Consistent Construction of Realistic One-Body Dens

arXiv:nucl-th/9511006v16Nov1995CONSISTENTCONSTRUCTIONOFREALISTICONE-BODYDENSITYMATRIXINNUCLEIA.N.Antonova,S.S.Dimitrovaa,M.K.Gaidarova,M.V.Stoitsova,M.E.Grypeosb,S.E.Massenb,K.N.YpsilantisbaInstituteofNuclearResearchandNuclearEnergy,BulgarianAcademyofSciences,Sofia1784,BulgariabDepartmentofTheoreticalPhysics,AristotleUniversityofThessaloniki,Thessaloniki54006,GreeceAbstractAphenomenologicalmethodbasedonthenaturalorbitalrepresentationisappliedtoconstructthegroundstateone-bodydensitymatrixwhichdescribescorrectlybothdensityandmomentumdistributionsin4He,16Oand40Canuclei.Theparametersofthematrixarefixedbyabestfittotheexperimentaldensitydistributionandtothecorrelatednucleonmomentumdistribution.Themethodallowsthenaturalorbitals,theoccupationprobabilitiesandthedepletionoftheFermiseatobeobtained.Ground-statecharacteristicsof4He,16Oand40Canuclei,suchasrmsradiiandmeankineticenergiesarecalculated,aswell.1IntroductionTheone-bodydensitymatrix(OBDM)playsanimportantroleinthenuclearstructuretheory.Itisknownthatthenuclearwavefunctioncontainsmuchmoreinformationthanitisaccessibletoobservation[1,2].Allmeasurablecharacteristicsofthenucleargroundstate(apartfromtheenergy)correspondtosingle-particleoperatorswhoseexpectationvaluescanbeexpressedintermsoftheOBDMelements.Thegroundstateenergycanbecalculatedbyasumruleanalysis[3,4,5]alsobymeansoftheOBDM.ThusthedeterminationoftheOBDMfromfirstprinciples[6]isoneofthemostessentialaimsofthenucleartheory.1Asshownin[7,8](anddiscussedin[9,10,11,12]),anappropriatecriterionfortheproximityoftheOBDMρ0(r,r′)correspondingtoasingle-Slaterdeterminantwavefunction(i.e.ρ20=ρ0,forinstanceintheHartree-Fockapproximation)totheOBDMρ(r,r′)ofthetrue(correlated)groundstateisthatthe”mean-squaredeviationperparticle”σ=A−1Tr[(ρ−ρ0)2](1)(whereAisthemassnumber)shouldbeminimal.Ithasbeenshownin[8]thatσisminimalwhenρ0(r,r′)correspondstosingleSlaterdeterminantwavefunctionconstructedwithnaturalorbitals,i.e.withthesingle-particlewavefunctionswhichdiagonalizetheOBDMρ(r,r′)[13].Aswasshownin[9],ifintheHartree-FockapproximationtheOBDMdiagonalelementsρHF0(r,r′)arefittedtoreproducecorrectlytheexactdensitydistributionρ(r)≡ρ(r,r)(i.e.ρHF0(r,r)≃ρ(r)),theextremumpropertyσ=σminleadsinevitablytoanincreasingdeviationbetweenthenon-diagonalelementsofthesetwomatrices(ρ(r,r′)andρHF0(r,r′)atr6=r′).Thnon-diagonalelementsarerelated,however,tothenucleonmomentumdistribution(NMD):n(k)=Zρ(r,r′)exp[ik(r−r′)]drdr′.(2)Thisleadstotheconclusion[9]thatgenerallythemean-fieldapproximationisunabletogivesimultane-ouslyacorrectdescriptionofthetwobasicnucleargroundstatecharacteristics,namelythedensityandmomentumdistributions.Thishasbeensupportedbythecalculationsinthecaseof4Henucleus[14].Theroleofthenucleon-nucleoncorrelationsinthenuclearsystemscanbeseenbytheanalysesofthequantityσ[10].WhileintheHartree-Fockapproximationσvanishesandthecontributionoftherandom-phaseapproximationtoσisabout0.002,forrealisticN-Nforcestheshort-andmedium-rangecorrelationeffectsleadtoavalueofσmin≃0.02÷0.03thatimpliesaFermiseadepletionofabout10÷15%.Thelatterisconfirmedbytheexperimentaldata[15,16].Theshort-rangeandtensorcorrelationsareresponsiblefortheexistenceofhigh-momentumcomponentsintherealisticmomentumdistributionsinnucleiwhichisnotthecaseinthemean-fieldapproximation.AphenomenologicalmethodtoconstructamorerealisticOBDMissuggestedin[1,2]forthecasesof16Oand40Canuclei,bothwithfractionaloccupationnumbersandfulloccupationofthefirstAnaturalorbitals.Thelatterareexpandedintermsofharmonic-oscillatorwavefunctions.Theexpansioncoeffi-cients,aswellastheoccupationnumbersarefixedbyabestfittotheexperimentaldensitydistributionsof16Oand40Caminimizingthermspercentagedeviationbetweentheexperimentalandtrialradialdensity.Asatisfactorydescriptionofg.s.properties,suchastheg.s.energyandrmsradii,isobtained2usingtheOBDMconstructedwithinthismethod,incontrastwiththepredictionsoftheHartree-Fockapproximation.Herewewouldliketoemphasizethataccordingtothediscussiongivenabove,theOBDMobtainedinthemethodfrom[1,2]cannotbearealisticonebecauseitisobtainedbyabestfitonlyofitsdiagonalelementstotheexperimentaldensitydistribution(whichisnotverysensitivetotheN-Ncorrelations)andisnotaimedtodescribethenucleonmomentumdistributionwhichisrelatedalsotothenon-diagonalelementsoftheOBDMandismostlyaffectedbythepresenceofshort-rangeandtensorN-Ncorrelationsinthenuclearsystem.Theaimofthisworkistoconstructtheg.s.OBDMin4He,16Oand40Canucleiinaconsistentwayfollowingthemethodfrom[1,2]butprovidinginanoptimalwayacorrectdescriptionofbothdensityandmomentumdistributionsinnucleiconsidered.Solvingthisproblem,weshowthat,inprinciple,theexpansionofthenaturalorbitalsintermsofharmonicoscillatorfunctions(inthetruncateds.p.space)cannotgiveacorrectdescriptionoftherealistichigh-momentumcomponentsofthenucleonmomentumdistribution.Thisimposestouseintheexpansionofthenaturalorbitalsanothersetofs.p.functions,forinstancethatonecorrespondingtoinfinitesquare-wellpotential.ThemethodsuggestedinthisworkenablesustoconstructamorerealisticOBDMwhichincludesthatpartofthenucleoncorrelationswhichare