Semiclassical evaluation of average nuclear one an

arXiv:nucl-th/0206065v27Apr2003SemiclassicalevaluationofaveragenuclearoneandtwobodymatrixelementsX.Vi˜nas1,P.Schuck2,M.Farine3andM.Centelles11Departamentd’EstructuraiConstituentsdelaMat`eria,FacultatdeF´ısica,UniversitatdeBarcelona,Diagonal647,08028Barcelona,Spain2InstitutdePhysiqueNucl´eaire,IN2P3–CNRS,Universit´eParis–Sud,91406Orsay-C´edex,France3ConsulatG´en´eraldeFrance`aCanton,339HuanShiDongLu,510098Guangzhou,Canton,ChinaAbstractThomas-Fermitheoryisdevelopedtoevaluatenuclearmatrixelementsav-eragedontheenergyshell,onthebasisofindependentparticleHamiltonians.One-andtwo-bodymatrixelementsarecomparedwiththequantalresultsanditisdemonstratedthatthesemiclassicalmatrixelements,asfunctionofenergy,wellpassthroughtheaverageofthescatteredquantumvalues.Fortheone-bodymatrixelementsitisshownhowtheThomas-Fermiapproachcanbeprojectedongoodparityandalsoongoodangularmomentum.Forthetwo-bodycasethepairingmatrixelementsareconsideredexplicitly.PACSnumber(s):21.10Dr,21.60.-n,31.15GyTypesetusingREVTEX1I.INTRODUCTIONThesolutionofthenuclearmany-bodyproblempresentsaformidablechallenge.Notonlybareandeffectivenucleon-nucleonforcesarenotcompletelyknown,butstillforthosegivenasgrantedonehastosolvethemany-bodyproblemofahighlyquantal,stronglyinter-acting,self-bound,andthereforeinhomogeneousFermisystem.Overtheyearssemiclassicaltechniqueshavehelpedtosolvethisproblem,forinstanceinregardtothelatteraspect.InpracticeitismainlytheThomas-Fermi(TF)methodanditsextensionsforthedescriptionofnuclearground-statepropertieswhichhasbeenconsidered(see[1]andreferencestherein).Thenucleardensityandkineticenergydensityarethemainingredientsofthisapproach.Thesemiclassicalapproximationoftengivesadirectphysicalinsight,yieldingtheshellaverageofthequantitiesunderconsiderationandprovidingtheirmaintrend(which,incertaincases,maybeobscuredbystrongshellfluctuations).Aknownexampleise.g.thenuclearbindingenergywhichcoincideswiththeliquiddroppartinthesemiclassicalapproach.Anotherquantityoflongstandinginterestistheaveragesingle-particleleveldensity.ItiswellknownthattheTFapproximationtotheleveldensity(including¯hcorrections)coincidesanalyticallywiththeStrutinskyaveragedquantalleveldensityfortheharmonicoscillator(HO)potential[2].FirstperformingthequantalcalculationandthentheaverageismorecumbersomethancalculatingtheshellaveragedirectlyviatheTFmethod.Thetechnicaladvantageofthelatterbecomessignificantinthedeformedcase[2]or,forinstance,whenonewantstogobeyondtheindependentparticledescriptiontoincludecorrelations[3].TheTFapproachisalsoveryhelpfulforthecalculationofsurfaceandcurvatureenergies.Actually,thelatterquantitycanonlybecorrectlyextractedinasemiclassicalprocedure[4].Ingeneral,however,webelievethatthetruevirtueoftheTFmethodshowsupnotonlyincalculatingaveragepropertiesintheindependentparticleapproximation,whereitcanreplacetheresultsobtainedthroughthemorecumbersomeStrutinskymethod[5],butratherinmany-bodyapplicationsgoingbeyondthemeanfieldorindependentparticlepicturewhereastraightforwardquantumsolutionforfinitesystemsmayreachitslimits.AcasewherewetreatedcorrelationeffectsinTFapproximationwas,asalreadymentioned,theleveldensityparameter[3].Pairingcorrelationsinfinitenucleihavealsoalreadysuccessfullybeentreatedinthepast[6].Thisisoneoftheaspectswhichweshallconsideragaininthisworkinmoredetail.InthisworkwewanttodwellonanaspectofThomas-Fermitheorywhichinthepasthasbeenexploitedonlyverylittle.Thisconcernstheevaluationofmatrixelementsaveragedoveracertainenergyintervalwhichmaybetypicallyoftheorderof¯hω,i.e.,theseparationofmajorshells.Itmustbepointedoutthatthisis,toourknowledge,thefirstattempttoevaluatenotonlyone-bodybutalsotwo-bodymatrixelementsintheTFapproximation.ThissemiclassicalcalculationprovidesthesmoothlyvaryingpartofthematrixelementsdroppingtheshelleffectsaccordingtotheideaoftheStrutinskyaveragingmethod[7].WewilldescribeseveraltestsoftheaccuracyoftheTFmethodforon-shelldensities.Inafirstpartwewilldevelopourapproachforthematrixelementsofsingle-particleoperators,forgivenparityandangularmomentum.Thisgoesalongsimilarlinesalreadydevelopedinthedomainofsystemswithchaoticbehaviour[8–11].Inasecondpartwewilladdressourmainobjective,whichistoshowthatthemethodalsoworksfortwo-bodymatrixelements.Some2preliminaryresultshavebeenpublishedpreviouslyinRef.[12].Asaspecificexamplewewilltreatthepairingmatrixelements.Letusgiveashortsummaryoftheapproachwearegoingtodevelop.ConsiderforexampletheexpectationvalueofasingleparticleoperatorˆOinsomeshellmodelstate|νi:Oν=hν|ˆO|νi=Tr[ˆO|νihν|].(1)InsteadofknowingOνquantumstatebyquantumstateitmaysometimesbeadvantageousandinstructivetoonlyknowhowthematrixelement(1)changesasafunctionofenergy.Wethereforeintroduceasingle-particlematrixelementaveragedovertheenergyshell:O(E)=Tr[ˆOˆρE],(2)wherewecallˆρEthedensitymatrixontheenergyshell.Itisrelatedwiththeso-calledspectraldensitymatrixandwillbedefinedimmediatelybelow.Thespectraldensitymatrixδ(E−ˆH)hasthecharacteristicdiscontinuousbehaviourduetothequantizationoftheeigenvaluesofthesingle-particleHamiltonianˆH.Itcanbewritten,however,asasumofasmo