26 April 2001

Physics Letters B 505 (2001) 71–74www.elsevier.nl/locate/npe

12Be molecular states in a microscopic cluster model

P. Descouvemont1, D. BayePhysique Nucléaire Théorique et Physique Mathématique, CP229 Université Libre de Bruxelles, B1050 Brussels, Belgium

Received 21 December 2000; accepted 19 February 2001Editor: J.-P. Blaizot

Abstract

The 12Be spectrum is investigated in the Generator Coordinate Method, using microscopic6He+ 6He andα + 8He wavefunctions. The model is consistent with recent experimental observations of molecular states, but predicts a strong mixingof both configurations, rather than a dominant6He+ 6He structure. A negative-parity band is also found. Electromagnetictransition probabilities and partial widths of molecular states are calculated. 2001 Elsevier Science B.V. All rights reserved.

Neutron-rich Be isotopes have been extensivelystudied in recent years [1]. The Borromean natureof 9Be and10Be is responsible for many interestingproperties and has been investigated by many authors(see Ref. [2] for recent works). On the other hand, the11Be nucleus has attracted much interest because ofthe well known parity-inversion effect. More recently,experiments aiming at investigate excited states ofBe isotopes have been developed with radioactivebeam facilities [3–6]. Exotic structures have beenfound in 9–12Be where molecular states appear athigh energies. New12Be states have been observed atEx = 8.6,10 and 14 MeV by Korsheninnikov et al. ina p+ 12Be experiment. A12Be breakup experimentby Freer et al. [4] indicates the existence of excitedstates with a significant decay to theα + 8He and6He + 6He channels. These results lead Freer et al.to the suggestion of6He + 6He molecular states,essentially based on the shape of the rotational band.These new states were subsequently supported by

E-mail address: pdesc@ulb.ac.be (P. Descouvemont).1 Directeur de Recherches FNRS.

Bohlen et al. [5] in a9Be(15N,12N)12Be experiment.These authors observe highly excited states in12Beand find out that their energies are in good alignmentin a J (J + 1) diagram. They could not, however,measure the widths and predict the structure of thesestates; spin assignments are only tentative.

A systematic study of Be isotopes by von Oertzen[7] predicts the existence of rotational bands present-ing a strongα clustering. Theα clustering is wellknown in the ground states of7Be and8Be and mayalso show up in excited states of heavier isotopes. Theantisymmetrized molecular dynamics (AMD) model[8] also suggests molecularlike states in light nuclei,and particularly in Be neutron-rich isotopes. On theother hand, the existence of6He + 6He molecularstates has been considered by Ito and Sakuragi [9] in asemi-microscopic coupled-channel calculation. Theseauthors derive an6He+ 6He interaction from foldingdensities and analyze the structure of12Be resonances;the α + 8He configuration, open at the threshold, ishowever not considered in that work.

In the present Letter, we investigate cluster states of12Be in the Generator Coordinate Method (GCM) [10].In this microscopic method, the 12-nucleon hamil-

0370-2693/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.PII: S0370-2693(01)00349-5

72 P. Descouvemont, D. Baye / Physics Letters B 505 (2001) 71–74

tonian reads

(1)H =12∑

i=1

Ti +12∑

i<j=1

Vij ,

where Ti is the kinetic energy of nucleoni, andVij an effective two-body interaction. Approximatesolutions of the Schrödinger equation associated to (1)are obtained in the cluster formalism, which assumesthat the 12 nucleons are distributed into two clusters.We consider here two clustering modes:6He + 6Heandα+8He; both channels are close to each other (thethresholds are located at 8.95 MeV for6He+ 6He and10.11 MeV forα + 8He). The GCM wave functions inpartial waveJπ are, therefore, defined as

Ψ JMπ =Aφ4φ8YMJ (ρ̂4−8)g

Jπ4−8(ρ4−8)

(2)+Aφ6φ6YMJ (ρ̂6−6)g

Jπ6−6(ρ6−6),

whereφn corresponds to the wave function ofnHe,gJπi−j to the radial function depending on the rela-

tive coordinateρi−j . Here and in the following, sub-scripts 4–8 and 6–6 refer to quantities related to theα + 8He and6He+ 6He configurations, respectively.In (2), A is the 12-nucleon antisymmetrizer whichensures the Pauli principle to be exactly taken intoaccount. The internal wave functionsφn are built inthe one-center harmonic-oscillator model with a fulls shell forα and(p 3

2)2 and(p 3

2)4 configurations for

6He and8He, respectively. It is well known that6Hepresents a halo structure and that accurate wave func-tions should be more extended than one-center shell-model wave functions. However, recent microscopicinvestigations using shell-model [11] or more elabo-rated 3-cluster functions [12] show thatα + 6He scat-tering properties at low energies are weakly dependenton the6He description.

The present model has been already applied to theα + 8He elastic scattering [11]. In Ref. [11], we ana-lyzed GCMα + 8He phase shifts and derived equiv-alent nucleus–nucleus potentials. From that work, wehave suggested the existence of 0+ and 0− molecu-lar bands, whose bandheads are located close to theα + 8He threshold. To investigate the possible exis-tence of 6He + 6He molecular states suggested byFreer et al. [4], we extend here this previous calcu-lation by including the6He+ 6He channel which, ofcourse, must be taken into account for a reliable in-vestigation of6He+ 6He states. Since this channel is

symmetric and involves zero-spin nuclei, it only af-fects positive-parity states. No change is expected forthe conclusions on negative-parity resonances.

A special attention must be paid to the choice ofthe nucleon–nucleon force. With standard interactionssuch as the Volkov [13] or Minnesota [14] poten-tials, the6He+ 6He threshold is overestimated by afew MeV. This problem arises from the shell-modeldescription of6He. Since6He+ 6He calculations in-volving 3-cluster wave functions of6He are currentlynot possible, it is necessary to compensate this draw-back by adapting the nucleon–nucleon force. As itwas done in the past for similar studies (see, for ex-ample, Ref. [15]), we use the VolkovV 2 force, withadditional Bartlett and Heisenberg terms. This intro-duces some flexibility and enables us to reproduce the6He + 6He andα + 8He thresholds simultaneously.The central force is defined by parametersw = 0.55,m = 0.45, b = −h = −0.2374, and is complementedby a zero-range spin–orbit force [16] with amplitudeS0 = 30 MeV fm5. As in Ref. [11], the oscillator pa-rameter is chosen asb = 1.65 fm forα, 6He and8He.The radial functionsgi−j (ρ) are expanded over a setof shifted gaussian functions which allows one to write(2) as a linear combination of projected Slater deter-minants. The calculation of resonance properties (en-ergy, partial widths) is performed in the microscopicR-matrix formalism [17].

In Fig. 1, we present the energy spectrum obtainedin 3 different conditions: single-channelα + 8He or6He + 6He, and the two-channel calculations. Hereand in the following, energies are expressed withrespect to theα + 8He threshold. In spite of differentclustering assumptions, the three spectra are fairlysimilar to each other for positive-parity states. Thiscan be understood from the energy curves displayedin Fig. 2. Energy curves give the energy of the systemfor a fixed distance between the clusters (see Ref. [16]for details). The 0+ and 2+ energy curves presenta minimum at small distance, which corresponds toshell-model states. The r.m.s. radii of the low-lying0+ and 2+ states are about 2.5 fm (see Table 1),characteristic of mass-12 nuclei; beyondJπ = 2+,the energy curves only present a shallow minimumnear 3 fm. A striking feature of Fig. 2 is that bothconfigurations give very close energies, the differencebeing essentially due to the threshold. This meansthat both bases should describe the same12Be states

P. Descouvemont, D. Baye / Physics Letters B 505 (2001) 71–74 73

Fig. 1. 12Be spectra with different conditions of calculation:single-channelα + 8He or 6He + 6He, and two-channel. E2transition probabilities (in WU) are given for the two-channelcalculation.

Fig. 2. Energy curves of theα + 8He (full curves) and6He+ 6He(dashed curves) systems.R is the distance between the clusters.

and that states with a strong6He + 6He clusteringare unlikely. Each spectrum of Fig. 1 shows a 0+rotational band. Such a band was already predictedin the single-channel calculation of Ref. [11], and nofurther band is found in the present model. We confirmthe existence of a negative-parity band starting neartheα + 8He threshold. The existence of such a band iswell established in10Be [6], and confirmed by similarcalculations [11,12].

Table 1Energies (in MeV), r.m.s. radii (in fm), dimensionless reducedwidths (in %, at 6 fm) and partial widths (in MeV) of12Be states

Jπ Ecm√(r2) θ2

4−8 θ26−6 Γ4−8 Γ6−6

0+ −8.70 2.55 1.3 0.4 0 0

2+ −5.97 2.52 1.3 0.4 0 0

0+ −1.85 2.97 19.1 9.4 0 0

2+ −0.26 3.03 21.8 9.7 0 0

4+ 3.03 3.50 18.7 12.0 0.26 0.02

6+ 11.5 17.6 13.4 0.81 0.55

1− −0.34 3.06 27.3 – 0 –

3− 1.30 2.92 18.0 – 0.03 –

5− 3.62 2.58 3.10 – 0.01 –

In Table 1, we gather some spectroscopic propertiesof 12Be: r.m.s. radii, dimensionless reduced widthsθ2,and partial widths for resonances. The 0+ and 2+ low-lying states presentθ2 values of the order of 1%, char-acteristic of weakly-deformed states. On the contrary,the states belonging to the molecular bands presentradii of about 3 fm, andθ2 values larger than 10%. Inpositive-parity, the strong mixing between theα+8Heand6He+ 6He channels is confirmed by the analysisof the partial widths. The reduced widths in both chan-nels are comparable, and it is not possible to assign adefiniteα+8He or6He+6He cluster structure to thesestates. The 0+ and 2+ states are predicted to be bound.ForJπ = 4+, the difference of theQ values makes the6He+6He partial width significantly lower than the to-tal width (Γ6−6/Γ = 0.07), in spite of similar reducedwidths. The 6+ member is suggested to have similarwidths in both channels.

In Fig. 1, we also give E2 reduced transition prob-abilities for the two-channel calculation. TheB(E2)value between the 2+1 and 0+1 states is 6.6 WU,whereas transition probabilities between molecularstates are strongly enhanced. This effect is well known,and arises from a similar deformed structure of thestates. Strong E2 transitions are a possible way to ob-serve a molecular band; this technique has been usedin the past for investigating12C + 12C molecular res-onances [18]. Notice that low transition probabilitiesare expected between states of different bands. This is

74 P. Descouvemont, D. Baye / Physics Letters B 505 (2001) 71–74

Fig. 3.12Be states predicted by the GCM (the width is indicated bya vertical bar) and experimental data of Refs. [4,5].

exemplified here with theB(E2) value between the 2+molecular state and the ground state (1.4 WU).

The present results are summed up in theJ (J + 1)diagram of Fig. 3. We plot the GCM states withthe experimental data of Freer et al. [4] and ofBohlen et al. [5]. As discussed previously, the GCMpositive-parity states can not be assigned to a definite6He + 6He structure, but to a mixing of6He + 6Heandα + 8He configurations. In their breakup exper-iment, Freer et al. [4] observe 4+, 6+ and 8+ statesin the 6He + 6He andα + 8He channels with rathersimilar energies. The difference being lower than theenergy resolution, the configuration mixing predictedby the GCM seems to be supported by the data. Theexperiment of Bohlen et al. [5] concludes that the ob-served12Be states are deformed, but can not separatetheα + 8He and6He+ 6He channels. ForJ � 6, thetheoretical energies deviate form a rotational behav-iour (for J = 8+ the GCM provides a 10 MeV broadresonance near 30 MeV). This can be explained fromtwo reasons:

(i) for largeJ values, energies are above several openchannels which are not included;

(ii) the halo structure of6He could play a more impor-tant role in high-spin states. Further calculationsare therefore necessary to improve the theoreticaldescription of these states.

In summary, we have performed a microscopic cal-culation of 12Be excited states, withα + 8He and6He+ 6He cluster structures. Our results are consis-tent with recent experiments suggesting a molecular

band in the12Be spectrum. We find that both config-urations are nearly equivalent and, consequently, thatmolecular states can not be considered as6He+ 6Hestates, but as a mixing of6He+ 6He andα + 8He con-figurations. As for10Be [11,12], the GCM suggestsa negative-parity band involving narrow resonanceswith a pureα + 8He structure, and not observed yet.The head of this band is found near theα+8He thresh-old.

Acknowledgement

This Letter presents research results of the BelgianProgram P4/18 on interuniversity attraction polesinitiated by the Belgian-state Federal Services forScientific Technical and Cultural Affairs.

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