Combinatorial Study of Nuclear Level Density for A

arXiv:nucl-th/9309025v124Sep1993COMBINATORIALSTUDYOFNUCLEARLEVELDENSITYFORASTROPHYSICALAPPLICATIONSBernardPICHOND´epartementd’AstrophysiqueRelativisteetdeCosmologieDARC-UPR176duCNRSObservatoiredeParis,SectiondeMeudon92195MeudonCedex–FranceAbstract:Wepresentsomeresultsandremarksbasedonacombinatorialapproachoftheevaluationofthenuclearleveldensity.First,weshowthatitispossibletoextractsomereliableinformationfromtheoutputoftheprogramwhoseroughdatapresentastrongstatisticalfluctuationfrombintobin.Thisincludessmoothingandevaluationofthedesiredquantities.Aftersomecommentsaboutthespinandparitydistributions,weconsiderthequestionofthenon-equipartitionofparities,mainly,atlowenergies.Finally,wepresentasimplemodeltoincludeandtestthiseffectinthecomputationofthermonuclearreactionrates.AcceptedforpublicationinNucl.Phys.A21:INTRODUCTIONNuclearleveldensitiesareoneofthesuitablequantitiesfornuclearphysicsand,especially,fornuclearreactionmodels(e.g.thestatisticalHauser-Feshbachtheory).Therefore,theyhavebeeninvestigatedalongtimeago,thefirstat-tempt,usingtheinfinitenon-interactingFermi-gasmodel,beingthatofBethein19361).Theneedtoevaluatereliablereactionsrates,andthusnuclearleveldensity,fornumerousnucleiarisesfromthespecificityofnuclearastrophysicswhichdealswithnucleiinthewholechartofnuclides,towardstheprotondripline(forα-porr-pprocess),nearbothsidesofthevalleyofstability(forpandsprocesses)ortowardstheneutrondripline(forther-process).Inallthesecases,veryfewreactionratesaremeasurableexperimentaly,especiallyattheenergiesofastrophysicalinterest.Foralltheothers,onlytheoreticalpredictionsarepossible.Forthispurpose,wemustdevelopsomeabinitio,butrathersimple,methods(inordertominimizethecomputationalwork)withtheminimumnumberofparameters.Thisisparticularlythecaseofthenuclearleveldensity.Theaimofthispaperistopresentacombinatorialmethodofcomputingtheleveldensitybyenumerationofallthepossibleconfigurations,andtoobtaineasilyasimplelawdescribingtheenergydependenceoftheleveldensity.Beyondthespindistribution,wefocusourattentionontotheparitydistributionwhich,partially,originatesthiswork.Thefinalintentistoevaluatethevariousuncertainties,constraintsordifficultiesrelatedwiththeleveldensityproblem,andtheirimpactonreactionratespredictions.Someofthepreviousattemptsarebrieflyreviewedandcommentedin§1.Ourattemptisdepictedin§2withthedescriptionofthemodelandthepreliminarytreatments.Someresults,includingthedeterminationoftheleveldensityparameterandacomparisonwiththeusualspindistributionfunction,areshownin§3.Theproblemofequipartitionoftheparitydistributionforthelowestenergylevelsispresentedin§4.Inourconclusions(§5),weconsidersomepossibledevelopmentsorapplications,afewofthem(mostlyastrophysicalones)beingnowinprogress.Thetechnicalpartofourconfigurationgeneratoralgorithmwillbedescribedelsewhere.Therearebasicallytwomainclassesofmethodsforevaluatingthenuclearleveldensities.Thefirstmethods,andthemostused,arebasedonthestatisticalapproachviathepartitionfunction(seee.g.refs.2),3)forreview,or4),5)forcomputationalprograms).Theothermethodisthecombinatorialapproachwhichhasbeenmuchlessinvestigated,mainlyduetotheextentofthecomputationsitrequires.Now,letusfocusonthislastone.ToourknowledgethefirstattemptisduetoHillmanandGrover6).Theyuseddifferentsetsofsingleparticlelevels,thepairingenergiesbeingevaluatedwithintheBCSapproximation.Inthesamespirit,arecursivemethodhasbeendeviced3byWilliams7),thatrepresentsaninterestingalternativewayofcomputingleveldensities(see§2-c).AnotherattemptwasdonebyFord8).HeusedWoods-SaxonorNilssonsin-gleparticlelevelswithanewcombinatorialalgorithmtogeneratetheconfigura-tions.ThisalgorithmworksaccordingtheM−scheme.Thisimpliesspecialandsup-plementaryassumptionsinordertoobtaintheJ−schemeconfiguration,andthus,thespindistributionand,moreimportantly,theenergyoftheconfigurationunderconsideration(see§2).2:THEPRESENTATTEMPTa—DescriptionofthemodelAspreviouslysaid,thecombinatorialmethodisbasedonasetofsingleparticlelevels,onanalgorithmtogenerateconfigurations,andonanevaluationofvariousphysicalquantitiesofeachconfigurationunderstudysuchasenergy,parityandspindistribution(duetodegeneracy).Theresultspresentedherehavebeenobtainedwiththeuseofsphericalsinglepar-ticlelevelsderivedwithintheframeworkofanenergydensityapproximationdescribedinref.9).Wedonotinvestigateheretheinfluenceontheleveldensityofachangeofthosesingleparticlespectra.Thisquestionwillbeexaminedelsewherealongwiththeconsiderationofdeformedsingleparticlespectra.Inordertogeneratealltheconfigurationsuptoagivenmaximumenergy(chosenlessorequalto30MeV),wehaveadaptedFord’salgorithmtoourspecificpurposes.Infact,thealgorithmdesignedbyFordexploresallthepossibleconfigurationsbyincreasingenergies.Hence,itavoidstheevaluationofunnecessaryphysicalquantitiesforconfigurationswithenergyfarinexcessoftheadoptedupperlimit.TheM−schemeusedinFord’salgorithm8)toclassifytheconfigurationshasbeenconvertedintothedesiredJ−scheme.Thistransformationiscomplicatedbythefactthatmany“M−configurations”leadtothesame“J−configuration”(seetheexamplegivenintable1).Wehavealsoaddedtotheorigi