What is the effect of the odd particle coupling to boson core for O 的奇数粒子耦合的玻色子的核心有什么影响课件

QuantumPhaseTransitionfrom

Sphericaltoγ-unstablefor

Bose-FermiSystem

MahmutBöyükata

KırıkkaleUniversity

Turkey

collabrationwithPadova–Sevillagroup10thINTERNATIONALSPRINGSEMINARONNUCLEARPHYSICS

NEWQUESTSINNUCLEARSTRUCTURE

VIETRISULMARE,MAY21–25,2010QuantumPhaseTransitionfrom17.9.202310thInternationalSpringSeminaronNuclearPhysicsThepresentationisstructuredasfollows.Introductionincludessomedefinitionsabout...QuntumPhaseTransitions(QPT)DynamicalSymmetries(DS)CriticalPointsSymmetry(CPS)Aim!...TheInteractingBoson-FermionModelHamiltonianTheIntrinsicFrameFormalismforodd-evenSystems aredescribed.TheResultsarepresented.Finally,Conclusionsaresummarized.

4.08.202310thInternationalSpQuantumPhaseTransitions:Thestudyofphasetransitionshasbeenthesubjectinmanyinvestigationsofthenuclearphysics.Phasetransitionscanbeclassifiedintotwoclasses.First-orderphasetransitionincludestwodifferentphases.Twocoexistingminimaoftheenergysurfacehavethesameenergy.Thesecondkindofphasetransitionsreceivethenameofcontinuous(second-order)phasetransition.Thephasechangeoccursverysoftlyinacontinuouswayfromonephasetotheotherasthecontrolparametersarevaried. FiguresfromIachello’sPresentation,Istanbul(2010)17.9.202310thInternationalSpringSeminaronNuclearPhysicsQuantumPhaseTransitions:4.08DynamicalSymmetries:Nucleicanbeclassified accordingtotheirshapes intheIBM.ThethreeIBM DynamicalSymmetries correspondtothethree analyticalsolutionsU(5),thevibrationalSU(3),therotationalO(6),γ-unstablelimits.

17.9.202310thInternationalSpringSeminaronNuclearPhysicsDynamicalSymmetries:4.08.202CriticalPointSymmetry(CPS)

:TheCriticalPointSymmetryhasbeenfirstlyintroducedbyIachello,TheCPSconceptapplieswhenaquantalsystemundergoestransitionsbetweendynamicalsymmetries(definiteshapes)Itisdesignedtoapplyatthecriticalpointoftheshapephasetransition.TheCPSproposeduptonowareknownas

E(5),transitionfromsphericaltoγ-unstableshapes(continuous),

X(5),fromsphericaltoaxiallydeformedshapes(firstorder),

Y(5)fromaxiallydeformedtotriaxialshapes(continuous),

FigurefromPRL,91(13),(2003)17.9.202310thInternationalSpringSeminaronNuclearPhysicsCriticalPointSymmetry(CPS)17.9.202310thInternationalSpringSeminaronNuclearPhysicsProlate

c=1,χ=-1,32Oblatec=1χ=+1,32γ-unstable;c=1,χ=0Spcerical

c=0

SphericalDeformed4.08.202310thInternationalSpPotentialEnergySurphase(PES)V(β,γ):Geometriccollectivedeformationsaredescribedbyintroducingtwocollectivevariables,calleddeformationparameters(β,γ).Theβvariablemeasures theaxialdeviationfromsphericityTheanglevariableγcontrols thedeparturefromaxialdeformation.

TheβandγplayasimilarroletotheintrinsiccollectiveshapevariablesintheBohrHamiltonian.17.9.202310thInternationalSpringSeminaronNuclearPhysicsPotentialEnergySurphase(PESStudiesonQuantumPhaseTransitionsinnuclei:TheQPTforboththeGCMandtheIBMhavebeenextensivelystudiedforeven-evennuclei.However,fewstudieshavebeendoneforodd-evennuclei.Recentstudiesofshapephasetransitionsinbose-fermisystems...Asupersymmetric(SUSY)approachhasbeenusedforthestudyofphasetransitionsforj=1/2,3/2,5/2.ThesupersymmetricextensionoftheCastentriangleforodd-Anuclei.ThecirclesindicatethelocationoftheBose-Fermisymmetries.PRC70,011305(R)(2004)17.9.202310thInternationalSpringSeminaronNuclearPhysicsStudiesonQuantumPhaseTransStudiesofshapephasetransitionsinodd-evennuclei...TheCPSE(5/4)symmetryhasbeenfirstlydiscussedbyIachelloforasinglej=3/2fermioncoupledtoabosoncorethatundergoesatransitionU(5)→O(6).withBohrHamiltonian(theGCM). PRL.95,052503(2005),alsowiththeIBFMHamiltonian.

PRC72,061302(R)(2005).AmorecomplexcaseoftheCPS,E(5/12)symmetry,hasbeendescribedfortherichercaseofafermionthatcanoccupysingle-particlestateswithj=1/2,3/2,5/2.U(5)→O(6).

PRL98,052501(2007),

PRC75,064316(2007).ForthetransitionfromsphericaltoaxiallydeformedshapeshasalsobeendescribedwiththeIBFM,j=1/2,3/2,5/2.U(5)→SU(3).PRC79,014306(2009).17.9.202310thInternationalSpringSeminaronNuclearPhysicsStudiesofshapephasetransitInthisstudy...

Wefocuson theeffectofthecouplingafermioninorbitofdefinitej

toabosoncorethatperformsthetransitionU(5)→O(6).

Theaimistounderstand:Howthecouplingoftheoddparticlemodifiesthegeometryimposedbythecore,Howeachoftheindividualcoupledstatesbehavesatthetransitionalregion,Howthecriticalpointisaffectedbytheinclusionoftheoddparticle.ThissituationisdescribedwiththeintrinsicframeformalismfortheIBFM.17.9.202310thInternationalSpringSeminaronNuclearPhysicsInthisstudy...4.08.202310thTheInteractingBoson-FermionHamiltonian:Ingeneral,IBFMHamiltonianiswrittenasHBisthebosonicpart,HFisthefermionicpart,VBFtermcouplesthebosonandfermion.

TheIBFMHamiltonianisparametrizedasfollows[U(5)→O(6);(χ=0)]

:17.9.202310thInternationalSpringSeminaronNuclearPhysicsTheInteractingBoson-FermionTheIBFMHamiltonian:ThebosonHamiltonian

[U(5);c=0andO(6);c=1,χ=0.]

Bychangingcbetweentwolimits,acontinuous(2nd-order)transitionisobservedwiththecriticalpointatTheboson-fermioninteraction17.9.202310thInternationalSpringSeminaronNuclearPhysicsTheIBFMHamiltonian:4.08.202TheIntrinsicFrameFormalism:AusefulwayoflookingatphasetransitionsistoapplytheconceptofintrinsicframetoassociatePEStoHamiltoniandependingonβandγ.WithintheIBM,theintrinsicstatefortheg.s.bandforaneven-evennucleusiswrittenasThebosoncreationoperatorisgivenbyTheenergysurfaceisobtainedbycalculatingtheexpectationvalueoftheHBintheintrinsicstate17.9.202310thInternationalSpringSeminaronNuclearPhysicsTheIntrinsicFrameFormalism TheIntrinsicFrameFormalismforodd-evennuclei:Intrinsicframestatesforthemixedboson-fermionsystemcanbeconstructedbycouplingtheoddsingleparticlestatestotheintrinsicstatesofthebosoncore.Toobtainthem,wefirstconstructthecoupledstatesThendiagonalizethetotalboson-fermionHamiltonianinthisbasis,givingasetofenergyeigenvaluesEn(β,γ).Herenisanindextocountsolutionsintheodd-evensystem.17.9.202310thInternationalSpringSeminaronNuclearPhysics TheIntrinsicFrameFormalismResults:Firstly,weconsiderthebosoncoreinO(6)limit,Forthis,thecontrolparameterc=1,χ=0intheHB.Thecorrespondingenergyfunctionisγ-independentasshowninFigure.Inthiscaseβ=0,78forNB=5.WewanttoconsiderinamixedBose-Fermisystem.

Forthis,asinglefermionwithj=9/2iscoupledtobosoncore.17.9.202310thInternationalSpringSeminaronNuclearPhysicsResults:4.08.202310thInternaResults:Forj=9/2,thepossiblemagneticcomponentsareK=−9/2,...,+9/2.Thereare10differentstates,werestrictto5sincethesymmetryK↔-K.InFig.(b),theodd-evenenergysurfacesareplottedasafunctionofβ.Statesarelabeledby thequantumnumberK.Theminimumofeachsurface ismarkedwithadot.K=7/2,9/2favoroblateshapesK=1/2,3/2,5/2favorprolateInFig.(c)Nilsson-likeschemeisobtained whenrepresentingtheodd-even energysurfacesrelativeto theenergyoftheeven-evensystem.17.9.202310thInternationalSpringSeminaronNuclearPhysicsResults:4.08.202310thInternaWhenmovingintothetransitionalregionbychangingthecfrom0to1.

InFig,theevolutionoftheenergysurfacesforthecoreandthedifferentKstatesintheodd-evensystemisgivenforasetofvaluesofc.Fromthefigureitisclearthatwhenthecoreisγ-unstable,

K=9/2,7/2

statesarealwaysfavoringoblate,K=5/2,3/2,1/2

favorprolate.17.9.202310thInternationalSpringSeminaronNuclearPhysicsWhenmovingintothetransitioComingtothesituationatthecriticalpoint.

Forthecaseunderstudy(NB=5) thecriticalpointfortheeven-even systemislocatedatc=0.625.InFig.(a)theenergysurface isplottedfortheeven-evencore.Itisseenthattheeven-evensystem isγ-independentandhasasphericalminimum.

InFig.(b),thebehaviouroftheodd-evenenergy surfacesareplottedatthecriticalpoint.

Theeven-evensurfaceisveryflatinasexpected forthecontinuouscriticalpointsituation.

Fig.(c)isaNilsson-likediagramof thesingleparticleenergiesrelativeto theeven-evencoreforthesamecase.17.9.202310thInternationalSpringSeminaronNuclearPhysicsComingtothesituationattheTheoverallresultsaresummarizedinFigwheretheminimainforthedifferentodd-evenstatesareplottedversusthecontrolparameterc.Positivevaluescorrespondtoprolatedeformation,whilenegativeonesmeanoblateshapes.Theeven-evencaseisplottedasareference.Thissplitsintotwolines inthedeformedregion sincefortheeven-evencase twodegenerateminimaappear.Fromthefigureonecanseethat alloverthetransitionthestateswithK=1/2,3/2,5/2prefertobeprolatewhileK=7/2,9/2areproducingoblateshapes.TheupperpanelisforNB=5, whiletheloweroneisforNB=15.17.9.202310thInternationalSpringSeminaronNuclearPhysicsTheoverallresultsaresummarTheoverallresultsaresummarizedinFigwheretheminimainforthedifferentodd-evenstatesareplottedversusthecontrolparameterc.Inbothcases,itisseenthattheoddsurfacestendtofollowthebehaviouroftheeven-evencore.However,forthesmallerNBvalue, thedeviationsfromtheeven-evencase arelargerforallK’s.AsNBgrows,thetransitionof theodd-evensystemgetscloser totheoneintheeven-evensystem.Inourcase,theγ-instabilityof thecoreallowstheoddstatestodrive theentiresystemtowardseither aprolateoraoblateshape dependingonthevalueofK.17.9.20