In the present discussion we concentrate on shape coexistence asobtained within a deformed single-particle field as well as startingfrom the spherical shell-model, incorporating deformationeffects via the residual proton-neutron quadrupole interaction. Wediscuss in particular the appearance of shape coexisting phenomena inthe Pb region. In a second part then, we present a number ofexperimental fingerprints that allow to recognize the appearance ofshape coexisting phenomena or of shape mixing through the use ofselective experiments (e.g. band structure, spectroscopic factors,static moments, E0 properties and alpha-decay)

De Beule, J

De Coster, C

Decroix, B

Heyde, Kris L G

Oros, A M

CERN Document Server

EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHCERN-EP/98-18116.11.1998SHAPE COEXISTENCE IN ATOMIC NUCLEI AND ITSSPECTROSCOPIC FINGERPRINTSK. Heyde1;2, J. De Beule1, B. Decroix1, C. De Coster1;3andA.M. Oros11Dept. of Subatomic and Radiation Physics, Proeftuinstraat 86, B-9000 Gent2present address : EP-ISOLDE, CERN CH 1211, Geneva 233Postdoctoral fellow of the Fund for Scientic Research-Flanders (Belgium)AbstractIn the present discussion we concentrate on shape coexistence asobtained within a deformed single-particle eld as well as starting fromthe spherical shell-model, incorporating deformation eects via theresidual proton-neutron quadrupole interaction. We discuss in particu-lar the appearance of shape coexisting phenomena in the Pb region. Ina second part then, we present a number of experimental ngerprintsthat allow to recognize the appearance of shape coexisting phenom-ena or of shape mixing through the use of selective experiments (e.g.band structure, spectroscopic factors, static moments, E0 propertiesand alpha-decay).(ISOLDE GENERAL)(Invited Talk Presented at the International Workshop on\Nuclear Models 1998 : Interacting Bosons ans Fermions",Camerino, Italy, 12.-14.10.98)1Def.Figure 1: Schematic potential energy surfaces for spherical, quadrupole de-formed and transitional (shape coexisting) nuclei, respectively.1 IntroductionEven though the study of the nuclear many-body problem, starting froma given two-body nucleon-nucleon interaction in order to derive a mean-eld and study detailed nuclear properties within a shell-model framework,can produce a large number of observable quantities (energies, transitionrates, ...), the introduction of model concepts to describe the atomic nucleusstill plays an important role in capturing the essential physics. Nuclearshape variables and the study of nuclear dynamics using collective shapevariables have proven to provide a very interesting framework for describingthe atomic nucleus and its internal excitations [1].When studying the properties of nuclei in various parts of the nuclearmass table, the total energy associated with various shapes and shape changes,can give interesting insight in the nuclear dynamics. Conversely, the nuclearshell structure, using a liquid-drop model plus Strutinsky corrections [2](shell and pairing corrections), leads to specic variations in the energy sur-face dened in the space of quadrupole deformation variables. So, at closedshells, a rather sti spherical shape results whereas for regions with manyvalence protons and neutrons outside of closed shells, a deformed quadrupoleequilibrium shape results. In certain mass regions, often called transitionalmass regions, various shapes can result and often may even coexist. In othercases (see eg. the Sr,Zr region), mixing between the various possible shapescan show up (see g. 1 for an illustration of the three major classes).2184 186 188 190 192 194 196 198 200 202 204 206A01234E x(MeV)2+4+6+8+10+12+0+0+2+4+6+0+Figure 2: Systematics of low-lying intruder bands in the even-even Pb nuclei.The heavy (thin) lines denote the prolate 4p-4h and oblate 2p-2h congu-rations ([6] and refs. therein.2 Nuclear Shape CoexistenceThe deformed single-particle model (Nilsson model, deformed Woods-Saxon,mean-eld Hartree-Fock approach, ...) is a most interesting avenue in or-der to study the appearance of various shapes in a given nucleus, albeitassociated with a dierent excitation energy region [1]. The strong up- anddownsloping orbitals just below and above the closed shells are the initialelements that, even in an intuitive way, show the natural tendency for nucleito acquire a deformed shape next to the spherical shape corresponding to theclosed shell itself. A large number of calculations have been carried out inthat respect but here we mainly concentrate on the Pb region. Calculationsof Bengtsson and Nazarewicz [3] and Nazarewicz [4] have clearly indicatedthe presence of slightly deformed oblate and more strongly deformed prolatebands to show up and mainly coexist in the neutron decient Pb region (seealso J.Wood et al. for a review on this subject and many references to theliterature [5]). In the Hg nuclei, the ground-band structure relates to theoblate shape while the intruder conguration connects to the prolate shape.396 100 104 108 112 116 120N-1.00.01.02.0E x(MeV)Hg prolatePt prolatePb prolatePo oblatePb oblateFigure 3: Energies of the 0+band-heads in the even-even Po, Pb, Hg andPt nuclei. The change of scale for the Pt nuclei assumes that the changebetween oblate and prolate ground-state happens between mass numberA=188 and A=186.In the Pt nuclei, the prolate conguration even becomes the lowest-lyingconguration in the mass region 178  A  186. In the Pb nuclei, one ob-serves both deformed shapes but here, the spherical conguration remainsthe determining factor for the lowest-lying congurations that have beenobserved. A collection of the most recent data on the Pb nuclei is shown ing. 2 [6]. In this gure, one clearly distinguishes the rapid lowering of therst excited 0+state (associated to the oblate shape) while from mass num-ber A=190 onwards, a well-deformed band structure shows up that can beconnected to the more strongly deformed prolate band. This very neutrondecient Pb region, encompassing both even-even and odd-mass nuclei [7]in the Pt, Au, Hg, Tl, Pb, Bi, Po isotopes forms a most active region forresearch with experiments carried out at a number of laboratories.In g. 3, we present the excitation energy for the lowest-lying excited 0+states in these nuclei and an interesting feature shows up. Irrespective of theproton number, the energy of these states drops towards the neutron mid-482 86 90 94 98 102 106 110 114 118 122 126012345672( j - 'j )epepD EpairD EMD EQNEUTRON NUMBEREXCITATION ENERGY (MeV)Figure 4: The various energy terms contributing to the energy of the 0+lowest intruder 2p-2h conguration. The separate unperturbed, pairing,monopole and quadrupole energy contributions are given [8].shell conguration at N=104. This experimental fact, related to the clear-cutobservation of particle-hole excitations at and near to the closed shell referto a possible alternative but equivalent formulation for the description ofshape coexisting phases near closed shells.A spherical shell-model approach has been formulated in which the un-derlying microscopic structure of low-lying intruder 0+states is largely con-nected to the presence of 2p-2h excitations across the proton closed shellat Z=82 (for the Pb nuclei) [8]. Similar and compelling evidence has beenobtained at the Z=50 shell closure too [5]. It was suggested that this maywell form a rather general basis for the formation of low-lying intrudingcongurations. In g. 4, the essential various energy terms are shown thatcontribute to the lowering of the rather high-lying unperturbed 2p-2h ex-citations in a shell-model framework. Besides the pairing energy gain andsmaller variations due to the nuclear monopole corrections originating in theproton-neutron interaction, the most dramatic lowering of the energy origi-5100 104 108 112 116 120 124Neutron Number N2.04.06.08.0ExcitationEnergy(MeV)Figure 5: The energy dependence of the lowest 2p-2h and 4p-4h intruder 0+excitations in the even-even Pb nuclei, according to the use of eqs. (1) and(2) [8].nates from the attractive quadrupole proton-neutron force. In the simpliedcase of a separable quadrupole-quadrupole force, the energy dierence be-tween the regular and intruder conguration can be depicted (see also g. 4)asEintr:(0+) = h0+Ij^Hj0+Ii   h0+GSj^Hj0+GSi ; (1)EQ' 2 (N)N; (2)where the latter expression is a very good approximation to the more re-alistic evaluation of the energy dierence. Here,  denotes the quadrupoleforce strength, Nthe number of neutron pairs and N, the number ofextra pairs formed in creating the intruder state. The use of the above pre-scription, discussed in detail in ref. [8] for the Pb nuclei, is illustrated, ing. 5, for the 2p-2h (oblate) states and the 4p-4h (prolate) states. In thereproduction of the data, only a single parameter i.e. the quadrupole forcestrength , shows up.696 100 104 108 112 116 120N0.01.02.03.0E x(MeV) Hg 2p-4h ?Pt 2p-6hPb 4p-4h Ex/2Po 4p-2hPb 2p-2h (a)96 100 104 108 112 116 120N0.01.02.03.0E x(MeV) Hg 4p-6h ? Ex/2Pt 2p-6hPb 4p-4h Ex/2Po 4p-2hPb 2p-2h (b)Figure 6: Illustration of the quadrupole energy scaling law (eq. (2)) as ap-plied to the Pb region. In part (a), we assume a 2p-4h structure for the0+intruder band in Hg nuclei whereas in (b) the assumption of a 4p-6hstructure is made in evaluating the scaled energy.7We can now make use of the result obtained within the spherical shell-model approach in order to suggest a possible explanation for the conspic-uous behaviour of the 0+excitation energy in the Pb region as a functionof neutron number as was illustrated in Figure 3. If indeed, the quadrupoleenergy gain of eq. (2) is the determining factor, then a scaling law shoulddescribe the energy of the various mp-nh intruder congurations relative tothe reference state for a given nucleus. This scaling factor is exactly thenumber of extra p-h pairs that are formed in creating the excited state (seeeq. (2)).So, by dividing the excitation energy of the 4p-4h excitations by a factor2, all excitation energies should ideally fall on a single curve. We presentresuls in g. 6a,b for two dierent assumptions concerning the excited 0+band in the Hg nuclei since the microscopic origin is not unambigously de-ned. The gure 6b illustrates that the origin as a 4p-4h excitation on topof the existing 2 holes in the Hg nuclei gives an even better illustration ofthe apparent scaling law.We nish this section by pointing out that the approach starting froma deformed single-particle mean eld, incorporating residual pairing cor-relations or starting from a spherical shell-model approach but includingthe important eects caused by the strong and attractive proton-neutronforce give very equivalent results. Also the application to the Pb region,at present forming the most documented region in the nuclear mass table,is not unique. Applications to the N=50, the N=82 mass regions, the Snregion but also to much lighter nuclei such as the N=20 nuclei have beencarried out [5].3 Experimental FingerprintsA number of very specic experimental ngerprints provide the necessaryelements to classify an observed excited band as a shape coexisting band(although quite often the presence of mixing between the normal and theintruding bands shows up through specic perturbations in standard energyand decay patterns).First, on the level of the excitation energy and the decay properties, itis clear that the band, corresponding to a much more quadrupole deformedshape, can show rotational features. Even more important is the presenceof very strong intraband B(E2) values and hindered or much reduced inter-band decay [5]. A recent study of the intruder band in114;116Sn is a verynice example and we illustrate part of the band, incorporating also decayproperties, in g. 7 (taken from reference [9]). This situation as well as theone in the other nearby even-even Sn nuclei shows overwhelming data asto the observation of a shape coexisting band exactly at the Z=50 protonclosed shell (in116Sn, the band extension all the way up to spin 20+has been811/2-19/2-15/2-978131231(12)W.u4.7(16)W.u27/2-23/2-82912938185891057144674247519/2-15/2-19/2-115Sn114Sn3.5(11)W.u<11W.u<55W.u26(7)W.u88813008+6+4+4+2+ 4+2+0+<21W.u15(3)W.u68257537593913141001120(48)W.u98(20)W.u50(50)W.u14651.6(7)W.u19(7)W.u4244.0(24)W.u1.6(3)W.u <2W.uFigure 7: Comparison of the low-lying intruder bands in114Sn and115Sn.The B(E2) values are also given in Weisskopf units (W.u.) (taken from [9]).observed [9]). The Hg and Pt nuclei as well as the more recently studied Pbnuclei are also very nice examples on this ngerprint of band structure.A second and equally important ngerprint is brought about by one-,two- and even cluster-transfer reactions. Those reactions can very selectivelypopulate nal states with a given particle-hole structure by appropriate se-lection of the initial nucleus and the transferred particle(s).In this context,it was shown clearly that e.g. in the Sn nuclei, one-nucleon transfer reac-tions starting with In or Sb as target nuclei, preferentially populate statesabove 4 MeV [5] (the energy gap at Z=50) and do not feed the intruderband structure whereas two-proton transfer (3He,n) reactions [10] startingfrom the even-even Cd nuclei, rather strongly populate the 0+band-headof these intruder bands. We like to stress the importance of getting moresystematic data on selective transfer into the proposed intruder bands sincethose experiments can give specic answers as to the underlying microscopicstructure for many of the observed bands. Unfortunately, in many cases, theparticular nuclei of interest are far from the region of stable nuclei and thoseexperiments cannot be carried out.A specic ngerprint is found in the study of static moments such asnuclear radii, electric quadrupole moments (indicating direct evidence for9quadrupole deformation) and also magnetic dipole moments. The lattercan play a selective role as they are ideal indicators for the single-particlestructure of a given state. In particular, the important work on systematicstudies of radii over many long mass chains has given unvaluable insight inthe deformation behaviour and the eect that shape changes may cause tothe atomic nucleus, even in its ground-state [11]. The experiments in theZ=40 region (Sr, Zr, ...) and also in the Pb region (Pb, Tl, Hg, Au andPt) have brought dramatic examples for shape coexistence and also for veryabrupt shape changes (see e.g. at N=60 in the mass A=100 region and nearthe neutron mid-shell N=104 in the Pb region in particular the Au odd-mass nuclei [11]). Quadrupole moments are known to a much lesser extentbut out of B(E2) rates in the intruder band, transition quadrupole momentscan be derived (see e.g. the odd-mass In and Sb nuclei, the even-even Hgand Pt nuclei [5]) and in all cases, the derived moment indicates quadrupoledeformations that are typical for strongly deformed nuclei and not at allwhat one would expect for nuclei at or near to closed-shell congurations.A most interesting ngerprint follows from studying E0 transitions andthe reduced 2values (similar to reduced matrix elements for the other elec-tromagnetic transitions). It has been shown that very large 2(E0) valuesare an optimal indication for strong mixing between states for which thewavefunctions are localized at largely dierent values of the deformationshape coordinate [12, 13]. By studying a particular two-level model thesalient features of such mixing already show up. More precise calculations,using either the nuclear shell-model or some other collective model approach,support this idea. We refer to a forthcoming paper of Wood et al. for anextensive discussion on E0 monopole properties in nuclei [14]. The essen-tial expression that relates the 2value to the dierence in mean-squaredeformation reads21;2= a21  a2342Z222  212; (3)where a indicates the mixing between the two unperturbed levels and weuse the denition for 2ias2i h0+ijXj2j2j0+ii : (4)In ref. [14] besides a number of interesting examples of how the observedE0 rates may well correlate with this simple expression, it is shown that onecan relate the 2value with the isotopic shifts. Here, we restrict ourselvesto illustrate E0 transitions in the mass A=100 region. Fig. 8 very clearlyillustrates the observation of the largest 2values observed in atomic nuclei,a result that most probably serves as a good ngerprint for strong shapemixing near N=58,60.1050 52 54 56 58 60 62 64102101SrZrMoRuPd2rx 103Neutron numberFigure 8: Experimental values for 2(E0) in the mass A=100 region. Refer-ences for the data can be found in [14].11Finally, -decay, much like the -particle transfer reactions, is able toprobe shape coexistence or shape mixing. The study of the very neutron-decient nuclei in the Pb region has been very intensively pursued over thelast years, in particular as the neutron-decient highly unstable nuclei have-decay as a major decay mode. For a compilation of the most recent results,we refer to the Ph.D. thesis of N.Bijnens where the intruding congurationsand more important, their mixing, can be studied over a large mass interval[15].4 ConclusionIn the present contribution, we have shown ample evidence for the presenceof shape coexisting congurations, in particular near closed shells in manyregions of the nuclear mass table. We have indicated that an approach whichstarts from a deformed single-particle eld, in order to evaluate the total en-ergy of the nucleus as a function of deformation, as well as an approach start-ing from a spherical shell model accentuating the proton-neutron quadrupoleforce can give a good account of those intruder structures. Moreover, bothmethods are equivalent. From the spherical shell-model approach, moreover,an interesting scaling law follows that is qualitatively able to describe themass behaviour of the excitation energy for the 0+band-head in many ofthese intruding congurations.In a second part, we have presented and illustrated a number of inter-esting spectroscopic ngerprints in order to obtain unambiguous and alsoselective evidence for the fact that a number of observed low-lying bandsindeed can be described as mainly shape coexisting congurations. We alsoshow the need to combine the outcome of a large number of dierent probes(band structure, E2 transitions, spectroscopic factors in transfer reactionsor -decay, the study of moments and E0 transitions if possible, ...) in orderto come to reliable conclusions about the precise nature of low-lying bandsin the vicinity of closed shells.AcknowledgementsThe authors like to thank the FWO and IWT for nancial support.One ofthe authors (KH) thanks CERN for nancial support. They are grateful formany discussions with a large group of people working in this eld of shapecoexistence,too many to mention them all here in this limited space.References[1] A.Bohr and B.Mottelson, Nuclear Structure, vol.2 (Benjamin,New-York), 197512[2] M.Brack, J.Damgaard, A.S.Jensen, H.C.Pauli, V.M.Strutinsky andC.Y.Wong, Revs.Mod.Phys. 44 (1972), 320[3] R.Bengtsson and W.Nazarewicz, Z.Phys. A334 (1989), 269[4] W.Nazarewicz, Phys.Lett. B305 (1993), 195[5] J.L.Wood, K.Heyde, W.Nazarewicz, M.Huyse and P.Van Duppen,Phys.Repts. 215 (1992), 101[6] G.D.Dracoulis, A.P.Byrne and A.M.Baxter, Phys.Lett. B432 (1998),37 and refs.therein[7] K.Heyde, P.Van Isacker, M.Waroquier, J.L.Wood and R.A.Meyer,Phys.Repts. 102 (1983), 291[8] K.Heyde, J.Jolie, J.Moreau, J.Ryckebusch and M.Waroquier,Nucl.Phys. A466 (1987), 189[9] A.Savelius et al. Nucl.Phys. A637 (1998), 491[10] H.W.Fielding et al., Nucl.Phys. A281 (1977), 389[11] E.Otten, in Treatise on Heavy-Ion Science, ed. A.Bromley (PlenumPubl. Co.,1989), vol8, 517[12] J.Kantele in Heavy-Ions and Nuclear Structure, Proc.of the XIV Sum-mer School Mikolajki, 1984, eds.B.Sikora and Z.Wilhelmi (HarwoodAcad. Publ., New-York, 1984), 391[13] K.Heyde and R.A.Meyer, Phys.Rev. C37 (1988), 2170[14] J.Wood, E.F.Zganjar, C.De Coster and K.Heyde, submitted for publ.[15] N.Bijnens, Ph.D.Thesis, University of Leuven, 1998, unpublished13

Shape coexistence in atomic nuclei and its spectroscopic fingerprints

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