Borromean structures in medium-heavy nuclei

D. Hove, D.V. Fedorov, H.O.U. Fynbo, A.S. Jensen, K. Riisager, N.T. ZinnerDepartment of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark

E. GarridoInstituto de Estructura de la Materia, IEM-CSIC, Serrano 123, E-28006 Madrid, Spain

(Dated: September 27, 2018)

Borromean nuclear cluster structures are expected at the corresponding driplines. We locate theregions in the nuclear chart with the most promising constituents, it being protons and α-particlesand investigate in details the properties of the possible Borromean two-α systems in medium heavynuclei. We find in all cases that the α-particles are located at the surface of the core-nucleus asdictated by Coulomb and centrifugal barriers. The two lowest three-body bound states resemble aslightly contracted 8Be nucleus outside the core. The next two excited states have more complexstructures but with strong components of linear configurations with the core in the middle. α-removal cross sections would be enhanced with specific signatures for these two different types ofstructures. The even-even Borromean two-α nucleus, 142Ba, is specifically investigated and predictedto have 134Te− α− α structure in its ground state and low-lying spectrum.

PACS numbers: 21.45.+v, 31.15.xj, 21.60.Gx

I. INTRODUCTION

Surprisingly large nuclear reaction cross sections werefound experimentally in 1985 [1, 2] for light nuclei col-liding with 11Li. Normal cross section were found for alllighter lithium isotopes. The interpretation and quali-tative understanding were almost immediately explainedas based on weak binding in relative s-waves [3]. Thestructure is in general called a nuclear halo, since thespatial extension is much larger than radii of ordinarynuclei [4]. The essence of theoretical descriptions is con-tained in a schematic model for two short-range inter-acting point-like particles as reviewed in [5]. Thresholdsfor binding nuclear clusters enhance the probability forfinding decoupled and spatially extended nuclear struc-tures. Driplines related to different nucleons or boundclusters of nucleons provide the best environment for thecorresponding ground state structures. The features ofsuch nucleon dripline nuclei are reviewed in [6]. Nucle-ons can be arranged in bound clusters and thereby formthe constituents for novel nuclear few-body structures asreviewed in [7–9].

Increasing the number of clusters progressively reducesthe possibility of large spatial extension when all particlesare distinctly separated. An effective centrifugal barrierconfines the mean square radius for a bound system, andoverlapping nuclei of finite size would couple intrinsic andrelative degrees of freedom. To prevent this collapse intoone much larger many-body system, the particles maycorrelate strongly into clusters and effectively reduce theactive degrees of freedom corresponding to much fewerwell-separated clusters of particles [5]. Already four par-ticles with infinitesimally small binding energy have finiteroot-mean-square radius, in sharp contrast to divergingradii of two- and three-body systems approaching zerobinding energy [10].

The repulsive centrifugal barrier combined with a

short-range attraction of given radius may leave an at-tractive pocket able to hold a bound state. This pre-vents occurrence of nuclear halos of large relative angu-lar momenta. The same mechanism opposes halos wherethe Coulomb repulsion dominates or contributes signifi-cantly. Thus, nuclear halos are most likely to occur forvery small binding energy, two- and three-body systems,relative s and p-waves, and small pairwise charge prod-ucts [11–13]. Two-body halos should then be searchedfor at their threshold for binding while still subject tothese conditions, both for ground and excited states.

Three-body cluster states are probably less frequentand potentially less pronounced than two-body halos.However, investigations of occurrence and properties areessentially all confined to light nuclei. For ground statesthe most promising structures appear to be for Bor-romean systems, since the unbound two-body subsystemsare prohibited from reducing the active degrees of free-dom to an effective two-body system. Then the condi-tions are positive pairwise cluster binding and very smallthree-body binding. The most obvious constituents areneutrons, protons, and α-particles. Heavier particles nec-essarily have both larger radii and charges, and thereforeless obvious constituents in a halo system.

Pairs of identical nucleons and α-particles are alwaysunbound, and the Borromean properties are therefore de-termined by the pairwise binding energies to the thirdparticle [14]. The positive binding of nucleon-core and acore-α systems dictate the position in the nuclear chartto be around the corresponding driplines. As we shalldiscuss later, combining nucleons and α-particles is thennot possible for neutrons while the proton dripline is suit-able for nuclei with neutron number N > 30. We shallnot deal with the individual light nuclei where the Bor-romean properties are thoroughly discussed in the avail-able literature [4, 8].

Two α-particles and a heavier core along the α-dripline

arX

iv:1

408.

6432

v2 [

nucl

-th]

15

Dec

201

4

2

can form a Borromean system. We shall in the presentpaper concentrate on two α-particles plus a mediumheavy core-nucleus which is a system so far very littlediscussed. The Borromean property is established exper-imentally for a few nuclei [15] as pointed out recently[16]. It remains to be seen whether α-particles in the endturn out to be substantially contributing constituents tothe structure of some low-energy states in medium heavynuclei. The minimum requirement is that the intrinsic α-particle degrees of freedom effectively decouple from allother nuclear degrees of freedom. Recent theoretical in-vestigations of large systems confirm the expectation thatα-particles are advantageous for nuclear matter densitiescorresponding to the tail of a nucleus [17, 18].

Explicit use of α-particle degrees-of-freedom is com-plementary to mean-field approximations, where correla-tions only appear through shell structures of single nu-cleons (or pairs) in deformed average fields. Such modelsprovide completely different basis states, but they arenot necessarily unable to describe the same features ofsome many-body states. In this context it is interestingto note that octupole deformation has been an impor-tant ingredient in descriptions of nuclei located close tothe α-dripline [19], where α-clustering is most likely tooccur in ground states. We assume α-particles as the ba-sic constituents with the inherent limits of validity thatonly very specific structures can be described. However,it may very well be states that cannot be described inmean-field or shell-models, or at most only with severedifficulties indicating that an inappropriate basis is cho-sen.

The α-clusterization, or more moderately α-correlations, should produce an otherwise largerbinding energy which then simply could be measuredas the revealing observable. The increased bindinghas to be compared to the surrounding nuclei, andthe signal would be contained in appropriate massdifferences [20, 21]. This would be in complete analogyto the odd-even staggering related to the “pairing gaps”[22]. Unfortunately, such a signal in the variationof the binding energy between neighbouring nuclei isextremely difficult to separate from the smooth back-ground variation which therefore necessarily has to beeliminated. The optimistic point of view would be thatα-correlations are more extended and vary smoothlyover smaller or larger regions of the nuclear chart.

Instead of futile searching for signals in the bindingenergies [21], we shall therefore directly calculate three-body properties from an assumption of the presence oftwo α-particles surrounding a heavier core-nucleus. Weare then able to study the solutions, test compatibilitywith the assumptions, and predict the properties of theemerging structures. We first in Sec. II outline which re-gions of the nuclear chart are most promising. In Sec. IIIwe briefly sketch the computational procedure and spec-ify the necessary parameters. This also includes the two-body α-core potential used in the initial three-body cal-culations. In Sec. IV 148Nd (140Ba + α+ α) is used as a

trial system to evaluate the general nature of such three-body α-Borromean systems. Dedicated calculations forthe Borromean 142Ba (134Te+α+α) nucleus are reportedin Sec. V, where comparisons are made to experimentalresults for α-dripline nuclei. Finally Sec. VI contains asummary and the conclusions.

II. DRIPLINES AND BORROMEAN REGIONS

Borromean systems are, almost by definition, most of-ten weakly bound, since all three two-body subsystemsmust be unbound and the same interactions are respon-sible for the three-body binding. This definition is ap-propriate for ground states of systems where the clusterdivision already is made. The total system may very wellhave much deeper-lying bound states of different struc-tures where the cluster division is completely inadequate.Thus the structures of interest here can appear as rela-tively highly excited states of the given nucleus.

It is only within the decided cluster structures thatthe corresponding three-body system is relatively weaklybound compared to the threshold of large-distance sep-aration of all three clusters. The weak cluster-bindingis compatible with large size and with cluster identitiesmaintained. Therefore, the most likely region for findingthree-body cluster states should be where the effectivecluster-cluster interaction provides small, positive or neg-ative, binding energies. Cluster driplines are then usefulin outlining regions where corresponding Borromean sys-tems should be more likely.

Let us now focus on two α-particles surrounding a core-nucleus. We want to find the α-dripline with vanishingα-separation energy, Sα, that is

Sα(A,X) = B(A+ 4, X)−B(A,X)−Bα, (1)

where B is the nuclear binding energy, N and Z are neu-tron and proton numbers, X = N − Z, A = N + Z, andBα = 28.295 MeV is the binding energy of an α particle.The dripline defined by Sα(A,X) = 0 can be estimatedby use of the liquid drop model, or specifically

BLD = avA− asA2/3 − acZ2

A1/3− aa

(A− 2Z)2

A(2)

= avA− asA2/3 − ac4A1/3

(A2 +X2 − 2AX

)− aa

X2

A,

where we use the parameter values [18]: (av, as, ac, aa) =(15.56, 17.23, 0.7, 23.285), all in MeV. Then Sα(A,X) = 0combined with Eqs. (1) and (2) results in the quadratic

3

0 30 60 90 120 150

0

30

60

90

N

Z

α

Neutron

Proton

60 90

60

Figure 1: (Colour online) The theoretical α, neutron,and proton driplines in the chart of nuclei, where nucleiwith a negative separation energy have been marked in

a corresponding colour. Marked in green are 140Ba,144Ce, and 148Nd, which represents nuclei of interest in

the α-dripline region.

equation

0 = c0 + c1X + c2X2 , (3)

c0 = 4av −Bα (4)

− as(

(A+ 4)23 −A 2

3

)− ac

4

((A+ 4)

53 −A 5

3

),

c1 =ac2

((A+ 4)

23 −A 2

3

), (5)

c2 = −ac4

(1

(A+ 4)13

− 1

A13

)− aa

(1

A+ 4− 1

A

).(6)

The two solutions, X±(A), to Eq. (3) are combined withZ = (A−X±(A))/2 and N = (A+X±(A))/2 to give theproton and neutron numbers of the α-dripline boundariesfor any nucleon number A. Similarly, it would be pos-sible to derive expressions for both neutron and protondriplines.

The results are shown in the nuclear chart in Fig. 1,where nuclei with experimentally known negative neu-tron, proton, or α-separation energies are marked. Thecomputed curves are in overall agreement with the mea-sured results, which is to be expected since the liquiddrop parameters are adjusted to achieve this goal. Thenegative neutron and proton binding energies are out-side the corresponding driplines. On the other hand, thenegative α-bindings occur between the legs of the twosolutions as emphasized by the many known α-unstableheavy nuclei marked in red.

If an isotope is marginally on the unstable side of adripline, it can rather likely form a Borromean system byadding two identical particles (neutrons, protons, or α’s)to the corresponding core-nucleus. This is often observedfor neutrons, but the repulsive Coulomb interaction maysometimes invalidate this expectation for protons and α-

particles. Thus, these nuclei are promising candidatesfor ground state cases of Borromean two-α systems. It isperhaps worth emphasizing that going away from the α-driplines into either α-unbound or α-bound nuclei wouldcorrespond to α-core ground state structures of negativeor positive binding energy, respectively.

Then the red nuclei, between the legs of the α-driplinecurves, should be simulated by a positive α-core energyeven for the ground state. Vice versa, outside the re-gion this two-body energy should be negative, and ex-cited states of two-body energy just above zero are thestrong candidates for the α-cluster states we are goingto investigate. In other words, both positive and nega-tive two-body energies are worth investigating, both asground states and as excited states.

From Fig. 1 we can also deduce which mixed speciesof nucleons and α-particles are most promising in con-nection with formation of Borromean states and the re-lated α-clusterization. First we notice that the neutron-proton-core system is excluded as a Borromean state dueto the bound deuteron. Borromean states involving nu-cleons in general only occur for ground states close to thecorresponding nucleon driplines. This excludes neutron-α-core systems since the neutron and α driplines neverintersect or come close to each other. When the neutron-core is unbound the α-core system is bound.

In contrast, proton-α-core systems are possible Bor-romean systems along the proton dripline for systemsheavier than about N ≈ 40 or Z > 45. This is especiallypromising in the region where proton and α driplines in-tersect each other as shown in Fig. 1. These structuresare interesting and should be investigated in details inthe future. It would involve both proton-core and α-coreeffective two-body interactions. At present we only em-phasize that this experimentally accessible region prob-ably would present a series of such Borromean systems.In addition, we notice that similar cluster structures mayappear as excited states in lighter nuclear systems.

III. METHOD AND PARAMETER CHOICES

The three-body calculations require two-body poten-tials between the three pairs of particles. For α-α-coresystems we only need to specify the α-α and α-core po-tentials. We treat all particles as point-like and the fi-nite sizes must then be accounted for through effectivepotentials. This also implies that the actual choice ofpotentials is less important. We can use the energy asthe crucial parameter and measure lengths relative tothe radius of the core-nucleus. Conclusions from individ-ual test cases are then more general. After definitions ofthe two-body potential we define notation and principalquantities in a brief sketch of the three-body method.

4

A. Two-body potentials

First we choose the α-α potential as the d-version ofthe Ali-Bodmer potential [23] as used previously in manyapplications [24]. This potential is angular momentumdependent without bound states while reproducing thelow-energy phase shifts very well. The measured energy,0.091 MeV, is reproduced, and the root-mean-square ra-dius of the corresponding resonance is calculated to be〈r2〉1/2 = 5.95 fm.

The second potential between α-particle and core mustnecessarily be phenomenologically adjusted. At the α-dripline the binding energy has to be vanishingly small,but not necessarily of the lowest state. In the presentwork the antisymmetry between nucleons in the core andin the α-particles are only accounted for by use of ashallow effective α-core potential or by exclusion of thedeepest-lying bound states. Thus, the Pauli principle isapproximately obeyed by occupying only states with verysmall binding energy. As shown in [25], where the halonuclei 6He and 11Li are investigated, for weakly boundsystems a shallow potential and a deep potential hold-ing Pauli forbidden states give rise to similar three-bodystructures provided that both potentials have the samelow-energy properties. We assume the intrinsic core-spinis zero and the total angular momentum is then exclu-sively from the orbital part. This implies that the po-tential is central and the decisive ingredient is the radialshape with corresponding size and strength. The natu-ral choice for medium-heavy nuclei is the Woods-Saxonpotential, V (r), with a constant central value and expo-nential fall off at larger radii, that is

V (r) = −V0

(1 + exp

(r −Ra

))−1

, (7)

where we use the diffuseness, a = 0.65, R(A) = r0A1/3 +

rπ + Rα, r0 = 1.16 fm, rπ = 1.4 fm, Rα = 1.7 fm.For A = 140 we arrive at R = 9.1 fm which shall beused throughout this paper. The remaining parameteris the strength, V0, which is tuned to give the desiredenergies for any choice of angular momentum and parity.The Coulomb potential is for a homogeneous sharp cut-off distribution of core-charge, Z = 56 and α-charge 2,with the resulting cut off radius, RC = 7.4 fm. Thissomewhat increased radius accounts for the finite sizesof both core and α-particle when a point-particle in apotential is assumed.

The variation of angular momentum allows us to in-vestigate the influence of non-vanishing partial waves onthe total three-body structure. It also allows compari-son between ground and excited states of both the sameand different angular momentum quantum numbers. Ex-cited states from the present effective potential may beunavoidable when deeper-lying levels are occupied andPauli-forbidden. The effect of changing the angular mo-mentum can be seen in Fig. 2, where the potential depthin each case is adjusted to produce the same energy. The

0 4 8 12 16

0

10

20

30

r [fm]

V(r

) [M

eV

]

J = 0

J = 2

J = 4

Figure 2: (Colour online) The two-body potentials forthree different angular momenta (and corresponding

potential depths). The potential depths are adjusted togive roughly the same energy as reflected by very

similar minima. The depth of the interaction potentialcan be found in Table I.

0 4 8 12 16

0

0.2

0.4

0.6

r [fm]

u(r

)

J = 0

J = 2

J = 4

Figure 3: (Colour online) The two-body radial wavefunctions, u, for three different angular momenta (and

corresponding potential depths).

minima are then almost identical, and the only differenceis the centrifugal barrier term deviating strongly at smalldistance.

The reduced radial ground state wave functions, u(r),corresponding to the potentials in Fig. 2 are shown inFig. 3. They are all spatially very similar and confinedto a rather narrow region around the common minimum.This behaviour is unlike that of a core and a neutronwhich typically has a very wide spatial extent with aslowly decreasing tail [5]. The narrow distribution is a

5

Table I: The two body energies and average sizes with aBa-140 core and an α-particle, where the angularmomentum, J , and potential depth, V0, have been

varied. Positive energies, E, correspond to unboundstates with negative binding. We denote ground, first

excited, second excited, and third excited states by (G),(F), (S) and (T), respectively. Energies are in MeV, and

distances are in fm.

V0 J State E 〈r2〉1/226.25 0 G. 0.1 7.0

F. 4.3 5.9S. 7.4 5.4

27.1 2 G. 0.0 7.1F. 4.9 6.4S. 9.1 6.4

28.8 4 G. 0.1 7.2F. 5.7 6.8

direct consequence of the potentials in Fig. 2, where theα-particle is confined by two very steep barriers on bothsides of the minimum.

It is especially worth noting that the Coulomb inter-action for s-waves is finite at r = 0 as obtained by ahomogeneous charge distribution within a sphere. Thefinite repulsion is still sufficient to push the wave func-tion out to the surface almost precisely as for finite an-gular momenta with the additional diverging short-rangerepulsion. This angular momentum independent result isonly achieved with a relatively large charge on the core-nucleus as for medium heavy or heavy nuclei.

The adjusted depths corresponding to J = 0, 2, and 4are collected in Table I. The resulting α-core energies andthe root-mean square radii are also given in this table forground and all excited states. Due to the large barriersaround the minimum these unbound states of positive en-ergy are sufficiently well defined to allow computation oftheir radii. The potential depths are adjusted with angu-lar momentum specifically to compensate the centrifugalbarrier and leave the energy of the weakest bound stateat essentially the same value. The intent is to have aslightly unbound two-body subsystem in the three-bodysystem. The deepest potential will be used as the α-corein the three-body system, as both the effect of changingangular momentum and the low lying excited states areof interest. Using the potential depth associated withJ = 4 in a calculation with J = 0 may clearly producemore bound states. The radii of the states in Table I in-crease with increasing energy, but only moderately. Theactual values would be strongly dependent on the radiusof the potential. The peaks in Fig. 3 occur at around7.2 fm which is about the size of the core plus α-particlecharge radius. Thus fortunately, but not surprisingly, theα-particle is located at the surface of the core.

B. Three-body formalism

The three-body calculations are carried out by useof the adiabatic hyper-spherical expansion method [26].First the Jacobi coordinates are defined as mass scaledvectors, x and y, between one pair of particles, and be-tween their center-of-mass and the third particle respec-tively. The relative orbital angular momenta, lx and ly,are related to this choice of Jacobi coordinates. Threedifferent choices are possible. The hyperspherical coordi-nates are defined by the hyperradius, ρ, and five hyper-angles, Ω. The (coordinate independent) definition of ρinvolves an arbitrary normalization mass, m, which hasno influence on the result and is only used for notationalconvenience.

We first solve the hyper-angular part of the Faddeevequations for fixed average radius ρ. Each partial wave ineach Faddeev component is expanded on the set of Jacobipolynomials from constants to the highest order definedby Kmax. This provides a set of angular eigenvalues,λn(ρ), and eigenfunctions, Φn(ρ,Ω), where all quantitiesdepend on ρ. The solution to these equations producethe effective potentials

Veff (ρ) =~2

2m

(λn(ρ) + 15/4

ρ2

), (8)

where λn(ρ) is the crucial ingredient. The total wavefunction, Ψ, is expanded on the complete set, Φn,

Ψ =∑n

ρ−5/2fn(ρ)Φn(ρ,Ω), (9)

where fn are the hyper-radial wave functions. They aredetermined by the coupled set of hyperradial equationsarising from insertion of Ψ into the Faddeev equations,that is(− ∂2

∂ρ2+

1

ρ2

(λn(ρ) +

15

4

)−Qnn −

2mE

~2

)fn(ρ),

=∑n′ 6=n

(2Pnn′

∂

∂ρ+Qnn′

)fn′(ρ) , (10)

where E is the energy. The coupling terms, P and Q, aregiven by

Pnn′(ρ) =

⟨Φn

∣∣∣∣ ∂∂ρ∣∣∣∣Φn′

⟩Ω

, (11)

Qnn′(ρ) =

⟨Φn

∣∣∣∣ ∂2

∂ρ2

∣∣∣∣Φn′

⟩Ω

, (12)

where the expectation values are over the hyperangles, Ω,for fixed ρ. The convergence with the number of includedadiabatic potentials is usually very fast, and only 4-6 arenecessary in Eq. (10).

IV. CORE PLUS TWO-ALPHA PROPERTIES

The previous section introduced the two-body poten-tials and the three-body formalism, which will be ap-

6

0 10 20 30 40

−10

0

10

20

30

40

50

Ve

ff(ρ

) [M

eV

]

ρ [fm]

Figure 4: (Colour online) Effective potentials, Eq. (8),calculated from the five lowest λn(ρ), for the case where

V0 = 28.8 MeV and J+ = 0+. The horizontal linescorrespond to the lowest, and first excited state of thelowest λn(ρ), when it is approximated by a harmonicoscillator. It is noteworthy how similar the effective

potentials are for the different λn functions.

plied in the present section. Here 148Nd is considered asa three body system consisting of a 140Ba core and twoα particles. These nuclei are chosen as they are at theedge of the α unstable region in Fig. 1. The purpose ofthis section is to study the nature of a general, relativelyheavy, α-Borromean system. As the same potential willbe used for all partial waves, and as this potential wasonly adjusted to create a slightly unbound two-body sys-tem, energy levels cannot be expected to be reproduced.Of particular interest are the distributions among boththe effective potentials and the partial waves, as well asthe spatial distributions. In Sec. V B a detailed fine tun-ing of the individual partial waves is included for a similarsystem (142Ba consider as 134Te + α + α) to reproduceboth energy levels and electric transition probabilities.

The calculations presented here treat the core as aninert particle with angular momentum and parity 0+.Therefore, the effects arising from excitations of the140Ba core, in the 148Nd case, or the 134Te core, in the142Ba case, into the 2+ excited state (at 0.60 MeV in140Ba and 1.28 MeV in 134Te) will not be considered.

The solutions are obtained in two steps. First the an-gular wave functions are calculated and second we solvethe coupled radial set of equations. In the first step bothangular wave functions, and radial potentials, and theircouplings are produced. In the first subsection, we dis-cuss the properties of these solutions, and in the secondsubsection we present the radial structure in simple geo-metric terms.

A. Angular three-body structure

The five lowest of the effective potentials given inEq. (8) are shown in Fig. 4 for angular momentumand parity, 0+, and a given appropriate strength, V0 =28.8 MeV, of the Woods-Saxon potential. The lowestminimum value is about −8 MeV and located close toρ ≈ 20 fm. The potential has a rather steep barrier ris-ing to about +11 MeV at roughly 27 fm, after which itdecreases slowly towards zero as ρ increases to infinity.The fall-off is proportional to 1/ρ because the Coulombpotentials are responsible for this long-range behaviour.This implies proportional fall-off for all potentials, sincethe large-distance Coulomb interactions are the same forall adiabatic potentials. The increase of the potential forsmall ρ is due to the centrifugal barrier behaviour of 1/ρ2,while the Coulomb potentials remain finite through theassumption of homogeneous charge distributions.

The higher-lying adiabatic potentials are remarkablysimilar to the lowest and each only shifted by about1.5 MeV, very crudely independent of ρ. The zeropoint motion of the best fit of the lowest potential bya one-dimensional oscillator is about 2.7 MeV (~ω ≈5.5 MeV). The first excited oscillator energy is then atabout 8.5 MeV above the oscillator bottom. The shiftedzero point in our potential is at about −8 MeV pro-ducing two oscillator estimates at about −5.0 MeV and+0.5 MeV as indicated by the horizontal lines in Fig. 4.

The distance between neighbouring adiabatic poten-tials is roughly about 1.5 MeV. The first excited state inthe lowest potential at about +0.5 MeV is then at aboutthe same position as the ground state of the fourth po-tential, which is estimated to be at about 4 × 1.5 MeV+2.7 MeV above the lowest minimum at −8 MeV, thatis +0.7 MeV. This implies that the wave functions of thelowest-lying two states in the 0+ spectrum can be ex-pected almost entirely built on individual potentials, un-less of course the couplings between the adiabatic poten-tials are unusually strong. The third excited state couldenergy-wise instead be composed of comparable compo-nents from first and fourth potentials.

Higher angular momentum potentials are rather simi-lar but with minima shifted upwards by the centrifugalbarrier amounting to roughly 0.4 MeV and 1.4 MeV for2+ and 4+, respectively. The large distance behaviouris essentially maintained, whereas the increase and even-tual divergence at short distance accelerate with angularmomentum as in Fig. 2. Finally, modest variation of thetwo-body potential strength will only displace the curvesslightly, and the effect is most noticeable at large dis-tances, where it has no effect on the bound state struc-ture.

The calculated energies are given in Table II for thepotential strength V0 = 28.8 MeV and different angularmomenta. Decreasing the attraction to V0 = 26.25 MeVonly one 0+ state (2+, 4+) is bound at −0.2 MeV. Afurther increase of strength to V0 = 27.1 MeV providestwo bound 0+ states at −1.8 MeV and −0.6 MeV, and

7

Table II: The three body energies of 148Nd(140Ba + α+ α), as well as average α-α and α-core

distances for different angular momenta withV0 = 28.8 MeV, a = 0.65 fm, R = 9.1 fm, RC = 7.4 fm.The weights of the contributing adiabatic potentials are

given for each state in the last five columns. Allenergies are in MeV and all distances are in fm.

Weights of potentials (%)

Jπ E 〈r2αα〉12 〈r2αc〉

12 1 2 3 4 5

0+ -4.9 4.8 6.9 95 4 0 1 00+ -3.7 12.0 7.0 7 92 1 0 00+ -2.6 10.9 7.0 3 1 94 2 00+ -0.8 10.2 7.2 17 3 2 74 42+ -4.5 4.7 7.0 95 5 0 0 02+ -3.3 12.0 7.0 7 89 3 1 02+ -2.3 9.6 7.0 2 4 22 70 22+ -0.1 10.9 7.4 2 1 9 22 664+ -3.5 4.9 7.0 95 4 0 0 04+ -2.4 12.4 7.1 7 84 9 0 04+ -1.6 9.7 7.0 1 10 81 6 24+ -0.7 8.8 7.0 0 1 10 78 10

two bound 2+ states at −1.3 MeV and −0.2 MeV. WhenV0 = 28.8 MeV, corresponding to Fig. 4, we find fourbound state solutions for each set of quantum numbers,0+, 2+ and 4+. Thus, the three-body bound states ap-pear much faster and more abundantly than the two-body α-core potentials in Fig. 2. The oscillator estimateof about −5.0 MeV and 0.5 MeV for the two lowest 0+

states built on the lowest adiabatic potential is ratheraccurate as only one corresponding bound state appearsat −4.9 MeV, see Table II. Unbound resonance states arenot computed until Sec. V B, where 142Ba is examined indetail.

The structure of these states is known through the cal-culated properties of the wave functions. We considerfirst the contributions from the different adiabatic po-tentials to the individual states. Only five potentials arenecessary to ensure accurate radial solutions. The rela-tive weights in Table II are remarkably simple with oneentirely dominating potential for all wave functions. Thetwo weakest bound 2+ states are the most fractionatedwith a division of (22%, 70%) and (22%,66%) on thirdand fourth, and fourth and fifth potentials, respectively.

In general, each potential then essentially carries thefull weight of a given state, such that the lowest poten-tial corresponds to the lowest energy, the second poten-tial and the second lowest energy are related, etc. Thisconfirms the main conclusion of one adiabatic potentialper state obtained from the estimate by use of an oscilla-tor approximation without couplings between potentials.The third excited 0+ state begins to have contributionsfrom both first and fourth adiabatic potentials. The ex-citation on the lowest potential competes with the lowestenergy on the fourth potential, and two configurationsarise. The second excited 2+ state is fractionated be-

Table III: The weights of each partial wave for thepotential specified in Table II. Here lx denotes the

relative angular momentum between the two particles,and ly denotes the angular momentum of the third

particle relative to the center of mass of the first twoparticles. The fifth column give the order, Kmax, of theJacobi polynomium used for the corresponding partialwave. The last four columns each describe ground (G),

first (F), second (S) and third (T) excited states.Components where all states have a weight less than

0.04 are omitted.

Jπ Jacobi lx ly Kmax G. F. S. T.0+ α-α 0 0 80 0.79 0.73 0.72 0.72

2 2 60 0.19 0.22 0.22 0.224 4 50 0.02 0.04 0.05 0.05

0+ α-c 0 0 100 0.43 0.51 0.04 0.071 1 80 0.39 0.48 0.09 0.062 2 60 0.10 0.01 0.85 0.013 3 50 0.00 0.00 0.00 0.79

2+ α-α 0 2 70 0.82 0.07 0.26 0.042 0 70 0.05 0.72 0.45 0.542 2 50 0.05 0.08 0.16 0.292 4 40 0.07 0.02 0.06 0.014 2 40 0.01 0.09 0.05 0.054 4 30 0.00 0.01 0.01 0.05

2+ α-c 0 2 70 0.18 0.25 0.02 0.002 0 70 0.19 0.23 0.03 0.001 1 50 0.39 0.50 0.03 0.002 2 50 0.06 0.01 0.05 0.231 3 40 0.04 0.00 0.38 0.073 1 40 0.05 0.00 0.37 0.073 3 40 0.01 0.00 0.03 0.352 4 40 0.00 0.00 0.01 0.054 2 40 0.00 0.00 0.02 0.044 4 30 0.00 0.00 0.03 0.05

4+ α-α 2 2 80 0.09 0.15 0.34 0.270 4 50 0.78 0.02 0.07 0.084 0 50 0.00 0.68 0.15 0.264 2 48 0.01 0.03 0.18 0.022 4 48 0.06 0.03 0.18 0.276 2 20 0.00 0.06 0.02 0.02

4+ α-c 2 2 80 0.40 0.51 0.03 0.021 3 80 0.20 0.26 0.02 0.003 1 66 0.22 0.22 0.03 0.002 4 68 0.01 0.00 0.00 0.204 2 68 0.01 0.00 0.00 0.214 0 50 0.04 0.00 0.41 0.030 4 50 0.03 0.00 0.42 0.034 4 50 0.00 0.00 0.01 0.065 1 40 0.00 0.00 0.01 0.091 5 40 0.00 0.00 0.01 0.10

tween third and fourth potentials, now because the po-tentials happen to be rather close-lying and the couplingsare therefore more effective.

The structure revealed by the partial wave decompo-sition of the wave functions is seen in Table III, whereonly the contributions amounting to more than 4% are

8

included. We give decompositions in both the two differ-ent Jacobi coordinate sets. Here it is worth noticing thatonly even angular momenta are allowed between the twoα-particles due to the identical boson characteristics. Wefirst emphasize that the small attractions, where the en-ergies approach all the way down to zero, all down to thepercent level produce the same partial wave decomposi-tion independent of the specific energy. We therefore onlyshow the results for one strength. The structures remainunchanged because essentially only the potential energiesare moved corresponding to a shifted energy scale.

0 4

8 12

16

0 4

8 12

16

0

0.04

0.08

0.12

0.16

r c,α

α2

r α,α

2 |

Ψ(r

α,α

,rc,

αα)|

2 [

fm-2

]

rα,α [fm]

rc,αα [fm]

0

0.04

0.08

0.12

0.16

0 3

6 9

12

0 5

10 15

20 25

0

0.04

0.08

0.12

0.16

r α,c

2 r

α,α

c2 |

Ψ(r

α,c

,rα

,αc)

|2 [

fm-2

]

rα,c [fm]

rα,αc [fm]

0

0.04

0.08

0.12

0.16

Figure 5: (Colour online) The probability distributionfor the 0+ ground state wave with the α-core potential

specified in Table II. Projected contour curves areshown at the bottom of each figure. The distance

variables correspond to the two different Jacobi sets,where top and bottom panels are for the first and

second Jacobi sets, respectively.

For the 0+ states we find more than 70% of α-α s-wavesin all solutions, while the d-waves absorb most of the re-maining probabilities. This is obviously consistent withthe stronger α-α attraction in the lowest partial waves.However, the α-core potential may prefer another com-peting structure. The two lowest 0+ states have roughlyequal amounts of α-core relative s- and p-waves, whereasthe third and fourth 0+ states are dominated by d- and f -waves, respectively. In combination with the results fromTable II this indicates that these higher-lying adiabaticpotentials are dominated by d- and f -waves.

The 2+ states must have non-zero angular momentumpartial waves. The most favourable structure seen in Ta-

ble III, is apparently s-waves between the two α-particlesbut a distribution for the α-core structure of equal s- andd-waves and twice as much p-waves. In contrast, the firstexcited state has dominating α-α d-waves and compara-ble to the ground state contributions from g-waves. Thesecond and third excited states have also α-α d-wavesas the largest components. The α-core contributions arenow moved to roughly equal p- and f -waves, and compa-rable d- and f -waves, respectively for second and thirdexcited state.

The 4+ states must have even larger finite angular mo-mentum contributions than the 2+ states. In Table IIIwe find again that s-waves dominate for ground state,while the first excited state is dominated by g-waves inthe α-α subsystem. In the α-core subsystem, these twostates have the largest contributions from d-waves androughly half as much for p- and f -waves. The third andfourth 4+ states have in the α-α subsystem more than50% of d-waves and roughly half as much g-waves. Inthe α-core subsystem s- and g-, and d- and g-wave com-ponents are about equal, respectively for the third andfouth 4+ states. The higher-lying states receive signifi-cant contributions from more partial waves than groundand excited states.

These rather complicated variations in structure forthe different states are dictated by minimizing the totalenergies. This involves combinations of the two-body in-teractions which in the present cases always prefer thelowest partial waves. The final results are then obtainedby combining the minimization with total angular mo-mentum conservation, partial wave couplings, and or-thogonality of all pairs of states.

B. Radial structure

The angular eigenvalues, λn, provide informationabout the crucial radial potentials, which in turn throughEq. (10) determine the radial wave functions, fn(ρ). Thelinear combination with the angular parts, Φn, fromEq. (9) gives access to complete information about eachof the solutions. We shall in particular be concernedwith probability distributions for the α-α and α-core dis-tances.

In Fig. 5 we show the probability distribution for the0+ ground state in two coordinate systems correspondingto the two different Jacobi coordinates. The structure isrelatively simple with only one peak at an α-α distanceof about 4 fm from the top panel and an α-core distanceof about 7 fm from the bottom panel of Fig. 5.

The probability distributions for the excited 0+ statesare shown in Fig. 6 for the first Jacobi set where the x-coordinate is between the two α-particles. In all these ex-cited states we find probability distributions between coreand α-particle almost identical to that of the ground stateas shown in the lower part of Fig. 5. Consequently, wedo not show these distributions. However, the identicaldistributions demonstrate that the α-particles strongly

9

0 4

8 12

16

0 4

8 12

16

0

0.02

0.04

0.06

0.08

r c,α

α2

r α,α

2 |

Ψ(r

α,α

,rc,

αα)|

2 [

fm-2

]

rα,α [fm]

rc,αα [fm]

0

0.02

0.04

0.06

0.08

0 4

8 12

16

0 4

8 12

16

0

0.02

0.04

0.06

0.08

r c,α

α2

r α,α

2 |

Ψ(r

α,α

,rc,

αα)|

2 [

fm-2

]

rα,α [fm]

rc,αα [fm]

0

0.02

0.04

0.06

0.08

0 4

8 12

16

0 4

8 12

16

0

0.02

0.04

0.06

0.08

r c,α

α2

r α,α

2 |

Ψ(r

α,α

,rc,

αα)|

2 [

fm-2

]

rα,α [fm]

rc,αα [fm]

0

0.02

0.04

0.06

0.08

Figure 6: (Colour online) The same as top panel ofFig. 5. for the three excited 0+ states.

prefer to be located at the surface of the core as forthe isolated two-body system with the wave functionshown in Fig. 3. The reason is that the α-core poten-tial overrules all other possible effects when determiningthe three-body structure. Apparently the α-α potentialis strongly attractive and the potential energy minimumis rather narrow and deep.

However, the α-α distribution varies from top to bot-tom in Fig. 6, and they are also different from the groundstate distribution. The first excited state shows a broaderdistribution around the α-α distance of 13 fm with amarginal reminiscence of a peak at the ground state lo-cation of about 4 fm. The second excited state has threepeaks at α-α distances of about 14 fm, 9 fm and 4 fm.The third excited 0+ state continues the trend by con-taining four peaks at α-α distances of 14 fm, 12 fm, 7 fm,and 4 fm. These different structures reflect the differ-

0 4

8 12

16

0 4

8 12

16

0

0.02

0.04

0.06

r c,α

α2

r α,α

2 |

Ψ(r

α,α

,rc,

αα)|

2 [

fm-2

]

rα,α [fm]

rc,αα [fm]

0

0.02

0.04

0.06

0 4

8 12

16

0 4

8 12

16

0

0.02

0.04

0.06

0.08

r c,α

α2

r α,α

2 |

Ψ(r

α,α

,rc,

αα)|

2 [

fm-2

]

rα,α [fm]

rc,αα [fm]

0

0.02

0.04

0.06

0.08

Figure 7: (Colour online) The same as top panel ofFig. 5. for the second and third excited 2+ states.

ent structures of the corresponding adiabatic potentials,which deliver the dominating contributions to each of theexcited states.

The probability distributions for the lowest two 2+

states are very similar to the 0+ distributions in Fig. 5.Also the α-core distributions are remarkably similar forthe computed higher-lying 2+ states. The α-α distri-bution for the second excited 2+ state in Fig. 7 revealsa much broader distribution. It is almost without peaksbut with a ridge stretching from α-core distances between10 fm and 4 fm with a corresponding increase of the dis-tance of the core from the α-α center of mass. The thirdexcited 2+ state is also shown in Fig. 7 exhibiting distinctpeaks at distances of about 15 fm, 11 fm and 4 fm.

The α-core distributions for the 4+ states are againalmost indistinguishable from the previously computeddistributions for the other angular momenta, such as theexample shown in the lower panel of Fig. 5. The α-αdistribution for the first excited 4+ state resembles thesame distribution for both first excited 0+ and 2+ states.The distribution for the second excited 4+ state is shownin Fig. 8, where the peak structure again is smeared outand the largest probability peak is at around 11 fm. Forthe third excited 4+ state in Fig. 8, the α-α distributiondisplays three peaks at around 14 fm, 10 fm and 4 fm.

In summary, the many different probability distribu-tions all have the remarkable property of one peak in theα-core distance. The corresponding interaction is strongand essentially a surface attraction due to centrifugal and

10

0 4

8 12

16

0 4

8 12

16

0

0.02

0.04

0.06

r c,α

α2

r α,α

2 |

Ψ(r

α,α

,rc,

αα)|

2 [

fm-2

]

rα,α [fm]

rc,αα [fm]

0

0.02

0.04

0.06

0 4

8 12

16

0 4

8 12

16

0

0.03

0.06

0.09

0.12

r c,α

α2

r α,α

2 |

Ψ(r

α,α

,rc,

αα)|

2 [

fm-2

]

rα,α [fm]

rc,αα [fm]

0

0.03

0.06

0.09

0.12

Figure 8: (Colour online) The same as top panel ofFig. 5. for the second and third excited 4+ states.

Coulomb potentials. In contrast the α-α distributions ex-hibit very large variations from one peak to several peaksor rather smeared out distributions. However, all thesedistributions have fortunately a sharp cut-off at smalldistances where the α-particles would beginning to over-lap.

The tempting interpretation in terms of simple geo-metric structures is then only meaningful when one nottoo broad peak contains a large fraction of the probabil-ity. It may still be rewarding to look at average distanceproperties as the root mean square radii given in TableII. Once more we emphasize that the α-core root meansquare radius is remarkably constant for all states. Wecan then conclude that the α-particles are located onspheres corresponding to this radius around the core.

The average α-α distances for the ground states of anyangular momentum are 4.8 fm which is similar to, al-though about 1 fm less than, the same quantity, 5.95 fmfor the two-α structure of 8Be. Combined with the 70%of s-waves in all these ground states we conclude thatthese three-body states resemble a core plus 8Be in itsground state. The first excited states of all three an-gular momenta are also very similar to each other butnow with more than 70% of α-α 2+ structure and witha much larger distance of about 12 fm. This structure isfar from any excited state of 8Be, and these structuresin fact resemble a linear structure with the core in themiddle.

The second and third excited states exhibit much more

complicated structures which cannot be collected intoone simple configuration. However, they can be de-scribed as containing three or maybe even four compo-nents each with different configurations. The resultingprobability distributions are more smeared out but both8Be like ground state structures, linear α-core-α chain-configurations, and intermediate structures are presentin each state.

V. OBSERVABLE CONSEQUENCES

The general structures for systems with weak bindingof one- and two α-particles are discussed in the previoussection. We shall here first compare to measured prop-erties of 148Nd, which was considered in Sec. IV as ageneral representative of α-Borromean structures in rel-atively heavy nuclei. In the second subsection we shalldiscuss the results obtained from fine-tuning the inter-action parameters to be appropriate for the only known(apart from 12C) even-even Borromean two-α nucleus,142Ba.

A. Properties of 148Nd (140Ba + α+ α)

The spectra in Table II for the lowest energies of the0+, 2+, and 4+ states present a rotational sequence,(0.0, 0.4, 1.4), with rigid body moment of inertia, I cor-responding to ~2/I ≈ 0.14 MeV. This implies a distance,rc,αα, of about 6.19 fm between the core and the cen-ter of mass of two α-particles, which is almost identicalto the value derived from Table II. Furthermore, this isin complete agreement with indistinguishable geometricproperties for these three 0+, 2+ and 4+ states shown inFig. 5 for the 0+.

The conclusion is that these states form a rota-tional band. The schematic rules for rotational B(E2)-transition probabilities are then obeyed for a core plus atwo-α structure rotating around their common center ofmass. The absolute values of the electromagnetic transi-tion probabilities are proportional to the intrinsic electricquadrupole moment, Q0, of the same structure. For a rel-atively heavy core we have Q0 ≈ 8er2

c,αα ≈ 307 e fm2,where 4 e is the charge of the combined two α-particles.The single-particle value, Qsp ≈ 75 e fm2, is about four(the charge) times smaller, provided the same radius isused in both estimates.

Let us now compare these numerical average results tomeasured values for 148Nd [27]. First, the observed exci-tation energies do not follow the simple rotational modelpredictions. The energies of the 4+ state is 2.5 timeslarger than the energies of the 2+ states. If anything thisis closer to the vibrational model value of 2 rather than3.3 valid for rotations. The vibrational picture does notmatch any better by combining the second 0+ and 2+

states.

11

Transition probabilities contain more detailed informa-tion about structures, but only rather uncertain dataare available for these nuclei. For 148Nd the availablemeasurements of B(E2) values are, B(E2; 0 → 2) =1.37 e2 b2, B(E2; 2 → 4) = 0.784 e2 b2, and thequadrupole moment, Q(2) = −1.46 e b, for the 2+ state.Transforming these transition values into the down goingprobabilities we get the ratio 0.78× 25/(1.37× 9) = 1.58which is comparable to 1.41 from the rotational modelbut also not too far from the vibrational value of 2. Thequadrupole moment is related to the intrinsic quadrupolemoment by Q0 = −Q(2)7/2 = 511 e fm2 where our modelvalue of 307 e fm2 is 1.7 times smaller than measured.

Considering the same effective potentials were used forall partial waves, an agreement within a factor of two isbetter than what could have been expected. This is inspite of the fact that the model forms a rotational spec-trum, while the data do not contain simple, strictly rota-tional or vibrational features. It is also worth noting thatthe simple model produced an energy spectrum where theenergy of the 4+ state is 2.5 times larger than the energyof the 2+ state, same as for 148Nd. This suggests that148Nd, and nuclei similar to it, might well be describedas two-α structures in their low-lying states.

The model with the same average parameters in theradial effective potential is independent of angular mo-mentum and known to be very inaccurate for odd paritystates. This average model can only marginally distin-guish between odd and even parity states, since the cen-trifugal barrier varies continuously with orbital angularmomentum. Only the Bose character of the α-particles isable to give small differences due to parity. On the otherhand, low-energy nuclear spectra with only very few ex-ceptions are dominated by the positive parity states whilenegative parity states are located at higher excitationenergies. In nuclear few-body models this feature is ac-counted for by partial wave (angular momentum and par-ity) dependent effective potentials. A proper compari-son to data therefore involves detailed input and carefulsearch for suitable nuclei where the few-body structureis possible.

B. Properties of 142Ba (134Te + α+ α)

The most tempting nuclei to investigate are Borromeantwo-α systems. Searching the available masses for candi-dates we find only one known even-even nucleus, 142Ba(α + α+134Te), of that structure. The exception of 12C(α + α + α) is special since the core also consists of anα-particle. Nuclear few-body models must assume de-coupling of intrinsic and relative cluster degrees of free-dom. Therefore the intrinsic degrees of freedom prefer-ably should be difficult to excite either by weak couplingsor by unreachable high excitation energy.

The present case has 134Te as the core where the lowestexcited state is a 2+ state at 1.279 MeV [28]. By adjust-ing the partial wave interactions the polarization is fully

Table IV: The Woods-Saxon depths, V0, reproducingthe resonance energies, Eres, for the different angularmomentum and parities, Jπ, in 138Xe (134Te + α) [29].

There is no known 5− state in the 138Xe excitationspectrum, so the same potential depth as for 3− was

used. The energy of the 0+ state is determined by the αseparation energy through Eq. (13). All energies are in

MeV.

Jπ 0+ 2+ 4+ 6+ 1− 3− 5− OtherEres 0.138 0.727 1.211 1.419 2.004 2.153 -V0 24.423 24.568 25.787 28.2 22.641 23.746 23.746 24.568

Table V: The same as Table II for 142Ba (134Te +α+α)using the potential depths specified in Table IV for theindividual partial waves. The column labelled Eexp isthe measured value from [30], where the energy of the0+ state is determined by the two α separation energy.

Two first excited states, indicated by (F ), are alsoincluded. The last five columns specifies the individual

weights of the five lowest potentials.

Weights (%)

Jπ Eexp Ecal Ecal − Eexp 〈r2αα〉12 〈r2αc〉

12 1 2 3 4 5

0+ -0.16 0.11 0.27 7.8 7.0 99 1 0 0 02+ 0.20 0.50 0.30 7.5 7.1 99 0 0 1 04+ 0.68 0.98 0.30 7.4 7.1 96 2 1 0 01− 1.17 1.11 -0.06 4.2 7.1 98 1 0 0 03− 1.13 1.37 0.24 4.3 7.1 96 2 1 0 12+(F ) 1.26 1.05 -0.21 5.9 7.1 0 98 1 1 06+ 1.31 0.90 -0.41 6.6 7.2 95 3 2 0 00+(F ) 1.38 1.59 0.22 9.3 7.1 2 96 1 0 0

included in the adopted effective potential on the two-body level. If present, an α-cluster structure should beseen as resonances in α-core scattering, that is as 138Xestates [29]. The energy of the 0+ ground state is thendetermined by the α separation energy

Eres(0+) = −

(B(138Xe)−B(134Te)−Bα

). (13)

This provides the depth of the radial potentials for eachangular momentum and natural parity. We choose thesame radial Woods-Saxon shape with the same radiusand diffuseness parameters as used above. We only adjustthe depth to reproduce the measured resonance energiesin 138Xe (α+134Te). The resulting values are given inTable IV.

The energies and sizes of the three-body eigenstatesare given in Table V, together with the distribution ofweights on the different adiabatic potentials, and the ex-perimentally measured energies [30]. Here the groundstate energy is determined by the two α separation en-ergy. Also included in Table V is the difference betweenthe calculated and the measured energies.

The absolute values of the calculated energies are seento be displaced by roughly 0.3 MeV for four of the five

12

lowest states. A slight displacement is not surprising asno attempt has been made to account for explicit threebody effects. A distinct three-body potential could beadded, but it would be an ad hoc addition adjusted to fitthe desired spectrum. More interesting are the relativedistances between individual levels in the calculated spec-trum, and they agree very well with relative distancesbetween the experimental measurements. The calculatedrelative distances between the 0+, and the 2+, 4+, and3− states only differ by about 0.03 MeV from the rela-tive distances in the experimental spectrum. The onlyexception among the lowest states is the 1− state, whichdoes not agree with the shifted spectrum. Its absolutevalue actually agrees more closely with the experimentalvalue. This agreement is most likely coincidental, andmerely the result of counteracting offsets. It may be ofinterest that in the symmetry classification of the corre-sponding states in the related system 12C [31] the lowest1− state appears in a different ”band” than the 0+, 2+,3−, and 4+ states.

For the three remaining, higher lying states, the devi-ations become more erratic. The difference between thecalculated and the experimental values is no longer closeto constant. This might indicate a limit to the modelgiven by the core excitation energy.

The weights of the individual potentials are as ex-pected from the results of the previous sections. Eachstate is dominated by a single adiabatic potential. Thelowest potential dominates for all non-excited states,while the second potential dominates for the first excitedstates. The coupling between different potentials mustthen be very weak, even when the potentials of each par-tial wave are adjusted individually.

The average α-core distance is again constant ataround 7.1 fm, which implies that the α particles arestill placed at the surface of a sphere around the core.The probability distributions between the core and theα-particle are also identical to the distribution seen inthe lower part of Fig. 5, and is therefore not included.The average α-α distances, on the other hand, are verydifferent from the ground state values in Table II, at leastfor the even parity states. However, the average valuesare somewhat misleading. In Fig. 9 the probability dis-tributions for the 0+, 2+, 4+, and 6+ states are shown.The same peak as in Fig. 5 at an α-α distance of roughly4 fm is seen for all three states. The large average valuesare caused by the appearance of a much smaller peak atan α-α distance of about 13 fm. This almost constitutesa line structure, with α particles on opposite sides of thecore.

The probability distribution for first excited 0+ and2+ states are seen in Fig. 10. The distribution of thefirst excited 0+ is similar to the top panel of Fig. 6, onlywith the large peak at the slightly smaller α-α distanceof 10 fm. The distribution of the first excited 2+ state isunusual compared with the first excited states examinedpreviously. It is almost identical to the distribution ofthe 2+ ground state seen in the second panel of Fig. 9.

On the other hand, the average distances for the oddparity states agree reasonably well with the earlier re-sults. The probability distribution for the 1−, and 3−

states are seen in Fig. 11. They are identical to the up-per part of Fig. 5. The single peak is then well describedby the average distance in Table V.

The contributions from the different partial waves arelisted in Table VI. The overall tendencies are the sameas in Table III, but the specific weights have changedslightly. The relative angular momentum between thetwo α particles is dominated by s-waves for all but the6+ state, although to a lesser extent than before. Par-ticularly interesting are the odd parity states which werenot included earlier. The 1− state is the only state whichhave a dominating contribution from p-waves to the rel-ative α-core angular momentum. This could be part ofthe explanation as to why the 1− state deviates fromthe other low lying states. Likewise, the 3− is the onlystate to have a significant f -wave contribution to the α-core angular momentum. For the 3− state there is aroughly even contribution from s-, p-, d-, and f -waves.The higher lying 6+ and 5− states have a much morescattered distribution of partial waves, in particular forthe α-core system. However, these results are less reli-able, as the states are outside the energy region, whichcan the model can reasonably be expected to cover.

The calculated dipole and quadrupole transition prob-abilities are presented in Table VII. Included are boththe results for the two-body 138Xe (134Te + α) and thethree-body 142Ba (134Te + 2α) systems. The transitionprobabilities are given by

B(Eλ; J → J ′) = Q20〈J0λ0|J ′0〉2, (14)

where 〈J0λ0|J ′0〉 is the Clebsch-Gordan coefficient cou-pling the states J and J ′. In the two-body system Q0 forthe quadrupole transition is

Q0 =

√5

16π2r2αc

zαem2c + zcem

2α

(mα +mc)2, (15)

while for the dipole transition Q0 is

Q0 =

√3

4πrαc

zαemc − zcemα

(mα +mc). (16)

Here mα and mc are the masses, zα and zc are the protonnumbers, and e is the elementary charge. The α-coredistance used is 7.1 fm, as given in Table V. There aretwo α particles in the three-body system, so twice the αmass, and twice the α charge is used. Also the distanceis replaced by the distance between the core and the α-αsystem, rc,αα. The value used is not the average valuefrom Table V, but the peak value of 6.8 fm from Figs. 9and 11.

Unfortunately, only a few experimental values areavailable at the moment, as seen in Table VII. Boththe dipole and the quadrupole transition probabilitiesare relatively small, but for different reasons. The in-trinsic particle degrees of freedom do not contribute to

13

Table VI: The same as Table III with 142Ba for thestates from Table V

Jπ Jacobi lx ly Kmax G F0+ α-α 0 0 80 0.80 0.62

2 2 60 0.18 0.304 4 50 0.02 0.07

0+ α-c 0 0 100 0.70 0.271 1 80 0.04 0.022 2 60 0.17 0.67

2+ α-α 0 2 70 0.62 0.062 0 70 0.20 0.442 2 50 0.11 0.462 4 40 0.04 0.02

2+ α-c 0 2 70 0.35 0.002 0 70 0.35 0.002 2 50 0.09 0.173 3 40 0.00 0.042 4 40 0.02 0.054 2 40 0.02 0.054 4 30 0.01 0.306 6 30 0.01 0.06

4+ α-α 2 2 80 0.040 4 50 0.634 0 50 0.182 4 48 0.08

4+ α-c 2 2 80 0.444 0 50 0.170 4 50 0.16

1− α-α 0 1 75 0.812 1 65 0.092 3 55 0.09

1− α-c 0 1 75 0.221 0 75 0.232 1 55 0.151 2 55 0.152 3 55 0.053 2 55 0.05

3− α-α 0 3 75 0.692 3 55 0.26

3− α-c 0 3 75 0.143 0 75 0.152 1 65 0.121 2 65 0.124 1 41 0.071 4 41 0.06

6+ α-α 0 6 100 0.236 0 90 0.062 4 90 0.062 6 90 0.526 2 80 0.08

6+ α-c 0 6 100 0.146 0 90 0.142 4 90 0.084 2 80 0.074 6 40 0.046 4 40 0.046 6 50 0.14

Table VII: The electric transition probability for thestates from Table V. The first column specifies the

nuclei, the second column the transition in question,and the third column the calculated result based on

Eq. (14). The available experimental values for 142Ba[30] and 138Xe [29] are presented in column four. Adash indicates the value was unavailable. The dipoletransition probabilities are in units of e2 b, and the

quadrupole probabilities are in e2 b2.

Nuclei B(Eλ; J → J ′) Model Experiment138Xe B(E2; 2→ 0) 0.075 0.076(20)

B(E2; 4→ 2) 0.11 −B(E2; 6→ 4) 0.12 −B(E2; 3→ 1) 0.097 −B(E1; 1→ 0) 0.0075 −B(E1; 1→ 2) 0.015 −B(E1; 3→ 2) 0.0097 −B(E1; 3→ 4) 0.013 −

142Ba B(E2; 2→ 0) 0.24 0.145(4)B(E2; 4→ 2) 0.34 0.188(12)B(E2; 6→ 4) 0.37 −B(E2; 3→ 1) 0.30 −B(E1; 1→ 0) 0.026 1.1(6) · 10−6

B(E1; 1→ 2) 0.052 2.0(10) · 10−6

B(E1; 3→ 2) 0.034 −B(E1; 3→ 4) 0.045 −

the rotational motion, because of the mass difference, sosmall values of the quadrupole transition probabilitiesare inherent in cluster rotations. Considering first thequadrupole transitions in 142Ba, the model values areseen to be around 1.8 times too large. It should be notedthat the distance enters in the fourth power, so chang-ing it slightly will have significant impact on the result.As this distance is dictated by Coulomb and centrifu-gal barriers, it does to some degree depend on the cho-sen parameters. This makes the agreement surprisinglygood. The ratio between the states is 0.24/0.34 = 0.71in the model, which is very much comparable to the ra-tio of 0.77 for the experimental values. Very few othercalculations are available, but specialized models, suchas the interacting boson model (IBM) [32], specificallydesigned to calculate transition probabilities, do exists.The few experimental B(E2) values for 142Ba are repro-duced more accurately by IBM, but the model struggleswith other, similar transitions for neighbouring nuclei.For 138Xe only one transition probability is known exper-imentally. The three-particle model value for this transi-tion is identical to the experimental value, although theexperimental uncertainty is quite large.

The absolute values of the dipole transitions are off bya factor 10−4, and does not resemble the experimentalvalues. This is not surprising, as the small values are aresult of the giant dipole resonances, which are not ac-counted for in this model. However, in spite of the largeexperimental uncertainties, the ratio between the tran-sition probabilities is still a relevant test of the model,

14

as the 1− → 0+ and 1− → 2+ transitions have almostequal branching ratios. The model’s transition probabil-ities have the ratio 0.026/0.052 = 0.5, which is very closeto the experimental ratio of 0.55.

Another possible and very relevant test of the three-body model is to estimate the charge radius, and com-pare it with the measured value. The experimen-tally measured ground state charge radii are 〈r2

ch〉1/2 =

4.895(8) fm, 〈r2ch−c〉1/2 = 4.757(4) fm, and 〈r2

ch−α〉1/2 =

1.676(3) fm for 142Ba, 134Te, and the α particle respec-tively [33]. The charge radius of the entire system can

be calculated as 〈r2ch〉 = Z−1

T

∑ZT

i=1〈r2i 〉, where ZT is the

total charge of the nucleus. For our three-body systemthis can be rewritten as

〈r2ch〉 =

ZcZT

(〈r2c 〉+ 〈r2

ch−c〉)

+ 2ZαZT

(〈r2ch−α〉+ 〈r2

α〉),

(17)

where Zc and Zα are the charges of the core and the αparticle, and 〈r2

c 〉 and 〈r2α〉 are the mean square radii of

the core and the α particle respectively. These expecta-tion values are calculated as in the three-body solution

〈r2c 〉 =

(2mα

mc + 2mα

)2

〈r2c,αα〉, (18)

〈r2α〉 =

(mc +mα

mc + 2mα

)2

〈r2α,cα〉. (19)

Using Eqs. (18) and (19) in Eq. (17) the result is〈r2ch〉1/2 = 4.96 fm, which is only 0.07 fm larger than

the measured value. It should be noted that neither thecharge distribution, nor the potential radius used in thethree-body calculations have been adjusted to reproducethis charge radius. Such a close agreement is much betterthan what could have been expected.

In summary, the low-energy spectrum of 142Ba is re-produced by the present three-body model as well as bycomparable light cluster models. Both the measured andthe calculated spectrum is neither rotational nor vibra-tional in character. However, the charge radius and thequadrupole transition probabilities are reproduced sur-prisingly well. The structure of states can be describedas two α particles just outside the surface of the core, andlocated either just over 4 fm apart (possibly as a 8Be), orat opposite sides of the core in an almost linear chain.

VI. SUMMARY AND CONCLUSION

We discuss the possibility of finding Borromean nuclearsystems with heavy constituents. Crudely speaking, twotwo-body systems each with Z2/A > 17 (squared chargeover mass numbers) do not bind, that is such pairs havenegative binding energy. They are then potential candi-dates for constituents in a Borromean system. However,this is impossible as a third nucleus first would have to be

similarly heavy in order not to bind, and second its addi-tion should produce a bound three-body system. There-fore it is hard to avoid light nucleons or α-particles butthey can still be combined with one heavy core-nucleus.

We sketch the driplines for nucleons and α-particles,and conclude that it is only possible to form a Borromeansystem with one medium heavy nucleus by combiningwith two α-particles, two protons, or one proton and oneα-particle. In the present investigation we focus on twoα-particles and a medium heavy core-nucleus. An α-αeffective potential is chosen to reproduce all low-energyscattering properties. The α-core effective potential ischosen in the same spirit to reproduce only the weakestbound two-body states. If the energy is zero this nucleusis at the α-dripline, and a positive binding energy couldallow more bound states where the weakest bound, orslightly unbound, is appropriate as α-cluster structure inan excited state. These states may be appropriate whenthe lower-lying α-core states are forbidden by the Pauliprinciple due to the same nucleonic constituents in bothcore and α-particle.

The core plus two-α calculations are carried out byuse of the hyperspherical adiabatic expansion method ofthe Faddeev equations. The total angular momentumdoes not have to be zero and the contributing individ-ual partial waves can as well be finite. Therefore we findbound state solutions for a few relatively small angularmomentum values. The adiabatic potentials are all re-markably similar with the same minimum and barrier po-sitions. They are repulsively diverging at small distances,then steeply increasing from the intermediate minimumtowards larger distances, and finally they decrease asCoulomb interactions at very large distance. The differ-ent adiabatic potentials are about 1.5 MeV apart fromeach other at the minimum, and their curvatures corre-spond to a zero point energy of several MeV.

The α-core potential is chosen to allow four three-bodybound states with energies varying from about −5 MeVup to almost zero for each angular momentum. Eachbound state is dominated at the level of more than 70%by one potential term. This means that the angularstructure of each bound state is directly related to oneadiabatic potential. Still, their partial wave decomposi-tions are much more complicated, but with s-waves asclearly dominating α-α structures for the lowest boundstate for all angular momenta. The second lowest state isdominated by the reverse (with respect to lx and ly) com-pared to the lowest state. The two highest-lying boundstates in contrast contain large fractions of α-α d-waves.This might indicate that the α-α system changes rela-tive structure from ground to first excited state of 8Be.However, their distance is too large for the attractive in-teraction to contribute, and this structure therefore hasto be attributed to angular momentum and parity con-servation.

The spatial distributions of α-particles around the corefor the different bound states are revealing. The firststriking result is that the probability distributions as a

15

function of α-core distances in all states are located inrather narrow distributions at distances correspondingto α-particles at the surface of the core. This is in con-trast to the much more varying α-α distance distribu-tions. The lowest bound states for all angular momentashow a spatial α-α distribution similar to the 8Be struc-ture, but with a slightly smaller average distance. How-ever, the higher-lying bound states clearly contain severalconfigurations, where the largest component often resem-bles a linear structure with the core between the two αparticles. In these excited states an 8Be-like structure ispresent but the other components are usually dominat-ing and the total probability distribution is much moresmeared out than for the two lowest bound states.

Measurable quantities like the energies should containinformation of α-correlations. However, this is exceed-ingly difficult to extract from the background of all othereffects contributing to the total energies. We thereforefocused on α-cluster structures resulting from strong α-correlations. These structures can be detected by scat-tering experiments where α-particle dripline nuclei arethe most obvious targets. Large cross sections for two-αremoval can be expected as measured in [34]. In moredetails, an 8Be structure should emerge from the lowestof our three-body bound states, and two non-interactingα-particles can be expected from the three higher-lyingof the four computed three-body states. Measuring thetransition matrix elements between different states willalso constitute a test of the model.

To compare in more details with measured quantitieswe followed the standard procedure in few-body nuclearphysics. We focused on the even-even Borromean two-αnucleus, 142Ba, where removal of two α-particles leavethe fairly inert core nucleus, 134Te. For each partialwave we construct effective potentials adjusted to re-produce the α-core, 138Xe, two-body resonances. Withthese potentials we calculated three-body energies andfound a very good agreement with the lowest states inthe known 142Ba spectrum, although the spectrum wasshifted slightly most likely due to the fact that pure three

body effects were not accounted for explicitly. The ra-dial structure showed that the α particles were placed onthe surface of the core. The relative angular momentumbetween the α particles was dominated by s-waves. Inaddition, both the electric quadrupole transition prob-abilities and the charge radius were reproduced ratherwell. Based on these finding we predict that the cor-responding structures of these low-lying states are twoα-particles in a 8Be configuration rotating with differ-ent angular momenta at the surface of a sphere aroundthe 134Te-core. This system is the most promising forexhibiting α clusterization in the ground state.

In summary, we investigated the structures of two α-particles surrounding a heavy core-nucleus in a three-body model. The assumptions are that α-clusters canbe found with significant probability in such nuclei. Weexpect the most promising places in the nuclear chartare at the α dripline where Borromean two-α structuresare experimentally established by mass measurements.These nuclei should have relatively large sizes in theirground states when the energies are close to zero. This isthe single most important feature characterizing spatiallyextended halo structures, which simultaneously enhancethe possibility for decoupling of core and α-particle de-grees of freedom. These three-body structures may alsoappear in excited states of nuclei where the α-particle isbound. Then the α and core degrees of freedom may bemixed in the ground states but decoupled in the excitedstate close to the α-threshold.

We have shown that Borromean two-α structures arepossible at the α dripline. The α-particles at the sur-face of the core-nucleus would produce rotational spectrawith the corresponding simple transition probabilities.Strongly enhanced α-removal cross sections would alsobe a signal. One interesting perspective is that similarproton-α-core structures should be characteristic featuresof ground states when proton and α driplines are closeto or intersect each other.

This work was funded by the Danish Council for In-dependent Research DFF Natural Science and the DFFSapere Aude program.

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17

0 4

8 12

16

0 4

8 12

16

0

0.04

0.08

0.12

r c,αα2 rα,α2 |Ψ

(rα,α

,rc,αα)|2 [

fm-2

]

rα,α [fm]

rc,αα [fm]

0

0.04

0.08

0.12

0 4

8 12

16

0 4

8 12

16

0

0.04

0.08

0.12

0.16

r c,αα2 rα,α2 |Ψ

(rα,α

,rc,αα)|2 [

fm-2

]

rα,α [fm]

rc,αα [fm]

0

0.04

0.08

0.12

0.16

0 4

8 12

16

0 4

8 12

16

0

0.04

0.08

0.12

0.16

r c,αα2 rα,α2 |Ψ

(rα,α

,rc,αα)|2 [

fm-2

]

rα,α [fm]

rc,αα [fm]

0

0.04

0.08

0.12

0.16

0 4

8 12

16

0 4

8 12

16

0 0.04 0.08 0.12 0.16

0.2

r c,αα2 rα,α2 |Ψ

(rα,α

,rc,αα)|2 [

fm-2

]

rα,α [fm]

rc,αα [fm]

0

0.04

0.08

0.12

0.16

0.2

Figure 9: (Colour online) The same as top panel ofFig. 5. for the lowest 0+, 2+, 4+, and 6+ 142Ba states.

18

0 4

8 12

16

0 4

8 12

16

0

0.04

0.08

0.12

r c,αα2 rα,α2 |Ψ

(rα,α

,rc,αα)|2 [

fm-2

]

rα,α [fm]

rc,αα [fm]

0

0.04

0.08

0.12

0 4

8 12

16

0 4

8 12

16

0 0.04 0.08 0.12 0.16

0.2

r c,αα2 rα,α2 |Ψ

(rα,α

,rc,αα)|2 [

fm-2

]

rα,α [fm]

rc,αα [fm]

0

0.04

0.08

0.12

0.16

0.2

Figure 10: (Colour online) The same as top panel ofFig. 5. for the first excited 0+ (above), and 2+ (below)

states of 142Ba.

0 4

8 12

16

0 4

8 12

16

0 0.04 0.08 0.12 0.16

0.2

r c,αα2 rα,α2 |Ψ

(rα,α

,rc,αα)|2 [

fm-2

]

rα,α [fm]

rc,αα [fm]

0

0.04

0.08

0.12

0.16

0.2

0 4

8 12

16

0 4

8 12

16

0 0.04 0.08 0.12 0.16

0.2

r c,αα2 rα,α2 |Ψ

(rα,α

,rc,αα)|2 [

fm-2

]

rα,α [fm]

rc,αα [fm]

0

0.04

0.08

0.12

0.16

0.2

Figure 11: (Colour online) The same as top panel ofFig. 5. for the lowest 1−, and 3− states of 142Ba.