2\alpha+t cluster feature of 3/2^{-}_{3} state in {}^{11}B

cluster feature of state in B

Yoshiko Kanada-En’yo Department of Physics, Kyoto University, Kyoto 606-8502, Japan    Tadahiro Suhara Matsue College of Technology, Matsue 690-8518, Japan
Abstract

We reanalyze cluster features of states in B by investigating the cluster distribution around a core in B, calculated with the method of antisymmetrized molecular dynamics (AMD). In the state, a cluster is distributed in a wide region around , indicating that the cluster moves rather freely in angular as well as radial motion. From the weak angular correlation and radial extent of the cluster distribution, we propose an interpretation of a cluster gas for the state. In this study, we compare the cluster feature in B() with the cluster feature in C(), and discuss their similarities and differences.

preprint: KUNS-2525

I Introduction

Many cluster structures have been found in light nuclei, such as Li with an cluster and Be with a cluster. In the excited states of C, various cluster structures have been discovered (for example, Refs. Fujiwara80 (); Freer:2014qoa () and references therein). In the early stages, the possibility of a linear 3 chain structure was proposed for C() by Morinaga et al. morinaga56 (); morinaga66 (). However, the linear chain structure of C() has been excluded by the decay width of this state suzuki72 (). Later theoretical studies with cluster models have revealed that C() is a weakly bound state with neither geometric structures of the linear chain nor triangle structures kamimura-RGM1 (); kamimura-RGM2 (); uegaki1 (); uegaki2 (); uegaki3 (); descouvemont87 (); kurokawa04 (); kurokawa05 (); Arai:2006bt (). Further extended models that do not rely on assumptions of the existence of clusters have also obtained similar results of the cluster feature in C Enyo-c12 (); KanadaEn'yo:2006ze (); Neff-c12 (); Chernykh:2007zz ().

In 2001, Tohsaki et al. proposed a new interpretation of C() Tohsaki01 (), i.e., the ” cluster gas” in which clusters are weakly interacting like a gas Funaki:2002fn (); Yamada:2003cz (); Funaki:2003af (); Funaki:2009zz (); Yamada:2011bi (). In such a dilute cluster system, particles behave as bosonic particles; therefore, this state has been discussed in relation to the condensation predicted in dilute infinite matter Ropke98 (). The cluster gas phenomenon has been attracting great interest because it is a new concept of the cluster state and is different from the traditional concept of geometric cluster structures, in which clusters are localized and have a specific spatial configuration. Indeed, in this decade, theoretical and experimental studies of nuclei, such as He, Be, B, and O, have included intensive searching for cluster gas states Kawabata:2005ta (); KanadaEn'yo:2006bd (); KanadaEn'yo:2007ie (); Itagaki:2007rv (); Wakasa:2007zza (); Funaki:2008gb (); Funaki:2010px (); Yamada:2010vc (); Ohkubo:2010zz (); Suhara:2012zr (); Ichikawa:2012mk (); Kobayashi:2012di (); Kobayashi:2013iwa ().

For B, a developed cluster structure in the state was predicted by a cluster model nishioka79 () and the method of antisymmetrized molecular dynamics (AMD) KanadaEn'yo:2006bd (); Suhara:2012zr (). The state at 8.56 MeV is assigned to this state because the experimentally measured and monopole transition strengths Kawabata:2005ta (); Kawabata:2004pc () and the transition strength of the mirror state Fujita04 () are reproduced by the calculation. Studies have found similarity between B() and C() in remarkable monopole transitions and also predicted a nongeometric cluster feature of the structure in B() similar to that of the cluster structure in C() Kawabata:2005ta (); KanadaEn'yo:2006bd (); Suhara:2012zr (). From the analogies to C(), B() was interpreted as a cluster gas.

However, the cluster gas in a system has not been fully understood. The following problems remain to be clarified. First, the parity of the state is negative and conflicts with the original idea of cluster gas, in which clusters occupy an orbit Yamada:2010vc () and form a positive parity state. How can we extend the cluster gas picture to the state containing a cluster with negative parity? Second, it is not obvious whether the cluster state shows a nongeometric feature because a state with three clusters is more bound in the system than in the system, resulting in a smaller size of B() than that of C() because of the deeper effective potential between and clusters than that between two clusters.

In this study, we reanalyze the B and C wave functions obtained by AMD calculations in previous studies KanadaEn'yo:2006bd (); KanadaEn'yo:2006ze () and investigate the motion of a cluster around . We pay particular attention to the angular motion of the cluster with respect to the orientation to judge whether the structures in B have a geometric feature with an angular correlation or a nongeometric structure with weak angular correlation similar to the gas state of C). We also discuss the analogy and differences between structures in B and structures in C. By considering the body-fixed plane of three clusters, we can connect the -wave motion of the cluster around to the -wave motion of the cluster around , and then extend the picture of the cluster gas to the system.

The paper is organized as follows. We describe the AMD and cluster models in Section II. Section III discusses cluster structures of states of B, in comparison with those of states of C. The paper concludes with a summary in section IV.

Ii Formulation

In Refs. KanadaEn'yo:2006bd (); KanadaEn'yo:2006ze (), states of B and states of C have been calculated with the AMD model. In the present study, we reanalyzed the AMD wave functions of B and C obtained in previous studies by using the and cluster model wave functions written by the Brink-Bloch (BB) model Brink66 (). In this section, we briefly explain the AMD model adopted in previous studies and describe the BB cluster wave function used in the present analysis.

ii.1 AMD model

The AMD model is a useful approach to describe the formation and breaking of clusters as well as shell-model states having non-cluster structures KanadaEnyo:1995tb (); KanadaEnyo:1995ir (). The applicability of the AMD method to light nuclei has been proven ENYOsup (); AMDrev (); KanadaEn'yo:2012bj (). In the previous studies of B and C, the variation after spin-parity projection (VAP) in the AMD framework was applied KanadaEn'yo:2006bd (); KanadaEn'yo:2006ze (). For the detailed formulation of the AMD+VAP, please refer to the previously mentioned references.

A Slater determinant of Gaussian wave packets gives an AMD wave function of an -nucleon system:

(1)
(2)
(3)
(4)

where and are spatial and spin functions of the th single-particle wave function, respectively, and is the isospin function fixed to be up (proton) or down (neutron). Accordingly, an AMD wave function is expressed by a set of variational parameters, . The width parameter is chosen to be a common value for all nucleons and it is taken to be fm for B and C.

In the AMD+VAP method, we perform energy variation after spin-parity projections in the AMD model space to obtain the wave function for the lowest state. Namely, the parameters and () of the AMD wave function are varied to minimize the energy expectation value, , with respect to the spin-parity eigenwave function projected from an AMD wave function, . For excited states, the variation is performed for the energy expectation value of the component of the projected AMD wave function , orthogonal to the lower () states.

In each nucleus, all AMD wave functions obtained for various states are superposed to obtain final wave functions for states by solving the Hill-Wheeler equation. To describe B and C, approximately, twenty independent AMD wave functions are adopted for basis wave functions in the superposition KanadaEn'yo:2006bd (); KanadaEn'yo:2006ze (). Because the number of basis AMD wave functions is finite, continuum states cannot be treated properly in the present AMD+VAP method. The method is a bound state approximation in which resonance states are obtained as bound states.

In the AMD model space, we treat all single-nucleon wave functions as independent Gaussian wave packets, and therefore, cluster formation and breaking are described by spatial configurations of Gaussian centers, . If we choose a specific set of parameters , the AMD wave function can be equivalent to a BB cluster wave function.

Note that the cluster-breaking component affects not only the ground state but also the excited cluster states of C, as shown in Refs. Chernykh:2007zz (); Suhara-new (). We should stress that the AMD model can describe various cluster structures while incorporating cluster-breaking effects.

ii.2 Cluster model

To analyze -cluster motion in the B wave functions obtained by the AMD+VAP model, we measure the position of the cluster around the core by using the BB wave function, which expresses a three-center cluster wave function with a specific spatial configuration of cluster positions.

The BB wave function for a cluster structure of B is described as

(5)

Here and are - and -cluster wave functions described by the harmonic oscillator and shell-model wave functions with the shifted center position , respectively. The width parameter of the harmonic oscillator is fm, same as the AMD wave functions for B and C in the previous studies.

When clusters are far from each other and the antisymmetrization effect between clusters is negligible, indicates cluster center positions around which the and clusters are localized. In other words, the parameters specify the spatial configuration of cluster positions. Note that if are close to each other, the antisymmetrization effect is strong and does not necessarily have a physical meaning of the cluster position. For instance, in the small distance limit between cluster centers , the BB wave function no longer describes localized clusters but rather it describes a shell-model wave function.

We consider a configuration of three clusters, and , on the plane at with . Two clusters are set at the distance with the orientation parallel to the -axis as , and the cluster is located at the distance from the center of the (see Fig. 1). We define the -cluster position relative to the center as

(6)

where the -cluster direction is chosen at an angle from the -axis (the orientation).

To measure cluster probability at a certain position, we calculate the overlap of the wave function, , for B() with the -projected BB wave function

(7)

for . In the present study, the intrinsic spin is set to the direction and is chosen so as to fix the -component of the orbital angular momentum to be .

If a cluster is located far from and the antisymmetrization effect between and is negligible, the relative wave function between the cluster and in the BB wave function is given by a Gaussian with , i.e., the cluster is well localized around . Therefore, the BB wave function can be regarded as the ’test function’ for the cluster, located at from the core with the - distance . The overlap of the test function with is roughly regarded as the wave function on the plane at , and its square stands for the -cluster probability at around the core.

In the present analysis, the normalization of is determined by the normalization of the constituent cluster wave functions and before the antisymmetrization and projections. Namely, norms of and are chosen independently of and are kept to be constant. They are determined to make the norm of the BB wave function to be a unit at the limit where cluster positions are far from each other and the antisymmetrization effect vanishes.

As cluster positions become close to each other, the norm of the BB wave function becomes smaller because of the antisymmetrization effect, which means that the cluster wave function, i.e., the -cluster probability, is suppressed by Pauli blocking effect between nucleons in different clusters.

Figure 1: Schematic figure of cluster structure.

For a system, we analyze the motion of an cluster around the core in a similar way to the cluster in the system. The BB wave function for a system is given by replacing the cluster in Eq. 5 with an cluster having a position of . A configuration of three clusters on the plane at is considered, and we define the distance parameter for the core and the position for the cluster around the in the same way as the case. To measure the cluster probability at a certain position around the core, we calculate the overlap of the wave function for C() with the -projected BB wave function. The overlap is regarded as the -cluster wave function on the () plane, and its square stands for the probability of an cluster at moving around the core.

Iii Results

iii.1 AMD+VAP wave functions for B() and C()

We analyzed the AMD+VAP wave functions for , , and states of B, and , , and states of C, which were obtained in previous studies KanadaEn'yo:2006bd (); KanadaEn'yo:2006ze ().

The adopted effective nuclear interactions is the MV1 force (case 3) MVOLKOV () of the central force, supplemented by the spin-orbit term of the G3RS force LS (). The interaction parameters used in Ref. KanadaEn'yo:2006bd () for B are , , and MeV, and those in Ref. KanadaEn'yo:2006ze () for C are , , and MeV. We slightly modified the parameter set for B from those for C to obtain a better reproduction of energy levels of B, as described in Ref. KanadaEn'yo:2006ze ().

The B wave functions are given by the superposition of eigenwave functions projected from 17 basis AMD wave functions obtained by the VAP calculations, whereas C wave functions are described by eigenwave functions projected from 23 basis AMD wave functions.

Figure  2 shows the energy levels for states of B and states of C. Compared with the experimental data, the interaction used in the previous studies tends to overestimate the relative energies to the threshold energies. We can improve the overestimation by changing the Majorana parameter of the MV1 force. To obtain deeper binding wave functions than the original results, we also perform calculations of B and C with the modified value in solving the Hill-Wheeler equation, using the same basis AMD wave functions as in the original studies. We call the original parametrization ”m62” and the modified one ”m60”. Figure  2 also shows the energy levels obtained with the m60 interaction. Interaction modification improves the energy positions of excited states. Because of the deeper binding, sizes of excited states and the radial motion of the -cluster are slightly shrunk in the m60 result, as shown later. However, the feature of the -cluster motion around in B is qualitatively unchanged by the modification from m62 to m60; therefore, in the present study, we mainly discuss the original m62 result.

Table 1 shows the calculated root mean square (rms) radii of matter density and rms charge radii. The experimental charge radii for the ground states are also listed. B(), C(), and C() have remarkably larger radii than the ground states because these states have developed cluster structures.

As discussed in Ref. KanadaEn'yo:2006ze (), C() has no geometric structure, but it is described by the superposition of various configurations of three clusters. This is consistent with calculations of the cluster model uegaki1 () and the fermionic molecular dynamics Chernykh:2007zz () and the cluster gas picture proposed by Tohsaki et al. Tohsaki01 (). In contrast to C(), for C(), the AMD calculation predicts a geometric cluster feature, having a large overlap with the cluster wave function with a chain-like configuration.

For B(), the AMD+VAP wave function shows a geometric feature quite similar to those of C(). This wave function is described by the superposition of various configurations of 2 and clusters, which means that three clusters are weakly interacting like a gas. Another analogy of B() to C() is strong monopole transition from the ground state Kawabata:2005ta (). This transition means that B() is understood by radial excitation, similarly to C(), rather than by angular excitation. Because of these analogies of B() to C(), an interpretation of the cluster gas state for B() was proposed in the previous studies.

Figure 2: Energies of states of B measured from the threshold and those of states of C measured from the threshold. Li+ and Be+ threshold energies are also shown. Theoretical values calculated using and those for are shown in comparison with the experimental energy levels.
m62 m60 exp.
matter charge matter charge charge
B 2.5 2.6 2.5 2.6 2.406 (0.291)
B 2.7 2.8 2.7 2.8
B 3.0 3.1 3.0 3.0

C
2.5 2.7 2.5 2.6 2.47 (0.022)
C 3.3 3.4 3.0 3.2
C 4.0 4.1 3.9 4.0
Table 1: The rms radii of matter density and rms charge radii calculated with the AMD+VAP using m62 and m60 interactions. The experimental charge radii are taken from Ref. nucl-data-charge-radius (). The unit is fm.

iii.2 and cluster systems

Because - and - effective interactions may give essential contributions to three-body cluster dynamics of and systems, we here describe properties of subsystems, and systems, obtained by the present nuclear interactions.

Li and Be are described well by and cluster models and they are regarded as weakly (quasi) bound two-body cluster states. The ground states of Li and Be are the and states, respectively. The experimental Li energy measured from the threshold is 2.47 MeV and the Be energy from the threshold is 0.093 MeV. These facts indicate that clusters are bound relatively deeper in the system than in the system.

We calculate Li and Be() with the generator coordinate method (GCM) using and cluster wave functions given by the BB model. The width parameter fm, same as for B and C wave functions, is used for and clusters. The distance parameters and for the generator coordinate is taken to be =1, 2, , 8 fm, and resonance states are obtained as bound states within a bound state approximation in a finite volume. In the GCM calculation, the - binding obtained in Li is deeper than the - binding in Be. The energies of Li and Be measured from the -decay threshold are 0.4 MeV and 1.4 MeV for the m62 case and 0.8 MeV and 1.2 MeV for the m60 case.

Figure 3 shows the energy projected from a single BB model wave function for and the energy for as functions of the intercluster distance . The figure also shows GCM amplitudes defined by the squared overlap of a -projected BB wave function with the GCM wave functions. Here the -projected BB wave function is normalized to have a unit norm. The energy curve shows the effective repulsion in a small distance region because of the Pauli blocking effect from the antisymmetrization. The effective repulsion is larger in the system because of the stronger Pauli blocking and it pushes clusters outward, as seen in the GCM amplitudes. Also in the system, clusters are developed spatially and distributed in an outer region because of the antisymmetrization effect in the inner region. Quantitatively, the system is relatively deeper bound than the system because of the weaker Pauli blocking as well as the attractive spin-orbit force.

Figure 3: (Upper) energy curve of the system for Li and that of the system for Be. The and projected energies are plotted as functions of the intercluster distance of the BB cluster wave function. (Lower) GCM amplitudes for the Li() state were obtained by the cluster GCM calculation, and those for the Be() state were obtained by the cluster GCM calculation.

iii.3 Cluster motion in B and C

In the previous study, we interpreted B() as the cluster gas state because of the analogies of B() to C() in cluster features and the monopole transition. One characteristic of these cluster states is that the nongeometric feature of three cluster configuration is different from C(), which has a large overlap with the open triangle configuration. If the system has a nongeometric cluster structure, the cluster around the has weak angular correlation, and it shows wide distribution in angular motion. Therefore, the distribution in the nongeometric cluster structure should be different from that in a geometric cluster structure, which concentrates at a certain angle, reflecting a specific configuration of cluster positions. In other words, the wide distribution of clusters in angular motion is an evidence of the nongeometric cluster structure, and it can be a probe for a cluster gas state, which is characterized by weak correlations in angular motion as well as in radial motion.

In the present analysis of cluster motion in and systems, we consider the body-fixed frame in which the configuration of cluster positions is parametrized by for the () position and for the - distance of the core, as explained before (see Fig. 1). In the body-fixed frame, the angular correlation is reflected by the angular distribution of the () cluster around the core.

iii.3.1 Energy surface

Figure 4 shows the energy surface of the wave function on the () plane for the position around a fixed core with the - distance and that of the wave function for the motion around a . The energy of the -projected BB wave function given as

(8)

for states of and states of are calculated with the m62 interaction.

In the energy surface of for the motion, the energy pocket at fm for the core with fm corresponds to the ground state of B. For the core with fm, the energy surface is soft in the fm region against the angular motion as well as the radial motion of the cluster. As shown later in this paper, B() contains significant components of in this soft region. In the and region along the -axis, the energy is relatively high, indicating that the linear configuration is unfavored. Also, in the energy surface of for the motion around the core, the energy surface is soft for the core with fm in the angular and radial motions, except for the and regions. It is shown that the and configurations for the linear structure are unfavored also in the system. From the softness of the energy surface, we expect that a cluster or an cluster can move around the rather freely in the large region, except for the and regions.

It should be noted that the state of has a node at the -axis and its component vanishes in the and configuration for the ideal linear structure. Although the energy of the state can be calculated for the small limit, the parameter does not have a physical meaning of the -cluster position in the fm region.

Figure 4: (Color online) Energy of the states projected from BB wave functions for the cluster and that of the states for cluster calculated with m62. This figure shows the energy surface on the plane for a fixed core with the distance , 4, and 5 fm.

iii.3.2 Cluster motion in B and C

To investigate cluster motion in B and C, we calculate the overlap between the BB wave function and the AMD+VAP wave function defined in Eq. 7. We focus on the dependence of to see the -cluster motion in B and to compare it with that of the -cluster motion in C.

We show the plot of the overlap for each for B(), B(), and B() in Fig. 5 and that for C(), C(), and C() in Fig. 6. The figures show the -cluster motion and the -cluster motion around a core with the fixed - distance . We also show the dependence of the for , 4, and 5 fm cases in Fig. 7, which shows the angular distribution of the cluster in B and the cluster in C.

In B(), a remarkable peak at fm for fm indicates that clusters are confined in the inner region to form a compact triangle configuration of . In contrast, B() has a small overlap with the component of cluster wave functions because this state is dominated by the excitation of the core and has a large overlap with the component rather than with . This result is consistent with the discussion in previous study that B() is the angular excitation, having the weak monopole transition from the ground state, and it has a radius as small as the ground state; therefore, it is not a cluster gas state. In B(), the cluster is not localized and its component is distributed in a wide area of and in the fm region. The component in the inner region near the core is suppressed because of the Pauli blocking between and clusters. Moreover, the component completely vanishes at the line because of the trivial nodal structure at the -axis of the motion in the orbit. Nevertheless, compared with the ground state, the B() wave function has a wide distribution of the cluster on the () plane, which means that the cluster is not localized, but it moves rather freely, excluding the and regions. Furthermore, it contains significant components for a large - distance fm of the core. This information indicates that two clusters in the core are bound more weakly in B() than in B().

Features of -cluster distributions in C() and C() are similar to the distributions in B() and B(), respectively. In C(), a significant peak of the overlap exists at fm for fm, indicating the large overlap with a compact triangle configuration of . In contract to the ground state, the cluster is not localized, but it is distributed in a wide area of and in C(). The component in the region close to the -axis, i.e., in the and regions, is suppressed because of the Pauli blocking effect from other clusters in the core and because the linear structure is energetically unfavored. As a result, the distribution in the fm region shows angular motion that is similar to that of the distribution in B(). One characteristic of the distribution in C() that is different from the distribution in B() is the significant distribution in the fm region. In this region, far from the core, the angular distribution of the cluster in C() becomes isotropic, showing -wave feature differently than that of B().

With C(), the distribution is concentrated in the fm and fm region. The distribution indicates a remarkably developed cluster structure, however, the angular motion of the cluster in C() is quite different from that in C(). C() shows a strong angular correlation corresponding to the geometric configuration of the chain-like structure. This result contrasts with the weak angular correlation in C(), in which the cluster is distributed in the wide region.

Note that the remarkable peak for the compact triangle configuration in the ground states of B and C originates in the antisymmetrization effect between clusters. In the region fm, the cluster wave function is almost equivalent to the shell-model wave function. Because the quantum effect is significant in this region, has less meaning of the localization or position of clusters in the classical picture.

We also performed the same analysis for the B and C wave functions obtained with the m60 interaction, and found that the cluster motion is qualitatively the same as the m62 case, as shown in Fig. 8 for the m60 result, which corresponds to Fig. 7 for the m62 result.

iii.3.3 Angular and radial motion in B() and C()

To discuss the angular motion of the cluster in B() and compare it with the cluster in C() in more detail, we show the dependence of the overlap for fm in each case in Fig. 9 for m62 and Fig. 10 for m60.

At fm, the distribution in B() has a peak structure at , which means that the angular motion of the cluster is restricted in the narrow region because of the Pauli blocking effect from clusters. At fm and 5 fm, the cluster is distributed widely. In particular, at fm, the distribution becomes almost flat in the wide region of . Because of the node structure of the motion in the orbit, the distribution vanishes at and in the component of the system. As a result, the cluster in fm moves almost freely in the wide region, except for and regions for the trivial nodes.

In C(), the angular distribution of the cluster at fm has a peak at , similar to the distribution in B(). As increases, the angular distribution becomes wide. However, because the linear configuration is energetically unfavored, the distribution is somewhat suppressed in and regions at , 5 fm. At fm, the angular distribution is almost flat in all regions, indicating that the cluster moves freely in angular motion with no angular correlation.

Both B() and C() contain significant cluster distribution in the fm region. In this region, the cluster around the moves almost freely in angular motion with weak angular correlation. The present result indicates that these two states, B() and C(), have a characteristic of a cluster gas in angular motion, if we tolerate the exclusion of the and regions.

Another characteristic of a cluster gas is a wide distribution of clusters in radial motion. The radial motion of the cluster around the at in B() and B() and that of the cluster in C() and C() are shown in Fig. 11. The distribution in B() is localized around fm and rapidly damps as increases. The suppression in the small region comes from the antisymmetrization effect from the . The distribution in C() has quite similar behavior to B(). In B() and C(), the () cluster around the is distributed in the outer region widely, compared to distribution in the ground states. The cluster distribution is suppressed in the small region, in particular, in the compact 2 core case with a small because of the orthogonality to the ground state as well as the antisymmetrization effect. The distribution in B() and the distribution in C() are outspread widely. In particular, the radial extent of the distribution in C() is remarkable, and that of the distribution in B() is relatively small. The relatively small extent in B() originates in the deeper - binding than the - binding because of the weaker Pauli blocking effect between and clusters. As a result, the radial spreading of the cluster in B() is less remarkable than that of cluster C().

The dependence of the overlap shows the - radial motion in the core. As seen in Fig. 11, in the ground states of B and C, the peak hight of the overlap rapidly decrease as increases, indicating that two clusters are tightly bound to form a compact core. In contract, B() and C() contain a significant cluster component for fm, indicating that the core in these states is a weakly bound cluster. The radial extent of the in B() is as large as that in C(). In other words, two clusters in B() behave as an cluster gas.

Figure 5: (Color online) Overlap of the BB wave function for cluster with the AMD+VAP wave functions for B(), B(), and B() calculated with the m62 interaction. The overlap is plotted on the plane for , 4, and 5 fm. The positions in the core are shown by black circles.
Figure 6: (Color online) Overlap of the BB wave function for a cluster with the AMD+VAP wave functions for C(), C(), and C(), calculated with the m62 interaction. The overlap is plotted on the plane for , 4, and 5 fm. The positions in the core are shown by black circles.
Figure 7: (Color online) dependence of overlap for B and C calculated with the m62 interaction in the case.
Figure 8: (Color online) Same as Fig. 7; However, it is for the m60 result.
Figure 9: (Color online) dependence of overlap for B() and C() calculated with the m62 interaction.
Figure 10: (Color online) Same as Fig. 9 but for the m60 result.
Figure 11: (Color online) dependence of for B(), B(), C(), and C() calculated with the m62 interaction.

iii.4 Analogies of and differences in and cluster gas states

As mentioned previously, B() and C() contain a significant cluster distribution in the fm region. In this region, the cluster around the core moves rather freely in the angular mode in a wide region, except for the and regions. For B(), the motion of the cluster in the component of the wave function elementarily has the node structure at the -axis. We know that the topological structure of the wave motion for the cluster is different from that of the wave motion for the cluster, experiencing neither node nor phase change in angular motion. However, when we consider the 2D cluster motion in the intrinsic frame, the angular motion of the cluster in the system can be associated with that of the cluster in the system, as follows.

The cluster distribution around the core tends to be suppressed at and because of the Pauli blocking effect from the core on the -axis and also because of the energy loss in the linear configuration. Let us consider the angular motion in the extreme case of . In the angular distribution of the cluster moving around the , the probability completely vanishes at and because of the Pauli exclusion principle between nucleons in different clusters. This situation is analogous to the angular distribution of the cluster in the wave with , in which the probability vanishes at and . It should be noted that the sign of the wave function is the same in and , but the sign of the wave function in is opposite to that in . The opposite sign for a wave function is a consequence of the negative parity of the total system, which originally comes from the negative parity of the lowest allowed orbit for the - intercluster motion, because the cluster consists of odd-numbered fermions. By taking the absolute value of the -cluster angular wave function, having two nodes at and around the in the system, we can obtain an angular distribution similar to the distribution around the in the system, which means that the one-dimensional motion in the angular mode around the fixed core can be mapped on the motion around . In three-dimensional space, the phase of the wave function of the orbit changes by in the rotation around the -axis. Again by taking the absolute value of the wave function, it may be possible to consider a mapping of the -cluster motion onto the -cluster motion, if the probability is suppressed at and .

In the system for C(), the probability is somewhat suppressed at and in the fm region. In this region, the angular motion of the cluster in B() has some association with the -cluster motion in C(), as described previously. That is, the cluster motion in both cases is characterized by the wide angular distribution, except for the and regions. However, this -cluster and -cluster association in the angular motion around the breaks down in the asymptotic region, far from the core. The cluster distribution is isotropic and goes to the ideal -wave motion in the asymptotic region, where the blocking effect from the core vanishes; by comparison, the cluster in a wave always has the node structure in the angular mode, even in the asymptotic region. Nevertheless, the cluster in B( is distributed in a wide region around the , indicating that the total system has a non-geometric cluster structure with weak angular correlation rather than a geometric structure. If we tolerate the exclusion of and regions and extend the concept of the cluster gas as a developed cluster structure with weak angular correlation, B() can be interpreted as a kind of cluster gas of .

Therefore we can state the following:

  • B() and C() are and cluster states with weak angular correlation, like a cluster gas.

In addition to the similarity of and difference in the angular correlation mentioned previously, the following similarities and differences exist in the cluster structure of B() and the cluster structure of C().

  • In the radial motion of the cluster around the , the distribution is outspread widely in B(), compared with the ground state, and it is similar to the radial motion of the cluster in C(). However, the radial extent of the distribution in B() is relatively smaller than that of the distribution in C) because the - binding is deeper than the - binding, which occurs because of the weaker Pauli blocking between and clusters.

  • The dependence of the overlap indicates that two clusters in B are bound as weakly as those in C.

In conclusion, B() is interpreted as a three-body cluster gas of in weak angular correlation and the radial extent of clusters. The cluster gas feature is more prominent in C() than in B().

Iv Summary

In this study, we reanalyzed cluster features of states of B and states of C, obtained with the AMD+VAP method in previous study. The cluster structures of B were compared with the cluster structures of C in the analysis of cluster distribution. We were particularly attentive to the -cluster motion around the core in the system. We considered the 2D cluster motion in the intrinsic frame, and investigated the cluster distribution on the 2D plane. To discuss the angular motion and radial motion of the cluster in B and those of the cluster in C, we studied the dependencies of the cluster distribution on the distance and the angle for the cluster position from the . In particular, we discussed the dependence to clarify the angular motion of the cluster around the .

B() and C() contain significant - and -cluster distributions in the radial distance fm region. In this region, the -cluster motion in B() is characterized by the wide angular distribution, except for the and regions. It is associated with the angular motion of the cluster in C(), where the distribution in the and regions is somewhat suppressed. By comparing the cluster structure in B() and the cluster structure in C(), the following similarities and differences are found.

  • In angular motion, B() and C() are and cluster states with weak angular correlation, like a cluster gas.

  • In radial motion of the cluster around the , the distribution is outspread widely in B(), compared with the ground state. However, the radial extent of the distribution in B() is less than that of the distribution in C) because the - binding is deeper than the - binding.

  • The dependence of the overlap indicates that two clusters in B are bound weakly as those in C.

In conclusion, B() is interpreted as a three-body cluster gas of in sense of weak angular correlation and radial extent of the clusters. The cluster gas feature is more prominent in C() than in B().

Acknowledgments

The authors would like to thank Prof. Schuck for fruitful discussions. The computational calculations of this work were performed using the supercomputers at YITP. This work was supported by JSPS KAKENHI Grant Numbers 25887049 and 26400270.

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