the islands of shape coexistence within the elliott and

The islands of shape coexistence within the Elliott and the proxy-SU(3) Models

Andriana Martinou1, Dennis Bonatsos1, T. J. Mertzimekis 2, K. E. Karakatsanis1,3,

I.E. Assimakis1, S. K. Peroulis1, S. Sarantopoulou1, and N. Minkov4

1Institute of Nuclear and Particle Physics, National Centre for ScientificResearch “Demokritos”, GR-15310 Aghia Paraskevi, Attiki, Greece

2Department of Physics, National and Kapodistrian University of Athens, Zografou Campus, GR-15784 Athens, Greece3Department of Physics, Faculty of Science, University of Zagreb, HR-10000 Zagreb, Croatia and

4Institute of Nuclear Research and Nuclear Energy,Bulgarian Academy of Sciences, 72 Tzarigrad Road, 1784 Sofia, Bulgaria

A novel dual–shell mechanism for the phenomenon of shape coexistence in nuclei within the ElliottSU(3) and the proxy-SU(3) symmetry is proposed for all mass regions. It is supposed, that shapecoexistence is activated by large quadrupole-quadrupole interaction and involves the interchangeamong the spin–orbit (SO) like shells within nucleon numbers 6-14, 14-28, 28-50, 50-82, 82-126,126-184, which are being described by the proxy-SU(3) symmetry, and the harmonic oscillator(HO) shells within nucleon numbers 2-8, 8-20, 20-40, 40-70, 70-112, 112-168 of the Elliott SU(3)symmetry. The outcome is, that shape coexistence may occur in certain islands on the nuclear map.The dual–shell mechanism predicts without any free parameters, that nuclei with proton number (Z)or neutron number (N) between 7-8, 17-20, 34-40, 59-70, 96-112, 146-168 are possible candidates forshape coexistence. In the light nuclei the nucleons flip from the HO shell to the neighboring SO–likeshell, which means, that particle excitations occur. For this mass region, the predicted islands ofshape coexistence, coincide with the islands of inversion. But in medium mass and heavy nuclei, inwhich the nucleons inhabit the SO–like shells, shape coexistence is accompanied by a merging of theSO–like shell with the open HO shell. The shell merging can be accomplished by the outer productof the SU(3) irreps of the two shells and represents the unificaton of the HO shell with the SO–likeshell.

I. INTRODUCTION

Ever since the beginning of Nuclear Physics afterRutherford’s discovery of the atomic nucleus severalquestions have arisen about its composition and thestructure. Some questions have been tantalizing nuclearscientists for decades, while some still do; especially thoserelated to the fundamental aspects of nuclear forces andthe dynamics governing the coexistence of protons andneutrons in the tiny volume of the nucleus.

The groundbreaking work by Mayer [1, 2] revealed theexistence of energy shells as the main feature of nuclearstructure. Particle excitations across such shells obeyquantum rules defined by the nuclear Hamiltonian. Verysoon after the formulation of the Shell Model it becameclear, that collective nuclear phenomena can have a di-rect correspondence to the single particle character, re-flected on proton and neutrons interactions providing themeans for a firm explanation of rotations and vibrations,recorded in experimental data in studies throughout thenuclear chart.

In this framework the shape of the nucleus has a cen-tral role in understanding the interplay of nuclear con-stituents at fundamental level. The presence of sym-metries, but also any violations of them, can be stud-ied experimentally by measuring the shape of nuclei inthe ground and excited states and taking advantage ofthe information becoming available through shape evo-lution among different structures in the same nucleus orin different nuclei. The correlation of shape evolutionto proton and neutron numbers is not yet fully under-

stood. The advent of radioactive beams has drasticallyexpanded the limits of the known nuclear chart, posingvarious new and important questions about nuclear struc-ture at the extremes.

The behavior of the nuclear forces is responsible for theoccurrence of phenomena like shape coexistence, in whichan excited band with a shape different from that of theground band resides close to the latter. Experimentally,shape coexistence was observed early in various types ofreactions, see e.g. [3], but experimental studies have beenintensified significantly in recent years.

The experimental interest was initially focused on rareearth isotopes, at masses A ≈ 150, such as Gd and Sm [4–7], but also in heavier masses near the magic lead nuclei[8, 9], an effort, that escalated significantly later on.

The discovery of the island of inversion around theneutron rich Mg isotopes, where the nuclear interactiondrives strong deformations, shuttering the “traditional”predictions of the shell model, has created new excite-ment on shape coexistence at the experimental level.Early spectroscopic studies (measurements of α spec-troscopic factors and E0 values) in Ar [10] and Ca iso-topes [11–13] provided strong hints for shape coexistencearound the upper end of the sd shell. The search forsimilar occurrences has been intense in the fp shell, aswell, where the protons and neutrons occupy orbitalswith similar Fermi energies, giving rise to strong proton–neutron correlations. Strong deformations, that correlateto shape coexistence phenomena in neutron–deficient Se[14] and Kr [15] nuclei, have been examined experimen-tally in line with similar studies in neighboring nuclei,such as Ge [16], Se [17, 18] and Sr [19] nuclei.

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This mid–weight mass regime around A ∼ 80 has beencentral in a series of experiments “east” and “west” of thevalley of stability. Due to the increasing use of radioac-tive beams, neutron–rich krypton isotopes have beenexamined thoroughly [20–22] to understand any poten-tial evolution of shape coexistence when neutron numbercrosses the magic number N = 50. Such studies have re-lied on Coulomb excitation reactions, which are known tofavor strongly E2 transitions, however they suffer ratherlow statistics when unstable beams are employed.

Shape coexistence is now recognized as a rather com-mon phenomenon potentially occurring in all nuclei [23].Despite the fact that several theoretical models have at-tempted to explain its dynamics (see the discussion laterin the present work), no explicit mechanism is knownto describe it fully at a fundamental level. The implicitstrong correlations in the fp shell, as well as the predic-tion of islands of shape coexistence in the heavier regionof the nuclear chart [23] has expanded the experimentalefforts towards consolidating the spectroscopic signaturesof shape coexistence in nuclei with A ∼ 120 − 180. Ex-perimental data in Mo [24], Ag [25], Cd [26], La [27], andrecent measurements of lifetimes in neutron–deficient nu-clei in the lead region [28, 29] provide strong evidenceof the onset of shape coexistence at neutron numbersnear to 108. Given the large gap of knowledge aboutcharacteristic experimental signatures of shape coexis-tence, such as lifetimes and reduced matrix elements, inneutron–rich Ytterbium and Hafnium nuclei, there re-main open questions on the universality of shape coexis-tence across the large shell forming above N = 82. Theexistence of odd parity bands next to the ground stateband, as well as the presence of a few low–lying 0+ states,hint the onset of shape coexistence.

According to the presently recognized understanding ofthe shape coexistence at certain proton or neutron num-ber, a multi particle–hole (mp–mh) configuration acrossthe closed shell or sub–shell appears to possess similarbinding energy, compared to the 0p–0h configuration.This can be explained by a specific correlation of energycontributions involving pairing, quadrupole and other in-teractions, providing the “intruder” mp–mh state with adeformation different from that of the ground state.

In the present work a novel dual–shell mechanism forshape coexistence is proposed. It is supposed that twotypes of open valence shells are active: the HO shell andthe SO–like shell. The coexistence of these two types ofshells is allowed at certain nucleon numbers, where thequadrupole-quadrupole interaction in the SO–like shell isgreater than the one of the HO shell.

As we shall see below, the new mechanism is fully com-patible with the particle–hole mechanism in the regionsof the nuclear chart, in which the latter has been ap-plied. The novel suggestion of the present work is, thatshape coexistence can only occur within certain islandsof the nuclear chart, fully compatible with the regions,in which shape coexistence has been seen experimentally[23, 30]. In contrast to the common belief, that shape

coexistence can appear at any place of the nuclear chart[23], we prove, that this is possible only within certainwell defined islands of the nuclear chart.

In order to determine the shores of these islands, wewill need the SU(3) symmetry of the three-dimensional(3D) isotropic harmonic oscillator [31, 32], used withinthe Elliott model [33–36], which is applicable in lightnuclei up to the sd shell, as well as the recently intro-duced proxy-SU(3) symmetry [37–39], which is applica-ble beyond the sd shell in medium-mass and heavy nu-clei. Shape coexistence is recognized to be related tosingle–particle energy gaps, which are affected by the de-formation [23]. Since it is a “deformation–driving” phe-nomenon (see section 6 of Ref. [23]), one should do thecalculations in a deformed basis and after that, projectedthe overall wave function on states with good angularmomentum. This task is easily done within the Elliottand the proxy-SU(3) symmetry, in which the cartesiansingle particle basis [39] is initially used for the intrinsicframe and as a second step the projection onto a rota-tional invariant wave function follows [35, 36, 40, 41].The Elliott and proxy-SU(3) symmetry and the will bereviewed in sections II and III. An application of the pro-jection technique will be demonstrated in section X B forthe calculation of the energy of the 0+2 of 16O.

The basic element of the new mechanism, namely par-ticle excitations among the HO shells defined by the 3Disotropic HO magic numbers 2, 8, 20, 40, 70, 112, 168,and the SO–like shells defined by the SO magic numbers6, 14, 28, 50, 82, 126, 184 [42] are described in sectionIV and a detailed example for the halo nucleus 11Be isgiven. In section V it is explained, how the the disso-lution of magic numbers emerges within certain areas ofthe nuclear chart, resulting in the merging of the HO andSO–like shells. The compatibility between the particle–hole mechanism and the present new mechanism is alsodiscussed in section V, before summarizing the main fea-tures of the new mechanism in section VI.

The two low–lying nuclear bands being candidates forshape coexistence are discussed in terms of their rele-vant symmetry features (the SU(3) irreducible represen-tations (irreps) in mathematical language) in section VII,while in section VIII the shores of the islands of shape co-existence are determined through a simple Hamiltonianbearing the SU(3) symmetry. The sudden onset of defor-mation in certain regions of the nuclear chart and its rolein clarifying the relation between shape coexistence andshape phase transitions [43–45] is discussed in section IX,while in section X specific examples of shape coexistenceappearing in seven different regions of the nuclear chartare presented. The conclusions based on the present find-ings and plans for further work are given in section XI.At last a nuclear chart with the possible islands of shapecoexistence, predicted by the dual–shell mechanism, ispresented in Fig. 25.

3

II. THE ELLIOTT SU(3) MODEL

The Elliott SU(3) symmetry [33, 34] can be consideredas a beyond-Shell-Model theory, in which the rotationalspectrum of deformed nuclei emerges naturally from thecoupling of the valence particles [34]. The Elliott SU(3)symmetry is valid in a valence proton or neutron shell,constituted by orbitals of the 3D-HO Hamiltonian withcommon number of quanta N . Such shells lie amongproton or neutron numbers 2, 8, 20, 40, 70, 112, 168,namely the HO magic numbers.

The Elliott SU(3) model uses the cartesian states|nz, nx, ny,ms〉 as building blocks [35, 36, 39, 46], wherems = ± 1

2 is the spin projection of the nucleon andnz, nx, ny are the oscillator quanta in the three cartesianaxes of the single particle 3D-HO Hamiltonian:

h0 =p2

2M+

1

2Mω2r2, (1)

with M,p, r, ω being the nucleon mass, the momentum,the spatial coordinate and the oscillator frequency re-spectively. The eigenvalues of the Hamiltonian (1) are[47]:

ε0 =

(N +

3

2

)~ω, (2)

where

N = nz + nx + ny. (3)

The eight generators of the SU(3) algebra are the l0,l±1, q±2, q±1, q0 [35, 46, 48], where l0, is the operator ofthe projection of the orbital angular momentum, l±1 arethe ladder operators of the algebra of the orbital angularmomentum and q±2, q±1, q0 are the five components ofthe algebraic quadrupole operator [49]. These generatorscan be expressed in terms of the well known creation andannihilation operators a†z ,a†x, a†y, az, ax, ay of a quan-tum in each cartesian direction [35, 46, 48]. Of utmostimportance is the q0, which measures the elongation ofthe z axis [34]:

q0 = 2nz − nx − ny. (4)

On the one hand the cartesian states |nz, nx, ny,ms〉are eigenstates of the single particle 3D-HO Hamiltonianh0 and of q0. The cartesian basis is suitable for the de-scription of the single–particle states of deformed nuclei[35], in which the quadrupole–quadrupole, two body in-teraction of the many nucleon system [48] :

QQ =∑

i,i′,m=±2,±1,0(−1)mqmiq−mi′ (5)

prevails. In the above the summation is for every valence(i, i′) nucleon pair and for every m component of thequadrupole operator.

On the other hand the cartesian states are not eigen-states of a) the single particle spin-orbit interaction l · s,where l, s are the orbital angular momentum and the spinof the nucleon respectively, and of b) the l2 interaction.These interactions are included in the Shell Model [1, 2]single particle Hamiltonian [50]:

h = h0 + υls~ωl · s + υll~ω(l2 − 〈l2〉N ), (6)

where υls, υll are strength parameters [50] (see Table Iof Ref. [37] for their values) and the term l2 − 〈l2〉N isusually used for the flattening of the HO potential.

The eigenstates of the Hamiltonian of Eq. (6) are ex-pressed in the spherical coordinate system as |n, l, j,mj〉,where n is the radial quantum number n = 0, 1, 2 with[51]:

N = 2n+ l, (7)

j is the total angular momentum and mj is its projec-tion on the z axis [52]. The |n, l, j,mj〉 states are theusual Shell Model orbitals, if one adds one unit in theradial quantum number n and represents the orbital an-gular momentum l = 0, 1, 2, ... by the small latin char-acters s, p, d, .... For instance the orbitals |n, l, j,mj〉:|0, 1, 32 ,

12 〉, |1, 2,

52 ,

32 〉 are labeled as 1p

j=3/2mj=1/2, 2d

j=5/2mj=3/2

in the Shell Model notation respectively.Despite the fact that the |nz, nx, ny,ms〉 are not eigen-

states of the Hamiltonian (6), they can be transformedinto them [39]. In Tables 1, 3, 5 of Ref. [39] ex-plicit expressions were given for the transformation ofthe |nz, nx, ny,ms〉 states to the |n, l, j,mj〉 orbitals. Assuch the cartesian orbital |1, 0, 0, 12 〉 transforms as:

|1, 0, 0,±1

2〉 = ∓ 1√

3|1p1/2±1/2〉+

√2

3|1p3/2±1/2〉 . (8)

The |1p1/21/2〉 orbital is the |n, l, j,mj〉 = |0, 1, 12 ,12 〉, while

the |1p3/21/2〉 is the |n, l, j,mj〉 = |0, 1, 32 ,12 〉, both of them

possessing N = 1 number of oscillator quanta.Nevertheless both types of bases |nz, nx, ny,ms〉,|n, l, j,mj〉 are eigenstates of the h0 of Eq. (1):

h0 |nz, nx, ny,ms〉 = ε0 |nz, nx, ny,ms〉 , (9)

h0 |n, l, j,mj〉 = ε0 |n, l, j,mj〉 , (10)

where ε0 is given by Eq. (2) and N is given by Eqs. (3),(7).

Since the nucleons are fermions and thus they obey thePauli exclusion principle [53, 54], the many–body nuclearwave functions are Slater determinants [55] of the single–particle states [35, 40, 56]. Initially in the Elliott SU(3)symmetry at Ref. [34] the many particle, intrinsic nu-clear wave functions X in the cylindrical coordinate sys-tem had been projected into the Ψ many particle wavefunctions in the spherical coordinates, which possess theorbital angular momentum L as a good quantum num-ber. Notice, that throughout the present work we use

4

capital letters for angular momentum quantum numbersreferring to the many–body system, while we use lowercase letters for angular momentum quantum numbers re-garding the single–particle states. It has been proven inRef. [34], that a single X wave function suffices, to gen-erate all the states of the rotational bands, which belongto the same SU(3) irrep.

It was later realized [35, 40], that the description ofthe rotational features of deformed nuclei becomes evensimpler in the cartesian coordinate system, in which themany–particle nuclear wave functions are labeled by Φ.The Φ many particle wave functions are being derived bythe single particle orbitals |nz, nx, ny,ms〉 in the carte-sian coordinate system. The L or J-projection of a sin-gle Φ wave function to wave functions, which possess theanugular momentum or the total angular momentum re-spectively as a good quantum number, derives all the nu-clear rotational states, which belong to the same SU(3)irrep (λ, µ) [35, 36, 40].

To conclude, the single–particle basis |nz, nx, ny,ms〉and the many–body wave function Φ, which is derivedfrom them, is the intrinsic basis, which describes de-formed nuclei in the Elliott SU(3) symmetry. An L orJ projection of the cartesian intrinsic wave function re-veals the rotational energy bands, which belong to thethe same SU(3) irrep [35, 36, 40].

The cartesian states are used to derive the SU(3) irreps(λ, µ). The subject of the calculation of (λ, µ) has beenexhausted in Refs. [33, 34, 57–61]. The important thingfor the dual–shell mechanism for shape coexistence is,that the low-lying nuclear properties are being derivedby the highest weight SU(3) irreps [38, 61]. This specialirrep emerges, when the valence nucleons fill the spatial,cartesian orbitals with the following specific order:

|nz, nx, ny〉 : |N , 0, 0〉 , |N − 1, 1, 0〉 , |N − 1, 0, 1〉 ,|N − 2, 2, 0〉 , |N − 2, 1, 1〉 , |N − 2, 0, 2〉 , ..., |0, 0,N〉 .

(11)

If this filling order is followed, then [34]:

λ =∑i

nz,i −∑i

nx,i, (12)

µ =∑i

nx,i −∑i

ny,i, (13)

where the summations are over every valence nucleon.For the many–body problem the relevant 3D-HO

Hamiltonian is the summation over every nucleon:

H0 =

A∑i=1

h0,i, (14)

where A is the mass number. The eigenvalues of H0 arelabeled as N0 [62]:

N0 =

A∑i=1

(Ni +

3

2

)~ω, (15)

with N given by Eqs. (3), (7).The overall algebraic quadrupole-quadrupole (QQ) in-

teraction in the Elliott SU(3) model is calculated as in[63, 64]:

QQ = 4C2 − 3L(L+ 1), (16)

where L(L + 1) is the eigenvalue of the square of theorbital angular momentum operator for all the valenceparticles [33, 48]:

L =∑i

li (17)

and C2 is the second order Casimir operator of SU(3)with eigenvalues [36, 65]:

C2 = λ2 + µ2 + λµ+ 3(λ+ µ). (18)

The C2 operator is linked to the nuclear quadrupole de-formation parameter β of the Bohr and Mottelson Col-lective Model [66] through the equation [67]:

β2 =4π

5(Ar2)2(C2 + 3), (19)

with r2 = 0.872A1/3 is the dimensionless mean squareradius (rms). Since usually C2 3, β2 is proportionalto C2, therefore one may treat the C2 as a measure ofdeformation. Due to fact that the Elliott SU(3) Modeltreats the algebraic QQ interaction, which has non zeromatrix elements among single–particle states with thesame number of quanta N , while the measured QQ in-teraction is the collective one, which has non zero matrixelements among single–particle orbitals with N ,N ± 2quanta [49, 64], the formula of Eq. (19) needs to be mul-tiplied by a scaling factor as in Ref. [38]:

A2

(S$ + Sν)2(20)

where S$, Sν is the size of the valence proton and neutronshell respectively.

A very simple SU(3) Hamiltonian, which includes onlythe 3D-HO and the QQ interaction and adequately de-scribes the L = 0 states, is the following [68]:

H = H0 −κ

2QQ, (21)

where the H0 was defined in Eq. (14), while the QQ isbeing calculated by Eqs. (16) and (18). The parameterκ is [68]:

κ =~ω2N0

, (22)

where in the numerator [56]:

~ω =41

A1/3MeV, (23)

5

while in the denominator the N0 is calculated with Eq.(15) by setting ~ω = 1.

The Elliott SU(3) symmetry is valid in a 3D-HO va-lence shell, which is constituted by orbitals with commonnumber of quanta N and lies among the 3D-HO magicnumbers 2, 8, 20, 40, 70, 112, 168. Since a significantspin–orbit interaction leads to the spin–orbit like (SO)magic numbers 6, 14, 28, 50, 82, 126, 184 one wouldexpect, that in the Elliott scheme either there is no spin–orbit interaction, or that this interaction is small. Nev-ertheless the spin–orbit interaction can and has to beapplied in the Elliott SU(3) scheme with the techniquesapplied in Refs. [35, 36, 40, 69].

It has also been proved, that the spin–orbit interaction[35, 36]:

Vs.o. = ξ∑i

li · si (24)

(with ξ being a parameter, while i counts every valencenucleon) becomes stronger as more valence nucleons areadded in the valence 3D-HO shell [40, 69]. At the secondhalf of a HO shell the spin–orbit interaction has grownso much, that one has to change either to a jj couplingscheme among the nucleons of the HO shell [40, 69], or toa spin–orbit like shell [1, 2]. In rotational nuclei the 3D-HO magic numbers are valid as long as the |n, l, j,mj〉orbitals with common N mix together. For instance aslong as the 2s1/2, 1d3/2 mix with the 1d5/2, the sd 3D-HOshell among magic numbers 8, 20 is being created. Butwhen a strong overall spin–orbit interaction, as definedin Eq. (24), applies in this shell, then the normal par-ity orbitals with N number of quanta: 2s1/2, 1d3/2 jointogether with the intruder parity orbitals with N + 1number of quanta: 1f7/2 and they create the SO–likeshell among magic numbers 14, 28.

III. THE PROXY-SU(3) MODEL

In the SO–like shells [42] among magic numbers 6, 14,28, 50, 82, 126, 182 (see Table 7 of [39]) the normal parityorbitals contain N quanta, while the intruders N + 1and as a result the Elliott SU(3) Model cannot have astraightforward application. In such SO–like shells theproxy-SU(3) symmetry can be applied instead [37–39,70]. The proxy-SU(3) symmetry had been introduced inRef. [37] and delivered at once parameter-free predictionsfor the nuclear deformation in Ref. [38].

The proxy-SU(3) Model restores the SU(3) symme-try in the notation of the asymptotic basis of the Nils-son Model [71] by replacing the intruder Nilsson orbitalsK[NnzΛ], where K,Λ are the projections of the to-tal and the orbital angular momentum respectively, bytheir ∆K[∆N∆nz∆Λ] =0[110] counterparts [37]. Thisreplacement is not altering the K,Λ, while it reducesby one unit the N and the nz of the intruder orbitals,which in other words means, that this replacement is

not affecting the quanta in the x − y plane. The in-spiration for this replacement was the observation ofan enhanced proton–neutron interaction between the∆K[∆N∆nz∆Λ] =0[110] counterparts [72, 73].

The proxy-SU(3) replacement of orbitals can also takeplace in the Shell Model basis [1, 2], where the in-truder Shell Model orbitals |n, l, j,mj〉 possessing N + 1quanta are replaced by their de Shalit–Goldhaber coun-terparts [74], identified by |n, l − 1, j − 1,mj〉 [39, 75].The transformation among the de Shalit– Goldhaber or-bitals proved to be a unitary transformation, similar innature to the one used in the pseudo-SU(3) symmetry[76]. However, in the proxy-SU(3) case, the unitarytransformation affects only the quanta in the z cartesiandirection and leaves the x−y plane intact [39]. As a resultthe proxy-SU(3) model restores the SU(3) symmetry byleaving the normal parity orbitals intact and performinga unitary transformation in the intruder parity orbitals,which affects the quantum number nz.

It is interesting, to understand, what the consequencesof the unitary transformation are [39], which applies inthe proxy-SU(3) symmetry, on the single particle energiesof a SO–like shell. The orbitals of the SO–like shells arepresented in Table 7 of Ref. [39]. For instance the 6-14

SO shell consists of the 1p1/2mj , 1d

5/2mj spherical Shell Model

orbitals. Such mixed parity shells do not possess SU(3)symmetry. But the SU(3) symmetry can be restored in

this shell, by applying on the intruder 1d5/2mj orbitals the

unitary transformation [39]:

U = az < a†zaz >−1/2, (25)

which reduces the quanta in the z cartesian axis by oneunit and normalizes the result. Such an action maps all

the 1d5/2mj orbitals (except 1d

5/2±5/2) to the 1p

3/2mj and so

the proxy shell consists by the 1p1/2mj , 1p

3/2mj orbitals and

possesses an SU(3) symmetry [39].This loss of one quantum in the z cartesian direction

of the intruder orbitals results to a modification of theHamiltonian, which describes a proxy shell, as discussedin Appendix A of Ref. [39]. If the single particle Hamil-tonian before the unitary transformation was that of Eq.(1), then the Hamiltonian, which describes the proxyshell after the unitary transformation must be:

h0,SO = h0 + εproxyδj,N+1/2, (26)

where the Kronecker δj,N+1/2 is affecting only the or-bitals with j = jmax = N + 1/2, within the proxy shell.

Thus from the orbitals 1p1/2mj , 1p

3/2mj of the 6-12 proxy shell

(see Table 7 of Ref. [39]) only the 1p3/2mj orbitals are af-

fected by the Kronecker delta function, which restores theloss of the energy from the replacement of the intruderorbitals by their proxies [37, 39]. The εproxy is the single–particle energy difference among the intruder orbital ofthe SO–like shell and the proxy orbital, which replaces it(see Table 7 of Ref. [39]). For instance the εproxy in the

6

6-14 SO–like shell is the single–particle energy difference

between the 1d5/2mj and the 1p

3/2mj orbitals.

The value of the εproxy has been given in Ref. [37] tobe:

εproxy =

(1− 2ε

3

)~ω, (27)

where ε is the deformation parameter in the asymptoticbasis of the Nilsson Model [71], connected with the defor-mation parameter δ of the Nilsson Model [71] through:

ε = δ +1

6δ2, (28)

with the deformation variable β of the Bohr and Mottel-son Model [66] being related to δ by [71]:

δ = 0.95β. (29)

The unit in Eq. (27) restores the loss of one quantumfrom the unitary transformation, which applies in theproxy scheme [39], while the 2ε

3 restores the damage inthe value of q0 of Eq. (4), due to the loss of one quantumin the z direction [39] (see section 1.3 of Ref. [60] andsection 7.3.1 of Ref. [64] for deeper understanding).

It is well known from Quantum Mechanics, that theaddition of a constant in the Hamiltonian is not alteringthe eigenstates, but simply lifts the eigenvalues by thisconstant, which means, that Eq. (10) is modified as:

h0,SO |n, l, j,mj〉 = ε0(|n, l, j,mj〉)SO |n, l, j,mj〉 , (30)

with

ε0(|n, l, j,mj〉)SO = ε0 + εproxyδj,N+1/2 (31)

where ε0 is given by Eqs. (2), (7). The Hamiltonian ofEq. (26) includes the j quantum number. As a conse-quence the |nz, nx, ny,ms〉 states are not eigenstates ofthe Hamiltonian of Eq. (26), while the |n, l, j,mj〉 are.

The change in energy of the spherical Shell Modelstates |n, l, j,mj〉 inevitably alters the expectation valueof the energy of the cartesian states |nz, nx, ny,ms〉. Ingeneral using the Hamiltonian of Eq. (26) the expecta-tion values of the cartesian orbitals of the proxy shellsare given by:

〈ε0(|nz, nx, ny,ms〉)SO〉 =

〈nz, nx, ny,ms|h0,SO|nz, nx, ny,ms〉 =∑n,l,j,mj

|〈n, l, j,mj |nz, nx, ny,ms〉|2 · ε0(|n, l, j,mj〉)SO,

(32)

where the summation is over every non–vanishing matrixelement 〈n, l, j,mj |nz, nx, ny,ms〉 of the transformationpresented in Ref. [39].

The effect of Eq. (26) on the many–particle Hamilto-nian is the following:

H0,SO = H0 +∑i

(δji,N+1/2)iεproxy, (33)

where H0 is given by Eq. (14) and the summation is forevery valence particle i. This operator depends on thetotal angular momentum of each valence particle ji. Letthe summation be labeled:

H0,proxy =∑i

(δji,N+1/2)iεproxy, (34)

since it carries the effect of the proxy replacement of or-bitals on the many–particle Hamiltonian.

The eigenvalues of H0,proxy have to be calculated inthe L-projected (or J-projected for odd mass nuclei)[35, 36, 40] wave functions of the Elliott Model, whichhave K,L,M as good quantum numbers with K beingthe band label and M is the projection of the angularmomentum L [35]. The L-projected wave functions read[35]:

Ψ(KLM) =P (KLM)

a(K,L)Φ, (35)

where a(K,L) are the expansion coefficients of the Slaterdeterminant Φ, which consists by the cartesian singleparticle states |nz, nx, ny,ms〉, in the L-projected wavefunction:

Φ =∑K,L

a(K,L)Ψ(KLK), (36)

and P (KLM) is the projection operator with explicitform given in the Appendix of Ref. [35]. The matrixelements of the projection operator are labeled by [35]:

A(KLK ′) = 〈Φ|P (KLK ′)|Φ〉 , (37)

which for the special case of K ′ = K become [35]:

A(KLK) = |a(K,L)|2. (38)

Analytic formulae for the a(K,L) coefficients are givenin Table 2A of Ref. [41]. Finally following Eqs. (35)-(38)the diagonal matrix elements of H0,proxy are:

〈Ψ(KLK)|H0,proxy|Ψ(KLK)〉 =

|a(K,L)|2 〈Φ|H0,proxy|Φ〉 . (39)

A sample calculation of the above quantity for 16O willbe given in section X B. In this sample calculation it willbe demonstrated, how easy it is, to do calculations inthe cartesian single–particle basis |nz, nx, ny,ms〉 (whichsuits to deformed nuclei) and then apply the L-projectiontechnique [35], within the proxy-SU(3) Model. The co-incidence of the calculation of the energy of the 0+2 state

(without any fitting) with the data for 16O will be out-standing.

A SO–like shell, with N quanta after the replacementof the intruder orbitals by their de Shalit–Goldhaberpartners [39, 74], has a U(Ω) symmetry, where Ω =(N+1)(N+2)

2 . The proxy-SU(3) irreps (λ, µ) are then cal-culated by the code of Ref. [57]. The various particle

7

distributions in the valence Shell Model space result to avariety of SU(3) irreps [61, 77]. Among them the highestweight irrep has been proven, to describe the low energynuclear properties [38, 61, 78], since this one correspondsto the most symmetric spatial wave function [61] and inaddition predicts the dominance of the prolate over theoblate shape [38]. The highest weight SU(3) irreps forthe HO magic numbers and the SO–like magic numbersusing the Elliott and the proxy-SU(3) symmetry respec-tively are listed in Tables 1, 2.

Consequently a 3D-HO valence nuclear shell, which iscreated by magic numbers 2, 8, 20, 40, 70, 112, 168,is described by the Elliott SU(3) irreps (λ, µ)HO, whilea SO shell, with magic numbers 6, 14, 28, 50, 82, 126,182 by the irreps (λ, µ)SO of the proxy-SU(3) symme-try. The eigenvalues of the C2 operator, as derived bythe two types of irreps, correspond to the deformation ofthe particle configuration within the two types of magicnumbers (see Eq. (19)), namely the HO and the SO–likemagic numbers. In the following we will suggest, thatthe phenomenon of shape coexistence emerges from aninterchange among the two types of magic numbers.

IV. PARTICLE EXCITATIONS

The state–of–the–art mechanism for shape coexistenceis the particle–hole (p–h) excitation mechanism, whichis reviewed in section IIA of Ref. [23]. The p–h excita-tion mechanism began with a suggestion of Morinaga inRef. [79] for the excited 0+2 state of 16O. He suggested,that this state could be understood as a 4 particle–4 holeexcitation from the p shell to the sd shell. This 4p–4hexcitation can also be understood as an excitation of analpha particle across the p shell [80].

In the spherical Shell Model approach of shape coex-istence, the Hamiltonian is the summation for every va-lence nucleon of all the single–particle energies plus thetwo–body nucleon–nucleon interactions (see Eq. (1) ofRef. [23]). The energy gaps [81] in this approach play akey role, since a reduction of the original valence spacemay lead to particle configurations with a different de-formation than the initial one. Thus while moving awayfrom closed shells, in areas where deformation arises, theknowledge of the effective single–particle energies, whichdetermine the shell gaps, is of major importance for theparticle–hole excitations.

In section II we have argued, that the intrinsic single–particle basis for the valence shell of deformed nucleiwithin the Elliott SU(3) symmetry is the cartesian basis|nz, nx, ny,ms〉, because such orbitals are eigenstates ofthe dominant q0,iq0,i′ interaction [82] and due to the fact,that rotational nuclear bands emerge from these states[35, 36]. In addition in section II in Eqs. (2), (3), (7) wehave given the formula to calculate the single–particleenergies of the Hamiltonian h0 (as defined in Eq. (1))in the two types of bases |n, l, j,mj〉, |nz, nx, ny,ms〉 ina 3D-HO valence shell among the HO magic numbers

2, 8, 20, 40, 70, 112, 168, which has the Elliott SU(3)symmetry. Furthermore in section III we have presentedthe proxy-SU(3) symmetry, which is valid in the SO–likevalence shells constituted by normal and intruder parityorbitals, among the magic numbers 6, 14, 28, 50, 82, 126,184 (see Table 7 of Ref. [39]). In such shells the singleparticle Hamiltonian h0,SO of Eq. (26) is valid, which isnot having the cartesian states |nz, nx, ny,ms〉 as eigen-states and as a result only the expectation values of Eq.(32) can be calculated.

In this section we will present the neutron single–particle energies of a nucleus with 7 neutrons in the El-liott and the proxy-SU(3) scheme. Specifically we willclarify, which orbitals are occupied, if a) the valence neu-trons obey the HO magic numbers and b) if they followthe SO–like magic numbers. Afterwards we will presentthe single–particle energies of these nucleons. Through acomparison of the single–particle energies it will becomeclear, that the particle configuration, which follows theSO shell 6-14, has excited single-particle energies, whencompared with the particle configuration, which followsthe HO shell 2-8. Thus for certain nucleon numbers thetransition of a particle from the HO shell to the neighbor-ing SO shell could be interpreted as a particle excitation.

Let us give an example, which suits to the case of thehalo nucleus 11

4Be7 [83, 84]. If the 7 neutrons of thisnucleus follow the HO magic numbers 2, 8, then the core

consists of 2 neutrons, which occupy the 1s1/2±1/2 orbitals.

The orbitals of a closed core can be described either in thespherical or in the cartesian basis, since both scenarioscontribute the same in the energy and in the nuclear wavefunction. For simplicity we choose, to describe the closedcore with the spherical basis. The valence shell amongmagic numbers 2-8, consists of the cartesian orbitals withN = 1 quanta (see Eq. (11)):

|nz, nx, ny,ms〉 : |1, 0, 0,±1

2〉 , |0, 1, 0,±1

2〉 , |0, 0, 1,±1

2〉 .

(40)

Therefore the 5 valence neutrons occupy the orbitals:

|nz, nx, ny,ms〉 : |1, 0, 0,±1

2〉 , |0, 1, 0,±1

2〉 , |0, 0, 1,+1

2〉 .

(41)

The eigenvalues in units ~ω of the Hamiltonian h0 of Eq.

(1), as given by Eqs. (2), (3), (7) for the 1s1/2±1/2 and for

the orbitals of Eq. (41) are 32 ,

52 ,

52 ,

52 respectively.

If the 7 neutrons of this nucleus follow the SO magicnumbers 6-14, then the core consists of the orbitals

1s1/2±1/2, 1p

3/2±1/2, 1p

3/2±3/2. The valence SO–like shell 6-14

consists of the orbitals 1p1/2±1/2, 1d

5/2±1/2, 1d

5/2±3/2, 1d

5/2±5/2.

The proxy valence shell, which results after the action of

the unitary transformation of Eq. (25) on the 1d5/2mj or-

bitals [39], consists of the orbitals 1p1/2±1/2, 1p

3/2±1/2, 1p

3/2±3/2,

with N = 1 quanta. Thus the proxy shell among magicnumbers 6-12 [39] consists of the cartesian orbitals of Eq.

8

TABLE I: The highest weight SU(3) irreps for the spin-orbit (SO) like magic numbers 2, 6, 14, 28, 50, 82, 126 according to theproxy-SU(3) symmetry and for the harmonic oscillator (HO) magic numbers 2, 8, 20, 40, 70, 112, 168 according to the ElliottSU(3) symmetry. The results have been obtained by the code of Ref. [57] and have been presented in [77]. Examples aboutthe calculation of the irreps have been given in Refs. [59–61]. An analytic formula for the calculation of the highest weightirreps is given in [58].

Particle Number (λ, µ)SO (λ, µ)HO Particle Number (λ, µ)SO (λ, µ)HO

2 (0, 0) (0, 0) 1 (0, 0) (0, 0)4 (0, 0) (2, 0) 3 (0, 0) (1, 0)6 (0, 0) (0, 2) 5 (0, 0) (1, 1)8 (2, 0) (0, 0) 7 (1, 0) (0, 1)10 (0, 2) (4, 0) 9 (1, 1) (2, 0)12 (0, 0) (4, 2) 11 (0, 1) (4, 1)14 (0, 0) (6, 0) 13 (0, 0) (5, 1)16 (4, 0) (2, 4) 15 (2, 0) (4, 2)18 (4, 2) (0, 4) 17 (4, 1) (1, 4)20 (6, 0) (0, 0) 19 (5, 1) (0, 2)22 (2, 4) (6, 0) 21 (4, 2) (3, 0)24 (0, 4) (8, 2) 23 (1, 4) (7, 1)26 (0, 0) (12, 0) 25 (0, 2) (10, 1)28 (0, 0) (10, 4) 27 (0, 0) (11, 2)30 (6, 0) (10, 4) 29 (3, 0) (10, 4)32 (8, 2) (12, 0) 31 (7, 1) (11, 2)34 (12, 0) (6, 6) 33 (10, 1) (9, 3)36 (10, 4) (2, 8) 35 (11, 2) (4, 7)38 (10, 4) (0, 6) 37 (10, 4) (1, 7)40 (12, 0) (0, 0) 39 (11, 2) (0, 3)42 (6, 6) (8, 0) 41 (9, 3) (4, 0)44 (2, 8) (12, 2) 43 (4, 7) (10, 1)46 (0, 6) (18, 0) 45 (1, 7) (15, 1)48 (0, 0) (18, 4) 47 (0, 3) (18, 2)50 (0, 0) (20, 4) 49 (0, 0) (19, 4)52 (8, 0) (24, 0) 51 (4, 0) (22, 2)54 (12, 2) (20, 6) 53 (10, 1) (22, 3)56 (18, 0) (18, 8) 55 (15, 1) (19, 7)58 (18, 4) (18, 6) 57 (18, 2) (18, 7)60 (20, 4) (20, 0) 59 (19, 4) (19, 3)62 (24, 0) (12, 8) 61 (22, 2) (16, 4)64 (20, 6) (6, 12) 63 (22, 3) (9, 10)66 (18, 8) (2, 12) 65 (19, 7) (4, 12)68 (18, 6) (0, 8) 67 (18, 7) (1, 10)70 (20, 0) (0, 0) 69 (19, 3) (0, 4)72 (12, 8) (10, 0) 71 (16, 4) (5, 0)74 (6, 12) (16, 2) 73 (9, 10) (13, 1)76 (2, 12) (24, 0) 75 (4, 12) (20, 1)78 (0, 8) (26, 4) 77 (1, 10) (25, 2)80 (0, 0) (30, 4) 79 (0, 4) (28, 4)82 (0, 0) (36, 0) 81 (0, 0) (33, 2)

(40). The one valence nucleon of the 6-12 proxy shell ac-cording to the occupancy order of Eq. (11) occupies the:

|nz, nx, ny,ms〉 : |1, 0, 0,+1

2〉 (42)

orbital.The single–particle energies ε0 of the Hamiltonian h0

for the occupied orbitals 1s1/2±1/2, 1p

3/2±1/2, 1p

3/2±3/2 of the

core in units ~ω are given by Eqs. (2), (3), (7) and re-sult to be 3

2 , 52 , 5

2 respectively. The expectation valueof the occupied state of Eq. (42) is given by Eq. (32).

The transformation coefficients for the specific orbital aregiven in Eq. (8) to be:

〈n, l, j,mj |nz, nx, ny,ms〉 : (43)

〈0, 1, 1

2,

1

2|1, 0, 0, 1

2〉 = − 1√

3, (44)

〈0, 1, 3

2,

1

2|1, 0, 0, 1

2〉 =

√2

3. (45)

Consequently the expectation value as given by Eqs.

9

TABLE I: (continued)


84 (10, 0) (34, 6) 83 (5, 0) (35, 3)86 (16, 2) (34, 8) 85 (13, 1) (34, 7)88 (24, 0) (36, 6) 87 (20, 1) (35, 7)90 (26, 4) (40, 0) 89 (25, 2) (38, 3)92 (30, 4) (34, 8) 91 (28, 4) (37, 4)94 (36, 0) (30, 12) 93 (33, 2) (32, 10)96 (34, 6) (28, 12) 95 (35, 3) (29, 12)98 (34, 8) (28, 8) 97 (34, 7) (28, 10)100 (36, 6) (30, 0) 99 (35, 7) (29, 4)102 (40, 0) (20, 10) 101 (38, 3) (25, 5)104 (34, 8) (12, 16) 103 (37, 4) (16, 13)106 (30, 12) (6, 18) 105 (32, 10) (9, 17)108 (28, 12) (2, 16) 107 (29, 12) (4, 17)110 (28, 8) (0, 10) 109 (28, 10) (1, 13)112 (30, 0) (0, 0) 111 (29, 4) (0, 5)114 (20, 10) (12, 0) 113 (25, 5) (6, 0)116 (12, 16) (20, 2) 115 (16, 13) (16, 1)118 (6, 18) (30, 0) 117 (9, 17) (25, 1)120 (2, 16) (34, 4) 119 (4, 17) (32, 2)122 (0, 10) (40, 4) 121 (1, 13) (37, 4)124 (0, 0) (48, 0) 123 (0, 5) (44, 2)126 (0, 0) (48, 6) 125 (0, 0) (48, 3)

(32), (44), (45) is:

〈ε0(|1, 0, 0, 1

2〉)〉SO =

1

3ε0(|1p1/21/2〉) +

2

3ε0(|1p3/21/2〉)SO, (46)

where according to Eqs. (2), (7):

ε0(|1p1/21/2〉) =

(N +

3

2

)~ω =

5

2~ω, (47)

and according to Eqs. (31), (7):

ε0(|1p3/21/2〉)SO =

(N +

3

2

)~ω + εproxy =

5

2~ω + εproxy.

(48)The +εproxy term in the above eigenvalue is the result ofthe replacement of the intruder orbitals by their proxies,which applies in the proxy SU(3) symmetry [37, 39]. Fi-nally the expectation value of Eq. (46) using Eqs. (47),(48) is:

〈ε0(|1, 0, 0, 1

2〉)〉SO =

5

2~ω +

2εproxy3

. (49)

For the ground state of 11Be we suppose, that the pro-tons follow the HO magic numbers, while the neutronsfollow the SO–like magic numbers, giving (λ, µ)$ = (2, 0)for protons and (λ, µ)ν = (1, 0) for neutrons (see TableI). The overall nuclear irrep is (λ, µ) = (3, 0), as will beexplained in section X A. Using this irrep and Eqs. (18)-(20), (28), (29) the deformation parameter of the ground

state band of 11Be is predicted to be ε = 0.36, which fromEq. (27) gives εproxy = 0.76~ω. Thus Eq. (49) becomes:

〈ε0(|1, 0, 0, 1

2〉)〉SO ≈ 3~ω. (50)

The results of this example are summarized in TableIII, where it becomes obvious, that the neutron config-uration in the 6-14 SO–like magic numbers has excitedsingle-particle energies comparing with those in the HO2-8 magic numbers.

Now a generalization can be made. A proxy SO–likeshell among magic numbers 6-12, 14-26, 28-48, 50-80,82-124, 126-184 (Table 7 of Ref. [39]) has excited single-particle energies comparing with those of a HO shellamong magic numbers 2-8, 8-20, 20-40, 40-70, 70-112,112-168 respectively. Thus one could say that, for cer-tain proton or neutron numbers, where both types ofshells are open and the single-particle states of the SOshell are excited comparing with those of the HO shell,then the transition from the HO shell to the relative SO–like shell could be interpreted as particle excitations. Theparticle numbers, for which this is possible, are presentedin Table IV.

In this section we focused on the case of 11Be, be-cause this was the simplest example one could present.Through this example it became evident, that for cer-tain nucleon numbers, which have been presented in Ta-ble IV, a transition from the HO shell to the open SOshell is equivalent to particle excitations. In section X Awe will present some predictions about 11Be within thedual–shell mechanism.

Returning to the case of 16O, for which the study ofshape coexistence began at 1956 [79], we could say, that

10

TABLE II: The continuation of Table I but for the spin-orbit (SO) like magic numbers 126, 184 using the proxy-SU(3) symmetry[37–39, 61, 70] and for the harmonic oscillator magic numbers 112, 168, 240 using the Elliott SU(3) symmetry [33, 34]. Thehighest weight irreps (λ, µ) have been calculated with the analytic formula of Ref. [58].


128 (12, 0) (50, 8) 127 (6, 0) (49, 7)130 (20, 2) (54, 6) 129 (16, 1) (52, 7)132 (30, 0) (60, 0) 131 (25, 1) (57, 3)134 (34, 4) (56, 8) 133 (32, 2) (58, 4)136 (40, 4) (54, 12) 135 (37, 4) (55, 10)138 (48, 0) (54, 12) 137 (44, 2) (54, 12)140 (48, 6) (56, 8) 139 (48, 3) (55, 10)142 (50, 8) (60, 0) 141 (49, 7) (58, 4)144 (54, 6) (52, 10) 143 (52, 7) (56, 5)146 (60, 0) (46, 16) 145 (57, 3) (49, 13)148 (56, 8) (42, 18) 147 (58, 4) (44, 17)150 (54, 12) (40, 16) 149 (55, 10) (41, 17)152 (54, 12) (40, 10) 151 (54, 12) (40, 13)154 (56, 8) (42, 0) 153 (55, 10) (41, 5)156 (60, 0) (30, 12) 155 (58, 4) (36, 6)158 (52, 10) (20, 20) 157 (56, 5) (25, 16)160 (46, 16) (12, 24) 159 (49, 13) (16, 22)162 (42, 18) (6, 24) 161 (44, 17) (9, 24)164 (40, 16) (2, 20) 163 (41, 17) (4, 22)166 (40, 10) (0, 12) 165 (40, 13) (1, 16)168 (42, 0) (0, 0) 167 (41, 5) (0, 6)170 (30, 12) (14, 0) 169 (36, 6) (7, 0)172 (20, 20) (24, 2) 171 (25, 16) (19, 1)174 (12, 24) (36, 0) 173 (16, 22) (30, 1)176 (6, 24) (42, 4) 175 (9, 24) (39, 2)178 (2, 20) (50, 4) 177 (4, 22) (46, 4)180 (0, 12) (60, 0) 179 (1, 16) (55, 2)182 (0, 0) (62, 6) 181 (0, 6) (61, 3)184 (0, 0) (66, 8) 183 (0, 0) (64, 7)

TABLE III: The list of the occupied orbitals for the neutron configuration of the halo nucleus 114Be7 for two sets of magic

numbers, namely the harmonic oscillator magic numbers (HO) and the spin-orbit like (SO) magic numbers. The single–particleenergies ε0 for the original orbitals have been calculated by Eq. (2), while the expectation value 〈ε0(|nz, nx, ny,ms〉)〉SO forthe proxy orbital |nz = 1, nx = 0, ny = 0,ms = 1

2〉 has been calculated by Eq. (32). See section IV for further discussion. In

this example when the neutron configuration flips from the HO shell to the SO–like, one neutron is excited from the 2-8 HOshell to the 6-14 SO–like shell.

Harmonic oscillator (HO) magic numbers Spin-orbit (SO) like magic numbersNeutrons Orbitals ε0 Orbitals 〈ε0〉

1st |1s1/21/2〉 1.5 |1s1/21/2〉 1.5

2nd |1s1/2−1/2〉 1.5 |1s1/2−1/2〉 1.5

3rd |nz = 1, nx = 0, ny = 0,ms = 12〉 2.5 |1p3/23/2〉 2.5

4th |nz = 1, nx = 0, ny = 0,ms = − 12〉 2.5 |1p3/21/2〉 2.5

5th |nz = 0, nx = 1, ny = 0,ms = 12〉 2.5 |1p3/2−1/2〉 2.5

6th |nz = 0, nx = 1, ny = 0,ms = − 12〉 2.5 |1p3/2−3/2〉 2.5

7th |nz = 0, nx = 0, ny = 1,ms = 12〉 2.5 |nz = 1, nx = 0, ny = 0,ms = 1

2〉 3

a change of the proton and neutron configuration fromthe HO shell among magic numbers 2-8 to the SO shellamong magic numbers 6-14 is equivalent to a 4 particleexcitation [68]. Specifically if the 8 protons and the 8neutrons of 16O inhabit the SO–like shells, then the 6-14

valence shell consists of 2 valence protons and 2 valenceneutrons. Thus the on hand mechanism, suggests indeedan excitation of an alpha particle across the p shell [80]to the mixed parity 1p1/2, 1d5/2 SO–like shell.

In the HO scheme the proton ($) and the neutron irrep

11

TABLE IV: In the first column are presented the proton or neutron numbers, for which both types of shells, namely the HOand the SO–like, are open. For these nucleon numbers the SO–like shell of the the third column is excited, when compared tothe HO shell of the second column. See section IV for further discussion.

Nucleon number HO shell excited SO shell6-8 2-8 6-14

14-20 8-20 14-2828-40 20-40 28-5050-70 40-70 50-8282-112 70-112 82-126126-168 112-168 126-184

(ν) irreps for 16O are (λ$, ν$)HO =(0, 0), (λν , µν)HO=(0, 0) (see Table I). The result of the outer product(λ, µ)$ ⊗ (λ, µ)ν [46, 85–87] is:

(λ, µ)HO = (λ$ + λν , µ$ + µν)HO = (0, 0). (51)

In the SO–like configuration for 16O (see Table I):

(λ$, ν$)SO = (2, 0), (52)

(λν , µν)SO = (2, 0), (53)

giving:

(λ, µ)SO = (λ$ + λν , µ$ + µν)SO = (4, 0). (54)

Thus following Eq. (19) with the scaling factor of Eq.(20) S$ + Sν = 12 the dual–shell mechanism predicts aless deformed shape with β = 0.12 coming from the HOconfiguration and a more deformed shape with β = 0.386from the SO–like particle configuration. The more de-formed shape originating from the SO–like shell gener-ates in the dual–shell mechanism the ground state bandof 16O with experimental deformation variable β = 0.364[88], while the less deformed shape originating from theHO shell matches with the 0+2 of 16O at 6.049 MeV [89].The transition B(E2) ↑ from the 0+1 to the 2+1 is used inRef. [88], to derive the experimental deformation variableβ = 0.364. The more deformed shape with irrep (4, 0),predicts a ground state band with levels L+ : 0+1 , 2

+1 , 4

+1

according to Eq. (23) of Ref. [34]. This sequence ofstates appears in the data [89] at 0 MeV, 6.9 MeV and10.4 MeV respectively. The observed E(2) transitionsamong them could support, that they consist the groundstate band. Their high energy values, could be predictedin future research by the additon of the pairing interac-tion into the Elliott SU(3) Hamiltonian, as in Ref. [90],since in N = Z nuclei with isospin T = 0 [89] the pairinginteraction is important. The energy of the 0+2 state of16O will be calculated in section X B within the dual–shellmechanism for shape coexistence. It will be explained insections VII and VIII, how it is possible, that the SO–like shell, although it has excited single–particle energiescomparing with those of the HO shell, reproduces theground state band. Actually the answer in this paradox,will deliver the condition, which predicts the islands ofshape coexistence on the nuclear map.

Another important consequence of the dual–shellmechanism is, that when the particles are placed in theSO–like shell, no holes are considered. For instance, whenthe particles are in the 6-14 SO–like shell, the previouslyfilled orbitals 1s1/2, 1p3/2 create a closed core, if one usesthe proxy-SU(3) symmetry [39] (see Table I of Ref. [82])and thus no hole irreps emerge.

This is actually a difference of the dual–shell mecha-nism with the particle–hole mechanism as realized in theSymplectic Model [68, 91–93]. Both approaches considerparticle excitations from the valence HO shell, but in theSymplectic Model, since only pure parity HO shells aretreated, the excitation of one particle from the 2-8 shellto the 8-20 shell generates one hole in the 2-8 shell.

Specifically for 16O the particle-hole excitation mech-anism predicts, that for the protons a 2p–2h excitationis described by a particle (λ, µ) irrep in the 8-20 shell(2, 0) ⊗ (2, 0) = (4, 0) and a hole irrep in the 2-8 shell(0, 1)⊗(0, 1) = (0, 2) [68]. The same stands for the 2p–2hneutron configuration of 16O [68]. Finally the particle–hole excitation mechanism predicts a 4p–4h (2 protonsand 2 neutrons) excitation from the 2-8 to the 8-20 shell.The proton and neutron irreps within the particle–holeexcitation mechanism for 16O are [68]:

(4, 0)⊗ (0, 2) = (4, 2) for 2 proton excitations, (55)

(4, 0)⊗ (0, 2) = (4, 2) for 2 neutron excitations, (56)

giving an overall nuclear irrep for protons and neutronstogether:

(4, 2)⊗ (4, 2) = (8, 4), (57)

as is indicated in Table I of Ref. [68].

The coexistence of the 0p–0h state with the 4p–4hstate for this example results to the shape coexistence in16O within the particle-hole excitation mechanism [68].The 0p–0h configuration is set to be the ground state at0 MeV, while the 4p–4h configuration represents the ex-cited 0+2 state at 4.9 MeV (see Table I of [68]), which issaid to be close to the data at 6.049 MeV [89] for thisstate of 16O.

12

V. THE DISSOLUTION OF MAGIC NUMBERS

The SO–like shells are being created by a significantspin–orbit interaction, as discussed in section II. Thecoexistence of the SO–like shell with the HO shell be-comes possible, because the significant single–particle en-ergy gaps at the major magic numbers dissolve due tothe deformation. When the open HO shell coexists withthe open SO–like shell shape coexistence becomes possi-ble. Each of the two types of active shells correspondsto a low–lying energy band. The resulting coexistingbands are characterized by different deformation param-eters (β, γ) of the Bohr and Mottelson Model [66], whichcan be calculated by the dual–shell mechanism.

Specifically the fading out of the major magic num-bers can be understood simply by looking at a Nilssondiagram [94] (which can be found in section “The Nils-son Model” of Ref. [95], or in Fig. 1 of Ref. [96]), wherethe single–particle energies are plotted versus the Nils-son deformation parameter ε or δ respectively [71]. Fora spherical nucleus with ε = 0 the nucleon numbers 28,50, 82 are definitely magic numbers, which means, thatthere are large energy gaps in the single–particle ener-gies above these nucleon numbers. But as deformationevolves (ε > 0) the single particle energies of the Nilssonorbitals are affected by the deformation (see section 1.3 ofRef. [60] and Ref. [97] for further understanding) and asa result at moderate deformation parameter ε ≈ 0.2 thelarge energy gaps at 28, 50, 82 have already faded out.Consequently the Nilsson diagrams alone tattle, that indeformed nuclei there are no major magic numbers.

While the Nilsson Model suffices for a qualitative ex-planation of the fading out of the major magic numbers atmoderate deformations, the same can be observed withinthe microscopic approach of Energy Density Function-als (EDF) derived from an effective interaction, in thecontext of the self consistent mean field theory. Morespecifically, constraining the quadrupole moment of thenuclear mean field one is able to examine the change ofthe calculated single–particle energies with axial defor-mation. Examples of Nilsson diagrams produced withthis method within the Relativistic Mean Field (RMF)have been reported in Ref. [98] and more recently in Ref.[99, 100]. For instance in Figs. 5.4-5.13 of Ref. [99] therespective diagrams for several isotopes of Hf, Os andNo are shown and the disappearance of major gaps inthe spectrum is clear.

The advantage of the RMF theory [101–105] is, thatit is a no core theory, i.e., all the nuclear shells partic-ipate in the calculation. Consequently shell merging isa build-in tool in the RMF theory. Another importantaspect of shape coexistence is, that the excited coexistingstates have short lifetimes, which means, that they arenot stable states or isomeric states. Therefore we havenot searched for shape coexistence in the various excitedlocal minima of the energy surface as a function of thedeformation variable β, which can be predicted by theEDF. We simply focused on the ground state predictions

of the theory and we observed there the peculiarities ofthe single–particle energies, which will enable shape co-existence to appear.

In addition, EDFs are extremely useful for calculat-ing properties of isotopes far from the valley of stability,where there is limited experimental input and the largenumber of nucleons prohibits Shell Model calculations. Inthe suggested dual–shell mechanism we are interested inthe energy gaps at the SO–like magic numbers and at theHO magic numbers and we are especially curious aboutthe situation in heavy isotopes. Hence, we have used thefunctional DDME2 of Ref. [106] within the numericalimplementation of the RMF model in the code of Ref.[107] to calculate the single–particle energies ε of the oc-cupied Nilsson orbitals K[NnzΛ] [71]. For an even–even

nucleus AZXN, with Z,N being the protron and neutron

number respectively, the energy gap at the ith nucleon isthe difference:

εgap = εi+2 − εi, (58)

where i = 1, 2, ..., N − 2 for neutron gaps, while i =1, 2, ..., Z − 2 for proton gaps.

In Figs.1 and 3 we present the predictions of the RMFtheory for the neutron single–particle energy gaps as de-fined in Eq. (58) for mercury and lead isotopes. In theseplots we clearly see, that the energy gaps at the majormagic numbers are mitigated and so the coexistence oftwo types of open neutron shells, namely the HO and theSO–like shell, is possible. Specifically a valley of reducedneutron magic numbers appears among 96 ≤ N ≤ 110,which matches exactly with the appearance of shape co-existence in the mercury isotopes (see Fig. 24, and Fig.10 of Ref. [23]).

Figs. 2 and 4 are the analogs of Figs. 1 and 3 forthe proton single–particle energy gaps in the Hg and Pbisotopes. We see, that in the case of proton gaps in Figs.2 and 4 a valley appears in the central area of 96 ≤N ≤ 110, similar to the one appearing in the neutrongaps in Figs. 1 and 3. This finding clearly suggests thatthe protons in the Hg and Pb series of isotopes are notindifferent spectators of the gradual addition of neutrons.The protons are strongly influenced by the addition ofneutrons and as a result a valley, in which the SO–likeproton magic numbers collapse and become comparableto the HO magic numbers, is created. It is within thisvalley, that shape coexistence is observed experimentally,as seen, for example, in Fig. 10 of Ref. [23]. The roleof the proton–neutron interaction in creating this valleycalls for further investigation.

Going into more detail, in Figs. 6 the proton Nilssonorbitals occurring in the DDME2 [106] calculation in the50-82 shell and in the beginning of the next shell are de-picted, drawn relatively to the Fermi energy εF in eachisotope. In Pb one would expect, that all orbitals of the50-82 shell (namely the 1h11/2, 1g7/2, 2d5/2, 2d3/2, 3s1/2

orbitals) should be occupied, while all orbitals above 82(the 1h9/2, 2f7/2 orbitals) should be empty. This is in-deed the case for N ≤ 96 and N ≥ 110. But within the

13

90 92 94 96 98 100 102 104 106 108 110 112 114 116

0

2

4

6

8

8-6 10-8 16-14 22-20 30-28 42-40 52-50 72-70 84-82 114-112

neut

ron

ener

gy g

aps

(MeV

)

N

Hg

FIG. 1: Relativistic Mean Field calculations [106, 107] of theneutron energy gaps as defined in Eq. (58) for the Hg isotopesversus the neutron number. The legend “8-6” signifies theenergy difference (see Eq. (58)) among the 6th and the 8th

neutron for each isotope, the “10-8” is the energy gap amongthe 8th and the 10th neutron etc. A valley of small energygaps appears among 96 ≤ N ≤ 110, where exactly shapecoexistence appears experimentally (see Fig. 10 of Ref. [23]and Fig. 24). For these mercury isotopes the major magicnumbers have been dissolved and so the open SO–like neutronshell among magic numbers 82-126 can coexist with the openHO neutron shell among magic numbers 70-112.

region 96 ≤ N ≤ 110 an interesting systematic effect isseen. The 1/2[400] and 3/2[402] Nilsson orbitals of 2d3/2,as well as the 11/2[505] orbital of 1h11/2, pop up abovethe Fermi energy, while the 1/2[541] and 3/2[532] orbitalsof 1h9/2 are sunk below the Fermi energy, thus providinga clear picture of proton excitations.

The present RMF results suggest, that the particle–hole mechanism for shape coexistence in heavy nuclei isfully justified microscopically within the 96 ≤ N ≤ 110region, in which the present dual–shell mechanism alsopredicts the presence of shape coexistence, as will be-come evident in section VIII. The extra hint providedby the present dual–shell mechanism is, that for theHg,Pb isotopes shape coexistence cannot exist outsidethe 96 ≤ N ≤ 110 region, as corroborated by the data(see Fig. 10 of Ref. [23], for example). In other words,the present dual–shell mechanism is not disproving theparticle-hole mechanism. In contrast, it corroboratesit within the nuclear regions in which the particle–holemechanism has been applied [23].

A more sensitive region of shape coexistence lies in theSr,Zr isotopes and in the lanthanides (Nd, Sm, ..., Yb),where a sudden onset of deformation is observed [108].This sudden onset of deformation is attributed to theFederman–Pittel proton–neutron pairs [109, 110]. Theonset of deformation due to the proton–neutron correla-tions leads to smaller proton energy gaps at: a) Z = 40

90 92 94 96 98 100 102 104 106 108 110 112 114 116

0

1

2

3

4

5

6

7

8

8-6 10-8 16-14 22-20 30-28 42-40 52-50 72-70 84-82

prot

on e

nerg

y ga

ps (M

eV)

N

Hg

FIG. 2: The same as Fig. 1 but for proton gaps.

90 92 94 96 98 100 102 104 106 108 110 112 114 116

0

2

4

6

8

8-6 10-8 16-14 22-20 30-28 42-40 52-50 72-70 84-82 114-112

neut

ron

ener

gy g

aps

(MeV

)

N

Pb

FIG. 3: The same as Fig. 1 but for the lead isotopes.

in the A ∼ 100 mass region, which includes the Sr,Zrisotopes and at b) Z = 64 in the A ∼ 150 mass region,which corresponds to the Nd, Sm, ..., Yb isotopes [108].Consequently the protons of the Sr, Zr may excite abovethe sub–shell closure at Z = 40, while the protons of thelanthanides can excite above the proton sub–shell closureat Z = 64 [108, 111]. This is the standard particle–holeexcitation mechanism for heavy nuclei.

But in the present dual–shell mechanism for shape co-existence we propose, that not only the proton gaps atZ = 40 and Z = 64 are reduced, due to the Federman–Pittel pairs, in the mass regions of A ∼ 100 and A ∼ 150respectively, but all the proton gaps are affected by theonset of deformation. Calculations of the proton gapswith the RMF theory [106] are presented in Figs. 7, 8.In the Sr isotopes especially for N = 58, as shown inFig. 8, the proton gap among the 28th and the 30th pro-

14

90 92 94 96 98 100 102 104 106 108 110 112 114 116

0

1

2

3

4

5

6

7

8

8-6 10-8 16-14 22-20 30-28 42-40 52-50 72-70 84-82

prot

on e

nerg

y ga

ps (M

eV)

N

Pb

FIG. 4: The same as Fig. 2 but for the lead isotopes.

48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 820

2

4

6

8

10

8-6 10-8 16-14 22-20 30-28 42-40 52-50 72-70

neut

ron

ener

gy g

aps

(MeV

)

N

Sn

FIG. 5: The same as Fig. 1 but for the tin isotopes, whichin general are not very deformed. Despite that, the tins110−120Sn manifest rotational energy bands and shape coex-istence (see Fig. 3.10 of Ref. [30] and Fig. 23).

tons is not major, thus a second possibility is now visible:that the protons of the 28-50 SO–like shell can interactwith the protons of the 20-40 HO shell. It is a matterof fact, that for the 96

38Sr58 there is the largest known E0transition for A > 56 (see Fig. 27 of Ref. [23]). Simi-larly in the Sm isotopes, especially around 150

60Sm88, theenergy gap among the 50th and the 52nd protons is mit-igated by the onset of deformation, as seen in Fig. 7.Thus the active protons of the Sm isotopes lie among theSO–like shell 50-82 and the 40-70 HO shell. Similarlyin the N ≈ 90 lanthanides there are observed strong E0transitions (Fig. 34 of [23] and Ref. [112]).

To resume, the dual–shell mechanism about shapecoexistence coincides with the particle–hole excitation

mechanism in the light nuclei, as discussed in section IV,where the nucleons traditionally are in the HO shells., butit adds a new aspect in the standard particle–hole exci-tation mechanism [111] in the heavier nuclei, where thenucleons traditionally are in the SO–like shells. Specifi-cally in the heavier nuclei the dual–shell mechanism sup-poses, that the protons (neutrons) of the SO–like valenceshell 28-50, 50-82, 82-126, 126-184 merge with the pro-tons (neutrons) of the still open HO shell 20-40, 40-70,70-112, 12-168 and that this is possible only when thethe proton (neutron) energy gaps at 28, 50, 82, 126 aredissolved by the large deformation.

In addition the effect of deformation on the single-particle energy gaps may cause the merging of the neu-tron HO shell with the relative neutron SO–like shell andthis is what we call “neutron induced” shape coexistence,but also may cause proton excitations in specific isotopes,as for example resulted from Figs. 6. For the Z ≈ 40and Z ≈ 64 regions we propose that the proton HO shellmerges with the relative proton SO–like shell and this iswhat we call “proton induced” shape coexistence. Theshell merging is caused by the deformation, which miti-gates the proton single-particle energy gaps, thus protonexcitations are also possible.

It should be emphasized, however, that the presentdual–shell mechanism is not implying that the particle–hole mechanism is not valid in heavy nuclei. It only im-plies that the particle–hole mechanism in heavy nucleican take place only within the N and Z regions sug-gested by the dual–shell mechanism, given in the nuclearchart of Fig. 25.

VI. THE DUAL–SHELL MECHANISM FORSHAPE COEXISTENCE

With the understanding, that an important spin–orbitinteraction is vital for the creation of the SO–like shells[40, 69], we suggest, that the natural mechanism forshape coexistence involves the following steps:a) The nucleus enters in a region with large QQ protonor neutron interaction of the SO–like shell [38], whichusually occurs before the HO shell closure at proton orneutron number 8, 20, 40, 70, 112, 168 (see Eq. (16) andFigs. 15-20).b) The large proton or neutron QQ interaction of theSO shell along with the spin–orbit interaction dissolvethe large single particle proton or neutron energy gapsas discussed in section V.c) Therefore two types of open, valence proton or neu-tron shells are active for the proton or neutron numbers ofTable IV, namely a HO shell and a SO–like shell. Shapecoexistence can be the result of the coexistence of thesetwo types of valence shells.

The above SU(3) mechanism for shape coexistence willbe used in section X to explain the parity inversion in Beisotopes [83, 84], the inversion of states in the Mg isotopes[114], the reappearance of the magic number at N = 40

15

90 92 94 96 98 100 102 104 106 108 110 112 114 116-10

-5

0

5

-F (M

eV)

N

1/2[550] 3/2[541] 5/2[532] 7/2[523] 9/2[514] 11/2[505]

Hg

(a)

90 92 94 96 98 100 102 104 106 108 110 112 114 116-10

-5

0

5

Pb

-F (M

eV)

1/2[550] 3/2[541] 5/2[532] 7/2[523] 9/2[514] 11/2[505]

N

96 110

(b)

90 92 94 96 98 100 102 104 106 108 110 112 114 116

-5

0

5

-F (M

eV)

Hg

N

1/2[400] 1/2[411] 3/2[402]

(c)

90 92 94 96 98 100 102 104 106 108 110 112 114 116

-5

0

5 Pb

-F (M

eV)

N

1/2[400] 1/2[411] 3/2[402]

(d)

90 92 94 96 98 100 102 104 106 108 110 112 114 116

-5

0

5

-F (M

eV)

Hg 1/2[541] 3/2[532]

N

(e)

90 92 94 96 98 100 102 104 106 108 110 112 114 116

-5

0

5 Pb

-F (M

eV)

1/2[541] 3/2[532]

N

(f)

FIG. 6: Relativistic Mean Field calculations [106, 107] for the single particle energies ε for protons in the lead and mercuryisotopes, reported relative to the proton Fermi energy εF in each isotope. We remark that the orbitals 1/2[541] and 3/2[532] of

2f7/2 of the 82-126 proton shell, which should be empty, are lying below the Fermi energy in the central region 98 ≤ N ≤ 108.On the contrary the 11/2[505], 1/2[400], 3/2[402], which were supposed to be filled, are empty. The large value of deformationmay cause up to 4 proton excitations from the 50-82 SO–like shell to the 82-126 SO–like shell. See section V for furtherdiscussion.

in the Ni isotopes [115], the shape coexistence in the Sn[30] and Hg [23] isotopes and the fission isomers at the Puisotopes [116]. Consequently the dual–shell mechanism isvalid across all mass regions.

VII. THE TWO LOW–LYING NUCLEARBANDS

Within the dual–shell mechanism two low–lying energybands are being predicted for each nucleus with shape co-existence. We will examine the simplified scenario, thatone band is derived by the pure highest weight SU(3)

16

86 88 90 92 940

1

2

3

4

5

6

7

prot

on e

nerg

y ga

ps (M

eV)

N

8-6 10-8 16-14 22-20 30-28 42-40 52-50 66-64

Sm

FIG. 7: The energy gaps for protons for the Sm isotopes ascalculated by the RMF theory [106, 107]. For instance thelegend “52-50” indicates the energy gap among the 50th andthe 52nd proton of the Sm isotopes. The large nuclear de-formation, due to the Federman-Pittel proton–neutron pairs[109, 110], is affecting all the energy gaps for protons, notonly the gap at Z = 64. Thus the one possibility is, thatsome of the protons of the Sm isotopes are able to exciteabove the sub-shell gap at Z = 64 [113], as is explained inRef. [108, 111]. Another possibility is, that since the pro-ton gap at among the 50th and the 52nd proton of the Smisotopes is dissolved from the large deformation (especiallyaround N = 88), then the protons of the open 40-70 HO shellare active along with the protons of the 50-82 SO–like shell.

48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82

0

1

2

3

4

5

6

7

8

8-6 10-8 16-14 22-20 30-28

prot

on e

nerg

y ga

ps (M

eV)

N

Sr

FIG. 8: The energy gaps for protons for the Sr isotopes ascalculated by the RMF theory [106, 107]. For instance thelegend “28-30” indicates the energy gap among the 28th andthe 30th proton of the Sr isotopes. The energy gap at the 28th

proton is mitigated before N = 60. Thus the protons of theopen 20-40 HO shell are active along with the protons of the28-50 SO–like shell.

irrep of the SO–like shell (λ, µ)SO and the other one isderived by the pure highest weight SU(3) irrep of the HOshell (λ, µ)HO. The question is: which one is the groundstate band and which one is the excited? As we shall seebelow, the answer lies in the symmetry of each nuclearwave function, which corresponds to each irrep.

In Ref. [61] it is exhibited, that a (λ, µ) irrep cor-responds to a nuclear wave function, which has λ + µsymmetric quanta and µ quanta which are neither sym-metric, nor antisymmetric. Thus the quantity λ+ µ is ameasure of symmetry. Also in Refs. [33, 40, 61, 117, 118]it is stated, that the most symmetric irrep lies lower inenergy.

Shape coexistence has long been related with particle–hole excitations (see section IV and Ref. [23]). TableIV present the nucleon numbers, for which the SO–likeshells have excited single–particle energies in comparisonwith those of the HO shells. We are further interested inthe nucleon numbers, for which the QQ interaction of theSO–like shell is intense. This interaction is necessary forthe dissolution of the major magic numbers as discussedin section V. Thus we shall focus on the nucleon numbersbefore the HO shell closures 8, 20, 40, 70, 112, 168 (seeEq. (16) and Figs. 15-20). For these nucleon numberswe shall test, which irrep (the (λ, µ)SO or the (λ, µ)HO)has more symmetric components λ + µ. The results arepresented in Figs. 9-14.

From the Figs. 9-14 becomes evident, that before theHO shell closure the most symmetric pure SU(3) irrep isthe (λ, µ)SO and thus this irrep has to derive the groundstate band. The less symmetric irrep is the (λ, µ)HO andthus it has to derive the excited band of a nucleus withshape coexistence. This suffices to justify the inversionof states around N = 8 and N = 20, which is manifestedin the Be and the Mg isotopes respectively. In theseisotopes the neutrons normally are in a HO shell, but assoon as they enter into a region with shape coexistence,the ground state band is being derived by the SO–likeshell and so inversion of states occurs. This subject willbe further discussed in sections VIII and X.

VIII. THE ISLANDS OF SHAPE COEXISTENCE

Shape coexistence is a phenomenon, which has beenmanifested in certain nuclei. A qualitative nuclear mapwith the islands of shape coexistence has been presentedin Fig. 8 of Ref. [23]. This mechanism indicates, thatshape coexistence may occur in certain islands on thenuclear map across all the mass regions. The key ob-servation is, that the two types of valence shells startto coexist, whenever the proton or neutron configurationmay change its valence shell from the SO–like to the HOand vice versa.

In section II we have argued, that a strong spin–orbitinteraction is necessary for the existence of the SO–likeshells. Afterwards in section IV we displayed the nucleonnumbers for which the SO–like shell has excited single–

17

5 6 7 8 9 10 11 12 13 14 15

0

1

2

3

4

5

6

HO SO

Z or N

6 8

FIG. 9: The symmetric quanta (λ + µ) of the nuclear wavefunction for the two types of valence shells, namely the SO–like and the HO. Particle excitations from the 2-8 HO shellto the 6-14 SO–like shell are possible for proton or neu-tron numbers within the interval 6-8, where both types ofshells are open (see Table IV). Just below 8 protons or neu-trons the quadrupole deformation of the SO–like shell alongwith the spin–orbit interaction may decrease the major magicnumbers, as discussed in section V. Just below 8 nucleons,the most symmetric irrep is the one of the SO–like shell:(λ + µ)SO ≥ (λ + µ)HO. Thus the (λ, µ)SO has to corre-spond to the ground state band and the less symmetric irrep(λ, µ)HO to the excited band.

particle energies in comparison with the HO shell. Insection V we argued, that a large QQ interaction of theSO–like shell is necessary for the dissolution of the majormagic numbers. All these phenomena together appearat nucleon numbers below the HO shell closure, whichmeans below 8, 20, 40, 70, 112, 168 protons or neutrons.In these regions the most symmetric irrep, which lies low-est in energy, is the (λ, µ)SO, as discussed in section VII.The question now is, how can it be possible, that the SO–like shell has excited single–particle energies and at thesame time lies lower in energy? The answer lies in thestudy of the QQ interaction and will lead us to the con-dition, according to which, one can identify the islandsof shape coexistence on the nuclear map.

With the use of the Hamiltonian of Eq. (21) the energydifference among the band–heads, that are being derivedby the two valence shells, is the eigenvalue of:

HHO −HSO = (H0,HO −H0,SO)

+κ

2(QQSO −QQHO), (59)

where H0,SO is given by Eq. (33) and H0,HO by Eq. (14).The term

∆H0 = H0,HO −H0,SO (60)

is the difference between the single–particle terms, while

12 14 16 18 20 22 24 26 28 30-2

0

2

4

6

8

10

12

14

16 HO SO

Z or N

14 20

FIG. 10: The same as Fig. 9 but for particle numbers among14 and 28. Particle excitations from the 8-20 HO shell tothe 14-28 SO–like shell are possible for proton or neutronnumbers among 14-20, where both types of shells are open(see Table IV). Calculations of the spin–orbit interaction inthe sd shell [40, 69] have shown that in the second half of the8-20 HO shell the strength of the spin-orbit force has grownvery much. Thus the 14-28 SO–like shell emerges. Just below20 protons or neutrons, the quadrupole deformation of theSO–like shell along with the spin–orbit interaction reduce thesingle–particle energy gaps, as discussed in section V. Justbelow 20 nucleons the most symmetric irrep is the one of theSO–like shell: (λ+µ)SO ≥ (λ+µ)HO. Thus the (λ, µ)SO hasto derive the ground state band and the less symmetric irrep(λ, µ)HO the excited band [61].

the

∆(QQ) = QQSO −QQHO (61)

is the difference between the quadrupole–quadrupole in-teractions. If N0,SO is the eigenvalue of H0,SO of Eq.(33) and N0,HO of Eq. (15) is the eigenvalue of H0,HO,then the eigenvalue of ∆H0 has a negative sign:

N0,HO −N0,SO ≤ 0, (62)

since the open SO–like shell 6-14, 14-28, 28-50, 50-82,82-126, 126-184 has excited single–particle energies, whencompared with those of the open harmonic oscillator shell2-8, 8-20, 20-40, 40-70, 70-112, 112-168 respectively asdiscussed in section IV. Furthermore as discussed in sec-tion VII the excited band has to correspond to a particleconfiguration in the HO shell, which means, that:

EHO − ESO ≥ 0, (63)

where EHO, ESO are the eigenvalues of the HamiltoniansHHO, HSO of Eq. (59) respectively. One can satisfy boththe conditions (62) and (63), when:

QQSO ≥ QQHO, (64)

18

25 30 35 40 45 50

0

5

10

15

20

25

HO SO

Z or N

28 40

FIG. 11: The same as Fig. 9, but for nucleon numbers among28 and 50.

45 50 55 60 65 70 75 80 85-5

0

5

10

15

20

25

30

35

40

HO SO

Z or N

50 70


and

QQSO −QQHO ≥ N0,SO −N0,HO ≥ 0. (65)

Consequently according to the inequality of Eq. (64) thestarting point of shape coexistence is at:

QQSO ≈ QQHO. (66)

Afterwards the condition (65) may narrow the range ofshape coexistence on the nuclear map.

In the Elliott Model [33, 34] the nuclear deformationis obtained from the SU(3) quantum numbers [57, 67].Therefore, the shape is determined by the distribution ofthe nucleons in the single–particle orbitals [60, 61]. Inthe ground state, due to the short–range and attractivenature of the nucleon–nucleon force, the particles result

80 90 100 110 120 130

0

10

20

30

40

50

60

HO SO

Z or N

82 112


120 130 140 150 160 170 180 190-10

0

10

20

30

40

50

60

70

80

HO SO

Z or N

126 168


in space symmetric compact packing, giving rise to thehighest weight SU(3) representations [61, 117]. In the HOand Elliott schemes the algebraic quadrupole nucleon–nucleon interaction defines the sequence of the orbitals,while in the spin–orbit scheme, described here by theproxy-SU(3) symmetry, the strong spin–orbit interaction[40, 69] plays an important role too, as discussed in sec-tions II and IV. Therefore, in general, one has two dif-ferent nucleon distributions, two different sets of SU(3)quantum numbers and two different deformations. Here,by Eq. (66) we put forward a conjecture, which says,that the shape coexistence appears in the special case,when the quadrupole deformation (given by the expecta-tion of the second order Casimir operator of SU(3)) is incoincidence for the HO and SO schemes.

If the protons or neutrons lie within a harmonic os-

19

cillator shell, the QQHO interaction of this shell can becalculated within the Elliott SU(3) symmetry. Especiallyfor the ground state of an even-even nucleus with L = 0Eqs. (16), (18) give:

QQHO = 4[λ2 + µ2 + λµ+ 3(λ+ µ)]HO

= 4C2,HO.

(67)

Similarly for a SO–like shell, using the proxy-SU(3) sym-metry one gets:

QQSO = 4[λ2 + µ2 + λµ+ 3(λ+ µ)]SO

= 4C2,SO. (68)

Figures 15-20 are plots of the eigenvalues of C2 versusthe proton or neutron number. In these figures it be-comes obvious, that the condition of Eq. (66) is satisfiedat proton or neutron numbers:

beginning 7, 17, 34, 59, 96, 145, (69)

which mark the beginning of shape coexistence across anisotopic or isotonic nuclear chain. The ending of shapecoexistence occurs at the harmonic oscillator shell closure

QQHO = 0, (70)

which happens at proton or neutron numbers:

end 8, 20, 40, 70, 112, 168. (71)

Figures about the β, γ deformation variables using thetwo types of open shells are presented in Sec. 4.5 of Ref.[119].

Regions on the nuclear map (see Fig. 25) with protonor neutron numbers 7-8, 17-20, 34-40, 59-70, 96-112, 145-168 have a SO–like proton or neutron shell, which pos-sesses a large value of QQSO interaction, which in turncauses a fading out of the major single–particle energygaps and thus two valence proton or neutron shells maycoexist, namely the HO and the SO–like shell. Withinthese islands on the nuclear map it is valid the condition(64).

Finally according to the dual–shell mechanism the is-lands of shape coexistence are expected to lie betweenproton or neutron numbers:

Z or N: 7−8, 17−20, 34−40, 59−70, 96−112, 145−168.(72)

Using the above nucleon numbers one may draw a mapwith the islands of shape coexistence as in Fig. 25. Var-ious nuclei in the colored areas of this map have experi-mentally observed shape coexistence [23, 30], while oth-ers do not have yet. In addition there is experimentalevidence for shape coexistence in doubly magic nuclei[120, 121], which is not predicted by the map of Fig.25. For the region of Ge [122] the dual–shell mecha-nism aligns with the findings of [45]. Consequently the

2 4 6 8 10 12 14

0

10

20

30

40

50

60

C2

Z or N

HO SO

7 8

FIG. 15: The eigenvalues of the second order Casimir oper-ator of SU(3) versus the proton (Z) or neutron number (N).An island of shape coexistence is predicted within proton orneutron numbers 7 − 8, where C2,SO ≥ C2,HO. This islandcorresponds to the parity inversion in 11Be [83, 84]. See sec-tion X for further discussion.

8 10 12 14 16 18 20 22 24 26 28 30

0

100

200

HO SO

C2

Z or N

17 2120

FIG. 16: The same as Fig. 15. An island of shape coexis-tence is predicted within proton or neutron numbers 17− 20.This island corresponds to the breaking of the magic numberN = 20 in the Mg isotopes [114]. See section X for furtherdiscussion.

map of Fig. 25 indicates, which nuclei have to be exam-ined experimentally and theoretically for shape coexis-tence within the dual–shell mechanism.

Returning to the question we addressed in the be-ginning of this section the answer is, that the parti-cle configuration of the SO–like shell, although it hasexcited single–particle energies (N0,SO ≥ N0,HO), lieslower in energy, because it also corresponds to a largerquadrupole-quadrupole interaction (QQSO ≥ QQHO),

20

20 25 30 35 40 45 50

0

100

200

300

400

500

600

HO SO

C2

Z or N

34 40

FIG. 17: The same as Fig. 15. An island of shape coexistenceis predicted within proton or neutron numbers 34− 40. Thisisland corresponds to the reappearance of the magic numberN = 40 at Ni isotopes [115]. See section X for further discus-sion.

40 50 60 70 80 90-200

0

200

400

600

800

1000

1200

1400

1600 HO SO

C2

Z or N

7059

FIG. 18: The same as Fig. 15. An island of shape coexistenceis predicted within proton or neutron numbers 59− 70. Thisisland corresponds to the shape coexistence in the Sn isotopes[30]. See section X for further discussion.

which decreases the energy bearing a negative sign inthe Hamiltonian of Eq. (21). This fact etches the is-lands of shape coexistence on the nuclear map as in Fig.25 according to the nucleon numbers of Eq. (72). Insection X we present, some emphatic cases of agreementbetween the predictions of the dual–shell mechanism forthe islands of shape coexistence and the data.

The condition of Eq. (66) is satisfied beyond the num-bers of Eq. (69) as seen in Figs. 15-20, but in thesenucleon numbers the QQSO has a small value and thusmajor single–particle energy gaps separate the SO–like

70 80 90 100 110 120 130

0

500

1000

1500

2000

2500

3000

HO SO

C2

Z or N

96 112

FIG. 19: The same as Fig. 15. An island of shape coexistenceis predicted within proton or neutron numbers 96−112. Thisisland corresponds to the shape coexistence in the Hg isotopes[23]. See section X for further discussion.

110 120 130 140 150 160 170 180 190

0

1000

2000

3000

4000

5000

6000 HO SO

C2

Z or N

145 168

FIG. 20: The same as Fig. 15. An island of shape coexistenceis predicted within proton or neutron numbers 145−168. Thisisland corresponds to the fission isomers in the Pu isotopes[116]. See section X for further discussion.

shell from the HO shell. Exceptions, in which the largedeformation of the HO shell dominates and dissolves themajor energy gaps, might appear [120]. Such exceptionsare warnings, that a detailed theoretical study has to beundertaken for each isotopic chain.

This coexistence of the SO–like shell with the HO shellcan explain, why inversion of states [123] occurs aroundN = 8, 20. In these mass regions the neutrons tradition-ally occupy the HO shells 2-8 and 8-20. But as soon asthe neutron number enters into an island of shape coex-istence, as defined by Eq. (72), the neutrons flip fromthe HO shells 2-8, 8-20 to the SO–like shells 6-14, 14-28

21

respectively. The passing of the neutrons from the HOshell to the SO shells is accompanied by excitations inthe single–particle energies. Therefore in light nuclei thisSU(3) mechanism is similar with the particle-hole mech-anism [23]. The term “inversion” implies, that althoughthe SO–like shell has excited single–particle energies, itlies lower in energy, due to the larger value of the QQinteraction.

But in heavier nuclei, such as the Sn [30] and Hg [23]isotopes are, the neutrons traditionally occupy the SOshells. In such cases the passing of the neutrons from theSO shells 50-82, 82-126 to the HO shells 40-70, 70-112 re-spectively cannot be interpreted as a particle excitation,since the SO shells have excited single–particle energiescomparing with HO shells, as presented in section IV.

In these mass regions a merging [124] of the proton(neutron) open SO–like shell with the open proton (neu-tron) HO shell can be treated through the outer productof the two SU(3) irreps:

(λ, µ)SO ⊗ (λ, µ)HO (73)

For instance in the Sn isotopes a merging of the 50-82shell with the 40-70 shell can be treated instead (seeFig. 5). The idea of shell merging has already beenapplied in nuclei lighter than Sn in Ref. [125]. Thefact, that the irreps of a proton or neutron configura-tion in a SO–like and in a HO shell have to be coupledby their outer product [46, 85–87] is further reinforced bythe experimentally observed monopole transition proba-bilities B(E0) among the two bands of shape coexistence[23, 112]. If the irreps of the two bands of shape coexis-tence were irrelevant, then the strong B(E0)s would nothave been justified in the SU(3) symmetry. Furthermorein the Bohr-Hamiltonian treatment of shape coexistenceof Refs. [126–128], the finite barrier among the two min-ima of the potential guarantees, that B(E0)s are allowedto occur among the coexisting shapes, while an infinitebarrier in the double well potential would not allow tun-neling among the two shapes.

The shell merging is accomplished within the SU(3)symmetry, by the outer product (coupling) of two irreps[46, 85–87], which derives numerous irreps as a result. Aphysical criterion, based on the first principles of nuclearphysics, has to be applied, in order to pick the right irrep,which describes the coexisting bands of a heavy nucleuswith shape coexistence. If we consider, that the SO shellis the outer valence shell and the open HO shell is partof the inner core, then the irrep coupling represents theeffect of the deformation (see Eqs. (18), (19)) induced bythe inner core on the deformation induced by the valenceshell. The coupling of a prolate irrep (λ > µ) with anoblate irrep (λ < µ) leads to a prolate–oblate shape co-existence. The coupling technique can be used for everynucleus with shape coexistence, in which the nucleonstraditionally occupy the SO shells. The conclusions ofsection VII corresponded to the pure highest weight ir-reps for protons (neutrons) within the SO–like or the HOshell. It is yet unknown, which will be the coupled irreps

and which one will be lowest in energy. This will be asubject of future research.

IX. THE SUDDEN ONSET OF DEFORMATION

It has long been understood, that the proton–neutroncorrelations enhance the deformation. The observationof an enhanced proton–neutron correlation was reportedby de Shalit and Goldhaber in Ref. [74], where the au-thors noticed, that when the last protons and the lastneutrons occupy specific orbitals, then the β transitionsbecome slower. This stabilization of the nucleus was at-tributed to the proton–neutron interaction. By the way,many years later, the de Shalit–Goldhaber pairs lead tothe unitary transformation, which is applied in the proxy-SU(3) symmetry [39]. Talmi in Ref. [129] stressed out,that the proton–neutron interaction is strong and attrac-tive. Federman and Pittel in Refs. [109, 110] studiedthe transition from the spherical to the deformed shape,when the last protons and neutrons occupy specific or-bitals, namely the Federman–Pittel pairs. The effect ofthe sudden onset of deformation on the phenomenon ofshape coexistence has been pointed out by Heyde, VanIsacker, Casten and Wood in Ref. [111]. Systematicsof nuclear observables, which are related with the on-set of deformation, were presented in a simple way byCasten in Ref. [108], using the NpNn scheme. So fruit-ful has been the study of the proton–neutron correla-tions, that triggered much work [37, 72, 73, 130–132] inthe years to come. Recently the relation of shape tran-sitions with shape coexistence has been investigated inRefs. [44, 133, 134].

In this work we will discuss again the results of theplots of Ref. [108], having in mind the predictions of thepresent mechanism. We choose to concentrate on Ref.[108], because of the rich collection of experimental datapresented in a simple way. The physical quantities used,are the ratio of the energy of the first J = 4+ state overthe energy of the first J = 2+ state E4+1

/E2+1, as well as

E2+1itself. The ratio E4+1

/E2+1varies from ∼ 2 for a vi-

brational nucleus, to ∼ 2.5 for a γ-soft asymmetric rotor,to ∼ 3.33 for a deformed symmetric rotor [108]. The en-ergy E2+1

varies from values near 1 MeV near closed shells

to ∼ 100 or ∼ 200 keV for well-deformed rotational nuclei[108]. In other words the higher is the ratio E4+1

/E2+1, or

the lower is E2+1, the more deformed is the nucleus.

By considering Figs. 1 and 2 of Ref. [108], showingthe E4+1

/E2+1ratio versus the proton and neutron num-

ber respectively, we observe, that the sudden onset ofdeformation at N = 60 is “more sudden” for Sr and Zr(Z = 38, 40), i.e., when the proton number lies withinthe island of shape coexistence with 34 ≤ Z ≤ 40 (seepresent Fig. 17). Again, in Figs. 3 and 4 of Ref. [108],which present the energy E2+1

versus N and Z respec-

tively, we observe a more violent onset of deformationfor Sr and Zr (Z = 38, 40), which lie within an island of

22

shape coexistence (34 ≤ Z ≤ 40, as seen in present Fig.17).

Similarly Figs. 12 and 13 of Ref. [108], showingthe E4+1

/E2+1ratio, indicate a more vivid onset of de-

formation at N = 90 when the proton number (Z =60, 62, 64, 66, 68, 70) lies within an island of shape coex-istence (59 ≤ Z ≤ 70, as seen in present Fig. 18). Thesame conclusion is drawn from Figs. 15 and 16 of Ref.[108], showing the energy E2+1

. In addition, Fig. 17 of

Ref. [108] indicates a sudden change in the trend of theenergy E2+2

after Ce (Z = 58), i.e., when the proton num-

ber enters the island of coexistence within 59 ≤ Z ≤ 70,as seen in present Fig. 18.

On the contrary Figs. 6-11 of section 2.2 of Ref. [108]indicate a smooth transition from the spherical to thedeformed shape in the A ' 130 region. In the cases of theXe,Ba,Ce isotopes (Z = 54, 56, 58) the proton numberlies outside the islands of shape coexistence, which arepredicted by the dual–shell mechanism (see expression(72) and present Fig. 25).

In general we observe, that the onset of deformationis more vivid, when the proton number lies within anisland of shape coexistence, which is predicted by thepresent mechanism. This more sudden transition fromthe spherical to the deformed shape could be related tothe coexistence of three SU(3) irreps (two for protons $and one for neutrons ν):

(λ, µ)$,HO ⊗ (λ, µ)$,SO ⊗ (λ, µ)ν,SO, (74)

when the proton number lies within an island of shapecoexistence, as was defined in Eq. (72), or it could berelated to four SU(3) irreps:

(λ, µ)$,HO ⊗ (λ, µ)$,SO ⊗ (λ, µ)ν,HO ⊗ (λ, µ)ν,SO (75)

when both the proton and neutron numbers lie withinan island of shape coexistence. These possibilities callfor further investigations.

Another important experimental fingerprint of the pro-ton induced shape coexistence is, that it is accompaniedby large electrical monopole transitions ρ2(E0). As anexample one may see Figs. 27, 31 and 34 of Ref. [23].In general in the proton induced islands of shape coex-istence, which are marked by the horizontal stripes onthe map of Fig. 25, one has to look for B(E0)s, whilein the neutron induced islands, which are marked by thevertical stripes on the map of Fig. 25, B(E2)s amongthe coexisting bands are more common.

X. PARADIGMATIC ISOTOPES WITH SHAPECOEXISTENCE

In the next we will demonstrate some exceptional man-ifestations of shape coexistence, which are predicted bythe dual–shell mechanism. The paradigms will beginfrom the Be and will end to the Pu, a fact that high-lights, that the dual–shell mechanism can be applied in

all mass regions. The reported phenomena will be theparity inversion, the inversion of states, the come-backof the HO magic numbers, the shape coexistence and thefission isomers, all of them being just the various faces ofshape coexistence in different mass regions.

A. The 11Be

The first island of shape coexistence, which is pre-sented in Fig. 15, appears in the light nuclei. A paradigmin this mass region is the 11

4Be7 halo nucleus, which isknown for the phenomenon of parity inversion [135]. Thelast unpaired neutron of 9

4Be5 lies in the 1p3/2 orbit andthus this isotope exhibits a ground state with negativeparity [136]. Similarly the 11

4Be7 nucleus should possess anegative parity ground state, too. But magnetic momentmeasurements [83] have revealed that 11

4Be7 has a posi-tive parity ground state. Furthermore, measurements ofthe nuclear charge radii in this isotopic chain [137] haveestablished the 11

4Be7 nucleus as a halo nucleus. Spec-troscopic factors led to the conclusion, that in 12Be thelast neutron pair occupies partially ( 2

3 ) the 2s1/2, 1d5/2

orbitals [138].This parity inversion can be justified by the first island

of shape coexistence of Fig. 15. The neutrons of the Beisotopes traditionally occupy the 2-8 harmonic oscilla-tor shell, namely the p shell. But as soon as the condi-tion (66) is satisfied, right at N = 7, the neutrons areexcited to the SO–like shell 6-14, which consists of the

1p1/2mj , 1d

5/2mj orbitals and thus a positive parity ground

state becomes possible. The orbitals occupied by neu-trons within the two sets of magic numbers have beenpresented in Table III and the calculation of the singleparticle energies within the Elliott and the proxy-SU(3)symmetry has been analyzed in section IV.

At this point, it has to be clear, that the total angularmomentum and the parity (J±) of a state of an even-odd nucleus within the Elliott SU(3) symmetry, is notthe J± of the last uncoupled neutron. The Elliott SU(3)model derives the J quantum number with the rules ofRef. [36]:

KS = S, (S − 1), ...,−S, (76)

K = KS +KL ≥ 0, (77)

KL = µ, µ− 2, ...,−µ, (78)

J = K,K + 1, ..., λ+ µ+ S, (79)

with the exception that if KL = KS = K = 0 then Jis even or odd with Jmax = λ + S. In the above S =12 is the nuclear spin of the odd mass nucleus, KS isthe nuclear spin projection, KL is the projection of theorbital angular momentum of the nucleus and K is thenuclear projection of the total angular momentum. As a

result the 2s1/2mj orbital is not necessary for the prediction

of the 1/2+ ground state of 114Be7 within the Elliott SU(3)

model. Instead using a proton ($) irrep (λ$, µ$)HO =

23

(2, 0) and a neutron (ν) irrep (λν , µν)SO = (1, 0) (seeTable I), the total irrep of the 11

4Be7 is (λ, µ) = (λ$ +λν , µ$ + µν) = (3, 0) [38], which gives [36]:

S =1

2, (80)

KS =1

2, (81)

KL = 0, (82)

K = KS +KL =1

2, (83)

J =1

2,

3

2,

5

2. (84)

Consequently the J of the ground state of this halo nu-

cleus, with the one last neutron lying in the 1p1/2mj , 1d

5/2mj

shell, is J = 12 within the dual–shell mechanism. A pos-

itive parity becomes possible due to the participation in

the SO–like shell of the 1d5/2mj orbital.

B. The 16O

The case of 16O has already been discussed in sectionIV, where an irrep (4, 0) has been predicted from theSO–like shell 6-14 to derive the ground 0+1 state, andan irrep (0,0) originated from the HO shell 2-8, matcheswith the excited 0+2 state. In section VIII we had argued,that although the 6-14 shell is excited in comparison withthe 2-8 shell, the first lies lower in energy, because it ischaracterized by larger QQ interaction. In this section wewill calculate the energy of the 0+2 state for this nucleususing the on hand mechanism. The energy of the 0+2 isthe eigenvalue of Eq. (59):

E0+2= (N0,HO −N0,SO) +

κ

2(4C2,SO − 4C2,HO), (85)

where Eq. (16) has been used for L = 0.The N0 of the HO shell is:

N0,HO = N0 (86)

of Eq. (15), while the N0,SO is the eigenvalue of Eq. (33):

N0,SO = N0 + 〈Ψ|H0,proxy|Ψ〉 , (87)

where Ψ is the L-projected Elliott wave function [35].Thus the eigenvalue of Eq. (62) is:

∆N0 = −〈Ψ|H0,proxy|Ψ〉 . (88)

The two protons and two neutrons of the 6-14 shellare placed in the cartesian state |nz = 1, nx = 0, ny = 0〉(with opposite spin projections ms = ±1/2 and isospinprojections mt = +1/2,−1/2 for protons and neutronsrespectively), according to the order of Eq. (11), pos-sessing (λ, µ) = (4, 0), as derived from Eqs. (12), (13).The resulting states, which are given in Eq. (8), form the

Slater determinant Φ. Following the notation of section2.6 of Ref. [40] the cartesian state |nz, nx, ny〉 = |1, 0, 0〉with mt = +1/2,ms = −1/2 for the first particle willbe labeled by φ+−(1). The Slater determinant for the 4particle system is:

Φ =1√4!

∣∣∣∣∣∣∣φ++(1) φ+−(1) φ−+(1) φ−−(1)φ++(2) φ+−(2) φ−+(2) φ−−(2)φ++(3) φ+−(3) φ−+(3) φ−−(3)φ++(4) φ+−(4) φ−+(4) φ−−(4)

∣∣∣∣∣∣∣ . (89)

The δj,N+1/2 = δj,3/2 term, which is included in

H0,proxy of Eq. (34), affects only the 1p3/2±1/2 component

of the |nz = 1, nx = 0, ny = 0,ms = ±1/2〉 states of Eq.

(8). Thus for the 4–particle excitation of the 16O:

〈Φ|H0,proxy|Φ〉 =8

3εproxy. (90)

The irrep (λ, µ) = (4, 0) of the 4 particles in the SO–like shell generates only the K = 0 band, following Eq.(78) with S = KS = 0 for an even–even nucleus. Con-sequently from Eqs. (39), (90) for the ground state withK = L = 0 we get:

〈Ψ|H0,proxy|Ψ〉 = |a(0, 0)|2 8

3εproxy. (91)

The coefficient a(K,L) for a µ = 0 case, as this one is, canbe taken from Table 2A of Ref. [41] to be a(0, 0) = 1√

5,

while the value of ~ω is calculated from Eq. (23) to be~ω = 16.27 MeV. By substituting the experimental valueof the deformation variable β = 0.364 for 16O [88] in Eqs.(28), (29) we derive, that the deformation parameter ofthe Nilsson asymptotic basis is ε = 0.36, which leads toεproxy = 0.76~ω using Eq. (27). The ∆N0 of Eq. (88)

for the 4–particle excitation in 16O is:

∆N0 = −〈Ψ|H0,proxy|Ψ〉 = −6.595 MeV. (92)

The quadrupole difference ∆(QQ) of Eq. (61) amongthe SO–like irrep (4, 0) for the 4–particle excitation andthe HO irrep (0, 0) for the 0–particle excitation reads [68]:

∆(QQ) =κ

2(4C2,SO − 4C2,HO) = 2κC2,SO, (93)

where κ is given by Eq. (22) and C2,SO = 28, C2,HO = 0are given by Eq. (18). Using N0 = 36 as derived fromEq. (15) for the 16 particles of this nucleus one gets that2κ = 0.452 MeV and so

∆(QQ) = 12.655 MeV. (94)

Finally by substituting Eqs. (92), (94) into (85), thepredicted energy for the 0+2 state of 16O within the dual–shell mechanism for shape coexistence is:

E0+2= 6.06 MeV, (95)

which is in very good agreement with the data at 6.049MeV [89].

24

In this sample calculation of the band-head of the co-existing band became obvious the competition betweenthe ∆N0 and the ∆(QQ) terms. Actually this competi-tion leads to the condition ∆(QQ) ≥ −∆N0 ≥ 0, whichderived the islands of shape coexistence in section VIII.It also became obvious, that the Elliott SU(3) and theproxy-SU(3) symmetry are capable of all types of calcu-lations concerning the nuclear shape and that the agree-ment with the data emerges naturally using global pa-rameters without any fitting.

C. The Mg isotopes

In Mg isotopes the neutrons traditionally occupy the 8-20 harmonic oscillator shell. But as the dual–shell mech-anism predicts, just after N = 17 (see Fig. 16) the neu-trons can flip from the 8-20 HO shell to the 14-28 SOshell. This procedure is able to explain the inversion ofstates [123] in the Mg isotopes and the breaking of themagicity of N = 20. According to Figs. 16 and 21 shapecoexistence and inversion of states in the Mg isotopes be-gin just after N = 17, while shape coexistence ends atthe harmonic oscillator shell closure at N = 20, and in-version of states endures as far as QQSO ≥ QQHO, whichis valid till N = 21 (see Fig. 21), predictions which alignwith the experimental facts [114, 139–141].

D. The Ni isotopes

The next island of inversion is indicated by Fig. 17,which supports, that the SO–like magic numbers are be-ing competed by the harmonic oscillator magic number[143] for isotopic chains with N ≥ 34. This prediction isin agreement with the data on the deformation variableβ in the Ni isotopes as plotted in Fig. 22.

E. Heavier nuclei

An island of shape coexistence is indicated by Fig. 18,which predicts the phenomenon among 59 ≤ N ≤ 70for an isotopic chain. The most impressive example forthis island of shape coexistence is the parabolic line ofthe excited K = 0+ bands in the 110−120Sn isotopes aspresented in Fig. 23 and in Ref. [30].

Another astonishing example of shape coexistence liesin the 176−190Hg isotopes (see Fig. 10 of Ref. [23]), whichmatches with the predictions of Fig. 19. The parabolicline of the excited K = 0+ bands begins at N = 96 andends at N = 110, just two neutrons below the harmonicoscillator shell closure at N = 112. The data for theparabolic and excited K = 0+ band for these isotopesare plotted in Fig. 24.

Finally Fig. 20 corresponds to the super–heavies andpredicts a comeback of the harmonic oscillator shell after

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30-20

0

20

40

60

80

100

120

140

160

180

200

220

C2

N

SO HO

17 21

island of inversion

Mg

FIG. 21: The eigenvalues of the SU(3) Casimir operator C2 forthe Mg isotopes. The 12 protons of the Mg isotopes lie in the6-14 SO–like shell [142] and thus posses (λ$, µ$)SO = (0, 0).For neutron numbers 17 ≤ N ≤ 20, the neutron configura-tion which is derived from the SO–like neutron shell, has tocorrespond to the ground state band (see section VII), whilethe less deformed configuration, coming from the HO neu-tron shell, has to correspond to the excited band, if only pureSU(3) irreps are considered. Inversion of states is predicted inthe 29−33

12Mg17−21 isotopes, which possess QQSO ≥ QQHO ac-cording to the condition of Eq. (64), while shape coexistenceis predicted in the 29−32

12Mg17−20 isotopes. These predictionsalign with the data of Refs. [114, 139, 140]. See section X Cfor further discussion.

N = 145. This region corresponds to the fission isomerscentered around 240

94Pu146 [116].

XI. DISCUSSION

In the present work a novel mechanism for the phe-nomenon of shape coexistence is introduced. Its mainfeatures are summarized here:1) The proposed mechanism is based on the SU(3) sym-metry. In light nuclei, up to the sd shell, the SU(3) sym-metry of the 3-dimensional (3D) isotropic harmonic oscil-lator (HO) is present, as used in the Elliott SU(3) model.Beyond the sd shell, the recently introduced proxy-SU(3)symmetry is used.2) The present mechanism is based on the interplay be-tween the HO magic numbers 2, 8, 20, 40, 72, 112, 168,and the spin-orbit (SO) like magic numbers 6, 14, 28, 50,82, 126, 184. The main element of the new mechanismare particle excitations occurring between the HO andSO sets of shells.3) These particle excitations lead to the dissolution ofthe magic numbers and to the merging of the two typesof shells.4) The main novel prediction of the present mechanism

25

28 30 32 34 36 38 40 42 44 46 48 50 520,0

0,1

0,2

0,3

SO HO data

N

Ni34

island of inversion

40

FIG. 22: The deformation parameter β as calculated from Eq.(19) with the scaling factor of Eq. (20) [38] for the Ni isotopesfor two valence neutron shells. The protons lie in the 28-50SO–like shell, with proton irrep (λ$, µ$) = (0, 0) (see TableI). The data on the deformation β [88, 144] indicate, thatjust after N = 34 the 20-40 HO shell becomes preferable forthe neutron configuration. The neutrons are predicted to flipback from the 20-40 shell to the 28-50 shell at the harmonicoscillator shell closure, which occurs at neutron number N =40. See section X D for further discussion.

is, that shape coexistence cannot appear everywhere onthe nuclear chart, but only within specific regions, calledislands of shape coexistence, the shores of which are de-termined through group theoretical arguments in a pa-rameter independent way.5) The present mechanism allows for a parameter-freeprediction of the energy of the 0+2 band being the shapecoexistence partner of the ground state band.6) The present mechanism suggests that shape coexis-tence, parity inversion, inversion of states, and fissionisomers are just shades of the same phenomenon, i.e.,the coexistence of the HO valence shell with the SO–likeshell.

In relation to the widely accepted particle–hole mecha-nism of shape coexistence, the following comments apply:1) The islands predicted by the present mechanism arefully compatible with the regions of the nuclear chart inwhich the particle–hole mechanism has been applied. Nocontradiction between the two mechanisms arises.2) The novel message from the present mechanism is,that the particle–hole mechanism cannot occur all overthe nuclear chart, but only within the specific regionspredicted by the present mechanism based on parameter–free SU(3) symmetry arguments.3) In light nuclei the two mechanisms coincide, since thepresent mechanism predicts particle excitations from theHO shell to the neighboring SO–like shell within certainislands on the nuclear map. In medium mass and heavynuclei, the present mechanism is based on shell merging.

58 60 62 64 66 68 700

2000

4000

6000

8000

10000

12000

14000

16000

Sn 30+ 28+ 26+ 24+ 22+ 20+ 18+ 16+ 14+ 12+ 10+ 8+ 6+ 4+ 2+ 0+

E (k

eV)

N

60 70shape coexistence

FIG. 23: One of the most striking manifestations of shapecoexistence appears in the Sn isotopes. An excited K = 0+

band, which is attributed to shape coexistence (see Fig. 3.10of Ref. [30]), appears within neutron numbers 60 ≤ N ≤ 70.Energy levels have been obtained by [145–150]. See sectionX E for further discussion.

96 98 100 102 104 106 108 1100

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

5500

6000

6500

Hg 22+ 20+ 18+ 16+ 14+ 12+ 10+ 8+ 6+ 4+ 2+ 0+

E (k

eV)

N

6-4-

96 110shape coexistence

FIG. 24: A peerless manifestation of shape coexistence ap-pears in the Hg isotopes with neutron numbers within 96 ≤N ≤ 110 (see Fig. 10 of Ref. [23]). The data have been col-lected from [151–158]. See section X E for further discussion.

The proposed mechanism is supported by results forthe single particle energies provided by Density Func-tional Theory (DFT) calculations. In particular:1) In the Pb and Hg isotopes both the neutron and theproton energy gaps above the HO and the SO magic num-bers collapse within the 96 ≤ N ≤ 110 region (as pre-dicted by the present mechanism), leading to shell merg-ing.2) Plotting the single particle energies relatively to theFermi energy in the Pb and Hg isotopes, one sees within

26

the 96 ≤ N ≤ 110 region (predicted by the present mech-anism) that some of the proton orbitals belonging to theshell above the 82 magic number sink below the Fermienergy, while some of the proton orbitals belonging to the50-82 shell pop up above the Fermi energy. These resultscorroborate the compatibility between the particle-holemechanism and the present shell merging mechanism,giving in parallel the message that the particle-hole mech-anism is applicable only within the islands predicted bythe present approach.

The proposed mechanism is rather general and appliesto all mass regions. It should be noticed that SU(3) isused as a classification scheme and not as a dynamicalsymmetry, therefore the method is applicable over thewhole nuclear chart and is not limited within regionsof highly deformed nuclei. For instance the dual–shellmechanism predicts the tin isotopes with shape coexis-tence, despite that these isotopes are not very deformed.Furthermore, its predictions are parameter-independent.

As a consequence the present work can be consideredas a first step towards a unified understanding of shapecoexistence and related phenomena. Detailed studies fordifferent isotopes and mass regions are called for, whichmight reveal the need for further elaboration of the mech-anism. The calculation of nuclear observables within thedual–shell and the comparison of the results with the

data and with the predictions of other nuclear modelsand mechanisms [126, 127, 159–165] could provide a bet-ter understanding of the realm of nuclear structure. Theproton–neutron interaction in the Elliott and the proxy-SU(3) schemes has to be studied in the future, since weexpect, that it will give us a more clear view of the is-lands of shape coexistence on the nuclear chart. Alsothe pairing interaction has to be included in the Hamil-tonian, so as to estimate accurately the energy levels ofthe coexisting bands.

Acknowledgements

Discussions with R. F. Casten, K. Blaum and J. Csehimproved considerably the manuscript. Financial sup-port by the Greek State Scholarships Foundation (IKY)and the European Union within the MIS 5033021 action,by the Bulgarian National Science Fund (BNSF) underContract No.KP-06-N48/1 and by the Tenure Track Pi-lot Programme of the Croatian Science Foundation andthe Ecole Polytechnique Federale de Lausanne and theProject No. TTP-2018-07-3554 Exotic Nuclear Structureand Dynamics with funds of the Croatian-Swiss ResearchProgramme is gratefully acknowledged.

[1] M. G. Mayer, Science 145, 999 (1964). doi 10.1126/sci-ence.145.3636.999.

[2] M. G. Mayer, Phys. Rev. 74, 235 (1948). doi10.1103/physrev.74.235.

[3] W. McLatchie, J. E. Kitching and W. Darcey,Phys. Lett. B 30, 529 (1969). doi 10.1016/0370-2693(69)90446-8.

[4] D. G. Burke, G. Lovhoiden and J. C. Wadding-ton, Phys. Lett. B 43, 470 (1973). doi 10.1016/0370-2693(73)90347-X

[5] P. Kleinheinz, R. K. Sheline, M. R. Maier, R. M. Di-amond and F. S. Stephens, Phys. Rev. Lett. 32, 68(1974). doi 10.1103/PhysRevLett.32.68.

[6] B. C. Smith, Phys. Rev. C 19, 1107 (1979). doi10.1103/PhysRevC.19.1107.

[7] R. G. Lanier, G. L. Struble, L. G. Mann, I. C. Oelrich,I. D. Proctor and D. W. Heikkinen, Phys. Rev. C 22,51 (1980). doi 10.1103/PhysRevC.22.51.

[8] J. H. Hamilton, K. R. Baker, C. R. Bingham, E. L.Bosworth, H. K. Carter, J. D. Cole, R. W. Fink, G.Garcia Bermudez, G. W. Gowdy, K. J. Hofstetter, M. A.Ijaz, A. C. Kahler, B. D. Kern, W. Lourens, B. Martin,R. L. Mlekodaj, A. V. Ramayya, L. L. Riedinger, W.D. Schmidt-Ott, E. H. Spejewski, B. N. Subba Rao, E.L. Robinson, K. S. Toth, F. Turner, J. L. Weil, J. L.Wood, A. Xenoulis and E. F. Zganjar, Izv. Akad. NaukSSSR, Ser.Fiz. 40, 18 (1976).

[9] J. D. Cole, A. V. Ramayya, J. H. Hamilton, H.Kawakami, B. van Nooijen, W. G. Nettles, L. L.Riedinger, F. E. Turner, C. R. Bingham, H. K. Carter,E. H. Spejewski, R. L. Mlekodaj, W. D. Schmidt-Ott,

E. F. Zganjar, K. S. R. Sastry, F. T. Avignone, K. W.Toth and M. A. Ijaz, Phys. Rev. C 16, 2010 (1977). doi10.1103/PhysRevC.16.2010.

[10] W. Bohne, K. D. Buchs, H. Fuchs, K. Grabisch, D.Hilscher, U. Jahnke, H. Kluge, T. G. Masterson andH. Morgenstern, Nucl. Phys. A 284, 14 (1977). doi10.1016/0375-9474(77)90682-0.

[11] R. Middleton, J. D. Garrett and H. T. Fortune,Phys. Lett. B 39, 339 (1972). doi 10.1016/0370-2693(72)90133-5.

[12] H. T. Fortune, M. N. I. Al-Jadir, R. R. Betts, J. N.Bishop, R. Middleton, Phys. Rev. C 19, 756 (1979). doi10.1103/PhysRevC.19.756.

[13] H. T. Fortune and A. E. L. Dieperink, Phys. Rev. C 19,1112 (1979). doi 10.1103/PhysRevC.19.1112.

[14] J. Ljungvall, A. Gorgen, M. Girod, J. -P. Delaroche, A.Dewald, C. Dossat, E. Farnea, W. Korten, B. Melon, R.Menegazzo, A. Obertelli, R. Orlandi, P. Petkov, T. Pis-sulla, S. Siem, R. P. Singh, J. Srebrny, Ch. Theisen, C.A. Ur, J. J. Valiente-Dobon, K. O. Zell and M. Zielinska,Phys. Rev. Lett. 100. 102502 (2008). doi 10.1103/Phys-RevLett.100.102502.

[15] T. J. Mertzimekis, N. Benczer-Koller, J. Holden, G.Jakob, G. Kumbartzki, K.-H. Speidel, R. Ernst, A. Mac-chiavelli, M. McMahan, L. Phair, P. Maier-Komor, A.Pakou, S. Vincent and W. Korten, Phys. Rev. C 64,024314 (2001). url 10.1103/PhysRevC.64.024314.

[16] K. Kumar, J. Phys. G: Nuclear Physics 4, 849 (1978).[17] K. Becker, G. Jung, K.-H. Kobras, H. Wollnik

and B. Pfeiffer, Z. Phys. A 319, 193 (1984). doi10.1007/BF01415633.

27

[18] J. Eberth, L. Cleemann and N. Schmal, ATOMKI Ko-zlem. 26, 12 (1984).

[19] F. Schussler, J. A. Pinston, E. Monnand, A. Moussa,G. Jung, E. Koglin, B. Pfeiffer, R. V. F. Janssens andJ. van Klinken, Nucl. Phys. A 339, 415 (1980). doi10.1016/0375-9474(80)90024-X.

[20] W. Korten, E. Bouchez, E. Clement, A. Chatillon, A.Gorgen, Y. Le Coz, Ch. Theisen, J. Wilson, J. M.Casandjian, G. de France, G. Sletten, T. Czosnyka, J.Iwanicki, M. Zielinska, C. Andreoiu, P. Butler, R. -D.Herzberg, D. Jenkins, G. Jones , F. Becker, J. Gerl, W.Catford and C. Timis, Nucl. Phys. A 746, 90c (2004).doi 10.1016/j.nuclphysa.2004.09.125.

[21] A. Gorgen, E. Clement, E. Bouchez, A. Chatillon,W. Korten, Y. Le Coz, Ch. Theisen, C. Andreoiu, F.Becker, B. Blank, A. Burger, P. Butler, J. M. Casand-jian, W. Catford, T. Czosnyka, P. Davies, S. P. Fox, G.de France, G. Georgiev, J. Gerl, H. Hubel, J. Iwanicki,D. G. Jenkins, F. Johnston-Theasby, P. Joshi, I. Matea,P. Napiorkowski, F. de Oliveira Santos, G. Sletten, C.Timis, R. Wadsworth and M. Zielinska, Acta Phys. Pol.B 36, 1281 (2005).

[22] E. Clement, A. Gorgen, W. Korten, E. Bouchez, A.Chatillon, J.-P. Delaroche, M. Girod, H. Goutte, A.Hurstel, Y. Le Coz, A. Obertelli, S. Peru, Ch. Theisen,J. N. Wilson, M. Zielinska, C. Andreoiu, F. Becker, P.A. Butler, J. M. Casandjian, W. N. Catford, T. Czos-nyka, G. de France, J. Gerl, R.-D. Herzberg, J. Iwanicki,D. G. Jenkins, G. D. Jones, P. J. Napiorkowski. G. Slet-ten and C. N. Timis, Phys. Rev. C 75, 054313 (2007).doi 10.1103/PhysRevC.75.054313.

[23] K. Heyde and J. L. Wood, Rev. Mod. Phys. 83, 1467(2011). doi 10.1103/revmodphys.83.1467.

[24] T. Thomas, V. Werner, J. Jolie, K. Nomura, T. Ahn, N.Cooper, H. Duckwitz, A. Fitzler, C. Fransen, A. Gade,M. Hinton, G. Ilie, K. Jessen, A. Linnemann, P. Petkov,N. Pietralla and D. Radeck, Nucl. Phys. A 947, 203(2016). doi 10.1016/j.nuclphysa.2015.12.010.

[25] M. D. Glascock, E. W. Schneider, W. B. Walters,R. A. Meyer, Z. Phys. A 283, 415 (1977). doi10.1007/BF01409524.

[26] P. E. Garrett, K. L. Green, R. A. E. Austin, G. C. Ball,D. S. Bandyopadhyay, S. Colosimo, D. S. Cross, G. A.Demand, G. F. Grinyer, G. Hackman, W. D. Kulp, K.G. Leach, A. C. Morton, C. J. Pearson, A. A. Phillips,M. A. Schumaker, C. E. Svensson, J. Wong, J. L. Woodand S. W. Yates, Acta Phys. Pol. B 42, 799 (2011). doi10.5506/APhysPolB.42.799.

[27] M. Ionescu-Bujor, S. Aydin, A. Iordachescu, N.Marginean, S. Pascu, D. Bucurescu, C. Costache, N.Florea, T. Glodariu, A. Ionescu, R. Marginean, C. Mi-hai, R. E. Mihai, A. Mitu, A. Negret, C. R. Nita,A. Olacel, B. SaygI, L. Stroe, R. Suvaila, S. Toma,A. Turturica, Phys.Rev. C 102, 044311 (2020). doi10.1103/PhysRevC.102.044311.

[28] B. Olaizola, A. B. Garnsworthy, F. A. Ali, C. An-dreoiu, G. C. Ball, N. Bernier, H. Bidaman, V. Bild-stein, M. Bowry, R. Caballero-Folch, I. Dillmann, G.Hackman, P. E. Garrett, B. Jigmeddorj, A. I. Kilic,A. D. MacLean, H. P. Patel, Y. Saito, J. Smallcombe,C. E. Svensson, J. Turko, K. Whitmore and T. Zidar,Phys. Rev. C 100, 024301 (2019). doi 10.1103/Phys-RevC.100.024301.

[29] M. Siciliano, I. Zanon, A. Goasduff, P. R. John, T.

R. Rodriguez, S. Peru, I. Deloncle, J. Libert, M.Zielinska, D. Ashad, D. Bazzacco, G. Benzoni, B.Birkenbach, A. Boso, T. Braunroth, M. Cicerchia, N.Cieplicka-Orynczak, G. Colucci, F. Davide, G. de An-gelis, B. de Canditiis, A. Gadea, L. P. Gaffney, F. Gal-tarossa, A. Gozzelino, K. Hadynska-Klek, G. Jaworski,P. Koseoglou, S. M. Lenzi, B. Melon, R. Menegazzo,D. Mengoni, A. Nannini, D. R. Napoli, J. Pakari-nen, D. Quero, P. Rath, F. Recchia, M. Rocchini, D.Testov, J. J. Valiente-Dobon, A. Vogt, J. Wiederholdand W. Witt, Phys.Rev. C 102, 014318 (2020). doi10.1103/PhysRevC.102.014318.

[30] J. L. Wood, K. Heyde, W. Nazarewicz, M. Huyseand P. van Duppen,Phys. Rep. 215, 101 (1992). doi10.1016/0370-1573(92)90095-H.

[31] B. G. Wybourne, Classical Groups for Physicists(Wiley, New York, 1974). isbn 9780140215793, lccn73017363

[32] M. Moshinsky and Yu. F. Smirnov, The HarmonicOscillator in Modern Physics (Harwood, Amsterdam,1996). isbn 9783718606214, lccn 99521805.

[33] J. P. Elliott, Proc. Roy. Soc. London Ser. A: Math. Phys.Sci. 245, 128 (1958). doi 10.1098/rspa.1958.0072.

[34] J. P. Elliott, Proc. Roy. Soc. London Ser. A: Math. Phys.Sci. 245, 562 (1958). doi 10.1098/rspa.1958.0101.

[35] J. P. Elliott and M. Harvey, Proc. Roy. Soc. Lon-don Ser. A: Math. Phys. Sci. 272, 557 (1963). doi10.1098/rspa.1963.0071.

[36] J. P. Elliott and C. E. Wilsdon, Proc. Roy. Soc. Lon-don Ser. A: Math. Phys. Sci. 302, 509 (1968). doi10.1098/rspa.1968.0033.

[37] D. Bonatsos, I. E. Assimakis, N. Minkov, A. Martinou,R. B. Cakirli, R. F. Casten, and K. Blaum, Phys. Rev.C 95, 064325 (2017). doi 10.1103/PhysRevC.95.064325.

[38] D. Bonatsos, I. E. Assimakis, N. Minkov, A. Marti-nou, S. Sarantopoulou, R. B. Cakirli, R. F. Casten,and K. Blaum, Phys. Rev. C 95, 064326 (2017). doi10.1103/PhysRevC.95.064326.

[39] A. Martinou, D. Bonatsos, N. Minkov, I. E. Assimakis,S. K. Peroulis, S. Sarantopoulou and J. Cseh, Eur.Phys. J. A 56, 239 (2020). doi 10.1140/epja/s10050-020-00239-0.

[40] C. E. Wilsdon, A survey of the nuclear s-d shell, us-ing the SU(3) coupling scheme, Ph.D. thesis, Uni-versity of Sussex (1965). Ethos ID: 477554, urlhttps://ethos.bl.uk/Home.do.

[41] J.D. Vergados, Nucl. Phys. A 111, 681 (1968). doi10.1016/0375-9474(68)90249-2.

[42] O. Sorlin and M.-G. Porquet, Progr. Part. Nucl. Phys.61, 602 (2008). doi 10.1016/j.ppnp.2008.05.001.

[43] K. Heyde, J. Jolie, R. Fossion, S. De Baerdemackerand V. Hellemans, Phys. Rev. C 69, 054304 (2004). doi10.1103/physrevc.69.054304.

[44] J.-E. Garcıa-Ramos and Kris Heyde, EPJ Web ofConferences 178, 05005 (2018). doi 10.1051/epj-conf/201817805005.

[45] F. H. Garcia, C. Andreoiu, G. C. Ball, A. Bell, A. B.Garnsworthy, F. Nowacki, C. M. Petrache, A. Poves,K. Whitmore, F. A. Ali, N. Bernier, S. S. Bhattachar-jee, M. Bowry, R. J. Coleman, I. Dillmann, I. Djianto,A. M. Forney, M. Gascoine, G. Hackman, K. G. Leach,A. N. Murphy, C. R. Natzke, B. Olaizola, K. Ortner,E. E. Peters, M. M. Rajabali, K. Raymond, C. E.Svensson, R. Umashankar, J. Williams and D. Yates,

28

Phys. Rev. Lett. 125, 172501 (2020). doi 10.1103/phys-revlett.125.172501.

[46] M. Harvey, Adv. Nucl. Phys. 1, ed. M. Baranger and E.Vogt (Plenum, New York, 1968). isbn 978-1-4757-0105-0.

[47] C. Cohen-Tannoudji, B. Diu, and Frank Laloe, Quan-tum Mechanics (Wiley, New York, 1991) vol. 1, com-plement BV II .

[48] H. J. Lipkin, Lie groups for pedestrians (Dovr, NewYork, 2002).

[49] V. K. B. Kota, SU(3) Symmetry in Atomic Nuclei(Springer, Singapore, 2020). doi 10.1007/978-981-15-3603-8.

[50] S. G. Nilsson and I. Ragnarsson, Shapes and Shells inNuclear Structure (Cambridge University Press, Cam-bridge, 1995).

[51] K. T. R. Davies and S. J. Krieger, Can. J. Phys. 69, 62(1991). doi 10.1139/p91-010.

[52] K. L. G. Heyde, The Nuclear Shell Model (Springer,Berlin, 1990). doi 10.1007/978-3-642-97203-4.

[53] E. Fermi and J. Orear, Nuclear Physics: A CourseGiven by Enrico Fermi at the University of Chicago,notes Compiled by Jay Orear (University of ChicagoPress, 1950).

[54] W. Pauli, in Writings on Physics and Philosophy(Springer, Berlin, 1994) p. 165.

[55] J. C. Slater, Phys. Rev. 34, 1293 (1929). doi10.1103/physrev.34.1293.

[56] P. Ring and P. Schuck, The Nuclear Many-Body Prob-lem (Springer, Berlin, 1980). isbn 978-3-540-21206-5.

[57] J.P. Draayer, Y. Leschber, S.C. Park and R. Lopez,Comp. Phys. Comm. 56, 279 (1989). doi 10.1016/0010-4655(89)90024-6.

[58] V. K. B. Kota, arXiv: 1812.01810 [nucl-th] (2018).[59] A. Martinou, D. Bonatsos, N. Minkov, I.E. As-

simakis, S. Sarantopoulou, and S. Peroulis,in Nuclear Theory vol. 37, ed. M. Gaidarovand N. Minkov (Heron, Sofia, 2018) p. 41. urlhttp://ntl.inrne.bas.bg/workshop/2018/proc.html.

[60] A. Martinou and D. Bonatsos, in Nuclear StructurePhysics, ed. A. Shukla and S. K. Patra (CRC, Taylorand Francis) in press. arXiv 1909.00233 [nucl-th] (2019).isbn 9780367256104.

[61] A. Martinou, D. Bonatsos, K. E. Karakatsanis, S.Sarantopoulou, I. E. Assimakis, S. K. Peroulis andN. Minkov, Eur. Phys. J A (2021) in press. arXiv2102.04409 [nucl-th] (2021).

[62] D. J. Rowe, J. Phys. G: Nucl. Part. Phys. 43, 024011(2016). doi 10.1088/0954-3899/43/2/024011

[63] J.P. Draayer and K.J. Weeks, Ann. Phys. 156, 41(1984). doi 10.1016/0003-4916(84)90210-0.

[64] J.P. Draayer, in Algebraic approaches to nuclear struc-ture: Interacting boson and fermion models, Contempo-rary Concepts in Physics vol. 6, ed. R.F. Casten (Har-wood, New York, 1993) p. 423. isbn 3-7186-0538-4.

[65] P.O. Lipas, in Algebraic approaches to nuclear struc-ture: Interacting boson and fermion models, Contempo-rary Concepts in Physics vol. 6, ed. R.F. Casten (Har-wood, New York, 1993) p. 47. isbn 3-7186-0538-4.

[66] A. Bohr and B. R. Mottelson, Nuclear Structure Vol. II(World Scientific, Singapore, 1998). doi 10.1142/3530.

[67] O. Castanos, J. P. Draayer and Y. Leschber, Z. Phys.A: At. Nucl. 329, 33 (1988). doi 10.1007/bf01294813.

[68] D. J. Rowe, G. Thiamova and J. L. Wood, Phys.

Rev. Lett. 97, 202501 (2006). doi 10.1103/phys-revlett.97.202501.

[69] M.C. Bouten, J.P. Elliott and J.A. Pullen, Nucl. Phys.A 97, 113 (1967). doi 10.1016/0375-9474(67)90774-9

[70] Dennis Bonatsos, Eur. Phys. J. A 53, 148 (2017). doi10.1140/epja/i2017-12346-x.

[71] S.G. Nilsson, Mat. Fys. Medd. K. Dan. Vidensk. Selsk.29, no. 16 (1955).

[72] R. B. Cakirli, K. Blaum and R. F. Casten, Phys. Rev.C 82, 061304 (2010). doi 10.1103/physrevc.82.061304.

[73] D. Bonatsos, S. Karampagia, R. B. Cakirli, R. F. Cas-ten, K. Blaum and L. Amon Susam, Phys. Rev. C 88,054309 (2013). doi 10.1103/physrevc.88.054309.

[74] A. de-Shalit and M. Goldhaber, Phys. Rev. 92, 1211(1953). doi 10.1103/physrev.92.1211.

[75] D. Bonatsos, H. Sobhani and H. Hassanabadi, Eur.Phys. J. Plus 135, 710 (2020). doi 10.1140/epjp/s13360-020-00749-2.

[76] O. Castanos, M. Moshinsky and C. Quesne, in GroupTheory and Special Symmetries in Nuclear Physics, ed.J. P. Draayer and J. Janecke (World Scientific, Singa-pore, 1992) p. 80. doi 10.1142/1468

[77] I. E. Assimakis, M.Sc. thesis, NationalTechnical University of Athens (2015). urlhttp://dx.doi.org/10.26240/heal.ntua.3240.

[78] D. Bonatsos, A. Martinou, S. Sarantopoulou, I. E. Assi-makis, S. K. Peroulis, and N. Minkov, Eur. Phys. J. ST229, 2367 (2020) doi 10.1140/epjst/e2020-000034-3.

[79] H. Morinaga, Phys. Rev. 101, 254 (1956). doi10.1103/physrev.101.254.

[80] J. Cseh, G. Riczu and J. Darai, Phys. Lett. B 795, 160(2019). doi 10.1016/j.physletb.2019.06.016.

[81] T. Otsuka, R. Fujimoto, Y. Utsuno, B. A. Brown, M.Honma and T. Mizusaki, in The Nuclear Many-BodyProblem 2001, NATO Science Series II: Mathematics,Physics and Chemistry vol. 53, ed. W. Nazarewicz andD. Vretenar (Springer, Berlin, 2001) p. 41.

[82] A. Martinou, D. Bonatsos, N. Minkov, T. J.Mertzimekis, I. E. Assimakis, S. Peroulis andS. Sarantopoulou, HNPS Advances in NuclearPhysics 26 (National Documentation Centre,Athens, 2019) p. 96. doi 10.12681/hnps.1804, urlhttp://dx.doi.org/10.12681/hnps.1804

[83] W. Geithner, S. Kappertz, M. Keim, P. Lievens, R.Neugart, L. Vermeeren, S. Wilbert, V. N. Fedoseyev,U. Koster, V. I. Mishin, V. Sebastian and ISOLDECollaboration, Phys. Rev. Lett. 83, 3792 (1999). , doi10.1103/physrevlett.83.3792.

[84] Y. Kondo, T. Nakamura, Y. Satou, T. Matsumoto, N.Aoi, N. Endo, N. Fukuda, T. Gomi, Y. Hashimoto, M.Ishihara, S. Kawai, M. Kitayama, T. Kobayashi, Y.Matsuda, N. Matsui, T. Motobayashi, T. Nakabayashi,T. Okumura, H.J. Ong, T.K. Onishi, K. Ogata, H.Otsu, H. Sakurai, S. Shimoura, M. Shinohara, T.Sugimoto, S. Takeuchi, M. Tamaki, Y. Togano andY. Yanagisawa, Phys. Lett. B 690, 245 (2010). doi10.1016/j.physletb.2010.05.031.

[85] A. Alex, M. Kalus, A. Huckleberry and Jan von Delft, J.Math. Phys. 52, 023507 (2011). doi 10.1063/1.3521562.

[86] S. Coleman, J. Math. Phys. 5, 1343 (1964). doi10.1063/1.1704245.

[87] D. Troltenier, A. Blokhin, J. P. Draayer, D. Rompf andJ. G. Hirsch, AIP Conf. Proc. 365, 244 (1996). doi10.1063/1.50225.

http://ntl.inrne.bas.bg/workshop/2018/proc.html

http://dx.doi.org/10.12681/hnps.1804

29

[88] S. Raman, C. W. Nestor and P. Tikkanen,At. Data Nucl. Data Tables 78, 1 (2001). urlhttps://doi.org/10.1006/adnd.2001.0858.

[89] D.R. Tilley, H.R. Weller and C.M. Cheves, Nucl. Phys.A 564, 1 (1993). doi 10.1016/0375-9474(93)90073-7.

[90] C. Bahri, J. Escher and J.P. Draayer, Nucl. Phys. A592, 171 (1995). doi 10.1016/0375-9474(95)00292-9.

[91] G. Rosensteel and D.J Rowe, Ann. Phys. 123, 36 (1979).doi 10.1016/0003-4916(79)90264-1.

[92] G. Rosensteel and D.J Rowe, Ann. Phys. 126, 343(1980). doi 10.1016/0003-4916(80)90180-3.

[93] D.J Rowe and G Rosensteel, Ann. Phys. 126, 198(1980). doi 10.1016/0003-4916(80)90380-2.

[94] E. Browne and R. B. Firestone, in Tables of Isotopes,7th ed., ed. V. S. Shirley and C. M. Lederer (Wiley, NewYork, 1978). isbn 978-0471041795.

[95] R.F. Casten, Nuclear Structure from a SimplePerspective (Oxford University Press, 2000). isbn9780198507246, lccn 89023064.

[96] V.K.B. Kota and U. D. Pramanik, Eur. Phys. J. A 3,243 (1998). doi 10.1007/s100500050174.

[97] I. E. Assimakis, D. Bonatsos, A. Martinou, S. Saran-topoulou, S. Peroulis,T. J. Mertzimekis and N. Minkov,HNPS Adv. Nucl. Phys. 26 (National DocumentationCentre, Athens, 2019) p. 9. doi 10.12681/hnps.1789.

[98] V. Prassa, T. Niksic and D. Vretenar, Phys. Rev. C 88,044324 (2013). doi 10.1103/physrevc.88.044324.

[99] K. Karakatsanis, Ph.D. thesis, Aristo-tle University of Thessaloniki, 2017. urlhttp://hdl.handle.net/10442/hedi/41280.

[100] K. E. Karakatsanis, G. A. Lalazissis, V. Prassa andP. Ring, Phys. Rev. C 102, 034311 (2020). doi10.1103/physrevc.102.034311.

[101] J. Boguta and A. R. Bodmer, Nucl. Phys. A 292, 413(1977). doi 10.1016/0375-9474(77)90626-1.

[102] B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. 16, 1(1986).

[103] P.-G. Reinhard, Rep. Prog. Phys. 52, 439 (1989). doi =10.1088/0034-4885/52/4/002.

[104] , P. Ring, Prog. Part. Nucl. Phys. 37, 193 (1996). doi10.1016/0146-6410(96)00054-3.

[105] J. Meng, H. Toki, S.G. Zhou, S.Q. Zhang, W.H. Longand L.S. Geng, Prog. Part. Nucl. Phys. 57, 470 (2006).doi 10.1016/j.ppnp.2005.06.001.

[106] G. A. Lalazissis, T. Niksic, D. Vretenar and P. Ring,Phys. Rev. C 71, 024312 (2005). doi 10.1103/phys-revc.71.024312.

[107] T. Niksic, N. Paar, D. Vretenar and P. Ring,Comp. Phys. Comm. 185, 1808 (2014). doi10.1016/j.cpc.2014.02.027.

[108] R. F. Casten, Nucl. Phys. A 443, 1 (1985). doi10.1016/0375-9474(85)90318-5.

[109] P. Federman and S. Pittel, Phys. Lett. B 69, 385 (1977).doi 10.1016/0370-2693(77)90825-5.

[110] P. Federman and S. Pittel, Phys. Rev. C 20, 820 (1979).doi 10.1103/physrevc.20.820.

[111] K. Heyde, P. Van Isacker, R.F. Casten and J.L. Wood,Phys. Lett. B 155, 303 (1985). doi 10.1016/0370-2693(85)91575-8.

[112] T. Kibedi and R.H. Spear, in AIP Conference Proceed-ings 769, 442 (2005). doi 10.1063/1.1945043.

[113] M. Ogawa, R. Broda, K. Zell, P. J. Daly, and P.Kleinheinz, Phys. Rev. Lett. 41, 289 (1978). doi10.1103/PhysRevLett.41.289.

[114] P. Himpe, G. Neyens, D.L. Balabanski, G. Belier,J.M. Daugas, F. de Oliveira Santos, M. De Rydt,K.T. Flanagan, I. Matea, P. Morel, Yu.E. Penionzhke-vich, L. Perrot, N.A. Smirnova, C. Stodel, J.C.Thomas, N. Vermeulen, D.T. Yordanov, Y. Utsunoand T. Otsuka, Phys. Lett. B 658, 203 (2008). doi10.1016/j.physletb.2007.11.017.

[115] F. Nowacki, A. Poves, E. Caurier and B. Bounthong,Phys. Rev. Lett. 117, 272501 (2016). doi 10.1103/phys-revlett.117.272501.

[116] P. G. Thirolf and D. Habs, Progr. Part. Nucl. Phys. 49,325 (2002) doi 10.1016/s0146-6410(02)00158-8.

[117] P. Van Isacker, D. D. Warner and D. S. Brenner,Phys. Rev. Lett. 74, 4607 (1995). doi 10.1103/phys-revlett.74.4607.

[118] P. Van Isacker, O. Juillet and B. K. Gjel-sten, Foundations of Physics 27, 1047 (1997). doi10.1007/bf02551152.

[119] A. Martinou, Ph.D. thesis, National Tech-nical University of Athens, 2018. urlhttp://hdl.handle.net/10442/hedi/43367.

[120] X.F. Yang, C. Wraith, L. Xie, C. Babcock, J. Bil-lowes, M.L. Bissell, K. Blaum, B. Cheal, K.T. Flana-gan, R.F. Garcia Ruiz, W. Gins, C. Gorges, L.K.Grob, H. Heylen, S. Kaufmann, M. Kowalska, J. Krae-mer, S. Malbrunot-Ettenauer, R. Neugart, G. Neyens,W. Nortershauser. J. Papuga, R. Sanchez and D.T.Yordanov, Phys. Rev. Lett. 116, 182502 (2016). doi10.1103/physrevlett.116.182502.

[121] X.F. Yang, C. Wraith, L. Xie, C. Babcock, J. Bil-lowes, M.L. Bissell, K. Blaum, B. Cheal, K.T. Flana-gan, and R.F. Garcia Ruiz, W. Gins, C. Gorges, L.K.Grob, H. Heylen, S. Kaufmann, M. Kowalska, J. Krae-mer, S. Malbrunot-Ettenauer, R. Neugart, G. Neyens,W. Nortershauser, J. Papuga, R. Sanchez and D.T. Yor-danov, Phys. Rev. Lett. 116, 219901 (2016).

[122] A. Gottardo, D. Verney, C. Delafosse, F. Ibrahim,B. Roussiere, C. Sotty, S. Roccia, C. Andreoiu, C.Costache, M.-C. Delattre, I. Deloncle, A. Etile, S.Franchoo, C. Gaulard, J. Guillot, M. Lebois, M.MacCormick, N. Marginean, R. Marginean, I. Matea,C. Mihai, I. Mitu, L. Olivier, C. Portail, L. Qi,L. Stan, D. Testov, J. Wilson and D.T. Yordanov,Phys. Rev. Lett. 116, 182501 (2016). doi 10.1103/phys-revlett.116.182501.

[123] T. Otsuka, A. Gade, O. Sorlin, T. Suzuki and Y.Utsuno, Rev. Mod. Phys. 92, 015002 (2020). doi10.1103/revmodphys.92.015002.

[124] J. D. Vergados, private communication (2018).[125] F. Nowacki and A. Poves, J. Phys.: Conf. Ser. 966,

012023 (2018). doi 10.1088/1742-6596/966/1/012023.[126] R. Budaca and A. I. Budaca, Europhys. Lett. 123,

42001 (2018). doi 10.1209/0295-5075/123/42001.[127] R. Budaca, P. Buganu and A.I. Budaca, Nucl. Phys. A

990, 137 (2019). doi 10.1016/j.nuclphysa.2019.07.006[128] P. Buganu, R. Budaca and A.-I. Budaca, EPJ Web

of Conferences 194, 01007 (2018). doi 10.1051/epj-conf/201819401007.

[129] I. Talmi,Rev. Mod. Phys. 34, 704 (1962). doi10.1103/revmodphys.34.704.

[130] J. Dobaczewski, W. Nazarewicz, J. Skalski and T.Werner, Phys. Rev. Lett. 60, 2254 (1988). doi10.1103/physrevlett.60.2254.

[131] R. B. Cakirli and R. F. Casten, Phys. Rev. Lett. 96,

30

132501 (2006). doi 10.1103/physrevlett.96.132501.[132] M. Duer, et al. (CLAS Collaboration), Phys.

Rev. Lett. 122, 172502 (2019). doi 10.1103/Phys-RevLett.122.172502.

[133] J. E. Garcıa-Ramos and K. Heyde, Phys. Rev. C 100,044315 (2019). doi 10.1103/physrevc.100.044315.

[134] J. E. Garcıa-Ramos and K. Heyde, arXiv: 2011.05581[nucl-th] (2020).

[135] H. Esbensen, B. A. Brown and H. Sagawa, Phys. Rev.C 51, 1274 (1995). doi 10.1103/physrevc.51.1274.

[136] D.R. Tilley, J.H. Kelley, J.L. Godwin, D.J. Millener,J.E. Purcell, C.G. Sheu and H.R. Weller, Nucl. Phys. A745, 155 (2004). doi 10.1016/j.nuclphysa.2004.09.059.

[137] W. Nortershauser, D. Tiedemann, M. Zakova, Z. And-jelkovic, K. Blaum, M. L. Bissell, R. Cazan, G. W.F. Drake, Ch. Geppert, M. Kowalska, J. Kramer, A.Krieger, R. Neugart, R. Sanchez, F. Schmidt-Kaler,Z.-C. Yan, D. T. Yordanov and C. Zimmermann,Phys. Rev. Lett. 102, 062503 (2009). doi 10.1103/phys-revlett.102.062503.

[138] A. Navin, D. W. Anthony, T. Aumann, T. Baumann, D.Bazin, Y. Blumenfeld, B. A. Brown, T. Glasmacher, P.G. Hansen, R. W. Ibbotson, P. A. Lofy, V. Maddalena,K. Miller, T. Nakamura, B. V. Pritychenko, B. M. Sher-rill, E. Spears, M. Steiner, J. A. Tostevin, J. Yurkonand A. Wagner, Phys. Rev. Lett. 85, 266 (2000). doi10.1103/physrevlett.85.266.

[139] M. Kowalska, D. T. Yordanov, K. Blaum, P. Himpe,P. Lievens, S. Mallion, R. Neugart, G. Neyens andN. Vermeulen, Phys. Rev. C 77, 034307 (2008). doi10.1103/physrevc.77.034307.

[140] W. Schwerdtfeger, P. G. Thirolf, K. Wimmer, D. Habs,H. Mach, T. R. Rodriguez, V. Bildstein, J. L. Egido, L.M. Fraile, R. Gernhauser, R. Hertenberger, K. Heyde,P. Hoff, H. Hubel, U. Koster, T. Kroll, R. Krucken, R.Lutter, T. Morgan and P. Ring, Phys. Rev. Lett. 103,012501 (2009). doi 10.1103/physrevlett.103.012501.

[141] K. Wimmer, T. Kroll, R. Krucken, V. Bildstein, R.Gernhauser, B. Bastin, N. Bree, J. Diriken, P. Van Dup-pen, M. Huyse, N. Patronis, P. Vermaelen, D. Voulot,J. Van de Walle, F. Wenander, L. M. Fraile, R. Chap-man, B. Hadinia, R. Orlandi, J. F. Smith, R. Lutter,P. G. Thirolf, M. Labiche, A. Blazhev, M. Kalkuhler,P. Reite, M. Seidlitz, N. Warr, A. O. Macchiavelli, H.B. Jeppesen, E. Fiori, G. Georgiev, G. Schrieder, S. D.Gupta, G. Lo Bianco, S. Nardelli, J. Butterworth, J. Jo-hansen and K. Riisager, Phys. Rev. Lett. 105, 252501(2010). doi 10.1103/physrevlett.105.252501.

[142] G. Saxena, D. Singh and M. Kaushik, Phys. Part. Nucl.Lett. 10, 220 (2013). doi 10.1134/s1547477113030114.

[143] C. Guenaut, G. Audi, D. Beck, K. Blaum, G.Bollen, P. Delahaye, F. Herfurth, A. Kellerbauer, H.-J. Kluge, D. Lunney, S. Schwarz, L. Schweikhardand C. Yazidjian,Eur. Phys. J. A 25, 33 (2005). doi10.1140/epjad/i2005-06-029-9.

[144] B. Pritychenko, M. Birch, B. Singh and M. Horoi,At. Data Nucl. Data Tables 107, 1 (2016). urlhttps://doi.org/10.1016/j.adt.2015.10.001.

[145] G. Gurdal and F.G. Kondev, Nucl. Data Sheets 113,1315 (2012). doi 10.1016/j.nds.2012.05.002.

[146] S. Lalkovski and F.G. Kondev, Nucl. Data Sheets 124,157 (2015). doi 10.1016/j.nds.2014.12.046.

[147] J. Blachot, Nucl. Data Sheets 113, 515 (2012). doi10.1016/j.nds.2012.02.002.

[148] J. Blachot, Nucl. Data Sheets 111, 717 (2010). doi10.1016/j.nds.2010.03.002.

[149] K. Kitao, Nucl. Data Sheets 75, 99 (1995). doi10.1006/ndsh.1995.1022.

[150] K. Kitao, Y. Tendow and A. Hashizume, Nucl. DataSheets 96, 241 (2002). doi 10.1006/ndsh.2002.0012.

[151] M.S. Basunia, Nucl. Data Sheets 107, 791 (2006). doi10.1016/j.nds.2006.03.001.

[152] E. Achterberg, O.A. Capurro and G.V. Marti,Nucl. Data Sheets 110, 1473 (2009). doi10.1016/j.nds.2009.05.002.

[153] E.A. McCutchan, Nucl. Data Sheets 126, 151 (2015).doi 10.1016/j.nds.2015.05.002.

[154] B. Singh, Nucl. Data Sheets 130, 21 (2015). doi10.1016/j.nds.2015.11.002.

[155] C. M. Baglin, Nucl. Data Sheets 111, 275 (2010). doi10.1016/j.nds.2010.01.001.

[156] C. M. Baglin, Nucl. Data Sheets 99, 1 (2003). doi10.1006/ndsh.2003.0007.

[157] F.G. Kondev, S. Juutinen and D.J. Hartley, Nucl. DataSheets 150, 1 (2018). doi 10.1016/j.nds.2018.05.001.

[158] B. Singh, Nucl. Data Sheets 99, 275 (2003). doi10.1006/ndsh.2003.0009.

[159] G. A. Lalazissis, S. Raman and P. Ring, At. Data Nucl.Data Tables 71, 1 (1999). doi 10.1006/adnd.1998.0795.

[160] T. Niksic, D. Vretenar, P. Ring and G. A. Lalazis-sis, Phys. Rev. C 65, 054320 (2002). doi 10.1103/phys-revc.65.054320.

[161] D. J. Rowe, Phys. Rev. C 101, 054301 (2020). doi10.1103/physrevc.101.054301,

[162] E. Caurier, F. Nowacki and A. Poves, Nucl. Phys. A693, 374 (2001). doi 10.1016/s0375-9474(00)00579-0.

[163] E. Caurier, J. Menendez, F. Nowacki and A. Poves,Phys. Rev. C 75, 054317 (2007). doi 10.1103/phys-revc.75.054317.

[164] E. Caurier, F. Nowacki and A. Poves, Phys. Rev. C 90,014302 (2014). doi 10.1103/physrevc.90.014302.

[165] T. Otsuka, Y. Tsunoda, T. Abe, N. Shimizu and P.Van Duppen, Phys. Rev. Lett. 123, 222502 (2019). doi10.1103/physrevlett.123.222502.

31

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