Quantum-Mechanical Analysis of Single-Particle Lev

arXiv:nucl-th/9711007v317Jan1998Quantum-MechanicalAnalysisofSingleParticleLevelDensity∗I.S¸tet¸cu†InstituteforNuclearPhysicsandEngineering”HoriaHulubei”,P.O.BoxMG-6,76900Bucharest,RomaniaAquantum-mechanicalcalculationofthesingle-particlelevel(s.p.l.)densityg(ε)iscarriedonbyusingtheconnectionwiththesingle-particleGreen’sfunction.TherelationbetweentheimaginarypartofGreen’sfunctionandsingle-particlewavefunctionsisusedseparatelyforthediscreteandcontinuousstates.Withinthebound-statesregiontheimaginarypartoftheGreen’sfunctioniscalculatedbyusingthewronskiantheorem.TheGreen’sfunctioncorrespondingtothecontinuumiswrittenbyusingtheregularandJostsolutionsoftheradialSchr¨odingerequation.Thesmoothpartoftherapidlyfluctuatings.p.l.densityiscalculatedbymeansoftheStrutinskiprocedure.Thecontinuumcomponentofthes.p.l.densityhasratherclosevalueswithineitherexactquantum-mechanicalcalculationswiththeWoods-Saxon(WS)potential,orThomas-FermiapproximationwithWSaswellasfinite-squarepotentialwells,providedthatthefree-gascontributionissubtracted.AsimilartrendisobtainedbymeansofthesimpleFGMformulaforthes.p.l.densityifthecontinuumeffectistakenintoaccount.PACS:21.10.Ma,21.10.Pc,21.60.CsI.INTRODUCTIONThenuclearleveldensityhasacentralroleinthestatisticalanalysisofnuclearreactions.In-formationconcerningnucleardensitiescanbeobtainedfromexperimentsbymeansoftheanalysisofneutronandchargedparticlesresonances,inelasticscattering,andparticleevaporationspec-tra,sothatthetheoreticalcalculationofthisquantityisquiteusefulforvalidationofnuclearmodels.Manyapproaches(e.g.[1–3]andreferencestherein)havebeendevelopedtocalculatethesingle-particlelevel(s.p.l.)densityonwhichisbasedthenuclearleveldensitycalculation,itses-tablishmentbeingyetconsideredadifficulttask.Aconstants.p.l.densityg∼A/14MeV−1wasusedinvariousstatisticalmodeldescriptionsofthenuclearreactions.AnincreasedcriticismhasbeenexpressedonthisconstantvaluewhichisinconsistentwiththenumberofAnucleonsinthenucleusandtheusualFermienergyF=38MeV.Thesemiclassicalapproximationscouldbethestartingpointofabasicanalysisofthes.p.l.density,includingtheenergydependence.First,theThomas-Fermiapproximationprovesade-quateforaspecialclassofpotentialwells[1]whichdescribethenuclearmeanfield(Woods-Saxon,harmonicoscillator,trapezoidalpotentialwell).However,forpotentialslikesquare-wellandinfi-nitesquarepotential,theresultsdeviatesignificantlyfromthequantum-mechanicalcalculation.Enhancedapproximationsaddthelowesttermsin¯hascorrectionstotheThomas-Fermiresults[1,2].∗ContributiontoEuropean(Int.)ConferenceonAdvancesinNuclearPhysicsandRelatedAreas,Thes-saloniki,Greece,8-12July1997;Rom.J.Phys.(inpress).†[e-mail]1Anexactquantum-mechanicalcalculationwasprovidedbyShlomo[1]byusingaGreen’sfunc-tionapproach.Aparticularcaseisthes.p.l.densityforafinitepotentialwell.Sinceitisincludingthefree-gasstates,onecancalculateandsubtractthiscomponentbyusingtheGreen’sfunctionassociatedwiththerespectivesingle-particleHamiltonian.Shlomofoundbymeansofbothsemi-classicalandquantummethodsthat,forarealisticfinitedepthpotentialwell,thes.p.l.densitydecreaseswithenergyinthecontinuumregion(thecontinuumeffect).Thepaperisorganisedasfollows.SectionIIisanintroductionofthegeneraltheoreticalaspectsconcerningthes.p.l.density,namelythedefinition,itsrelationwiththeGreen’sfunction,theStrutinskismoothingprocedure,andsemiclassicalmethodsusuallyinvolved.SectionIIIgivesageneraldescriptionofthesingleparticleGreen’sfunctiontheoryandamethodforthecalculationofitsimaginarypart.ThemodelparametersforthemeanfieldpotentialandnumericalresultsarediscussedinsectionIV.Finally,theconclusionsaredrawninsectionV.II.BASICFORMALISMFORSINGLE-PARTICLELEVELDENSITYA.Thequantum-mechanicalsingle-particleleveldensityAbriefpresentationofthes.p.l.densitydefinition,Green’sfunctionandrelationbetweenthemaregiveninthissection.Thes.p.l.densityisdefinedas[1]g(ε)=Tr(δ(ε−ˆH)),(1)whereˆHisthecorrespondingsingle-particleHamiltonian(meanfield)ˆH=ˆP22m+V(r).(2)TheeventualboundstatesoftheHamiltoniangivenbyEq.(2)canbeobtainedfromthetimeindependentSchr¨odingerequationˆH|nαi=εn|nαi,αbeingthedegeneracygivenbyotherquantumnumbers.Itissupposedthatthenormalisationfortheeigenvectorscorrespondingtoboundstatesishn′α′|nαi=δnn′δαα′,(3)sothatthes.p.l.densityfollowingEq.(1)isgB(ε)=Xnαδ(ε−εn).(4)Inthecaseofafinitepotentialwellwhichhasboundaswellascontinuousstatesthes.p.l.densitycanbeconsideredforthetworegionsofthespectrumg(ε)=gB(ε)+gC(ε),wheregB(ε)andgC(ε)arethecontributionsgivenbytheboundandcontinuusstatesrespectively.Thecontinuumcontributionaredeterminedfromthescatteringphaseshiftsδlj(ε)by[1,4]gC(ε)=1πXlj(2j+1)dδlj(ε)dε.2ItcanbedeterminedalsofromEq.(1)buttheeigenvectorscorrespondingtothecontinuumstatescannotbenormalisedsimilarlytotheboundstatescase.Therelation(3)becominghε′α′|εαi=δ(ε−ε′)δ(α−α′),(5)itresultsgC(ε)=Xα′Zdε′δ(ε−ε′)hε′α′|ε′α′i,(6)wherethesumafterα′canbeanintegralifthisparameteriscontinuous.Forthemomentwewillassumethatthestatesarenotdegeneratesothatthes.p.l.densityinthewholeenergyrangehastheformg(ε)=Xnδ(ε−εn)+Zdε′δ(ε−ε′)hε′|ε′i.(7