A New Nonlinear Liquid Drop Model. Clusters as Sol

arXiv:nucl-th/9612006v12Dec1996NonlinearLiquidDropModel.CnoidalWavesAndreiLudu1,AureliuS˘andulescu2andWalterGreinerInstitutf¨urTheoretischePhysikderJ.W.GoetheUniversit¨at,D-60054FrankfurtamMain,GermanyAbstractByintroducinginthehydrodynamicmodel,i.e.inthehydrodynamicequa-tionsandthecorrespondingboundaryconditions,thehigherordertermsinthedeviationoftheshape,weobtaininthesecondordertheKortewegdeVriesequation(KdV).Thesameequationisobtainedbyintroducingintheliquiddropmodel(LDM),i.e.inthekinetic,surfaceandCoulombterms,thehighertermsinthesecondorder.TheKdVequationhasthecnoidalwavesassteady-statesolutions.Thesewavescoulddescribethesmallanharmonicvibrationsofspheri-calnucleiuptothesolitarywaves.Thesolitonscoulddescribethepreformationofclustersonthenuclearsurface.Weapplythisnonlinearliquiddropmodeltothealphaformationinheavynuclei.Wefindanadditionalminimuminthetotalenergyofsuchsystems,correspondingtothesolitonsasclustersonthenuclearsurface.Byintroducingtheshelleffectswechoosethisminimumtobedegener-atedwiththegroundstate.Thespectroscopicfactorisgivenbytheratioofthesquareamplitudesinthetwominima.PACSnumbers:23.60.+e,21.60.Gx,24.30.-v,25.70.ef.1PermanentAddress:BucharestUniversity,Bucharest-Magurele,PO.BoxMG-5211,Romania;e-mail:[e-mail]2PermanentAddress:InstituteofAtomicPhysics,DepartmentofTheoreticalPhysics,Bucharest-Magurele,P.O.BoxMG-6,Romania;e-mail:[e-mail],CaleaVictoriei125,71102Bucharest,Romania;E-mail:[e-mail]11IntroductionItiswellknownthatliquiddropmodel,asacollectivemodelofthenucleusdescribesexcelentlythespectraofsphericalnucleiassmallvibrations(harmonicinthelinearapproximationoranharmonicinhigherapproximations)aroundtheirshape.Ontheotherhanditisknownthatonthenuclearsurfaceofheavynucleiclosetothemagicnuclei(208Pb,100Sn)alargeenhancementofclusters(alpha,carbon,oxigen,neon,magnesium,silicon)existswhichleadstotheemissionofsuchclustersasnaturaldecays[1,2].Itisalsoclearthattraditionalcollectivemodels[3]arenotabletogiveacompleteexplanationofsuchnaturaldecays,i.e.theystilldidnotcompletelyanswertothemainphysicalquestion:whyshouldnucleonsjointogetherandspontaneousformanisolatedclusteronthenuclearsurface?Onlybytheintroductionintheshellmodelofmanybodycorrelationeffectswecouldformanisolatedbump,stableintime.Itispossibletodescribetheformationofsuchclustersinacollectivemodel[4,5]?Inthepresentpaperbyintroducingthenonlinearitiesintheliquiddropmodelwesuccededtogiveapositiveanswertothisproblem.Inanonlinearliquiddropmodelwecandescribesimultaneously,bycnoidalwavesthetransitionsfromsmallvibrationstotheformationofsolitons.Theexperimentaldiscoveryandthetheoreticalfoundationofsolitons[6]asnon-dispersivelocalizedwavesmovinguniformly,leadtoapowerfulltheoryofclassicalfieldequationswithsuchsolutions[7-9].Alsoitwaspossibletoexplainthecorrespondencebetweenclassicalsolitonsolutionsandtheextended-particlestatesofthequantizedversionofthetheory.Thisleadtoageneralisationofthesemiclassicalexpansionofquantummechanicsinquantumfieldtheory.Inthelastcasesolitons(andbreathersorinstantonsolutions)arenon-perturbative.Ofcourse,theabovemethodsworkonlyiftheinitialphysicalmodelallowsfirsttheexistenceofsomenon-trivialclassicallocalizedsolutions.Fromthemathematicalphysicspointofviewsolitons,assolutionsofnon-linearevolutionequations,areisolatedwaveswhichpreservetheirshapeandhavefiniteandlocalizedenergydensity.Recentlyhasbeenshownthatsolitonsmaybealsorelevant2innuclearphysics[4,5]orparticlephysics[10].Alsoitwasrealizedthatmanyfieldtheoreticalmodelsforparticleinteractions,andevenforquantumextendedparticles[11],possesssolitonorbreathersolutionsandthatthesolitonsoughttobeinterpretedasadditionalparticle-likestructuresintheory.ThetravelingsolutionsoftheKdVequation[7]andsingularsolutionshavingpolesat±∞.Recently,inordertodescribethequasimolecularspectra,wehaveintroducedin[5]aone-dimensionalsolitonmodelforthecluster(alphaparticle)andtherotator-vibratormodelforthenucleus.Anexcelentagreementwiththeexperimentaldatawasobtained.Weconcludethatthenonlineartermsleadtonewqualitativepictureoftheliquiddropmodel,i.e.theharmonicoscillationsgrowintoanharmoniconeswhichcanleadtoastablesolitonconfiguration.Boththepotentialpictureandthephasespaceportraitssupportthisbehaviour.WestressthatalltheseresultsareembeddedintoaHamiltonianformalism.Inthepresentpaperwefirstintroduceinthehydrodynamicalmodelthehighertermsinthedeviationofthesphericalshape.WehaveshownthatinthethirdorderweobtaintheKortewegdeVriesequation(KdV).Inthefollowingchapter3byintroducingintheliquiddropmodelthethirdordertermsinthedeviationoftheshapeweobtainedthesameKdVequationasinthenonlinearhydrodynamicalmodel.WeshouldliketostressthatthesenonlinearequationsareHamiltonianeqautionswhichdescribethetotalenergyofthesystem.Chapter4containsthenonlinearsolutions:cnoidalwavesandthesingularsolutionsforKdVequation.Lastchapterdiscusstheapplicationofnonlinearliquiddropmodeltothealphapreformationfactors.2ThenonlinearhydrodynamicmodelTherearetwopossiblewaystodescribetheclassicaldynamicsofa