K ( ) K + M K ^ K n ~ Y L f ( ^ R pB ) ~ Y L i ( ^ R dA ) o K; M K n ~ Y ` f (^ r nA ) ~ Y ` i (^ r np ) o KM K ; (D.43) 162 toarriveat: T QM Q m f = (4 ˇ ) 2 ^ I 2 p k i k f ( ) 3 I p + j i + J d + j f m f ^ j i X L i J P i X L f J P f i L i L f e i ( ˙ L i + ˙ L f ) ^ J P i ^ J P f Z ˚ j f ( r nA ) ˜ L f J P f ( R pB ) V ( r np ) ˚ j i ( r np ) ˜ L i J p i ( R dA ) R pB R dA X gh h L f L i ( g ) J P f J P i ( h ) QM Q j L f J P f ( I p ) L i J P i ( J d ) QM Q i X h 0 h I p J d ( j i ) L f L i ( h 0 ) hm h j I p L f ( J P f ) J d L i ( J P i ) hm h i I n ( ˘ n ) I n ( ˘ n ) 00 X m g m h C QM Q gm g hm h X m i m h 0 C hm h j i m i h 0 m h 0 X KM C KM j f ; m f j i m i n ~ Y L f ( ^ k f ) ~ Y L i ( ^ k i ) o gm g h ` f ` i ( K ) I n I n (0) KM j ` f I n ( j f ) ` i I n ( j i ) KM i n I p ( ˘ p ) I p ( ˘ p ) o 00 ( ) h 0 m h 0 h 0 K m h 0 ; M ^ h 0 X M K ( ) K + M K ^ K n ~ Y L f ( ^ R pB ) ~ Y L i ( ^ R dA ) o K; M K n ~ Y ` f (^ r nA ) ~ Y ` i (^ r np ) o KM K d R pB d r nA d˘ n d˘ p : (D.44) Nextwesumover h 0 and m h 0 ,sothat h 0 = K ,and m h 0 = M .ThenEq.(D.44)becomes: 163 T QM Q m f = (4 ˇ ) 2 ^ I 2 p k i k f ( ) 3 I p + j i + J d + j f m f ^ j i X L i J P i X L f J P f i L i L f e i ( ˙ L i + ˙ L f ) ^ J P i ^ J P f Z ˚ j f ( r nA ) ˜ L f J P f ( R pB ) V ( r np ) ˚ j i ( r np ) ˜ L i J p i ( R dA ) R pB R dA X gh h L f L i ( g ) J P f J P i ( h ) QM Q j L f J P f ( I p ) L i J P i ( J d ) QM Q i h I p J d ( j i ) L f L i ( K ) hm h j I p L f ( J P f ) J d L i ( J P i ) hm h i I n ( ˘ n ) I n ( ˘ n ) 00 X m g m h C QM Q gm g hm h X m i C hm h j i m i K M X KM C KM j f ; m f j i m i n ~ Y L f ( ^ k f ) ~ Y L i ( ^ k i ) o gm g h ` f ` i ( K ) I n I n (0) KM j ` f I n ( j f ) ` i I n ( j i ) KM i n I p ( ˘ p ) I p ( ˘ p ) o 00 ( ) K + M ^ K X M K ( ) K + M K ^ K n ~ Y L f ( ^ R pB ) ~ Y L i ( ^ R dA ) o K; M K n ~ Y ` f (^ r nA ) ~ Y ` i (^ r np ) o KM K d R pB d r nA d˘ n d˘ p : (D.45) Theintegralsover d˘ n and d˘ p give ^ I p ^ I n ,sothat: 164 T QM Q m f = (4 ˇ ) 2 ^ I p k i k f ( ) 3 I p + j i + J d + j f m f ^ j i ^ I n X L i J P i X L f J P f i L i L f e i ( ˙ L i + ˙ L f ) ^ J P i ^ J P f Z ˚ j f ( r nA ) ˜ L f J P f ( R pB ) V ( r np ) ˚ j i ( r np ) ˜ L i J p i ( R dA ) R pB R dA X K 1 ^ K 2 X gh h L f L i ( g ) J P f J P i ( h ) QM Q j L f J P f ( I p ) L i J P i ( J d ) QM Q i h I p J d ( j i ) L f L i ( K ) hm h j I p L f ( J P f ) J d L i ( J P i ) hm h i X m g m h C QM Q gm g hm h X m i M ( ) M C hm h j i m i K M C KM j f ; m f j i m i X M K ( ) M K h ` f ` i ( K ) I n I n (0) KM j ` f I n ( j f ) ` i I n ( j i ) KM i n ~ Y L f ( ^ k f ) ~ Y L i ( ^ k i ) o gm g n ~ Y L f ( ^ R pB ) ~ Y L i ( ^ R dA ) o K; M K n ~ Y ` f (^ r nA ) ~ Y ` i (^ r np ) o KM K d R pB d r nA (D.46) WenextconsiderthesumoverClebsch-Gordancots,anduseEq.(11),p.245,of[93]: 165 X m i M ( ) M C hm h j i m i K; M C KM j f ; m f j i m i = X m i M ( ) M C hm h j i m i K; M ( ) j f + m f ^ K ^ j i C j i ; m i j f ; m f K; M = X m i M ( ) M C hm h j i m i K; M ( ) j f + m f ^ K ^ j i ( ) K M ^ j i ^ j f C j f m f j i m i K; M =( ) j f + m f + K ^ K ^ j f X m i M C hm h j i m i K; M C j f m f j i m i K; M =( ) j f + m f + K ^ K ^ j f hj f m h m f : (D.47) InsertingEq.(D.47)intoEq.(D.46),thensummingover h and m h ,weobtain T QM Q m f = (4 ˇ ) 2 ^ I p k i k f ( ) 3 I p + j i + J d +2 j f ^ j i ^ j f ^ I n X L i J P i X L f J P f i L i L f e i ( ˙ L i + ˙ L f ) ^ J P i ^ J P f Z ˚ j f ( r nA ) ˜ L f J P f ( R pB ) V ( r np ) ˚ j i ( r np ) ˜ L i J p i ( R dA ) R pB R dA X K ( ) K ^ K X g h L f L i ( g ) J P f J P i ( j f ) QM Q j L f J P f ( I p ) L i J P i ( J d ) QM Q i h I p J d ( j i ) L f L i ( K ) j f m f j I p L f ( J P f ) J d L i ( J P i ) j f m f i X m g C QM Q gm g j f m f h ` f ` i ( K ) I n I n (0) KM j ` f I n ( j f ) ` i I n ( j i ) KM i n ~ Y L f ( ^ k f ) ~ Y L i ( ^ k i ) o gm g X M K ( ) M K n ~ Y L f ( ^ R pB ) ~ Y L i ( ^ R dA ) o K; M K n ~ Y ` f (^ r nA ) ~ Y ` i (^ r np ) o KM K d R pB d r nA : (D.48) Wenextreorganizethesums: 166 T QM Q m f = (4 ˇ ) 2 ^ I p k i k f ( ) 3 I p + j i + J d +2 j f ^ j i ^ j f ^ I n X K ( ) K ^ K h ` f ` i ( K ) I n I n (0) KM j ` f I n ( j f ) ` i I n ( j i ) KM i X L i J P i X L f J P f i L i L f e i ( ˙ L i + ˙ L f ) ^ J P i ^ J P f h I p J d ( j i ) L f L i ( K ) j f m f j I p L f ( J P f ) J d L i ( J P i ) j f m f i X g h L f L i ( g ) J P f J P i ( j f ) QM Q j L f J P f ( I p ) L i J P i ( J d ) QM Q i X m g C QM Q gm g j f m f n ~ Y L f ( ^ k f ) ~ Y L i ( ^ k i ) o gm g (D.49) Z ˚ j f ( r nA ) ˜ L f J P f ( R pB ) V ( r np ) ˚ j i ( r np ) ˜ L i J p i ( R dA ) R pB R dA X M K ( ) M K n ~ Y L f ( ^ R pB ) ~ Y L i ( ^ R dA ) o K; M K n ~ Y ` f (^ r nA ) ~ Y ` i (^ r np ) o KM K d R pB d r nA : Weplace ^ k i inthe^ z -directionsothat n ~ Y L f ( ^ k f ) ~ Y L i ( ^ k i ) o gm g = i L f + L i X ~ M i ~ M f C gm g L f ~ M f L i ~ M i Y L f ~ M f ( ^ k f ) Y L i ~ M i ( ^ k i = i L f + L i X ~ M f C gm g L f ~ M f L i 0 Y L f ~ M f ( ^ k f ) ^ L i p 4 ˇ ~ M f m g = i L f + L i C gm g L f m g L i 0 Y L f m g ( ^ k f ) ^ L i p 4 ˇ : (D.50) ThenEq.(D.49)becomes: 167 T QM Q m f = (4 ˇ ) 3 = 2 ^ I p k i k f ( ) 3 I p + j i + J d +2 j f ^ j i ^ j f ^ I n X K ( ) K ^ K h ` f ` i ( K ) I n I n (0) KM j ` f I n ( j f ) ` i I n ( j i ) KM i X L i J P i X L f J P f i 2 L i e i ( ˙ L i + ˙ L f ) ^ L i ^ J P i ^ J P f h I p J d ( j i ) L f L i ( K ) j f m f j I p L f ( J P f ) J d L i ( J P i ) j f m f i X g h L f L i ( g ) J P f J P i ( j f ) QM Q j L f J P f ( I p ) L i J P i ( J d ) QM Q i X m g C QM Q gm g j f m f C gm g L f m g L i 0 Y L f m g ( ^ k f )(D.51) Z ˚ j f ( r nA ) ˜ L f J P f ( R pB ) V ( r np ) ˚ j i ( r np ) ˜ L i J p i ( R dA ) R pB R dA X M K ( ) M K n ~ Y L f ( ^ R pB ) ~ Y L i ( ^ R dA ) o K; M K n ~ Y ` f (^ r nA ) ~ Y ` i (^ r np ) o KM K d R pB d r nA : Wecannowbreaktheremainingcouplingstoobtain: 168 T QM Q m f = (4 ˇ ) 3 = 2 ^ I p k i k f ( ) 3 I p + j i + J d +2 j f ^ j i ^ j f ^ I n X K ( ) K ^ K h ` f ` i ( K ) I n I n (0) KM j ` f I n ( j f ) ` i I n ( j i ) KM i X L i J P i X L f J P f i 2 L i e i ( ˙ L i + ˙ L f ) ^ L i ^ J P i ^ J P f h I p J d ( j i ) L f L i ( K ) j f m f j I p L f ( J P f ) J d L i ( J P i ) j f m f i X g h L f L i ( g ) J P f J P i ( j f ) QM Q j L f J P f ( I p ) L i J P i ( J d ) QM Q i X m g C QM Q gm g j f m f C gm g L f m g L i 0 Y L f m g ( ^ k f ) Z ˚ j f ( r nA ) ˜ L f J P f ( R pB ) V ( r np ) ˚ j i ( r np ) ˜ L i J p i ( R dA ) R pB R dA X M K ( ) M K X M f M i i L f + L i C K; M K L f M f L i M i Y L f M f ( ^ R pB ) Y L i M i ( ^ R dA ) X ~ m f ~ m i i ` f + ` i C KM K ` f ~ m f ` i ~ m i Y ` f ~ m f (^ r nA ) Y ` i ~ m i (^ r np ) d R pB d r nA : (D.52) R pB isanotherindependentvariable.Weplace ^ R pB inthe^ z -direction.Inthatcase, M f =0, Y L f M f ( ^ R pB )= ^ L f = p 4 ˇ ,and M i = M K .Eq.(D.52)isthento: 169 T QM Q m f = 4 ˇ ^ I p k i k f ( ) 3 I p + j i + J d +2 j f ^ j i ^ j f ^ I n X K ( ) K ^ K h ` f ` i ( K ) I n I n (0) KM j ` f I n ( j f ) ` i I n ( j i ) KM i X L i J P i X L f J P f i 3 L i + L f + ` f + ` i e i ( ˙ L i + ˙ L f ) ^ L i ^ L f ^ J P i ^ J P f h I p J d ( j i ) L f L i ( K ) j f m f j I p L f ( J P f ) J d L i ( J P i ) j f m f i X g h L f L i ( g ) J P f J P i ( j f ) QM Q j L f J P f ( I p ) L i J P i ( J d ) QM Q i X m g C QM Q gm g j f m f C gm g L f m g L i 0 Y L f m g ( ^ k f ) Z ˚ j f ( r nA ) ˜ L f J P f ( R pB ) V ( r np ) ˚ j i ( r np ) ˜ L i J p i ( R dA ) R pB R dA X M K ( ) M K C K; M K L f 0 L i ; M K Y L i ; M K ( ^ R dA )(D.53) X ~ m f ~ m i C KM K ` f ~ m f ` i ~ m i Y ` f ~ m f (^ r nA ) Y ` i ~ m i (^ r np ) R 2 pB d R pB dR pB r 2 nA sin dr nA d˚: Sinceweare ^ R pB inthe^ z -direction,integratingover d R pB resultsonlyinafactorof 4 ˇ .Wealsotheothervectorstobeinthe xz -plane,whichmeansthattheintegralover d˚ providesanadditionalfactorof2 ˇ :IntroducingtheseintoEq.(D.53)weobtain: 170 T QM Q m f = 32 ˇ 3 ^ I p k i k f ( ) 3 I p + j i + J d +2 j f ^ j i ^ j f ^ I n X K ( ) K ^ K h ` f ` i ( K ) I n I n (0) KM j ` f I n ( j f ) ` i I n ( j i ) KM i X L i J P i X L f J P f i 3 L i + L f + ` f + ` i e i ( ˙ L i + ˙ L f ) ^ L i ^ L f ^ J P i ^ J P f h I p J d ( j i ) L f L i ( K ) j f m f j I p L f ( J P f ) J d L i ( J P i ) j f m f i X g h L f L i ( g ) J P f J P i ( j f ) QM Q j L f J P f ( I p ) L i J P i ( J d ) QM Q i X m g C QM Q gm g j f m f C gm g L f m g L i 0 Y L f m g ( ^ k f ) Z ˚ j f ( r nA ) ˜ L f J P f ( R pB ) V ( r np ) ˚ j i ( r np ) ˜ L i J p i ( R dA ) R pB r 2 nA R dA X M K ( ) M K C K; M K L f 0 L i ; M K Y L i ; M K ( ^ R dA ) X ~ m f ~ m i C KM K ` f ~ m f ` i ~ m i Y ` f ~ m f (^ r nA ) Y ` i ~ m i (^ r np )sin dR pB dr nA : (D.54) Eq.(D.54)isvalidforageneral ` i and ` f .Sinceweareinterestedinapplyingtheformalism to( d;p )weuse ` i =0deuteron.Therefore Y ` i ~ m i (^ r np )=1 = p 4 ˇ ,~ m i =0,~ m f = M K ,and K = ` f .IntroducingthisintoEq.(D.54)wearriveat: 171 T QM Q m f = 32 ˇ 3 ^ I p p 4 ˇk i k f ( ) 3 I p + j i + J d +2 j f + ` f ^ j i ^ j f ^ ` f ^ I n h ` f 0( ` f ) I n I n (0) ` f M j ` f I n ( j f )0 I n ( j i ) ` f M i X L i J P i X L f J P f i 3 L i + L f + ` f + ` i e i ( ˙ L i + ˙ L f ) ^ L i ^ L f ^ J P i ^ J P f h I p J d ( j i ) L f L i ( ` f ) j f m f j I p L f ( J P f ) J d L i ( J P i ) j f m f i X g h L f L i ( g ) J P f J P i ( j f ) QM Q j L f J P f ( I p ) L i J P i ( J d ) QM Q i X m g C QM Q gm g j f m f C gm g L f m g L i 0 Y L f m g ( ^ k f ) X M K ( ) M K C ` f ; M K L f 0 L i ; M K Z ˚ j f ( r nA ) ˜ L f J P f ( R pB ) V ( r np ) ˚ j i ( r np ) ˜ L i J p i ( R dA ) R pB r 2 nA R dA Y L i ; M K ( ^ R dA ) Y ` f M K (^ r nA )sin dR pB dr nA : (D.55) WecanfurthersimplifyEq.(D.55)byusing,Eq.(5),p.334,Eq.(1),p.357Eq.(1),andEq.(1), p.299,of[93]: 172 h ` f 0( ` f ) I n I n (0) ` f M j ` f I n ( j f )0 I n ( I n ) ` f M i = ^ ` f ^ 0 ^ j f ^ I n 8 > > > > > < > > > > > : ` f 0 ` f I n I n 0 j f I n ` f 9 > > > > > = > > > > > ; = ^ ` f ^ 0 ^ j f ^ I n 8 > > > > > < > > > > > : j f I n ` f ` f 0 ` f I n I n 0 9 > > > > > = > > > > > ; = ^ ` f ^ 0 ^ j f ^ I n ( ) I n + ` f + ` f + I n 1 ^ ` f ^ I n 8 > < > : j f I n ` f 0 ` f I n 9 > = > ; = ^ ` f ^ 0 ^ j f ^ I n ( ) I n + ` f + ` f + I n 1 ^ ` f ^ I n ( ) j f + I n + ` f 1 ^ I n ^ ` f =( ) 3 I n + ` f + j f ^ j f ^ ` f ^ I n : (D.56) Since ` f isaninteger,( ) 2 ` f =1,andweobtain: 173 T QM Q m f = 32 ˇ 3 ^ I p p 4 ˇk i k f ( ) 3 I p +3 I n + j i + J d +3 j f ^ j i ^ ` 2 f X L i J P i X L f J P f i 3 L i + L f + ` f + ` i e i ( ˙ L i + ˙ L f ) ^ L i ^ L f ^ J P i ^ J P f h I p J d ( j i ) L f L i ( ` f ) j f m f j I p L f ( J P f ) J d L i ( J P i ) j f m f i X g h L f L i ( g ) J P f J P i ( j f ) QM Q j L f J P f ( I p ) L i J P i ( J d ) QM Q i X m g C QM Q gm g j f m f C gm g L f m g L i 0 Y L f m g ( ^ k f ) X M K ( ) M K C ` f ; M K L f 0 L i ; M K (D.57) Z ˚ j f ( r nA ) ˜ L f J P f ( R pB ) V ( r np ) ˚ j i ( r np ) ˜ L i J p i ( R dA ) R pB r 2 nA R dA Y L i ; M K ( ^ R dA ) Y ` f M K (^ r nA )sin dR pB dr nA : Expandnowexpandthe9js,asinEq.(5),p.334,of[93] h L f L i ( g ) J P f J P i ( j f ) QM Q j L f J P f ( I p ) L i J P i ( J d ) QM Q i =^ g ^ j f ^ I p ^ J d 8 > > > > > < > > > > > : L f L i g J P f J P i j f I p J d Q 9 > > > > > = > > > > > ; h I p J d ( j i ) L f L i ( ` f ) j f m f j I p L f ( J P f ) J d L i ( J P i ) j f m f i = ^ j i ^ ` f ^ J P f ^ J P i 8 > > > > > < > > > > > : I p J d j i L f L i ` f J P f J P i j f 9 > > > > > = > > > > > ; ; (D.58) Finally,insertingEq.(D.58)intoEq.(D.57)givesustheformfor T QM Q m f whichweimple- mentinNLAT: 174 T QM Q m f = 32 ˇ 3 p 4 ˇk i k f ( ) 3 I p +3 I n + j i + J d +3 j f ^ j f ^ J d ^ ` f X L i J P i X L f J P f i 3 L i + L f + ` f + ` i e i ( ˙ L i + ˙ L f ) ^ L i ^ L f ^ J 2 P i ^ J 2 P f 8 > > > > > < > > > > > : I p J d j i L f L i ` f J P f J P i j f 9 > > > > > = > > > > > ; X g ^ g 8 > > > > > < > > > > > : L f L i g J P f J P i j f I p J d Q 9 > > > > > = > > > > > ; X m g C QM Q gm g j f m f C gm g L f m g L i 0 Y L f m g ( ^ k f ) X M K ( ) M K C ` f ; M K L f 0 L i ; M K Z ˚ j f ( r nA ) ˜ L f J P f ( R pB ) V ( r np ) ˚ j i ( r np ) ˜ L i J p i ( R dA ) R pB r 2 nA R dA Y L i ; M K ( ^ R dA ) Y ` f M K (^ r nA )sin dR pB dr nA ; (D.59) Theobservableisthetialcrosssection,whichaswesawinEq.(D.28),isgivenby d˙ d = k f k i i f 4 ˇ 2 ~ 4 ^ J 2 B ^ J 2 d ^ J 2 A ^ j 2 f X m f QM Q T QM Q m f T QM Q m f : (D.60) 175 AppendixE ChecksoftheCodeNLAT Toperformthecalculationsinthisthesis,thecode\NonlocalAdiabaticTransfer"(NLAT) waswrittentocalculate( d;p )transferreactionswiththeinclusionofnonlocality.Inorder toensurethatthecodeworksproperly,multiplecheckswereperformed,andarediscussed inthefollowingsections.Whenmakingcomparisonstolocalcalculations,wewillcompare withthecode FRESCO [104]. LocalElasticScattering First,welookatthelocalelasticscatteringdistribution.InFig.E.1weshowthischeckfor thereaction 209 Pb( p;p ) 209 Pbat E p =50 : 0MeV.Thesolidlineisalocalcalculationusing NLAT,thedottedlineisanonlocalcalculation,butwith =0 : 05fmsothatitreducesto thelocalcalculation,andthedashedlineis FRESCO .Weused =0 : 05fmratherthan =0fmsincewewouldhavenumericalproblemswithdividingbyzeroifweset exactly equaltozero.Forthesecalculations,weusedastepsizeof0 : 01fm,amaximumradiusof 30fm,andincludedpartialwavesupto L =20.Thesecalculationsareconvergedinthata smallerstepsizeormorepartialwavesdoesnotchangetheresultsofthecalculation. 176 FigureE.1:ntialelasticscatteringrelativetoRutherforedasafunctionofscattering angle. 209 Pb( p;p ) 209 Pbat E p =50 : 0MeV:ThesolidlineisobtainedfromNLAT,thedotted lineisobtainedfromNLATandsetting =0 : 05fm,andthedashedlineisfrom FRESCO . NonlocalElasticScattering Next,welookatthenonlocalelasticscatteringdistribution.InFig.E.2wepresent 208 Pb( n;n ) 208 Pbat E n =14 : 5MeV.Thesolidlineisanonlocalcalculationwith =0 : 85 fmusingNLAT.Thedashedlineisthedigitizedresultsofthesamecalculationfromthe paperofPereyandBuck[1].Thetwocalculationsagreequitewell,indicatingthatNLAT calculateselasticscatteringwithanonlocalpotentialproperly.ThecalculationsofPerey andBuckweredigitizedfromtheirpaper,soanydiscrepanciesbetweentheresultsshown hereandtheirsisaresultoferrorsinthedigitizingprocess. 177 FigureE.2:tialelasticscatteringasafunctionofscatteringangle. 208 Pb( n;n ) 208 Pb at E p =14 : 5MeV:ThesolidlineisobtainedanonlocalcalculationusingNLAT,andthe dashedlineisthenonlocalcalculationpublishedbyPereyandBuck[1]. BoundStates Next,weexaminetheboundwavefunctions.InFig.E.3weshowthe n + 48 Caboundwave functionaswellasthedeuteronboundwavefunction.Forthe n + 48 Cawavefunctions, thesolidlineisobtainedfromalocalcalculationwithNLAT,thedottedlineisanonlocal calculationwith =0 : 05fm,andthedashedlineisobtainedfrom FRESCO .Forthe deuteronboundwavefunction,thedot-dashedlineresultsfromalocalcalculationusing NLAT,andtheopencirclesarefrom FRESCO .Forallcalculationsweusedastepsizeof 0 : 01fm,amatchingradiusof1 : 5fm,andamaximumradiusof30fm.Thismodelspace producesconvergedwavefunctions. 178 FigureE.3: n + 48 Caboundwavefunction,andthedeuteronboundwavefunction. n + 48 Ca: ThesolidlineisobtainedfromNLAT,thedottedlineisobtainedfromNLATandsetting =0 : 05fm,andthedashedlineisfrom FRESCO .Deuteron:Dot-dashedlineisdeuteron boundwavefunctionobtainedfromNLAT,andtheopencirclesareobtainedwith FRESCO . AdiabaticPotential Next,wechecktheadiabaticpotential.InFig.E.4weshowthelocaladiabaticpotential for d + 48 Caat E d =20MeVcalculatedwiththeCH89globalopticalpotential[23].The comparisoniswiththecode TWOFNR [24].Panel(a)istherealpartoftheadiabatic potential,and(b)istheimaginarypart. InFig.E.5weshowelasticscatteringnormalizedtoRutherfordfor 48 Ca( d;d ) 48 Caat E d =20MeVwhenusingtheadiabaticpotential.Whiletheadiabaticpotentialisnot suitableforaccuratelydescribingelasticscattering,thiscomparisonistoshowthatNLAT calculatestheadiabaticpotentialproperly,andcorrectlydoesthescatteringcalculation. Thenonlocalcalculationused =0 : 1sinceaccuracywaslostwithasmaller duetoinac- curaciesincalculatingthenonlocalintegral.Theagreementbetweenthenonlocalandlocal 179 FigureE.4:Thelocaladiabaticpotentialfor d + 48 Caat E d =20MeVcalculatedwithNLAT andwith TWOFNR [24].(a)Realpart,(b)Imaginarypart. calculationsdemonstratesthatthenonlocaladiabaticintegralisbeingcalculatedproperly sinceitreducestothelocalcalculationinthelimitof ! 0,asitshould.Thesolidline isacalculationdonewithNLATusingthelocaladiabaticpotential,thedottedlineisa nonlocalcalculationwith =0 : 1fminthenucleonopticalpotentials,andthedashedline isalocalcalculationdonewith FRESCO .Forthesecalculations,weuseda0 : 01fmstep size,amaximumradiusof30fm,andpartialwavesupto L =20. Transfer Next,wechecktheT-matrixcalculation.InFig.E.6weshowDWBAtransfercrosssections for 132 Sn( d;p ) 133 Snat E d =50MeV.ThesolidlineisacalculationusingNLAT,andthe dashedlineisacalculationdoneusing FRESCO .Theagreementwith FRESCO demon- stratesthatNLATcalculatestheT-matrixfor( d;p )transferreactionsproperly.Therefore, aslongasthewavefunctionsgoingintotheT-matrixarecorrect,thecorrectcrosssection willbecalculated.Thepreviouscheckshavedemonstratedthatthenonlocalwavefunctions 180 FigureE.5: 48 Ca( d;d ) 48 Caat E d =20MeV.Thesolidlineiswhenusingthelocaladiabatic potential,thedottedlineiswhendoinganonlocalcalculationwith =0 : 1fminthenucleon opticalpotentials,andthedashedlineisacalculationdonein FRESCO . beingcalculatedarecorrect,sowecantrustthatthetransferresultswhenusingnonlocal potentialswillbecorrectaswell.Forthiscalculationweusedastepsizeof0 : 01fm,a maximumradiusof30fm,andpartialwavesupto L =30. FigureE.6: 132 Sn( d;p ) 133 Snat E d =50MeV.SolidlineisalocalDWBAcalculationwith NLAT,thedashedlineisacalculationdonewith FRESCO . 181 NonlocalSource Finally,weexaminethenonlocalsource.InSec.Eweshowedthatthenonlocaladiabatic sourceiscalculatedaccuratelyfor ˇ 0fm.Forlarger values,acheckwithMathematica [116]wasdone,andtheresultsofthiscomparisonareshowninTableE.1.Forthiscompar- ison,weusedanalyticexpressionsforthewavefunctionsthatmimickedthebehaviorofthe numericalwavefunctions.Fortheboundwavefunctionweused ˚ ( r )= 2 r +3 e 0 : 3 r ; (E.1) forthescatteringwavefunctionweused ˜ ( R )= sin(4 R ) 6 R i sin(3 R ) 5 R ; (E.2) andthe V np ( r )potentialwasacentralGaussian: V np ( r )= 72 : 15 e r 1 : 494 2 : (E.3) Thereisoneadditionalcomplication,namely,inordertocalculatetheT-matrixaccu- rately,wewouldlikeour d + A scatteringwavefunctiontobecalculatedinstepsof0 : 01fm. Todothis,weneedtoknowoursourceterm S ( R )(the rhs ofEq.(2.49))instepsof0 : 01fm aswell.However,itrequiresasignitamountofcomputertimetocalculate S ( R )with 182 L R Mathematica NLAT =0 : 45fm 0 0.05 13.70+ i 16.53 13.71+ i 16.54 0 2.00 1.69+ i 0.78 1.69+ i 0.78 0 5.00 0.41- i 0.29 0.41- i 0.29 1 0.05 1.67- i 2.30 1.70- i 2.33 1 2.00 2.86+ i 1.31 2.86+ i 1.30 1 5.00 0.71- i 0.50 0.71- i 0.51 5 0.05 0.00- i 0.0002 -0.015+ i 0.016 5 2.00 4.07+ i 1.62 4.08+ i 1.62 5 5.00 1.30- i 0.91 1.29- i 0.92 =0 : 85fm 0 0.05 24.00- i 23.61 24.01- i 23.62 0 2.00 2.96+ i 1.10 2.95+ i 1.10 0 5.00 0.71- i 0.33 0.71- i 0.33 1 0.05 6.52- i 6.61 6.55- i 6.64 1 2.00 5.08+ i 1.90 5.08+ i 1.89 1 5.00 1.23- i 0.56 1.23- i 0.57 5 0.05 0.007- i 0.0007 -0.02+ i 0.01 5 2.00 8.91+ i 3.29 8.92+ i 3.28 5 5.00 2.33- i 1.06 2.32- i 1.08 TableE.1:Thenonlocaladiabaticintegral, rhs ofEq.(2.49),calculatedwithMathematica andNLATusinganalyticexpressionsforthewavefunctionsandpotentials. suchagrid.Therefore,inpractice, S ( R )iscalculatedinstepsgreaterthan0 : 01fm,and thenlinearinterpolationisusedtoconstruct S ( R )instepsof0 : 01fm.Tosavecomputer time,wewouldlikethestepsizewecalculate S ( R )withtobeaslargeaspossiblewhile stillmaintainingthedesiredlevelofaccuracy.InFig.E.7weshow 208 Pb( d;p ) 209 Pbusing variousstepsizesfor S ( R ).Itisseenthatthelargertwostepsizesagree,whilethestepsize of0 : 01fmdisagreeswiththeothertwocalculations.Infact,allcalculationswithastepsize rangingfrom0 : 02 0 : 05fmagree,andonlywhenastepsizeof0 : 01fmwasuseddidwe disagreement.Thisrequiredfurtherinvestigationtodeterminewhichcalculationiscorrect. Whencalculatingthewavefunctionnumericallyforhighvaluesoftheangularmomen- tum, L ,thereareneartheoriginduetothelargecentrifugalbarrier.Toremedy thisproblem,whatisoftendoneisthewavefunctionissetequaltozeroneartheori- 183 FigureE.7:Angulardistributionsfor 208 Pb( d;p ) 209 Pbat E d =50MeVobtainedbyusing tstepsizestocalculatethe rhs ofEq.(2.49).Thesolidlineusesastepsizeof0 : 01 fm,thedashedlineastepsizeof0 : 03fm,andthedottedlineastepsizeof0 : 05fm. gin.Thisisdonewithaparameterwewillcall\CutL".Fromtheorigintoadistanceof (StepSize) (CutL) ( L ),thewavefunctionissetequaltozero.Forallthecalculations doneinthisstudy,weusedCutL=2.Itwassuspectedthatthediscrepancybetweenthe calculationwithastepsizeof0 : 01fmandtheothertwoinFig.E.7wasbecauseCutLwas notbigenough.Thiscancauseproblemsifwetrytocalculatethewavefunctionbelowa verylargecentrifugalbarrier,becausenumericalinaccuracieswillpropagatetotherestof thewavefunctionaswecontinuetointegrateoutward. TooutwhichcalculationinFig.E.7iscorrect,weincreasedtheCutLparameterfor the0 : 01fmstepsizecalculation.TheresultsareshowninFig.E.8.Whenweincreasedthe CutLparameterofthe0 : 01fmcalculation,fromCutL=2toCutL=3,theresultingangular distributionisnowinagreementwiththeothertwocalculations.Therefore,thediscrepancy wasindeedduetoantvalueforCutL.Thisproblemwasinvestigatedforallofthe reactionsstudiedinthisthesis.Inallcases,astepsizeof0 : 05fmforthe rhs ofEq.(2.49) 184 withCutL=2wasttoobtainconvergedresults. FigureE.8:Angulardistributionsfor 208 Pb( d;p ) 209 Pbat E d =50MeVobtainedbyusing tstepsizesandvaluesofacutparameter(CutL)tocalculatethe rhs ofEq.(2.49). Thesolidlineusesastepsizeof0 : 01fmwithCutL=2,thedashedlineastepsizeof0 : 01fm withCutL=3,andthedottedlineastepsizeof0 : 05fmwithCutL=2. 185 AppendixF MirrorSymmetryofANCs Directprotoncaptureatlowrelativeenergiesneededforastrophysicsarealwaysperipheral duetotheCoulombbarrier.Atthelimitsof E ! 0thesereactionsareuniquelydetermined bytheasymptoticnormalizationcot(ANC)ofthesingleprotonoverlapfunctionof thenucleus[41].ItisforthisreasonthattheANCmethod[41]hasbeenputforthasan indirectwayofobtainingprotonradiative-capturecrosssectionsfromtheANCsextracted fromexperiments,suchastransfer. Insomeastrophysicalenvironments,protoncapturemayoccuronproton-richnuclei. ObtainingthenecessaryANCexperimentallyinordertounderstandtheseastrophysically importantreactionsmaybecultorimpossiblesincetheexperimentwouldrequireproton- richradioactivebeams.However,anindirecttechniquehasbeenproposed[44]whichuses informationaboutthemirrorsysteminordertoextractthenecessaryANC.Themirror nucleusisasthenucleuswithinterchangednumbersofprotonsandneutrons.While anexperimentmaynotbeabletobeperformedontheproton-richnucleusofinterest, experimentsonthemirrorsystemcansometimesbeperformedwithstablebeams,andthus, withmuchhigheraccuracy. In[44,4,5],theratio, R ,oftheprotontoneutronANCsquaredisdeterminedfora widerangeoflightnucleiwithinamicroscopicclustermodel(MCM).In[44]ananalytic derivationoftheratio, R o ,ispresented.TheratioobtainedfromtheMCMcalculationsis infairagreementwiththepredictionsoftheanalyticformula[4,5].Inthiswork,wewant 186 toexploretheofcouplingsinducedbydeformationsofthecoreandcoreexcitations. Thereasonforrelyingonchargesymmetryargumentsratherthanjustcalculatingthe ANCdirectlyisduetolargeuncertaintiesinthetheoreticalpredictionsofANCs.The individualANCsin[44,4,5]arestronglydependentonthe NN interactionused,butthe ratioofANCswasfoundtobeindependentofthechoiceofthe NN interaction,withina fewpercent. TheoreticalConsiderations Weconsiderthe A = B + x modelusedin[117],whichstartsfromaneHamiltonian representingthemotionofthevalencenucleon( x = n;p )relativetothecore, B : H A = T r + H B + V Bx ( r ;˘ ) ; (F.1) where T r istherelativekineticenergyoperator,and H B istheinternalHamiltonianofthe core. V Bx istheeinteractionbetweenthecoreandthevalencenucleonwhichdepends onthe B x relativecoordinate, r ,andtheinternaldegreesoffreedomofthecore, ˘ .In themodelof[117], V Bx istakentobeadeformedWoods-Saxonpotential V Bx ( r )= V WS 1+exp r R ( ;˚ ) a 1 (F.2) where V WS isthedepth,andmaydependontheorbitalangularmomentum, ` .Theradius, R ,isangledependent,andgivenby: 187 R ( ;˚ )= R WS 2 4 1+ Q X q =2 q Y q 0 ( ;˚ ) 3 5 (F.3) where q characterizedthedeformationofthecore,andthus,thestrengthofthecoupling betweenthevarious B + x Asusual,theradiusisgivenby R WS = r WS A 1 = 3 with A themassnumberofthe B + x system.Wealsoincludethetypicalundeformed spin-orbitpotentialdescribedinChapter3. The B + x wavefunctionisexpandedineigenstatesofthecore, I ˇ B ,withspin I ,parity ˇ B ,andeigenenergy I ˇ B : J ˇ = X n`jIˇ B n`j ( r ) Y `j ( ^ r I ˇ B ( ˘ ) : (F.4) Inthisexpansion,wefactorizetheradialpart, n`j ,andthespin-angularpart, Y `j for convenience.Thequantumnumbers n and j correspondtotheprincipalquantumnumber andtheangularmomentumobtainedfromcouplingtheorbitalangularmomentum, ` ,with thespin, s ,respectively.Withthisexpansion,thecoupled-channelsequationforeach is givenby[117]: h T ` r + V ii ( r ) i i ( r )+ X j 6 = i V ij j ( r )= x J ˇ i i ( r ) ; (F.5) where i representsallpossible( n`jI ˇ B )combinations, x J ˇ isthebindingenergyinthe A = B + x system,andthepotentialmatrixelements V ij aregivenby 188 V ij ( r )= h i ( ˘ ) Y i ( ^ r ) j V Bx ( r ;˘ ) jY j ( ^ r j ( ˘ ) i : (F.6) Wetake i fromtherotationalmodelwithparametersphenomenologically.The solutionofEq.(F.5)isfoundbyimposingbound-stateboundaryconditionsandnormalizing J ˇ tounity.See[117,118]formoredetails. Inthismodel,thenormof i relatesdirectlytoaspectroscopicfactor, S x i : S x i = Z 1 0 j i j 2 r 2 dr (F.7) whiletheANC, C x i ,isdeterminedfromtheasymptoticbehaviorof i : i ( r ) ! r !1 C x i W x i ;` +1 = 2 (2 i r )(F.8) with i = p 2 Bx j J ˇ i j = ~ 2 and Bx isthereducedmass.Here, W istheWhittaker functionwith x i theSommerfeldparameter[92]. Asanexampletoillustratethemodel,considerthemirrorpair 17 Oand 17 F.Thecore forbothnucleiis 16 O,whichhasa0 + groundstate,andtwolow-lying2 + and3 states, whichstronglycoupletothegroundstatethrough E 2and E 3transitions,respectively.If weincludethe0 + and2 + statesof 16 Oinourmodelspace,thentheground,5 = 2 + stateof 17 Oand 17 Fwouldcontainnotonlya1 d 5 = 2 valencenucleoncoupledtothegroundstate, butalso,forexample,a2 s 1 = 2 nucleoncoupledtotheexcited2 + state. 189 Inthisstudy,wecomparetheprotonANCs, C p i ,withtheneutronANCs, C n i ,through theratio R = C p i C n i 2 : (F.9) TheratioofANCscalculatedinourmodelisthencomparedwiththeanalyticformula derivedin[44,119] R o = F ` ( p i R N ) p i R N j ` ( n i R N ) 2 ; (F.10) where F ` and j ` areregularCoulombandsphericalBesselfunctions,respectively,[92],and R N =1 : 25 A 1 = 3 istheradiusofthenuclearinterior,ofwhich R o isnotstronglydependent. WewillcomparetheratiooftheANCsfromourcalculationswiththevalueobtainedfrom thisrelation. Results RatioforSpMirrorPartners SinceweareinterestedintheANCsforeachmirrornuclei,andthesedependstronglyonthe energyofthesystemrelativetothreshold,itisimportantthatwereproducetheexperimental separationenergiesexactly.Wedothisbyadjustingthedepthsof V Bn and V Bp toreproduce exactlythecorrespondingbindingenergies.Allcalculationsareperformedwiththeprogram 190 Nuclei I ˇ B n`j RR o R MCM 8 Li/ 8 B3 = 2 ; 1 = 2 1 p 3 = 21 : 04 0 : 041.121 : 08 13 C/ 13 N0 + ; 2 + 1 p 1 = 21 : 19 0 : 021.201 : 14 17 O/ 17 F(g.s.)0 + ; 3 1 d 5 = 21 : 18 0 : 011.221 : 19 17 O/ 17 F(e.s.)0 + ; 3 2 s 1 = 2693 16799736 17 O/ 17 F(g.s.)0 + ; 2 + 1 d 5 = 21 : 219 0 : 0041.221 : 19 17 O/ 17 F(e.s.)0 + ; 2 + 2 s 1 = 2756 23799736 23 Ne/ 23 Al0 + ; 2 + ; 4 + 1 d 5 = 2(1 : 852 0 : 014) 10 4 2 : 06 10 4 2 : 96 10 4 27 Mg/ 27 P0 + ; 2 + ; 4 + 2 s 1 = 240 : 1 1 : 843.744 : 3 TableF.1:RatioofprotontoneutronANCsforthedominantcomponent:Comparisonof thiswork R withtheresultsoftheanalyticformula R 0 Eq.(F.10)andtheresultsofthe microscopictwo-clustercalculations R MCM [4,5]includingtheMinnesotainteraction.The uncertaintyin R accountforthesensitivitytotheparametersof V Bx . FACE [120],andalldetailsofthecalculationsforeachmirrorpaircanbefoundin[25]. OurresultsaresummarizedinTableF.1.Fromtheprotonandneutronwavefunctions calculatedfromEq.(F.5),wedeterminetheANCsandtheratio R .Foreachcase, R cor- respondsto r WS =1 : 25fm, a =0 : 65fm,and V so =6MeV.Theuncertaintythe rangeobtainedwiththegeometry r WS =1 : 2fm, a =0 : 5fm,and V so =8MeV.Ourresults for R arecomparedtothevaluesobtainedwiththeanalyticformula R o ofEq.(F.10),and thoseobtainedwithintheMCM,wheretheyassumedtwoclustersandusedtheMinnesota interaction, R MCM [4,5]. Fornearlyallcases, R , R o ,and R MCM areallinfairagreement.However,there wereafewcaseswheretherewerediscrepancies.For 23 Ne/ 23 Al,itisimportanttonote thatinourcalculationsweimposerealisticbindingenergies,whereasintheMCMresults, bindingenergiescansometimestly.Since R dependsstronglyonthebinding energies,thiscancauselargebetweenourvaluesandthoseof[5].Thevalues for R o presentedinTableF.1alsoassumetheexperimentalbindingenergies,therefore, between R and R o mustberelatedtothefailureofthesimpleanalyticrelations. 191 Suchisthecasefor 17 O/ 17 F(e.s.)whenthe3 excitedstateisincludedinthemodelspace. TestingModelindependence Theusefulnessoftheratiomethodisthattheratio R shouldbemodelindependent.This wasdemonstratedinSectionFwhereformostcasesstudied,theratioobtainedfromthis study, R ,usingoursimplemodelwasinfairagreementwiththeratioobtainedwiththe muchmoresophisticatedmicroscopicclustermodel, R MCM .Inthissubsection,weusethe deformationparameterasafreevariabletotunetheamountofcouplingbetweenthevarious Withtheof 23 Ne/ 23 Aland 27 Mg/ 27 Ppairsbeingverysimilar tothe 17 O/ 17 Fsystemsinitsgroundandexcitedstates,respectively,weconcentrateofthe threelightercases. Wendnotintheratio R forboththe 8 Li/ 8 Band 13 C/ 13 Nmirror pairs.Inthesecases,themaincomponentsofthewavefunctionare p waves,eveninthe includingcoreexcitation.For j 2 j =0 : 0 0 : 7,theresultingrangeofvaluesfor R are1 : 038 1 : 044for 8 Li/ 8 Band1 : 201 1 : 251for 13 C/ 13 N.Thisconstancyisobtainedeven thoughthevariationin 2 leadstotchangesinthespectroscopicfactor: S x 1 p 3 = 2 goesfrom1to0 : 75for 8 Li/ 8 B,while S x 1 p 1 = 2 decreasesdownto0 : 32for 13 C/ 13 N. Thesituationfor 17 O/ 17 Fers.Weconsidertheseparateofincludingthe3 stateandthe2 + state.Letusconsidertheinclusionof 16 O(0 + ; 3 ).Likefor 8 Li/ 8 Band 13 C/ 13 N,thevariationin R wassmall,eventhoughover30%ofthe5 = 2 + ground-statewave functionisinacore-excitedat 3 =0 : 7.Forthis 3 ,the1 = 2 + excited-state wavefunctionisalmostexclusivelyinthe 16 O(0 + ) 2 s 1 = 2 ( S x 2 s 1 = 2 ˇ 95%).As aresult,thechangeinthecorrespondingratiowaslimitedtolessthan1%. Wenextconsidertheinclusionof 16 O(0 + ; 2 + ).Inthiscase,the d 5 = 2 groundstateadmixes 192 FigureF.1:Neutronandprotonspectroscopicfactorsfor 17 Oand 17 F,respectively,consid- eringthe 16 Ocoreinits0 + groundstateand2 + excitedstate:(a)5 = 2 + groundstate and(b)1 = 2 + excitedstate.Figurereprintedfrom[25]withpermission. FigureF.2:RatioofprotonandneutronANCsfor 17 Oand 17 F,respectively,including 16 O(0 + ; 2 + ):(a)5 = 2 + groundstateand(b)1 = 2 + excitedstate.Figurereprintedfrom [25]withpermission. withan s 1 = 2 componentwiththecoreinitsexcitedstate.Forthe1 = 2 + excitedstateof 17 O, the s 1 = 2 componentcoupledtotheground-stateofthecoreadmixeswith d componentswith thecoreinits2 + excitedstate.Likeinallcases,theenergiesofthetwoloweststatesin 17 Oand 17 Fwerebysimultaneouslyadjustingthedepthsofthepotentialforeach 2 .Forboththe5 = 2 + and1 = 2 + states,thespectroscopicfactors,ofthecomponentwith thecoreinthegroundstateexperiencesalargereductionatlarge 2 ,asisseeninFig.F.1. 193 Forthegroundstate(5 = 2 + ),theprotonandneutronspectroscopicfactorsvarytogether, whilefortheexcitedstate(1 = 2 + ),thereismoreadmixtureintheneutronsystemthanfor theprotonsystem.ThisisinatbehavioroftheANCratios. InFig.F.2wepresenttheratio R aswellasamodratiocompensatingforthe changesinthespectroscopicfactors R = R S n =S p .Theanalyticprediction, R o isalso shownbythehorizontaldashedline.Forthe5 = 2 + groundstate,neither R nor R deviate muchfromthevalueat 2 =0,correspondingtothesingle-particleprediction,asseenin Fig.F.2a.Bothoftheseratiosareclosetotheanalyticprediction, R o .Onthecontrary,for the1 = 2 + excitedstate, R showsalargevariationpartlycausedbythebetween neutronandprotonspectroscopicfactors,asseeninFig.F.2b.ThefeaturesseeninFig.F.2 canbeextrapolatedto 23 Aland 27 P,since,asmentionedbefore,theformerhasastructure verysimilartothatof 17 F(g.s.),whilethelatterexhibitsthesamecomponentsas 17 F(e.s.). In[4,5]coreexcitationisexploredwithintheMCM.Eveninthesestudiestherewas growingdisagreementbetween R MCM and R 0 asmorecorestateswereexplicitlyincluded inthemodelspace.ThiswasunderstoodintermsofthelongrangeCoulombquadruple termwhichwasaddedtotheHamiltonianintheprotoncase,atermnotconsideredin thederivationof R 0 ,norinourpresentcalculations.Here,however,wenotonlyseea deviationfrom R 0 ,butalsoastrongdependenceonthedeformationparameterforsome cases.Thereforeweconcludethesourcefordeviationsfrom R 0 andthebreakdownofthe constantratioconceptisinducedbythenuclearquadrupleterm,whichispresentinboth neutronandprotonsystems. Thesurprisingresultsforthe1 = 2 + mirrorstatesledtoseveraladditionaltestswhich isolatedthecauseforthelargecouplingdependenceon R .Therearethreeessentialingredi- ents:lowbinding,theexistenceofan s -wavecomponentcoupledtothegroundstateofthe 194 core,andatadmixturewithothertions.Itappearsthatwhenallthree conditionsaremet,thebetweentheneutronandprotonwavefunctionsincrease aroundthesurface,exactlywherethenuclearquadrupleinteractionpeaks.Thisresultsina strongerofcouplingontheneutronsystemcomparedtotheprotonsystem,inducing in S n relativeto S p ,whichareinthecouplingdependenceon R .Our testsshowthattheisindependentofwhetherthewavefunctionshaveanode. Conclusions Theproposedindirectmethodforextractingprotoncaptureratesfromneutronmirrorpart- nersreliesontheratiobetweenasymptoticnormalizationcotsofthemirrorstate beingmodelindependent.In[25],wetestedthisideaagainstcoredeformationandexcita- tion.Weconsideredacore+ N modelwherethecoreisdeformedandallowedtoexcite,and appliedittoavarietyofmirrorpairs( 8 Li/ 8 B, 13 C/ 13 N, 17 O/ 17 F, 23 Ne/ 23 Al,and 27 Mg/ 27 P) andweexploredhowthemirrorstatesevolveasafunctionofdeformation. Formostcases,theratiooftheANCofmirrorstateswasfoundtobeindependentofthe deformation,andthecalculatedratioofANCsagreedwellwiththesimpleanalyticformula. 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