Extended Thomas-Fermi approximation to the one-bod

arXiv:nucl-th/9902039v117Feb1999ExtendedThomas-Fermiapproximationtotheone-bodydensitymatrixV.B.SoubbotinNuclearPhysicsDepartment,PhysicalResearchInstitute,St.PetersburgUniversity,St.Petersburg,RussiaX.Vi˜nasDepartamentd’EstructuraiConstituentsdelaMat`eria,FacultatdeF´ısica,UniversitatdeBarcelona,Diagonal645E-08028Barcelona,SpainFebruary6,2008AbstractTheone-bodydensitymatrixisderivedwithintheExtendedThomas-Fermiapproxi-mation.ThishasbeendonestartingfromtheWigner-Kirkwooddistributionfunctionforanon-localsingle-particlepotential.Thelinksbetweenthisnewapproachtothedensitymatrixwithformeronesavailableintheliteraturearewidelydiscussed.Thesemiclassi-calHartree-FockenergyatExtendedThomas-Fermilevelisalsoobtainedinthecaseofanon-localone-bodyHamiltonian.NumericalapplicationsareperformedusingtheGognyandBrink-Boekereffectiveinteractions.ThesemiclassicalbindingenergiesandrootmeansquareradiiarecomparedwiththefullyquantalonesandwiththoseobtainedusingtheStrutinskyaveragedmethod.11IntroductionTheone-bodydensitymatrix(DM)ρ(r,r′)=Pαφ∗α(r)φα(r′)orequivalentlyitsWignertrans-formthedistributionfunctionf(R,p)(definedbelow),playsacrucialroleintheHartree-Fock(HF)calculations.Ifzero-rangeSkyrmeforces[1]areused,onlythediagonalpartoftheDMisneeded.However,fullknowledgeofρ(r,r′)(orf(R,p))isnecessaryifoneconsidersfinite-rangeeffectivenuclearforceswhicharederivedfromG-matrixcalculationsinnuclearmatterthroughthelocaldensityapproximation[2,3,4,5]orpostulatedempiricallywiththeirparametersfittedtoreproducesomepropertiesofnuclearmatterandfinitenuclei[6,7].Thefullcalculationofthedensitymatrix(orthedistributionfunction)isnotaneasytaskandrequiressomecomputationaleffort[7,8].Consequently,approximationswhichsimplifythecalculationand,atthesametime,showmoreclearlythephysicalcontentoftheDMareinorder.Thesimplestoneistoreplacethenon-diagonalpartoftheDMbyitsvalueinnuclearmatter(Slaterapproach).Finitesizeeffectsareaddedusingthedensitymatrixexpansion(DME),eitherthatduetoNegeleandVautherin(NV)[9,10]orthemodifiedexpansionduetoCampiandBouyssy(CB)[11].Veryrecently,theCBapproachhasbeenappliedtoHFcalculationsoffinitenuclei[12]usingadensity-dependentversionoftheM3Yinteraction[4,5].Ontheotherhand,semiclassicalmethods[13]areveryusefulfordescribingnuclearpropertiesofaglobalcharactersuchasbindingenergiesornucleardensitiesandtheirmoments.ConcerningthenucleargroundstatepropertiesatHFlevel,semiclassicalapproachesarebasedontheWigner-Kirkwood(WK)¯h-expansionofthedistributionfunctionwhichforasetofnucleonsmovinginalocalexternalpotentialV(r)uptosecondorderisgivenby[13]:fWK(R,p)=Θ(λ−HW)−¯h28mΔVδ′(λ−HW)+¯h224m[(∇V)2+1m(p.∇)2V]δ′′(λ−HW)+O(¯h4),(1)whereλisthechemicalpotentialandHWistheWignertransform[13]oftheone-bodyHamil-tonian,whichreadsHW=p22m+V(R).(2)Thesemiclassicaldistributionfunctionf(R,p)isarepresentationofthetruephase-spacefunc-tionintermsofdistributionsandisveryefficientinordertoobtainsemiclassicalexpectationvaluesbyintegralsoverthewholephase-space[14,15].2ThemainpurposeofthispaperistoderivetheexplicitexpressionoftheDMintheEx-tendedThomas-Fermi(ETF)approximation[16]startingfromtheveryrecentlypresentedWKexpansionupto¯h2orderofthedistributionfunctionfornon-localpotentials[15].Ononehand,wewanttoestablishalinkbetweentheNV(andCB)expansionsoftheDMwiththissemi-classicalapproachand,ontheother,toapplythisETFDMtoderivetheexchangeHFenergywhenfiniterangeforcesareused.Thepaperisorganizedasfollows:InthefirstsectionwecomparethesemiclassicalETFdensitymatrixwiththeformerapproximationsofNVandCBinthecaseofalocalpotential.InthesecondsectionwederivethedensitymatrixandtheHFenergyintheETFapproximationforanon-localpotential.WealsoperformrestrictedHFvariationalcalculationsforsomeselectedsphericalnucleiusingtheGogny[7]andBrink-Boeker[6]effectiveforces.WecomparetheseHFETFresultswiththoseobtainedquantally,withthoseobtainedwiththeStrutinskyaveragemethod[19]andwiththosewhichresultfromtheNVandCBapproachestotheDM.Inthelastsectionwegiveourconclusionsandoutlook.TechnicaldetailsconcerningthecalculationoftheDMandHFenergyintheETFapproachforanon-localpotentialaregivenintheAppendix.2ExtendedThomas-FermiDensityMatrixThefirststepistoperformtheinverseWignertransformof(1)toobtainthesemiclassicalWKdensitymatrixincoordinatespace.ThedefinitionusedherefortheWignertransformoftheone-bodydensitymatrixis[13]:f(R,p)=Zdse−ips/¯hρ(R+s2,R−s2),(3)whereR=(r1+r2)/2,s=r1−r2andpare,respectively,thecentre-of-mass,therelativecoordinatesandthephase-spacemomentum.AftersomelengthybutstraightforwardalgebrathesemiclassicalDMintermsofRandsatWKlevelisgivenby:ρ(R,s)=gk3F6π23j1(kFs)kFs+g24π2ΔkF[j0(kFs)−kFsj1(kFs)]+g48π2(∇kF)2kF[j0(kFs)−4kFsj1(kFs)+k2Fs2j2(kFs)]3−g48π21kF∇[kF∇kFss]ss[−3kFsj1(kFs)+k2Fs2j2(kFs)]+O(¯h4),(4)wherekF=q2m¯h2(λ−V(R))isthelocalFermimomentum,jl(kFs)arethesphericalBesselfunctionsandgstandsforthedegeneracy.Thisexpression,althoughwritteninaslightlydifferentway,coincideswiththeonesobtainedpreviouslybyDreizlerandGross[17]andJennings[18].Thefirsttermoftheexpansion(4)correspondstotheSlaterapproach,whereasthe¯h2termsarethepartthattakeintoacco