INTERPLAYBETWEENSINGLE-PARTICLEANDCOLLECTIVEMOTIONWITHINNUCLEARDENSITYFUNCTIONALTHEORYByChunliZhangADISSERTATIONSubmittedtoMichiganStateUniversityinpartialentoftherequirementsforthedegreeofPhysics|DoctorofPhilosophy2016ABSTRACTINTERPLAYBETWEENSINGLE-PARTICLEANDCOLLECTIVEMOTIONWITHINNUCLEARDENSITYFUNCTIONALTHEORYByChunliZhangNucleardensityfunctionaltheory(DFT)canbeemployedtostudypropertiesofgroundstates(g.s.)andselectedexcitedstatesofnucleianywhereinthenuclearchart.Thefocusofthisworkisonthedescriptionofsingle-particle(s.p.)andcollectivemotioninnucleiusingnuclearDFT.Sincenuclearcollectivephenomenaresultfromacoherentmotionofindividualnucleons,thesharpdistinctionbetweenthesetwomodescannotbemade.Forexample,nuclearrotationleadstothealignmentofangularmomentumwithrotationalfrequency,whichresultsinthevariationofoccupationsins.p.orbitals.Spontaneousleadstonotonlylargegeometricalrearrangements,butalsoimpactstheinternalshellstructure.Thisdissertationisdividedintothreeparts.Inthepart,Ishallintroducethenuclearmodelused.Inthesecondpart,thegeneralformalismofnuclearDFTanditsmainingredient,theenergydensityfunctional(EDF)willbeoutlined.Inthelastpart,theappli-cationsofnuclearDFTwillbepresented.First,westudythenuclearshapesandassociatedrotationalbandsfornucleiwithAˇ110;yrast,near-yrastbandstructures,angularmomen-tumalignmentswithrotationalfrequency,andtransitionquadrupolemomentsareanalyzedandcomparedtoexperimentaldata.Then,basedontheKerman-Onishicondition,weper-formsystematictilted-axis-crankingcalculationsfortriaxialstronglydeformed(TSD)bandsin160Yb,explainthenatureoftheseTSDbands,andpredictpossiblecollectivebehaviorathighspin.Nextweexploreclusterstructuresinlightnucleiusingthenovelconceptofthenucleonlocalizationfunction(NLF).TheNLFisthenusedtostudytheinternalstructureevolutionandemergenceoffragmentsinheavynucleialongtheirpredictedpathways.WethenshowthattheNLFcanbeemployedtoidentifyfragmentswellbeforescissionin240Pu.Thelastsectioncontainstheconclusionsofthisdissertationandperspectivesforfuturework.Thisdissertationisdedicatedtomygrandparents.ivACKNOWLEDGMENTSFirstofall,Iwouldliketothankmyadvisor,Prof.WitoldNazarewicz,forhisoutstandingguidanceandinsightfuladvice.TheattitudetowardscienresearchwhichIlearnedfromhimwillbeofendlessbtomylife.Iwouldliketoexpressmygratitudetoallmycolleaguesfortheiradviceandhelpwithbothresearchandlife,especiallyYueShi,BastianSchuetrumpf,ErikOlsen,YannenJagana-then,KevinFossezandSiminWang.IwouldliketothankNobuoHinohara,JhilamSadhukhanandNicolasSchunckfortheirgeneroushelpwithmyresearchandendlesspatiencewithallmyquestions.TheworkpresentedinthisdissertationwascarriedoutatTheUniversityofTennessee,Knoxville(UTK)from2012to2014andMichiganStateUniversity(MSU)from2014to2016.IwouldliketothankthemembersatUTKandMSUfortheirhelp.IwouldliketothankZacharyMattesonandErikOlsenforreadingandhelpwitheditingmydissertation.IwouldliketothankFilomenaNunes,ScottBogner,AlexandraGade,andCarloPier-marocchiforservingonmydoctoralcommittee.Iamverygratefultomyparentsfortheirunconditionalloveandsupport.Iamverygratefultomyhusband,ZhilingDun,forhiscompanyandencouragement.vTABLEOFCONTENTSLISTOFTABLES....................................viiiLISTOFFIGURES...................................ixChapter1Introduction...............................11.1Nucleardensityfunctionaltheory........................21.2Nuclearshapesandcollectivemotion......................31.3Nucleonicclustering................................4Chapter2TheoreticalTool:NuclearDFT...................62.1Generalconsiderationandformalism......................62.2Skyrmeenergydensityfunctional........................72.2.1Localdensities..............................82.2.2SkyrmeEDFformalism.........................102.3methodsbasedonnuclearDFT...................132.3.1Hartree-FockandHartree-Fock-Bogoliubovequations.........152.4NuclearDFTsolvers...............................18Chapter3RotationofTriaxialNuclei......................203.1Introduction....................................203.2Triaxialdeformation...............................223.3Bandcrossingandangularmomentumalignment...............233.4Tilted-axis-crankingcalculationsandtheKerman-Onishicondition......243.5ProjectA:TriaxialrotationintheAˇ110region...............263.5.1Ground-statepotentialenergysurfaces.................273.5.2Rotationalproperties...........................293.6ProjectB:Descriptionoftriaxialstronglydeformedbandsin160Yb.....363.6.1Tilted-axis-crankingcalculationsfor160Yb...............363.7Summary.....................................39Chapter4NucleonLocalization..........................404.1Introduction....................................404.2Spatiallocalization................................414.3ProjectC:Nucleonlocalizationinnuclei....................434.3.1NLFswithintheaxialharmonicoscillatormodel............434.3.2NLFsinlightnuclei............................464.3.3NLFsinheavynuclei...........................504.4ProjectD:Identifyingfragments.....................564.4.1Fissionfragmentdistribution......................574.4.2Fragmentsidennfor240Pu....................58vi4.5Summary.....................................61Chapter5ConclusionandOutlook........................63Chapter6ListofPublicationsandMyContributions............65APPENDICES......................................67AppendixAHohenberg-Kohntheorems........................68AppendixBSpatialLocalization...........................71BIBLIOGRAPHY....................................74viiLISTOFTABLESTable3.1:Bohrquadrupoledeformationparameters2andcalculatedinHFB+UNEDF0forthegroundstatesof106;108Moand108;110;112Ru...27Table3.2:Thein160Ybstudiedinthiswork.Eachisdescribedbythenumbersofoccupiedstatesinthefourparity-signature(ˇ;r)blocks,intheconventionofRef.[1].........37viiiLISTOFFIGURESFigure3.1:PESinthe(Q20;Q22)planeincrankedHFB+UNEDF0for106Moand108Mo.Left:standardpairingstrengths.Right:pairingstrengthsincreasedby5%,seetext.Thebetweencontourlinesis0.5MeV....................................27Figure3.2:SimilartoFig.3.1,butfor108Ru,110Ru,and112Ru..........28Figure3.3:One-q.p.Routhiandiagramfor106Mo(left)and112Ru(right)ob-tainedwithcrankedHFB+UNEDF0.Theparityˇandsignaturerastheeigenvalueof^Ry=exp(iˇ^Jy))ofindividuallev-elsareindicatedinthefollowingway:ˇ=+;r=+i{solidline;ˇ=+;r=i{dottedline;ˇ=;r=+i{dot-dashedline;ˇ=;r=i{dashedline.ThethinlineindicatestheFermienergy.29Figure3.4:Angularmomentumalignmentfor106Moand108;112Ru.CrankedHFB(~!<0:3MeV)andcrankedHF(~!>0:3MeV)calculationsarecomparedtoexperiment[2{4]...................30Figure3.5:Summaryofequilibriumdeformationsofthelowestˇ=+;r=1bandsin106;108MocalculatedwithcrankedHFB+UNEDF0(groundband)andcrankedHF+UNEDF0(alignedbands).Therotationalfrequencyisvariedfromzeroto~!=0:6MeV.Thecorrespondingrangeofangularmomentumisindicated.Thealignedbandsareaccordingtothenumberofoccupiedhigh-Nintruderlevels(N=5and4forneutronsandprotons,respectively)........31Figure3.6:SimilartoFig.3.5,butfor108;110;112Ru................32Figure3.7:DiabatictotalRouthiansurfacesfor106Moat~!=0:5MeVcal-culatedwithCHF+UNEDF0forthefollowing(a)ˇ(9;9;12;12)(17;17;15;15)(ˇ4454inFig.3.5(a));(b)ˇ(10;10;11;11)(17;17;15;15)(ˇ4654);and(c)ˇ(10;10;11;11)(16;16;16;16)(ˇ4656).................................34Figure3.8:Transitionquadrupolemomentsfor106;108Moand108;110;112Rucal-culatedincrankedHFB(opencircles)andcrankedHF(dots)com-paredtoexperiment.TheQtvalueatI=2istakenfromRef.[5]andthehigh-spinvaluesfromRef.[6].................35ixFigure3.9:TotalRouthiansin160YbcalculatedwithintheSHFmethodasfunc-tionsofthetiltinganglefortheeTSDlistedinTable3.2.Solidanddashedlinesmarkwithpositiveandnegativeparity,respectively....................38Figure4.1:Left:particledensityˆn"(innucleons/fm3)forprolatein8Beand20Newith=2,andforanoblatein12Cwith=0:9.Right:correspondingmaskedNLFsasdescribedinSec.4.2.WhitedottedlinesarethecontourlinesC=0:9oftheoriginal(4.3)..........................45Figure4.2:SimilartoFig.4.1,butfor36Ar,16Oand24Mg............46Figure4.3:Left:neutrondensityˆn(innucleons/fm3)fortheHFground-stateof8BecomputedwiththefunctionalUNEDF1-HFB.Right:corre-spondingmaskedNLFsasdescribedinSec.4.2.WhitedottedlinesarethecontourlinesC=0:9oftheoriginal(4.3).....47Figure4.4:(a)TotalHFenergyof20Neversusquadrupoledeformationparam-eter2.Thepredictedlowestminimumisnormalizedto0.Theneutrondensities(innucleons/fm3)atthetwolocalminimaIandIIareshowninpanels(b)and(d),respectively,andthecorrespondingNLFsareplottedinpanels(c)and(e).................48Figure4.5:SimilartoFig.4.4,butfor36Ar....................49Figure4.6:SimilartoFig.4.3butforthegroundstateof20Neat2=0:35and3=0:57....................49Figure4.7:Thepotentialenergycurvesof232Thand240PucalculatedwithUN-EDF1alongthepathways[7,8].Thefurtherdis-cussedinFigs.4.8and4.12aremarkedbysymbols.Theirquadrupoleandoctupolemoments,Q20(b)andQ30(b3=2)respectively,areindi-cated....................................51Figure4.8:Nucleonicdensities(innucleons/fm3)andspatiallocalizationsfor232ThobtainedfromHFBcalculationswithUNEDF1forecourationsalongthepathwaymarkedinFig.4.7........52Figure4.9:Nucleonicdensities(innucleons/fm3)andspatiallocalizationsforthegroundstateof132Sn(left)and100Zr(right)..............53xFigure4.10:Neutron(left)andproton(right)NLFfor232Th(bluethickline),100Zr(greenline),and132Sn(redline)alongthezaxis(r=0).Thesecond,andthirdpanelscorrespondtotheinthethird,fourthandfthcolumnsofFig.4.8,respectively.....54Figure4.11:SimilartoFig.4.8butfortheof264Fm.Thequadrupolemomentsoftheonsaredenotedontopofeachcolumn..55Figure4.12:SimilartoFig.4.8butfortheof240PuindicatedinFig.4.7..................................56Figure4.13:Outerturningline(thicksolidline)andscissionline(dashedline)onthepotentialenergysurfaceof240Pu.Symbolsindicatetheselectedforwhichsubsequentresultsareshown.........58Figure4.14:One-dimensionalplotsofCq"(solidlines)andˆq"(dashedlines)alongthesymmetryaxisforthe(a)and(b)ofFig.4.13.Theblacklinesareresultsforthefragmentedsystems,whilethegreenandredlinesdenotetheresultsforfragments............59Figure4.15:Contoursofˆn(toprow)andCn"of240Pu(secondrow).ThelastrowgivesacomparisonofonedimensionalCn"(thicksolidlines)withˆn(thickdashedlines).Thethreecolumnscorrespondto(1),(5),and(11)ofFig.4.13......................60Figure4.16:SameasFig.4.15,butforprotons....................60xiChapter1IntroductionTheatomicnucleusisastrongly-correlated,self-boundmany-bodysystem,whichexhibitsavarietyofcollectiveandsingle-particlenon-collectivephenomena.Incollectivemotion,manynucleonsareinvolved,forexample,nuclearvibration,nuclearrotation,whichcouldresultinexternalobservablepropertychanges.Whileinsingle-particlemotionindivid-ualnucleonsmoveindependently,internalstructuresgovernedbyshellareimpacted,whichcancausetheparticle-holeexcitation,pairbreakingandclusteringofnucleons.Nu-merousmicroscopictheoreticalapproacheshavebeendevelopedtostudynuclearstructure.Theycanbegroupedintothreegeneralcategories:abinitiomethods[9{13],shellmodel(orninteraction)theories[14,15]andnuclearDFTbasedontheself-consistentmeanapproach[16,17].Inthisdissertation,themaintopicofstudyiscollectivemotion,including(i)nuclearrotationbasedontriaxialonsthatexhibitstrongquadrupolecollectivityand(ii)spontaneousinheavynucleithatisassociatedwithquadrupoleandoctupolecollectivedynamicsthroughlarge-amplitudecollectivemotion.SincenuclearDFTdescribesnuclearsystemsintermsoflocaldensities,theresultingspontaneoussymmetrybreaking[18,19]canbeemployedtonucleargeometries.Ontheotherhand,thereisalwaysasubtleinterplaybetweencollectiveands.p.motionatthesameenergyscale.Forexample,astherotationalfrequencyincreases,thealignmentofangularmomentumwithinthecollectiverotationalbandcanleadtoband-crossing[18,20,21],1atwhichthelowest(yrast)willchangerapidly.Inself-consistentmethodsbasedonnuclearDFT,thechangeofs.p.canbeexpressedthroughoccupationsofs.p.orbitals.Theresultingstructuralchangescanbedescribedusingthecrankingapproach,wherethenuclearmotionisdescribedintherotatingreferenceframe.Bythesametoken,inspontaneousasthesystemevolvesalongthepathway,itsinternalshellstructureisimpacteddramatically.Here,ourstrategyistointroducethenucleonlocalizationfunction[22],whichwillhelptovisualizenucleonclusteringandcorrelationsimprintedbyshellThedissertationisorganizedasfollows.NuclearDFTisintroducedinChap.2.Appli-cationstotriaxialstructuresareexploredinChap.3.InChap.4,thenucleonlocalizationfunction[23,24]isemployedtoillustratetheemergenceofclusteringstructuresinlightnucleiandfragmentsinheavynuclei.Finally,conclusionsandprospectsforfutureworkaresummarizedinChap.5.Abriefintroductiontothesechaptersisgiveninthefollowing.1.1NucleardensityfunctionaltheorySincethenucleusisaself-boundsystemwhichmaydisplaybothindividualnucleonexcita-tionsandcollectiveexcitationsatthesameenergyscale,itisachallengetodescribeitusingasingletheoreticalframework.NuclearDFTincorporatesnucleoncorrelationsbyintroducingcomplexdensitydependenceintoEDFs,whilecollectivemotioncanbeconsideredexplicitlybyintroducingLagrangemultiplierstoconstrainthecollectivecoordinates,suchasnuclearmultipolemomentsandangularmomentum.NuclearDFTisatool;ithasbeenemployedtoavarietyofphenomena,includingshapedeformations[17,25],neutron/protondrip-lines[26],two-protondecay[27],(large-amplitude)collectivemotion[17,28,29],and2nuclearpasta[30,31]inneutronstars.Sincethescopeofourresearchrangesfromlighttoheavynuclei,nuclearDFTisourtoolofchoice.1.2NuclearshapesandcollectivemotionTheatomicnucleuscanexhibittshapes,forexample,spherical,prolate/oblate,tri-axial,pear-like,etc.Thecollectiverotationofasphericalnucleusisimpossibleduetotheindistinguishabilityofrotatedwhileadeformednucleuscanexhibitcollectiverotationalmotionthatresultsinthepresenceofrotationalbands[18,32].Themostcom-moncaseofthisisanaxiallydeformednucleuswithprolateoroblateshape,withangularmomentumbuiltalongtheaxisthatisperpendiculartothesymmetryaxisofthesystem.Anotherlesscommoncaseisthatofthetriaxialnucleus.Here,thecollectiveangularmo-mentumcanbealignedinanydirectionastheaxialsymmetryofthesystemisbroken.Experimentprovidesveryindirectinformationabouttriaxials.Theoretically,veryfewnucleiarepredictedtobetriaxiallydeformedintheirgroundstates.Theoretically,themacroscopic-microscopicmodel[33]andtraditionalcrankedshellmodel[34,35]havebeenemployedtostudynuclearshapesandrotations.However,unlikerela-tivistic[6,36]andnon-relativistic[37,38]models,thoseapproachesarenotself-consistent,andthisconstitutesamajorproblem.Inmyresearch,nuclearDFTwithrecentlyoptimizedEDFsisappliedtostudytriaxialshapesandrotationalmotionintheAˇ110regionandhigh-spinbandsin160YbinterpretedintermsofTSDstructures.BysolvingthecrankedHartree-FockandHartree-Fock-Bogoliubovequations,characteristicfeaturesofrotationalbands,suchasband-crossingsandshapechanges,willbestudied.31.3NucleonicclusteringTheappearanceofclusterstatesinatomicnucleiisaubiquitousphenomenonwithmanyimplicationsforbothnuclearphysicsandastrophysics[39{43].Whileseveralfactorsareknowntocontributetoclustering,acomprehensivemicroscopicunderstandingofthisphe-nomenonstillremainselusive.Clusterscanbeenergeticallyfavoredduetothelargebindingenergypernucleoninconstituentclusters,suchasparticles.Thisbinding-energyargumenthasoftenbeenusedtoexplainpropertiesof-conjugatenuclei[44],clusteremission[45,46]and[47],andtheappearanceofagasoflightclustersinlow-densitynuclearmatter[48{50]andintheinteriorregionofheavynuclei[51].Anotherimportantfac-toristhecouplingofclusterstatestodecaychannels,whichexplains[52,53]theiroccurrenceatlowexcitationenergiesaroundcluster-decaythresholds[54].Themicroscopicdescriptionofclusterstatesrequirestheuseofanadvancedmany-body,open-systemframework[52,53,55]employingrealisticinteractions,andtherehasbeentprogressinthisarea[56{60].Theoretically,theBloch-Brinkalpha-clusterandantisymmetrizedmoleculardynamicsmodels[39{41]havebeenusedtodescribeclusterstates.Theformeradoptsmulti-centerclusterwavefunctionsandisparametrizedbythegeometryofclustercenterpositions;itismainlyusedin-conjugatenucleiassumingtheexistenceofclusterstates.ThelatterisbasedonaSlaterdeterminantconsistingofGaussianwavepackets.Foraglobalcharacterizationofclusterstatesinlightandheavynuclei,agoodstartingpointisnuclearDFT.Here,clusterstateshaveasimpleinterpretationintermsofquasi-molecularstructures.Sincethemapproachisrootedinthevariationalprinciple,thebinding-energyargumentfavorsclusteringincertaincharacterizedbylarge4shellofconstituentfragments[61{68];thecharacteristicsofclusterstatescanbeindeedtracedbacktothesymmetriesandgeometryofthenuclear[69,70].5Chapter2TheoreticalTool:NuclearDFTInthischapter,thetheoreticalmodelusedinmyresearchisdescribed.Insection2.1,weshalloutlinethetheoreticalfoundationsofnuclearDFT;thesingle-referenceSkyrmeEDFusedinthisworkisintroducedinsection2.2.Insection2.3,theself-consistentSkyrmeEDFmethodispresented,andIexplainhowthemequationscanbesolved.Finally,thenuclearDFTsolversusedinmyresearch,andtheircharacteristicsandlimitations,arediscussedinsection2.4.2.1GeneralconsiderationandformalismTheHamiltonianofamany-bodysystemcanbewrittenas:^H=^T+^V+^Uext;(2.1)whereVistheinter-particleinteractionandUextistheexternalTheeigenstatesoftheinteractingsystemjicanbeobtainedby,e.g.,diagonalizingtheHamiltonian,andtheone-bodylocalnucleonicdensitycanbewrittenas:ˆ(r)=NZdr2drAjr;r2;;rA)j2:(2.2)DFTisrootedinthetheoremsformulatedbyHohenbergandKohn[71].Itwasoriginally6usedtoinvestigateelectronicstructureinquantumchemistryandcondensedmatterphysics,and-inparalleldevelopments-wasextendedtonuclearphysicsaswell.HohenbergandKohndemonstrated[71]thatthereexistsauniversalfunctionalofthedensity,F[ˆ(r)],independentofexternalpotential,Uext,suchthatthetotalenergy,EF[ˆ(r)]+RUext(r)ˆ(r)dr,hasitsminimumattheexactg.s.associatedwiththeexternalpotential(seeAppendixAforthedemonstration).Therefore,everythingwouldbesimpleaslongasweknewtheuniversalfunctional.Hy-pothetically,onecouldscanvariousexternalpotentialsandthefunctionalE[ˆ(r)]withrespecttothedensityˆ(r).However,suchaprocedureisimpossibleaswedonotknowtheexactinter-nucleoninteractioninmedium.Thereexistvariousapproximationschemestoconstructthefunctionalfortheenergy,e.g.,theSkyrmeEDF,theGognyEDF,EDFfrometheory(EFT),etc.Theyarederivedfromthezero-rangeSkyrmeforce[72,73],theGognyforce[74{76]andusingtheEFTframework[77],respectively,eventhoughE[ˆ]inDFTdoesnothavetobeassociatedwithanynuclearinteraction.Inmywork,theSkyrmeEDFisthefunctionalofchoice,primarilyduetoitssimplicityandsuccessinexplainingthedata;itwillbeintroducedinthenextsection.2.2SkyrmeenergydensityfunctionalForthezero-rangeSkyrmeforce,thetotalenergyofthenucleus,asafunctionalofone-bodydensityandpairingmatrices,isgivenby[78]E[ˆ;~ˆ]=ZH(r)dr;(2.3)7whereH(r)isareal,scalar,isoscalar,time-evenfunctionaloflocaldensitiesandtheirderivatives.TheSkyrmeEDF(2.3)canbedecomposedintothekineticenergydensity,interactionenergydensity,pairingenergydensity,Coulombenergydensity,andadditionalcorrections,suchasthecenter-of-masscorrection[78]:H(r)=HKin(r)+HInt(r)+Hpairing(r)+HCoul(r)+Hcor(r):(2.4)BeforewepresenttheformofeachterminEq.(2.4),weshallintroducethelocaldensitiesused.2.2.1LocaldensitiesInthemeaapproximation,theone-bodynonlocaldensityandpairingdensityareasˆ(rs;r0s0)=hjayr0s0arsji;(2.5a)~ˆ(rs;r0s0)=2s0hjar0s0arsji;(2.5b)respectively,whereayrsandarscreateandannihilateaneutron(=n)orproton(=p)atthespacepointrwithspins,andjiisamany-bodywavefunction.Byexpandingthenonlocalparticledensityandpairingdensityinspinspace,oneobtains:ˆ(rs;r0s0)=12ˆ(r;r0)ss0+12s(r;r0)˙ss0;(2.6a)~ˆ(rs;r0s0)=12~ˆ(r;r0)ss0+12~s(r;r0)˙ss0;(2.6b)8where˙ss0=(˙xss0;˙yss0;˙zss0)arespinPaulimatricesandˆ(r;r0),s(r;r0),~ˆ(r;r0)and~s(r;r0)arederivedasˆ(r;r0)=Xsˆ(rs;r0s);(2.7a)s(r;r0)=Xss0ˆ(rs;r0s0)˙ss0;(2.7b)~ˆ(r;r0)=Xs~ˆ(rs;r0s);(2.7c)~s(r;r0)=Xss0~ˆ(rs;r0s0)˙ss0:(2.7d)SincetheSkyrmeEDFisexpressedasafunctionaloflocaldensitiesandtheirderivatives,thefollowingnucleonicdensitiesneedtobeconsidered[79,80]:1.Particle-densityˆ(r)andspindensitys(r):ˆ(r)=ˆ(r;r);(2.8a)s(r)=s(r;r):(2.8b)2.Kineticdensity˝(r)andvectorkineticdensityT(r):˝(r)=[rr0ˆ(r;r0)]r=r0;(2.9a)T(r)=[rr0s(r;r0)]r=r0:(2.9b)93.Momentumdensityj(r)andspin-currenttensor$J(r):j(r)=12i[(rr0)ˆ(r;r0)]r=r0;(2.10a)J(r)=12i[(rr0)s(r;r0)]r=r0:(2.10b)4.Pairingdensity~ˆ(r):~ˆ(r)=~ˆ(r;r):(2.11)Foreachoftheselocaldensities,isoscalar(T=0)andisovector(T=1)densitiesareintroduced.Forexample,isoscalarandisovectorkineticdensitiesareas˝0(r)=˝n(r)+˝p(r);˝1(r)=˝n(r)˝p(r);(2.12)respectively.Then,eachterminEq.(2.4)canbespasafunctionaloftheselocaldensities.2.2.2SkyrmeEDFformalismThekineticenergydensitycanbewrittenas:HKin(r)=~22m11A˝0(r);(2.13)where11Astemsfromasimpleapproximationtothecenter-of-masscorrection[81].TheinteractionenergydensityisderivedfromtheSkyrmeenuclearpotential10[72,73],andcanbewrittenas[80]HInt(r)=Xt=0;1(˜event(r)+˜oddt(r));(2.14)wheretheisospinindex,t,labelsisoscalar(t=0)andisovector(t=1)densities(foradetailedderivationoftheinteractionenergydensity,onecanrefertotheappendixin[82]).The˜evenand˜oddtermsgeneratethetime-evenandtime-oddmeanelds,respectively[80]:˜event(r)Cˆtˆ2t+Cˆtˆtˆt+C˝tˆt˝t+CJt$Jt2+CrJtˆtrJt;(2.15a)˜oddt(r)Csts2t+Cststst+CTtstTt+Cjtj2t+Crjtst(rjt):(2.15b)ThecouplingconstantCˆtcontainsadensitydependentterm,whichiswrittenasCˆt=Cˆt0+CˆtDˆ:(2.16)Thus,thestandardSkyrmeinteractionenergydensityisbymeansofthefollowingparameters:fCˆt;Cˆt;C˝t;CJt;CrJtgt=0;1fCst;Cst;CTt;Cjt;Crjtgt=0;1and:(2.17)Oneshouldnotethatwhentimereversalsymmetryisconserved,thetime-oddtermsvanish.Thepairingchannelcanbeparametrizedbyadensity-dependentdelta-pairingforcewithmixedvolumeandsurfacefeatures[83],V(n;p)pair=V(n;p)0112ˆ0(r)ˆc(rr0);(2.18)11whereV(n;p)0isthepairingstrengthforneutrons(n)andprotons(p),ˆ0(r)istheisoscalarlocaldensity,andˆcisthesaturationdensity,atˆc=0:16fm3.Theresultingpairingenergydensitycanbewrittenas[78]Hpairing(r)=X=n;pV02h112ˆ0(r)ˆci~ˆ2(r);(2.19)where~ˆisthelocalpairingdensity.TheCoulombenergycanbedividedintothedirectterm,HDirCoul,andexchangeterm,HExcCoul.ThedirecttermtakestheusualformHDirCoul(r)=e22Zdr1ˆp(r)ˆp(r1)jrr1j;(2.20)andtheexchangetermisusuallycalculatedintheSlaterapproximation[84,85]:HExcCoul(r)=34e23ˇ1=3ˆ4=3p:(2.21)Sofar,theSkyrmeEDFformalismhasbeenpresented.Byparameters(2.17)toselectedexperimentdata,thefollowingSkyrmefunctionalshavebeenoptimized:SkM[86]SkP[87]SLy4[88]SV-min[89]UNEDF0[78]UNEDF1[90]UNEDF1-HFB[91]UNEDF2[92].Ofthese,theUNEDFfamilywasoptimizedintheframeworkofHartree-Fock-Bogoliubovtheory,whichconsiderstheandpairingchannelsimultaneously.122.3methodsbasedonnuclearDFTGiventheenergyfunctional,onecanconstructtheg.s.densityusingtheKohn-Shamtheorem[93],whichstatesthatforanymany-bodyinteractingsystem,thereexistsanon-interactingsystemwithaKohn-ShampotentialVKS,whichgivesthesamedensityastheoriginalsystem.Thequestionis,givenanEDF,howcanonebuildtheKohn-Shampotential?WeassumethattheHamiltonianoftheoriginalsystemandtheone-bodyKohn-Shampotentialcanbewrittenas^H=^T+^V+^Uext;(2.22)^HKS=AXi=1^h(i);i=fri;Si;tig;^h=^t+^vKS;(2.23)respectively.IntheKohn-Shamapproximation,thesingleparticlewavefunctionsandenergiescanbeobtainedbysolvingtheone-bodyScodingerequation:^h'i="i'i;(2.24)andtheg.s.wavefunctionofthesystemjicanberepresentedasaSlaterdeterminantofsingleparticleorbitalswiththelowestenergies.Thecorrespondingone-bodydensityisgivenby,ˆ0(r)=AXi=1j'i(r)j2:(2.25)Iftheg.s.wavefunctionfortheoriginalinteractingsystemisji,thentheg.s.energy13becomes:E0=hj^Hji=hj^Hji+(hj^Hjihj^Hji)=hj^Tji+hj^Vji+hj^Uextji+(hj^T+^Vjihj^T+^Vji):(2.26)Here,weintroduce:T=hj^Tji;(2.27a)EH=hj^Vji;(2.27b)Eext=hj^Uextji;(2.27c)Exc=hj^T+^Vjihj^T+^Vji:(2.27d)Then,E0=T+EH+Eext+Exc;(2.28)whereTisthekineticenergyofthenon-interactingsystemandExcrepresentstheexchangeandcorrelationenergies.Applyingthevariationalprinciple:E0jˆ=ˆ0=Xi(h'ij^tj'ii+h'ij^tj'ii)jˆ=ˆ0+(EH+Eext+Exc)jˆ=ˆ0=Xi"ih'ij'iihZvKSˆdrijˆ=ˆ0+(EH+Eext+Exc)jˆ=ˆ0=0;(2.29)14Oneobtains:vKS=Eextˆ+EHˆ+Excˆ=Uext+vH+vxc:(2.30)Therefore,givenE[ˆ],thecorrespondingKohn-Shampotentialcanbefoundbyapplyingthevariationalprinciple.Then,bydiagonalizingtheresultingonecanconstructtheg.s.wavefunction,one-bodydensity,andtotalenergyusingtheSlaterdeterminantbasedontheKohn-Shamorbits.2.3.1Hartree-FockandHartree-Fock-BogoliubovequationsThedequationsincludetheHartree-Fock(HF)equations(whichdonotincludepairingcorrelations)andtheHartree-Fock-Bogoliubov(HFB)equations(whichaccountforthepairingchannel).VariationoftheSkyrmeEDFwithrespecttoˆand~ˆresultsintheSkyrmeHFBequations[94]:0B@h~h~hh1CA0B@UkVk1CA=0B@UkVk1CAEk;(2.31)wherehistheHFand~histhepairinghn=~22m+even0+odd0+even1+odd1);(2.32a)hp=~22m+even0+odd0even1odd1)+UCoul;(2.32b)~hq=V01V1ˆˆ0~ˆq:(2.32c)15Themomentum-dependentmeanaregivenby[79,95]event=r[Mt(r)r]+Ut(r)+12i$r˙$Bt(r)+$Bt(r)$r˙;(2.33a)oddt=r[(˙Ct(r))r]+˙t(r)+12i(rIt(r)+It(r)r);(2.33b)whereUt=2Cˆtˆt+2Cˆtˆt+C˝t˝t+CrJtrJt+U0t;(2.34a)t=2Cstst+2Cstst+CTtTt+Crjtrjt;(2.34b)Mt=C˝tˆt;(2.34c)Ct=CTtst;(2.34d)$Bt=2CJt$JtCrJt$rˆt;(2.34e)It=2Cjtjt+CrJtrst:(2.34f)InEq.(2.34a),thetermU0trepresentstherearrangementtermsresultingfromthedensitydependenceofthecouplingconstants.Forcalculationswithconstraints,theRouthianE0is[80]:E0=E+Emult+Ecran+Enumb;(2.35)i.e.,equaltothesumoftheenergyandthetermsresponsibleforconstraints,hereincludingthemultipole,crankingandparticle-numberconstraints.16Forthemultipoleconstraints,thestandardquadraticformistaken:Emult=XC(h^QiQ)2;(2.36)whereh^Qiaretheaveragevaluesofthemass-multipole-momentoperator,Qaretheconstraintvaluesofthemultipolemoments,andCaretheconstraints.Thecrankingconstraintsareassumedasthesimplelinearform:Ecran=!yh^Jyi;(2.37)where^Jyistheoperatorofthecomponentofthetotalangularmomentumalongtheyaxis(duetotheassumedsymmetryinthecodesused),Jyisthetargetvalue,andtherotationalfrequency!yisthecorrespondingLagrangemultiplier.Theparticle-numberconstraintsaregivenasEnumb=nh^Nniph^Npi;(2.38)whichensurethecorrectneutronandprotonnumbers.pandnaretheneutronandprotonFermienergies.VariationoftheRouthianE0resultsintheconstrainedHFBequation,wheretheHFinEq.2.31isreplacedbyh0=h+2XC(h^QiQ)^Q!y^jy(2.39)17BysolvingtheHFBequations,theresultingmatrixWy,0B@UyVyVTUT1CAtransformstheparticleoperatorscy;cintothequasiparticleoperatorsy;through0B@y1CA=Wy0B@ccy1CA:ThentheHamiltonianHcanbeapproximatedasH=H0+XkEkykk+fhigher-orderg:(2.40)TheeigenstatesofHarethequasi-particlevacuum(withtheenergyH0),onequasi-particlestateswithenergiesH0+Ek,twoquasi-particlestates,andsoon.SincetheHFBequationsarenon-linear(duetothedensity-dependenceofthepotentials),theyareoftensolvedbyiterativediagonalizationmethods.Itisworthnotingthat,duetotheself-consistentsymmetryoftheHFBequations,ifsomecertainsymmetryisexpectedinthesolution,oneshouldalwaysstartfromadensityinitializedwiththissymmetry.Thislimitsthesizeofmodelspaceandreducesthecomputational2.4NuclearDFTsolversSolvingtheHFandHFBequationsforcomplexexoticnucleiornucleiwithsuper-deformedcanbebothandtime-consuming.Inthisdissertation,twoDFT18solvers,hfoddandhfbtho,havebeenemployed:1.hfbtho[94,96]hfbthosolvestheHFBequationsinthecylindricalharmonicoscillator(HO)basisorinthetransformedHObasisobtainedbyapplyingalocalscaletransformation[97,98]ontheHOfunctions.ItimplementsallthegeneralSkyrmefunctionals,andisabletodomultipleconstraintcalculations.However,axialandtime-reversalsymmetriesareexplicitlyimposedhere.2.hfodd[1,80,99{103]hfoddismoreversatilethanhfbtho.ItsolvestheHF,HF+BCSandHFBequationsintheHObasiswiththeSkyrme,GognyorYukawaforce,anditallowsonetobreakallgeometricandtimereversalsymmetries.hfoddhasbeenemployedtoperformnumerouscalculations,suchasconstrainedcrankingcalculations[104]andstudiesofspontaneousattemperatures[7].BothsolvershavebeenaugmentedwithOpenMulti-Processing(OpenMP)andMessagePassingInterface(MPI)routines.19Chapter3RotationofTriaxialNuclei3.1IntroductionAmongthepredictedtriaxialnuclei,neutron-richMoandRuisotopeswithAˇ110areofthemostinterest,astheyexhibitshapechangesandshape-coexistencephenomena[105].Withincreasingneutronnumber,triaxialdeformationsareexpectedtoappearintheirgroundstatesduetotheoccupationofthe1h11=2and1ˇg9=2intruderorbitals[33].Experimentally,theclearestsignatureoftriaxialshapescomesfromthe-rayspec-troscopyofrotatingnuclei.Sinceevidenceforrotational-likebehaviorintheveryneutron-richeven-evenZr-Pdregionwasreportedin[106],therehavebeennumerousexperimentsdevotedtoshapetransitionsandrotationalpropertiesofnucleiinthisregion.Forexample,thedeformedin103;104;107Zrand107;108Mowerestudiedin[107],andatran-sitionfromsphericaltotriaxialshapesin104Ruwasstudiedin[108].Afterthat,moreandmoreevidencefortriaxialdeformationinneutron-richMoandRuisotopesstartedtoappear,including(i)thesteadydecreaseofthe-bandband-headenergyin110Ruand112Ru[109];(ii)thecollectivetriaxialbehaviorofRuandMoisotopesthroughthespectroscopyofsionfragments[2{4,110{112];(iii)themeasurementofthequasi-bandin110Mo[113,114],and,(iv)thetriaxialdeformationsin104Ruand110MoextractedfromE2andM1matrixelementsobtainedbyCoulombexcitationstudies[115].Theoretically,triaxialgroundstateshavebeeninvestigatedwithtmodels.For20example,inRef.[33],triaxialg.s.minimawerepredictedinneutron-richMoisotopeswithN=62-66usingthemacroscopic-microscopicapproach.InRef.[116,117],thelargestshellduetotriaxialdeformationswerefoundaround108Ru.InRef.[37,38],po-tentialenergysurfacecalculationswithHFandinteractingbosonmodelsshowedshallowtriaxialminimaforisotopeswithN=64-70.InRef.[36],self-consistentHFBcalculationswiththeGognyD1Sinteractionpredictedtriaxialdeformationsfortheeven-evenisotopes104-110Moand104;106Ru.Thesetpredictionscomefromtheextremesoftnessoftriaxialpotentialenergysurfacesofnucleiinthisregion.Intheproject,motivatedbythenewexperimentalresultsforthetransitionquadrupolemomentsofrotationalbands[6],weusednuclearDFTwiththeEDFUNEDF0[78]tode-scribeyraststructuresinMoandRuisotopes.Ourcalculationspredicttriaxialg.s.de-formationsfor106;108Moand108;110;112Ru,andtransitionquadrupolemomentsthatareconsistentwithexperiment[29].Triaxiallydeformednucleicanexecutenotonlyprincipal-axis-cranking(PAC),wheretherotationisaboutoneoftheprincipalaxes,butalsotilted-axis-cranking(TAC)[19,118],wheretherotationalaxistiltsawayfromtheprincipalaxis(PA).Thestandardtheoreticalmodeltodescribecollectiverotationisthecrankingmodel,inwhichthecrankingterm,!1J1[18,32,119],isaddedtotheldHamiltonian.Inself-consistentPACcalcula-tions,thenucleusisguaranteedtostayinthePAsystem.However,inTACcalculations,spuriousprocessionalmotionhastobepreventedbyaddinglinearconstraints,whichgivezeromatrixelementsoftheinertiatensor.AsshownbyKermanandOnishi[120],thecorrespondingLagrangemultipliersdependontherotationalfrequencies,angularmo-mentaandquadrupolemomentsofthesystem,andtheactualrelationisgivenbyEq.(3.6)in[120]and(3.10)below.21Inthesecondproject,withtheproperlinearconstraints,wevetheKerman-Onishiconditioninself-consistentcalculations.Wehaveshownthatthesolver-adaptedLagrangemultiplierisconsistentwiththeKerman-Onishiconditiontohighpre-cision.IthenappliedthisresultinPACandTACcalculationswithlinearconstraintsfor160Yb[104]usingtheSkyrmeHFapproach(similartothecalculationsfor158Er[121]).Asanillustrativeexample,belowIshallexplainthenatureoftwoTSDbandsof160Ybandalsopredictthepossiblelarge-amplitudecollectivemotionathighspins.3.2TriaxialdeformationBeforewecaninvestigatenuclearshapesandtheirassociatedcollectivemotion,weneedtointroducetheparametrizationofthenuclearsurface,whichisgivenintermsofthelengthofaradiusvectorpointingfromtheorigintothesurface[122]:R=R(;˚)=R01+00+1X=1X=Y(;˚);(3.1)whereR0istheradiusofaspherewiththesamevolume.Thequadrupoleshapesaredescribedbyeparameters2.Byasuitablerotation,onecantransformtothebody-systemcharacterizedbythreeaxes,whichcoincidewiththeprincipalaxesofthemassdistribution.Fivequadrupoledeformationparameterscanbethusreducedtotwoindependentquadrupolevariables20and22andthreeEulerangles.Forconvenience,theso-calledBohrquadrupoledeformationsareintroducedthrough20=2cos;22=1p22sin:(3.2)22Inhfodd,themultipolemomentsareasQ=a3A4ˇR0;(3.3)whereAisthemassnumber,R0canbeapproximatedasp5=3hr2i1=2,wherehr2i1=2isther.m.s.radius,a20=p16ˇ=5anda22=p32ˇ=5[1].BycombiningEq.(3.2)and(3.3),onecanget:2=rˇ51Ahr2iqQ220+Q222;(3.4a)tan=Q22Q20;(3.4b)whichwillbeusedinthefollowing.2representstheelongationofthenucleus,andisthetriaxialdeformationthatrangesbetween(0;60)intheabsenceofrotation.Underrotation,canbeextendedtotherangeof(120;60).3.3BandcrossingandangularmomentumalignmentInthecrankingmodel,thetotalHamiltonianintherotatingreferenceframecanbewrittenasH!=H!1J1:(3.5)Byapplyingthevariationalprincipleintherotatingframe,thecrankedHFBequationbe-comes0B@h!j1h++!j11CA0B@UkVk1CA=0B@UkVk1CAE!k;23where!istherotationalfrequency,whichisalsotheLagrangemultipliertoconstraintheexpectationvalueoftheangularmomentum.TheE!kcanbereferredtoasthesinglequasiparticleRouthianenergy.Astherotationalfrequency!increases,thesinglequasiparticlelevelscrossatsomefrequency!=!(e.g.seeFig.3.3).Beyondthecrossingthevacuumnchangesfromtheg-band(withtheband-headintheg.s.tothes-band(withtheband-headinthetwoquasi-particleThebandcrossingwillleadtoasuddenchangeinangularmomentum.Inthenextsection,theshapechangesandangularmomentumalignmentwillbeinves-tigatedfor106;108Moand108;110;112Ru.3.4Tilted-axis-crankingcalculationsandtheKerman-OnishiconditionInself-consistentHFTACcalculations,whentherotationalaxismovesawayfromthePA,thenucleushastostayinthePAreferenceframe.Thiscanberealizedbyaddinglinearconstraints,whichguaranteethattheresultingmatrixelementsoftheinertiatensor(orquadrupolemoment)vanish.Using^Qijxixj;(3.6)thefollowingconditionshouldbehj^Bji=0and^Bk=^Qij(i;j;kcyclic):(3.7)24Thus,theresultingRouthiancanbewrittenas^H0=^H!^J^B;(3.8)wherearethreeLagrangemultipliers,andtherotationalfrequencies!aredeterminedfromtheangular-momentumcondition:J=hj^Jji:(3.9)KermanandOnishiprovedthattheLagrangemultiplierscorrespondingtolinearconstraintsdependonangularmomenta,rotationalfrequencies,andquadrupolemoments[120].Thisrelation,referredtoastheKerman-Onishiconditioninthefollowing,isgivenbyk=(!J)kDiDj(i;j;kcyclic);(3.10)whereDi=hj^Dijiand^Di^Qii.Consequently,nonzerovaluesofimplythat!andJarenotparallel.TheKerman-Onishiconditioncanalsobewrittenas:1x2x3=+!2j3!3j2p36<hQ22i+12hQ20i=Q21L021=Q21;(3.11a)2x1x3=!3j1!1j3p36<hQ22i+12hQ20i<Q21L021<Q21;(3.11b)3x1x2=!1j2!2j12<hQ22i=Q22L022=Q22:(3.11c)Inourcalculations,weonlyallowtherotationalaxistobetiltedfromthex2axisto25thex1-x2plane.Inthiscase,onlyoneofthethreeconstrains,L22=Q22,isactive.Theothertwoconstraints,h=Q21i=0andh<Q21i,areautomaticallymetbyenforcingx3-T-simplexsymmetry[123].3.5ProjectA:TriaxialrotationintheAˇ110regionAllthecalculationsforthisprojectwereperformedwithhfodd(version2.49t)[103],andthewavefunctionswereexpandedin800sphericalharmonicoscillator(HO)basisstateswithanoscillatorfrequencyof~!=49:2MeV/A1=3.Wehavetestedthatsuchabasisprovidesaveryreasonableprecisionfortheobservablesstudied.Intheparticle-holechannel,weemployedtheglobalEDFUNEDF0[78].Inthepairingchannel,beforethebandcrossing[18,32],wetookthezero-rangedensity-dependentpairingforce(2.18)withtheLipkin-NogamicorrectionforparticlenumberTheoriginalpairingstrengthswere(V0;Vˇ0)=(170:374;199:202)MeVfm3,withacutoenergyinthequasiparticle(q.p.)spectrumofEcut=60MeV.Inthepresentcalculation,thestrengthsofthepairingforceforneutronsandprotonswereincreasedby5%toproducethekinematicmomentofinertiaoftheg.s.bandof106Mo.Asdiscussedbelow,thepotentialenergysurfaces(PES)arenotsensitivetosuchavariationofpairingstrengths.Beyondthebandcrossing,pairingwasneglectedinourcalculationasthestatingpairingcorrelationsarereducedbyangularmomentumalignment;thisisknownastheCoriolisanti-pairing[124].Inmultidimensionalpotentialenergysurfacecalculations,constraintsareimposedontheexpectationvaluesofmultiplemoments.WeusedtheaugmentedLagrangianmethod[125]toperformtheconstrainediterations.ThetotalRouthiancalculationswerecomputedwithin26thePACapproach[19].3.5.1Ground-statepotentialenergysurfacesFigure3.1:PESinthe(Q20;Q22)planeincrankedHFB+UNEDF0for106Moand108Mo.Left:standardpairingstrengths.Right:pairingstrengthsincreasedby5%,seetext.Thebetweencontourlinesis0.5MeV.Table3.1:Bohrquadrupoledeformationparameters2andcalculatedinHFB+UNEDF0forthegroundstatesof106;108Moand108;110;112Ru.106Mo108Mo108Ru110Ru112Ru20.190.180.160.160.151618242524WestarttheexplorationofshapedeformationwithHFBcalculationsofg.s.potentialenergysurfacesfor106;108Moand108;110;112Ru.TheresultsareshowninFig.3.1and3.2,respectively.Inboththeleftpanelsrepresenttheresultswiththeoriginalpairingstrengths,whiletherightpanelsshowtheresultswithincreasedpairingstrengths.Triaxialdeformedgroundstatesarepredictedinallcases.Itturnsoutthata5%changeinthepairingstrengthsdoesnotpracticallythePESs,causingonlyaslightincrease27Figure3.2:SimilartoFig.3.1,butfor108Ru,110Ru,and112Ru.insoftness.Inparticular,thetriaxialminimaappearingat(Q20;Q22)ˇ(8:0-9:5;2:0-3:0)barehardlybypairing.Thecorrespondingg.s.quadrupoledeformations(2;)aredisplayedinTable.3.1.For106;108Mo,wepredictedthetriaxialg.s.minimaat(2;)ˇ(0:19;17).Similarresultswereobtainedinthemacroscopic-microscopiccalculationsofRefs.[33,117,126]andHFB+D1Scalculations[36].For108;110;112Ru,wealsopredicttriaxialg.s.minima;thisisconsistentwithRefs.[33,127]andHF+SIIIcalculationsofRef.[109].Triaxialg.s.shapesfor108;110Ruwerealsoobtainedinthesurvey[117]but112Ruwascalculatedtobeaxial.28Figure3.3:One-q.p.Routhiandiagramfor106Mo(left)and112Ru(right)obtainedwithcrankedHFB+UNEDF0.Theparityˇandsignaturerastheeigenvalueof^Ry=exp(iˇ^Jy))ofindividuallevelsareindicatedinthefollowingway:ˇ=+;r=+i{solidline;ˇ=+;r=i{dottedline;ˇ=;r=+i{dot-dashedline;ˇ=;r=i{dashedline.ThethinlineindicatestheFermienergy.3.5.2RotationalpropertiesTheangularmomentumalignmentpatternofMoandRunucleiisgovernedbytheh11=2andˇg9=2high-jq.p.excitations,whichgiverisetostrongshapepolarization[33].Figure3.3showsself-consistentcrankedHFB+UNEDF01-q.p.Routhiandiagramsversusrotationalfrequencyfor106Moand112Ru,respectively.Inbothcases,thealignmentof(h11=2)2andˇ(g9=2)2pairsoccursatsimilarrotationalfrequenciesof~!ˇ0:3MeV.Athigherrotationalfrequencies,atransitionisexpectedfromtheg.s.bandtoaligned(h11=2)2andˇ(g9=2)22-q.p.bands,andthentoa4-q.p.(h11=2)2ˇ(g9=2)2band.ThesetwoconsecutivecrossingsareditofollowincrankedHFBcalculations,asthiswouldrequireaextension[21,32,128]ofthecurrentframework.SuchanextensionishighlynontrivialincrankedHFBastheself-consistentassociatedwithalignedareexpectedtotlyfromthoseoftheg.s.29band[33].Moreover,pairingcorrelationsinthealignedbandsarequenchedandthiscausesnumericalinstabilitiesaroundthebandcrossing.Therefore,toprovideinterpretationofthetransitionquadrupolemomentsathigherangularmomenta,wecarryoutcrankedSkyrmeHFcalculationswithoutpairingat~!>0:3MeV.Inthiscase,diabaticcanbebythenumberofsingle-particleRouthiansoccupiedinthefourparity-signatureblocks[100].Sp,eachneutronandprotonrationisbyfouroccupationnumbers[N++;N+;N+;N]representingthenumberofparticlesNˇ;roccupyingsingle-particlestatesofgivenˇandr.ThelowesttotalRouthianwithˇ=+andr=1isassociatedwiththeyrastFigure3.4:Angularmomentumalignmentfor106Moand108;112Ru.CrankedHFB(~!<0:3MeV)andcrankedHF(~!>0:3MeV)calculationsarecomparedtoexperiment[2{4]Theangularmomentumalignment(totalangularmomentumasafunctionofrotationalfrequency)isshowninFig.3.4for106Mo,108Ru,and112Ru.Belowthepredictedbandcrossingat~!ˇ0:3MeV,ourcalculationsreproduceexperimentwell.(Note,however,thatourpairingstrengthswereadjustedtomatchthekinematicmomentofinertiaof106Mo.)30Thebandcrossing,associatedwiththealignmentoftheh11=2neutronpair,isseenin108;112Rudataslightlybelow~!=0:4MeV,andistlydelayedin106Mo.Thepre-dictedalignedabovethebandcrossinghasafairlytshapeascomparedtothatoftheg.s.band,anditistofollowtheg.s.bandat~!>0:3MeV.Figure3.5:Summaryofequilibriumdeformationsofthelowestˇ=+;r=1bandsin106;108MocalculatedwithcrankedHFB+UNEDF0(groundband)andcrankedHF+UNEDF0(alignedbands).Therotationalfrequencyisvariedfromzeroto~!=0:6MeV.Thecorrespondingrangeofangularmomentumisindicated.Thealignedbandsareaccordingtothenumberofoccupiedhigh-Nintruderlevels(N=5and4forneutronsandprotons,respectively).Toinvestigatetheevolutionofnuclearshapeswithrotation,wecomputetheequilib-rium2anddeformationsforlow-lyingˇ=+;r=1bandsin106;108Mo(Fig.3.5)and31Figure3.6:SimilartoFig.3.5,butfor108;110;112Ru.32108;110;112Ru(Fig.3.6).Inallcasesconsidered,thetriaxialpairedg.s.bandundergoessmallcentrifugalstretchinginthedirectionof2.Forinstance,inthecaseof108Ru,2increasesfromthevalueof0.15at~!=0to0.17at~!=0:3MeV.Athigherspins(10I36),whenpairingisneglectedinourcalculations,itisusefultolabelmany-bodybythenumberofoccupiedintruderlevels,i.e.,Nosc=4protons(primarilyg9=2)andNosc=5neutrons(primarilyh11=2).Forinstance,thealignedˇ(9;9;12;12)(17;17;15;15)in106Mo(shownbycirclesinFig.3.5(a))canbedenotedasˇ4454,andthesamelabelappliestotheˇ(9;9;12;12)(18;18;15;15)in108Mo(shownbyup-trianglesinFig.3.5(b)).Thequadrupoledeformations2ofalignedbandsarepredictedtobeintherangeof0:1220:16,whichrepresentsareductionascomparedtotheshapesofpairedground-statebands.Thealignedbandsremaintriaxialwithvaluesaround30upto~!=0:6MeV.Thisisconsistentwithearlierresults[33]whichemployedacrankedmacroscopic-microscopicapproach.Atthehighestrotationalfrequenciesconsidered,ourcalculationspredicttheappearanceofalignedtriaxialwith>0,whicheventuallyterminateatoblateshapes(ˇ60),see,e.g.,Fig.3.5(b).Tostudythestabilityofttriaxialminimaathighspins,weanalyzerelateddiabatictotalRouthiansinthe(Q20;Q22)plane.InFig.3.7weshowthetotalRouthiansurfacesat~!=0:5MeVfortheselectedlow-lyingalignedin106ModiscussedinFig.3.5(a).Forallthosethecollectivetriaxialminimumwithbetween30and15appearslowestinenergy.Fortheˇ4454showninFig.3.7(a),wealsopredictanoncollectiveoblatestatewithI=34thatrepresentsaterminationpointofthe>0band.Toeliminatespuriousminimathatareunstablewithrespecttotheangularmomentum33Figure3.7:DiabatictotalRouthiansurfacesfor106Moat~!=0:5MeVcalculatedwithCHF+UNEDF0forthefollowing(a)ˇ(9;9;12;12)(17;17;15;15)(ˇ4454inFig.3.5(a));(b)ˇ(10;10;11;11)(17;17;15;15)(ˇ4654);and(c)ˇ(10;10;11;11)(16;16;16;16)(ˇ4656).orientation,wealsoinvestigatethedependenceoftheRouthiansontheangularmomentumtiltinganglewithrespecttotheaxisofrotation(y-axis).Tothisend,weusedtheTACformalismofRefs.[104,121].Thecalculationswereperformedforthealignedbandsin106Mo.At~!<0:5MeV,thetotalRouthiansoftriaxial(<0)ˇ4454,ˇ4555,andˇ4654ofFig.3.5(a)showaminimumat=0.At~!ˇ0:5MeV,theRouthiansbecomeverysoftin,indicatingalarge-amplitudecollectivemotioninthisdirection.Thisinstabilityisnotpresentforthe(ˇ=;r=1)ˇ(9;9;12;12)(18;17;15;14)(ˇ4455),whichshowsapronouncedminimumat=90associatedwith>0.Thisresultisconsistentwiththedeformation-drivingofalignedh11=2neutronsorbitalsdiscussedinRef.[33].34Figure3.8:Transitionquadrupolemomentsfor106;108Moand108;110;112RucalculatedincrankedHFB(opencircles)andcrankedHF(dots)comparedtoexperiment.TheQtvalueatI=2istakenfromRef.[5]andthehigh-spinvaluesfromRef.[6].Thetransitionquadrupolemomentsalongtheyrastbandin106;108Moand108;110;112RuareshowninFig.3.8asafunctionofrotationalfrequency.Atlowrotationalfrequencies~!<0:3MeV,thereisagradualincreaseofQtwith!duetothecentrifugalstretchingseeninFigs.3.5and3.6.Asdiscussedearlier,athigherfrequenciescrankingcalculationsareperformedwithoutpairing.WhilethisapproximationseriouslythepredictedangularmomentumalignmentshowninFig.3.4,theequilibriumshapesobtainedinthecrankedHFmethodarereasonableapproximationstothoseobtainedinthefullcrankedHFBframework[129,130],andreproduceexperimentalQt-valuesforaligned[131,132].AsseeninFig.3.8,thepredictedtransitionquadrupolemomentsinalignedbands35areslightlyreducedwithrespecttothelow-spinregionduetothedeformationreductionassociatedwiththealignedh11=2andˇg9=2pairs.Thisreductionisgenerallyconsistentwithexperiment,exceptperhapsfor110Ru,wheretheoryoverestimatesthemeasuredQtvaluesabove~!=0:3MeV.TofurtherillustratetheimportanceofthedegreeoffreedominthedescriptionofthebandstructuresintheMoandRuisotopes,wehavealsocarriedouttriaxial-projected-shell-modelcalculationsfor106Mowitharangeofvalues.Theobtainedresults,carriedoutbymycollaborators,paintthesamepictureascrankedHFBandHFcalculations,andstronglyfavorthetriaxialinterpretation[29].3.6ProjectB:Descriptionoftriaxialstronglydeformedbandsin160YbIntheprevioussection,TACcalculationshavebeenperformedforthealignedbandsin106Mo,butnodetailsofthisprocedurewereprovided.Inthissection,toexplaintheexperi-mentallyobservedTSDbandsin160Yb[133]andtoillustratetheKerman-Onishicondition,TACcalculationsareperformedwiththecrankedHFapproach.3.6.1Tilted-axis-crankingcalculationsfor160YbTheTACcalculationswereperformedbyusingthesymmetry-unrestrictedsolverhfodd(version2.49t).Tobeconsistentwiththepreviouswork,weusedtheSkyrmeEDFSkM[86]andthewavefunctionwasexpandedin1000deformedHObasisstateswithfrequenciesof~x=~y=10:080MeV(uptoNx=Ny=15HOquanta)and~z=7:418MeV(upto36Nz=20HOquanta).Table3.2:Thein160Ybstudiedinthiswork.Eachcisdescribedbythenumbersofoccupiedstatesinthefourparity-signature(ˇ;r)blocks,intheconventionofRef.[1].LabelminimumparityATSD1[23;23;22;22]ˇ[16;18;18;18]+BTSD1[23;24;21;22]ˇ[16;18;18;18]CTSD1[23;24;21;22]ˇ[18;18;17;17]DTSD3[23;23;22;22]ˇ[18;18;17;17]+ETSD3[23;23;22;22]ˇ[17;17;18;18]+For160Yb,ourcalculationsshowthattwocompetingPACminimawithsimilarvaluesof2andjjbutoppositevaluesofappear,whichindicatesunstablePACsolutionsandthusthepossibleappearanceofrotationalonganaxiswhichdoesnotcoincidewiththePA.Therefore,beforetheTACcalculations,weperformextensivePACcalculationssoastodeterminedeformationsofvariousminima.SimilartothePACcalculationsin158Er[134],wefoundthatthegenerallyhavethreetypicaldeformations,namely,(Qt;)˘(9eb;9-14)(TSD1),(Qt;)˘(10:8-12:2eb;10)(TSD2),and(Qt;)˘(10:0-10:5eb;13)(TSD3).Rangesofdeformationsindicateshapechangeswithrotationalfrequency.Figure3.9showsthetotalRouthiansofein160YbcalculatedwithSkyrmeHFasfunctionsofthetiltedangle,,astheanglebetweenthex2-axisandtherotationalaxisinthex1-x2plane.ThecorrespondingandparitiesaregiveninTable3.2.At=90,theQ22valuechangessignandTSD1becomesTSD2.Itcanbeseenthat,at!=0:5MeV,theRouthiansofthebandsAandBareverysoftagainst.ForbandA,evenaminimumwith6=0or90develops.Insuchasituation,onemayexpectlarge-amplitudecollectivemotionwitharotationalaxisthateasilychangesitsdirection.Astherotationalfrequencyincreases,theenergyoftheTSD2bandsincreases37Figure3.9:TotalRouthiansin160YbcalculatedwithintheSHFmethodasfunctionsofthetiltinganglefortheeTSDlistedinTable3.2.Solidanddashedlinesmarkwithpositiveandnegativeparity,respectively.rapidlyandthesecobecomeunphysicalsaddlepoints.Atthesametime,forthefrequenciesconsidered,theenergiesoftheTSD3bandsareclosetoorevenbelowtheenergiesoftheTSD1andTSD2bands.383.7SummaryInthischapter,weappliednuclearDFTtostudytriaxialshapesinthemedium-heavynuclei106;108Moand108;110;112Ru,andTSDbandsin160Yb.Triaxialg.s.deformationswerepredictedfor106;108Moand108;110;112Ru.Observedhigh-spinbehaviorofthesenucleiisalsoconsistentwithtriaxialrotation.However,thepredictedtriaxialgroundstateminimaarefairlyshallow,andthisisperhapswhyinsomecalculations,e.g.,thecrankedrelativisticHartree-BogoliubovmodelofRef.[6],axialrationsmaybeslightlyfavored.Particularly,inourwork[29],thetriaxialprojectedshellmodelwasalsoemployedtoexplaintheobservedbandstructuresbyassumingstabletriaxialshapes,whichhasgivenconsistentresultswithournuclearDFTcalculations.Inaddition,withtheKerman-Onishiconditionassumed,weemployedtheSkyrmeHFmethodtoperformTACcalculationsfor160Yb.TheresultsexplainedthenatureoftheTSD1andTSD2bandsin160Ybandpredictedlarge-amplitudecollectivemotion,whichmightappearatrotationalfrequency!ˇ0:5MeV.Frommassdistributionsalone,multipolemomentsonlytellusaboutgeometricshapesofnuclei,leavingoutdetailedinformationontheirshellstructure.Inthenextchapter,weshallintroducethenucleonlocalizationfunction,whichalsoprovidesspinformationons.p.orbits.39Chapter4NucleonLocalization4.1IntroductionThedegreeofclusteringinnucleiistoassessquantitativelyinDFTasthesingleparticlewavefunctionsarespreadthroughoutthenuclearvolume;hence,theresultingnu-cleonicdistributionsarerathercrudeindicatorsofclusterstructuresastheirbehaviorinthenuclearinteriorisfairlyuniform.Therefore,inthischapter,weutilizeatmeasurecalledspatiallocalization,whichisamoreselectivesignatureofclusteringandclustershellstructure.Spatiallocalization,originallyintroducedfortheidenoflocalizedelec-tronicgroupsinatomicandmolecularsystems[22,135{139],hasrecentlybeenappliedtocharacterizeclustersinlightnuclei[23].Thischapterisorganizedasfollows:Sec.4.2givesanintroductiontothespatiallocalizationformalism,thentworelatedprojectsarediscussedinSec.4.3and4.4.Intheproject,toillustratethebasicconceptsofnucleonlocalization,weemploythedeformedharmonicoscillator(DHO)modelandnuclearDFTtostudyclusterstructuresindeformedlightnuclei.Thenweapplythismeasuretotrackthedevelopmentoffragmentsinheavynuclei(232Th,264Fmand240Pu)withtheircharacteristicoscillationpatterns[24,31].Spontaneous(SF)isatypeofslowbutlarge-amplitudecollectivemotion,andonlyhappensinheavynuclei,whichmakesnuclearDFTanexcellenttooltodescribeit.40Asreportedinpreviouswork[7,140{144],thepathwayfromthegroundstatesofcompoundnucleitoseparatedfragmentscanbedescribedbyseveralcollectivecoordinates(usuallydeintermsofmultipolemoments),andmayalsobebypairingandexcitationenergy.Therefore,determiningthemulti-dimensionalPESisusuallythestepofthecalculation.Furthermore,sinceSFisaquantumprocess,allpathwaysshouldbeconsideredtheoreticallywithtprobabilities[8].ThesetwoconsiderationsmakeSFcalculationscomputationallyexpensive.Inthesecondproject,weemploynuclearDFTtocalculateboththedensitydistributionandspatiallocalizationdistributionontheouter-turningline[140]andthescissionlinewherehappensfor240Pu.Withthestructureinformationobtainedviaspatiallocalization,fragmentscanbeidenbeforescission.4.2SpatiallocalizationThespatiallocalizationmeasurewasoriginallyintroducedinatomicandmolecularphysicstocharacterizechemicalbondsinelectronicsystems.Italsoturnedouttobeusefulinvisualizingclusterstructuresinlightnuclei[23].Thelocalizationmeasurecanbederivedbyconsideringtheconditionalprobabilitiesofanucleonwithinadistancefromagivennucleonatrwiththesamespin˙("or#)andisospinq(norp).Asdiscussedin[22,23],theexpansionofthisprobabilitywithrespecttocanbewrittenasRq˙(r;)ˇ13 ˝q˙14jrˆq˙j2ˆq˙j2q˙ˆq˙!2+O(3);(4.1)41whereˆq˙,˝q˙,jq˙,andrˆq˙aretheparticledensity,kineticenergydensity,currentdensity,anddensitygradient,respectively.AdetailedderivationcanbefoundinAppendixB.ThedensitieshavebeeninSec.2.2.1.Byusingthecanonicalbasis[145],theycanbere-expressedthroughthecanonicalHFBorbitals (r˙):ˆq˙(r)=X2qv2j (r˙)j2;(4.2a)˝q˙(r)=X2qv2jr (r˙)j2;(4.2b)jq˙(r)=X2qv2Im[ (r˙)r (r˙)];(4.2c)rˆq˙(r)=2X2qv2Re[ (r˙)r (r˙)];(4.2d)withv2beingthecanonicaloccupationprobability.Thus,theexpressionintheparenthesesofEq.(B.6)canserveasalocalizationmeasure.Unfortunately,thisexpressionisneitherdimensionlessnornormalized.AnaturalchoicefornormalizationistheThomas-Fermiki-neticenergydensity˝TFq˙=35(6ˇ2)2=3ˆ5=3q˙.ConsideringthatthespatiallocalizationandRq˙areinaninverserelationship,adimensionlessandnormalizedexpressionforthelocalizationmeasurecanbewrittenasCq˙(r)="1+˝q˙ˆq˙14jrˆq˙j2j2q˙ˆq˙˝TFq˙2#1:(4.3)Inourwork,timereversalsymmetryisconservedandjq˙vanishes.AvalueofCclosetooneindicatesthattheprobabilityoftwonucleonswiththesamespinandisospinatthesamespatiallocationisverylow.Thusthenucleon'slocalizationislargeatthatpoint.Inparticular,nucleonsmakingupthealphaparticleareperfectlylocalized[23].AnotherinterestingcaseisC=1=2,whichcorrespondstoahomogeneous42Fermigasasfoundinnuclearmatter.Whenappliedtomany-electronsystems,thequantityCisreferredtoastheelectronlocalizationfunction,orELF.Innuclearapplications,themeasureoflocalization(4.3)shallthusbecalledthenucleonlocalizationfunction(NLF).TheaboveoftheNLFworkswellinregionswithnon-zeronucleonicdensity.Whenthelocaldensitiesbecomeverysmallinregionsoutsidetherangeofthenuclearmeannumericalinstabilitiescanappear.Ontheotherhand,whentheparticledensityisclosetozero,localizationisnolongerameaningfulquantity.Consequently,fornuclei,wemultiplytheNLFbyanormalizedparticledensityC(r)!C(r)ˆq˙(r)=[max(ˆq˙(r))].4.3ProjectC:NucleonlocalizationinnucleiInthisproject,weemployedtheaxialDHOmodelandnuclearDFT.ForthelightnucleidiscussedinSec.4.3.2,wherepairingcanbeneglected,wesolvedtheconstrainedHFproblemwiththefunctionalUNEDF1-HFB[91](exceptforthediscussioninoctupoledeformed20Ne.Fordetails,seeSec.4.3.2).InthediscussionofheavynucleiinSec.4.3.3,pairingcorrelationsareimportant.Therein,wesolvedtheconstrainedHFBproblemwiththeUNEDF1functionaloptimizedfor[90]inthepresenceofpairingtreatedbymeansoftheLipkin-NogamiapproximationasinRef.[146].Bothhfbtho[96]andhfodd[103]wereusedtosolvetheHFandHFBequations.4.3.1NLFswithintheaxialharmonicoscillatormodelSincetheaveragepotentialinlightnucleicanbefairlywellapproximatedbythatofadeformedHO[147{149],manypropertiesofthesesystemscanbecharacterizedintermsoftheHOshellstructure.Quantummechanically,theunusualstabilityofclusterstates43inlightnucleicanbeattributedtostrongshellthatarepresentindeformedsingleparticleorbitals.Indeed,inthedeformedHOmodelthestrongestleveldegeneracyoccurswhentheratioofoscillatorfrequenciesisarationalnumber;thisresultsintheappearanceofsupershells[70,148,149].Consequently,thedeformedHOmodelcanserveasaroughguidetodescribeclusterstates[61,64,70,150].Tothisend,westudytheNLFsusingthewavefunctionsoftheaxialHOwithfrequencies!?and!z.Fortheintegervaluesof!?=!z(prolateshapes),asupershellstructureappearsthatisassociatedwiththe-foldSU(3)dynamicalsymmetryoftherationalHO[70].Forinstance,for=2,thesuperdeformedmagicnumbersare2;4;10;16;:::,etc.Foroblateshapes,thedegeneracypatternoftherationalHOisent[151].Forinstancefor=0:9themagicnumbersare2;6;8;14;:::[149,151].IntheexamplesbelowforN=Znuclei,weonlyshowdensitiesandNLFsforonecombinationofspinandisospinastime-reversalandisospinsymmetriesareconserved.InFig.4.1weshowparticledensitiesandNLFsforsuperdeformedin8Beand20Newith=2,andforanoblatein12Cwith=0:9.AllofthesecorrespondtoclosedshellsofthedeformedHO.Inthecaseof8Be,boththedensityandtheNLFrevealtwoclearcenters.TheNLFvaluesatthesecentersareclosetoone,whichmeansthatthenucleonsarehighlylocalized,implyingthepresenceofclusters.For20Ne,theclusterstructureistoseeintheparticledensityplot.However,theNLFclearlydemonstratesthepresenceoftwoclustersatthetipsandaringstructurearoundz=0associatedwithanoblate-deformed12Cnucleus.Figures4.1(e)and(f)indeedshowthatthe=0:9gurationin12Cexhibitsaverysimilarlocalizationpattern,exceptthatthelocalizationlevelishigher.Thisiseasytounderstandasthewavefunctionsofclustersin20Nehaveanon-zerooverlapwiththeringstructureof12C,andthisdecreases44Figure4.1:Left:particledensityˆn"(innucleons/fm3)forprolatein8Beand20Newith=2,andforanoblatein12Cwith=0:9.Right:correspondingmaskedNLFsasdescribedinSec.4.2.WhitedottedlinesarethecontourlinesC=0:9oftheoriginalnition(4.3).theleveloflocalization.InFig.4.2,weshowthreemoreexamplesofelongatedsin36Ar,16Oand24Mg.Intherow,weshowahyperdeformed(=3)in36Ar.Whiletheparticledensityhardlyshowsclustering,thelocalizationshowslargevalues,especiallyatthetipsofthenucleus.Thestructureinbetweencorrespondstoadeformed28Siandalsoexhibitsclusterstructuresatz=0andzˇ2fm.ThewhitedottedlinerepresentstheC=0:9contouroftheoriginallocalizationmeasure.Inthesecondandthirdrows,inorder45Figure4.2:SimilartoFig.4.1,butfor36Ar,16Oand24Mg.tosimulate-chaingurations,wechooseaccordingtotheparticlecontent.Whileaseparationintoparticlesistoseeintheparticledensityplot,especiallyfor24Mg,theNLFclearlyrevealsfourmaximafor16Oandsixmaximafor24Mg,withlocalizationsclosetoone.Thismeansthatthenucleonsareverylocalizedforeachspin/isospincomponent,implyingthepresenceof-chain4.3.2NLFsinlightnucleiInordertocomparethedeformedHOresultswitharealisticnuclearmodel,wecarryoutnuclearDFTcalculationsfor8Be,20Neand36Ar.ForlightnucleiwithN=Z,resultsfor46neutronsandprotonsareverysimilarastheCoulombcontributionissmall.Therefore,weonlyconsiderneutrondensitiesandspin-upNLFs.Figures4.3,4.4and4.5showourresultsofconstrainedHFcalculationswithUNEDF1-HFBassuminggoodparity.Figure4.3:Left:neutrondensityˆn(innucleons/fm3)fortheHFground-stateof8Becom-putedwiththefunctionalUNEDF1-HFB.Right:correspondingmaskedNLFsasdescribedinSec.4.2.WhitedottedlinesarethecontourlinesC=0:9oftheoriginal(4.3).Figure4.3depictsˆnandCn"fortheground-stateof8Be.BothquantitiesareverysimilartotheHOresultsofFig.4.1(a)and(b),againrevealingthepresenceoftwoclusters.Figure4.4showsHFresultsfor20Neforbothitsaxiallydeformedwith2ˇ0:38andahyperdeformedminimumwith2ˇ0:9.ThevaluesofˆnandCn"attherstlocalminimumaresimilartotheHOresultsofFigs.4.1(c)and(d).Again,theparticledensitydoesnotshowanypronouncedclusterstructure,whilethespatiallocalizationshowsthepresenceofthe-12C-structure.Asthequadrupoledeformationincreases,thelocalizationofthethreeclustersbecomesmorepronounced.Ourresultsfor8Beand20NecomparewellwiththeHF-SkI3calculationsofRef.[23].Inbothcases,unmaskedNLFsexhibitnumericalartifactsinthelow-densityregion,becausethenumeratoranddenominatorinEq.(4.3)arebothclosetozero.Figure4.5showstheHFenergyof36Arasafunctionofthequadrupoledeformation47Figure4.4:(a)TotalHFenergyof20Neversusquadrupoledeformationparameter2.Thepredictedlowestminimumisnormalizedto0.Theneutrondensities(innucleons/fm3)atthetwolocalminimaIandIIareshowninpanels(b)and(d),respectively,andthecorrespondingNLFsareplottedinpanels(c)and(e).2.Thethreelocalminimaarepredictedat2ˇ0:1,0:5,and0:8.ThecorrespondingneutrondensitiesandNLFsarealsodisplayedinFig.4.5(clusteringin36ArhasalsobeenstudiedintheDFTcalculationsofRef.[68]).Theweakly-deformedgroundstateat2ˇ0:1doesnotshowanystructureinthedensity.ItsNLFexhibitsamaximuminthecenterandanenhancementatthetips.Thisdistributionconstitutesauniquetoftheshellstructureof36Arthatisclearlymissinginthedensityplot.ConIIislessdeformedthanthatcalculatedwiththeHOinFig.4.2.However,itsNLFissimilar.Inparticular,the48Figure4.5:SimilartoFig.4.4,butfor36Ar.localizationenhancementatthetipsrevealsthepresenceofalphaclustering.Thecentralstructureshowstworingsoftheenhancedsurfacelocalization.UnlikeshapeII,shapeIIIhasamoreuniformNLF,withthelocalizationpeakedaroundthenuclearsurfacewherethecontributionfromonespsingle-particleorbitalislikelytodominate.Figure4.6:SimilartoFig.4.3butforthegroundstateof20Neat2=0:35and3=0:57.IfparityisnotconstrainedinHFcalculations,thegroundstateof20Neisexpected49tobeetric[152].InRefs.[153{155],theKˇ=0in20Neispredictedtohaveacluster-likestructurewithalargeoctupoledeformation,whiletheKˇ=0+groundstateissituatedbetweencluster-likeandshell-likeToillustratethisspecialcase,weusehfoddtoperformaconstrainedHFBcalculationfor20Neattheoctupoledeformation3ˇ0:57[155],whichisbforobtainingthestablesolution,whilethepairingstrengthschosenareverysmall.Theoctupoledeformation3isdthroughthecorrespondingaxialoctupolemomentQ30:3=4ˇ5q35Q30Ahr2i3=2:AsseeninFig.4.6,theintrinsicof20Nehasarathercompactshape.AcleardipintheNLFatthecentermanifestsaregionwhereallwavefunctionsoverlap.Aslightenhancementofthespatiallocalizationnearthetipisvisible.Whiletheground-stateof20Neshowsafainttraceofan-16Ostructure,thelocalizationisratherlow;hence,thequasimolecularpicturedoesnotapplyfortheUNEDF1-HFBSkyrmefunctionalused.4.3.3NLFsinheavynucleiBasedontheexamplesshowninthelastsection,weconcludethattheNLFisanexcellenttoolforvisualizingclusterstructuresinlightnuclei.Inthissection,wewillapplythistooltomonitorthedevelopmentandevolutionoffragmentsin232Th,264Fmand240Pu.SincetheCoulombinteractionistinheavynuclei,theisospinsymmetryisbroken.Therefore,bothneutronandprotonresultswillbediscussedhere.Figure4.7showsthepotentialenergycurvesof232Thand240Pualongthemostprobablepathwaypredicted,respectively,inRefs.[7]and[8].ThecalculationsareperformedwiththeUNEDF1functional.Bothcurvesshowsecondaryminimaassociatedwithsuperde-formedisomers.For232Th,apronouncedsoftnessisobservedatlargequadrupolemomentsQ20ˇ150200b.Inthisregionofcollectivespace,ahyperdeformedthird50Figure4.7:Thepotentialenergycurvesof232Thand240PucalculatedwithUNEDF1alongthepathways[7,8].ThecfurtherdiscussedinFigs.4.8and4.12aremarkedbysymbols.Theirquadrupoleandoctupolemoments,Q20(b)andQ30(b3=2)re-spectively,areindicated.minimumispredictedbysomeSkyrmefunctionals[7].Inthenextstep,weconsiderealongthepathwaytoperformdetailedlocalizationanalysis.Figure4.8showsneutronandprotondensitiesandNLFsfor232Thalongthepathway.Thecolumncorrespondstotheground-stateconwherethedensitiesdonotshowobviousinternalstructures.However,theneutronNLFshowsthreeconcentriccirclesandtheprotonNLFexhibitstwomaximaandanenhancementatthesurface.Thesecondcolumncorrespondstotheisomer.HeretwocenterdistributionsbegintoforminbothNLFs.Asdiscussedin[7],thedistributionsshowninthethirdcolumncanbeassociatedwithaquasimolecular\third-minimum"inwhichonefragmentbearsastrongresemblancetothedoublymagicnucleus132Sn.Thefourthcolumnrepresentstheclosetothescissionpoint,wheretwowell-developedfragmentsarepresent.51Figure4.8:Nucleonicdensities(innucleons/fm3)andspatiallocalizationsfor232Thob-tainedfromHFBcalculationswithUNEDF1forealongthepathwaymarkedinFig.4.7.Asseeninthelastcolumn,atlargerelongationsbeyondscission,thenucleusbreaksupintotwofragments,onesphericalandanotheronestronglydeformed.Tostudytheevolutionoffragments,weperformedHFBcalculationsforthepre-sumedfragments132Snand100Zr.TheresultsareshowninFig.4.9.Thenucleus132Snisadoubly-magicsystemwithacharacteristicshellstructure.ExceptforasmalldepressionatthecenterofprotondensityinFig.4.9(c)(left),thenucleonicdensitiesarealmostconstantintheinterior.Ontheotherhand,theNLFsshowcharacteristicpatternsofconcentricringswithenhancedlocalization,inwhichtheneutronNLFexhibits52Figure4.9:Nucleonicdensities(innucleons/fm3)andspatiallocalizationsforthegroundstateof132Sn(left)and100Zr(right)..oneadditionalmaximumascomparedtotheprotonNLF;thisisduetotheadditionalclosedneutronshell.Asonecansee,unlikeinatomicsystems[22],thetotalnumberofshellscannotbedirectlyreadfromthenumberofpeaksintheNLF,becausetheradialdistributionsofwavefunctionsbelongingtotnucleonicshellsvaryfairlysmoothlyandarepoorlyseparatedinspace.Nevertheless,eachmagicnumberleavesastronganduniqueimprintonthespatiallocalization.For100Zr,thecalculationisperformedattheprolatewithQ20=10b,whichcorrespondstothelighterfragmentpredictedin[7].Again,whiletheparticledensitiesarealmostconstantintheinterior,theneutronNLFshowstwoconcentricringsandtheprotonNLFexhibitstwocentersintheinteriorandoneenhancedringatthesurface.ThecharacteristicpatternsseenintheNLFsoffragmentscanbespottedduringtheevolutionof232ThinFig.4.8.Toshowitmoreclearly,Fig.4.10displaystheNLFsofthethreemostelongatedof232Thalongthez-axisinFig.4.8andcomparesthemtothoseof132Snand100Zr.ToavoidnormalizationproblemswepresentNLFsgivenbyEq.(4.3),i.e.,withoutapplyingthedensityformfactor.InFigs.4.10(a)and(b),neutron53Figure4.10:Neutron(left)andproton(right)NLFfor232Th(bluethickline),100Zr(greenline),and132Sn(redline)alongthezaxis(r=0).Thesecond,andthirdpanelscorrespondtotherationsinthethird,fourthandcolumnsofFig.4.8,respectively.andprotonlocalizationsatthecenterarearound0.5,whichisclosetotheFermigaslimit.Thisisexpectedforafairlyheavynucleus.Intheexterior,thelocalizationsoftwodevelopingfragmentsmatchthoseof100Zrand132Snfairlywell.Inpanels(c)and(d),theNLFsof232Thgrowintheinterior;thisdemonstratesthatthenucleonsbecomelocalizedattheneckregion.Finally,inpanels(e)and(f),thefragmentsareseparatedandtheirNLFsareconsistentwiththelocalizationsof100Zrand132Sn.Asanotherillustrativeexample,weshowinFig.4.11thedistributionscomputedfor54Figure4.11:SimilartoFig.4.8butfortheof264Fm.Thequadrupolemomentsofthetionsaredenotedontopofeachcolumn.264Fm,inwhichsymmetricwaspredicted.AstheconstrainingquadrupolemomentQ20getslarger,theparticledensitiesbecomeincreasinglyelongated.AneckdevelopsatQ20ˇ145b,andthescissionpointisreachedatQ20ˇ265babovewhich264Fmsplitsintotwo132Snfragments.Bycomparingtotheresultsfor132SninFig.4.9,onecanseethegradualdevelopmentofthe132Snclusterswithinthenucleus.Finally,letusconsidertheimportantcaseof240Pu.Recently,amicroscopicmodelingofmassandchargedistributionsinspontaneousofthisnucleuswascarriedoutinRef.[8].Togiveaninsightintotheevolutionof240Pualongitspathway,inFig.4.12weillustratetheNLFsof240Pu.ThetransitiontothepathwaybeginsatQ20ˇ95b.ItisseenthattwonascentfragmentsstartdevelopingatthisAtlargerelongationsinternalparityisbrokenandtwofragmentsareformedwithdistinctshellimprintsinthecorrespondingNLFs.Inthelastcolumn,theringsofenhancedlocalization55Figure4.12:SimilartoFig.4.8butfortherationsof240PuindicatedinFig.4.7.arealmostclosed,andthefragmentsarenearlyseparated.Threeexamples,232Th,264Fmand240Pu,showinaratherdramaticfashionthattheNLFscanserveasexcellenttsofboththeformationandevolutionofclusterstructuresinnuclei.4.4ProjectD:IdentifyingfragmentsIntheprevioussection,nucleonspatiallocalizationwasemployedtodescribetheclusteringsubstructuresinlightandheavynucleisuccessfully.Inthisproject,itwillbeappliedtoidentifyfragmentsin240Pu.564.4.1FissionfragmentdistributionInasemi-classicalapproximationforspontaneousthepenetrationprobabilityfromtheinner-turningpoint,A,totheouter-turningpoint,B,isgivenbyP=(1+exp[2S(L)])1;(4.4)whereS(L)istheleastactionintegralcalculatedalongtheone-dimensionalpathL(s)preselectedinthemultidimensionalcollectivespace:S(L)=ZsBsA1~p2M(s)(V(s)E0)ds;(4.5)whereV(s)andM(s)aretheepotentialenergyandinertiaalongthepathL(s),respectively.Vcanbeobtainedbysubtractingthevibrationalzero-pointenergyEZPEfromthetotalHFBenergyEtot.EZPEcanbeestimatedbyusingtheGaussianoverlapapproximation[156].IntegrallimitssAandsBaretheclassicalinnerandouterturningpoints,respectively,byV(s)=E0onthetwoextremesofthepath.Thecollectivegroundstateenergy,E0,wasassumedtobeequalto1MeV.dsistheelementoflengthalongL(s).TheexpressionforMis[8,157{159]M(s)=XijMijdqidsdqjds;(4.6)whereMijarethecomponentsofthemulti-dimensionalcollectiveinertiatensor,whichcanbecalculatedbyusingthenon-perturbativecrankingapproximation[160].Theqiarecollectivevariables.57Fromouter-turningpointstoscissionpoints,thetime-dependentpathcanbeapproximatedbysolvingthedissipativeLangevinequations[161,162]:dpidt=pjpk2@@xi(M1)jk@V@xiij(M1)jkpk+gijj(t);dxidt=(M1)ijpj;(4.7)wherexi=qiqiisthedimensionlesscoordinate,qiisthescalingparameter[141,160],pirepresentsthemomentumconjugatetoxi,ijisthedissipationtensor,giji(t)istherandom(Langevin)forcewithj(t)beingatime-dependentstochasticvariablewithaGaussiandistribution,andgijistherandom-forcestrengthtensor.Thisprojectisbasedontheworkin[8],andusethesameparameterstherein.4.4.2Fragmentsidenfor240PuFigure4.13:Outerturningline(thicksolidline)andscissionline(dashedline)onthepotentialenergysurfaceof240Pu.Symbolsindicatetheselectedforwhichsubsequentresultsareshown.58Figure4.14:One-dimensionalplotsofCq"(solidlines)andˆq"(dashedlines)alongthesymmetryaxisforthe(a)and(b)ofFig.4.13.Theblacklinesareresultsforthefragmentedsystems,whilethegreenandredlinesdenotetheresultsforfragments.Theouterturninglineandscissionlinefor240PuareshowninFig.4.13.Thescissionlinecorrespondstothedeformationwherespatialnucleondensitiesofthefragmentsarewellseparated.WecalculatetheNLFsonthisscissionline.Correspondingone-dimensionalplotsofCq˙andˆq˙alongthesymmetryaxisareshowninFig.4.14forthetwoasindicatedinFig.4.13by(a)and(b).Evidently,boththeCq˙andˆq˙ofthefragmentedsystemsareindistinguishablefromthoseofthecorrespondingfragmentscalculatedseparately.TheoscillationsinCq˙uniquelydeterminetheclusterstructureoffragmentsthatareabsentinthescalarparticledensityobtainedwithˆq˙.Therefore,theNLFcanbeusedasamoreappropriateprobetoidentifythefragments.Thescenariochangesappreciablyattheouterturningline.Wetake(1),(5)and(11)asexamplestoillustratethis.OnlyatlowervaluesofQ30(1)ofFig.4.13),wheretheouterturninglineisveryclosetothescissionpoint,awelld59Figure4.15:Contoursofˆn(toprow)andCn"of240Pu(secondrow).ThelastrowgivesacomparisonofonedimensionalCn"(thicksolidlines)withˆn(thickdashedlines).Thethreecolumnscorrespondto(1),(5),and(11)ofFig.4.13.Figure4.16:SameasFig.4.15,butforprotons.neckcanbevisualizedfromthenucleondensityasshowninFig.4.15and4.16forneutronsandprotons,respectively.However,itisprematuretopredictthefragmentswithˆat(5)and(11).Next,wecomputeCq˙forallthreeAsshowninFig.4.15and4.16,Cq˙calculatedforeachoftheshowsremarkably60oscillatingpatternsandrevealsmorestructureinformationascomparedtothedensitiesofthesystem.Inongoingwork,wewillintegrateovertheouterpartsofthedensityforthreecon-toobtaintheverynascentfragments.Bycomparingthelocalizationscalculatedforthefragmentswiththoseforsystems,theneckswillbeclearly4.5SummaryInthiswork,wepresenteddevelopmentspertainingtothetheoreticaldescriptionofnucleonicclusteringinlightandheavynucleiusingtheconceptofnucleonlocalization.FollowingaschematicHOanalysis,wecarriedoutself-consistentDFTcalculationsforlightN=Znucleiandheavysystems.Wedemonstratedthatnucleonlocalizationisasuperbindicatorofclusteringinlightandheavynuclei;thecharacteristicpatternsofNLFscanserveastsofthesingle-particleshellstructureassociatedwithclusterInparticular,theNLFsof8Be,12C,16O,20Neand24Mgcanserveasexcellentindicatorsofclustering.Here,wefoundthattheresultsofrealisticHFcalculationsforNLFsarenottlytfromtheresultsofthedeformedHOmodel.ThisresultsuggeststhatthedetailsoftheEDFareperhapsnotthatimportantforthestructureofclusterinlightnuclei,asthegeometricpropertiesofs.p.orbitsrobustlyfollowtheHOdescription.WhilethecharacteristicoscillatingpatternsoftheNLFmagnifyclusterstructuresinlightnuclei,shellofnascentfragmentsinnucleialsoleaveastrongimprintonthelocalization.OurDFTanalysisofevolutionof232Th,264Fmand240Pudemonstratesthatthefragmentsareformedfairlyearlyintheevolution,andtheidenofthesefragmentsbeforescissionisverypromising.61Wealsoevaluatedthenucleonlocalizationlevelof240Pualongtheouter-turningline.Foreachconsideredontheouter-turningline,astochasticLangevindynamicssimulationwillbeperformedtotheaveragepathstothescissionline.Thiswillallowustoidentifyfragmentsbeforescission.62Chapter5ConclusionandOutlookThisdissertationfocusedonseveralapplicationsofnuclearDFTtocollectivestatesinatomicnuclei.TwoapplicationsofnuclearDFTtonuclearrotationwerepresented.First,shapechangeswithrotationintriaxial106;108Moand108;110;112Ruwerestudiedaswellasangularmomen-tumalignmenttorotationalfrequency.Thecomputedtransitionquadrupolemomentswerecomparedtoexperiment.Second,for160Ybwithtriaxialdeformationathighspins,nuclearDFTwasemployedtostudystronglydeformedbandsthroughTACcalculations.TheKerman-OnishiconditionwasusedtoconstrainthenucleustostayinthePAsystem.Theresultsshowedthat,at~!ˇ0:5MeV,thesoftRouthiancurvewithrespecttothetiltedangleisindicativeofthepossibleappearanceoflarge-amplitudecollectivemotionassociatedwiththedirectionofangularmomentumvector;withincreasingrotationalfrequency,oneofthetwocompetingrotations(attiltedangle0and90)becamefavoredenergeticallyandtheotherbecameanunphysicalsaddlepoint.Tofurtherunderstandthepossiblelarge-amplitudecollectivemotion,methodsbeye.g.thegeneratorcoordinatemethod[18],mustbeemployed.Asshowninthethirdproject,spatiallocalizationprovedtobeanexcellentindicatorofclusterstructuresandwavefunctioncorrelations.Apossiblefutureresearchtopiccouldbetheevaluationofnucleonlocalizationinnucleiunderrotation,whichisexpectedtoshow63changesresultingfromangularmomentumalignmentinthepresenceofbrokentimereversalsymmetryandthecurrenttermintheNLF.Inourlastproject,weemployedspatiallocalizationtoidentifyfragmentsbeforescissionin240Pu.Thenucleonlocalizationsatselectedpointsontheouter-turninglineandscissionlinewerecomputed.Thedegreeoflocalizationinfragmentsalongthepathwayiscloselyrelatedtothefragments'shellstructures.Therefore,theectivenessofidentifyingfragmentswithspatiallocalizationisstillunderinvestigation.Insummary,thisdissertationhasshownthatnuclearDFTcanbesuccessfullyemployedtostudycollectiveandsingle-particleinnuclei.However,theapplicationsarenotlimitedtothosepresentedhere.Otherapplicationsincludethestudyofthezero-energyNambu-Goldstonemode[163,164],neutrinolessdouble-betadecaywithbeyDFT[165,166]andnuclearpastaphases[30,167]withtime-dependentDFT,etc.Theexperiencehasgivenusthectoexploretheseproblems.64Chapter6ListofPublicationsandMyContributions1:YueShi,C.L.Zhang,J.Dobaczewski,andW.Nazarewicz,\Kerman-Onishicondi-tionsinself-consistenttilted-axis-crankingmecalculations",Phys.Rev.C88,034311(2013).PerformedPACandTACcalculationstoexplainTSDbandsin160Yb.ProducedFig.4inthepaper.2:C.L.Zhang,G.H.Bhat,W.Nazarewicz,J.A.Sheikh,andYueShi,\Theoreticalstudyoftriaxialshapesofneutron-richMoandRunuclei",Phys.Rev.C92,034307(2015).PerformedallthenuclearDFTcalculations.ProducedtheplotsshowingnuclearDFTresults.Wrotethedraftofthepaper(exceptforSec.II(B)andIII(B)).3:C.L.Zhang,B.Schuetrumpf,andW.Nazarewicz,\Nucleonlocalizationinlightandheavynuclei",arXiv:1607.00422(2016).Implementedtheaxialharmonicoscillatormodeltocalculatethenucleonlocalization.Implementednucleonlocalizationextensionsinhfbthoandhfodd.Performedallthecalculationsshowninthispaper.65DevelopedaPythoncodetoproducealltheplotsshowninthispaper.Wrotethedraftofthepaper.4:B.Schuetrumpf,C.L.Zhang,andW.Nazarewicz,\Clusteringandpastaphasesinnucleardensityfunctionaltheory",arXiv:1607.01372(2016).Performednucleonlocalizationcalculationsforlightandheavynucleiwithhfbthoandhfodd.ProducedalltheplotsinSec.5.5:JhilamSadhukhan,C.L.Zhang,W.NazarewiczandNicolasSchunk,\Nucleonlocal-ization{anewtooltoidentifyfragments",inpreparation.Providedthenucleonlocalizationmoduleinhfoddtoperformedrelatedcalculations.Producedtheplotsshowinglocalizationresults.6:R.NavarroPerez,N.Schunck,R.Lasseri,C.L.ZhangandJ.Sarich\AxiallydeformedsolutionoftheSkyrme-Hartree-Fock-Bogolyubovequationsusingthetransformedhar-monicoscillatorbasis(III)hfbthov3.00:anewversionoftheprogram",inprepa-ration.Implementedthenucleonlocalizationmoduleinhfbtho.Providedthetestedinputandoutputforlocalizationcalculations.66APPENDICES67AppendixAHohenberg-KohntheoremsHohenberg-KohntheoremI:Therecannotbetwoexternalpotentials,Uext;1andUext;2(Uext;26=Uext;1+const)thatgivethesameelectrondensity.(i.e.givenanexternalpotential,thedensityisuniquelydetermined.)ProofI:Thiscanbeprovenwithreductioadabsurdum.Withtwoexternalpotentials,theHamiltonianscanbewrittenas:^H1=^T+^V+^Uext;1;(A.1a)^H2=^T+^V+^Uext;2:(A.1b)BydiagonalizingtheHamiltonians,thegroundeigenstatesaregivenasj1iandj2i,re-spectively,withgroundeigenvaluesE1andE2.Assumingthatthegroundstateisnon-degenerateandthesamedensity,ˆ(r),isgivenbytwoexternalpotentials,then,E1h1j^H1j1i<h2j^H1j2i=E2+Zdr[Uext;1(r)Uext;2(r)]ˆ(r);(A.2)E2h2j^H2j2i<h1j^H2j1i=E1Zdr[Uext;1(r)Uext;2(r)]ˆ(r):(A.3)68AddingA.2andA.3,weobtainE1+E2<E2+E1;(A.4)whichisacontradiction.Thus,thetheoremhasbeenproven.Forthedegeneratecase,thetheoremshouldbeslightlymoed:therecannotbetwoexternalpotentials,Uext;1andUext;2(Uext;26=Uext;1+const)thatgivethesameclassofelectrondensities.Moredetailscanbefoundin[168].Hohenberg-KohntheoremII:AuniversalfunctionalfortheenergyE[ˆ(r)]canbeintermsoftheelectrondensity,andisminimizedwhenˆ(r)isthegroundstatedensity.ProofII:Theenergyoftheinteractingsystemcanbewrittenas:E=hj^Hji=hj^T+^V+^Uextji:(A.5)Sincethedensityisuniquelydeterminedandweassumeanon-degenerategroundstate,thegroundstatewavefunctionisalsouniquelydetermined.Thus,theenergyintermsofthewavefunctioncanbetransformedintothefunctionalofthedensity:E[ˆ(r)]=F[ˆ(r)]+ZUext(r)ˆ(r)dr:(A.6)Thetermisauniversalfunctionalbecausethetreatmentofboththekineticandpotentialenergiesisuniversaloveralltheinteractingsystems.Whenˆ(r)isgroundstatedensity,thegroundstatewavefunctionisuniquelydetermined69asjg:s:i,thenE[ˆg:s:]=hg:s:j^Hjg:s:i<h2j^Hj2i=E[ˆ2];(A.7)whereˆ2isatdensityfromthegroundstate,andj2iisthecorrespondingwavefunction.Therefore,theenergyisminimizedwhenˆistheexactgroundstatedensity.70AppendixBSpatialLocalizationTheconditionalprobabilityofanucleonatr1whenweknowwithcertaintythatanothernucleonwiththesamespinandisospinisatris[23]Rq˙(r;r1)=ˆq˙(r1)jˆq˙˙(r;r1)j2ˆq˙(r):(B.1)Sinceweareonlyinterestedinthelocalizationpartofthisprobability,itisttoconsiderthelocalshort-rangebehavioroftheconditionalprobability,whichcanbeobtainedbyperformingasphericalaveragingoverashellofradiusaboutthereferencepointr.WeperformtheTaylorexpansioninthecoordinateaboutthereferencepointr[169]:Rq˙(r;r+)=e1Rq˙(r;r1)jr1=r:(B.2)ThesphericalaverageoftheTaylorexpansionisgivenashe1i=14ˇZe1d=sinh(r1)r1=1+13!2r21+15!4r41+17!6r61+:(B.3)SinceRq˙(r;r1)jr1=r=0;(B.4)71andr21jˆ˙(r;r1)j2ˆ˙(r)jr1=r=r2ˆ˙(r)2˝˙(r)+12jrˆ˙(r)j2ˆ˙(r)+2j2˙(r);(B.5)onecanobtainRq˙(r;)ˇ13 ˝q˙14jrˆq˙j2ˆq˙j2q˙ˆq˙!2+O(3):(B.6)Ifthisconditionalprobabilityissmall,itmeansthatonenucleonwithspinandisospinishighlylocalizedinthespace.Therefore,thespatiallocalizationhasaninverserelationwiththeprobability.Tomakethelocalizationnormalizedanddimensionless,thespatiallocalizationcanbeasCq˙(r)="1+˝q˙ˆq˙14jrˆq˙j2j2q˙ˆq˙˝TFq˙2#1:(B.7)Proofof(B.5):r21jˆ˙(r;r1)j2jr1=r=r1[r1ˆ˙(r;r1)ˆ˙(r;r1)+ˆ˙(r;r1)r1ˆ˙(r;r1)]jr1=r=[r21ˆ˙(r;r1)ˆ˙(r;r1)+2r1ˆ˙(r;r1)r1ˆ˙(r;r1)+ˆ˙(r;r1)r21ˆ˙(r;r1)]jr1=r;(B.8)72where,1stterm+3rdterm:=[r21ˆ˙(r;r1)ˆ˙(r)+ˆ˙(r)r21ˆ˙(r;r1)]jr1=r=ˆ˙(r)[r21ˆ˙(r;r1)+r21ˆ˙(r;r1)]jr1=r=ˆ˙(r)[r21ˆ˙(r;r1)+r2ˆ˙(r;r1)]jr1=r=ˆ˙(r)(r21+r2)ˆ˙(r;r1)jr1=r=ˆ˙(r)[(r1+r)22r1r]ˆ˙(r;r1)jr1=r=ˆ˙(r)[r2ˆ˙(r)2˝˙(r)]];(B.9)and,2ndterm:=2r1ˆ˙(r;r1)r1ˆ˙(r1;r)jr1=r=12[r1ˆ˙(r;r1)+r1ˆ˙(r1;r)]212[r1ˆ˙(r;r1)r1ˆ˙(r1;r)]2jr1=r=12[r1ˆ˙(r;r1)+rˆ˙(r;r1)]212[r1ˆ˙(r;r1)rˆ˙(r;r1)]2jr1=r=12[rˆ˙(r)]212[2ij˙(r)]2=12[rˆ˙(r)]2+2j˙(r)2;(B.10)then,r21jˆ˙(r;r1)j2ˆ˙(r)jr1=r=r2ˆ˙(r)2˝˙(r)+12jrˆ˙(r)j2ˆ˙(r)+2j2˙(r):(B.11)73BIBLIOGRAPHY74BIBLIOGRAPHY[1]J.DobaczewskiandP.Olbratowski,\Solutionoftheskyrme{hartree{fock{bogolyubovequationsinthecartesiandeformedharmonic-oscillatorbasis.(iv)HFODD(v2.08i):anewversionoftheprogram,"Comput.Phys.Commun.,vol.158,no.3,pp.158{191,2004.[2]H.Huaetal.,\Triaxialityandthealignedh112neutronorbitalsinneutron-richZrandMoisotopes,"Phys.Rev.C,vol.69,p.014317,Jan2004.[3]Q.H.Luetal.,\Structureof108;110;112Ru:Identicalbandsin108;110Ru,"Phys.Rev.C,vol.52,pp.1348{1354,Sep1995.[4]C.Y.Wu,H.Hua,D.Cline,A.B.Hayes,R.Teng,D.Riley,R.M.Clark,P.Fallon,A.Goergen,A.O.Macchiavelli,andK.Vetter,\Evidenceforpossibleshapetransitionsinneutron-richRuisotopes:Spectroscopyof109;110;111;112Ru,"Phys.Rev.C,vol.73,p.034312,2006.[5]S.Raman,C.Malarkey,W.Milner,C.N.Jr.,andP.Stelson,\Transitionprobability,b(e2)",fromthegroundtothe2+stateofeven-evennuclides,"At.DataNucl.DataTables,vol.36,no.1,pp.1{96,1987.[6]J.Snyderetal.,\High-spintransitionquadrupolemomentsinneutron-richMoandRunuclei:Testingsoftness?,"Phys.Lett.B,vol.723,no.1,pp.61{65,2013.[7]J.D.McDonnell,W.Nazarewicz,andJ.A.Sheikh,\Thirdminimainthoriumanduraniumisotopesinaself-consistenttheory,"Phys.Rev.C,vol.87,p.054327,May2013.[8]J.Sadhukhan,W.Nazarewicz,andN.Schunck,\Microscopicmodelingofmassandchargedistributionsinthespontaneousof240Pu,"Phys.Rev.C,vol.93,no.1,p.011304,2016.[9]J.Carlson,\Green'sfunctionmontecarlostudyoflightnuclei,"Phys.Rev.C,vol.36,no.5,p.2026,1987.[10]S.C.PieperandR.B.Wiringa,\Quantummontecarlocalculationsoflightnuclei,"arXivpreprintnucl-th/0103005,2001.75[11]P.Naatil,J.Vary,andB.R.Barrett,\Large-basisabinitiono-coreshellmodelanditsapplicationto12C,"Phys.Rev.C,vol.62,no.5,p.054311,2000.[12]P.Naatil,S.Quaglioni,I.Stetcu,andB.R.Barrett,\Recentdevelopmentsinno-coreshell-modelcalculations,"J.Phys.G:Nucl.Part.Phys.,vol.36,no.8,p.083101,2009.[13]B.R.Barrett,P.Naatil,andJ.P.Vary,\Abinitionocoreshellmodel,"Prog.Part.Nucl.Phys.,vol.69,pp.131{181,2013.[14]N.Michel,W.Nazarewicz,M.P 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