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Microscopic Study of Nuclear Level Densities and Thermal Properties in^56,57 Fe

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Microscopic Study of Nuclear Level Densities and Thermal Properties in^56,57 Fe

    Microscopic Study of Nuclear Level Densities and Thermal Properties in^56,57

    Fe

    Commun.Theor.Phys.(Beijing,China)48(2007)PP?531533

    ?InternationalAcademicPublishersVo1.48,No.3,September15,2007 Micr.sc.picStudy.fNuc1eaFLeve1DensitiesandThermalPropertiesin56'57Fe z.KDogDo,N.Ghahramani,andAMoradi

    PhysicsDepartment,ShirazUniversity,Slfiraz71454,Iran

    (ReceivedAugust8,2006;RevisedSeptember26,2006)

    AbstractWecalculateleveldensitiesfor.,57FenucleiusingBCShamiltonianwithinclusionofpairinginteraction?

    Teresujts0fcajcujatj0nss

    0wthatthestepstructureobservedexperimentallyhasbeensupportedbymicroscopic j1e0fAlsotheS

    shapedenergyandentropyasfunctionoftemperaturehavebeenobtainedintheoreticalcalculations?

    Struuresin

    ecuryesafeinterpretedasfingerprintsofbreakingCooperpairsandquenchingofpairingcorrelations?

    PACSnumbers:21.10.k,21.10.Ma

    Keywords:leveldensity,..,wFe.pairingphasecorrelation

    1Introduction

    Nuclearleveldensityisimportantinthecalculations

    ofreactioncrosssections.Inastrophysics,therateof

    neutroncapturereactionsinrand8processesDoepro-

    portionaltothenuclearleveldensities.Neutroncapture

reactionsDoeimportantinthesynthesisofelementsheav

    ierthanlron.ThereforenuclelinirongroupareofpDotic

    ulDointerest.

    Leveldensitiesareaffectedbypairing.Underthein

    fiuenceofashortrangeattractiveforce,nucleonsprefer toformpairs.Nucleonpairs(Cooperpairs)cai1breakup athighertemperature.ThethermalbreakingofCooper pairsleadstoincreasingleveldensityandentropy. Experimentalistsrecentlydevelopedamethodtoex

    tractleveldensitiesfrommeasured,yspectra,socalled

    0sktypetechnique.1?2lThismethodhasbeenrecently appliedtoextractleveldensitiesin0FeandFeiso- topesusing(.He,n,y)and(.He,.He,y)reactionson57Fe target.3,4JTheobservedstepstructureintheleveldensi

    tiesofbothisotopesisinterpretedaspairbreaking. Inthisworkthenuclearleveldensitiesandrelated propertiessuchasentropyandcaloriccurvearestudied intheframeworkofsuperconductingtheory.Section2 presentsthegeneraltheory.InSec.3wediscussthecalcu

    lationalresultsfor56Feand57Fenuclei.Section4contains themainconclusionsofourwork.

    2Theory

    IntheBCStheorythepairinggapAisobtainedby usingthefollowinggapequation:[J

    2

    ——=

    G

    ?1——2AE

    k

    (1)

    whereEk:,/i.isthequasipDoticleenergy, Ckisthesingle-pDoticleenergy,and^=(1+eEs,)the

    Fermi-DiracquasipDoticleoccupancy.Thepairingforce strengthGischosensoastoproducetheexperimental Email:kargar~susc.ac.irorkargarz~yahoo.corn oddevenmassdifferenceatzerotemperature.Thechem

    icalpotentialisdeterminedbythenumberconstraint ?=[1Ek(12A)]

    Theenergyandtheentropyaregivenby

    E:

    J

    1-Ck

    

    --

    #(12)]_A2,

    S=一?lnA(1-fk)ln(1^)],

    (2)

    (3)

    (4)

    Finally,thestatedensityandtheleveldensityofasys

    ternofNneutronsandZprotonsatexcitationenergy c,(:E+)canbeobtained,

    (?,Z,U)=

    p(N,Z,U)=

    eS

    (27r)./D/.'

    (?,Z,U)

    (27r)/'

    whereDisa3x3determinantwithitselementsgiv termsofthesecondderivativesoftheentropyand

thespincutofffactor.

    3ResultsandDiscussion

    (5)

    (6)

    enln

    0.2is

    Calculationshavebeencarriedoutfor56Feand57Fe nuclei.Thegroundstategapparameteristakenfrom Refs.8and9andSeegresingleparticlelevelsDoe

    used.10Theparametersofnuclearleveldensityarecho- sensoasthebestfittotheexperimentaldatayields. InFig.1,weshowthecalculatedtemperaturedepen- denceofexcitationenergyf0r50Feand57Fenuclei.The arrowsindicatethetransitionfromthesuperconducting statetothenormalstate.Thedipsinthecaloriccurves, ortheSshapecurvesforenergyasfunctionoftempera- ture,reflectthestructuralchangeofthesystems. Figure2showsthetemperaturedependenceofentropy for50Feand57Fenuclei.TheS.shapedentropycanbein

    dicationofbreakingoftheCooperpairs.Also,theexcess

    Z.Kargar,N.Ghahramani,andA,MoradiVo1.48 ofentropyforodd-systemisseensinceforoddsystem

    manylow-lyingstatescanbereachedwithoutneedingto breakapair.

    Excitationenergy(MeV)

    Fig.1Caloriccurvesfor.,Fenuclei

    TemperatureT(MeV)

    Fig.2Dependenceoftheentropyupontemperature for0.,5Fenuclei.

    InFigs.3and4wedisplaytheresultsofcalculations forthelogarithmofnuclearleveldensityfortheisotopes 56Feand57Feincomparisonwiththecorrespondingex. perimentaldata.Asitcanbeenseen.thetheoreticalre- suitsshowastep-likestructureforbothnuclei.Theagree. mentbetweentheoryandexperimentisgood.Thestep structureisinterpretedasthebreakingofCooperpairs andquenchingofpairinteraction.

    For50FenucleusA+A=2.43MeV.Also,theex?

    aminationofSeegersingleparticlestatesshowsthatthe energydifferencebetweenthegroundstateenergyand thefirstexcitedstateiS0.6MeV.So3.03MeVenergyiS neededtobreakonepairandexciteoneofthemtothe nextlowerstate.Thisisinverygoodagreementwiththe observedstepthatisstartedat3MeVexcitationenergy. FDr57Fellucleusthecalculationsaredoneinthesalne way.Besides.thebackshiftedmodewaF,appliedfortiffs

    odd.Anucleus.[11,12]andweobtainedtheleveldensityof 5Fenucleusfromtheleveldensityof56Fenucleusby shiftingtheenergyscaleof?=1.2MeV.Asitisshown

    inFig.4thestartofthestepiSaround2MeV.T.heoh- servedstepstartsat2MeV.Theagreementbetweenthe experimentalleveldensitywithamicroscopictheoryin. cludingthepairinginteractioniSexcellent.Alsoaswe haveexpectedforodd.Anucleithenuclearleveldensity shiftedbackwardwithenergyof?.

    ExcitaaonenergyF(MeV)

    Fig.3Comparisonoftheexperimentalleveldensityof Fcwiththetheoreticalcalculations.Theexperimental

dataweretakenfromRefs,3and4.

    ExcitationenergyE'(MoV)

    Fig.4Comparisonoftheexperimentalleveldensityof Fewiththetheoreticalcalculations.Theexperimental dataweretakenfromRefs.3and4.

    4Conclusions

    Inthisworkwehaveshownthatthemicroscopicthe? oryincludingpairinginteractionprovidespreciseinfor

    mationofthenuclearleveldensity.Thisindicatesthe >:eJn1|E.L

    ..>elIlsclemJ

    Alu

    ^8k一?coo>o-1

NO.3MicroscopicStudyofNuclearLevelDensitiesandThermalPropertiesin56?57Fe533

    importantroleofpairinginteractionintheoreticalinves

    tigations.Also,theSshapeenergyandentropyasfunc

    tionsoftemperatureareasignatureofthetransitionfrom stronglypairedstateatlowtemperaturetotheunpaired oneathighertemperature,i.e.thepairingphasetransi

    tion.

    References

    Aprimaryaccountofthisworkispresentedin ND2004.[131

    Acknowledgments

    WeacknowledgethesupportoftheResearchCouncil ofShirazUniversitythroughGrantNo.83-GR-SC49.

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