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Unified studies of chemical bonding structures and resonant scattering in light neutron-excess systems, 10,12Be

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 Rep. Prog. Phys. 77 096301 (http://iopscience.iop.org/0034-4885/77/9/096301) View the table of contents for this issue, or go to the journal homepage for more

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Reports on Progress in Physics Rep. Prog. Phys. 77 (2014) 096301 (46pp)

doi:10.1088/0034-4885/77/9/096301

Review Article

Unified studies of chemical bonding structures and resonant scattering in light neutron-excess systems, 10,12Be Makoto Ito1,2,3 and Kiyomi Ikeda2,3 1

Department of Pure and Applied Physics, Kansai University, Yamatecho, 3-3-35, Suita, Japan Research Center for Nuclear Physics (RCNP), Osaka University, Mihogaoka 10-1, Suita 567-0047, Japan 3 RIKEN Nishina Center for Accelerator-based Science, RIKEN, Wako,351-0198, Saitama, Japan 2

E-mail: [email protected] and [email protected] Received 4 December 2009, revised 28 April 2014 Accepted for publication 15 May 2014 Published 15 September 2014 Abstract

The generalized two-center cluster model (GTCM), which can treat covalent, ionic and atomic configurations in general systems with two inert cores plus valence nucleons, is formulated in the basis of the microscopic cluster model. In this model, the covalent configurations constructed by the molecular orbital (MO) method and the atomic (or ionic) configuration obtained by the valence bonding (VB) method can be handled in a consistent manner. The GTCM is applied to the light neutron-rich system 10,12 Be = α + α + Xn (X = 2, 4). The continuous and smooth changes of the neutron orbits from the covalent MO states to the ionic VB states are clearly observed in the adiabatic energy surfaces (AESs), which are the energy curves obtained with a variation of the α–α distance. The energy levels obtained from the AESs nicely reproduce the recent observations over a wide energy region. The individual spectra are characterized in terms of chemical-bonding-like structures, such as the covalent MO or ionic VB structures, according to analysis of their intrinsic wave functions. From the analysis of AESs, the formation of the mysterious 0+2 states in 10,12 Be, which have anomalously small excitation energies in comparison to a naive shell-model prediction, is investigated. A large enhancement in a monopole transition from a ground MO state to an ionic α + 6,8 He VB state is found, which seems to be consistent with a recent observation. In the unbound region, the structure problem, which handles the total system of α + α + Xn (X = 2, 4) as a bound or quasi-bound state, and the reaction problem, induced by the collision of an asymptotic VB state of α + 6,8 He, are combined by the GTCM. The properties of unbound resonant states are discussed in close connection to the reaction mechanism, and some enhancement factors originating from the properties of the intrinsic states are predicted in the reaction observables. Keywords: neutron-rich system, excited states, cluster structures (Some figures may appear in colour only in the online journal)

protons and neutrons. In nuclei, the saturation of the density and the bonding energy per nucleon are important properties [1]. These two quantities vary only smoothly with mass number and are approximately constant from light systems to heavy ones. The saturation can be attributed to the nature

1. Introduction 1.1. Mean field picture and cluster picture

Nuclei are finite quantum many-body systems comprised of nucleons,

1.1.1. Mean-field structures and saturation in nuclei.

0034-4885/14/096301+46$88.00

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In the mean field of nuclear systems, there is a strong spin–orbit interaction, l · s, which means the interaction between a nucleon’s orbital angular momentum (l) and a nucleon’s spin (s). Inclusion of the spin–orbit interaction in the levels in the WS potential yields the modified level scheme shown on the right-hand side in figure 1. In the results with the l · s interaction, energy levels bunch up, and such a level scheme is called the ‘nuclear shell structure’, in which there are energy gap among the bunched levels. Filling up the level scheme with nucleons according to Pauli’s exclusion principle, the WS + (l · s) model can generate magic numbers, such as 2, 8, 20, 28, . . . , as shown in the rightmost part in figure 1. A nucleus with a magic number reveals an unusual stability with large separation energy for a nucleon [1]. This magic number is basically the same concept as the shell closure of electron shells inside of the central Coulomb interaction, although, in the Coulomb field, electron levels with different parities degenerate. The ground state of nuclear systems with the same number of neutrons (N ) and protons (Z) can be nicely understood in the basis of the shell structure produced by the one-body mean field with the spin–orbit force except for a few systems. Figure 1. Energy levels of the single-particle states in a one-center mean-field potential. The lines on the left-hand side show the levels in the harmonic oscillator (HO) potential, while those at the center position represent the levels in the Woods–Saxon (WS) potential. The levels on the right-hand side are the results of WS plus the one-body spin–orbit (l · s) potential. The number in the rightmost part means the magic number. See text for details.

1.1.2. Cluster structures in N = Z systems.

In nuclei, the bonding energy per nucleon is almost saturated; hence, we can draw an analogy between nuclei and liquid drops from the viewpoint of this saturation of the bonding energy. This analogy means that a nucleus can easily dissociate into subunits, which are called clusters, with a small energy difference. Therefore, cluster formations and their associations are considered as elementary and important features in nuclear systems because the clustering phenomena are based on the saturation property of the bonding energy. The ground states of nuclear systems are well understood in terms of a shell model picture, but cluster structure, in which a nucleus is decomposed into subunits, is one of the characteristic features appearing in the excited states of light nuclear systems [6–8]. In particular, the most famous subunit is the α particle (the 4 He nucleus), and a typical example of α cluster structures is the α + α structures in 8 Be. In 8 Be, the yrast states of J π = 0+ , 2+ and 4+ can be nicely described by the rotational band of two α particles [9, 10]. Other examples of α cluster structures are the 3α structure in 0+2 of 12 C at Ex = 7.65 MeV, called the Hoyle state [11, 12], α + 12 C in 0+2 of 16 O at Ex = 6.05 MeV [13], and α + 16 O in 0+4 of 20 Ne at Ex = 8.03 MeV [14]. The α particle is quite inert because of its large nucleon separation energy of ∼20 MeV, hence it is considered to be a building block for light N = Z systems. The cluster structures are considered to appear according to the threshold rule [15]. The threshold rule for the 4N nuclei is summarized as an Ikeda diagram which has proved to be very powerful in identifying 4N cluster structures (α,12 C,16 O, 20 Ne, etc). The diagram for the lighter mass systems is shown in figure 2. It illustrates various cluster structures which could exist in excited states of light nuclei based on the hypothesis that particular cluster structures will emerge for excitation energies near the corresponding threshold energy decaying into the respective cluster configuration. Since there are several combinations of subunits in a nucleus, cluster structures can

of a nucleon–nucleon force, which has a complicated state dependence of a two-nucleon pair, such as nucleons’ spin, isospins and parity. In a ground state of nuclei with saturation, a simple structure is realized: nucleons independently move in an average field produced by the other nucleons [1]. The basic properties of ground states are nicely explained by a onecenter mean-field model, such as the Hartree–Fock method [2], or by the configuration interactions [3] based on mean-field structures. A mean-field potential should be determined in a selfconsistent manner based on the Hartree–Fock method, but it is usually replaced by a simple one-center potential, such as the harmonic oscillator (HO) potential [4] or the Woods–Saxon (WS) [5] potential, which has a functional form of the Fermi distribution. Single-particle energy levels of nucleons moving inside of the potentials are shown in figure 1. The levels in the HO potential (the levels on the left-hand side in figure 1) are labeled by the principal quantum number (n) and the orbital angular momentum (l), and their energy is given by n,l = (N + 3/2)¯hω, with the total oscillator quanta N = 2(n − 1) + l. The levels with a definite N we call an oscillator shell. The oscillator shells only contain either even or odd l values; that is, one oscillator shell contains only states with the same parity (s, d, g, . . . or p, f, h, . . . ), and they are degenerate in the level with a constant N . This accidental degeneracy of the HO potential is removed from the WS potential (the middle levels in figure 1), but levels with the same parity still form the groups, and the shell structures generated in the HO potential are not so strongly modified in the result of the WS potential. 2

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mean field, where they are trapped in a common S-wave orbital [16–18]. This new multi-α state realized by the THSR wave function is called the ‘nuclear α-particle condensation state’ [18]. The THSR wave function is successful in describing the S-wave motion in the multi-α states, such as 8 Be = 2α [10], 12 C = 3α [16, 19, 20], 16 O = 4α [16, 20, 21], and 20 Ne = 5α [16], α + 16 O [22]. The rotational excitation of the multi-α particles can also be described by the excited THSR wave function, in which an orbital motion of an α particle is excited to the D-wave orbit inside of the bosonic mean field [18, 23]. Moreover, the analogous states to the 3α condensation in 12 C are established in the non-4N systems with one neutron excess or one proton deficiency, such as 13 C = 3α + n [17] or 11 B = α + α + T [24]. The multi-α structure is also an important issue in the molecular resonance (MR) phenomena induced by slow collisions of light-ion systems [25, 26]. Figure 2. Ikeda diagram in the lighter mass region. The numbers in parenthesis represent the threshold energy dissociating the ground state into the respective cluster configuration. The threshold energies are in MeV.

In the last two decades, developments in experimental techniques have helped studies on unstable nuclei, which are proton-rich (N  Z) or neutron-rich (N  Z) systems and unstable to β decays, to progress extensively. Various neutron-rich nuclei, which do not exist in nature, are artificially synthesized in laboratories, and their exotic structures attract much scientific interest [27]. In the mean-field picture, the Fermi surface of the neutrons’ and protons’ potentials becomes different, and this difference leads to anomalous features of neutron-excess systems. For instance, a typical example of exotic phenomena is the halo and skin formations of the excess neutrons [28]. In the neutron-drip region, the separation energy of neutrons becomes quite small, and hence neutrons’ single-particle orbits penetrate outside the mean-field potential due to the quantum tunneling effect. Furthermore, the breaking and shift of the magic number in neutron-excess nuclei has been reported in recent studies based on a large shell-model calculation [29]. Cluster degrees of freedom are quite important also in the excited states of neutron-excess systems, although in a ground state the mean-field structure with an unbalanced Fermi surface is still realized. In neutron-excess nuclei, the cluster cores are surrounded by the excess neutrons, and excess neutrons play a glue-like role among the weakly coupled cluster cores. When neutrons are added to cluster states existing around the respective threshold energy, the single-particle wave function of neutrons spreads over cluster cores so as to reduce their kinetic energy. In addition, there is an energy gain due to the attractive interactions between clusters and neutrons. Therefore, the excitation energies of cluster states with neutron glue become much lower than those without the glue. This energy change leads to the formation of the so-called intruder states, which have a well developed cluster structure in the lower bound region [30–34]. In the analysis of the cluster structure with neutron glue, therefore, it is natural to introduce a model focusing on the coupling between the cluster and valence neutron degrees of freedom. In particular, the molecular orbital (MO) model is successful in describing the low-lying states of light neutronrich nuclei [30, 31, 35, 36]. In this model, valence neutrons perform single-particle motions in the mean field generated by

1.1.3. Cluster structures in neutron-excess systems.

change from level to level, and they coexist in the same nucleus with energy intervals of possible cluster decay thresholds [15]. These cluster structures have been established by accumulating the laborious comparisons of the experimental observations with theoretical calculations based on microscopic cluster models [6–8]. Microscopic cluster models assume several clusters, each of which is a group of nucleons trapped by an HO potential. In this model, the anti-symmetrization among all nucleons is completely taken into account. Due to this anti-symmetrization, cluster-model wave functions smoothly change to the shell model ones when clusters completely overlap [6–8]. Therefore, this model can naturally handle cluster structures in excited states as well as formation of a mean-field structure in a ground state. In analysis by the model, cluster states are interpreted in terms of the ‘higher nodal states of clusters’ relative wave function’ with respect to a ground state. Specifically, in cluster states, a relative motion of an α particle and a residual nucleus is excited, and the relative wave function has additional nodes compared with that in a ground state, in which clusters strongly overlap with each other and a mean-field structure is realized [6–8]. Because a cluster state appears around the threshold energy in general, a large part of the total bonding energy is consumed in forming the clusters, and decomposed clusters are weakly coupled to each other. In addition, the clusters mainly perform the S-wave relative motions so as to reduce the centrifugal potential arising from the kinetic energy. This weak-coupling feature in the dominant S-wave motion leads to a dilute and spatially extended density distribution due to the quantum tunneling effect. If a 4N system is decomposed into a possible multi-α configuration, in which a pair of the α particles interacts with the S wave dominantly, the radial extension is much more prominent. In recent theoretical studies, a new type of microscopic multi-α wave function is proposed by Thosaki, Horiuchi, Schuck and Re¨opke, which is called the THSR wave function [16–22]. In the THSR formalism, the multi-α particles, to good approximation, can be viewed to move in their own bosonic 3

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cluster cores, and it is similar to covalent electrons forming chemical bonds in di-atomic molecules. The MO model has been mainly applied to Be [30, 31, 37] and B [35, 37] isotopes built on α + α in 8 Be plus valence nucleon (N ) structure. In Be isotopes, many kinds of abnormal feature difficult to explain with the one-center mean-field picture are observed, and such observations can be naturally understood by the MO formation around two α cores. In recent studies, furthermore, studies based on the MO picture are extended to heavier systems, such as C isotopes (3α + n + n. . . ) [36, 38] and Ne ones (α + 16 O + n + n. . . ) [32, 39]. For instance, a stabilization of 3α equilateral triangular structure [40] and α + 16 O cluster structures [32] by the effects of neutron glue is pointed out in the lower excited state of 14 C and 22 Ne, respectively. 1.2. Interesting issues in Be isotopes

Figure 3. Energy levels and threshold energies for various configurations in Be isotopes. In this figure, all the energies are plotted with respect to the lowest thresholds of the α + x He + n configuration, which is taken to be the zero energy.

1.2.1. Anomalous observations in Be isotopes and the success of the MO model. Although the ground state of a nucleus has

a simple mean-field structure as just mentioned, Be isotopes are exceptional systems, in which cluster structures appear in a ground state. We can clearly understand anomalous features in level schemes of these isotopes shown in figure 3. Typical evidence of the two-center structure in Be isotopes can be seen in the systematics of even Be isotopes (8,10,12 Be), for instance. The anomaly appears in the level spacing of the 0+ states. The ground 0+ state in 8 Be is identified at the energy of 0.092 MeV, just above the α + α threshold, as shown in the leftmost levels. In this unbound 0+ state, two α particles are weakly coupled to each other, and they finally decay within a finite lifetime. In 10 Be (N = 6, Z = 4), the 0+2 state appears at Ex = 6.18 MeV, which can be generated by two excess neutrons’ excitation from (1p3/2 )2 → (1p1/2 )2 . However, such two-neutron excitation requires about 10 MeV excitation energy because of the large energy gap of 1p3/2 –1p1/2 , about 5 MeV, in the one-center mean-field model. Thus, the 6 MeV excitation energy of 0+2 is difficult to explain in the one-center mean-field picture. Furthermore, the excitation energy to 0+2 in 12 Be (N = 8, Z = 4) is much smaller than that in 10 Be, say about 2 MeV [41, 42]. Such a low-lying 0+2 state, called the intruder state, is extremely difficult to explain in the one-center meanfield picture. This nucleus has four protons and eight neutrons, the magic number, and the ground state is obtained by filling up the orbits shown in figure 1 with nucleons according to the one-center mean-field picture. In forming the excited 0+2 state, two neutrons in the highest occupied orbits, 1p1/2 , are excited to the lowest unoccupied orbits, 1d5/2 , because of the parity conservation. Since, in the one-center mean field, the energy gap of 1d5/2 –1p1/2 reaches about 7 MeV (see figure 1), the excitation energy of the resultant 0+2 state must exceed 10 MeV. Thus, the observation of the 0+2 state at Ex = 2.24 MeV is difficult to explain in a simple mean-field model. Other measurements also report the breaking of the N = 8 magic number in 12 Be [43, 44]. Anomalous features beyond the one-center mean-field picture can also be confirmed in the systematics in odd Be isotopes (9,11 Be). In 9 Be, the ground state has a spin

parity of 3/2− , which is consistent with the prediction of a mean-field picture, but the first excited state, which exists at Ex = 1.68 MeV, just above the α +α+ n threshold, has the spin parity of 1/2+ , which is the opposite parity to the expectation from the one-center mean field. In a mean-field picture, a last neutron filling the 1p3/2 orbit excites to the 1p1/2 orbits having minus parity, but this naive expectation is inconsistent with the observation. In 11 Be, we can see a parity inversion between the ground 1/2+ state and the first excited 1/2− state (0.32 MeV), which are formed below the 10 Be + n threshold. According to the expectation in one-center mean field, a last neutron must occupy the 1p1/2 and 1d5/2 orbits for the ground and first excited states, respectively. This configuration leads to the formation of the ground (first excited) state with the minus (plus) parity. These results strongly suggest that the shell gap appearing at the N = 8 magic number has disappeared or is quenched in Be isotopes. The properties of the ground and low-lying states in Be isotopes (9 Be, 10 Be, . . . ) can be naturally explained by adding neutrons to the α + α cluster in 8 Be. Therefore, these isotopes are considered as not simple one-center mean-field systems but two-center superdeformed systems, which build on an α + α rotor of 8 Be. Since the α–α and α–n interactions are quite weak, the α–α clustering and the neutrons’ singleparticle orbits are strongly coupled to each other. Therefore, the α correlation prominently appears in the low-lying states. In the basis of the two-center picture, excess neutrons are considered to perform single-particle motions inside of the strongly deformed mean field generated by two α particles. Such a two-center picture is nicely modeled by the so-called MO model, and there is a long history of studies of Be isotopes based on the MO model [30, 31, 35, 37, 45–48]. The formulation of the MO model can be classified into two categories: one is the two-center shell model (TCSM), and the other one is based on the linear combination of atomic orbitals (LCAO) method. In TCSM, excess neutrons perform single-particle motions in orbits generated by the two-center 4

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HO potential [45]. This model is convenient in understanding anomalous properties in Be isotopes intuitively. We will explain some applications of the TCSM to Be isotopes in the following section. In the LCAO treatment [30, 31, 35, 37, 46– 48], the neutrons’ orbits are basically handled as covalent orbits, associated with electrons’ covalent bonds in quantum chemistry: specifically, the neutrons’ orbits around two α particles are constructed by LCAOs as ϕL ± ϕR , where ϕL and ϕR mean the atomic orbital (AO) around the left-hand and right-hand centers, respectively. From the late 1960s, the MO models based on the LCAO treatment were introduced in problems of nuclear systems, such as the exchange interactions in nuclear scattering [46, 111] and the structure of the N = Z nuclei [47]. The first applications to the neutron-rich Be and B isotopes were performed from the 1970s [35, 48]. In the late 1990s, the MO picture was employed in interpreting the rotational band structure of Be and B isotopes [37], and the theoretical framework of the MO model was improved [30, 31]. Let us briefly explain the LCAO treatment in Be isotopes, which have the structure of α + α + n + n + . . .. Since the four nucleons inside of the individual α particles occupy 1s1/2 orbits, the 1p AOs are used in constructing the covalent orbits. There are three directions in 1p orbits according to the value of the magnetic quantum number m = ±1, 0, and the covalent orbits − can be produced as πm=±1 = ϕL (m = ±1) + ϕR (m = ±1) + and σm=0 = ϕL (m = 0) + ϕR (m = 0). The former and latter covalent orbits have a good magnetic quantum number of m = ±1 and m = 0, respectively. The π − orbit, which is generated from the |m| = 1 AO, has one node in the direction perpendicular to the α–α axis, while the σ + orbit with m=0 has two nodes along the α–α axis. The neutrons’ occupation of the σ + orbit elongates the α–α distance so as to reduce the neutron’s kinetic energy. The low-lying band structures in even Be, including the mysterious 0+2 states in 10,12 Be, can be nicely described by the neutrons’ occupation of the MO orbits of π − and σ + [30, 31, 35, 37, 49]. The energy levels with abnormal parities in odd Be isotopes can also be nicely understood by the MO model. In recent developments of theoretical studies, the approach of the MO model has been justified to give a realistic physical picture in these isotopes. For example, two α structures in 8 Be have been practically reproduced by ab initio calculation, the Green function Monte Carlo [9], which starts with a realistic nucleon– nucleon force. Furthermore, in recent studies of the antisymmetrized molecular dynamics (AMD), in which nucleons’ orbits are handled by Gaussian wave packets and there is no assumption of α cores basically, the MO structure with two α cores is actually realized in the low-lying states in Be and B isotopes [33]. These results obtained from various microscopic studies strongly support the foregoing microscopic model approaches with the α clusters in these isotopes.

Figure 4. Energy levels of the single-particle states in the two-center mean-field potential [45]. The abscissa represents the distance between two potentials, while the ordinate show the energy of the single-particle state. The levels at zero distance correspond to the levels in the one-center potential shown in figure 1. Rmin denotes the position of two α cores, where the total energy of Be isotopes becomes its minimum value. See text for details. Figure reproduced from [7] with permission. Copyright 2011 The Physical Society of Japan.

anomalous observations. In this section, we show global features of a schematic two-center system and discuss an essential mechanism to describe the properties of Be isotopes. In order to see features of two-center systems, in figure 4 we show the single-particle energies generated inside of the twocenter mean-field potential. At the distance of R = 0, where the centers of two potentials completely overlap, the energy levels are the same as those in a one-center mean field, while the levels are shifted in the case of finite R. In Be isotopes, the optimal distance, where the energy of a total system becomes almost its lowest, is not R = 0 but a finite R. In figure 4, the optimal distance is shown by Rmin , and the value of Rmin is about 3.5∼ 4.5 fm. In the one-center limit of R → 0, there is a large energy gap between the 1p1/2 orbit and the 1d5/2 orbit, and this gap leads to the formation of the N = 8 magic number. Around Rmin , however, the level ordering of 1d5/2 –1p1/2 is inverted, and the energy gap is only about 1 MeV although the original gap is about 10 MeV in the one-center limit. At a finite distance, the levels shifted from 1p3/2 and 1d5/2 are − − + and σ1/2 , respectively. The orbits of π3/2 and labeled by π3/2 + σ1/2 are basically the same as the covalent orbits constructed by the LCAO treatment in the MO model. These level shifts can explain the formation of the mysterious 0+ in even Be isotopes, which are generated in a much lower-energy region than expected in the one-center mean field. In figure 5, an example of 12 Be is shown. As shown in the left-hand panel, the ground state can be constructed by filling the levels at Rmin with nucleons. The four protons and four neutrons, corresponding to the nucleon number of 8 Be + 4 − 4 ) (σ1/2 ) configuration, which (N = Z = 4), form the (σ1/2 can be obtained by filling the lowest two orbits in figure 4. This

1.2.2. Two-center picture in low-lying states of Be isotopes.

We have summarized experimental observations, which are difficult to understand in the one-center mean field picture, and mentioned that the MO model in two-center systems with the α + α cores is a powerful tool to explain the 5

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Figure 5. Single-particle structures of 12 Be in the basis of the two-center levels at Rmin shown in figure 4. The left and right panels represent the nucleons’ occupation for the ground and excited 0+ states, respectively. The four neutrons and the four protons ± occupying the σ1/2 orbits are almost equivalent to the 2α wave function. See text for details.

Figure 6. Threshold energies and energy spectra observed above the α decay energy in 10,12 Be. The α emission energy is taken to be the position of zero energy. The thresholds are shown by the dotted lines. In 10 Be, the shaded area represents the broad strength of the J π = 0+ state, while the solid line is a candidate for a resonant level. The individual 0+ levels observed in 12 Be are shown by the solid lines. Each of the solid lines in 12 Be has a finite width of about 1 MeV or less on average.

configuration for the eight nucleons has a large overlap with the 2α wave function with (1s1/2 )4 due to the anti-symmetrization effect among nucleons. In the ground state, the four excess neutrons occupy the − + π3/2 and σ1/2 orbitals, which are the shifted levels from the − + –σ1/2 1p3/2 and 1d5/2 orbitals, respectively. Since the π1/2 energy gap, corresponding to 1p1/2 –1d5/2 in the one-center limit, is reduced to about 1 MeV, the excitation energy for − 2 + 2 the two-neutron excitation of (σ1/2 ) → (π1/2 ) amounts to Ex ∼ 2 MeV, which is almost same as the observed energy of the 0+2 state. This situation is shown in the panels of figure 5. Therefore, the two-center picture can naturally describe the formation of the intruder 0+ state in 12 Be. In a similar manner, the energy of the 0+2 state in 10 Be can also be reproduced by the same two-center picture. The parity inversion phenomena in odd Be isotopes can be qualitatively explained by a reduction − of the energy gap between the negative parity orbit of π3/2 and + the positive parity orbit of σ1/2 .

of about 3 MeV. The lowest α + 6 Heg.s. threshold exists at 7.41 MeV, while other He dimer channels, such as α + 6 He(2+1 ) and 5 Heg.s. + 5 Heg.s. , open at 9.21 MeV and at 9.98 MeV, respectively. There is a threshold of the four-body decay channel, α+α+2n, at 8.39 MeV, which is close to the thresholds of the He dimer channels. The threshold spacing in 12 Be is almost the same as that in 10 Be, as shown in the right-hand part of figure 6. The decay into α + 8 Heg.s. opens at 9.0 MeV, and other thresholds for decays into subunits of He isotopes appear with a small energy interval of about 1–2 MeV: 10.1 MeV for 6 Heg.s. + 6 Heg.s. , 11.9 MeV for 6 Heg.s. + 6 He(2+1 ) and 13.2 MeV for 5 Heg.s. + 7 Heg.s. . The energy intervals of these thresholds are about 1 MeV, and hence the threshold energies reveal a strong degenerate feature. If we assume that the threshold rules established in N = Z systems is still valid in the neutron excess N > Z systems, the threshold diagram in figure 6 predicts that various He dimers, such as x He +y He (x + y = 4(2) in 12 Be (10 Be)), appear with a strong degenerate-like behavior. In fact, recent experiments strongly support the degenerate feature of the resonant levels above the α decay threshold; specifically, prominent resonance structures have been observed around the threshold energy of the He-dimer channels in 10,12 Be. For example, in reactions of three-body sequential decays or two-body breakup reactions induced by high-energy RI beams, many resonant states are observed in the decay channel of 10 Be → α + 6 Heg.s. [53–55] and 12 Be → α + 8 Heg.s. , 6 Heg.s. +6 Heg.s. [56–62]. In 12 Be, spins of the observed resonant states are clearly identified by the multipole decomposition analysis [57, 58], and the resonant levels of the observed J π = 0+ states are shown in figure 6 by the solid lines. According

Lowlying states in Be isotopes have been studied by sophisticated MO models in the past three decades. Experimental data on 10,12 Be are the most abundant of all of the isotopes [41–44, 50–52], and comparisons of theoretical calculations of the MO models and experiments have been deeply discussed in these two nuclei. Thus, the MO picture is the best established in these two systems. In 10,12 Be, furthermore, a wide variety of cluster structures beyond the MO structures is expected to appear above the threshold of an α emission. In figure 6, threshold energies of 10,12 Be above an α decay energy are plotted by the dotted lines. In these systems, the thresholds of various ‘nuclear dimers of He isotopes’, which are produced by neutron rearrangements around two α cores, appear with a small energy interval. All the threshold energies in 10 Be, plotted in the lefthand part of figure 6, are confined within the energy width 1.2.3. Extension of the study to highly excited states.

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Be gives the linear combination of the partitions, (8 He +α) + ( He +5 He) + (6 He +6 He) + (5 He +7 He) + (α+ 8 He). In general, 5∼8 He are not ground states but polarized states, which contain excited states as well as a ground state. Therefore, not only He dimers with even–even partitions (α + 8 He and 6 He +6 He) but also a dimer with an odd–odd partition (5 He +7 He) are essential in the formation of the MO configuration. In contrast, if we take a partition with a specific neutron atomic configuration, the atomic or ionic He-dimer configuration corresponding to the VB state can be produced. In order to handle the formation of the covalent MO and ionic He-dimer structure in a consistent manner, all the possible partitions are employed as a basis functions, and their mixing amplitudes are optimized according to the variation principle. This is the basic idea of the generalized two-center cluster model (GTCM), which is developed in a series of our recent studies [68–76]. Specifically, a wave function of a total system is constructed by a superposition of all the possible AOs in two centers (x He +y He, x + y = constant), and their mixing amplitudes are optimized according to the variational principle. Due to the variational optimization, the total wave function automatically generates MOs or He dimers, which depend on conditions such as excitation energy, spins and so on. The MO bases are optimal configurations at small α–α distance and are valid for describing spatially compact states. Thus, the MO states should be classified as ‘internal states’, which are confined in the region of the compact α–α distance. In contrast to this compact property, in the excited states, the α–α distance must be extended because of its orthogonality to the internal MO states. This spatial extension leads to the formation of the ‘external states’ with the He dimers, which corresponds to the asymptotic binary channels. Therefore, nuclear structures can be divided into two types, the internal MO and the external He-dimer states. By applying GTCM, it is possible to analyze the structure changes from the viewpoint of the transition between the internal MO states and the external He-dimer states. Furthermore, the scattering problem can be easily considered in the framework of GTCM because the external He-dimer states are just the same as the basis function in the quantum scattering problem. According to the prediction of the threshold rule, the He-dimer states are expected to appear as weakly bonding or resonant states; the imposition of the scattering boundary condition is essential. In a series of our recent studies, we have applied the GTCM to 10,12 Be [68, 69, 71–75] and performed unified studies of nuclear structures and reactions. In the study of 12 Be [71–75], global features in structural changes from MO states in a bound region to He dimers in a continuum are predicted [71]. In subsequent studies, the reaction mechanism of the formation and decay scheme of resonances through two-neutron transfer, α + 8 Heg.s. → 6 Heg.s. + 6 Heg.s. , is investigated [72, 74]. In studies of 10 Be, similar structure changes have been investigated, and formation of a parity doublet of the α + 6 He structure has been discussed in connection to the Landau–Zener non-adiabatic transition induced in slow scattering of 6 He by an α target [69]. In the present paper, we report the structural transition in 10,12 Be from the internal MO states to the external Hedimer states. The structure transitions are analyzed in the

to a recent experiment [62], the resonant state, which strongly decays into α + 8 Heg.s. , has been confirmed at Ex ∼ 10.3 MeV. These observed J π = 0+ levels, which have energy intervals of less than 1 MeV, correspond nicely to the threshold energies. In a recent experiment on 10 Be, a broad 0+ state, which may be a sign of the overlap of several resonances, has been found above the α threshold as shown by the shaded area in the lefthand part of figure 6 [63]. In addition, a candidate for 0+ resonance has just been identified in the recent experiment, which is shown by the solid line at E = 4.4 MeV [64]. Furthermore, in 10,12 Be, it is possible to measure cross sections of the 6,8 He scattering with an α target. Experiments of low-energy 6,8 He reactions have been conducted with an intense helium isotope beam provided by the ISOL technique [65, 66]. Data on the α + 6 Heg.s. reactions have been accumulated. In the elastic scattering, for example, some characteristic enhancements are observed in differential cross sections [66] and in the excitation function [65]. The signature of the resonance has been confirmed in the inelastic scattering, α + 6 Heg.s. → α + 6 He(21+ ) [64]. In contrast, low-energy 8 He scattering on an α target was performed, and recently an excitation function and angular distributions of α + 8 He have been measured at GANIL [67]. Therefore, the direct excitation of resonant states, which are confirmed in the decay experiments of 10,12 Be, will be possible by low-energy α + 6,8 He scattering. In view of the information on the excellent results of the MO model in low-lying states and the recent progress of experiments on highly excited resonances, applications of a suitable α cluster model to the highly excited states in 10,12 Be are strongly stimulated. In the covalent MO states, the neutron’s orbit spreads over two α cores, while He-dimer states, such as 6 Heg.s. + 6 Heg.s. or α+ 6 Heg.s. [84, 85], correspond to the atomic (or ionic) configurations, where the electrons are trapped at one of the cores. The ionic and atomic configurations are usually handled by the valence bonding (VB) method or Heitler–London method, which was originally proposed in describing the chemical bonding in H2 molecules [83]. In this method, the electrons are localized at one of the core nuclei, and exchanges of electrons belonging to the different cores are taken into account. Therefore, the MO states and the atomic (or ionic) ones are very different in the distributions of the neutrons’ orbits; the former orbit spreads over two α cores, while the orbit is localized at one of two α cores in the latter state. Thus, it is very interesting to extend the studies by an α cluster model to the highly excited resonances.

7

1.2.4. Applications of generalized two-center cluster model to Be isotopes. In MO configurations, a total wave function

is constructed by a Slater determinant composed of covalent neutrons, which is an anti-symmetrized direct product of single-particle orbitals φ as φ(1)φ(2) · · · φ(N ) for N neutrons. As we have just explained, a neutron’s covalent orbital φ is constructed by the LCAO method as φ = ϕL ± ϕR , with the left-hand (ϕL ) and right-hand (ϕR ) AOs. According to LCAO treatments, whole products of MOs can be expanded in terms of all the possible partitions, which are combinations of the numbers of the left-hand AOs and right-hand AOs. For example, the LCAO decomposition of the MO configuration in 7

Rep. Prog. Phys. 77 (2014) 096301

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basis of the ‘adiabatic states’ (ASs), obtained by solving the eigenvalue problem at a fixed α–α distance, and the ‘adiabatic energy surfaces’ (AESs), which are the series of adiabatic energies. The intrinsic structures of the resonant levels above the α threshold are characterized in terms of the ASs, and the formation and the decay of the resonance are directly combined with the scattering problem of 6,8 He by an α target. In particular, we compare the result for 10 Be with that for 12 Be and focus on the difference of these two systems. 12 Be has two additional neutrons in comparison to 10 Be, and these extra neutrons give rise to a wider variety than the case of 10 Be in both intrinsic structures and reaction mechanism. The organization of the present article is the following. In section 2, we formulate GTCM by extending the cluster model, which employs the microscopic two-center shell-model wave function. The comparison of the GTCM basis function with the electrons’ chemical bonding state, such as the MO or VB basis, is explained. GTCM and a standard MO model are compared in section 3, and advantages in GTCM are demonstrated. The AESs are defined in this section. The features of the AESs in 10,12 Be are analyzed in section 4. In AESs, we can observe a smooth transition from the internal MO to the external He-dimer states, and energy levels calculated from AESs nicely reproduce recent experiments. We show that, in the calculated energy levels, various chemical-bondinglike structures are realized. From the analysis of AESs, in section 5, formations of the mysterious 0+2 states in 10,12 Be, which appear in a much smaller-excitation-energy region than the expectation of a naive one-center mean-field picture, are investigated. The formation mechanism of the chemical bonding structures, which are realized as the resonant levels above the α decay thresholds, is investigated in section 6. The chemical bonding states in the resonances are classified on the basis of the excitation degrees of freedom. In section 7, the monopole transition from the ground 0+1 to the excited 0+ is investigated for 10,12 Be, and the strong enhancement of the monopole transition is discussed in connection to the recent experiments. In section 8, the structure studies are combined with the reaction problem. We discuss the enhancement factor observed in the α + 8 Heg.s. and α + 6 Heg.s. resonant scattering in connection to the chemical bonding structure.

and mean-field states by considering a simple example of 12 Be = α + 8 He in the basis of the HO potential. We consider a simple one-dimensional Gaussian function such as
1/4 2ν 2 e−νz , (1) ϕ0 (z) = π

2.1.1. Properties of the shifted Gaussian.

which has a peak at the origin. This is the ground state (n = 0) wave function in the one-dimensional HO potential. If the peak position of this wave function is shifted to z = S, we can obtain the following expansion:
1/4 ∞  2ν 2 e−ν(z−S) = CνS (n)ϕn (z). (2) ϕ0 (z − S) = π n=0 Here, ϕn (z) is the HO wave function for the nth excited states with the center of the origin (z = 0), while CνS (n) denotes the mixing amplitude of ϕn (z). The squared magnitude of CνS (n) gives the probability of finding the system in a specific nth excited state, and its explicit form is the so-called Poisson distribution, 2 n e−νS  . (3) |CνS (n)|2 = νS 2 n! From the expression of the Poisson distribution in equation (3), we can easily evaluate the expectation value of the oscillator quantum number, n, ˆ of the shifted Gaussian as ∞  ˆ 0 (z − S) = n|CνS (n)|2 = νS 2 . n ˆ ≡ ϕ0 (z − S)|n|ϕ n=0

(4) According to the property of the Poisson distribution, √ νS 2 . the standard deviation of n becomes n = Equations (2)–(4) mean that a shifted Gaussian is expressed by the mixture of the higher-n states in the original HO potential. Thus, the coherent superposition of the single-particle orbits always occurs to form a spatial localization of the singleparticle wave function around r ∼ S. Since the expectation value of the mixing quanta, n, is proportional to the squared magnitude of the shifted distance S, the localization at a large distance enhances the mixing amplitude of the higher n. It is quite easy to extend the shifted Gaussian to the threedimensional function. The three-dimensional Gaussian wave packet is
3/4 2ν 2 e−ν r , (5) φ1s (r ) = ϕ0 (x)ϕ0 (y)ϕ0 (z) = π and we consider its shift of the peak position to r = S = (0, 0, S) as
3/4 2ν 2 e−ν(r−S ) . φ1s (r − S ) = ϕ0 (x)ϕ0 (y)ϕ0 (z − S) = π (6)

2. Theoretical framework 2.1. Relation between the cluster picture and the shell-model picture

In the previous section, a global feature of a two-center system has just been discussed, and we have confirmed that the twocenter picture can naturally explain various kinds of anomalous property observed in Be isotopes. In this section, we show a detailed structure of a wave function in a two-center cluster state, and the relation between the cluster state and the meanfield state is discussed. In the mean-field picture nucleons move around one center in a self-consistent mean field, while in the cluster picture nucleons form groups of subunits such as α clusters and move keeping the subunits. Therefore, cluster states can basically correspond to a multi-center mean-field structure. Here we demonstrate the relation of the cluster

This function has a mixture of the oscillator quanta n in the z direction, which is similar to the one-dimensional case. Equation (6) is known as the Brink orbital, and this orbital is the basis in unified handling of the shell-model state and the cluster states [77]. 8

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In the second line, ψR8 (α, 4N ) represents the 8 He wave function localized on the right-hand side, and hence ψR8 ≡ ψR (α) 4q=1 ϕq (R). In the last line of equation (13), ψˆ 4 (ξα ) 8 (ξ6 ) are the internal wave functions for the α and 8 He and ψˆ m nuclei, respectively, while the first exponential part with the reduced mass µ and the second one with the total mass A correspond to the wave functions for the relative and centerof-mass motions, respectively. The wave function of the 8 He cluster contains the four neutron configurations, m = (m1 , m2 , m3 , m4 ), and each mq is a set of a neutron’s atomic orbit, (R, pdq , σq ). It should be stressed that S is not a dynamical coordinate but a variational parameter called the generator coordinate. The basis function m (S ) has a peak at R = S , and hence S (= |S |) is often called the distance parameter.

The clustermodel wave function can be constructed from the product of the Brink orbitals. Let us discuss the properties of the clustermodel wave function by showing the example of 12 Be. If this nucleus is decomposed into the binary cluster state of α + 8 He, the 12-body wave function becomes [8, 77]

2.1.2. Example of a cluster-model wave function.

12 (r1 . . . r12 , SL , SR )   = J A ψL (α)ψR (α) 4q=1 ϕq (R, pdq , σq , n) .

(7)

The total wave function of 12 is divided into three parts: the α cluster on the left (ψL (α)) and right (ψR (α)) sides and four neutrons. The explicit form of the wave function of the α particles is (8) ψL (α) ≡ 4 (r1 . . . r4 , SL ) = 4i=1 φ1s (ri − SL ) χσi ,τi 4 ψR (α) ≡ 4 (r5 . . . r8 , SR ) = i=1 φ1s (ri − SR ) χσi ,τi . (9) In this equation, ri represents the position vector for the ith nucleon, and all of ri and the vectors of SL,R are measured from an arbitrary center. Individual α clusters are designated by the direct product of the Brink orbital φ1s (r − S ) in the HO potential at the center of SL and SR , and χσi ,τi means the wave function of the spin–isospin part with σ =↑, ↓ and τ = p, n. The symbol A denotes the anti-symmetrization operator, while J represents the normalization constant. In equation (7), ϕq represents the wave function of the qth neutron (q = 1–4) localized on the right-hand α particle, ϕq (R, dq , σq , n) ≡ φ1pdq (rq − SR ) χσq ,n

2.1.3. Configuration mixing to produce a cluster-model wave function. The wave function in equation (7) is divided into

an α particle and 8 He with a distance parameter, S. Since the anti-symmetrization among 12 nucleons is completely taken into account, this wave function smoothly goes to a simple mean-field state with the (1s)4 (1p)8 configuration in the shellmodel limit of S → 0. Therefore, the shell-model limit wave function has a total oscillator quantum, N = 8. In contrast, the α + 8 He wave function with a finite S is generated by the configuration mixing of various nucleons’ particle–hole excitations with the higher oscillator configurations in a onecenter mean field. By employing equations (7)–(12), we can estimate the order of magnitude of the configuration mixing in the shell-model calculation, which has to produce the cluster states. To calculate the mixing weight, we calculate the expectation value of the total oscillator quantum N ,

(10)

with the 1p-wave version of the Brink orbital
 √ 2ν 3/4 2 φ1pd (r − SR ) = 2 ν ud · (r − SR )e−ν(r−SR ) . π (11)

N  = 12 (r1 . . . r12 , SL , SR )|N |12 (r1 . . . r12 , SL , SR ). (14)

In equation (11), the direction of the 1p wave is designated by the unit vector of ud denoting the direction of (u1 , u2 , u3 ) = (xˆ , yˆ , zˆ ). Since there are three directions, x, y, z, in the 1p shell, in principle, we can set an arbitrary direction for the four excess neutrons. In the present demonstration, we consider a simple paired configuration perpendicular to the α–α axis as (pd1 , pd2 , pd3 , pd4 ) = (px , px , py , py ) and (σ1 , σ2 , σ3 , σ4 ) = (↑, ↓, ↑, ↓). The corresponding expression of the wave function is

4q=1 ϕq (R, dq , σq , n) = ϕ1 (R, x, ↑, n)ϕ2 (R, x, ↓, n) ×ϕ3 (R, y, ↑, n)ϕ4 (R, y, ↓, n).

with the definition of N=

12  

ni (k),

(15)

k=x,y,z i=1

where k and i represent the oscillator direction and the nucleon number, respectively. If the distance between α and 8 He is sufficiently large, all the single-particle orbits are orthogonal with each other. Thus, the matrix element of equation (14) becomes the following simple summation:

(12)

In a simplified treatment, the origin of the shift vectors, SL and SR , is taken to be the center of mass system of α + 8 He, and the relative shift vector S is defined by S = SR − SL . In addition, the direction of S is set to be the z axis, and hence S = (0, 0, S), SL = (0, 0, SL ) and SR = (0, 0, SR ). SL and SR can be parametrized by S: SL = −2/3S and SR = 1/3S. By introducing the coordinate vectors for the relative motion (R) and the center-of-mass motion (Rcm ), the basis function in equation (24) can be finally reduced to the following form [8]:   12 (r1 , . . . r12 , S ) = J A ψL (α)ψR8 (α, 4N )  2 2 8 (13) ∼ A ψˆ 4 (ξ1 ) ψˆ m (ξ2 ) e−µν(R−S ) e−Aν Rcm .

N  =

12  

φi (r − Si )|n(k)|φi (r − Si ).

(16)

k=x,y,z i=1

Here the integration over the spin and isospin functions for a single nucleon has been already performed, and hence the matrix elements contain only the spatial integration. In this condition, the summation in equation (16) becomes   20 N  = 2 + 2 + 4νSL2 + 4νSR2 = 4 + νS 2 . 9

(17)

In the first line of equation (17), the first two numbers of 2 arise from the first and second summations in equation (16), which 9

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count the oscillator quanta in the x and y directions. Since a pair of the excess neutrons occupies the 1px and 1py orbitals, around the right-hand α cores, the total oscillator quantum for the excess neutrons is equal to 2 + 2. In the third line of equation (17), the last term originates from the development of two α cores, which keep the separation of S. The number of 4 originates from the condition that an α particle contains four nucleons. The result of equation (17) means that the expectation value of the mixing oscillator quanta can be parametrized by two values, the distance parameter S and radius parameter of the HO potential, ν, which is taken to be ν = 0.23 fm−2 in standard calculations for lighter systems. If the relative distance parameter for α + 8 He is S = 4 fm, corresponding to the surface touching distance, N  reaches about 12. One general property of cluster states is that the composite clusters are weakly coupled with each other. Thus, the tail of the cluster’s relative wave function is much extended. If the distance parameter is S ∼ 6.5 fm, where there is a large amplitude in the clusters’ relative wave function as shown in the realistic calculation, we obtain N ∼ 26. The√standard deviation around N can be estimated to be N = N ∼ 4, and hence the range of the mixing quanta amounts to about 30. This order of magnitude for the mixing quanta seems to be much larger than the computational scale in the present ab initio calculation, the no-core shell model [78–81], for instance. In a mean-field approach, huge model space is required in handling the clustering phenomena. The estimation of the total oscillator quantum in equation (17) is just based on a finite shift of the center of the four nucleons. In this estimation, the weak-bonding property of the cluster relative wave function is not completely taken into account. As was pointed out in the introduction, cluster states are weak-bonding or resonant states, which appear at the threshold energy for the decays into the corresponding subunits. If we consider the weak-bonding property of the cluster wave function, the total quanta greatly exceed the present estimation. In a weak-bonding cluster system, the tail of the relative wave function of the clusters is much more extended than the tail in the ground state because of the quantum tunneling effect. Moreover, the cluster wave function smoothly connects to the scattering state if a cluster state is realized as a resonant state. The superposition of the many particles plus many holes in a uniform mean field is expected to make it difficult to describe such an extended tail or the scattering boundary for the coherent four-particle motion. In fact, the order of magnitude of the oscillator quantum has been estimated in the 3α wave function in 12 C [82]. According to the analysis in [82], the oscillator quantum has been found to go beyond N ∼ 100.

cluster model according to the concept of the two-center covalent orbits. Brief explanations of GTCM have already been given in [68, 71, 73]. In the cluster-model wave function of α + 8 He in equation (7), the orbits for the four excess neutrons occupy the atomic orbits localized on the right-hand α particle. This state just corresponds to the chemical bonding state constructed by the VB method [83], where valence electrons localized at individual ions and their exchange effects are completely taken into account. In contrast, the covalent orbits for valence neutrons, which spread over two α cores, are constructed by the MO method [83], which can be achieved by LCAO. Therefore, both the cluster-model wave function and covalent orbits are analogous to a standard technique to describe the electrons’ orbits in a two-center molecule. In the internal region, where two α cores overlap but their separation is finite, neutrons favor the covalent orbits spreading over two α particles. Thus, we construct the covalent orbit from the atomic Brink orbit. Since four nucleons (two neutrons and two protons) inside individual α particles fill the 1s1/2 orbit, the covalent orbit for excess neutrons must be constructed from the 1p-wave atomic orbits. In the LCAO treatment, the covalent orbit is obtained by superposing the left-hand AO ϕ(L) ˜ and the right-hand AO ϕ(R) ˜ as ϕ(L) ˜ ± ϕ(R). ˜ There are three pairs of the covalent orbits around two α cores constructed by the LCAO treatment. The explicit expressions of the LCAO covalent orbits are given by ± = ϕ(L, ˜ +1, ↑) ∓ ϕ(R, ˜ +1, ↑) πK=+3/2

(18)

∓ σK=+1/2 = ϕ(L, ˜ 0, ↑) ± ϕ(R, ˜ 0, ↑)

(19)

± = ϕ(L, ˜ +1, ↓) ∓ ϕ(R, ˜ +1, ↓), πK=+1/2

(20)

where ϕ(A, ˜ lz , σ ) is the 1p-wave AO localized around one of the α clusters, which is labeled by the center A(= L or R), the orbital direction of lz (= ±1, 0), and the nucleon’s spin σ (=↑ or ↓). In equations (18), (19) and (20), each of the 1pwave AOs, ϕ(A, ˜ lz , σ ), can be expressed by the Brink orbits in equations (10) and (11), ϕ(A, ˜ ±1, σ ) = ∓ϕ(A, x, σ ) − i · ϕ(A, y, σ ) ϕ(A, ˜ 0, σ ) = ϕ(A, z, σ ).

(21) (22)

All covalent orbits include the distance parameter between the two α cores, S. The relative coordinate between the α cores is taken to be the z axis for the definition of the K quantum number, which is a projection of the total angular momentum to the z axis, and all of the orbits have good K quantum numbers. The extension of AO to the higher shell-model orbits, such as 2s, 1d, 2p, . . . , can also be done in a straightforward manner, although in the present studies AOs are restricted to the lowest unoccupied orbit, 1p, for simplicity. A schematic picture of the LCAO single-particle levels is shown in figure 7. In equations (18), (19), and (20), the orbits with a lower sign are called the ‘bonding orbits’, while those with the upper sign are called the ‘anti-bonding orbits’. In general, energies of the bonding orbits are lower than those of the anti-bonding orbits because the latter orbits have an

2.2. Framework of GTCM

In the previous sections, we show a basic structure of the clustermodel wave function in a simple two-center form, and global behaviors of covalent orbits in the two-center potential. In this section, we formulate the GTCM by extending the microscopic

2.2.1. Extended basis function in a two-center system.

10

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S =0

S> 0

1s0d

1/2

1p0f

1/2

+

Review Article

−

π

to be a basis function in solving an eigenvalue problem of a total system. The basis function in the GTCM has a generalized form of equation (12) in the configuration of excess neutrons; the centers and the directions of the neutrons’ orbits are flexible in equation (23), while the orbits are restricted in a standard cluster-model wave function in equation (12). The complete form of the basis function, which contains the α cores’ degree of freedom explicitly, is given by   4  

ϕq (mq ) (24) m (S ) = A ψL (α)ψR (α)  

+

−

+

1s0d

3/2

0p

1/2

−

σ

−

σ

+

+

q=1

+

1/2

1s0d

−

π

Jm K (S)

3/2

−

α

π

α

+

S Figure 7. Schematic picture of the single-particle levels around two α cores. The solid and dashed lines represent the bonding and anti-bonding orbitals, respectively. The naive shell-model orbit, corresponding to MO in the zero limit of the α–α distance (S → 0), is shown in the left-hand part of the single-particle energy levels. Reproduced from [76]. © 2012 The American Physical Society.

additional node along the α–α axis in comparison with the former orbits. In figure 7, the location of the single-particle level is shown schematically by the solid and dashed levels for the bonding and anti-bonding orbits, respectively. In the + orbit becomes case of a finite α–α distance (S > 0), the σ1/2 − . an intruder state, which has a lower energy than that of π1/2 In the left-hand part of the single-particle levels, we show the naive shell-model orbits, corresponding to the limit of S = 0. As can be confirmed from figure 7, we consider the covalent orbits around two α cores up to the 2p1f configuration for the excess neutrons in the present treatment, in which the AO is restricted to the 1p shell. Various MO configurations can be obtained, such as − 2 − 2 − 2 + 2 ) (σ1/2 ) , (π3/2 ) (π1/2 ) , and so on, by filling the MO (π3/2 levels in the case of 12 Be with four excess neutrons, for instance. If a configuration interaction among various MO configurations is performed, such a calculation corresponds to the shell-model calculation inside of a two-center mean field. However, a configuration interaction among MOs can be reduced to the coupled channels among the two atomic bases (or VB bases). This is because general MO configurations, which are the product of the orbits in equations (18)–(20), are constructed by the linear combination of the product function of equations (21) and (22). This is a basic idea in the GTCM. Specifically, in the framework of the GTCM, we can set a direct product of the 1p Brink atomic orbits, 4

ϕq (Aq , dq , σq ),

(25)

As has already been explained in the previous section, the α-cluster ψn (α) (n = L, R) is expressed by the (1s)4 configuration of the HO potential centered at the left (L) or right (R) side with the relative distance S [8]. ϕq (mq ) represents the 1p-wave AO for the j th neutron in the Cartesian coordinate, and {mq } are indices of AO (A, d, τ ). In the lefthand side of equation (24), m represents a set of AOs for the four neutrons, m = (m1 , m2 , m3 , m4 ). In equation (25), the intrinsic basis functions m (S ) are fully anti-symmetrized by the operator of A, and hence this basis function smoothly goes to the shell-model limit in the case of S → 0. The basis function is projected to the eigenstate of the total spin parity J π and its intrinsic angular projection K π in equation (25) by the projection operator PˆKJ . The projected basis function depends only on the magnitude of S . The J π projection is essential in handling nuclear systems because basis functions do not necessarily keep good symmetries on spatial rotation and spatial reflection.

−

0p

S

= PˆKJ m (S ). π

The wave function of 12 Be is finally given by taking a superposition over the relative distance parameter S and m as   π Jπ Jπν Cm (S) Jm (S). (26) ν = dS

2.2.2. Total wave function and the eigenvalue problem.

m

The total wave function in equation (26) is feasible to describe the vibrational motion of two α cores as well as the rotational motion of the whole system because of the superposition of the distance parameter S. The vibrational motion corresponds to the higher nodal excitation in the α–α relative wave function. The basis function in equations (25) and (24) is constructed from the covalent MO picture in the two-center system, but the total wave function in equation (26), which is the superposition of the basis, can reproduce not only MO configurations but also the asymptotic binary channels composed of the He dimers, such as α + 8 He (A1 A2 A3 A4 = LLLL or RRRR in equation (24)), 6 He +6 He (LLRR or RRLL), and 5 He +7 He (LLLR or RRRL) [84, 85]. A detailed explanation of the consistent treatment of the MOs and the He dimers will be presented in a later section. Jπν The coefficients Cm (S) in equation (26) for the νth eigenstate are determined by solving a coupled channel GCM (generator coordinate method) equation [8]  π  π π Jm (S)|Hˆ − EνJ |νJ = 0. (27)

(23)

q=1

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The bound states below the particle decay threshold are calculated by a diagonalization procedure, while the scattering boundary condition is explicitly imposed above the threshold. The details of the scattering boundary condition will be explained in the later sections. If we fix the distance parameter S for the α–α core and diagonalize the Hamiltonian shown in equation (30), we obtain the energy eigenvalues for a given S. Namely, we solve (Hˆ − EµJ (S))AS (S) = 0  Jπµ Jπµ Jπµ AS (S) = DmK (S) mK (S). π

Jπµ

Table 1. Comparisons of the threshold energies in the calculation with those in the experiment. In the upper panel the result of 10 Be is shown, while that of 12 Be is shown in the lower panel. All the energies are measured from the energy of the calculated ground state (units of MeV). The parameters of Volkov No 2 are M = 0.643 and B = H = 0.125, while the strength of G3RS is taken to be +3000 MeV and −2000 MeV for the repulsive and attractive parts, respectively. The radius parameter of HO b (ν = 1/2b2 ) is fixed to be 1.46 fm.

(28)

Channel in 10 Be α + 6 Heg.s. α + 6 He(2+1 ) 5 Heg.s. + 5 Heg.s. Channel in 12 Be α + 8 Heg.s. 6 Heg.s. + 6 Heg.s. 6 Heg.s. + 6 He(2+1 ) 5 Heg.s. + 7 Heg.s.

(29)

mK π The µth eigenvalue EµJ (S) is a function of the relative distance π parameter S, and a sequence of EµJ (S) forms the energy π Jπµ surfaces. The energies EµJ (S) and wave functions AS (S)

correspond to the so-called ‘AESs’ (adiabatic energy surfaces) and ‘ASs’ (adiabatic states), respectively, in atomic physics.

12  i

tˆi − Tˆc.m. +

12  i