Cluster formation in neutron-rich Be and B isotopes

Evaluating the formation of clusters in finite and infinite nuclear systems is a funda-mental problem in nuclear physics, and the variation in clustering in the isotope chain up to the neutron drip-line is one such intriguing phenomenon. Recently, experiments performed using the ( p, pα ) reactions have revealed a negative correlation between the α -cluster formation and neutron number in Sn isotopes, showing a trend opposite to that theoretically predicted for Be and B isotopes. Although the observed charge radii of the Be and B isotopes suggest the formation of clusters as the neutron drip-line is approached, these radii themselves are not a physical quantity that can directly probe the cluster structure. Here, we directly investigated the cluster formation in Be and B isotopes to elucidate the possibility of clustering as the neutron drip-line is approached. It has been shown that the sum of the S -factors (i.e., S ( α ), S ( 6 He), and S ( 8 He)) increases as the neutron drip-line approaches, in agreement with those of the previous studies. The results indicate that the excess neutrons contribute to the formation of 6 He and 8 He clusters as well as α clusters. Thus, S ( α ) is not sufficient to estimate the enhancement of clustering in Be and B isotopes.


Introduction
Cluster formation is a universal phenomenon exhibited at all levels constituting the hierarchy of matter. In the sub-atomic regime, the α-cluster structure plays an important role in both finite and infinite nuclear systems. In dilute nuclear matter, the formation of α clusters is theoretically predicted to depend on the nuclear density as well as the symmetry energy [1][2][3][4]. Notably, in finite nuclei, cluster formation is expected to be hindered by the growth of a neutron skin [5]. Therefore, it is important to clarify the correlation between the neutronskin thickness and the cluster formation. Recently, it was shown that the proton-induced α knockout reaction (p, pα) is useful for assessing the α-cluster formation at the nuclear surface [6]. Using this reaction, the negative correlation between the neutron-skin thickness ORIGINAL UNEDITED MANUSCRIPT and the α-cluster formation at the nuclear surface of Sn isotopes has been reported [7]. A similar negative correlation was also theoretically shown in neutron-rich C isotopes [8].
In contrast to the trend observed in Sn isotopes, the α-cluster formation in light nuclei, such as Be and B isotopes, is predicted to enhance as the neutron drip-line approaches [9][10][11][12]. The enhancement of the α-cluster formation in Be isotopes as the neutron drip-line is approached is theoretically well explained by the molecular orbital model, which assumes that the excess neutrons bond with two α particles, similar to the covalent electrons in a molecule [13][14][15]. For example, in 12 Be, the two excess neutrons are promoted into the sdshell across an N = 8 shell gap and enhance the clustering compared with that in 10 Be. The observed increase in the charge radii of the Be and B isotopes suggests the development of clusters as the neutron drip-line is approached [16][17][18].
However, direct observation of the enhancement in the clustering is a significant challenge. Therefore, the study based on the physical quantities that can directly probe the cluster structure is highly desirable. Consequently, we investigated the cluster formation probability, which can be experimentally measured via (p, pα) reactions, at the nuclear surfaces of Be and B isotopes through antisymmetrized molecular dynamics (AMD) calculations. We focused on the 6 He and 8 He clusters that are expected to emerge because of the excess neutrons near the neutron drip-line as well as the α clusters.
This paper is organized as follows. In section 2, the theoretical framework of AMD is briefly explained. In section 3, we present the numerical results and discuss the clustering in Be and B isotopes. Finally, section 4 summarizes this work.

Hamiltonian and model wave functions
The microscopic Hamiltonian for A-body system is expressed as, wheret i andt c.m. denote the nucleon and center-of-mass kinetic energies, respectively. The Gogny D1S parameter set [19] is employed to represent the effective nucleon-nucleon interactionv NN ij , and the Coulomb interactionv C ij is approximated by a sum of seven Gaussians.
The model wave function of the A-body system is a parity-projected Slater determinant, whereP π denotes the parity projection operator, and ϕ i is the i-th nucleon wave packet expressed by the deformed Gaussian [20], Here, the isospin part η i is fixed to either a proton or neutron. The Gaussian widths (ν x , ν y , ν z ), centroids Z iσ , and spin directions a i and b i are the parameters of the model wave function (2) and are determined by the frictional cooling method, which minimizes the 2/12 ORIGINAL UNEDITED MANUSCRIPT sum of the Hamiltonian and constraint potential, In this case, a sufficiently large value of the constraint potential strength v β is selected to ensure that the quadrupole deformation parameter ⟨β⟩ [21] of the model wave function is equal to the input value β. Further, we obtain the optimized wave function Φ π (β), which has the minimum energy for each given value of β. After the energy variation, the optimized wave functions are projected to the eigenstate of the angular momentum, where D J M K (Ω) andR(Ω) denote the Wigner's D-function and the rotation operator, respectively. Then, the projected wave functions which have different values of the deformation parameter β, are superposed as, This is the wave function of the generator coordinate method (GCM) [22], where the deformation parameter β is employed as the generator coordinate (Eq. (6)). This GCM wave function can approximately well describe the bound state as well as the unbound state with a bound state approximation. The coefficients g jK ′ ,α and the eigenenergy E α are determined by solving the Hill-Wheeler equation [22], The GCM wave functions can be used to derive the properties of Be and B isotopes, such as the distribution radii and degree of clustering.

Reduced width amplitude
To evaluate the degree of clustering in Be and B isotopes, we calculated the reduced width amplitude (RWA), which is defined as the overlapping between the A-body GCM wave function and the reference state composed of the cluster with a mass C 1 and the daughter nucleus with a mass A − C 1 . Thus, the RWA is the probability amplitude of a cluster at a distance r = a from the daughter nucleus, where Φ 0+ C1 is the fully-antisymmetrized wave function of the cluster C 1 with spin-parity 0 + ; Φ Jπ A−C1 and Φ Jπ A are the fully-antisymmetrized wave functions of the ground states of the daughter and mother nuclei. The spin-parity J π is 0 + and 3/2 − for the Be and B isotopes, respectively. In this study, we only considered the orbital angular momentum of the intercluster motion, that is, ℓ = 0. In the practical calculations, Eq. (10) was evaluated by using the Laplace expansion method [23].

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The degree of clustering may be evaluated by the spectroscopic factor (S-factor), which is the squared integral of the RWA, Notably, S(C 1 ) is not normalized to unity because of the antisymmetrized effects between the cluster S(C 1 ) and the daughter nucleus. We also calculated the root mean square (RMS) radius of the RWA between the cluster C 1 and the daughter nucleus, defined by, where an expectation value of the squared intercluster distance a is normalized by the S-factor, S(C 1 ).

Structure of the ground states of Be isotopes
Be isotopes are well known to have an enhanced two α-cluster core surrounded by valence neutrons; such cores have been studied using numerous cluster models [24][25][26][27][28][29]. In the molecular orbital model, the excess neutrons occupy the molecular orbitals and bond with the α particles [13][14][15]30]. 10 Be is dominated by the π 2 configuration, in which the two excess neutrons occupy the π orbitals corresponding to the spherical p-orbitals in the limit of zero distance between the two α clusters. The two additional excess neutrons in 12 Be occupy the σ orbitals which correspond to the spherical sd-shell in the limit of zero distance between the two α clusters; further, the π 2 σ 2 configuration is dominant in 12 Be. Similarly, the two additional excess neutrons in 14 Be occupy the π orbitals, and the π 4 σ 2 configuration is dominant. It is considered that the σ-bond structure separates the α particles, while the π-bond structure brings them closer [26,29]. In an ordinary spherical shell model, 10 Be and 14 Be are dominated by the normal configuration (i.e., the 0ℏω configuration). In contrast, 12 Be is dominated by the 2ℏω configuration, in which two of the valence neutrons occupy the orbitals in the sd-shell beyond the N = 8 shell gap. The ground-state properties of 10 Be, 12 Be, and 14 Be deduced by our calculations are summarized in Table 1. The point proton or charge distribution radii of the Be isotopes increase toward the neutron drip-line in accordance with the enhancement of clustering. In Fig. 1, both the observed and calculated RMS charge radii increase from 10 Be toward the neutron drip-line, although our calculations systematically overestimate the radii. This is because the Gogny interaction used in this study tends to overestimate the radii of the s-shell nuclei, in particular that of the α particle, which is an important ingredient of the clustered ground states of Be isotopes. The calculated RMS neutron and matter radii also show a similar trend. The large quadrupole deformation parameter β also supports the presence of two α-cluster cores with large intercluster distances. Moreover, the calculated principal neutron quantum numbers, N n , of 10 Be and 14 Be are approximately 4 and 10, respectively, which indicate the 0ℏω configuration, whereas for 12 Be, N n ≃ 8, which corresponds to the 2ℏω configuration. Table 1 Calculated quadrupole deformation parameter β; the RMS proton, neutron, and matter distribution radii of the Be isotopes (unit: fm); and the principal neutron quantum numbers N n . The approximated shell model (SM) and molecular orbital (MO) configurations [30] are also listed.

Cluster formation in the ground states of Be isotopes
Here, we discuss the enhancement of clustering in Be isotopes based on the calculated RWAs. Figure 2(a) shows the calculated α RWA of the ground state of A Be for the |α ⊗ A−4 He(0 + 1 )⟩ channel. The amplitudes are suppressed and oscillate in the nuclear interior because of the Pauli exclusion principle. The effect of Pauli exclusion is approximately described by the Wildermuth-Tang rule [31], which asserts that the nodal quantum number n and orbital angular momentum ℓ of the RWA must satisfy the condition 2n + ℓ ≥ 4 in 10 Be and 2n + ℓ ≥ 6 in 12,14 Be. Thus, the α RWAs with ℓ = 0 have two and three nodes in 10 Be and 12,14 Be, respectively. Note that the condition 2n + ℓ ≥ 6 for 12 Be implies the dominance of the 2ℏω configuration, that is, the breaking of the N = 8 magic number. The amplitudes are peaked at the nuclear exterior (r ≳ 3.0 fm), indicating the cluster formation at the nuclear surface. The peak position is more outward for 12,14 Be than for 10 Be, and this result is consistent with the enhanced cluster formation by the excess neutrons occupying the σ orbitals. This trend is also evident from the calculated RMS radius of the RWA, a rms , shown in Fig. 3. The a rms (α) of 12 Be is larger than that of 10 Be owing to the cluster development in 12 Be. Additionally, the a rms (α) of 14 Be is slightly shorter than that of 12 Be, because the two neutrons of 14 Be occupying the π orbitals attract the two α-cluster cores.
However, the calculated S-factor for α clustering, shown in Fig. 4, contradicts the aforementioned analysis. The S-factor decreases as the neutron number increases, suggesting the suppression of α-cluster formation by the excess neutrons. The associated mechanism may be explained as follows: Because of the glue-like role played by the valence neutrons in 12 Be, ) a (fm) 10 Be(π 2 ) 12 Be(π 2 σ 2 ) 14 Be(π 4 σ 2 ) 2.0 4.0 6.0 8.0 (b) 6 He cluster a (fm) 12 Be(π 2 σ 2 ) 14 Be(π 4 σ 2 ) Fig. 2 The calculated α (panel (a)) and 6 He (panel (b)) RWAs of the Be isotopes with orbital angular momentum ℓ = 0.  Calculated α and 6 He S-factors and the sum of these S-factors as a function of the neutron number of the Be isotopes.

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this Be isotope has a mixing of α+ 8 He and 6 He+ 6 He configurations. As a result, the S(α) of 12 Be becomes smaller than that of 10 Be, although the sum of S(α) and S( 6 He) is larger than that of 10 Be. Similarly, 14 Be may be an admixture of α+ 10 He and 6 He+ 8 He configurations. To verify this hypothesis, the calculated 6 He RWAs of 12 Be and 14 Be are shown in Fig. 2(b). In both 12 Be and 14 Be, the peak height of the 6 He RWA is comparable to that of the α RWA, and the peak position slightly shifts inward, which is reflected by the RMS radii of the RWAs for the α and 6 He clusters shown in Fig. 3. Thus, the magnitude of S( 6 He) is comparable with that of S(α). The sum of S(α) and S( 6 He) (i.e., S sum ) does not decrease with increasing neutron number but increases slightly. Thus, taking into account the clustering of 6 He as well as α, it can be stated that Be isotopes have a well-developed cluster structure.

Structure of ground states of B isotopes
Although the structures of B isotopes have been theoretically studied by many authors [10][11][12][32][33][34], the physical quantities, which facilitate a direct analysis of clustering, have not been discussed. Here, we explain our calculated ground-state properties of B isotopes. Table 2 lists the calculated quadrupole deformation parameters β and γ, and the RMS proton, neutron, and matter distribution radii of the B isotopes. The principal neutron quantum numbers, N n , are approximately 4, 6, 10, 14, and 18 for 11,13,15,17,19 B, respectively, which 6 19 B is oblately deformed (γ p,n ≈ 60 • ), while the spin-fixed AMD calculations [12] yielded a prolate deformation. This difference significantly affects the cluster formation probabilities as discussed later in this paper. Figure 5 shows the calculated RMS point proton radii, which are in reasonable agreement with the experimental values, except for 11 B. The magnetic moments µ and electric quadrupole moments Q p of 11−17 B shown in Fig. 6 and 7 are also in good agreement with the observed values and do not exhibit a strong dependence on the neutron number. Thus, the present calculation results successfully describe the ground states of B isotopes.

Cluster formation in B isotopes
In this subsection, we discuss the cluster formation in B isotopes based on the intrinsic density distributions, RWAs, radii of the RWAs, and S-factors.

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Cluster formation in B isotopes can be evaluated from Fig. 8, which shows the proton, neutron, and matter intrinsic density distributions of the Be and B isotopes. The peak positions of the proton density distributions are marked by points, and the distances d between them are shown in panel (c). All the Be isotopes exhibit a prolate shape, and the distance d increases as the neutron number increases, which is consistent with the growth of clustering toward the neutron drip-line. For the B isotopes, the S-factors are expected to be smaller than those of the Be isotopes, because the distance d is smaller than that of the Be isotopes. The S-factor increases as a function of the neutron number until 17 B. The isotope 19 B has an oblate shape and does not exhibit clustered structures. Thus, we expect that the clustering is enhanced until 17 B, but the corresponding S-factors are always smaller than those of the Be isotopes.
Next, we discuss the RWAs in the |α⊗ A−4 Li(3/2 − 1 )⟩ channel with A = 11, 13, and 15 as shown in Fig. 9(a). The RWAs of the B isotopes are peaked at the nuclear exterior and suppressed in the interior because of the Pauli exclusion satisfying the Wildermuth-Tang rule [31]. The peak height is much smaller than that of the Be isotopes, suggesting the presence of less developed clustered structures as expected. Similar to the case of Be and C isotopes [8], the S(α) decreases as the neutron number increases as shown in Fig. 10 because of the excess neutrons, which cause the mixing of the α+ 9 Li and 6 He+ 7 Li configurations in 13 B and that of the 6 He+ 9 Li and 8 He+ 7 Li configurations in 15 B. The calculated 6 He and 8 He RWAs are shown in Fig. 9(b) and (c), respectively. Evidently, in 13 B, the peak height of the 6 He RWA is comparable to that of the α RWA. In 15 B, the peak heights of the 6 He and 8 He RWAs are larger than that of α. Similarly, the excess neutrons in 17 B contribute to the formation of the 8 He+ 9 Li configuration. These features are reflected in the S-factors depicted in Fig. 10 as well as the radii of the RWAs shown in Fig. 11. Clearly, some of the α, 6 He, and 8 He cluster structures are enhanced in 11,13,15,17 B. In Fig. 10, S sum shows the sum of S(α), S( 6 He), and S( 8 He) and is not hindered by the excess neutrons in the isotopes up to 17 B.
In 19 B, the corresponding RWA and S-factor indicate a less developed 8 He-cluster structure, which is expected from the intrinsic densities. Therefore, the behavior of S sum is consistent with the enhancement of clustering, determined from the intrinsic density distributions in Fig. 8. This figure indicates a certain amount of cluster development at S sum values smaller than that of Be isotopes.

Summary
In this study, AMD calculations were performed that predicted the enhancement of α-cluster formation in Be and B isotopes as the neutron drip-line is approached, which is opposite to the experimental observations of Sn isotopes. Experimental investigation of the cluster formation based on the physical quantities that can directly probe the cluster structure is a considerable challenge. Thus, we theoretically evaluated the cluster formation in Be and B isotopes to clarify the possible clustering as the neutron drip-line is approached. We investigated the cluster formation probability at the nuclear surface by analyzing the 6 He and 8 He clusters that are expected to emerge because of the excess neutrons as well as the α clusters. The AMD framework successfully described the ground-state properties of both the Be and B isotope chains. We estimated the enhancement of clustering from the intrinsic density distributions in the Be and B isotopes as well as by using the molecular 8 (c) 11   . Note for 11 B that the largest distance between one peak and the mean position of the other peaks is adopted and shown as an arrow.
orbital model for the Be isotopes. Although the calculated α spectroscopic factors (i.e., S(α)) show a negative correlation with the neutron number, the sum of S(α) and S( 6 He) for the Be isotopes and that of S(α), S( 6 He), and S( 8 He) for the B isotopes are not being hindered as the neutron drip-line approaches except for 19 B. This result is consistent with the enhancement of clustering estimated from the intrinsic density distributions. Thus, the results of this study reveal the possibility of cluster formation as the neutron drip-line is approached in Be and B isotopes. These results are based on the discussion which explains the mechanism of cluster formation in Be isotopes and shows the consistency between the Calculated RMS radii of the RWAs for α, 6 He, and 8 He clusters of the B isotopes.
cluster distances of the intrinsic density distributions and cluster formations in B isotopes. Consequently, the results indicate that 6 He and 8 He clusters are formed along with α clusters owing to the excess neutrons surrounding the α-cluster cores.