Time-dependent relativistic mean-field theory

arXiv:nucl-th/9709061v126Sep1997TIMEDEPENDENTRELATIVISTICMEAN-FIELDTHEORYDarioVretenar∗Physik-DepartmentderTechnischenUniversit¨atM¨unchenAbstractTherelativisticmean-fieldtheoryprovidesaframeworkinwhichthenuclearmany-bodyproblemisdescribedasaself-consistentsystemofnucleonsandmesons.Inthemean-fieldapproximation,theself-consistenttimeevolutionofthenuclearsystemdescribesthedynamicsofcollectivemotion:doublegiantresonances,nuclearcom-pressibilityfrommonopoleresonances,regularandchaoticdynamicsofisoscalarandisovectorcollectivevibrations.1IntroductionandoutlineofthemodelRelativisticmean-field(RMF)modelshavebeensuccessfullyappliedincalculationsofnu-clearmatterandpropertiesoffinitenucleithroughouttheperiodictable.Intheself-consistentmean-fieldapproximation,detailedcalculationshavebeenperformedforavarietyofnuclearstructurephenomena[1].InthepresentworkwereviewtheapplicationsofRMFtothedynamicsofcollectivevibrationsinsphericalnuclei.Inrelativisticquantumhadro-dynamics[2],thenucleusisdescribedasasystemofDiracnucleonswhichinteractthroughtheexchangeofvirtualmesonsandphotons.TheLagrangiandensityofthemodelisL=¯ψ(iγ·∂−m)ψ+12(∂σ)2−U(σ)−14ΩμνΩμν+12m2ωω2−14~Rμν~Rμν+12m2ρ~ρ2−14FμνFμν−gσ¯ψσψ−gω¯ψγ·ωψ−gρ¯ψγ·~ρ~τψ−e¯ψγ·A(1−τ3)2ψ.(1)TheDiracspinorψdenotesthenucleonwithmassm.mσ,mω,andmρarethemassesoftheσ-meson,theω-meson,andtheρ-meson,andgσ,gω,andgρarethecorrespondingcouplingconstantsforthemesonstothenucleon.U(σ)denotesthenonlinearσself-interactionU(σ)=12m2σσ2+13g2σ3+14g3σ4,(2)1AlexandervonHumboldtFellow,onleaveofabsencefromUniversityofZagreb,Croatia1andΩμν,~Rμν,andFμνarefieldtensors[2].ThecoupledequationsofmotionarederivedfromtheLagrangiandensity(1).TheDiracequationforthenucleons:i∂tψi=α−i∇−gωω−gρ~τ~ρ−e(1−τ3)2A!+β(m+gσσ)+gωω0+gρ~τ~ρ0+e(1−τ3)2A0#ψi(3)andtheKlein-Gordonequationsforthemesons:∂2t−Δ+m2σσ=−gσρs−g2σ2−g3σ3(4)∂2t−Δ+m2ωωμ=gωjμ(5)∂2t−Δ+m2ρ~ρμ=gρ~jμ(6)∂2t−ΔAμ=ejemμ.(7)Intherelativisticmean-fieldapproximation,thenucleonsdescribedbysingle-particlespinorsψi(i=1,2,...,A)areassumedtoformtheA-particleSlaterdeterminant|Φi,andtomoveindependentlyintheclassicalmesonfields.Thesourcesofthefields,i.e.den-sitiesandcurrents,arecalculatedintheno-seaapproximation[3]:thescalardensity:ρs=PAi=1¯ψiψi,theisoscalarbaryoncurrent:jμ=PAi=1¯ψiγμψi,theisovectorbaryoncurrent:~jμ=PAi=1¯ψiγμ~τψi,theelectromagneticcurrentforthephoton-field:jμem=PAi=1¯ψiγμ1−τ32ψi.ThesummationisoveralloccupiedstatesintheSlaterdetermi-nant|Φi.Negative-energystatesdonotcontributetothedensitiesintheno-seaapproxi-mationofthestationarysolutions.Itisassumedthatnucleonsingle-particlestatesdonotmixisospin.Thegroundstateofanucleusisdescribedbythestationaryself-consistentsolutionofthecoupledsystemofequations(3)–(7),foragivennumberofnucleonsandasetofcouplingconstantsandmasses.Thesolutionforthegroundstatespecifiespartoftheinitialconditionsforthetime-dependentproblem.Thedynamicsofnuclearcollectivemotionisanalyzedintheframeworkoftime-dependentrelativisticmean-fieldmodel,whichrepresentsarelativis-ticgeneralizationofthetime-dependentHartree-Fockapproach.Foragivensetofinitialconditions,i.e.initialvaluesforthedensitiesandcurrents,nucleardynamicsisdescribedbythesimultaneousevolutionofAsingle-particleDiracspinorsinthetime-dependentmeanfields.FrequenciesofeigenmodesaredeterminedfromaFourieranalysisofdynamicalquan-tities.Inthismicroscopicmodel,self-consistenttime-dependentmean-fieldcalculationsareperformedformultipoleexcitations.Anadvantageofthetime-dependentapproachisthatnoassumptionaboutthenatureofaparticularmodeofvibrationshastobemade.Retar-dationeffectsforthemesonfieldsarenotincludedinthemodel,i.e.thetimederivatives∂2tintheequationsofmotionsforthemesonfieldsareneglected.Thisisjustifiedbythelargemassesinthemesonpropagatorscausingashortrangeofthecorrespondingmesonexchangeforces.Negativeenergycontributionsareincludedimplicitlyinthetime-dependentcalcu-lation,sincetheDiracequationissolvedateachstepintimeforadifferentbasisset[3].2Negativeenergycomponentswithrespecttotheoriginalground-statebasisaretakenintoaccountautomatically,evenifateachtimesteptheno-seaapproximationisapplied.Thedescriptionofnucleardynamicsasatime-dependentinitial-valueproblemisintrinsi-callysemi-classical,sincethereisnosystematicproceduretoderivetheinitialconditionsthatcharacterizethemotionofaspecificmodeofthenuclearsystem.Thetheorycanbequantizedbytherequirementthatthereexisttime-periodicsolutionsoftheequationsofmotion,whichgiveintegermultiplesofPlanck’sconstantfortheclassicalactionalongoneperiod[4].Forgiantresonancesthetime-dependenceofcollectivedynamicalquantitiesisactuallynotperiodic,sincegenerallygiantresonancesarenotstationarystatesofthemean-fieldHamiltonian.Thecouplingofthemean-fieldtotheparticlecontinuumallowsforthedecayofgiantresonancesbydirectescapeofparticles.Inthelimitofsmallamplitudeos-cillations,however,theenergyobtainedfromthefrequencyoftheoscillationcoincideswiththeexcitationenergyofthecollectivestate.InRefs.[3,4,5]wehaveshownthatthemodelreproducesexperimentaldataongiantresonancesinsphericalnuclei.2Dynamicsofcollectivevibrations2.1D