AQUANTALTRANSPORTTHEORYFORNUCLEARCOLLECTIVEMOTIONT

AQUANTALTRANSPORTTHEORYFORNUCLEARCOLLECTIVEMOTION:THEMERITSOFALOCALLYHARMONICAPPROXIMATIONHelmutHOFMANNPhysik-Department,T30,TUMiinchen,85747Garching,GermanyELSEVIERAMSTERDAM-LAUSANNE-NEWYORK-OXFORD-SHANNON-TOKYOs__-_Fl!iBPHYSICSREPORTSELSEWIERPhysicsReports284(1997)137-380Aquanta1transporttheoryfornuclearcollectivemotion:ThemeritsofalocallyharmonicapproximationHelmutHofmannPhysik-Department,T30,TUMiinchen,85747Garching,Grrman~ReceivedNovember1996;editor:G.E.BrownContents1.Introduction1.1.Abriefhistoricalviewonnuclearcollectivemotion1.2.Thescopeofthepresentwork2.Theapproximationscheme2.I.Theutilityofpropagators2.2.ThebeautyofGaussiansolutions2.3.Numericalsolutionsforglobalmotion2.4.Locallinearization3.Themanyfacetsofharmonicmotion3.1.Averagedynamics3.2.Thecollectivedegreesoffreedomasquantumobjects3.3.Dynamicalfluctuationsofthecollectivedegreesoffreedom3.4.Theinvertedoscillator4.Alongtheroadtoreality4.1.AnappropriateHamiltonianforintrinsicmotion4.2.Microscopicoriginofmacroscopicdamping4.3.Thermodynamicproblems4.4.Averagedynamicsinmulti-dimensionalspaces5.Consequencesandapplicationstorealisticsituations1401411441451451471541561591595.1.Strengthdistributionofisoscalarvibrationsatfinitetemperature5.2.Transportcoefficientsforslowcollectivemotion5.3.Onthemacroscopiclimitofcollectivemotion5.4.Dynamicsofmeta-stablecollectivestates6.Concludingremarks6.1,Ashortsummaryofthemethodandthemostbasicfeatures6.2.RemarksonLandau-VlasovandBUUapproaches1806.3.OpenproblemsAppendixA191213228A.1,Thequasi-staticpictureofthermodynamics,adaptedtonuclearcollectivevariablesA.2.Linearresponsetheorywithinwalkingdistance228A.3.ExploitingthetechniqueofGreen’sfunctions233A.4.TheNakajima-Zwanzigprojection255technique275AS.WignertransfonnsA.6.TheStrutinskysmoothingprocedureReferences2792792892993103213213243263213273303463543613673730370-1573/97/$32.00Copyright01997ElsevierScienceB.V.AilrightsreservedPIISO370-1573(97)00006-9H.HofinannIPhysicsReports284(1997)137-380139AbstractAtransporttheoryisdevelopedforcollectivemotionofsystemssuchasanatomicnucleus,whichmaybeconsideredasatypicalrepresentativeofaself-boundmicro-system.Albeitforpragmaticreasons,collectivevariablesareintroducedasshapeparameters,self-consistencywithrespecttothenucleonicdegreesoffreedomhasbeenimplementedatvariousimportantstages.ThisfeatureleadstosubsidiaryconditionswhichareobeyedlocallyforboththeaveragemotionaswellasforthequantizedHamiltonianconstructedthroughaBohm-Pinesprocedure.Furthermore,self-consistencygovernsthedefinitionofthetransportcoefficientsappearingintheequationsforcollectivemotion,Thelatterisassociatedtothetimeevolutionofthedensityincollectivephasespace,forwhichtheconceptoftheWignerfunctionisemployed.Globalmotionisdescribedbypropagatingthesysteminsuccessivetimelapswhicharemacroscopicallysmall,butmicroscopicallylarge.Thisenablesonetoexploitlinearizationproceduresandtotakeadvantageofthebenefitsoflinearresponsetheory.Amicroscopicdampingmechanismisintroducedbydressingtheenergiesoftheindependentparticlemodelbycomplexself-energies,theparametersofwhicharedeterminedfromopticalmodelconsiderations.Numericalevaluationsoftransportcoefficientsaredescribedandtestedforthecaseoffissioninthelightofrecentexperimentalfindings,ThetheoryallowsonetoextendbothKramers’pictureofthisprocessaswellashisequationforthedensitydistributionintothequantumregime.Thesequantumeffectsareseentoshowupatsmallerexcitations,saybelowTNl-l.5MeVincasethattheconceptoftemperaturemaybeused.ThisextensionispossibledowntoacriticaltemperatureT,belowwhichforunstablemodesasencounteredformotioninbarrierregions,diffusioncoefficientscannolongerbedefined.Forlargedampingther,isonlyslightlygreaterthantheso-called“crossovertemperature”knownfromconventionaltreatmentsofdissipativetunneling,butther,issmallonthenuclearscale.Lastbutnotleast,relationsareestablishedtootherapproaches,suchas“dissipativediabaticdynamics”,oronesbasedonrandommatrixmodelsaswellasthoseemployingwallfrictionandtwobodyviscosity.PACS:5.4O.fj;6.6O.+w;21.60.-n;24.10.PaKeybclords;Brownianmotionforself-boundsystems;Quanta1dynamicsoffluctuations;Nucleardissipation;CollectiveandnucleonicmotionatfiniteT140H.HojinannIPhysicsReports284(1997)137-3801.IntroductionThegoalofanytransporttheoryistoreducethedescriptionofthetimeevolutionofacomplexsystemtooneofasmallsubsetofitsdegreesoffreedom.Thedynamicsoftheresidualsetisnotconsideredindetailbutinsomeaveragesenseonly.Oftenthefirstclassofvariablesisreferredtoasthecollectiveor“macroscopic”ones,whereastherestisreferredtoasthe“intrinsic”system.Thenotion“macroscopic”indicatesthatinmanycasesthesevariablesrepresentmacroscopicallylargequantitieswhosedynamicscanbepicturedasatransportofmatterinarealsense.Letusbrieflydescribetwomajorexamplesofthistype.ConsiderfirstthetypicalcaseofBrownianmotionwhereaheavyobjectofmassMisimmersedinsomefluidconsistingofmoleculesofmassmM.Themacroscopicvariablesinthiscasearegivenbythevectorsforpositionandmomentumoftheparticle.Usually,theimpactofitsdynamicsonthefluidisofnointerest.Itisofgreatimportance,howeve