2.1 Resonating Group Method

Microscopic cluster models have been employed to study the cluster structures of light nuclei since the early 1960s. Generally, the Hamiltonian of the A-nucleon system can be written as

where ${\overrightarrow{p}}_i$ is the momentum vector of the nucleon i, ${\overrightarrow{r}}_i$ and ${\overrightarrow{r}}_j$ are the space coordinates of the nucleons i and j, respectively, Vij is the nucleon-nucleon interaction potential which contains both the Coulomb and nuclear terms (central and spin-obit terms) [44], and m represents the nucleon mass. Taking into consideration the computational complexity of solving the Schrödinger equation with large nucleon numbers (A>4), the cluster approximation, under which the A-nucleon system is divided into two separated sub-clusters, may be adopted. This approximation was introduced by Wheeler in the resonating group method (RGM) [3].

The RGM wave function for two-cluster systems can be expressed as [45]

where ${\overrightarrow{r}}_1$ and ${\overrightarrow{r}}_2$ are the coordinates of the center-of-mass (c.m.) of the two ingredient clusters, $\overrightarrow{r}={\overrightarrow{r}}_1-{\overrightarrow{r}}_2$ is the relative coordinate, ${\varphi }_i({\overrightarrow{r}}_i)$ represents the internal wave function, and $\chi (\overrightarrow{r})$ is the relative wave function describing the inter-cluster relative motion which can be deduced by the energy variation [46]. A_T is the total antisymmetrizer of all the nucleons that is given by

where 𝒜i represents the antisymmetrizer of the nucleons inside one cluster and 𝒜 describes the antisymmetrizer between the two clusters. It may be noted that the RGM calculations can provide an explanation not only for the cluster states but also for the single-particle states [47], on account of the total antisymmetrizer 𝒜_T, which makes it possible for a resonant exchange of nucleons between the groups.

Y.C. Tang and M. LeMere have applied the RGM calculations to investigate the bound and resonant states of ¹⁷O, ¹⁸F, ¹⁹F and ²⁰Ne, with a predominant X+¹⁶O configuration [48]. The calculated levels agree quite well with the previous experimental results [49], which indicate the existence of X + ¹⁶O cluster structure in these nuclear systems.

2.2 Generator Coordinate Method

In spite of the success of the RGM in depicting the inter-cluster relative motion, there is still an enormous challenge in dealing with heavier systems or multichannel problems which would entail tedious calculations in separating the internal and relative coordinates with antisymmetrization. This difficulty could not be resolved until the adoption of the generator coordinate method (GCM) [45, 50] with the Bloch-Brink cluster wave function.

The relative wave function $\chi (\overrightarrow{r})$ in GCM is expanded over some Gaussian functions that are centered at different generated coordinates. As a consequence, the total GCM wave function $\psi ({\overrightarrow{r}}_1\mathrm{,}{\overrightarrow{r}}_2\mathrm{,}\overrightarrow{r})$ can be expressed as a superposition of the Slater determinants [50], which signifies that the computational complexity can be reduced to such an extent that the investigation of the cluster structures in heavy systems becomes a feasible issue.

The one-cluster Slater determinants with A₁ nucleons located at a common position $\overrightarrow{S}$ can be defined as

where the single-nucleon wave function ${\phi }_1({\overrightarrow{r}}_1\mathrm{,}\overrightarrow{S})$ (neglecting the spin and isospin components) is a harmonic-oscillator function. Therefore, Eq. (4) can be rewritten as [51]

where ${\varphi }_1({\overrightarrow{r}}_1)$ is the same as that in Eq. (2) and b is the oscillator parameter which is uniform for all the nucleons. Thus, the two-center wave function located at ${\overrightarrow{r}}_1$ and ${\overrightarrow{r}}_2$ of the A-nucleon system can be written as

where $\overrightarrow{R}={\overrightarrow{r}}_1-{\overrightarrow{r}}_2$ is the generator coordinate whose origin is located at the c.m. and N is the normalization factor. With the combination of Eq. (5) and Eq. (6), the Slater determinants can be rewritten as

where ${\psi }_{\text{cm}}(\overrightarrow{R})$ describes the motion of the c.m. and $\Gamma (\overrightarrow{R}\mathrm{,}\overrightarrow{R})$ is the radial wave function. The calculation of the GCM matrix elements becomes more concise with the separation of the motion of the c.m. as only simple Gaussian functions are involved in Eq. (7) [52].

In the case of overlapped clusters, the wave function in Eq. (7) would be equivalent to the shell-model wave function owing to the antisymmetrization of the nucleons among the clusters. However, if the distance between the clusters is large enough to ignore the inter-cluster antisymmetrization, the wave function in Eq. (7) effectively describes the real cluster structure.

P. Descouvemont performed a simultaneous analysis of the ^9,10,11Be isotopes with the 2α+Xn multi-cluster GCM calculation and anticipated the decrease in the alpha-clustering effect from ⁹Be to ¹¹Be [53]. In a similar study, he predicted a strong mixing of the ⁶He+⁶He and ⁴He+⁸He configurations, rather than a dominant ⁶He+ ⁶He structure, in ¹²Be [54].

2.3 Orthogonality Condition Method

The orthogonality condition method (OCM) was proposed by S. Saito [45, 55] as a simplified approach to deal with the role of the Pauli principle in the interaction between clusters. It gives rise to the characteristic energy-independent, but angular-momentum-dependent, inner oscillation for the solution of the relative wave function $\chi (\overrightarrow{r})$ in Eq. (2). In OCM calculations, the relative wave function between the clusters is required to be orthogonal to the Pauli forbidden states (PFS), ensuring that the interaction takes effect only in the physical space orthogonal to the PFS under antisymmetrization. Considering the Pauli principle, the projection operator is defined as follows [55]

where |φi〉 are the PFS which can be derived from the internal configurations of the component clusters. It is obvious that Λ|φi〉=0. Hence, the Schrödinger equation for the OCM can be deduced as [45]

where E_rel denotes the relative energy in the c.m. system, T is the kinetic operator and V_eff is the local potential regarded as the average potential between the two clusters. Since |φi〉 are redundant solutions of Eq. (9), it can be rewritten as

where the relative wave function $|\chi (\overrightarrow{r})\rangle$ permanently complies with the orthogonality condition $\langle {\phi }_i|\chi (\overrightarrow{r})\rangle =0$.

The OCM was applied extensively to study the nuclear clustering structures. For example, T. Matsuse and M. Kamimura [56] concluded that the first Kπ=0^- band in ²⁰Ne had a strong α-clustering character whereas the ground state band possessed the shell-model like structure preferably, based on the quantitative computation of the α-width and spectroscopic factors with OCM.

A certain amount of progress was made towards resolving the multi-cluster problem by H. Horiuchi [57], wherein the binding energy and excited levels of ¹²C were reproduced by using an extended 3α OCM model. It was found that the ground state had a shell-model-like configuration whereas the excited 0⁺ state had a well-developed molecular-like structure. Various multi-cluster OCM calculations have been carried out [58] since then.

T. Yamada [59] adopted a four-body 3α+n OCM theory to study the changes in the configurations of ¹³C brought about by the additional valence neutrons and determined the monopole transition strength of the low-lying states below the 3α+n threshold. It was clear that the monopole matrix elements of the four lowest 1/2^- states above the ground state were much stronger than the single-particle state, indicating the dominant ⁹Be+α cluster structures in these states. The latest experiment of ¹³C conducted by J. Feng et al. [40] has established an enhanced monopole matrix element of (6.3±0.6)fm² for the 14.3 MeV (${1/2}_5^{-}$) state, which is in good agreement with the prediction of the OCM calculations.

In addition, T. Yamada extended the OCM calculations to the four-α-particle condensate state in ¹⁶O, and the results of the calculations in six of the lowest Jπ=0⁺ states are in excellent agreement with the experimental spectrum [60]. The $0_6^{+}$ state residing in 2 MeV above the four-α-particle breakup threshold has a large α condensate fraction of 61% as well as a large component of $\alpha +{}^{\text{12}}{\text{C(0}}_{\text{2}}^{+}\text{)}$ configuration. This brings a new perspective on the experimental endeavor to explore the α-particle condensation state in ¹⁶O.

2.4 Molecular Orbit Model

As the α+α dumbbell cluster structure of ⁸Be is well established [17], it is logical to treat the neutron-rich beryllium isotopes as typical molecular structures with valence neutrons moving around the two cluster-cores. In the molecular orbit model (MO), the orbits of the valence neutrons are constructed from a succinct linear combination of the cluster orbits (LCCO) [21, 23, 61].

Neglecting the spin-orbit coupling, the spatial part of the MO function ∣λ⟩ is expanded by the orthogonal basis vector ∣α⟩, constructed with the cluster orbits. ∣λ⟩ can be defined as

where Cλα can be determined by using the Hartree-Fock equation and the basis vector ∣α⟩ can be written as

where |m;A〉 are the cluster orbits for which the harmonic oscillator wave functions are applied for convenience, and DαmA are determined by a classification using an irreducible representation of an assumed point group symmetry and an orthogonalization among orbits belonging to the same representation [17].

S. Okabe and Y. Abe succeeded in extending the MO model to the odd-A system ⁹Be [21, 61], proposing that the degree of clustering in ⁹Be depends crucially on the motions of the valence neutrons. The properties of the low-lying levels of ⁹Be are reproduced satisfactorily with the MO model. Itagaki et al. [35] carried out excellent work in describing the neutron-rich carbon isotopes with the well-developed 3α-cluster MO model, suggesting configurations of two neutrons in the σ orbits and two in the π orbits, for the excited states in ¹⁶C.

2.5 Generalized Two-center Cluster Model

Although the MO model was reasonably successful in reproducing the unusual low-lying states of the light neutron-rich nuclei, there was a difficulty in describing the highly excited states above the particle decay threshold. M. Ito et al. [24, 62, 63] introduced the generalized two-center cluster model (GTCM) where the valence neutrons are localized around one of the α-cluster cores and can be described as a single-particle motion in an atomic-like orbit. The representative results with GTCM were given for ¹⁰Be and ¹²Be. For example, the basis function for ¹⁰Be can be defined as

where $P_K^{J^{\pi }}$ is the projection operator with angular projection K, total spin projection J, and total parity projection π; S is the relative distance parameter between two α-clusters; ψ_L(α) and ψ_R(α) represent the α-cluster wave function located at the left and right side, respectively; and φ(m) and φ(n) are the atomic orbits of the valence neutrons. The total wave function is finally given by taking a superposition over S and K as given below:

where $C_i^K(S)$ is the coefficient which can be deduced from the GCM model:

When the relative distance parameter S is fixed, the energy eigenvalue can be calculated by using Eq. (15).

GTCM describes the formation of the molecular orbits as well as the asymptotic cluster states depending on their relative distance [62], as shown in Fig. 1. In S ≥ 6 fm, the calculated energy surfaces and energies of the cluster-coupling wave function are almost uniform, however, in S ≤ 4 fm, the wave functions of the molecular orbits are expressed as a linear combination of various cluster-coupling wave functions where the valence neutrons move around the alpha cores with a strong coupling configuration. The GTCM calculations also suggest the existence of Jπ=6⁺ and Jπ=7^- high spin states, which is in agreement with the measurements of M. Freer using two neutron transfer reactions ¹²C(¹²Be, ¹⁰Be^*→⁴He+⁶He)¹⁴C [31].

Figure 1:Full-screen View

Dynamical transition from the molecular orbits to the cluster-coupling states in the adiabatic energy surfaces. On the left side, the surfaces from A to H have a predominant molecular component for the valence neutrons of ${({\pi }_{3/2}^{-})}^2$, ${({\sigma }_{1/2}^{+})}^2$, ${({\pi }_{1/2}^{-})}^2$, ${({\pi }_{3/2}^{+})}^2$, $({\pi }_{1/2}^{+}{\sigma }_{1/2}^{+})$, $({\pi }_{1/2}^{-}{\sigma }_{1/2}^{-})$, ${({\sigma }_{1/2}^{-})}^2$, and ${({\pi }_{1/2}^{+})}^2$, respectively. On the right side, the solid curves represent the energies of the α+⁶He cluster states and the dotted curves show the energies of the ⁵He+⁵He cluster states. The figures are taken from Ref. [62]

A direct extension from ¹⁰Be (α+α+2n) to ¹²Be (α+α+4n) was conducted successfully about a decade ago [63], which developed a covalent super-deformation with a hybrid configuration of both the molecular-orbit and atomic-orbit configurations in the unbound region above the particle-decay thresholds of ¹²Be. Furthermore, after comparing the monopole transition strength of the GTCM with that of the single-particle model, M. Ito deduced that the total strength was comparable to or slightly larger than the single-particle strength which is considered to be one of the typical characteristics of the clustering structure of nuclei. This enhanced monopole transition strength was observed in Z.H. Yang’s breakup-reaction experiment for ¹²Be at RIBLL1 [26], the details of which are discussed in Sec. 3.

M. Ito also predicted, within the GTCM calculations [24, 25], that the 4⁺ states with various cluster structures of ⁴He + ⁸He, ⁶He + ⁶He, and ⁵He + ⁷He were located in the energy region of 13.4 MeV to 17.4 MeV. This coincides with the latest experimental observations of a broad (∼1 MeV) Jπ=4⁺ resonant state at the excitation energy of 14 ~ 15 MeV, presented by M. Freer et al. in experiments using ⁴He(⁸He, ⁴He) and ⁴He(⁸He, ⁶He) reactions [64].

2.6 Antisymmetrized Molecular Dynamics

The antisymmetrized molecular dynamics model (AMD) is based on single- nucleon wave functions. As a consequence, this theoretical framework can represent not only the cluster structures but also the shell-model-like configurations [15].

The AMD wave function for the A nucleon system is represented as a Slater determinant of single-nucleon Gaussian wave functions,

where ϕi describes the single-particle wave function of the ith nucleon written as the product of spatial (φZi), intrinsic spin (χi) and isospin (τi) wave functions as follows:

where 𝒁i denotes the center of the Gaussian wave packet and ξi indicates the spin orientation. The width parameter ν is fixed as a common and optimum value for the system under consideration. If all the Gaussian functions center around a common position, the AMD wave function in Eq. (16) will resemble the shell-model-like wave function. On the other hand, if all the Gaussian functions center around several positions, the AMD wave functions will produce the multi-cluster structures.

To obtain the expectation values of the observable operators such as the Hamiltonian and radii with energy variation [65], the AMD wave function should be projected to both the parity and angular-momentum eigenstates. In the conventional AMD calculation, the energy variation is done after the parity projection (VAP) and before the angular-momentum projection (VBP) [66]. The parity-projected wave function can be expressed as

where P^± = (1±Pr)/2 denotes the parity projection operator. Using the good-spin wave function Ф^±(Z), the angular-momentum projected wave function can be defined as

where ${\mathcal{P}}_{MK}^J\equiv {\displaystyle \int }\text{d}\Omega D_{MK}^{J_{*}}(\Omega )\mathcal{R}(\Omega )$ is the total angular-momentum projection operator. Its expectation values are numerically calculated by a summation over the mesh points of the Euler angles Ω, where R(Ω) represents the rotation operator and $D_{MK}^{J_{*}}(\Omega )$ is the Wigner’s D-function. The correct K quantum number for each spin parity can be deduced from the principle that the energy expectation value for the spin parity eigenstate ${\Phi }_{MK}^{J\pm }(Z)$ is a minimum [34].

The wave function $\Phi (Z_1^{J\pm })$ of the lowest $J_1^{\pm }$ state, which can be expressed as ${\Phi }_{MK}^{J\pm }(Z_1^{J\pm })$, is calculated by varying the parameters 𝒁i and ξi in Eq. (17) and Eq. (18), in order to obtain the smallest expectation values of the Hamiltonian [67]. The wave functions $\Phi (Z_n^{J\pm })$ are superposed to be orthogonal to the lower states as given in Eq. (21) [68, 69]:

The final results are constructed by diagonalizing the Hamiltonian matrix $\langle \Phi (Z_i^{J\pm })\left|H\right|\Phi (Z_j^{J\pm })\rangle$ formed from all the involved intrinsic states.

Y. Kanada-En’yo explored the structure of the excited states of ¹⁰Be by the method of VAP in the basic framework of AMD [68]. The density distribution of the intrinsic wave functions for the band-head states of ¹⁰Be was determined as shown in Fig. 2, wherein a well-developed 2α cluster configuration surrounded by the valence neutrons can be seen.

Figure 2:Full-screen View

(Color online) Density distribution of the intrinsic wave functions for the band-head states of ¹⁰Be. The distributions for matter, proton, and neutron densities are illustrated in the left, middle and right panels. The figures are taken from Ref. [70]

In order to understand the motion of the valence neutrons around the core clusters, the single-particle wave functions should be determined from the traditional AMD wave function [71].As the single-particle wave functions φi in Eq. (16) are not orthonormal, another base wave function $\widetilde{{\phi }_{\alpha }}$ must be constructed as follows:

where λα and ciα are the eigenvalues and eigenvectors of the overlap matrix Bij ≡ 〈φi|φj〉. Thus, the single-particle Hamiltonian can be calculated and diagonalized for the Hamiltonian operator as $H\equiv {\displaystyle {\sum }_i}\widehat{t_i}+{\displaystyle {\sum }_{i<j}}{\widehat{V}}_{ij}+{\displaystyle {\sum }_{i<j<k}}{\widehat{V}}_{ijk}$, which is a sum of three terms, namely, the kinetic term, two-body interaction term and three-body interaction term, respectively. As a result, the AMD-HF singe-particle energies, parity components as well as the angular momenta can be calculated.

The AMD program was extended to deal with the 3α-cluster structures in the neutron-rich carbon isotopes by T. Baba and M. Kimura [41, 42, 72, 73]. With the help of the AMD-HF method, T. Baba determined the single-particle wave functions of the valence neutrons, leading to the explicit identification of the so-called π-bond or σ-bond molecular configurations. For ¹⁴C, three different clustering structures with distinct magnitudes of deformation and various valence neutron configurations, namely, the triangular, π-bond linear chain and σ-bond linear chain configurations, were exhibited as shown in Fig. 3. The calculated energies and α reduced widths of the π-bond linear chain state are close to the resonances observed in the ⁴He + ¹⁰Be scattering experiments [36, 74].

Figure 3:Full-screen View

(Color online) Density distribution of the positive [(a)-(d)] and negative parity states [(e)-(h)] of ¹⁴C. Figs. (a, e, h) represent non-cluster structures, (b, f) display the triangular configurations, (c, g) show the π-bond linear chain configurations, and (d) exhibits the σ-bond linear chain configuration. The contour lines show the proton density distributions. The colored plots show the single-particle orbits occupied by the most weakly bound neutron. The open boxes show the centroids of the Gaussian wave packets describing the protons. The figures are taken from Ref. [41]

Furthermore, an analysis of the decay patterns by T. Baba suggests that the σ-bond linear chain state decays almost exclusively to the ~6 MeV states in ¹⁰Be whereas the other resonant states decay mostly to the first excited state (3.4 MeV, 2⁺) as well as the ground state in ¹⁰Be. This decay pattern is considered to be a strong characteristic of the σ-bond linear-chain formation, which is consistent with the experimental observations of J. Li et al. at CIAE [38].

For ¹⁶C, the AMD calculations show the existence of both the triangular and linear-chain configurations [72]. The energy spectrum of ¹⁶C reconstructed from the ⁶He + ¹⁰Be breakup reaction [43] shows a rough peak at about 20.6 MeV, which was interpreted as a possibility for a linear-chain configuration. The need for further experimental investigation of the various cluster states in ¹⁶C is imperative.

Recently, P. Arumugam et al. [75] developed the RMF approach to investigate the clustering phenomena in light-mass stable and uncommon nuclei. The RMF calculations of the spatial distribution of the nuclear density and deformation parameters show the existence of α-clustering and halo structures. However, the limitations of the RMF method are evident as it adopts the oscillator basis, which is improper for large deformations and weak-binding. The space reflection symmetry of the RMF method places further restrictions on describing a nucleus with an asymmetric form, e.g., triaxial shapes.

3.1 Inelastic excitation and decay

In order to explore the cluster structures in ¹²Be, Z.H. Yang et al. carried out a breakup reaction using a 29 MeV/nucleon ¹²Be secondary beam with an approximate intensity of 3000 pps, off a carbon target [26, 27]. The detection was concentrated at most forward angles by using a zero-degree telescope. Owing to the well-developed self-calibration methods for the double-sided silicon detectors (DSSD) [80], the energy resolutions were good enough to get an unambiguous particle identification of the breakup fragments ⁴He, ⁶He, and ⁸He. Two helium fragments in coincidence can, therefore, be clearly selected. As a consequence, the excitation energy spectra for ¹²Be can be reconstructed with the ⁶He + ⁶He and ⁴He + ⁸He decay channels, as shown in Fig. 4.

Figure 4:Full-screen View

(Color online) Excitation energy spectra of ¹²Be reconstructed with the (a) ⁶He + ⁶He and (b) ⁴He + ⁸He decay channels, compared with those reported in Ref. [31], (c) and (d). The green arrows show the respective cluster decay thresholds. The figures are taken from Ref. [26]

For the ⁶He + ⁶He decay channel, two peaks at 11.7 MeV and 13.3 MeV are clearly observed whose respective spin-parity, 2⁺ and 4⁺, was verified with the distorted wave Born approximation (DWBA) calculation. The events near the corresponding decay threshold of 10.1 MeV can be expressed as the potential 0⁺ band-head state, which needs more concrete observation. For the ⁴He + ⁸He decay channel, three peaks at about 10.3 MeV, 12.1 MeV, and 13.6 MeV are identified. The spin-parities of the first and last states are assigned to 0⁺ and 4⁺, respectively, with the model-independent angular correlation analysis [30], which are in good agreement with the GTCM calculations [25]. Due to the low statistics, the spin-parity of the 12.1 MeV state is only tentatively assigned to 2⁺, according to the GTCM predictions. These observed states, except for the new 11.7 MeV and 10.3 MeV states, agree quite well with those reported by M. Freer [26] and R. J. Charity [81]. The molecular rotational bands can then be constructed from these experimental measurements, as shown in Fig. 5, whose ħ²/2J values are approximately 0.15 MeV. The large value of the moment of inertia J is closely linked to the strongly deformed structure, which is a characteristic evidence for the cluster formation in ¹²Be, as indicated by the clustering criterion Criterion I.

Figure 5:Full-screen View

(Color online) Energy-spin systematics for the resonant states in ¹²Be associated with the ⁶He + ⁶He and ⁴He + ⁸He configurations. The vertical axis is defined with respect to the separation energy of the ⁴He + ⁸He channel. The figures are taken from Ref. [27]

From a least-square fitting of the observed spectrum with the Breit-Wigner line shape function with angular momentum L=0 [82], the total decay width of the 10.3 MeV excited state in ¹²Be is deduced to be Γ_tot=1.5(2) MeV, in conformance with the predictions of a particularly large width for a cluster band head just above the threshold [69]. The dominating decay channels for this resonant state are the binary helium cluster channels and the neutron or γ emission, which can be written as Γ_tot ≈ Γ_He + Γ_Be. The beryllium branch ratio is taken as Γ_Be/Γ_tot= 0.28(12) from A. A. Korsheninnikov’s measurements [83]. Therefore, the helium branch ratio can be deduced as Γ_He/Γ_tot=0.72(12) and the partial cluster decay width is Γ_He=1.1(2) MeV. Based on the single-channel R-matrix approach [84], the reduced decay width is ${\gamma }_{\text{He}}^{\text{2}}=0.50(9)\text{MeV}$ and the dimensionless cluster spectroscopic factor can be expressed as ${\theta }_{\text{He}}^{\text{2}}=0.53(10)$ [28]. This is comparable to that for typical clustering states such as the 4⁺ state in ¹⁰Be (${\theta }_{\text{He}}^{\text{2}}=0.66~2.23$) [85], providing a strong affirmation for the clustering structure in ¹²Be, as indicated by the clustering criterion Criterion II.

The isoscalar monopole transition strength M(IS,0) can be deduced from the multipole-decomposition analysis of the inelastic scattering cross section [86]. As shown in Fig. 6, the distorted wave Born approximation (DWBA) calculations ${(\frac{\text{d}\sigma }{\text{d}\Omega })}_{L=\mathrm{0,}\text{DWBA}}$, with a normalization factor of a₀=0.034(10), are perfectly consistent with the experimental differential cross sections ${(\frac{\text{d}\sigma }{\text{d}\Omega })}_{\text{exp}}$. The factor a₀ can be corrected as 0.075(24) considering the 0⁺ state (10.3 MeV) in the entire excitation spectrum. The energy-weighted sum rule (EWSR) for isoscalar transitions can be written as [87]

Figure 6:Full-screen View

(Color online) Experimental differential cross sections compared with the DWBA calculations for the 10.3 MeV state in ¹²Be. The figures are taken from Ref. [27]

where m is the nucleon mass, A is the nuclear mass number, and R_rms is the root-mean-square matter radius of the nucleus. With Eq. (23), M(IS,0) can be deduced as 7.0±1.0 fm², which is comparable to the typical clustering states such as the Hoyle state in ¹²C (5.4±0.2 fm²) [60] and in accordance with the GTCM calculations. It gives a strong confirmation of the clustering structure in ¹²Be as mentioned in the clustering criterion Criterion III.

In conclusion, a distinct clustering structure is justified for the 10.3 MeV (0⁺) state in ¹²Be, based on the three clustering criteria outlined above.

In view of the considerable success in identifying the cluster states in neutron-rich beryllium nuclei, it is reasonable to extend the clustering investigations to neutron-rich carbon isotopes and oxygen isotopes. J. Feng et al. conducted a breakup reaction using a 65 MeV ¹³C primary beam onto a self-supporting ⁹Be target [40]. The purpose was to determine the spin-parity and isoscalar monopole transition strength of the 14.3 MeV resonant state in ¹³C.

The experimental and calculated differential cross sections with the FRESCO code [88] for the 14.3 MeV state in ¹³C are shown in Fig. 7. It is obvious that only the calculation with Jπ=(1/2)^- generates the oscillatory character of the experimental results, whereas the Jπ=(5/2)^- and Jπ=(7/2)^- configurations do not agree with the observations. As a result, the spin-parity of the 14.3 MeV resonant state in ¹³C can be assigned to Jπ=(1/2)^-, in agreement with the ${(1/2)}_5^{-}$ state in ¹³C as predicted by the OCM calculations [59]. The isoscalar monopole transition strength M(IS,0) of this resonant state can be determined by following the same procedure as applied for ¹²Be. The resulting value is 6.3±0.6 fm², even larger than the theoretical prediction of 3.3 fm² within the OCM approach. This enhanced monopole transition strength is an indication of the cluster structure in ¹³C, according to the clustering criterion Criterion3 3.

Figure 7:Full-screen View

(Color online) Experimental and calculated differential cross sections with FRESCO code within the CDCC framework for the 14.3 MeV state in ¹³C. The figures are taken from Ref. [40]

With an objective of investigating the near-threshold cluster resonance in ¹⁴C, H.L. Zang et al. conducted a breakup reaction using a 25.3 MeV/nucleon ¹⁴C secondary beam with an approximate intensity of 2×10⁴ pps, off a CH₂ target [39]. By gating on the Q_ggg peak at about -12.0 MeV for ¹⁰Be on the ground state, the resonant states of ¹⁴C can be reconstructed with the ⁴He + ¹⁰Be(g.s.) decay channel, as shown in Fig. 8, together with the previous observations of M. Freer [77].

Figure 8:Full-screen View

(Color online) Resonant states of ¹⁴C deduced from the ⁴He + ¹⁰Be(g.s.) decay channel, measured in the (a) previous and in the (b) present experiments. The figures are taken from Ref. [39]

In Fig. 8, the resonant states at 14.8 MeV and 15.6 MeV are distinctly identified in the previous and present measurements, whose spin-parities are assigned to 5^- and 3^-, respectively. Due to the low statistics of another clearly observed 14.1 MeV peak, it is difficult to determine the spin-parity information by the angular correlation or differential cross-section methods. However, by comparing the observed relative yields between 14.1 MeV and 15.6 MeV states, the spin of the 14.1 MeV resonant state is constrained to be 0, 1, and 2. This is compatible with the prediction of a 0⁺ band-head at about 14 MeV for a π-bond linear-chain configuration [73]. It would be interesting to study this state further with a direct determination of its spin and cluster-decay branching ratio.

3.2 Transfer Reactions

Transfer reactions are also considered to be an important approach to explore the cluster structures in neutron-rich light nuclei. For example, W. Jiang et al. performed an experiment using a 45 MeV ⁹Be primary beam with an approximate intensity of 7 enA, off a self-supporting ⁹Be target [29]to study the high-lying excited states in ¹⁰Be. The α + ⁶He and t + ⁷Li decay channels were detected simultaneously by the invariant mass as well as missing mass methods.

The experimental Q-value spectrum of three different decay channels, namely, ⁹Be(⁹Be, ⁸Be→α+α), ⁹Be(⁹Be, ¹⁰Be^*→α+⁶He), and ⁹Be(⁹Be, ¹⁰Be^*→ t+⁷Li), is shown in Fig. 9, by selecting the coincidence events in detectors. The missing mass spectrum for ¹⁰Be can be extracted by gating on the Q-value peaks in Fig. 9(a). Owing to the appropriate energy calibration and event selection, the four measured excited states as well as the ground state are consistent with the previous observations [89].

Figure 9:Full-screen View

Experimental Q-value spectrum for 3 coincident events: (a) ⁹Be(⁹Be, ⁸Be→α+α), (b)⁹Be(⁹Be, ¹⁰Be^*→α+⁶He), and (c) ⁹Be(⁹Be, ¹⁰Be^*→ t+⁷Li). The figures are taken from Ref. [29]

By gating on the peaks in Fig. 9(b), the energy spectrum for ¹⁰Be can be obtained from α + ⁶He coincident events by the standard invariant mass method, where a new state is located at 15.6 MeV in agreement with the proposed 6^- member of the 1^- (5.96 MeV) rotational band [90].

By gating on the peaks in Fig. 9(c), the energy spectrum for ¹⁰Be can be reconstructed from t + ^7Li coincident events in a similar manner. The 18.55 MeV excited state is found to decay to the ground state of ⁷Li as well as the 0.478 MeV state with a relative branch ratio of 0.93±0.33 and the 18.15 MeV excited state decays predominately to the ground state of ⁷Li, which is inconsistent with the previous experimental result [32]. This abnormal decay pattern of the high-lying excited states in ¹⁰Be may indicate some unique cluster structures that require further theoretical investigations.

The clustering investigations with transfer reactions were extended to the neutron-excess carbon isotopes as well. For example, J. Li et al. recently succeeded in searching for promising evidence of the intriguing σ-bond linear-chain structure in ¹⁴C from the transfer reaction ⁹Be(⁹Be, ¹⁴C^*→⁴He+¹⁰Be)⁴He with Q=17.3 MeV [38]. The experimental Q-value spectra with a very clean Q_ggg peak and two additional clean peaks are shown in Fig. 10, which makes it possible to clearly select the final states of ¹⁰Be fragments.

Figure 10:Full-screen View

Experimental Q-value spectrum for three data sets: (a) identified ¹⁰Be and identified α, (b) identified ¹⁰Be and unidentified α, and (c) identified α and unidentified ¹⁰Be. Spectrum (d) is taken from a previous experiment [91]. The figures are taken from Ref. [38]

By gating on different Q-value peaks in Fig. 10, the excitation spectra of ¹⁴C can be obtained from the α + ¹⁰Be fragments by the invariant mass method. The excited states located at 16.5 MeV, 19.8 MeV, 20.8 MeV, 21.4 MeV, and 22.5 MeV agree quite well with the previous experimental results [91]; moreover, a new resonant state at 23.5(1) MeV is clearly identified which decays dominantly to the first excited state in ¹⁰Be and must be further investigated.

The relative branching ratio of the 22.5(3) MeV resonant state in ¹⁴C has been quantitatively determined as shown in Fig. 11. This state is found to decay strongly to the 6 MeV states in ¹⁰Be, which corresponds well with the band-head at 22.2 MeV of the σ-bond linear-chain molecular band within the framework of AMD, as predicted by T. Baba [41].

Figure 11:Full-screen View

(Color online) Comparison of the relative decay branching ratio obtained from the experiment [38] from the theoretical prediction [41].

The theoretical predictions suggest a pure decay branching ratio to the ~6 MeV states whereas the experimental observations show a small fraction of the branching ratio to the ground state and the first excited state in ¹⁰Be, as illustrated in Fig. 11. This may be inferred as the existence of other configurations, except for the σ-bond linear-chain structure in the 22.5(3) MeV state. Out of the four contiguous states in the ~6 MeV region, namely, 5.958 MeV(2⁺), 5.96 MeV(1^-), 6.18 MeV(0⁺), and 6.26 MeV(2^-), only the 6.18 MeV(0⁺) state has a pure σ-bond configuration whereas the other three states possess a π-bond configuration [22]. Therefore, it can be presumed that the predominantly collected final state in ¹⁰Be should be the 6.18 MeV state with spin-parity Jπ=0⁺. This special structural link between the parent and daughter nucleus is another important clustering criterion (Criterion IV) as shown qualitatively in Fig. 12.

Figure 12:Full-screen View

(Color online) Selective decay patterns of high-lying excited states of ¹⁴C due to the structural link.

3.3 Resonant Scattering Reactions

Resonant scattering with projectiles travelling through a light thick target was studied by several groups to investigate the nuclear clustering in some resonant states. The targets are required to be thick enough to stop the beam projectiles so that the detectors can be placed at zero degrees, covering the large backward angles in the c.m. frame [92]. As the beam particles slowdown in the target, a resonant state may be created inside the projectile-target composite system when the c.m. energy coincides with the corresponding resonant energy. The excited composite system subsequently decays into the projectile and target nuclei. In general, the beam particles should be heavier than the target nucleus, in order to apply detection in inverse kinematics. As different resonances would be produced at different positions within the target, corrections must be made for the detection solid angle for each resonance. In addition, the effects of inelastic scattering must be taken into account, which may lead to false identification of a resonant energy.

M. Freer et al. reported a resonant ⁶He + ⁴He scattering experiment to investigate the molecular clustering structure in the resonances of ¹⁰Be [85]. When the c.m. energy, contributed by the energy of the ⁶He projectile, is equal to a certain resonance-energy of ¹⁰Be, a composite nucleus is produced which subsequently decays into the fragments ⁶He + ⁴He. These fragments can be coincidentally observed by suitably placed detectors.

The angular distribution of the c.m. for the 10.15 MeV resonance was experimentally determined as shown in Fig. 13. The simulation results with a single Breit-Wigner formula demonstrate that the oscillatory behavior of the experimental angular distribution can only be reproduced with a spin-parity of Jπ=4⁺, which is considered as the third member of a rotational band with a 6.18 MeV($0_2^{+}$) band head state. The ħ²/2J value of this rotational band is about 0.20 MeV, which is comparable with that of the 10.3 MeV(0⁺) rotational band in ¹²Be,suggesting a large deformation in the 10.15 MeV resonant state [26]. Moreover, the alpha decay width (Γ_α) was deduced to be 0.10∼0.13 MeV from the measured distributions and the branching ratio was Γ_α/Γ_tot = 0.35 ~ 0.46. The corresponding dimensionless cluster spectroscopic factor (${\theta }_{\alpha }^{\text{2}}=0.66\sim 2.23$), which is comparable to that of the 10.3 MeV(0⁺) cluster resonance in ¹²Be. Such a large deformation of the rotational band and large alpha decay width confirm a well-developed α∶2n∶α molecular structure in ¹⁰Be.

Figure 13:Full-screen View

(Color online) Experimentally determined c.m. distribution (data points), compared with simulations for the decay of a resonance with spin 4 (blue-solid line), 2 (red-dashed line), and 6 (black-long-dashed line). The inset shows the energy-spin systematics of the rotational band, of which the present 4⁺ state is a member. The figures are taken from Ref. [85]

Similar methods were developed by several groups to study the cluster structures in neutron-rich ¹⁴C. M. Freer et al. performed another resonant scattering reaction with ¹⁰Be and ⁴He to obtain the resonant spectra [36]. The associated spin-parity information was discussed in detail with an analysis of the angular distribution. Monte Carlo simulations were applied to determine the energy loss processes of the decay fragments and the solid angle associated with the interaction position in the thick target. The experimental angular distribution for the 18.8 MeV resonance was compared with that of the simulation results, as illustrated in Fig. 14. It can be seen that the experimental data is in agreement with a spin-parity of Jπ=5^-. Similar angular distribution analysis was extended to the high-lying 17.2 MeV and 20.9 MeV resonances as well, whose spin-parity is proposed to be Jπ=3^- and Jπ=6⁺, respectively.

Figure 14:Full-screen View

(Color online) The experimental angular distributions for 18.8 MeV resonance at 32 MeV beam energy are compared with that of the simulations with Jπ=4⁺, 5^-, 6⁺. The figures are taken from Ref. [36]

The experimental excitation function above 16.5 MeV can be analyzed by the R-matrix method [84] and the resonance parameters such as the resonant energy, spin-parity and α-decay widths can be determined. The R-matrix analysis proposes that these resonant states have large α-decay widths which are close to the Wigner limit, indicating a strongly deformed ¹⁰Be + ⁴He cluster structure in ¹⁴C.

In order to collect more evidence for the cluster configurations in ¹⁴C, A. Fritsch et al. carried out further ¹⁰Be + ⁴He resonant scattering experiments with the newly developed AT-TPC detector [94], which covers a larger range of c.m. angles for both elastic and inelastic scattering measurements [74]. Fig. 15 shows the trajectories of the ¹⁰Be + ⁴He resonant scattering event and the kinematical correlation of the θ_lab between ¹⁰Be and ⁴He. The resonance parameters can be obtained by the R-matrix method. Two resonances in ¹⁴C located at 15.0 MeV and 19.0 MeV were assigned a spin-parity of 2⁺ and 4⁺, and a spectroscopic factor of 30% and 20%, respectively.

Figure 15:Full-screen View

(Color online) The trajectories of a ¹⁰Be + ⁴He resonant scattering event and the kinematical correlation of the θ_lab between ¹⁰Be and ⁴He. The figures are taken from Ref. [74]

It is worth noting that a recent ¹⁰Be + ⁴He resonant scattering experiment was reported by H.Yamaguchi et al. [96], in which the rotational band with a linear-chain configuration in ¹⁴C was suggested. The elastic scattering excitation function in this experiment is illustrated in Fig. 16. R-matrix calculations with a single channel involved were performed to determine the resonance parameters.

Figure 16:Full-screen View

(Color online) Elastic scattering excitation function fitted with R-matrix method. The figures are taken from Ref. [96]

Based on the AMD calculation, T. Suhara and Y. Kanada-En’yo proposed a prolate band ( J^π=0⁺, 2⁺, 4⁺) with a linear-chain cluster structure at a few MeV about the ¹⁰Be + ⁴He threshold [95]. By the angular distribution and R-matrix analyses, spin-parities of 0⁺, 2⁺, and 4⁺ were assigned to the three resonances located at 15.1 MeV, 16.2 MeV, and 18.7 MeV,respectively, which are in excellent agreement with the predictions. By fitting the energy spin systematics, the newly identified resonances can be interpreted as a rotational band, whose ħ²/2J value is approximately 0.19 MeV, indicating a strongly deformed structure in conformance with the linear-chain configuration.

Many experiments have been conducted with the resonant scattering method to investigate the clustering structure in ¹⁴C up to now. However, there is no consistency in these results, which indicates that more measurements using different methods are necessary.