Fractional dynamic symmetries and the ground state

FractionaldynamicsymmetriesandthegroundstatepropertiesofnucleiRichardHerrmannGigaHedron,Farnweg71,D-63225Langen,GermanyE-mail:[e-mail]-andCaputodenitionofthefractionalderivativeweusethefractionalextensionsofthestandardrotationgroupSO(3)toconstructahigherdimensionalrepresentationofafractionalrotationgroupwithmixedderivativetypes.Anextendedsymmetricrotormodelisderived,whichpredictsthesequenceofmagicprotonandneutronnumbersaccurately.Thegroundstatepropertiesofnucleiarecorrectlyreproducedwithintheframeworkofthismodel.PACSnumbers:21.60.Fw,21.60.Cs,05.30.Pr1.IntroductionTheexperimentalevidencefordiscontinuitiesinthesequenceofatomicmasses,and-decaysystematicsandbindingenergiesofnucleisuggeststheexistenceofasetofmagicprotonandneutronnumbers,whichcanbedescribedsuccessfullybysingleparticleshellmodelswithaheuristicspin-orbitterm[1],[2].ThemostprominentrepresentativeisthephenomenologicalNilssonmodel[3]withananisotropicoscillatorpotential:V(xi)=3Xi=112m!2ix2i~!0(2~l~s+l2)(1)Althoughthesemodelsareexibleenoughtoreproducetheexperimentalresults,theylackadeepertheoreticaljustication,whichbecomesobvious,whenextrapolatingtheparameters,,whichdeterminethestrengthofthespinorbitandl2termtotheregionofsuperheavyelements[5].Henceitseemstemptingtodescribetheexperimentaldatawithalternativemethods.Typicalexamplesarerelativisticmeaneldtheories[6],[7],wherenucleonsaredescribedbytheDirac-equationandtheinteractionismediatedbymesons.Althoughaspinorbitforceisobsoleteinthesemodels,dierentparametrizationspredictdierentshellclosures[8],[9].ThereforetheproblemofatheoreticalfoundationofmagicnumbersremainsanopenquestionsinceElsasser[10]raisedtheproblem75yearsago.Afundamentalunderstandingofmagicnumbersforprotonsandneutronsmaybeachievediftheunderlyingcorrespondingsymmetryofthenuclearmanybodysystemisdetermined.Thereforeagrouptheoreticalapproachseemsappropriate.arXiv:0806.2300v2[physics.gen-ph]12Aug2008Fractionaldynamicsymmetriesandthegroundstatepropertiesofnuclei2Grouptheoreticalmethodshavebeensuccessfullyappliedtoproblemsinnuclearphysicsfordecades.Elliott[11]hasdemonstrated,thatanaveragenuclearpotentialgivenbyathreedimensionalharmonicoscillatorcorrespondstoaSU(3)symmetry.LowlyingcollectivestateshavebeensuccessfullydescribedwithintheIBM-model[12],whichcontainsasonelimitthevedimensionalharmonicoscillator,whichisdirectlyrelatedtotheBohr-MottelsonHamiltonian.Inthispaperwewilldeterminethesymmetrygroup,whichgeneratesasingleparticlespectrumsimilarto(1),butincludesthemagicnumbersrightfromthebeginning.Ourapproachisbasedongrouptheoreticalmethodsdevelopedwithintheframeworkoffractionalcalculus.Thefractionalcalculus[13]-[16]providesasetofaxiomsandmethodstoextendthecoordinateandcorrespondingderivativedenitionsinareasonablewayfromintegerorderntoarbitraryorder:fxn;@n@xng!fx;@@xg(2)Thedenitionofthefractionalorderderivativeisnotunique,severaldenitionse.g.theFeller,Fourier,Riemann,Caputo,Weyl,Riesz,Grunwaldfractionalderivativedenitionscoexist[17]-[25].Adirectconsequenceofthisdiversityisthefact,thatthesolutionse.g.ofaonedimensionalwaveequationdiersignicantlydependingonthespecicchoiceofafractionalderivativedenition.Untilnowithasalwaysbeenassumed,thatthefractionalderivativetypeforanextensionofafractionaldierentialequationtomulti-dimensionalspaceshouldbechosenuniquely.Incontrasttothisassumption,inthispaperwewillinvestigatepropertiesofhigherdimensionalrotationgroupswithmixedCaputoandRiemanntypedenitionofthefractionalderivative.Wewilldemonstrate,thatafundamentaldynamicsymmetryisestablished,whichdeterminesthemagicnumbersforprotonsandneutronrespectivelyandfurthermoredescribesthegroundstatepropertiesofnucleiwithreasonableaccuracy.2.NotationWewillinvestigatethespectrumofmultidimensionalfractionalrotationgroupsfortwodierentdenitionsofthefractionalderivative,namelytheRiemann-andCaputofractionalderivative.Bothtypesarestronglyrelated.StartingwiththedenitionofthefractionalRiemannintegralRIf(x)=8:(RI+f)(x)=1()Zx0d(x)1f()x0(RIf)(x)=1()Z0xd(x)1f()x0(3)where(z)denotestheEuler-function,thefractionalRiemannderivativeisdenedastheproductofafractionalintegrationfollowedbyanordinarydierentiation:R@x=@@xRI1(4)Fractionaldynamicsymmetriesandthegroundstatepropertiesofnuclei3Itisexplicitelygivenby:R@xf(x)=8:(R@+f)(x)=1(1)@@xZx0d(x)f()x0(R@f)(x)=1(1)@@xZ0xd(x)f()x0(5)TheCaputodenitionofafractionalderivativefollowsaninvertedsequenceofoperations(4).AnordinarydierentiationisfollowedbyafractionalintegrationC@x=RI1@@x(6)whichresultsin:C@xf(x)=8:(C@+f)(x)=1(1)Zx0d(x)@@f()x0(C@f)(x)=1(1)Z0xd(x)@@f()x0(7)Appliedtoafunctionsetf(x)=xnusingtheRiemannfractionalderivativedenition(5)weobtain:R@xxn=(1+n)(1+(n1))x(n1)(8)=R[n]x(n1)(9)wherewehaveintroducedtheabbreviationR[n].FortheCaputodenitionofthefractionalderivativeitfollowsforthesamefunctionset:C@xxn=8:(1+n)(1+(n1))x(n1)n00n=0(10)=C[n]x(n1)(11)wherewehaveintroducedthe