Application of the generalized two-center cluster model to 10Be

A generalized two-center cluster model (GTCM), including various partitions of the valence nucleons around two alpha-cores, is proposed for studies on the exotic cluster structures of Be isotopes. This model is applied to the 10Be = alpha + alpha + n + n system and the adiabatic energy surfaces for alpha-alpha distances are calculated. It is found that this model naturally describes the formation of the molecular orbitals as well as that of asymptotic cluster states dependeing on their relative distance. In the negative-parity state, a new type of the alpha + 6He cluster structure is also predicted.

A generalized two-center cluster model (GTCM), including various partitions of the valence nucleons around two α-cores, is proposed for studies on the exotic cluster structures of Be isotopes. This model is applied to the 10 Be=α + α + n + n system and the adiabatic energy surfaces for α-α distances are calculated. It is found that this model naturally describes the formation of the molecular orbitals as well as that of asymptotic cluster states depending on their relative distance. In the negative-parity state, a new type of the α+ 6 He cluster structure is also predicted. Recent experiments by Freer et al., [1] revealed the existence of the interesting resonant states, which dominantly decay to 6 He g.s. + 6 He g.s. and α+ 8 He g.s. channels in the 10 to 25 MeV excitation energy interval of 12 Be. Similar resonant states, decaying to He-isotopes such as 6 He and 8 He, have also been observed in other Be isotopes of 10 Be [1] and 14 Be [2]. These experiments strongly suggest the existence of exotic cluster states consisting of the respective He-isotope clusters.
Theoretically, Kanada-En'yo and her collaborators [3] have studied a wide range of light nuclei within the antisymmetrized molecular dynamics (AMD), and predicted many kinds of cluster structures in isotope chains of the elements Be, B and C. In particular, much attention has been concentrated on Be isotopes in which a motion of valence neutrons couples to the two-α structure of 8 Be. Based on the molecular orbital model (MOM), Itagaki et al. has extensively studied the low-lying states of Be isotopes to clarify the roles of valence neutrons on a development of clusterization. The molecular orbitals, such as π and σ orbitals associated with the covalent binding of atomic molecules, have been shown to give a good description for the low-lying states of Be-isotopes as emphasized in the AMD studies [4,5].
On the other hand, Descouvemont and Baye [6] have applied a traditional cluster model ( 6 He+ 6 He) + (α+ 8 He), assuming the substructures of 6 He g.s. and 8 He g.s. , to illustrate the resonances observed in 12 Be. However, it is plausible to assume 6 He and 8 He nuclei to be stable and frozen clusters because the energy for twoneutrons separation is quite small in both nuclei. Thus, it is properly considered that the valence neutron of 6 He and 8 He are associated with companion clusters when they approach each other, which is similar to the covalent binding of valence neutrons in the low-lying states.
The purpose of this letter is to study the molecular structures in low-lying and high-lying states in a unified way by taking into account couplings between a valence neutron's motion and a relative motion of the α clusters. To achieve this purpose, we propose a new framework of a generalized two-center cluster model (GTCM) where we can describe atomic-orbital motions of valence neutrons around individual α clusters on the same footing with molecular orbitals, being a single-particle motion around two α-cores. We apply this model to the 10 Be=α+α+n+n and discuss its applicability to studies of the molecular structure in both the low-lying and high-lying states.
The basis functions of GTCM for 10 Be are given as where C J π K m,n (S) means the product of d J π K m · d J π K n . The α-cluster wave function ψ i (α) (i=L, R) is given by the (0s) 4 configuration in the harmonic oscillator (HO) potential with the relative distance-parameter S. The position of an α-cluster is explicitly specified as the left (L) or right (R) side. A single-particle state for valence neutrons around one of α clusters is given by an atomic orbitals, ϕ(i, p n , τ ) with the subscripts of a center i (=L or R), a direction p n (n=x, y, z) of 0p-orbitals and a neutron spin τ (=↑ or ↓). In Eq. (2), the index m(n) is an abbreviation of the atomic orbital (i, p n , τ ). The basis functionΦ π m,n ( 10 Be; S) with the parity π is projected to the eigenstate of the total spin J and its intrinsic angular projection K by the projection operatorP J K . Various linear combinations of Φ J π K m,n (S) can be shown to reproduce not only molecular orbital configurations but also cluster-model states of 4 He+ 6 He and 5 He+ 5 He.
First, we illustrate that the molecular-orbital configuration of (σ + ) 2 can be constructed from a linear combination of the basis function for instance. Since σ + orbital is expressed as ϕ(L, p z , τ ) − ϕ(R, p z , τ ), the wave function of 10 Be with (σ + ) 2 of valence neutrons are written as follows: This expression means that a 10 Be wave function with the molecular orbitals can be described by a linear combination, such as ( 6 He-α)+( 5 He-5 He)+(α-6 He), because the clusters of 5 He(=α+n) and 6 He(=α+2n) are given by respectively. This linear combination can be exactly constructed from basis functions ofΦ π m,n ( 10 Be; S). Next, we show that the basis functions of Eq. (2) can also describe the cluster-model states in which neutron's orbitals have a definite spin around one of α-cores.
Thus, the 3/ 2 − states of 5 He with the 0p 3/2 neutron is writ- Similarly, the 6 He(I π ) clusters with a definite intrinsic spin-parity I π , such as [(0p 3/2 ) 2 ] 0 + , can be constructed from a certain linear combination of m,n A{ψ i (α)·d m ϕ(m) d n ϕ(n)}. Therefore, the wave function of Eq. (1) can describe the clustermodel states of [α ⊗ 6 He(I π )] and [ 5 He(I π1 1 ) ⊗ 5 He(I π2 2 )]. The wave function of 10 Be is finally given by taking a superposition over the relative distance-parameter S and the intrinsic angular projection K as with β ≡ (m, n). The coefficients C J π K β (S) are determined by solving a coupled channel GCM (Generator Coordinate Method) equation [7]: To see the coupling properties, we solve Eq. (5) in a step by step. First, we solve Eq. (5) at a fixed S. Namely, we solve The eigenvalue E J π (S) is a function of the relative distance-parameter S, and then we call the solutions of energies and wave functions "adiabatic energy surfaces" and "adiabatic eigenstates", respectively. The calculated adiabatic energy surfaces for the J π =0 + state are shown by open circles in Fig. 1. Here, for the nucleon-nucleon interaction, we adopted the Volkov No.2 with the Majorana parameter m=0.576 and without the Bartlett and Heisenberg exchanges. Due to the reduction of the majorana parameter, the total binding energy is gained and it becomes easy to see the continuum states. We also employed the G3RS interactions for the spin-orbit parts. The radius parameter b of HO wave functions for α clusters and valence neutrons is taken as 1.44 fm.
At the asymptotic distance (S→∞), where two α-cores are completely separated, we can define the asymptotic channels such as [ 4 He+ 6 He(I)] L and [ 5 He(I 1 )+ 5 He(I 2 )] IL , in which individual clusters have intrinsic spins (I 1 , I 2 ) and coupled with the channel spin I (I=I 1 + I 2 ) and the relative one L. We call the coupling scheme of these asymptotic channels as "a clustercoupling scheme". The solid and dotted curves shown in the right part of Fig. 1 are the expectation values < H > of the [ 4 He+ 6 He(I)] L and [ 5 He(I 1 )+ 5 He(I 2 )] IL clustercoupling schemes, respectively From Fig. 1, we can see that in S ≥ 6 fm, the calculated energy surfaces are completely the same as those of cluster-coupling wave functions. This means that, in this region, the valence neutrons are localized at one of α-cores and rotation of two clusters is de-coupled to each other. At the asymptotic region, therefore, each nucleus keeps its isolated states and weakly coupled to each other. On the other hand, in S ≤ 6 fm, the energies for the cluster-coupling scheme deviate from the adiabatic energy surfaces.
We study the adiabatic eigenstates in an internal region (S 4 fm) in the view of the molecular orbital formation. As shown in an example of (σ + ) 2 in Eq. (3), the wave function of molecular orbitals is expressed by a linear combination of different kinds of cluster-wave functions, where each cluster has no good angular momenta. In the molecular orbitals, the valence neutrons are moving around two α-cores with a specific direction in respect to the α-α axis. Such a configuration of the system is called as "a strong-coupling scheme". The overlap of the adiabatic eigenstates with the wave functions of molecular orbitals identifies the dominant components in the adiabatic eigenstates as shown in Fig. 1. The adiabatic eigenstates connected by the thin-solid curves at the internal region have the common dominant-components of various kinds of molecular orbitals. 3/2 ) 2 , (σ + 1/2 ) 2 , (π − 1/2 ) 2 and (π + 3/2 ) 2 , respectively, while E, F, G and H (π + 1/2 σ + 1/2 ) and (π − 1/2 σ − 1/2 ), (σ − 1/2 ) 2 and (π + 1/2 ) 2 , respectively.  Table I shows the dominant molecular-orbital configurations in the adiabatic eigenstates along to the lowest surface A of Fig. 1. The squared amplitudes listed in Table I have the meaning of the probability of finding the system in a molecular orbital, because an orthogonality among the different molecular orbitals is good in spite of the anti-symmetrization effect. This is due to the fact that the molecular orbitals are similar to the deformed shell-model orbitals in the internal region.
We can see that the eigenstate has a dominant component of (π − 3/2 ) 2 at S=2 and 4 fm. On the other hand, at the asymptotic region (S ∼ 8 fm), the molecular orbitals are strongly mixed with each other, and the [α+ 6 He(0 + 1 )] L=0 channel dominates. Thus, the clustercoupling scheme becomes a good basis state for describing the system at the external region, while the strong-coupling one at the internal region. At an intermediate region (S=5∼6 fm), we find that the dominant component of the lowest surface A is changed from The same as Fig. 1 but for the negative parity states (J π =1 − ). In the internal region of S ≤4.5 fm, the lowest, second, third and fourth curves connected by a thin-solid line show the surfaces having a dominant component of (π − 3/2 σ + 1/2 )K=1, (π − 1/2 σ + 1/2 )K=1, (π − 1/2 σ + 1/2 )K=0, (π + 3/2 π − 1/2 )K=0, respectively. The surface with a thick curve has a dominant component of the [α+ 6 He(0 + 1 )]L=1 channel.
Finally, the GCM equation (5) is solved by employing the basis states ranging from S=1 fm to S=9 fm with the mesh of 0.5 fm. We obtain the result that the lowest three GCM solutions have energy gains of about 1∼2 MeV, and the dominant amplitudes around the respective local minimums in the adiabatic energy surfaces A, B and C. Therefore, these solutions of the 0 + 1 , 0 + 2 and 0 + 3 states are concluded to have the molecular-orbitals configuration of (π − 3/2 ) 2 , (σ + 1/2 ) 2 and (π − 1/2 ) 2 , respectively [4]. Similarly, we calculate the J π =1 − state. The calculated energy surfaces are shown in Fig. 2. We focus on the lowest five surfaces and investigated their intrinsic structure, because there is no definite local-minimums in higher surfaces. As shown by two vertical dotted-lines in Fig. 2, we find the dynamical transition from the strongcoupling scheme to the cluster-coupling one in the adiabatic surfaces. However, the first excited surface connected by a thick curve has an almost pure-component of the [α+ 6 He(0 + 1 )] L=1 channel. Thus, this excited surface is made by the pure cluster-coupling state of this channel in a wide range of the α-α distance.
The energy gain of the lowest three solution due to GCM is about 1∼3 MeV depending on the states. In the lowest and the third 1 − states, we find that the wave functions distribute around the respective local minimums of the adiabatic surfaces. Thus, the intrinsic struc- 3/2 σ + 1/2 ), respectively, while that with the white circles does (σ + 1/2 ) 2 . The solid symbols have a developed α+ 6 He cluster structure. The inset is a same figure but for the observed states [1,9]. The spins for the states in parentheses are tentatively suggested in Ref. [1].
tures of the 1 − 1 and 1 − 3 states are explained with the molecular orbitals of (π − 3/2 σ + 1/2 ) K=1 and (π − 1/2 σ + 1/2 ) K=1 , respectively. The 1 − 2 state has the main components of the first excited surface in Fig. 2, which is interpreted in terms of the cluster-coupling states of [α+ 6 He(0 + 1 )] L=1 . To see band structures in 10 Be, we solve GCM Eq. (5) for the higher spins with a natural parity (−1) J . The calculated bands are shown in Fig. 3 with an inset showing the respective observed states [1,9]. The moment of inertia of individual bands is well reproduced. Furthermore, our model predicts the existence of the higher spin states which were suggested by a recent experiment [1]. In the (σ + 1/2 ) 2 band shown by the white circles, we find the enhancement of the [α+ 6 He(2 + 1 )] L=4 state at a maximum spin (solid circle). In the present calculation, all the excitation energies is higher than those of the observed states. This can be improved by optimizing the nucleonnucleon force and the radius b to reproduce the cluster's threshold energies. The reproduction of threshold energies is quite important for the study on the decay width. The quantitative analysis including the decay width will be given in forthcoming papers.
In summary, we proposed a generalized two-center cluster model (GTCM) and discussed its application to the 10 Be=α+α+n+n system with J π =0 + and 1 − . The adiabatic energy surfaces depending on the α-α distance were calculated. The adiabatic eigenstates have the molecular orbital configuration at an internal region, while they becomes the cluster-coupling states at an external one. The middle distances correspond to the transitional region having an intermediate coupling-scheme between the internal regions and the external ones. It should be noticed that, in our model, both schemes of the molecular-orbitals and cluster-coupling schemes can be naturally described in an equal footing without any difficulty relevant to the double-projection procedure.
Finally, we solved the coupling between the relative motions of clusters and the intrinsic motion of valence neutrons. The low-lying 0 + states and the lowest 1 − one have the respective molecular-orbital configurations, which are consistent with previous studies based on MOM [4] and AMD [5]. In addition, our model predicts the possible appearance of the [α+ 6 He(2 + 1 )] L=4 cluster-structures in the lowest 6 + state. Furthermore, we theoretically obtain the second 1 − state with the [α+ 6 He(0 + 1 )] L=1 structure and expect to observe it experimentally. To see correspondence with experiments, it is necessary to analyze resonant states above threshold energies. We are now going to tackle this problem.
In conclusion, we can say that GTCM well describes the low-lying molecular orbitals obtained by other theoretical models [4,5,8]. Furthermore, this model naturally reproduce the asymptotic cluster-coupling states which are described by the traditional cluster model [6] in an equal footing. These results indicate that the present GTCM is applicable to the study of low-lying and highlying states in 10 Be. Since the α+α+2n model for 10 Be is a special case (x=2) of a general α+α+xn model, it is very easy to apply GTCM to systematic studies of resonances observed in excited Be-isotopes as well as their low-lying states. In particular, its application to 12 Be = α+α+4n is very interesting because of their accumulated experimental results [1,2]. A direct extension to the α+α+4n model for 12 Be is is now under progress.