The manifestation of $SU(4)$ symmetry within an interacting boson model including particle-like and hole-like $\pi$- and $\nu$-bosons is shown for light nuclei around the Z=N=8 shell. We also present a consistent description of the particle-hole (intruder spin or $I$ spin) multiplets in the Extended Interacting Boson Model (EIBM) and of $\pi$-$\nu$ ($F$ spin) multiplets in the IBM-2 as a breaking of this $SU(4)$ symmetry

De Beule, J

De Coster, C

Decroix, B

Heyde, Kris L G

Physics Letters B

Decroix, Bruno

De Beule, Joachim

DE COSTER, C

Heyde, Kristiaan

Ghent University Academic Bibliography

SU(4) symmetry in the extended proton-neutron interacting boson model: multiplets and symmetry breaking.

B. Decroix

J. De Beule

C. De Coster

K. Heyde

Crossref

SU(4) symmetry in the extended proton-neutron interacting boson model: multiplets and symmetry breaking

CERN Document Server

EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHCERN-EP/98-15806/10/98SU(4) SYMMETRY IN THE EXTENDED PROTON-NEUTRONINTERACTING BOSON MODEL : MULTIPLETS ANDSYMMETRY BREAKING.B. Decroix(1), J. De Beule(1), C. De Coster(1);and K. Heyde(1;2).(1)Institute for Theoretical Physics, Vakgroep Subatomaire en Stralingsfysica, Proeftuinstraat 86, B-9000Gent (Belgium)(2)present address: EP-ISOLDE, CERN, CH 1211 Geneva 23 (Switzerland)Postdoctoral research fellow of the Fund for Scientic Research - Flanders (Belgium)AbstractThe manifestation of SU(4) symmetry within an interacting boson model including particle-like andhole-like - and -bosons is shown for light nuclei around the Z=N=8 shell. We also present a consistentdescription of the particle-hole (intruder spin or I spin) multiplets in the Extended Interacting BosonModel (EIBM) and of - (F spin) multiplets in the IBM-2 as a breaking of this SU(4) symmetry.(ISOLDE GENERAL)(Accepted for publication in Physics Letters B.)1The Interacting Boson Model has been introduced to describe quadrupole collective phenomena observed inmedium-heavy and heavy nuclei. The building blocks are nucleon-pairs with angular momentumL= 0+and2+which are mapped onto s- and d-bosons, respectively. In the original Interacting Boson Model (IBM-1),only one kind of s- and d-bosons are considered [1, 2].Using the charge of the nucleons as a supplementary but essential degree of freedom, the IBM-2 was con-structed enabling a classication of states with the introduction of a new quantum label F , associated withthe F spin algebra SUF(2). The symmetry of the spatial part and of the charge part of the wave functionis determined by this F spin label. The total wave function, being the product of an orbital wave func-tion with the F spin wavefunction, is symmetric and thus allows to consider states with possible valuesF = N=2; N=2  1; :::; j Fzj with Fz=N N2. IBM-1 is incorporated in IBM-2 by the condition on the wavefunctions : F = Fmax, hence demanding a totally symmetric orbital wave function. All the other possiblestates have a 'mixed F spin symmetry'.In an attempt to describe particle-hole excitations across the closed shell in the IBM-picture, an ExtendedInteracting Boson Model (EIBM) was proposed [3] in which a particle or hole character gets assigned to thes- and d-bosons. The symmetries in this model can be developed in analogy with the IBM-2, where now weintroduce the I spin (intruder spin) algebra SUI(2). By doing so we can consider states with a 'mixed Ispin symmetry', corresponding with a classication I < Imaxwhere Imax= N=2 and N the total numberof bosons. Applications of this model have shown to be successful in describing the coexistence of dierentsymmetries for the groundstates and the 2p-2h intruder states in both the Z=50 and Z=82 region [4, 5]. Sincethe EIBM states are restricted to those members of the I spin multiplet where the neutrons are consideredas the reference state, i.e. the I spin label is restricted to the proton conguration only and we deman!d I= 0, it shows to be ve ry important to construct a full Extended Proton-Neutron Interacting BosonModel (EIBM-2) by introducing both the particle-hole character and the charge of the nucleons as a degree offreedom for the bosons on equal footing. By doing so a new reduction scheme, containing an SU (4) algebra,has been suggested [6].In the present article we present an example of this SU (4) symmetry in the Z=N=8 mass region. Also, theEIBM-2 allows us to describe the assumption I= 0 in both [4, 5] in a consistent way within the model.In the same way it is shown how the F spin multiplets in IBM-2 applications are consistent with an SU(4)symmetry breaking.By introducing both the particle-hole character and the charge character of the bosons, the EIBM-2 wasconstructed [6]. A rst possible reduction scheme for the dynamical algebra U (24) is :UL(6)  G OL(3)U (24)  SU (4) :(1)The Gstands for the IBM-1 reductions of U (6) [1].The generators for the SU (4) algebra can be considered as follows :(i) The generators of the SU (2) algebra for the intruder spin (I spin) for the -bosons and for the -bosons :^I()z= 1=2(^N()p ^N()h); (2)^I()+= syp;sh;+ dyp;~dh;; (3)^I() = syh;sp;+ dyh;~dp;; (4)with  = ; .(ii) The generators of the SU (2) algebra for the F spin for the particle bosons and for the hole bosons :^F()z= 1=2(^N() ^N()); (5)2^F()+= sy;s;+ dy;~d;; (6)^F() = sy;s;+ dy;~d;; (7)with  = p; h.(iii) The generators of the SUB(2) algebra :^Bz= 1=2(^N() ^N()); (8)^B+= syp;sh;+ dyp;~dh;+ syh;sp;+ dyh;~dp;; (9)^B = syp;sh;+ dyp;~dh;+ syh;sp;+ dyh;~dp;: (10)We denote these algebras respectively by SUI()(2); SUI()(2); SUF(p)(2); SUF(h)(2) and SUB(2).Taken together with the SUB(2) generators, the algebras SUI(2) and SUF(2), generated by symmetrisingthe operators in (, ) and (p,h) degrees of freedom respectively, now form an SU (3) algebra. So we canconsider a second reduction in EIBM-2 :UL(6)  G OL(3)U (24)  SU (4)  SU (3)  (SUI(2)  SUF(2)):(11)The above symmetries, as depicted in equations (1) and (11), can now be used to identify a number ofmultiplets according to the SU (4) and SU (3) symmetries. We denote the conguration of a nucleus with acertain number of particle-like and hole-like - and -bosons as (N()p; N()h; N()p; N()h) and can visualizethis in a schematical way as presented in gure 1.e e...e e e...eu u...u u u...uph Figure 1:Conguration of a nucleus with a given number of proton (particles and holes) and neutron (particlesand holes) pairs : N()p; N()h; N()p; N()hrespectively.An SU (4) multiplet is dened as all the eigenstates in all the nuclei for a conguration with a constanttotal number of bosons N =P;N()which are connected by the 10 ladder operators^I();^F();^B. If weconsider the action of the operator^B+on a nucleus which has a conguration described as in gure 1, wegenerate the conguration (N()p+ 1; N()h; N()p; N()h  1) + (N()p; N()h+ 1; N()p  1; N()h) (see gure 2).It is obvious that the same conguration can be established by action of the operator^I()+^F(h)++^I() ^F(p)+. Tond the equivalent operators for the action of the ladder operator^B we point out that it is the hermitianconjugate of^B+. Here, an important feature has to be pointed out : when one of! the ladder operators^Bacts on a state of a nucleus with a certain conguration as indicated in gure 1, the result will be thesum of two eigenstates of two dierent nuclei, as shown in gure 2. In order to avoid any further problemswe will use the convention that this 'mixing of states belonging to dierent nuclei' does not inuence thedenition of the multiplet. Of course, when considering the Hamiltonian and its symmetries, we have toconsider the implications of this. Using this particular convention, it becomes clear that all members of acertain SU (4) multiplet are connected by use of only 8 ladder operators associated with^I()and^F()spin.3Another implication of this convention is that the SU (4) multiplet can also be identied as identical with theSU (3) multiplet and also with the SUI(2) SUF(2) multiplet. For this reason, we shall only speak of SU (4)multiplets, characterized by a total number of bosons N .euQQQQQk+eu+Figure 2: The conguration generated by acting with the operator^B+on the above conguration, i.e. theresulting conguration (N()p+ 1; N()h; N()p; N()h  1) + (N()p; N()h+ 1; N()p  1; N()h).In order to describe a system with a constant number of particle-like and hole-like - and -bosons, we canwrite the following Hamiltonian :^H =X;^H()+X0;00;0;00^V(0;00)0;00; (12)where ; 0; 00= p; h and ; 0; 00= ; , 06= 00or 06= 00. By using the creation and annihilationoperators for the four types of bosons and considering up to second order, one obtains :^H()= E;0+Plm;l0m0;lm;l0m0by;lmb;l0m0+1=2Plimiu;(li;mi)by;l1m1by;l2m2b;l3m3b;l4m4;^V(0;00)0;00=Plimiw(0;00)0;00;(li;mi)by00;l1m1b00;l2m2by0000;l3m3b0000;l4m4:(13)Since this Hamiltonian conserves all N(), it satises :[^H;^I()z] = [^H;^F()z] = [^H;^Bz] = 0: (14)This also implies :[^H;^Iz] = [^H;^Fz] = 0: (15)All these conditions express particle-number conservation which is fundamental to describe eigenstates of anucleus with a certain conguration.We now call SU (4) a dynamical symmetry for the system described by the Hamiltonian (12) if the followingconditions are satised :[^H; (^I())2] = [^H; (^F())2] = [^H;^B2] = 0: (16)We note that, since for example [(^I())2;^B2] 6= 0, we cannot have eigenstates of the Hamiltonian with bothgood I()spin and good B spin.We call SU (4) a real symmetry for the system described by the Hamiltonian (12) if SU (4) is not only adynamical symmetry but if also the following conditions are fullled :[^H;^I()] = [^H;^F()] = [^H;^B] = 0: (17)Since it has been shown that^B+=^I()+^F(h)++^I() ^F(p)+; (18)^B =^F(h) ^I() +^F(p) ^I()+: (19)4we can conclude that SU (4) is a real symmetry for the system described by the Hamiltonian (12) when SU (4)is a dynamical symmetry and when the hamiltonian commutes with the 8 ladder operators^I()and^F().These are exactly the ladder operators needed to dene the SU (4) multiplet.Similarly, we call SU (3) a dynamical symmetry for the system described by the Hamiltonian (12) if thefollowing conditions are satised :[^H;^I2] = [^H;^F2] = [^H;^B2] = 0; (20)and SU (3) is called a real symmetry for the system described by the Hamiltonian (12) if not only SU (3) isa dynamical symmetry but if the following conditions are also satised :[^H;^I] = [^H;^F] = [^H;^B] = 0: (21)We remark that SU (4), as a real symmetry,immediately implies SU (3) as a real symmetry. Technically, thereremains a problem : Applying^I;^For^Bon an eigenstate belonging to one nucleus leads to the mixingof two eigenstates belonging to two dierent nuclei. This means that, though SU (3) can be considered as areal symmetry, the application of the ladder operators is ill-dened in the context of symmetries. Therefore,we shall restrict ourselves to examining the SU (4) as real or dynamical symmetry.We shall now illustrate the above, more general discussion, showing (i) experimental evidence for an SU (4)symmetry at the N=Z=8 shell closure, and (ii) discuss the appearance of the EIBM and IBM-2 structures asresulting from SU (4) symmetry breaking.0200040006000800010000Energy[keV]16O( )24Mg - 16O( ) -- 16O( ) assym.rotorK =0+0+2+4+6+(0)(1368)(4122)(9528)(6049)(6917)(10356)(16275)(12049)(13020)(16844)(21648)(14032)(15260)(18600)(23879)Figure 3: Comparison of the K= 0+N = 4 SU (4) multiplet members in16O and24Mg. Thereby we canput the (2p-2h)(2p-2h), (4p-4h), (4p-4h) excitations in16O (denoted respectively by16O(),16O(),16O()) and (4p)-(4p) (24Mg) band members in a given multiplet.(i) In the nucleus16O, there is experimental evidence for a K= 0+band and a K= 2+band associatedwith 4p-4h excitations, featuring the properties of an asymmetric rotor [7, 8]. The microscopic structureassociated herewith is a (2p-2h)(2p-2h) excitation [9]. In the EIBM-2 context, this means that these levelscan belong to the SU(4) multiplet with N = 4. If the SU(4) symmetry is a real symmetry for this multiplet,the same structure should be observed in the nucleus2412Mg12too. Also the levels associated with the (4p-4h)5020004000600080001000012000Energy[keV]16O( )24Mg - 16O( ) -- 16O( ) asymm.rotorK =2+2+3+4+6+(4238)(5235)(8439)(12002)(9844)(11080)(13869)(17555)(16352)(16817)(19375)(23000)(18202)(21500)Figure 4: Comparison of the triaxial K= 2+band for the same SU (4) multiplet members as mentioned ingure 3.and (4p-4h) excitation of168O8belong to the same multiplet and are good candidates to establish SU(4) asa real symmetry. Results are presented in gure 3 when comparing the K= 0+4p-4h multiplets in16O((2p-2h)(2p-2h); (4p-4h); (4p-4h)) with the (4p)-(4p) groundstate K= 0+band in24Mg. Levels are!tentatively associated with the (4p-4h) and the (4p-4h) congurations according to the SU (4) multipletstructure. Similar comparisons are carried out for the K= 2+triaxial bands as appearing in16O and in24Mg (gure 4). The comparison of experimental data gives good indications for the presence of a rather wellestablished SU (4) symmetry in these light nuclei.(ii) Since SU (4) can be reduced to the algebras SUI()(2); SUI()(2); SUF(p)(2) or SUF(h)(2), the EIBM andthe IBM-2 can be considered as an SU (4)-symmetry breaking while one of the mentioned SU (2) algebrabecomes the real symmetry for the system. The I spin multiplet members as mentioned in [4, 5] are anexample of SUI()(2) symmetry. By doing so, it is obvious why we then consider I()= 0. The algebraicstructure describing these multiplets is then given by :U()L(6)  G OL(3)U (24)  U()(12)  UI()(2)(22)where the generators of U (24) are [6] by;;lmb0;0;l0m0with ; 0= p; h describing the particle or holecharacter of the bosons, ; 0= ;  describing the charge character of the bosons and l; l0= 0; 2 for s-and d-bosons, while m =  l; :::; l and m0=  l0; :::; l0. The generators for U()(12); U()L(6) and UI()(2) arerespectively by;;lmb0;;l0m0;Pby;;lmb;;l0m0andPl;mby;;lmb0;;lm. The algebras Gand OL(3) arethose commonly used in the IBM-1 reduction [1]. One can e.g. describe the complementary symmetry ofSUI()at neutron closed shell nuclei N=50, N=82.The other symmetries, for example F(p)spin or F(h)spin, can be described using a complete analogousreduction scheme. Each of these symmetries reects the breaking of the SU (4) symmetry into a symmetryof a specic SU (2) subalgebra.In conclusion, we have shown how the Extended Proton-Neutron Interacting Boson Model (EIBM-2) allowsthe multi-particle-multi-hole excitations in168O8and the regular low-lying bands in2412Mg12to be describedas members of the same N = 4 SU (4) multiplet in the N=Z=8 mass region. Moreover, a consistent way has6been established in order to describe both the EIBM and the IBM-2 as resulting from an SU (4) symmetrybreaking. Other examples of realization of the SU (4) symmetry will be examined in other mass regions andresults will be published in a forthcoming article.AknowledgementsThree of the authors (B.D. C.D.C. and K.H.) thank the IWT, FWO and IIKW for nancial support. Oneof them (KH) gratefully acknowledges nancial support from CERN. We also thank P. Van Isacker and J.L.Wood for inspiring discussions.References[1] F. Iachello and A. Arima, The interacting boson model (Cambridge Univ. Press, Cambridge, 1987).[2] A. Frank and P. Van Isacker, Algebraic Methods in molecular and nuclear structure physics (John Wileyand Sons, Inc. New York, 1994).[3] C. De Coster, K. Heyde, B. Decroix, P. Van Isacker, J. Jolie, H. Lehmann, J.L. Wood, Nucl. Phys. A600(1996) 251.[4] H. Lehmann, J. Jolie, C. De Coster, B. Decroix, K. Heyde, P. Van Isacker, Nucl. Phys. A621 (1997),767.[5] C. De Coster, B. Decroix, K. Heyde, J.L. Wood. H. Lehmann, J. Jolie, Nucl. Phys. A621 (1997), 802.[6] B. Decroix, J. De Beule, C. De Coster, K. Heyde, A. Oros, P. Van Isacker, Phys. Rev. C Vol. 57 (1998),2329.[7] S. E. Larsson, G. Leander, S. G. Nilsson, I. Ragnarsson and R. K. Sheline, Phys. Lett. B47 (1973), 422.[8] S. Aberg, I. Ragnarsson, T. Bengtsson and R. K. Sheline, Nucl, Phys. A391 (1982), 327.[9] F. D. Becchetti et al., Nucl. Phys. A344 (1980), 336.7

SU(4) symmetry in the extended proton-neutron interacting boson model: multiplets and symmetry breaking

Abstract

Similar works

Full text

Available Versions

Ghent University Academic Bibliography

Crossref

CERN Document Server