Microscopic investigation of the 12Be spectroscopy

Nuclear Physics A 836 (2010) 242–255

www.elsevier.com/locate/nuclphysa

Microscopic investigation of the 12Be spectroscopy

M. Dufour a, P. Descouvemont b,∗,1, F. Nowacki a

a IPHC Bat27, IN2P3-CNRS/Université de Strasbourg, BP28, F-67037 Strasbourg Cedex 2, Franceb Physique Nucléaire Théorique et Physique Mathématique, C.P. 229, Université Libre de Bruxelles (ULB),

B 1050 Brussels, Belgium

Received 19 October 2009; received in revised form 1 February 2010; accepted 2 February 2010

Available online 11 February 2010

Abstract

We analyze the 12Be spectrum with two microscopic methods. Microscopic cluster calculations are per-formed within the Generator Coordinate Method (GCM) framework. An extended two-cluster model whichincludes the 8He(0+,2+) + α, and 6He(0+,2+) + 6He(0+,2+) channels is developed. It represents anextension of a previous work by Descouvemont and Baye where only the 0+ ground states of 8He and 6Hewere taken into account. The GCM calculations lead to a good description of the molecular band suggestedby Freer et al. and by Saito et al. The 0+

2 isomeric state and the 2+2 state are also well described by the GCM.

New bands of positive and negative parities are proposed. This approach is complemented by No-Core ShellModel (NCSM) calculations, in order to clarify the structure of the ground state and the hypothesis of anassociated rotational band. The GCM and NCSM calculations do not support the existence of a band builton the ground state.© 2010 Elsevier B.V. All rights reserved.

Keywords: 12Be spectroscopy; Molecular states; GCM; NCSM

1. Introduction

The description of light exotic nuclei such as 12Be is an interesting challenge for experimentsas well as for theoretical calculations. In spite of significant progresses of experimental tech-niques, the understanding of the 12Be nucleus remains incomplete (see, for example, Refs. [1–5]

* Corresponding author.E-mail addresses: [email protected] (M. Dufour), [email protected] (P. Descouvemont),

[email protected] (F. Nowacki).1 Directeur de Recherches FNRS.

0375-9474/$ – see front matter © 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.nuclphysa.2010.02.002

M. Dufour et al. / Nuclear Physics A 836 (2010) 242–255 243

Fig. 1. (Color online.) 12Be experimental spectrum (only states with a definite spin/parity assignment are shown). Thresh-olds are indicated by dotted lines. Experimental data are from Refs. [3,10,6,7]. The 6+ states represented here correspondto states observed in the 6He + 6He channel [3].

and references therein). In particular, several experimental states do not have a definite spin/parityassignment. Neutron and gamma widths are in general not measured.

In Fig. 1, we display experimental 12Be states with a definite spin/parity assignment. Belowthe 11Be + n threshold located at 3.17 MeV, four states are known: the 0+

1 ground state, the2+

1 state at 2.10 MeV, the 0+2 state at 2.24 MeV [6], and the 1−

1 state at 2.68 MeV [7]. The 0+2

level is believed to be an isomeric state [6]. Above the 6He + 6He threshold, 4+, 6+ and 8+states have been identified in the breakup of 12Be into the 6He + 6He and 8He + α channels byFreer et al. [3]. They are believed to be members of a molecular band. The existence of thesestates is also supported by Bohlen et al. [8], but some of them have not been confirmed in arecent experiment by Charity et al. [9]. The 0+ and 2+ members have been identified later in the6He + 6He channel, at 10.9 MeV and 11.3 MeV, respectively, by Saito et al. [10].

In the shell-model approach [11], the structure of the ground state and of the low-lying statesis linked to the understanding of the breakdown of the N = 8 shell closure. In order to explainthe β-decay half life of the ground state, Barker pointed out as early as in 1976 the necessityto introduce non-p-shell configurations in the wave function [12]. Since then, several experi-ments and theoretical works have discussed and confirmed this interpretation (see, for example,Refs. [4,6,7,13] and references therein).

Studies with sophisticated models such as microscopic cluster models [14–16] or the no-coreshell model (NCSM) [17] are necessary in order to investigate such complicated states. Severalcluster models have already been developed in order to analyze the 12Be nucleus, in particularthe antisymmetrized molecular dynamic (AMD) model of Kanada-En’yo and Horiuchi [15] andthe molecular-like model of Ito et al. [16]. A recent work by Romero-Redondo et al. [18] onlyfocuses on low-lying states with nonmicroscopic three-body wave functions. On the other side,

244 M. Dufour et al. / Nuclear Physics A 836 (2010) 242–255

the NCSM approach has been intensively used in recent years for the study of light nuclei (seeRef. [19] and references therein).

In the present work, we start with an analysis of the low-lying 0+ and 2+ states, as well as ofthe first 4+ level. Then we extend the calculation to other 12Be states, in particular to molecularstates. For the GCM [20], we have developed a two-cluster model which allows the inclusion ofthe 8He(0+,2+) + α, and 6He(0+,2+) + 6He(0+,2+) channels simultaneously. It generalizesa previous GCM study [14] where only channels involving the 6He and 8He ground states wereincluded. Recently, the GCM wave functions have been significantly improved [21,22], in partic-ular by increasing the number of channels in the variational basis. The addition of new channelsinvolving the 2+ excited states of 6He and 8He is expected to improve the description of the 12Benucleus.

To complement the GCM, NCSM calculations are performed with the code ANTOINE de-veloped in Strasbourg by Caurier et al. [23]. The NCSM method is known to be a powerful tool,expected to give accurate wave functions. To our knowledge, it is the first NCSM investigationof the 12Be. Our purpose is to clarify the structure of the ground state and to discuss the possibleexistence of a rotational ground-state band, as proposed in the literature [15,16]. Such a hypoth-esis could be justified only if a breaking of the N = 8 shell closure is confirmed. In that case, the0+ and 2+ wave functions would be clearly dominated by non-p-shell configurations.

The present paper is organized as follows. In Section 2, we briefly present the theoreticalframework of the GCM and of the NCSM calculations. Section 3 is devoted to the results ob-tained with both approaches. More precisely, we review the results obtained for the 0+, 2+, and4+ states, and the positive- and negative-parity bands obtained with the GCM. Summary andconcluding remarks are presented in Section 4.

2. Theoretical framework

2.1. General presentation

The GCM and the NCSM are two different methods which aim at solving the A-nucleonSchrödinger equation. In both cases, the microscopic Hamiltonian of the system is written as

H =A∑

i

Ti +A∑

i>j=1

Vij +A∑

i>j>k=1

Vijk + · · · , (1)

where Ti is the kinetic energy of nucleon i, Vij is a two-body nucleon–nucleon (NN) interaction,Vijk is a three-body nucleon–nucleon (3N) interaction. In the present work, the interaction islimited to two-body terms.

The GCM is well adapted to the description of cluster states, in particular for light nuclei. Itcan be used in order to calculate the properties of bound states and resonances. Information onthe wave functions in terms of reaction channels can be deduced and compared to experiment.However, the Hamiltonian involves an effective interaction whose parameters are fitted on someproperties of the considered nuclear system. Furthermore, the choice of the channels involved inthe GCM variational basis restricts the model to states which present this structure.

In the NCSM, very large Slater-determinant harmonic-oscillator (HO) bases and effective in-teractions are considered [17]. These effective interactions are derived from realistic forces andadapted for finite model spaces through a particular unitary transformation. Wave functions arethen expected to be accurate, but states presenting a strong clustering remain difficult to describe


with this model. Indeed, in spite of considerable advances in computer facilities, the calculationsremain limited by the size of the model space. The necessity to introduce a 3N force or more(4N , . . . ) is now established in order to get highly accurate spectra [19]. However, genuine ex-pressions of these potentials remain under study [19]. Another drawback of the NCSM approachis to treat all the solutions of the Schrödinger equation as bound states. In particular, particlewidths cannot be directly calculated. The GCM and NCSM approaches are then complementaryand interesting to compare with each other.

In the present work, we also discuss the existence of rotational bands. This classification maybe somewhat ambiguous as the structure of different states may be complicated in microscopicmodels. In the following, we consider that states belong to a same band if they are aligned ina J (J + 1) diagram, if the electromagnetic transitions are strong compared to transitions withother states, and if the intrinsic quadrupole moments Q0 are similar. For an E2 transition betweenstates Ji and Jf , Q0 is linked to the B(E2) transition probability through

B(E2, Ji → Jf ) = 5

16π

∣∣〈JiK20|Jf K〉∣∣2Q2

0, (2)

where K is the projection over the intrinsic axis [24].

2.2. Conditions of the GCM calculations

The two-body part of the Hamiltonian is an effective interaction defined as

Vij = V EVIij + V SO

ij + V Coulij , (3)

where V EVIij , V SO

ij and V Coulij are the nuclear [25], spin–orbit [26] and Coulomb two-body inter-

actions, respectively. With the amplitude S0 of the spin–orbit force, and the two free parametersof the V EVI

ij interaction (B = −H and W +M = 1, see Ref. [25]), the Hamiltonian involves three

parameters which are determined in order to optimize the 12Be description.The 12Be wave functions are expressed as sums of antisymmetrized basis functions involv-

ing two clusters. A multi-channel wave function in partial JMπ is given by a superpositionof 8He + α and 6He + 6He components, each channel involving various excited states. Moreprecisely, the wave function is given by

Ψ JMπ12Be =

∑

i

Ψ JMπ8He(i)+α

+∑

j,k

Ψ JMπ6He(j)+6He(k)

, (4)

where the sums run over 0+ and 2+ internal states of 6He and 8He. The internal wave functionsare defined in the harmonic-oscillator model with a common oscillator parameter b. In a two-cluster theory, each component of Eq. (4) is written as

Ψ JMπc =

∑

�I

A[Y�(Ωρ) ⊗ [

φI1π11 ⊗ φ

I2π22

]I ]JMgJπ

c�I (ρ), (5)

where c refers to the partition and to the excitation level of the colliding nuclei, (I1, I2) arethe corresponding spins, I is the channel spin, and � the relative angular momentum. In thisequation, φ

I1π11 and φ

I2π22 are the internal wave functions of the clusters, gJπ

c�I (ρ) is the radialfunction depending on the relative coordinate ρ, and A is the A-nucleon antisymmetrizer. In the


GCM, the radial functions are expanded over Gaussian functions which makes Eq. (5) equiva-lent to a combination of projected Slater determinants, well adapted to numerical calculations.The Gaussian functions are centered at different locations called the generator coordinates. Thewave functions of bound states and of resonances are determined with the R-matrix method[27–29]. This method ensures the exact asymptotic behavior of the wave functions. It is basedon the separation of the configuration space into two regions: the internal region (of radius a)where the nuclear interaction and the antisymmetrization between the clusters are properly takeninto account, and the external region where they are negligible. For relative distances ρ � a, thewave function (5) is described by the GCM, whereas the external definition of gJπ

c�I (ρ) involvesCoulomb wave functions. A precise treatment of asymptotic boundary conditions is crucial inorder to describe accurately resonances and states with a strong clustering. Further detail con-cerning the GCM formalism and the R-matrix method can be found in Refs. [28,22,29] andreferences therein.

In the GCM, the basis states are not orthogonal to each other. This prevents a straightforwardanalysis of the wave function in terms of components as in the shell model framework. Neverthe-less, an equivalent information can be deduced from the study of dimensionless reduced widths,defined as

θ2c�I = γ 2

c�I

γ 2W

= a3

3

∣∣gJπc�I (a)

∣∣2, (6)

where γc�I is the reduced width amplitude in channel c�I , and is proportional to the wave func-tion at the channel radius a. In this expression, γ 2

W = 3h2/2μa2 is the Wigner limit [30]. A largevalue of θ2

c�I indicates a channel important for the description of the wave function.In the present model, (I1, I2) are equal to the 0+ ground states and 2+ excited states of 6He

and 8He. The 0+ internal wave functions of 6He and 8He are built in a one-center harmonic-oscillator model with (p3/2)

2 and (p3/2)4 configurations, respectively and the 2+ states, with

(p3/2)(p1/2) and (p3/2)3(p1/2)

1 configurations, respectively. The choice of harmonic-oscillatororbitals is of course an approximation which does not fully account for the halo structure of 6Heand 8He. This approximation is however necessary for two-cluster calculations, and is expectedto be suitable for the 12Be spectroscopy (see Ref. [14]).

The channel radius is equal to a = 9.0 fm. We use a set of generator coordinates rangingfrom 2.7 fm to 9.0 fm with a step of 0.9 fm. The harmonic-oscillator parameter common toall clusters is chosen as b = 1.65 fm. For the nucleon–nucleon interaction, two parameter setshave been selected. Both sets reproduce exactly the experimental energy difference between the6He + 6He and 8He + α thresholds (1.16 MeV). Set I (S0 = 37.2 MeV fm5, M = 0.43, B =−0.2722) reproduces the binding energy of the 0+

1 ground state, and set II (S0 = 22.5 MeV fm5,M = 0.43, B = −0.2416) the experimental energy of the 0+ molecular band-head [10]. Set I isused in the calculation of the 0+, 2+, and 4+ states. As in Ref. [14], the same set is used to studynegative-parity states. Set II being more adapted to cluster states, we keep it for all the otherpositive-parity states. Throughout the paper, the notation E4−8 corresponds to the c.m. energyexpressed with respect to the 8He + α threshold.

Some properties (energies and r.m.s. radii) of 4,6,8He are given in Table 1. As expectedfrom simple shell-model wave functions, the binding energies of 6He and 8He are rather farfrom experiment. However the important quantities in cluster models are the difference betweenthreshold energies. As mentioned above, these differences are reproduced by an appropriatechoice of the effective nucleon–nucleon interaction.


Table 1Absolute binding energies EB , excitation energies Ex (in MeV), and r.m.s. matter radii (in fm) of 4,6,8He in the presentshell model.

Set I Set II Exp. [31–34]

EB(α) −24.95 −28.30EB(6He,0+) −17.65 −17.72 −29.27Ex(6He,2+) 3.23 3.41 1.80EB(8He,0+) −11.51 −11.66 −31.41Ex(8He,2+) 3.62 1.53 3.1√

r2(α) 1.75 1.63 ± 0.03√r2(6He) 2.08 2.33 ± 0.04√r2(8He) 2.22 2.49 ± 0.04

Fig. 2. (Color online.) Schematic representation of the NCSM model spaces. Major shells involved in the calculation upto Nmax = 8 are labeled from (1) to (10). 2hΩ excitations are shown explicitly. A typical 8hΩ excitation is suggestedby the arrow to indicate the limit of the calculations.

2.3. Conditions of the NCSM calculations

A detailed description of the NCSM approach as it is implemented in this work can be foundin Refs. [17,35,23]. Let us just remind here the main features. In an NCSM calculation, themany-body problem is replaced by a diagonalization in a finite model space whose dimension isexpected sufficiently large to give converged results. The size of the HO basis, and consequentlythe dimension of the matrix, is controlled by the parameter Nmax. It represents the maximum HOquanta of the many-body excitation above the lowest model space (Nmax = 0) defined from themajor HO shells involved in the description of the ground state in a simple shell-model picture[11]. For example, in the case of 12Be nucleus, the Nmax = 0 model space corresponds to the 0s

and 1p major shells where the 0s shell is filled for neutrons and protons. The Nmax = 2 modelspace is obtained by including to the Nmax = 0 model space all the states that can be reached bytwo quanta of HO excitation (2hΩ), where Ω corresponds to the frequency of the HO potential.Excitations towards the sd and fp major shells are then taken into account. Larger Nmax modelspaces are obtained in the same way. A schematic representation of the NCSM model spaces isgiven in Fig. 2.

The frequency Ω is a crucial parameter of an NCSM calculation. Indeed, the method devel-oped to define the effective Hamiltonian in the NCSM framework leads to an explicit dependence


on the choice of hΩ [17]. It can be shown that the NCSM method becomes independent of theHO frequency and is convergent to the exact result with increasing model-space size. In practice,series of calculations are performed for different frequencies to test the convergence.

The calculations are performed up to Nmax = 8 with two different high-precision realistic NN

interactions: the CD-Bonn potential [36] and the ‘Inside Nonlocal Outside Yukawa’ (INOY) [37]potential. Both interactions represent recent developments in the theory of nuclear forces. Ac-cording to Ref. [38], they have quite different and complementary behaviors, and the sensitivityof 12Be energies should be tested.

The CD-Bonn interaction is nonlocal in momentum space. It is adjusted to reproduce theNN -scattering data with high precision. It has been previously applied in the NCSM approachto study p-shell nuclei such as, for example, the 12C nucleus [35]. We follow the argumentationof Ref. [35] to choose the optimal HO frequency as hΩ = 15 MeV.

The nonlocal INOY potential respects the local behavior of the usual NN interactions at largedistances but presents a nonlocality at short distances. It is also able to reproduce the binding en-ergy of 3H and 3He nuclei. This interesting property is expected to simulate a 3N force behavior.The HO frequency is chosen here as hΩ = 18 MeV according to Ref. [38].

To solve numerically the problem, we use the NCSM version of the shell-model codeANTOINE [23]. This code works in the M-scheme for model spaces involving many majorshells. It uses the Lanczos algorithm for diagonalization. Our largest calculations involve matrixdimension of ≈ 1.4 × 108.

3. Results

3.1. The 0+1 , 2+

1 , and 4+1 states

We focus here on the energies and wave functions of the 0+, 2+, and 4+ states. GCMcalculations are performed with set I which reproduces the 0+

1 binding energy. In these con-ditions, the energy difference between the 0+

1 and 2+1 is of the same order of magnitude as

compared to the NCSM calculations (≈ 3.4 MeV, see Fig. 3). By analyzing the reduced widths,we find that the wave functions are very similar. This is confirmed by the large value of theB(E2,2+

1 → 0+1 ) = 12.6 e2 fm4. This result is comparable to the AMD calculation of Kanada-

En’yo and Horiuchi [15] (14 e2 fm4). Several 4+ states are obtained with the GCM. Neverthelesssmall B(E2) values indicate that none of them can be identified as a member of a possibleground-state rotational band.

The GCM study is complemented by NCSM results. Fig. 3 shows the NCSM spectra obtainedwith the CD-Bonn potential (full lines) and the INOY potential (dotted lines) for different model-space sizes. Table 2 gives the total energies and an analysis of the corresponding wave-functioncomponents [23]. These components, as well as the excitation energies of the 2+ and 4+ states,weakly depend on the interaction.

As shown in Fig. 3, the 2+1 excitation energy is clearly converged with both interactions. The

NCSM calculation overestimates the experimental value by about 1 MeV. A fair convergence ofthe absolute energies (see Table 2) is also obtained with the INOY interaction. The convergenceis less good with the CD-Bonn force. The corresponding wave functions of the 0+

1 and 2+1 states

are very similar. They are characterized by dominant 0hΩ components (≈ 60%). It is interestingto notice that the other contributions, mainly due to 2hΩ and 4hΩ excitations, are of the sameorder of magnitude (≈ 15%).


Fig. 3. (Color online.) Energies of the 0+ , 2+ and 4+ states (with respect to the 0+ energy, in MeV). For the NCSMcalculations: full lines correspond to the energies obtained with the CD-Bonn potential and dotted lines with the INOYpotential for different values of Nmax.

The evolution of the 4+1 energy as a function of Nmax presents the typical behavior of a

state dominated by higher hΩ excitations. Of course the theoretical properties of this level arequalitative only since it is treated at the bound-state approximation in the NCSM. In order toestimate its converged energy, we use an extrapolation method proposed by Forssén et al. [38].By assuming an exponential dependence on Nmax, we find Ex = 15.21 MeV for CD-Bonn andEx = 13.75 MeV for INOY. Consequently, this state cannot be assigned to a possible 4+ levelat Ex = 5.70 MeV [1]. For Nmax = 8, the wave function is dominated by 2hΩ (≈ 60%) compo-nents. Keeping in mind that a 4+ state cannot be generated by s, p shells, we can conclude thatNCSM calculations do not suggest a band build on the ground state, as suggested in Refs. [15,16].

This result is also supported by B(E2) calculations (obtained without effective charge).We find B(E2,2+

1 → 0+1 ) = 4.6 e2 fm4 for CD-Bonn and 3.5 e2 fm4 for INOY, and B(E2,

4+1 → 2+

1 ) = 0.04 e2 fm4 for both potentials. These values are smaller than in the GCM. Al-though Nmax = 8 does not provide full convergence (for Nmax = 6, the 2+

1 → 0+1 transitions

probabilities are 4.2 and 3.1 e2 fm4), the strong difference between the 2+ and 4+ wave functionsis confirmed. According to Ref. [23], the NCSM values are probably underestimated. A measure-ment of transition probabilities in 12Be would be helpful to clarify the situation.

In shell-model approaches the breakdown of the N = 8 shell closure is traditionally analyzedfrom the 0hΩ component [12,13,7,6,4]. Our result (≈ 60%) is larger than the value of Navinet al. [13] who find 32% from an analysis of 11Be + n spectroscopic factors. However the inter-pretation of Navin et al. is derived from traditional shell-model calculations based on a Warburtonand Brown effective interaction [39]. This force is especially adapted to a valence space muchsmaller than in the present NCSM. The conditions of calculation are therefore very different.


Table 2Energies (in MeV) and components of the 0+

1 ,2+1 ,4+

1 wave functions with the CD-Bonn and INOY potentials.

CD-Bonn INOY

Nmax = 2

Jπ E 0hΩ 2hΩ E 0hΩ 2hΩ

0+1 −67.9 0.71 0.29 −70.8 0.76 0.24

2+1 −64.6 0.71 0.29 −67.7 0.76 0.24

4+1 −35.3 0 1 −36.4 0 1

Nmax = 4

Jπ E 0hΩ 2hΩ 4hΩ E 0hΩ 2hΩ 4hΩ

0+1 −64.1 0.66 0.19 0.15 −67.3 0.71 0.15 0.14

2+1 −60.9 0.66 0.18 0.16 −64.3 0.71 0.145 0.145

4+1 −35.6 0 0.75 0.25 −37.1 0 0.77 0.23

Nmax = 6

Jπ E 0hΩ 2hΩ 4hΩ 6hΩ E 0hΩ 2hΩ 4hΩ 6hΩ

0+1 −61.4 0.61 0.18 0.13 0.08 −65.9 0.66 0.16 0.11 0.07

2+1 −58.0 0.62 0.17 0.13 0.08 −62.6 0.67 0.15 0.11 0.07

4+1 −38.5 0 0.66 0.19 0.15 −42.5 0 0.68 0.19 0.13

Nmax = 8

Jπ E 0hΩ 2hΩ 4hΩ 6hΩ 8hΩ E 0hΩ 2hΩ 4hΩ 6hΩ 8hΩ

0+1 −59.5 0.59 0.16 0.13 0.07 0.05 −65.8 0.61 0.16 0.13 0.06 0.04

2+1 −56.2 0.59 0.15 0.14 0.06 0.06 −62.5 0.63 0.14 0.13 0.06 0.04

4+1 −39.9 0 0.59 0.20 0.13 0.08 −46.5 0 0.60 0.20 0.13 0.07

Consequently, even is the breaking closure is weaker in our calculation, a direct comparison withsuch approach cannot be done straightforwardly.

In conclusion, the present GCM and NCSM calculations do not support the hypothesis of arotational ground-state band as proposed in Refs. [15,16].

3.2. Other positive-parity states of 12Be

With the GCM, we obtain states which can be interpreted as members of the molecular bandof Freer et al. [3] and of Saito et al. [10] (see Fig. 4). Widths, B(E2) and intrinsic quadrupolemoments are given in Table 3. We also present the dimensionless reduced widths (see Eq. (6)) ata = 9 fm. We remind that the 0+ head-band energy is exactly reproduced by the interaction.

The analysis of the dimensionless reduced widths shows that the 0+ wave function is dom-inated by the 6He(0+) + 6He(0+) channel. The theoretical 2+ and 4+ energies are in goodagreement with the results of Saito et al. [10] and of Freer et al. [3], respectively. The wave func-tions are dominated by the 6He(0+) + 6He(0+) and 8He(0+) + α ground-state configurations.We also find a 6+ state with a dominant 6He(0+) + 6He(0+) structure. Our calculation thereforesupports the existence of a molecular band, as proposed in the experiments of Freer et al. [3]and of Saito et al. [10]. The large reduced widths support the molecular structure of this band.In addition, we confirm the existence of a 6+ state which has not been observed in the recentexperiment of Charity et al. [9].


Fig. 4. (Color online.) Positive-parity 12Be states predicted by the GCM (full symbols) and experimental candidates(open symbols [3,10,6]).

Table 312Be energies (in MeV), total widths in (MeV), dimensionless reduced widths (6) (in %, at a = 9 fm), E2 reducedtransition probabilities (J → J − 2, in e2 fm4), and intrinsic quadrupole moments Q0 (in e fm2). Experimental data aretaken from Refs. [3,10,6].

Jπ Exp. GCM

Ex E4−8 E4−8 Γ θ24−8 θ2

6−6 B(E2) Q0

Molecular band0+ 10.9 1.95 1.95 0.365 2.2 36.92+ 11.3 2.35 3.05 0.760 13.5 13.6 27.4 37.14+ 13.2 4.15 4.96 0.98 19.4 11.6 50.3 42.16+ 16.1 7.05 9.17 0.16 0.5 2.3 77.9 49.9

K = 0+2 band

0+ 2.24 −6.71 −6.38 0 3.4 × 10−2 6.2 × 10−3

2+ 4.56 −4.39 −4.81 0 4.4 × 10−2 7.7 × 10−3 19.1 31.04+ −1.85 0 4.2 × 10−2 8.4 × 10−3 23.6 28.86+ 2.26 7.8 × 10−4 5.0 × 10−1 1.0 × 10−2 29.7 30.8

Our calculations extend the previous GCM investigation of Descouvemont and Baye [14] byintroducing excited configurations. Let us remark that the 0+ and 2+ states were not knownexperimentally when Ref. [14] was published. We also confirm that molecular states can beconsidered as a mixing of 6He + 6He and 8He + α configurations, and that the mixing dependson J .

The 0+2 and 2+

2 states are well reproduced by the GCM. Indeed, the energy difference withexperiment is less than 0.5 MeV for both states (see Fig. 4 and Table 3). According to Refs. [15,16,5], we confirm that these states belong to a same 0+

2 band. Contrary to the works of Kanada-En’yo and Horiuchi [15] and of Ito et al. [16], we propose other band members. However, we donot confirm the assignment of the experimental peaks at 9.6 MeV and 18.95 MeV as the 4+ and6+ members of the 0+

2 band suggested in Ref. [5]. The calculation shows that the wave functionsof the 0+

2 -band members are dominated by the 8He + α channels.The B(E2) value for the 0+

2 → 2+1 transition gives 1.2 e2 fm4, lower than the experimental

value of 7.0±0.6 e2 fm4 [6]. This result is obtained without effective charge and is very sensitiveto the details of the 0+ and 2+ wave functions. Let us remark that all the results concerning the
2 1


Table 4Theoretical E2 reduced transition probabilities (in e2 fm4)between states of the K = 2+ positive-parity band.

Jπii

Jπf

fB(E2)

3+ 2+ 41.54+ 3+ 17.45+ 4+ 12.74+ 2+ 12.35+ 3+ 21.6

Fig. 5. (Color online.) Negative-parity 12Be states predicted by the GCM. Full circles and full diamonds correspond toGCM calculations. Crosses are experimental data from Ref. [5].

0+2 isomeric state and the associated band are new as compared to the previous GCM work of

Descouvemont and Baye [14].We propose a new K = 2+ band with non-natural-parity states (see Fig. 4). The states have in

common to follow a straight line in the J (J +1) representation and to present large E2 transitionprobabilities (see Table 4). The wave functions have a dominant structure in the 8He+α channels.

3.3. Negative-parity states of 12Be

The present GCM calculations confirm the negative-parity K = 0−1 band found in the previous

work of Descouvemont and Baye [14] near the 8He + α threshold (see Fig. 5). B(E2) are givenin Table 5. We remind here that the 6He(0+) + 6He(0+) channel is forbidden in negative-paritystates.

In addition, a new negative-parity K = 1− band, which could correspond to the tentatively as-signed band seen in a three-neutron stripping reaction on 9Be [5], is obtained by the present GCMcalculations (see Fig. 5). As expected, the wave functions have a dominant 8He + α structure.B(E2) are given in Table 5.

4. Summary and conclusion

In this work, we have investigated the 12Be spectrum with two complementary microscopicmethods. We have first performed GCM calculations with an extended two-cluster model whichincludes the 8He(0+,2+) + α, and 6He(0+,2+) + 6He(0+,2+) channels. It represents an ex-


Table 5Theoretical E2 reduced transition probabilities in (e2 fm4) between states of the K = 0−

1and K = 1− bands.

Jπii

Jπf

fB(E2) Q0

K = 0−1

2− 1− 101 62.85− 3− 98.7 57.2

K = 1−2− 1− 15.92− 3− 10.14− 3− 4.14− 2− 14.65− 3− 17.1

tension of a previous work [14], where only ground-state configurations were considered. Theintroduction of the 2+ states of 8He and 6He confirms the importance of excited channels inthe GCM variational basis. Within this framework, we get a more accurate description of themolecular band of Freer et al. [3] and of Saito et al. [10]. As new results, let us mention thedescription of the 0+

2 isomeric state, the 2+2 state and the proposal of 4+ and 6+ band members.

We also suggest a new K = 2+ band and a negative-parity band K = 1− that could be assignedto experimental states.

In addition, we have applied the NCSM model in order to clarify the structure of the 0+ groundstate and the possible existence of an associated rotational band [15,16]. We find ≈ 60% of 0hΩ

components, larger than the result of Navin et al. [13]. We consider that our calculations haveestablished a baseline of results at the pure two-body realistic-interaction level. The evolution ofthe 12Be spectrum with a 3N term will be very interesting to study in the future.

From a more general point of view, we have performed calculations within two very differentframeworks. Such investigations could also be very interesting for other nuclei. In the case ofthe 12Be nucleus, both approaches lead to the same conclusion concerning the non-existence ofa rotational band built on the ground state. A measurement of the B(E2,2+

1 → 0+1 ) would be

very interesting to compare more precisely the GCM and NCSM approaches. Nevertheless, themicroscopic cluster model appears to be better adapted to describe molecular 12Be resonances.Progresses in the NCSM theory including improvements of the Hamiltonian and larger modelspaces are suitable in order to get a similar predictive power at higher excitation energies.

Acknowledgements

We would like to thank Professor P. Navrátil to provide us with the matrix elements necessaryfor the NCSM calculations and Professor A.P. Zuker for many useful discussions on realisticinteractions. This text presents research results of the IAP program P6/23 initiated by the Belgian-state Federal Services for Scientific, Technical and Cultural Affairs.

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Microscopic investigation of the 12Be spectroscopy