Largely deformed states of {}^{13}B

Largely deformed states of B

Yoshiko Kanada-En’yo Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, Japan    Yusuke Kawanami111The present affiliation is TOHOKUSHINSHA FILM CORPORATION., Yasutaka Taniguchi Department of Physics, Kyoto University, Kyoto , Japan    Masaaki Kimura Institute of Physics, University of Tsukuba, Tsukuba , Japan

The excited states of B were studied with a method of antisymmetrized molecular dynamics(AMD). The theoretical results suggest that the intruder states with large deformations construct the rotational bands, and , starting from 5 MeV and 8 MeV, respectively. The neutron structure of the is analogous to the intruder ground state of Be. In the predicted band, we found very exotic structure with a proton intruder configuration. This proton intruder state has a larger deformation than superdeformation. The band-head state is assigned to the (4.83 MeV), which was experimentally suggested to be the proton intruder state because of the strong production via the proton-transfer to the Be() state in the He(Be,B) experiments.

I Introduction

In the recent progress of experimental and theoretical researches of unstable nuclei, various exotic phenomena have been discovered. As well known, one of the attractive subjects in neutron-rich nuclei is the breaking of neutron magic number such as and suggested in neutron-rich -shell and -shell nuclei. The breaking of the shell in Be has been well known for a long time because of the parity inversion of the ground state, . For Be, various experimental researches have been recently achieved iwasaki00 (); navin (); pain () to study the properties of the ground band, and the breaking of shell closure has been established. Concerning the mechanism of the shell breaking, the recent observationpain () of the significant -wave component in the Be ground state is the direct probe for the deformation, which should be one of the essential factors for the breaking in Be isotopes suggested in theoretical calculations otsuka96 (); ITAGAKI (); Enyo-be11 (); Enyo-be12 (); Oertzen-rev (). In the theoretical side, many kinds of microscopic calculations have been performed to investigate neutron-rich nuclei with the breaking shell closure. In case of neutron-rich Be isotopes, various properties have been successfully described by many groups from a point of view of cluster (see references in Ref.Oertzen-rev ()). As a result, it is considered that molecular orbital structure with large deformation is a key for the breaking of magic number in neutron-rich Be.

In the molecular orbital picture, Be isotopes are described by 2 clusters and valence neutrons which occupy the molecular orbitals formed around the coreITAGAKI (); Oertzen-rev (); Okabe77 (); SEYA (); OERTZEN (). The molecular orbitals are given by a linear combination of -orbitals around the clusters, and they are associated with the orbitals in two-center shell modeltwocenter (). In the spherical limit, the negative-parity -orbitals are the lowest for the valence neutrons. On the other hand, with the development of the clustering, so-called orbital, which is the longitudinal positive-parity orbital, comes down because its kinetic energy decreases. Finally, the inversion of the and orbitals occurs in a system with well-developed clustering. The inversion of the molecular orbitals corresponds to the parity inversion of the ’’ and ’’ orbitals because the and orbitals in the molecular orbital model are associated with the and orbitals in the deformed shell model, respectively. The intruder state of Be is written by the configuration with two neutrons in the -orbital. If the structure develops enough in Be, the intruder state may become the ground state. This necessarily causes the significant mixture of -wave component, which is consistent with the recent measurements pain (). Moreover, most of the low-lying states in neutron-rich Be isotopes can be well described by the molecular orbital structureITAGAKI (); Enyo-be11 (); Enyo-be12 (); Oertzen-rev (); SEYA (); OERTZEN (); ARAI (); DOTE (); Enyo-be10 (); OGAWA (); ITO04 (), and a variety of cluster states has been predicted in excited states of Be and Be Enyo-be11 (); Enyo-be12 (); Oertzen-rev (); OERTZEN (); Ito00 (); Descouvemont01 (). These facts indicate that cluster aspect is one of essential features in light unstable nuclei as well as in light stable nuclei. In particular, cluster structure is favored in neutron-rich Be, where a variety of exotic structure arises due to the the formation of 2 clusters and the molecular orbitals.

Let us turn to structure of B isotopes, which have a proton number larger by one than Be isotopes. In contrast to the situation of Be, B is considered to have the normal ground state with a neutron -shell closed configuration. Comparing with the intruder configuration of the ground state in Be, this shows that the magic number is restored in B due to the additional proton. Thus, the additional proton gives drastic structure change of the ground state, however, it is still natural to expect that intruder states may appear in the excited states of B. If a cluster structure can develop also in B as well as Be, the intruder states may lie in low excitation energy region, because such states can be stabilized in a similar way to the ground state of Be. The intruder states may have large deformation and construct rotational bands. Cluster features in neutron-rich B isotopes have been systematically studied in a chain of B isotopes with a molecular orbital model(MO) SEYA () and a method of antisymmetrized molecular dynamics(AMD)ENYObc (); ENYOsup (). These studies have been concentrated on the structure of ground states, and have shown that B is most spherical among B isotopes, while developed cluster structure has been predicted in further neutron-rich B like B and B. These results suggest a trend of two-center cluster structure in systems.

In this paper, we investigated deformed states of B by performing microscopic calculations of the ground and excited states of B. We focused on their cluster aspect. In particular, the molecular orbital structure is our major interest. In the present study we applied an AMD method, which is known to be a powerful approach to investigate cluster structure of unstable nucleiENYObc (); ENYOsup (); AMDrev (). The present method is the same as that applied to the studies of Be,Be, and BeEnyo-be11 (); Enyo-be12 (); Enyo-be10 (). Namely, we performed variation after spin-parity projection within the framework of AMDEnyo-c12 (). This method has been proved to successfully describe various properties of the ground and excited states of Be isotopes.

The paper is organized as follows. In the next section, we briefly explain the theoretical method of the present work. Results and discussions are given in III. Finally, we give a summary in IV.

Ii Formulation

We performed energy variation after spin parity projection(VAP) within the AMD model space, as was done in the previous studiesEnyo-be12 (); Enyo-be10 (); Enyo-c12 (). The detailed formulation of the AMD method for nuclear structure study is described in ENYObc (); ENYOsup (); AMDrev (); Enyo-c12 (). In particular, the formulation of the present calculations is basically the same as that described in Enyo-be12 (); Enyo-be10 ().

An AMD wave function is a Slater determinant of Gaussian wave packets;


where the th single-particle wave function is written by a product of spatial(), intrinsic spin() and isospin() wave functions as,


and are spatial and spin functions, and is iso-spin function which is fixed to be up(proton) or down(neutron). We used a width parameter fm, which is chosen to be the optimum value for B. Accordingly, an AMD wave function is expressed by a set of variational parameters, .

For the lowest state, we varied the parameters and () to minimize the energy expectation value of the Hamiltonian, , for the spin-parity projected AMD wave function; . Here, is the spin-parity projection operator. Then we obtained the optimum solution of the parameter set; for the lowest state. The solution for the th state are calculated by varying so as to minimize the energy of the orthogonal component to the lower states.

After the VAP calculations for the states with respect to various , and , we obtained the optimum intrinsic wave functions, , which approximately describe the corresponding states. After the VAP, we superposed the spin-parity eigen wave functions projected from all the obtained AMD wave functions. Namely, we determined the final wave functions for the states as,


Iii Results

We adopted the same effective nuclear interaction as that used in the study of Be isotopes Enyo-be11 (); Enyo-be12 (), which consists of the central force, the spin-orbit force and the Coulomb force. The Majorana parameters in the MV1 force are , and the strengths of the spin-orbit force are MeV. The VAP calculations of AMD using these interactions reproduce well the breaking of neutron magic number in Be and Be. With this interaction the calculated binding energies of Be and B are 61.9 MeV and 76.4 MeV, which underestimate the experimental values, 68.6 MeV and 84.5 MeV, respectively, however, we adopted this parametrization because the energy levels of the excited states in Be, Be and Be are reasonably reproduced.

The basis AMD wave functions were obtained by the VAP for the ground and excited states of B. The number of the basis AMD wave functions in the present calculations are 23. The initial wave function in the energy variation was randomly chosen for states. For states, we started the variational calculation from the initial wave function projected from the obtained wave function for the states. These independent AMD wave functions were superposed to calculate the final wave functions.

iii.1 Energies and deformation

B is a nucleus with neutron magic number , and its ground state is the with normal configuration of -shell closure. Above the ground state of B, it is experimentally known that many states exist in the excitation energy MeV region with high level density. Unfortunately, spins and parities of most of these states are unknown. Recently, the state at 4.83 MeV has been assigned to be a state by He(Be,B) experimentsOta07 (). Because of its strong production via proton-transfer to the Be() state, this excited state is suggested to be a proton intruder state.

The calculated energy levels of the negative- and positive-parity states of B are shown in Fig. 1. In addition to the ground state, we obtained many excited states with various in the region MeV. These states may correspond to the observed levels in this energy region. In the excited states, we found three largely deformed bands, , and (solid lines). These bands are composed of intruder states or well-developed cluster states. In particular, the band-head state of the is the proton intruder state with a large deformation, and hence this should be assigned to the experimental (4.83 MeV) state. The band was obtained by the spin-parity projection and the diagonalization of the obtained wave functions, though the VAP calculations were not performed for the corresponding states. This band is dominantly the -Li cluster state. The intrinsic structures of these deformed states are discussed later in detail. As for other excited states (disconnected filled circles), intrinsic deformation of the major AMD wave function , which dominates the final wave functions , is small or as large as normal deformation at most. These excited states are regarded to be dominated by or and neutron configurations.

Figure 2 shows density distribution and deformation parameters of the major AMD wave functions, , , , , and , which were obtained by the VAP for the corresponding states. The ground state () has the most spherical shape(Fig. 2(b)), due to the neutron -shell closure. This is consistent with the previous work by AMD ENYObc (). In the state(Fig. 2(a)), a three-center cluster core structure appears. The core clusters are an with two valence neutrons, a triton and an . This state is approximately regarded as the -limit cluster state though the spatial cluster development is somehow contained. The similar three-center cluster structure is found also in the state. In the and the states (Fig. 2(c) and (d)), we found remarkably deformed structure with developed cluster cores. These states are the members of the band which starts from MeV. It is interesting that such a largely deformed band appears only 5 MeV above the ground state, even though this nucleus has neutron magic number . The state of this band is on the yrast line. Moreover, it is striking that a further large deformation arises in the state(Fig. 2(e)), which is the band-head state of the band. The deformation of this state exceeds the value for superdeformation and is close to the value 0.9 for hyperdeformation. In the state, we obtained the well-developed cluster structure like Li+(Fig. 2(f)). In the final wave functions after the superposition, these two components of the largely deformed state (Fig. 2(e)) and the Li+ cluster state (Fig. 2(f)) constitute the rotational band, , with a mixing of them. The , , states are dominated by in about 90%, 65% and 60%, respectively. On the other hand, the and contains major percentage of with remarkable Li+ cluster structure, while other states in the band are the mixture of these two components. It indicates that the weak-coupling cluster feature is enhanced in high spin region of the band. The negative-parity states projected from the Li+ cluster structure construct the band.

Figure 1: Excitation energies of the negative- and positive-parity states of B calculated by superposition of the basis wave functions. Filled circles are the energies of the states, for which the VAP calculations were done. Open circles are the energies of the states, which were obtained by diagonalization of Hamiltonian by superposing wave functions, but the VAP calculations were not performed for the corresponding states.

Figure 2: Density distribution of the ground and excited states of B. The intrinsic density of the dominant AMD wave function in the is shown. The density is integrated along the axis perpendicular to the plane. The deformation parameter Enyo-oppo () of matter density for the intrinsic state are also given. The box size is 10 fm.

iii.2 Cluster feature in the and

In this subsection, we discuss cluster features of the deformed bands, and , by analyzing the single-particle orbitals in the intrinsic states. Hereafter, we mainly analze the major AMD wave functions, , and obtained by VAP for the , and states, and compare them with that for the intruder ground state of Be.

First, we give single-particle energies in Fig. 3. In these deformed states, the level structure of the single-particle wave functions shows a feature of the structure rather than that of spherical shell structure. That is to say, the lowest four proton orbitals and the lowest four neutron orbitals form the 2 core, while the higher orbitals correspond to one valence proton and four valence neutrons.

Next, we illustrate the density distribution of the single-particle wave functions for the valence nucleons in Fig. 4. The total densities of protons and neutrons are also given as well as total matter densities in the figure. Generally speaking, the matter densities show two-center structures in all these deformed states. However, the behavior of the valence nucleons are different among these three states, , and .

In the state (Fig. 4(a)), two valence neutrons occupy an approximately positive-parity orbital, and the other two neutrons and a proton are in orbitals with dominant negative-parity component. Since the negative- and positive-parity orbitals of the valence nucleons are associated with the -orbitals and -orbitals, respectively, we can roughly describe the states in the band by the neutron excited configurations. Let us turn to the molecular orbital features. The positive-parity orbital of the last two neutrons is largely deformed and has nodes along the direction(longitudinal axis). This orbital well corresponds to the so-called orbital in the molecular orbital picture ITAGAKI (); Oertzen-rev (); Okabe77 (); SEYA (); OERTZEN (). It has been already known that the -like orbital of valence neutrons appear in various Be isotopes (see references in Oertzen-rev ()). In the AMD studyEnyo-be12 (), it has been revealed that the ground state of Be is dominated by the intruder state with two neutrons in the orbital, which reduce kinetic energy due to the developed -core structure. In Fig. 4(4), density distribution and single-particle orbitals of Be() are shown. As seen in the figure, the last two neutrons in the Be() state occupy the orbital. The point is that the neutron structure of the B() band is very similar to that of the Be(). Therefore, we conclude that the is the band of the intruder neutron states, and interpreted as Be()+, where the Be() has the intruder configuration and the additional proton strongly couples to the deformed core. It is also interesting that the additional proton in the normal -shell affects a change of the deformation of total density, which results in the smaller deformation of the B() than the Be().

In the band-head state of the band (Fig. 4(b)), excitation occurs in the proton shell. Namely, the last proton occupy a -like orbital, which is quite similar to that of the highest neutron orbital in the band and also that in the Be(). This is extraordinary configuration because excitations are naively expected in the neutron side in case of neutron-rich nuclei. It can be understood by the lowering mechanism of the orbital due to the developed two-center structure. In the two-center shell modeltwocenter (), the energy of the orbital comes down with the increase of two-center distance . On the other hand, the negative-parity levels() originating in the orbitals split and four of the negative-parity levels go up as the two-center distance increases. As a result, the inversion of the positive-parity orbital and the negative-parity orbitals occurs. Finally, in the large distance region fm, the orbital becomes the fifth orbital which is the lowest one for valence nucleons around the core. This situation is realized in the state, where the two-center structure with the distance fm was obtained in the present calculation. It is also associated with the hyperdeformation, where the declined positive-parity orbital becomes the fifth lowest orbital in the Nilsson’s deformed shell model. In fact, the deformation of the state exceeds the value for the superdeformation. On the other hand, four valence neutrons are localized in one side of the two-center structure. It is contrast to the molecular orbital feature of the valence proton, which is moving around the whole system. It seems that spatial correlation of four valence neutrons is so strong that form a shell closure. As a result, totally 8 neutrons separate into two groups consisting of 2 and 6 neutrons with a weak-coupling feature. We here give a comment on similarity of the neutron structure between the and states. Comparing the neutron structure of the with that of the , the profile of the total neutron density of the is similar to that of the described by the neutron configuration. Therefore, we can propose an alternative interpretation in a mean-field picture that the state is roughly described by the configuration with the proton and the neutron excitations. In the present calculation, we suggest that such the exotic state, the proton intruder state of neutron-rich nuclei, may exist as the lowest state at MeV. Recently, the state at 4.83 MeV has been assigned to be a state by He(Be,B) experimentsOta07 (). Because of strong production via the proton-transfer to the Be() state and analysis of angular dependence, Ota et al. suggested this state to be a proton intruder state. The present prediction of the proton intruder configuration in the state is consistent with this observation though the theoretical excitation energy of the present calculation is slightly higher than the experimental value.

In the state in the band, all the valence proton and the valence four neutrons are localized around one of the cores(Fig. 4(c)). It indicates that the molecular orbital aspect disappears, while the weak-coupling feature of -Li clustering is enhanced. Because of the spatial localization, the orbitals of the valence nucleons have no definite parity. In this case, the unnatural parity ’’ of the total system is carried by the parity asymmetric -Li structure. It is in different situation from the state with the intruder proton configuration, where the parity ’’ originates in the proton excitation of the single-particle orbital from negative to positive one. As mentioned before, after the superposition of the AMD wave functions, the proton intruder state and the Li cluster state are mixed to each other to contribute the structure change depending on in the band. The former component is dominant in the low spin states, while the latter component is significant in high spin states.

Figure 3: Single-particle energies in the , and states. The energies are calculated by diagonalizing Hartree-Fock-like single-particle Hamiltonian within the single AMD wave function DOTE () which dominates the state.

Figure 4: Distribution of the proton, neutron and matter density, and density distribution of the valence nucleons. The left three panels show the distribution and deformation parameter of matter, proton and neutron density. In the right three panels, density distribution of the highest proton orbitals, the highest and the third neutron orbitals are given. The single-particle orbitals are extracted by diagonalizing Hartree-Fock-like single-particle Hamiltonian within a single AMD wave function DOTE (). The results are those calculated for the intrinsic AMD wave function , which dominates the state. The AMD results for the ground state of Be are also shown for comparison. The box size is 10 fm.

Iv Summary

The excited states of B were studied with a method of antisymmetrized molecular dynamics(AMD). We obtained the largely deformed states which construct the rotational bands, , and . In these deformed states, we found various kinds of cluster aspect including molecular orbital features in the two-center structure.

The band is the deformed one with molecular orbital structure. This band is described by the intruder neutron configuration, and interpreted as Be()+, where the Be() has the intruder configuration and the additional proton strongly couples to the deformed Be core. The excited neutron orbital is regarded as the orbital in the molecular orbital picture. Experimentally, there is a report which has been suggested the excited state at 10 MeV to be a high spin state of the neutron configuration Kalpakchieva00 (), which would be a member of this band.

We found the proton intruder structure in the band with remarkably large deformation. The deformation of the band-head state is larger than that for the superdeformation. In the molecular orbital picture, the last proton is described by the orbital. This is very exotic state with the proton intruder configuration in neutron-rich nuclei. In the present calculations, it was suggested that the band starts from the lowest state at MeV. We assigned this state to the recently observed state at 4.83 MeV, which has been suggested to be the proton intruder state by Ota et al. in the He(Be,B) experiments. In the high spin states of the band, +Li-like cluster structure well develops.

It is striking that such the deformed states with highly excited configurations or well-developed cluster states appear in the energy region compatible to the normal excited states, even though B has the neutron-shell closure in the ground state. Especially the molecular orbital, , is formed in the deformed states of B as well as neutron-rich Be isotopes. The cluster aspect of the deformed states in B can be understood in natural extension of cluster structure of Be isotopes. That is to say, the formation of the -cluster core and the role of the valence nucleons(one proton and four neutrons) are key in the largely deformed states of B. It is challenging to investigate possible exotic structure with clustering in excited states of further neutron-rich B isotopes like B and B.


The authors would like to thank Dr. Ota and Prof. Shimoura for the valuable discussions. In fact, this study has been motivated by their suggestion of the “proton intruder state”. They are also thankful to members of Yukawa Institute for Theoretical Physics(YITP) and Department of Physics in Kyoto University. The computational calculations in this work were performed by the Supercomputer Projects of High Energy Accelerator Research Organization(KEK) and also the super computers of YITP. This work was supported by Grant-in-Aid for Scientific Research Japan Society for the Promotion of Science and a Grant-in-Aid for Scientific Research from JSPS. It is also supported by the Grant-in-Aid for the 21st Century COE ”Center for Diversity and Universality in Physics” from MEXT. Discussions during the workshop YITP-W-06-17 on Nuclear Cluster Physics held in YITP were useful to complete this work.


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