Resonating group method

The RGM was proposed by Wheeler in 1937 [12]. At that time, the nucleon- and cluster-based models were competing to account for the observed properties of nuclei. Within the RGM approach, nucleons in a nucleus are configured into many groups, with the number of groups and numbers of protons and neutrons in each group being defined in a selective way. The spin of each group is constrained such that all spins must sum to the total spin of the nucleus. In addition, different configurations can be added together to present the complete w.f. [12]. The RGM is distinguished by its definition of the c.m. of each group and the relative coordinates between these c.m. positions (inter-cluster coordinates), upon which the interactions between the groups (cluster interactions) and the w.f. for the group–c.m. distribution (cluster w.f.) can be defined. The real cluster w.f. (probability distribution of the groups) can then be obtained according to the standard energy minimizing method, assuming the known Hamiltonian, which is also configured in line with the group definitions. In general, RGM focuses on addressing the cluster interactions and the cluster w.f. while treating the internal interaction and structure of each cluster (group) as a known basis. Hence, the selection of the group configurations should not be arbitrary. Instead, the selection should be performed in accordance with the real structure possibilities, as reflected by the existing observations and solid predictions. Otherwise, the results from the calculation might be different from reality due to the limited knowledge of the interactions (Hamiltonian) and unavoidable truncation of the configuration possibilities (state space). The RGM provides a flexible means by which to describe a complex many-body system that can suitably mix single-particle-type and cluster-type structures for use with nuclei that are far from the β-stability line. However, their method of use is still an unsolved challenge associated with experimental hints, theoretical background knowledge, and physical understanding [15, 59]. In addition, the full anti-symmetrization among all neutrons or protons of a system would cause large difficulties in real calculations. This constraint could partially be relaxed via the selection of a simple group configuration and the application of numerical methods, such as the generator coordinate method.

Here, we use a simple example to illustrate the RGM application [67, 15]. A system with A nucleons can be divided into two clusters with A₁ and A₂ nucleons. Then, the corresponding RGM w.f. can be expressed as: $\text{Φ}(A_1\mathrm{,}{A}_2\mathrm{,}r)=\mathcal{A}\chi (r)\varphi (A_1)\varphi (A_2)\mathrm{,}$ (1) Where the relative coordinate $r=X_{A_1}-X_{A_2}$, with $X_{A_1}=\left(1/A_1\right){\displaystyle {\sum }_{i=1}^{A_1}{r}_i}$ and $X_{A_2}=\left(1/A_2\right){\displaystyle {\sum }_{i=1}^{A_2}{r}_i}$ being the c.m. coordinates of the two ingredient clusters A₁ and A₂, respectively. Here, 𝒜 is the anti-symmetrization operator for all nucleons. The internal w.f.s φ(A₁) and φ(A₂) are pre-solved, using, e.g., in the harmonic oscillator single-particle model, which is represented only by its internal coordinates. The w.f. χ(r) that governs the relative motion between the clusters can be determined via the application of the energy variation principle after configuring the Hamiltonian as the internal terms and a relative motion term [15]. In this framework, it is necessary to define the internal coordinates for each cluster: $\begin{array}{ll}{\xi }_{1i}\hfill & =r_i-X_{A_1}\mathrm{,}i=\mathrm{1,...}{A}_1\mathrm{,}\hfill \\ {\xi }_{2i}\hfill & =r_i-X_{A_2}\mathrm{,}i=A_1+\mathrm{1,,...}A\mathrm{.}\hfill \end{array}$ (2) Numerous studies have applied RGM, including those that examined the 3α configuration of the Hoyle state in ¹²C [19]; X + ¹⁶O configurations for ¹⁷O, ¹⁸F, ¹⁹F, and ²⁰Ne [68, 69]; α + ⁶He configuration for ¹⁰Be [70]. One example for dripline nuclei can be found in Ref. [59], in which the extremely neutron-rich nucleus ⁸He is configured as both ⁶He(shell-like) + 2n^* (dissociable 2n cluster) and ⁴He(shell-like) + 2n^* + ²n(dineutron). After the energy minimization calculation, the dineutron-condensation-like structure is well revealed in the 0⁺₂ state of ⁸He, which gives rise to an important instruction for upcoming experiments [60].

Generator coordinate method

The GCM was originally proposed by Griffin and Wheeler to address the collective motion of the nuclear system [13]. This method aims to address overall coordinates, such as the deformation parameter of a nucleus, at several discrete values to generate the basis of the structure w.f.. These basis w.f. are then weighted and summed (integrated) to represent the structure w.f. for the system. The unknown weight factors, as a function of the generator coordinate, are determined according to the variational principle under the operation of the appropriate Hamiltonian. Hence, the complex calculations using the continuous variations of the dynamic coordinates can be simplified and extended to relatively heavier systems.

The GCM has naturally been combined with the RGM by using the inter-cluster coordinates as the generator coordinates [71, 67]. In the case of ^9-11Be nuclei, the analysis using GCM reveals a decrease of the α-cluster contents from ⁹Be to ¹¹Be. In addition, the results of this analysis predict a strong mixing of the ⁶He + ⁶He and ⁴He + ⁸He configurations in ¹²Be nuclei. Recently, the combination of RGM and GCM has become the fashion for studying the structural configurations of nuclei far from the β -stability line. An exemplar is presented in Ref. [59], which demonstrates a case of ²n clustering⁸. The w.f. of the 0⁺ states (⁸He) is given in terms of the RGM + GCM: $\begin{array}{c}{\text{Ψ}}_{{}^{\text{8}}\text{He}\left({\text{0}}^{+}\right)}={\displaystyle \sum _i{c}_i}{\mathcal{P}}_{00}^{0+}{\text{Φ}}_{{}^{\text{6}}\text{He}+2n^{*}}\left(\kappa ;\lambda \mathrm{,}{d}_{\lambda }\mathrm{,}{b}_{\lambda }=b_{\alpha }\right)\\ +{\displaystyle \sum _j{c}_j}{\mathcal{P}}_{00}^{0+}{\text{Φ}}_{\text{DC}}\left(\lambda \mathrm{,}{d}_{\lambda }\mathrm{,}{b}_{\lambda };\beta \mathrm{,}{b}_n\right)\mathrm{.}\end{array}$ (3) Here, ${\mathcal{P}}_{00}^{0+}$ denotes the projection operator; κ assigns the neutron configuration in the ⁶He core; i = ｛λ, d_λ｝_i and j = ｛λ, d_λ, b_λ;β, b_n｝ are the selected computational values for the generator coordinates corresponding to the dissociation strength of the 2n^* cluster (λ), distance between the ⁶He core and 2n^* cluster (dλ); size of the 2n^* cluster (bλ), volume for the ²n-pair movement (β); and size of the ²n-pair (b_n). The weight coefficients c_i,j are determined by the diagonalization of the Hamiltonian. The characteristics of these 0⁺ states can be analysed via their overlap with the well-formed dineutron condensation w.f. ${\text{Φ}}_{\text{DC}}$: $\begin{array}{l}{\mathcal{N}}_{\text{DC}}^{{}^{\text{8}}\text{He}}\left(\beta \mathrm{,}{b}_n\right)\\ ={\left|\langle {\mathcal{P}}_{00}^{0+}{\text{Φ}}_{\text{DC}}\left(\lambda ={\lambda }_0\mathrm{,}{d}_{\lambda }=d_0\mathrm{,}{b}_{\lambda }=b_{\alpha };\beta \mathrm{,}{b}_n\right){\text{Ψ}}_{{}^{\text{8}}\text{He}\left({\text{0}}^{+}\right)}\rangle \right|}^2.\end{array}$ (4)

This overlap is equivalent to the projection of the eigenstate w.f. onto the RGM clustering basis. The BEC character of the ⁸He(0⁺₂) state obtained from this RGM + GCM calculation is displayed in Fig. 2. This work demonstrates the flexibility and powerful ability of using the RGM combined with the GCM.

Fig. 2Full-screen View

(Color online) The overlaps of the ⁸He(0⁺₂) state with the dineutron-condensation w.f. on the β-bn plane. Panels (a) and (b) show the results of the m55 and m59 interaction parameter sets, respectively. These parameter sets have different Majorana, Bartlett, and Heisenberg parameter values in the central force. The figures are taken from Ref. [59]

Notably, when extending the inter-cluster coordinate to a large range, the GCM can be applied to scattering problems. The first treatment of the scattering problem using GCM was reported in 1970 by Horiuchi [71]. Since then, significant progress has been achieved in this direction [15, 72-75].

Orthogonality condition method

The calculations in the RGM + GCM approach can be simplified further using the orthogonality condition method (OCM), which was initially proposed by Saito in 1968 [14]. The idea is to exclude the states for the relative motion of the clusters that are occupied by the internal structure of the clusters themselves, namely $\langle {\phi }_i|\chi \left(r\right)\rangle =0$ for φi, representing the internal states of the clusters, and χ(r), which is the between-cluster relative motion w.f.. The detailed formulation of the OCM can be found in Ref. [67].

OCM was proven to be highly effective in describing the complex nuclear cluster structure and scattering. In the 1970s, Matsuse and Kamimura conducted quantitative computations using OCM and indicated that the first $K^{\pi }{=0}^{-}$ band of ²⁰Ne exhibits a strong α-clustering character, whereas the ground state band predominantly displays shell-model-like structures [76]. Notably, Horiuchi extended the OCM to address the multi-cluster problems and reproduced the binding energy and excited levels of ¹²C, thus revealing the coexistence of the shell model-like and molecular-like structures [77]. In recent years, OCM has been applied to the structure of the ¹³C nucleus to investigate the influence of the additional valence neutron upon the 3α core [78-80]. Moreover, Yamada has extended the OCM calculations to a large multi-cluster model space and revealed that the $0_6^{+}$ state of ¹⁶O, which is approximately 2 MeV above the 4α breakup threshold, exhibits a significant 4α-condensation character. Similar work was also conducted for the 2α + t configuration in ¹¹B [81].

Molecular orbit model and generalized two-center cluster model

The molecular viewpoint of nuclear structure was introduced by Wheeler in 1937 [82], the same year the release of the RGM [12]. This is natural, as the interactions between groups (clusters) are quite similar to molecular bonds in atomic physics. The molecular picture supports the excessive binding energy of the α-conjugate nuclei, which is roughly proportional to the number of bonds between α-particles [11]. This concept was further developed and applied to light nuclei to account for the energy-level spectra, which could be classified into the molecular rotational bands together with the proposal of the “threshold rule (Ikeda diagram)” [23, 21, 83, 84].

The MO model was proposed for nuclei with excessive nucleons in addition to cluster cores, particularly unstable neutron-rich or proton-rich nuclei [85, 29, 30, 39, 3, 86]. The basic idea is to exploit the multicenter potential, corresponding to the cluster cores, within which the valence nucleons move around one core or multiple cores, similar to the valence electrons playing roles in ionic or covalent bonding between atoms, respectively. The MO states may occur at energies below the cluster-separation threshold and create bound (lower-lying) molecular structures [3, 86]. Here, we illustrate a simple case of a two-center system, such as that for ^9,10Be. The molecular w.f. can be formed by the linear combination of nuclear orbitals (LCNO): $\begin{array}{l}{\text{Φ}}_{\text{LCNO}}^{K\mathrm{,}\text{Π}}\left(R\mathrm{,}{r}_n\right)=N(R)\frac{1}{\sqrt{\left(1+{\delta }^{K\mathrm{,}p}\left(R\right)\right)}}\\ \times \left[C_1{\text{Ψ}}_{{\alpha }_1}{\text{Ψ}}_{{\alpha }_2}{\text{Φ}}_{n\mathrm{,}{l}_1}^K\left(r_{n{\alpha }_1}\right)+{(-1)}^p{C}_2{\text{Ψ}}_{{\alpha }_1}{\text{Ψ}}_{{\alpha }_2}{\text{Φ}}_{n\mathrm{,}{l}_2}^K\left(r_{n{\alpha }_2}\right)\right],\end{array}$ (5) where 𝑹 is the vector connecting the c.m. of the two α cores, r_n is the vector between the c.m. of the 2α system and the valence neutron, r_n,ai the vector between the c.m. of each α core and the valence neutron (Fig. 13 described in Ref. [3]), Ψ_ai is the internal w.f. of each α core, ${\text{Φ}}_{n\mathrm{,}li}^K$ the single-particle w.f for the valence neutron around each core (⁵He state) with an li orbital angular momentum of and K projection of its total angular momentum onto the direction of 𝑹, and ${\delta }^{K\mathrm{,}p}$ is the nonorthogonality factor determined by the overlap of the two single-particle w.f.s associated with α 1 and α 2. The phase (-1)^p defines the gerade (g, positive sign) and ungerade (u, negative sign) properties, which give rise to the parity of the entire molecular system ∏ = (-1)^p+l. In addition, Ci gives the amplitudes for the sharing of valence neutrons between the two α-centers, with an extreme case of ionic bonding for C₁ = 1 and C₂ = 0, and vice versa. Neglecting the effect of the spin of the valence neutron, K becomes m, which represents the projection of li (=1 in the case of ^{9, 10}Be) onto the symmetry axis. A value of m = 0 would yield a π-bond configuration, corresponding to the valence neutron mostly distributed outside the symmetry axis, whereas m = 1 would yield a π-bond configuration for the valence neutron distributed mainly along the symmetry axis, as depicted in Fig. 3. By taking the distance between the 2α core as a generator coordinate and by using more li shell orbitals, the molecular states can be calculated as shown in Fig. 4.

Fig. 3Full-screen View

(Color online) Illustration of the spacial overlap of two ⁵He(p_3/2) orbits corresponding to m for π-bond (upper part) and m for σ-bond (lower part) molecular orbitals, respectively. This figure was taken from Ref. [3]

Fig. 4Full-screen View

(Color online) Molecular orbitals for a two-center system. At R_min, the α–α–n potential attains its minimum. The quantum numbers (K, parity, and gerade and ungerade (g and u) properties) for various two-center orbits are indicated. The orbitals become ionic at large values of R. This figure was taken from Ref. [3]

Okabe and Abe reproduced the properties of the low-lying levels of ⁹Be [29, 30] using the MO model. Itagaki et al. predicted the exotic behaviors in excited states of ¹⁶C in which two neutrons occupy a σ orbit while another two neutrons occupy a π orbit, thus offering new insights into neutron-rich carbon isotopes [87].

Since the MO model is limited in its descriptions of the low-lying states of light neutron-rich nuclei, an extended molecular-type model, namely, the GTCM, was introduced by Ito et al. [40, 88, 89]. This method combines the covalent configuration, similar to the LCNO, with the ionic configuration and solves the equation of state as a function of the distance between the two centers (which is used as a generator coordinate) in a unified way. The model can then be applied to predict highly excited resonant states in which the two correlated centers may become two quasi-independent clusters, with each valence particle being combined into one as an ionic system. These resonant states then undergo decays, according to the cluster partial widths. Taking ¹²Be (α+α+4n) as an example, the basis function within the GTCM approach can be expressed in the form [40, 89]: ${\text{Φ}}_m^{J^{\pi }K}\left(S\right)={\mathcal{P}}_K^{J^{\pi }}\mathcal{A}{｛{\psi }_{\text{L}}\left(\alpha \right){\psi }_{\text{R}}\left(\alpha \right){\displaystyle \prod _{j=1}^4}{\phi }_j\left(m_j\right)｝}_S$ (6) with 𝒜 representing the antisymmetrization operator for all nucleons, ${\mathcal{P}}_K^{J^{\pi }}$ being the projection onto the eigenstate with a total spin of J, projection of K, and total parity of π. The relative distance parameter between the 2α centers is S, which is variational as a generator coordinate. The w.f. for the α core, denoted by Ψ_i(α) (i = L or R), has a (0s)⁴ configuration of its harmonic oscillator centered at the left (L) or right (R) cluster. The shell-like single-particle w.f. is described by φ_j(m_j) (j = 1 to 4). The symbol m represents the set $\left\{m_1\mathrm{,}{m}_2\mathrm{,}{m}_3\mathrm{,}{m}_4\right\}$, where each mj term represents a set of indices for a shell-like orbit $\left\{l_j\mathrm{,}{i}_j\mathrm{,}{\tau }_j\right\}$ with an orbital angular momentum of lj, associated center of ij (L or R), and spin projection of τj (↑ or ↓).

The total w.f. is obtained by taking a superposition over S, 𝒎, and K as: ${\text{Φ}}^{J^{\pi }}={\displaystyle \int }\text{d}S{\displaystyle \sum _{m\mathrm{,}K}{C}_{mK}}\left(S\right){\text{Φ}}_m^{J^{\pi }K}\left(S\right)\mathrm{,}$ (7) The summation includes all covalent and ionic configurations because of all the combinations of ij (L or R). The amplitude C_mK(S) can be determined by solving the coupled channel GCM equation: $0={\displaystyle \int }\text{d}S{\displaystyle \sum _{mK}{C}_{mK}}\left(S\right)\langle {\text{Φ}}_{\text{m"}}^{J^{\pi }K\text{'}}\left(S^{\text{'}}\right)|H-E^{J^{\pi }}|{\text{Φ}}_m^{J^{\pi }K}\left(S\right)\rangle \mathrm{.}$ (8) GTCM naturally captures the formation of covalent molecular orbits as well as the ionic cluster states, depending on the relative distance S. Fig. 5 shows the GTCM predicted Jπ = 0⁺ states in ¹²Be [90]. The low lying states are basically dominated by the covalent configurations, whereas the higher lying states are closer to the ionic configurations because of the expansion of the system size (larger S). Particularly, the predicted $0_3^{+}$ state, configured as ⁴He + ⁸He, has a large monopole transition strength from the g.s., which is an observable imprint of the cluster-structure formation in a 0⁺ state with a relatively low excitation energy [90]. This prediction has been confirmed by the following experiment [46].

Fig. 5Full-screen View

(Color online) The intrinsic structures and energy positions for the 0⁺ excited states of ¹²Be are shown by the illustrations on the left side, whereas the histograms on the right show the respective monopole transition strengths. The horizontal dashed lines show the indicated cluster-separation thresholds. The bottom-right arrow represents the energy position of the ground state. The blue-dashed arrow represents the single-particle excitation, whereas the red-solid arrow indicates the cluster excitation in adiabatic conjunction form. This figure was taken from Ref. [90]

Antisymmetrized molecular dynamics and fermion molecular dynamics

In the 1960s, the Brink w.f. was proposed to describe the nuclear structure using the wave-packet concept rather than the orbit configurations [22]. Basically, each nucleon or a group of nucleons is presented by a Gaussian function centered at a position R. Many Rs values can become the generator coordinates, and various wave-packet configurations can be combined and fixed, according to the weights determined by the GCM equation. The wave-packet basis is in favor of describing the clustering system within a limited state space, whereas the orbit configurations are better suited to account for the single-particle behaviors in a centralized mean field. Brink-type w.f.s have since been accepted and extended by many cluster-oriented models, such as AMD and THSR [32, 25].

In the AMD model, each nucleon is described by a Gaussian wave packet, which can be regarded as the extreme of the Brink w.f.. The model originated from the quantum molecular dynamics (QMD) approach used to simulate the heavy-ion reactions, but was implemented by applying the full antisymmetrization among all nucleons to account for the real nuclear structure [91, 32, 33]. FMD is similar to AMD, but FMD incorporates a variable width parameter for the Gaussian-type wave packet [92]. Both AMD and FMD consistently suggest that the second 0⁺ state of ¹²C (the Hoyle state) has a prominent gas-like 3α configuration with an extended size [92-94].

The AMD w.f. for a nucleus is expressed as a Slater determinant composed of single-nucleon Gaussian wave functions: ${\text{Φ}}_{\text{AMD}}(Z)=\frac{1}{\sqrt{A!}}\mathcal{A}\{{\phi }_1\mathrm{,}{\phi }_2\mathrm{,}\dots \mathrm{,}{\phi }_A\}\mathrm{,}$ (9) where $1/\sqrt{A!}$ is a normalization factor and φi describes the single-particle w.f. of the ith nucleon, which is a product of a spatial part (${\varphi }_{Z_i}$), an intrinsic spin part (χi), and an isospin part (τi): ${\phi }_i={\varphi }_{Z_i}\cdot {\chi }_i\cdot {\tau }_i\mathrm{.}$ (10) The spatial wave function, denoted by ${\varphi }_{Z_i}$, is given by: ${\varphi }_{Z_i}\left(r_j\right)={\left(\frac{2\nu }{\pi }\right)}^{4/3}\text{exp}｛-\nu {\left(r_j-\frac{Z_i}{\sqrt{\nu }}\right)}^2｝\mathrm{,}$ (11) where Z_i presents the center of the Gaussian wave packet. The size parameter ν is common to all nucleons. The intrinsic spin can be expressed in the form: ${\chi }_i=\left(\frac{1}{2}+{\xi }_i\right){\chi }_{\uparrow }+\left(\frac{1}{2}-{\xi }_i\right){\chi }_{\downarrow }\mathrm{,}$ (12) where the ξi parameter provides the specific gravity of components with different spin orientations. If all the Gaussian functions in the AMD w.f. (Eq. 11) are centered around a common position, the resulting w.f. resembles a shell-model-like w.f.. These results indicate a distribution of nucleons with individual wave packets localized around a single center. However, if the Gaussian functions are centered around multiple positions, the AMD w.f. gives rise to a multi-cluster structure that is equivalent to the Brink w.f..

The Hamiltonian of the AMD can be given by: $\widehat{H}={\displaystyle \sum _{i=1}^A\frac{{\widehat{p}}_i^2}{2m_i}}+{\displaystyle \sum _{i<j}{V}_{ij}^{\left(2\right)}}+{\displaystyle \sum _{i<j<k}{V}_{ijk}^{\left(3\right)}}-T_{\text{c}\text{.m}\text{.}}\mathrm{,}$ (13) where A denotes the total number of nucleons, T_c.m. denotes the center-of-mass kinetic energy, and $V_{i\mathrm{,}j}^{\left(2\right)}$ and $V_{i\mathrm{,}j\mathrm{,}k}^{\left(3\right)}$ denote the two-body and three-body interactions, respectively. The two-body part can be specified as: $\begin{array}{ll}{V}_N^{\left(2\right)}\hfill & =\left(\omega +b{\widehat{P}}_{\delta }+h{\widehat{P}}_{\tau }-m{\widehat{P}}_{\delta \tau }\right)\left[V_{\text{A}}{e}^{-\frac{r_{ij}}{a_A}}+V_{\text{R}}{e}^{-\frac{r_{ij}}{a_{\text{R}}}}\right]\mathrm{,}\hfill \\ V_{LS}^{\left(2\right)}\hfill & =\left(U_A{e}^{-\frac{r_{ij}}{a_A}}+U_{\text{R}}{e}^{-\frac{r_{ij}}{a_{\text{R}}}}\right)\widehat{P}\left({}^{3}O\right)\widehat{L}\cdot \left({\widehat{S}}_i+{\widehat{S}}_j\right)\mathrm{,}\hfill \\ V_{ijk}^{\left(2\right)}\hfill & ={\nu }_3\left(r_i-r_j\right)\delta \left(r_i-r_k\right)\mathrm{.}\hfill \end{array}$ (14) where $V_N^2$ is the nucleon-nucleon interaction, $V_{LS}^2$ the coupling term for intrinsic orbits, ${\widehat{P}}_{\sigma }$ is the spin exchange operator, and ${\widehat{P}}_{\tau }$ the isospin exchange operator. The subscripts A and R correspond to the attractive and repulsive interactions, respectively. Last, $\widehat{P}\left({}^{3}O\right)$ is the projection operator. The two-body interaction parameters are usually taken from the Volkov, Minnesota, parameter sets [95, 96] or the widely used Gogny forces. Here, $V_{ijk}^3$ represents the density-dependent three-body force, and ν₃ indicates its strength. Strong correlations between nucleons within the cluster structure emphasize the significance of multi-body correlations in understanding the clustering structure. The energy of the system can then be calculated according to: $\begin{array}{l}E=\frac{\langle {\text{Φ}}_{\text{AMD}}\left|H\right|{\text{Φ}}_{\text{AMD}}\rangle }{\langle {\text{Φ}}_{\text{AMD}}|{\text{Φ}}_{\text{AMD}}\rangle }\mathrm{.}\hfill \end{array}$ (15) The positions of wave packets vary under the influence of nuclear forces until they reach a stable configuration that corresponds to the minimum energy state of the system. However, at this stage, the w.f. obtained using Eq. 9 does not satisfy the requirements of parity or angular momentum symmetry. To restore their symmetry, it is necessary to obtain the eigenwave functions of spin and angular momentum via spin parity projection: $\begin{array}{c}{\text{Φ}}^{\pm }\left(Z\right)={\mathcal{P}}^{\pm }{\text{Φ}}_{\text{AMD}}\left(Z\right)=\frac{1\pm {\mathcal{P}}_r}{2}{\text{Φ}}_{\text{AMD}}\left(Z\right)\mathrm{,}\\ {\text{Φ}}_{MK}^{J\pm }\left(Z\right)={\mathcal{P}}_{MK}^J{\text{Φ}}^{\pm }\left(Z\right)={\displaystyle \int }\text{d}\text{Ω}{D}_{MK}^{J*}\left(\text{Ω}\right)\mathcal{R}(\text{Ω}){\text{Φ}}^{\pm }\left(Z\right)\mathrm{,}\end{array}$ (16) where ${\mathcal{P}}^{\pm }=(1\pm {\mathcal{P}}_r)/2$ and ${\mathcal{P}}_{MK}^J\equiv {\displaystyle \int }\text{d}\text{Ω}{D}_{MK}^{J*}\left(\text{Ω}\right)\mathcal{R}(\text{Ω})$ are the parity projection and total angular momentum projection operators, respectively; Ω denotes the solid angle; $\mathcal{R}(\text{Ω})$ denotes the rotation operator; and $D_{MK}^{J*}\left(\text{Ω}\right)$ the standard Wigner’s D function. The K number for each spin parity is determined based on the principle that the energy expectation value for the spin parity eigenstate ${\text{Φ}}_{MK}^{J\pm }\left(Z\right)$ is minimized [94]. Moreover, the AMD basis w.f. starts with a random initial distribution of nucleons. Many such bases must be superposed to produce various eigenstates of the nucleus, in a way similar to the application of GCM [33].

Over the past thirty year, AMD has been developed to provide not only the energy eigenvalues and nuclear radii but also other observables, such as the cluster decay widths, magnetic dipole and electric quadrupole moments, and electromagnetic transition probabilities. Recently, AMD has successfully reproduced experimental properties of various nuclei, including Be, C, O, and Ne isotopes [33], and it has become a powerful tool for investigating the cluster and molecular structures in atomic nuclei [17]. Here, we illustrate typical examples that are closely related to the latest experimental findings.

In 1998, Kanada-En’yo used the AMD model to reproduce all energy levels below 15 MeV in ¹²C together with the related E2 transition rates and β-decay strengths [97]. Subsequently, the AMD model was extended to investigate the excited states of ¹⁰Be using the variation after projection technique [98]. Remarkably, without assuming preformed 2α clusters, AMD calculations unveiled a notable 2α + 2n cluster structure in many intrinsic states of ¹⁰Be [98]. To extract the behaviors of the valence nucleons, the single-particle Hamiltonian under the AMD framework can be defined in the following form [99]: $\begin{array}{c}{h}_{\alpha \beta }=\langle {\tilde{\phi }}_{\alpha }\left|\widehat{t}\right|{\tilde{\phi }}_{\beta }\rangle +{\displaystyle \sum _{\gamma =1}^A\langle {\tilde{\phi }}_{\alpha }{\tilde{\phi }}_{\gamma }|{\widehat{\nu }}^N+{\widehat{\nu }}^C|{\tilde{\phi }}_{\beta }{\tilde{\phi }}_{\gamma }-{\tilde{\phi }}_{\gamma }{\tilde{\phi }}_{\beta }\rangle }\\ +\frac{1}{2}{\displaystyle \sum _{\gamma \mathrm{,}\delta =1}^A\langle {\tilde{\phi }}_{\gamma }{\tilde{\phi }}_{\delta }\left|{\tilde{\phi }}_{\alpha }^{*}{\tilde{\phi }}_{\beta }\frac{\delta {\widehat{\nu }}^N}{\delta \rho }\right|{\tilde{\phi }}_{\gamma }{\tilde{\phi }}_{\delta }-{\tilde{\phi }}_{\delta }{\tilde{\phi }}_{\gamma }\rangle }\mathrm{.}\end{array}$ (17) The single-particle orbits can be expressed as ${\tilde{\varphi }}_s={\displaystyle {\sum }_{\alpha =1}^A{f}_{\alpha s}{\tilde{\phi }}_{\alpha }}$, where f_as represents the eigenvectors of $h_{\alpha \beta }$. Within this framework [98, 39], the two valence neutrons in ¹⁰Be occupy molecular orbits around the 2α clusters with band heads of $K^{\pi }{=0}^{+}{\mathrm{,1}}^{-}{\mathrm{,0}}_2^{+}$, corresponding to the two valence neutrons occupying the π², $\sigma \pi$, and σ² bonds, respectively.

Later, Baba et al. extended the AMD calculation to ¹⁶C, ¹⁴C, and ¹⁴O nuclei [65, 66, 100, 101]. They identified 3-cluster structures with various valence-neutron configurations, as shown in Fig. 6. Panels (a) and (b) show the ground-state rotational bands of ¹⁶C [66], which exhibit approximately spherical structures. Panels (c) and (d) display asymmetric triangular rotational band structures, with $0_2^{+}$ and $2_5^{+}$ as the band head members, respectively. Panels (e) and (f) depict linear chain structures with a ${\left(3/2_{\pi }^{-}\right)}_{\pi }^2{\left(1/2_{\sigma }^{-}\right)}_{\sigma }^2$ configuration. The reversed moment of inertia for this band is $\frac{{\hslash }^2}{2\mathcal{I}}\sim 112\text{ keV}$, which agrees with the experimental result of $\frac{{\hslash }^2}{2\mathcal{I}}\sim 122\text{ keV}$ reported by Liu et al. in 2020 [49]. Fig. 7 shows the functional relationship between the positive parity energy levels and the quadrupole deformation parameter β [101]. The black circles, triangles, and squares represent the structures of the ground state, triangular, and ${\left(3/2_{\pi }^{-}\right)}^2{\left(1/2_{\sigma }^{-}\right)}^2$ linear chain rotational bands, respectively. In addition, in the higher excitation energy region near β=1.6, a peculiar linear chain structure with a pure σ configuration may exist [101], namely ${\left(1/2_{\sigma }^{-}\right)}^2{\left(1/2_{\sigma }^{+}\right)}^2$. The energy of the band-head is $E_x\doteq 31.72\text{ MeV}$ (0⁺) with a reversed moment of inertia $\frac{{\hslash }^2}{2\mathcal{I}}$ of ～50 keV, which is significantly smaller than that of the ${\left(3/2_{\pi }^{-}\right)}^2{\left(1/2_{\sigma }^{-}\right)}^2$ configuration [101]. This state was suggested in Ref. [49] to correspond with the tentatively observed 27.2 MeV state.

Fig. 6Full-screen View

(Color online) Density distributions of the AMD-predicted positive-parity states in ¹⁶C, with (a, b) ground, (c, d) triangular, and (e, f) linear chain configurations. The contour lines show the proton density distributions. The color plots display the single-particle orbits occupied by four valence neutrons. The lower panels show the two most weakly bound neutrons, whereas the upper panels show the other two valence neutrons. Open boxes show the centers of the proton wave packets. This figure was taken from Ref. [66]

Fig. 7Full-screen View

(Color online) Calculated energy curves (J^π=0⁺) as a function of the quadrupole deformation parameter β for ¹⁶C. Four different cluster structures are marked by circles, triangles, squares, and diamonds. The corresponding threshold energies are shown by dashed lines. This figure is taken from Ref. [101]

Various exotic clustering configurations have been predicted [65, 100, 99] and observed [102, 103] for ¹⁴C, including triangular, π-bond, and σ-bond linear chain structures. Baba et al. analyzed the associated decay patterns. Their results indicated that the σ-bond linear chain structure primarily decays into the ～6 MeV states of ¹⁰Be, whereas other resonant states mainly decay into the ground and first excited states (3.4 MeV, 2⁺), as illustrated in Fig. 8. These selective decay paths provide an important means by which to identify the existence of the σ-bond linear chain configuration, which was effectively used in later experiments conducted by Li et al. [48] and Han et al. [50].

Fig. 8Full-screen View

(Color online) Schematic illustrating the selective decay pattern of the rotational band members of the π and σ linear chain structures in ¹⁴C, as calculated by AMD [65, 99]

A theoretical comparison was made between the mirror nuclei ¹⁴C and ¹⁴O, regarding their molecular structures [100]. Fig. 9 shows the excitation energies of the 0⁺ states in ¹⁴C and ¹⁴O, corresponding to the ground, triangle, and linear chain configurations. The shift between a pair of states, mostly caused by Coulomb interactions, also known as the Thomas–Ehrman Shift (TES) [104-106], is related to the expansion (in terms of size and deformation) of the corresponding systems. The measurement of the TES for ¹⁴C and ¹⁴O, in comparison to the calculated values, would provide new evidence for the existence of the intriguing linear chain structure in exotic nuclei.

Fig. 9Full-screen View

(Color online) The comparison of excitation energies between the mirror-pair nuclei ¹⁴C and ¹⁴O for J^π=0⁺ states. The dashed lines represent theoretical thresholds. This figure was taken from Ref. [100]

THSR wave function and container model

According to the Ikeda diagram [23], the α conjugate nuclei may be transformed into a multi-α clustering system at the vicinity of the corresponding separation energy. Owing to the small relative energy (decay energy) in this situation, the relative motion of these α-clusters could all be in s-wave. These results have stimulated the idea of α-condensation in nuclei, as proposed by Röpke et al. in 1998 [24]. Subsequently, the THSR w.f. was developed to investigate the 3α condensation in ¹²C [25]. This w.f. can be regarded as a variant of the Brink w.f.. Whereas the Brink w.f. allows the subsystems (wave packets) to remain at random positions inside the nucleus, the THSR w.f. requires all subsystems (each with a spin of 0) to move in s-wave around the center of mass of the entire system [25], similar to the case of BEC [25, 26, 15, 17].

As an example, the THSR w.f. for a 3α system can be expressed as: $\begin{array}{c}{\text{Ψ}}_{3\alpha }^{\text{THSR}}=\mathcal{A}｛\exp \left[-\frac{2}{B^2}\left(X_1^2+X_1^2+X_3^2+\right)\right]\text{Φ}\left(3\alpha \right)｝\\ \exp \left(-\frac{{\xi }_3^2}{B^2}\right)\mathcal{A}\left[\exp \left(-\frac{4{\xi }_1^2}{3B^2}-\frac{{\xi }_2^2}{B^2}\right)\text{Φ}\left(3\alpha \right)\right]\mathrm{,}\end{array}$ (18) where $\mathcal{A}$ is the antisymmetrization operator, B is a parameter characterizing the overall size of the nucleus, X_i is the c.m. of the ith α-cluster, and $\text{Φ}\left(3\alpha \right)={\varphi }_{{\alpha }_1}{\varphi }_{{\alpha }_3}{\varphi }_{{\alpha }_3}$ is the 3α w.f.. The Jacobi coordinates ξ are related to X according to ξ₁=X1-1/2(X₂-X₃), ξ₁=X₂-X₃, and ${\xi }_3=\frac{1}{3}\left(X_1+X_2+X_3\right)$. Eq. 18 can be directly applied to the ${}^8\text{Be}\left(0_1^{+}\right)$ configuration, in which $\mathcal{A}\exp [{\xi }_2^2/B^2]{\varphi }_{{\alpha }_2}{\varphi }_{{\alpha }_3}$ describes the ${}^8\text{Be}\left(0_1^{+}\right)$ subsystem and $\exp [4{\xi }_1^2/3B^2]{\varphi }_{{\alpha }_1}$ represents the remaining α-particle [17, 15]. By utilizing the THSR w.f., the Hoyle state in ¹²C can be accurately reproduced [25]. The THSR w.f. has also been utilized to investigate BEC-like structures in heavier nuclei, such as ¹⁶O [25, 15, 17]. Moreover, the THSR w.f. has been extended to incorporate different size parameters B in different dimensions, thus enabling the description of nuclear deformations, including possible linear chain structures [107, 15].

The THSR w.f. has been widely adopted as a standard approach to evaluate the intrinsic components of BEC-like structures in any quantum state. For instance, the Hoyle state of ¹²C obtained via the general RGM/GCM possesses an almost 100% overlap with an extended single THSR w.f. [108, 107, 15] ${\left|\langle \varphi (3\alpha \text{RGM}/\text{GCM})|\varphi (3\alpha \text{THSR})\rangle \right|}^2\approx 100\%\mathrm{.}$ (19) The inversion–doublet band (headed by Jπ=0⁺ and Jπ=0^- states) for the asymmetric ¹⁶O + α structure in ²⁰Ne can also be described by a single THSR-type w.f. with different size parameters for the core ¹⁶O and overall ²⁰Ne system [109, 110, 15]. These results imply that the THSR w.f. incorporating adjustable size parameters may represent some fundamental physical characteristics of the clustering states [15].

A container model based on the THSR w.f. was recently proposed [15, 38]. Unlike traditional cluster models, such as the Brink wave packet, the container model focuses on the dynamic evolution of containers to capture cluster motion. The container size evolves from smaller to larger values as the excitation energies increase. The cluster motion inside the mean-field potential is characterized by using the size parameter under the condition of full antisymmetrization among all nucleons. This approach has been studied and documented by Funaki et al. [15, 38]. The w.f., which is generally nonlocalized, can result in localized density distributions for clusters due to the Pauli blocking effect [37, 15]. With its relatively simple picture and parameter settings, the container model offers a relatively simple conceptual framework and parameter settings, thereby showing promise as a powerful alternative to traditional methods, such as the RGM and Brink-GCM, for describing cluster dynamics [15, 16]. As an example, Fig. 10 schematically illustrates the α + ⁸Be cluster structure in ¹²C and the α + ¹²C cluster structure in ¹⁶O under the “container” framework. In this model, 2α or 3α clusters are confined in a “container” characterized by the parameter β₁, while the remaining α cluster is confined in a larger “container” characterized by β₂ (where β₂ is related to the parameter B, as defined in Eq. 18.

Fig. 10Full-screen View

(Color online) Schematic showing the α + ⁸Be and α + ¹²C systems within the “container” picture, in which multi-α (2α or 3α) and another α are in different “containers” characterized by the size parameters β₁ and β₂, respectively

In 2023, Zhou et al. reported the microscopic five-body calculations for ²⁰Ne [27], thus revealing a 0⁺ state located approximately 3 MeV above the 5α threshold, which could correspond to the state observed in recent experiments [111]. This state exhibited the ${}^{16}\text{O}\left(0_6^{+}\right)+\alpha$ structure, thereby indicating distinct characteristics of a 5α condensate state. The calculated partial α decay width from the 5α condensate state to the 4α condensate state was as high as 0.7 MeV due to the increased excitation energy. The monopole transition M(E0) between the condensate state and the ground state was ～1 fm². These characteristics are all similar to those of 3α and 4α condensate states.

α-cluster formation and decay from heavy nuclei

α clustering in heavy nuclei is essentially connected to α decay, which is generally described as the quantum tunneling of the preformed α particles via a semi-classical model initiated by George Gamow in the 1920s [112, 113, 9]. Various theoretical approaches have been developed to predict the half-life of α decay [114-125]. However, the preformation probability of the α-cluster within a heavy nucleus still cannot be reproduced from the shell-type calculations [126]. These results are most likely attributable to the difficulties associated with accounting for the four-nucleon correlation in a complex multi-nucleon environment. Generally, this probability can only be obtained empirically by using the experimental decay probability and subtracting the calculated barrier penetrating factor [119, 127-130].

Recently, a novel microscopic approach for describing the α formation was proposed by Röpke et al. [131, 132] and developed by Xu et al. [124, 133]. The model aimed to create the quartetting w.f. (QWFA) for the four valence nucleons upon the remaining core of the nucleus. The introduction of the c.m. coordinates (new degree-of-freedom) for the quartet system (RGM–Brink-type) should have captured the main feature of the cluster-formation dynamics [121, 124, 131-134]. In QWFA, the w.f. of four valence nucleons is divided into two components [133]: the c.m. motion relative to the core and the intrinsic motion of the four nucleons relative to their c.m. [124]: $\text{Ψ}\left(R\mathrm{,}{s}_j\right)={\phi }^{\text{intr}}\left(s_j\mathrm{,}R\right){\text{Φ}}^{\text{com}}\left(R\right)\mathrm{,}$ (20) where R is the c.m. coordinate of the quartet and s_j = S, s, s’ are the Jacobi–Moshinsky coordinates for the quartet nucleons relative to R [121], as sketched in Fig. 11. The Hamiltonian for this quartet system contains the kinetic energy and the interaction potential which depends on both R and s_J. The coupled equations can then be established for φ^intr(s_j, R) and Ф^com(R). The complex nonlocal interaction between the quartet and core can be approximately replaced by an effective local potential, which largely simplifies the calculation without changing the basic features of the potential.

Fig. 11Full-screen View

(Color online) Sketch of quartet coordinates for two protons at positions r₁ ↑ and r_{2 ↓} as well as two neutrons at r₃ ↑ and r_4↓. This figure was taken from Ref. [121].

Two different regions for R<R_sep and R>R_sep are considered to describe the α-decay process [135, 136, 121]. When R < R_sep, the intrinsic w.f. of the quartet represents four independent nucleons in quasi-particle states within the shell model framework. When R > R_sep, the nature of the intrinsic w.f. undergoes a transformation, signifying the emergence of a bound state that exhibits characteristics similar to those of an α-particle [124, 119]. Considering that the α-like state can only exist at densities lower than the critical value ρ_c(R_c)=0.02917 fm^-3 and dissolves at higher densities, the α-cluster preformation probability Pα can be calculated by integrating the bound-state w.f. Φ(r) from the critical radius Rc to infinity [121]: $P_{\alpha }={\displaystyle {\int }_{R_c}^{\infty }}{d}^{3}R{\left|\text{Φ}\left(R\right)\right|}^2\Theta \left[{\rho }_c-{\rho }_B\left(R\right)\right]\mathrm{,}$ (21) where ρB(R) represents the baryon density at a distance of R from the core center. The probability of finding an α-particle at certain position R can be expressed as the square of the overlap between the quartet w.f. ${\phi }_{\text{quartet}}^{\text{intr}}$ and free α-particle w.f. ${\phi }_{\alpha }^{\text{intr}}$ (THSR-type), as exhibited in Fig.12. Notably, the calculated α-cluster preformation factors and α-decay widths within the QWFA approach for a series of heavy nuclei are in good quantitative agreement with experimental observations (Fig. 2 of Ref. [133]). Hence, the QWFA can capture the main feature of the α-cluster preformation mechanism inside a heavy nucleus and thus provides a convincing means by which to solve a long-standing problem.

Fig. 12Full-screen View

(Color online) Overlaps between the intrinsic quartet and α-particle w.f.s as a function of R for the α + doubly magic core systems ²⁰Ne, ⁴⁴Ti, ¹⁰⁴Te, and ²¹²Po. This figure was taken from Ref. [121]

Missing mass method and cluster production cross-sections

Experimental studies of cluster structure in light nuclei often start with inclusive measurements that typically require simple detection systems. The method relies on the assumption of a two-body system in both the initial and final channel of the reaction, i.e., A(a, b)B or a + A→ b + B. With the properties of the beam (a) and target (A) particles being known and by measuring the energy and emission angle (position) of the projectile nucleus (b), the effective mass of the assumed recoil nucleus, denoted by mB*, can be deduced according to the conservation of energy and momentum in the reaction process. This process is equivalent to the deduction of the effective reaction Q-value described in the textbook [137]. The difference between this deduced effective mass (mB*) and its g.s. mass (mB) is called the MM of the reaction, or the excitation energy of the nucleus B. Typical examples for the production of the excited states in ¹⁴C can be found in Refs. [138, 3]. In these works, various reaction mechanisms are employed to populate the excited states in ¹⁴C, including the two-neutron-transfer reactions ¹²C(¹⁴N, ¹²N), ¹²C(¹⁵N, ¹³N), and ¹²C(¹⁶O, ¹⁴O), two-proton-transfer reaction ¹⁶O(¹⁵N, ¹⁷F), and multi-nucleon-transfer reaction ⁹Be(⁷Li, d). The two-nucleon-transfer reactions are mostly in favor of populating the shell-like states, whereas the multi-nucleon transfer reaction can strongly populate the cluster-type states, owing to the associated multi-particle, multi-hole configurations. Different reaction Q-values, corresponding to different momentum mismatches, in the direct reactions may introduce different momentum transfers into the final particles which, in turn, may be placed into different spin-value ranges. Based on these carefully designed reaction tools and combined with theoretical model calculations, the shell-like states can be classified into various deformation schemes. The remaining states, particularly those that are significantly populated only in the multi-nucleon transfer reaction, could be grouped into cluster-based molecular rotational bands. Fig. 13 shows the states in ¹⁴C populated from the multi-nucleon-transfer reaction ⁹Be(⁷Li, d). The cluster-like states are grouped into two molecular rotational bands, corresponding to the parity-inversion-doublets [138, 3]. Notably, this double-band configuration at high excitation energies resulting from the reflection-asymmetric configuration, such as ¹⁰Be + α in ¹⁴C, constitutes strong evidence for the molecular (cluster) structure formation in atomic nuclei [21]. The energy gap between the two bands can be estimated according to the asymmetric structure configuration. Similar exemplars for the study of cluster structures in nuclei via inclusive MM measurement can also be found in works such as Ref. [139] for ²⁰O (See Fig. 14).

Fig. 13Full-screen View

(Color online) Deuterium spectrum from the multi-nucleon-transfer reaction ⁹Be(⁷Li, d). The structured backgrounds shown are three- and four-body continua or measured backgrounds with ¹²C and ¹⁶O targets, as depicted by the full and dashed lines and indicated as ¹⁷O^* and ²¹Ne^*, respectively. This figure was taken from Refs. [138, 3]

Fig. 14Full-screen View

(Color online) Proposed prolate-type α-cluster bands in ¹⁴C. These are states that cannot be attributed to single-particle states. Their energies are plotted as a function of their assumed spins, J(J+1), forming the parity-inversion-doublet rotational bands related to the reflection-asymmetric configuration, such as ¹⁰Be + α in ¹⁴C. This figure was taken from Ref. [138]

The inclusive measurements may also incorporate the experimental determination of the cluster production cross-section in various reaction processes. An early typical study of this kind was reported in Ref. [140], where the He-cluster breakup and neutron removal cross-sections for ^{10-12, 14}Be were reported. It is interesting to see the persistence of the He-cluster structure within the neutron-rich Be isotopes up to the dripline nucleus ¹⁴Be, as was demonstrated by comparing the measured He-cluster breakup cross-section with the phase-space estimation. In an inverse kinematics experiment, using A and a to denote the projectile and the target, respectively, the aforementioned MM measurement of the recoil nucleus b can be combined with the selected cluster production related to the breakup of the nucleus B in order to probe the cluster production cross-section as a function of the excitation energy. A representative example can be found in Ref. [141], in which the MM measurement of the triton particle from the ¹⁹F(³He, t)¹⁹Ne reaction was made in coincidence with the proton or α decayed from the possible resonant states of ¹⁹Ne. As such, the α-cluster production is observed as a function of the ¹⁹Ne-excitation energies. This provides crucial information about the cluster-formation probability along the excitation axis, comprising both direct breakup and resonance-decay two-step processes, as depicted in Fig. 15.

Fig. 15Full-screen View

(Color online) Triton spectrum measured from the ¹⁹F(³He, t) reaction with a beam energy of 25 MeV [141]. Events with α-particle detection (blue) or proton detection (red) in coincidence are also shown. The result of the fit function is shown in red, and individual peaks are highlighted in yellow. The background originating from the (³He, d) reaction has been shown in blue. This figure was taken from Ref. [141]

Invariant mass method

The invariant mass (IM) method is dedicated to the resonant states, often at higher excitation energies that decay into fragments. As described above, the MM measurement is inclusive and sensitive to all kinds of states characterized by such as single-particle motion, collective motion, and cluster structures. Particularly, at high excitation energies the spectrum is generally dominated by a continuum associated with a high-level density. To improve the sensitivity to the clustering states and avoid as much of the overwhelmingly strong continuum background as possible, it is essential to measure the cluster-decay process [49]. The latter can be used to reconstruct the excitation energy of the reaction-produced cluster-like mother nucleus via the IM method, to determine the spins of the resonances based on the model-independent angular correlation analysis, extract the spectroscopic factor of the cluster structure in an observed resonant state, deduce the strength of a typical cluster excitation mode (e.g. the monopole-transition strength), and connect the unknown structures of the mother nucleus to the known structures of the detected daughter fragments via selective decay paths [46, 142, 49, 50]. Here, we highlight some useful quantities for the measurements and data analyses.

3.2.1

Kinematics

The IM method lies in a two-step process, namely, the resonance excitation followed by the cluster decay, as expressed by a(A, B^*→c+d)b in inverse kinematics and depicted in Fig. 16. Using relativistic expressions, the sum of the squares of the four-dimensional momentum (Pμ) is a Lorentz invariant quantity in different inertial reference frames. For a resonance-decay process, the IM method can be expressed as: $P^{\mu }{P}_{\mu }=\left(\frac{E}{c}\mathrm{,}p\right)=\frac{E^2}{c^2}-p\cdot p=m_{{\text{B}}^{*}}^2{c}^2\mathrm{,}$ (22) where m_B* represents the total mass of the mother nucleus (resonance) B^*, $m_{{\text{B}}^{*}}^2{c}^2$ is the square of the four-dimensional momentum in the c.m. system, E=∑_iE_i and p=∑_ip_i are the sums of the total energy and total three-dimensional momentum, respectively, of all the decay fragments in the laboratory system. The values of Ei and p_i can be deduced from experimental measurements.

Fig. 16Full-screen View

(Color online) The schematic of the reaction a(A,B^* →c+d)b. The two-body collision process is presented in the red-line box, whereas the cluster-decay of a resonant state is displayed in the green-line box

The relative energy of the decay system, such as that shown in Fig. 16, is defined by: $\begin{array}{l}{E}_{\text{rel}}=m_{B\text{*}}{c}^2-m_c{c}^2-m_d{c}^2\mathrm{.}\hfill \end{array}$ (23) The excitation energy of the nucleus B can then be deduced from: $\begin{array}{l}{E}_x=m_{B*}{c}^2-m_B{c}^2=E_{\text{rel}}+E_{\text{th}}\mathrm{.}\hfill \end{array}$ (24) with E_th=(m_cc²+m_dc²)=m_Bc² being the separation energy of the cluster system.

Similar to the description in the MM method, the Q-value of the reaction–decay process, as illustrated in Fig. 16 and corresponding to the mass deficit of the initial two-body channel and the final three-body channel, can also be deduced from the measured energy–momentum of the particles involved. If such a measured Q-value is smaller than the g.s. Q-value, then excitation can be expected in some of the final nuclei.

3.2.2

Useful observations

In the analysis of reaction–decay processes, as depicted in Fig. 16, some quantities (observables) are useful and often employed to clarify the true process or structural effects, such as the energy–momentum (E–P) plot, monopole transition strength, angular correlation analysis, cluster spectroscopic factor. A brief outline of these quantities is given below.

E–P plot for target component selection

In a real experiment, some composite materials may have to be used for targets, such as (CH₂)_n and (CD₂)_n. In such cases, the accurate differentiation of the recoil target nucleus is crucial to obtain the clear Q-value spectrum and the excitation-energy spectra [143, 144]. The E–P plot is an effective method for selecting the target component based on the measured quantities [145, 50]. Defining x and y from the measured quantities: $\begin{array}{l}x=p_b^2/2u\\ y=T_b-Q=T_A-T_c-T_d\end{array}$ (25) where u is the known atomic mass unit, and $p_b^2$ can be deduced from the momentum of the beam and the two decay particles, according to the momentum conservation. For the recoil nucleus with mass number Ab, the kinetic energy can be expressed as $T_b=\frac{1}{A_b}\frac{p_b^2}{2u}$. The two-dimensional plot of y versus x, known as the E–P plot, follows the linear relation $y=x/A_b-Q$, with the slope parameter 1/A_b characterizing the actual target nucleus and the intercept of the respective Q-value. Fig. 17 shows the E–P plot for a ¹⁴C + (CH₂)_n experiment [146] in which three bands with a slope of approximately 1 correspond to an H target but the decay fragment ¹⁰Be in its ground, $2_1^{+}$, and ～6 MeV excited states, respectively. The red line in the Fig. 17, with a slope of ～1/12, indicates the contribution from the carbon target. Thus, the cut on E–P plot provides a target component selection. The quality of the E–P plot depends on the energy resolutions of the detected particles and the beam particles. When using RIBs with a large beam-energy spread, it would be difficult to make a specific cut on the E–P plot. Also, in the case of the target–nucleus breakup during the reaction, which often occurs in cases involving deuteron or carbon targets, the resolution of the E–P plot deteriorates.

Fig. 17Full-screen View

(Color online) E–P plot of the reaction ¹⁴C + (CH₂)_n. See the Section 3.B on the EP-plot for details. This figure was taken from Ref. [146]

Monopole transition strength

As described in Section 2.D for GTCM, the enhanced monopole-transition strength between two nearby 0⁺ states is an imprint of the cluster formation [90]. Experimentally, the extraction of the monopole-transition matrix element from either the electromagnetic process [M(E0)] or nuclear process [M(IS,0)] is an important aspect of a cluster structure study. These matrix elements are defined in the following forms [147, 148]: $\begin{array}{l}M(\text{E0})=\langle f\left|{\displaystyle \sum _{i=1}^A\frac{1+{\tau }_{3i}}{2}}{\left(r_i-R_{\text{c}\text{.m}\text{.}}\right)}^2\right|\text{g}\text{.s}\text{.}\rangle ,\\ M(\text{IS}\mathrm{,0})=\langle f\left|{{\displaystyle \sum _{i=1}^A\left(r_i-R_{\text{c}\text{.m}\text{.}}\right)}}^2\right|\text{g}\text{.s}\text{.}\rangle ,\end{array}$ (26) where τ₃ is the isospin projection operator and f denotes the excited 0⁺ states. The energy-weighted sum rule (EWSR) for the isoscaler transition S(IS,0) is given by [149]: $\begin{array}{l}S(\text{IS}\mathrm{,0})={{\displaystyle \sum _f\left|M(\text{IS}\mathrm{,0})\right|}}^2{E}_f=\frac{2{\hslash }^2}{m}A{R}_{\text{rms}}^2\mathrm{.}\hfill \end{array}$ (27) The isoscalar monopole transition is generally treated in the framework of the breathing mode oscillation, which can be analyzed within the optical model [150, 148]. The transition potential G₀(r) is expressed as[150]: $\begin{array}{l}{G}_0\left(r\right)=-{\alpha }_0^U\left[3U\left(r\right)+r\frac{\text{d}U\left(r\right)}{\text{d}r}\right]\mathrm{,}\hfill \end{array}$ (28) where U(r) is the standard optical potential and ${\alpha }_0^U$ is the amplitude parameter. The transition matrix element can be acquired via a multipole decomposition (MD) analysis of inelastic differential cross-sections [151, 152]: $\begin{array}{l}{\left(\frac{\text{d}\sigma }{\text{d}\text{Ω}}\right)}_{\text{exp}}={\displaystyle \sum _L{a}_L}{\left(\frac{\text{d}\sigma }{\text{d}\text{Ω}}\right)}_{L\mathrm{,}\text{DWBA}}\mathrm{,}\hfill \end{array}$ (29) where ${\left(\frac{\text{d}\sigma }{\text{d}\text{Ω}}\right)}_{\text{exp}}$ are the experimental data and ${\left(\frac{\text{d}\sigma }{\text{d}\text{Ω}}\right)}_{L\mathrm{,}\text{DWBA}}$ represents the cross-section calculated under the distorted-wave Born approximation (DWBA) for a transferred angular momentum L. Here, aL can be obtained via a fitting procedure corresponding to the fraction of EWSR for a monopole (L = 0) or multipole (L>1) transition [153].

In 2014, an experimental method by which to extract the monopole transition strength using inelastic scattering data was established [46]. Fig. 18 shows the experimental differential cross-sections for the 10.3 MeV (0⁺₃) state of ¹²Be. The cross-sections were reproduced by the DWBA calculations with a normalization factor of a₀ = 0.034(10), which is associated with the EWSR fraction (6727.9 fm⁴MeV²). The deduced monopole transition matrix element (M(IS)) is 7.0±1.0 fm², which is comparable to those of typical cluster states [154, 155, 90].

Fig. 18Full-screen View

(Color online) Experimental differential cross-sections compared with the DWBA calculations for the 10.3 MeV state in ¹²Be, as reconstructed by the ⁴He + ⁸He decay channel. This figure was taken from Ref. [153]

Angular correlation analysis

Spin is always an important quantity in nuclear structure characterization. In the case of a cluster or molecular structure, spin-parity determination is a precondition to addressing the excitation and decay modes as well as allocating the members in a rotational band. In early inclusive MM experiments, the spin of a produced recoil nucleus associated with transferred orbital angular momentum [137] is typically determined by measuring the angular distribution of the probe. However, this approach is valid only for direct reactions (scattering) to low lying states, and it is questionable when applied to highly excited resonant states. Fortunately, the resonance-decay provides another independent and effective means by which to determine the spin of the mother nucleus, regardless of its production mechanisms [44, 46, 49, 102, 103, 142, 153, 156-158].

By assuming a two-step process for the resonance production and decay, as depicted in Fig. 16 for the resonant nucleus B^*, the conservation of the total angular momentum adheres to: $\begin{array}{c}{l}_i+J_A+J_a=J_{B^{*}}+J_b+l_f\\ J_{B^{*}}=J_c+J_d+l^{\text{'}}\mathrm{,}\end{array}$ (30) where l_i and l_f(l’) represent the orbital angular momentum of the incident and decay channels, respectively, and J_i (i = A, a, B^*, b, c, d) denotes the spin of the corresponding nucleus. In the Cartesian coordinate system, the conservation of the angular momentum in the z (beam) direction can be expressed as m_i+m_A+m_a=m_B*+m_b+m_f and m_B*=m_c+m_d+m’, where mi = 0 as the z direction is the same as that in the beam direction. The double-differential cross-section can be expressed as the product of the transition amplitudes for the formation ($T_{m_b{m}_{B*}}^{m_a{m}_A}\left({\text{Ω}}^{*}\right)$) and decay ($T_{m_c{m}_d}^{m_{B*}}\left({\text{Ω}}_{\psi }\right)$) [159, 160]: $\begin{array}{l}\frac{{\text{d}}^2\sigma }{\text{d}{\text{Ω}}^{*}\text{d}{\text{Ω}}_{\psi }}\propto {\displaystyle \sum _{m_a{m}_A{m}_c{m}_d{m}_b}{\left|{\displaystyle \sum _{m_{B*}}}{T}_{m_b{m}_{B*}}^{m_a{m}_A}\left({\text{Ω}}^{*}\right)T_{m_c{m}_d}^{m_{B*}}\left({\text{Ω}}_{\psi }\right)\right|}^2\mathrm{.}}\hfill \end{array}$ (31) If the spins of the decaying daughter nuclei (c and d), projectile nucleus (A), target nucleus (a), and recoil nucleus (b) are all zero, then the spin of the compound nucleus B^* is determined by the change of the orbital angular momentum before and after the reaction. In this case, Eq. 31 can be simplified as: $\begin{array}{l}\frac{{\text{d}}^2\sigma }{\text{d}{\text{Ω}}^{*}\text{d}{\text{Ω}}_{\psi }}\propto {\left|{\displaystyle \sum _{m_{B*}}}{T}_{m_{B*}}^{J_{B*}}\left({\text{Ω}}^{*}\right)Y_{J_{B*}}^{m_{B*}}\left({\text{Ω}}_{\psi }\right)\right|}^2\mathrm{.}\hfill \end{array}$ (32) In the strong absorption approximation, the incident channel is dominated by a peripheral partial wave li. Then [159]: $\begin{array}{c}{T}_{m_{B*}}^{J_{B*}}\left({\text{Ω}}^{*}\right)={\displaystyle \sum _{m_f}C{Y}_{l_i}^0}\left(\mathrm{0,0}\right)Y_{l_f}^{m_f}\left({\theta }^{*}\mathrm{,}{\varphi }^{*}\right)\\ =C^{\text{'}}{Y}_{l_i}^0\left(\mathrm{0,0}\right)Y_{l_f}^{m_f=-m_{B*}}\left({\theta }^{*}\mathrm{,}{\varphi }^{*}\right)\mathrm{,}\end{array}$ (33) where C and C’ are given by $\langle J_{B*}\mathrm{,}{m}_{B*}\mathrm{,}{l}_f\mathrm{,}{m}_f|l_i\mathrm{,0}\rangle$ and $\langle J_{B*}\mathrm{,}{m}_{B*}\mathrm{,}{l}_f\mathrm{,}{m}_f=-m_{B*}|l_i\mathrm{,0}\rangle$, respectively. The peripheral partial wave angular momentum of the incident channel is $l_i=r_0\left(A_p^{1/3}+A_t^{1/3}\right)\sqrt{2\mu E_{\text{c}\text{.m}\text{.}}}$.

According to the relation between the primary reaction and sequential decay processes, angular correlations can be divided into in-plane and out-of-plane correlations. The correlation angles are defined in Fig. 1 of Ref. [159]. In-plane correlation refers to the alignment between the decay plane and the reaction plane, where the azimuthal angle χ is constrained around 0 or 180. The double-differential cross-section only depends on the scattering angle θ^* and decay angle ψ. In some experiments [142, 157-166], owing to the geometric arrangement of the detectors, both the reaction and decay planes closely align with the detector plane, thereby satisfying the conditions for in-plane correlation. When the spins of the initial and final-state particles are zero, the events of in-plane correlation exhibit a linear “ridge” structure in the θ^*–ψ plot, which is commonly employed in spin analyses [158-160, 162-164]. This linear “ridge” structure is described by the following function: $\begin{array}{l}W\left({\theta }^{*}\mathrm{,}\psi \right)\approx W\left({\theta }^{*\text{'}}=\mathrm{0,}{\psi }^{\text{'}}=\psi -{\theta }^{*}/a\right)\mathrm{.}\hfill \end{array}$ (34) Here, a=J_B*/(l_i-J_B*) is related to the spin J_B* of the compound nucleus B^* and orbital angular momentum li of the incident channel. When the θ^*–ψ correlation structure of the differential cross-section in the θ^*–ψ plane is projected onto ${\theta }^{*\text{'}}=0$, it follows the $|P_{J_B{}_{*}}\left(\cos {\psi }^{\prime }\right){|}^2$ distribution.

The reason for the appearance of the linear “ridge” structure in the in-plane correlation is that when the spins of all the initial and final particles are zero, the spin J_B* of B* can be determined by the change of orbital angular momentum (J_B*=l_i-l_f). Moreover, there is a strong coupling between J_B* and the final-state orbital angular momentum l_f in their orientations (m_B*=-m_f). Rae and Marsh also provided rigorous derivations of this phenomenon from a quantum mechanical perspective [160, 167]. Hence, substituting Eq. 33 for the transition amplitude of compound nucleus formation into Eq. 32 for the differential cross-section: $\begin{array}{l}\frac{{\text{d}}^2\sigma }{\text{d}\text{Ω}*\text{d}{\text{Ω}}_{\psi }}\propto \\ {\left|{\displaystyle \sum _{m_{B*}}{C}^{\text{'}}}{Y}_{l_i}^0\left(\mathrm{0,0}\right)Y_{l_f}^{m_f=-m_{B*}}\left(\theta *\mathrm{,0}\right)Y_{J_{B*}}^{m_{B*}}\left(\psi \mathrm{,0}\right)\right|}^2\mathrm{.}\end{array}$ (35) When the spins of the projectile, target, and recoil nucleus are non-zero, the spin J_B* of the compound nucleus B* may be influenced not only by the change of orbital angular momentum but also by the change of spins between nuclei that are involved before and after the reaction. According to Eq. 30, the expressions for the transition amplitude (Eq. 33) and differential cross-section (Eq. 35) include the coupled terms $Y_{l_{\text{f}}}^{m_{\text{f}}\ne -m_{B\text{*}}}\left(\theta *\mathrm{,0}\right)Y_{J_{B\text{*}}{m}_{B\text{*}}}\left(\psi \mathrm{,0}\right)$ with m_B*≠m_f. These coupled terms introduce additional data points that fill in between the peaks of the linear “ridge” structure, thereby reducing or even completely erasing the prominence of the “ridge” pattern. Some experiments with a significant number of nonzero spins for the projectile, target, or recoil nucleus have revealed clear in-plane angular correlation structures [164, 142, 157]. Hence, the proportion of data points contributed by these coupled terms is critical.

In the case of out-of-plane correlation, the angular difference between the decay plane and the reaction plane is significant, indicating a large deviation of the χ angle from 0 or 180. For instance, a wider range of θ^* and χ angles could be covered by employing detectors with larger sensitive areas or placing them closer to the target [159]. Another example comes from the use of a zero-degree detector that covers θ*～0° as well as the entire ranges of θ^* and χ angles [49, 46, 168, 153]. In the case of spin-0 for all initial and final particles, the “ridge” structure on the θ^*–ψ plot deviates from linearity [159], making it unsuitable for spin analysis. To enhance the structure in cases of out-of-plane correlation, additional constraints (projections) are typically needed. There exist two experimental cases that may greatly simplify the analysis of the angular correlation [159]. The first involves the zero-degree measurements (θ*～0°), where the motion of the compound nucleus B* and relative motion between B^* and b are almost aligned along the z-axis, resulting in m_f = 0 and m_B*=m₀=0 (where ψ is not necessarily equal to 0). Therefore, the differential cross-section distribution follows the square of a Legendre polynomial and is solely correlated with ψ in the form: $\frac{{\text{d}}^2\sigma }{\text{d}{\text{Ω}}^{*}\text{d}{\text{Ω}}_{\psi }}\propto {\left|P_{J_{B\text{*}}}\left(\cos \psi \right)\right|}^2\mathrm{.}$ (36) In resonant scattering experiments with no recoil particle b, the condition of zero-degree measurement (θ*～0) is automatically satisfied. This result is helpful in the determination of the spin [102, 103], although the background counting is often high due to the mixing of the non-resonant processes. Another case involves verifying whether the spin of the compound nucleus, denoted by J_B*, is 0. If so, then the decay process is independent of the decay angle and the distribution on Ψ is naturally uniform. Otherwise, if J_B* is nonzero, it becomes necessary to select data within the reaction plane and apply rigorous angular corrections to accurately determine the spin of the resonant state [49, 46, 168, 153]. The determination of the spins of members of the linear chain rotational band in ¹⁴C provides an excellent example of this scenario [50].

Cluster spectroscopic factor

In a nuclear state, the cluster structure might be mixed with other configurations, such as shell-like contents. Therefore, it is important to quantify the probability of having the cluster structure in an energy eigenstate. This has been realized in the literature by using the cluster spectroscopic factor within the R–matrix framework [169]. First the cluster-decay partial width, which is the product of the spectrum-extracted (BW form) total width of the resonance and the fraction of the IM-reconstructed cluster cross-section over the total MM cross section, should be determined experimentally. Within the single-channel R–matrix approach, this measured partial cluster width, denoted by Γ_α(E), can be related to the reduced width ${\gamma }_{\alpha }^2$ and barrier penetrability factor Pl(E), according to [170, 171]: $\begin{array}{l}{\text{Γ}}_{\alpha }\left(E\right)=2{\gamma }_{\alpha }^2{P}_l\left(E\right)\mathrm{,}\hfill \end{array}$ (37) with $\begin{array}{l}{P}_l\left(E\right)=\frac{ka}{{\left(F_l\left(ka\right)\right)}^2+{\left(G_l\left(ka\right)\right)}^2}\mathrm{.}\hfill \end{array}$ (38) Here E is the decay energy (or relative energy), and a is the channel radius. The latter is generally given by $a=r_0\left(A_1^{1/3}+A_2^{1/3}\right)$ with r₀≈1.4 fm (although this may vary for exotic structures). In addition, F_l(ka) and G_l(ka) represent regular and irregular Coulomb wave functions, respectively [170, 171]. The ${\gamma }_{\alpha }^2$ term is usually compared with the Wigner limit ${\gamma }_W^2$ to give the dimensionless reduced width θ², which is close to 1 for a strong cluster structure: $\begin{array}{l}{\theta }_{\alpha }^2=\frac{{\gamma }_{\alpha }^2}{{\gamma }_W^2}\mathrm{,}\text{with}{\gamma }_W^2=\frac{3{\hslash }^2}{2\mu a^2}\mathrm{.}\hfill \end{array}$ (39) where ${\theta }_{\alpha }^2$ is also known as “the cluster spectroscopic factor” and represents the probability for finding the corresponding clusters in the resonant state. As references, ${\theta }_{\alpha }^2\left({}^{\text{8}}\text{Be,g}\text{.s}\text{.}\right)\approx 0.45$ [19] and ${\theta }_{\alpha }^2\left({}^{\text{12}}{\text{Be,0}}_{\text{3}}^{+}\left(\text{10}\text{.3 MeV}\right)\right)\approx 0.53$ [168].

3.2.3

Experimental investigations of cluster structures

Over the past three decades, there have been a growing number of experiments studying nuclear cluster structures by utilizing the IM method. The detection systems were mostly composed of position-sensitive silicon detectors with advanced data acquisition systems [172-174] to measure multiple particles in coincidence and with high resolution. These experiments encompass a wide range of reaction types, including the inelastic scattering, transfer reaction, and resonant scattering. The important criteria for identifying the cluster (molecular) structure in a nucleus include the extremely large moment of inertia deduced according to the energy-spin systematics for a rotational band, relatively large cluster decay width or cluster spectroscopic factor, enhanced characteristic (e.g., isoscalar monopole) transition strength, and selective decay paths. All these criteria depend on the cluster decay measurement and spin determination. Because a single experiment may not be able to obtain all these quantities, complementary works are generally required. We outline a few selected experimental works here to demonstrate the major progress made regarding IM measurements.

Studies on cluster structure in beryllium isotopes

The neutron-rich Be isotopes possess an inherent two-center feature, considering the unambiguous 2α structure of ⁸Be. The clustering configurations and the molecular rotational bands in ¹⁰Be have been intensively been studied since the 1990s [3, 44, 45]. Here, we focus on ¹²Be to demonstrate the recent experimental progress.

In 1999, Freer et al. conducted an inelastic breakup experiment for ¹²Be at 31.5 MeV/u on a (CH₂)_n target [175]. They utilized silicon telescopes to measure helium fragments and reconstruct ¹²Be resonant states via the ⁴He + ⁸He and ⁶He + ⁶He decay channels. They successfully observed a few cluster states above 13.2 MeV and with spins J exceeding 2. Fig. 19 (c) and (d) display the IM spectra for these decay channels. Subsequently, several experiments employed the IM method to further investigate the highly excited cluster states in ¹²Be. However, despite all efforts, the identification of the band head state, which should be closer to the cluster separation threshold, remained elusive throughout the following decade [176].

Fig. 19Full-screen View

(Color online) Excitation energy spectra of ¹²Be reconstructed from the (a, c) ⁶He+⁶He and (b, d) ⁴He+⁸He decay channels. The spectra in the upper panels (a, b) are from Ref. [46], and those in the lower panels (c, d) are from Ref. [143]

Significant progress was made by Yang et al. in 2014 through a novel inelastic scattering experiment of ¹²Be at 29 MeV/u on a carbon target [46, 153]. A major difference compared with the previous experiment was the use of a zero-degree telescope composed of high-performance silicon-strip detectors that substantially improved detection efficiency at energies close to the threshold, as displayed in Fig. 19(b) (dotted line). This experiment enabled identification of the band head (10.3 MeV) of the ⁴He + ⁸He molecular rotational band. Furthermore, the spin-parity of this state was firmly assigned as 0⁺, based on the angular correlation analysis, associated isoscalar monopole transition strength, and extracted cluster spectroscopic factor. All these criteria consistently support the dominant ⁴He + ⁸He cluster structure in this prominent and near-threshold resonance [46, 153, 168].

Studies on cluster structures in carbon isotopes

Owing to the possibility of 3α cores, more abundant cluster structures can be expected for carbon isotopes, including the triangle and linear chain configurations [3, 41, 65, 66, 99, 177-179]. Compared with prior work on beryllium isotopes with 2α cores, experimental investigations of carbon systems are more challenging. Here, the decay fragments, such as ^10,12Be, may possess bound excited states. Hence, the conversion from the measured relative energy spectrum to the physically meaningful excitation energy spectrum necessarily depends on the specific states of the decaying fragments, corresponding to different Q-values (or total energies in the final channel) [48-50]. A series of breakup reaction experiments (Fig. 16) that were dedicated to the simultaneous reconstruction of the relative energy and reaction Q-value have been performed to search for the exotic cluster structures in carbon isotopes [48, 158, 180-185]. These measurements generated consistent resonant energies but little information on the spin-parities, primarily due to the limited resolutions or statistics [183, 186]. Traditional resonant scattering experiments using the thick-target (⁴He gas) technique played an important role in the study of Be isotopes but were not able to measure the reaction Q-values and hence may have overlapped the excitation-energy spectra for different threshold energies [102, 103, 47]. Therefore, the high resolution Q-value spectrum is crucial when studying the cluster structure in carbon or heavier isotopes, especially when determining the correct application of the decay path analysis [49, 156].

The linear chain structure was initially proposed for the 0⁺₂ state of ¹²C (the Hoyle state) [51]. Previous studies have revealed the instability of the chain structure in ¹²C [177, 99]. Hence, it is crucial to search for linear chain molecular bands in ^{14, 16}C. One major step forward was made by Liu et al. in their work on the inelastic scattering of ¹⁶C at 23.5 MeV/nucleon on (CD₂)_n targets [49, 156]. As illustrated in Fig. 16, in previous IM experiments, it is typical to measure only two of the three final particles (particles b, c, and d) [184, 157]. The energy of the remaining particle is deduced according to the conservation of momentum by considering the beam energy. Because the PF-type RIB used to have a large energy spread, the obtained Q-value spectrum did not have a sufficient energy resolution [184]. In novel new inelastic scattering experiment for ¹⁶C conducted by Liu et al. [49, 156], all three final particles were coincidentally detected using high-resolution silicon detectors. Hence, the beam energy could be deduced event-by-event from the final particles, based on the momentum conservation. This way, the obtained Q-value spectrum resolution does not depend on the RIB quality and instead depends only on the detector performance. In addition, a state-of-the-art analysis method was developed to recover events with two nearby fragments hitting adjacent strips of the detector. The zero-degree telescope composed of multilayer silicon-strip detectors played an essential role in this experimental use of RIB in inverse kinematics. The high resolution Q-value spectra have enabled the clear selection of the decay paths from the ¹⁶C resonances to various states of the final fragment ¹⁰Be (+ ⁶He) or ¹²Be (+ ⁴He). Fig. 20 shows the observed relative decay strengths for different decay paths. The excellent agreement between the observed and predicted characteristic decay patterns gives, for the first time, strong evidence for the existence of the linear chain configuration in ¹⁶C. In addition, the experimental determination of the spin-0 for the 16.5 MeV band head provides further support for the chain-type molecular rotational band. These assigned band members correspond to the ${\left({3/2}_{\pi }^{-}\right)}^2{\left({1/2}_{\sigma }^{-}\right)}^2$ valence neutron configuration. A tentative state at approximately 27.2 MeV also hints at the possible existence of the pure σ-bond configuration, although this phenomenon requires further investigation.

Fig. 20Full-screen View

(Color online) Relative cluster decay widths from the resonant states in ¹⁶C to the ¹²Be (+ ⁴He) and ¹⁰Be (+ ⁶He) final channels, as obtained from the experiment reported in Ref. [49] (left panel), compared with calculations performed via the AMD approach [66] (right panel). This figure was taken from Ref. [49]

Subsequently, a similar inelastic scattering experiment was also conducted for ¹⁴C at 23.0 MeV/nucleon on (CH₂)_n targets [50]. Proton targets were chosen here, at the expense of lower c.m. energy, to provide a better energy resolution and avoid the target breakup effect as significantly occur for deuteron target. The complex experimental setup, centered around the zero-degree telescope composed of multilayer-silicon-strip detectors, is schematically shown in Fig. 21. Again, three-fold coincidence measurements were realized for all final particles (⁴He and ¹⁰Be decay fragments plus the recoil proton), providing an almost background-free and high-resolution Q-value spectrum as well as well-established resonant states in ¹⁴C. One state at 13.9 MeV was firmly identified for the first time. The contribution of this work is its determination the spin-parities of the 13.9 MeV, 14.9 MeV, and 17.3 MeV states based on the clear angular correlation analyses and differential cross-section analyses (Fig. 22). These states agree excellently with the members of the predicted linear chain molecular rotational band with the π²-bond configuration.

Fig. 21Full-screen View

(Color online) Schematic view of the detection system for the ¹⁴C experiment. The ¹⁴C beam particles (red line) are tracked by three parallel plate avalanche chambers (PPACs), and the interested final particles (red arrows) are detected by different telescopes consisting of double-sided silicon strip detectors (DSSDs and ADSSDs), large-size silicon detectors (SSDs), and CsI(Tl) scintillators. This figure was taken from Ref. [50]

Fig. 22Full-screen View

(Color online) Angular correlation and differential cross-section spectra for the 15.6 MeV, 13.9 MeV, 14.9 MeV, and 17.3 MeV resonances in ¹⁴C. The data are shown as black squares. (a–d) Each measured angular correlation spectrum is compared to a Legendre polynomial of order J = 3, 0, 2, or 4 (represented by the blue-dotted lines), respectively, plus a constant background (represented by the gray-dashed line). (e–h) Data for the differential cross-sections fitted using the DWBA calculations under different spin-parity assumptions. This figure was taken from Ref. [50]

There are also several experimental works devoted to the study of cluster structures for other carbon isotopes, such as ¹¹C [187], ¹³C [79], and its mirror nucleus ¹³N [188]. Abundant cluster (molecular) configurations have also been unveiled in these nuclei.

Studies on BEC-like structures

As indicated in section II.F, the concept of BEC-like states in nuclei was initially raised and accepted in the literature for the $0_2^{+}$ state of ¹²C, which is commonly known as the “Hoyle state” [18]. A comprehensive summary of measurements related to the properties of the Hoyle state are provided in Ref. [19]. However, many of these measurements are indirect and require theoretical models to make connections with the internal structure. For instance, the inelastic scattering form factor is sensitive to the size of the final excited state, and these results provide strong evidence for the presence of an expanded gas-like structure within the Hoyle state [3, 16, 92, 189-191].

It is natural to anticipate more BEC-analog structures in other nuclei, such as ¹⁶O, ²⁰Ne, and ²⁴Mg. However, experimental studies on these heavier systems face more challenges due to the requirement of simultaneous detection and identification of more bosonic particles in the final state. Recently, the 0⁺₆ state at 15.1 MeV in ¹⁶O, located immediately above the 4α separation threshold at 14.4 MeV, has attracted significant attention, owing to its predicted BEC-analog structure [38, 192, 193]. Previous experiments [194-196] relied on only partial α-PID, resulting in high background levels in the E_x spectra. This background contamination has hindered the possible 4α resonant states [194-197].

In a recent inelastic excitation decay experiment for ¹⁶O [28], four α-particles in the final channel were fully detected and clearly identified. By selecting events with 3α forming the Hoyle state of ¹²C and the remaining one as a valence particle, the reconstructed IM spectrum of ¹⁶O exhibits four narrow resonances located immediately above the α + ${}^{12}\text{C}\left(0_2^{+}\right)$ decay threshold, as shown in Fig. 23. These observed 4α resonant states are in good agreement with the results of theoretical calculations [198], which suggest the existence of a rotational band having the ${}^{12}\text{C}\left(0_2^{+}\right)+\alpha$ molecular configuration and band members of 0⁺, 2⁺, 4⁺, and 6⁺ at energies similar to those of the present observations. Within this molecular band, the 0⁺ band head can naturally be considered as a BEC-analog state. Further measurements to directly determine the spins of these resonances in ¹⁶O will be of great value.

Fig. 23Full-screen View

(Color online) Schematic illustrating the observed ${}^{12}\text{C}\left(0_2^{+}\right)+\alpha$ resonant states in ¹⁶O. The detection system consisted of eight sets of charged particle telescopes, symmetrically installed on both sides of the beam axis (illustrated here with only two sets). Four narrow resonances immediately above the 15.1 MeV state were firmly identified after selecting the ${}^{12}\text{C}\left(0_2^{+}\right)+\alpha$ decay channel. This figure was taken from Ref. [28]

Some pioneering experiments have also been performed to probe multi-α, BEC-like states in heavier systems [199], including ²⁰Ne [200, 111], ²⁴Mg [201-203], ²⁸Si [204, 205]. The measurement of α-cluster formation and decay from heavy nuclei have also been advanced in recent years [206-209].

A new form of BEC in the nuclear system is related to the ²n cluster in low-density environments. After the discovery of the two-neutron (2 n) halo in the mid-1980s, the concept of neutron halo was proposed, based on the idea of valence-neutron clustering in neutron-rich nuclei, such as ¹¹Li [52, 210]. It has been gradually realized that ²n-formation can be enhanced at the surface of neutron-rich nuclei or in the inner crust layers of neutron stars [53-58]. Moreover, ²n is defined as a spatially compact entity with a size of 2～5 fm, comprising a spin-singlet neutron pair in an internal s-orbital that behaves like a boson. Therefore, in very neutron-rich nuclei, numerous excessive neutrons at the surface of the nucleus might form a ²n-BEC-analog state. Experimentally, it is interesting to investigate the ⁸He system, which might be the lightest one composed of multi-²n in the BEC configuration [59]. Recently, an inelastic scattering experiment was performed using an intense ⁸He beam at 82.3 MeV/nucleon [211]. The decay products ⁶He+2n were detected with high precision (Fig. 24). One important contribution of this work is the clarification of the excitation-decay mechanism via the application of a novel angular correlation analysis method that enabled the significant reduction of background events resulting from direct breakup processes [60]. The emergence of a prominent peak at a relative energy of 4.54 (6) MeV was then observed with a high significance level, as displayed in Fig. 25 (a). Combining these results with the spin determination, monopole-transition analysis, and 2n-correlation analysis, this resonant state can be classified as the $0_2^{+}$ state of ⁸He [Fig. 25 (b)], which possesses the predicted BEC-like configuration [59, 60]. This groundbreaking observation is encouraging for the future exploration of ²n-BEC states in heavier neutron-rich nuclei and neutron stars.

Fig. 24Full-screen View

(Color online) Schematic of the population and decay of the ⁸He(0⁺₂) state. The ⁸He(0⁺₂) state [94, 59] is populated via an isoscalar monopole transition induced by a carbon target nucleus and then decays predominantly via the emission of a neutron pair. The inset shows the overlap of the w.f. of the predicted ⁸He(0⁺₂) state, as calculated using the microscopic α + 4 n model, with the condensate-like α+²n+²n (THSR) w.f. specified by the nucleus size parameter B and dineutron (²n) size parameter bn. This figure was taken from Ref. [60]

Fig. 25Full-screen View

(Color online) (a) Relative energy Er spectra for (CH₂)_n and carbon targets (c.m. scattering angle θ_c.m.<2), fitted using a Gaussian peak and second-order polynomial background, modified by the acceptance. (b) Angular distribution of the 4.5-MeV state for the (CH₂)_n target. This figure was taken from Ref. [60]

Resonant scattering and thick target method

The resonant scattering method can be regarded as a variant of the IM method. It typically utilizes a thick gas target in which the beam particle loses energy as a function of the track depth. When the center-of-mass energy of the beam–target system coincides with the corresponding resonant energy of the compound nucleus, the reaction cross-section may suddenly increase. Then, the unstable compound nucleus decays back to the projectile and target nuclei. To satisfy the conditions of the inverse kinematics, the beam particle should be heavier than the target. This allows the light fragments to penetrate a sufficient depth into the gas to reach the particle detectors [19, 202]. The excitation energy of the compound nucleus can then be deduced by measuring the energy of the fragments. In these experiments, silicon detectors are commonly placed in a gas-filled target chamber. Fig. 26 illustrates a typical setup for a resonant scattering experiment.

Fig. 26Full-screen View

(Color online) Illustration of the experimental setup for a typical resonant scattering experiment. The detection system consisted of one DSSSD and two PPACs. This figure was taken from Ref. [212]

Using resonant scattering setups, several experiments have been conducted to investigate nuclear cluster structures, including the ${}^{\text{6}}\text{He}{+}^{\text{4}}\text{He}$ configuration in ¹⁰Be [44], ¹⁵O + α in ¹⁹Ne [212], and ¹⁴C + α in ¹⁸O [213]. Usually, the R–matrix calculation is required to fit the spectrum and extract the resonance parameters (configurations, energies, spin-parities, widths, etc.), as demonstrated in Fig. 27.

Fig. 27Full-screen View

(Color online) Excitation function measured from the ¹⁴C + α resonant scattering [213]. The solid curve shows the best fit using the R–matrix calculation

The limited resolution of the reaction position within the thick target and interference from inelastic scattering can introduce large uncertainties in the determination of the resonant states. The situation may become more serious when the final fragments have bound excited states, as the determination of the reaction position might be mixed with the determination of the excited states of the final particle [49, 50]. This problem can be resolved by employing the active target time projection chamber (AT-TPC) which may independently recognize the reaction position without relying on particle energy detection [214-216]. These results are due to the tracking capability of the TPC in addition to the energy measurements. Fig. 28 shows a schematic for a typical AT-TPC setup.

Fig. 28Full-screen View

(Color online) Schematic of the cross-section of a prototype AT-TPC. The AT-TPC is equipped with a cylindrical field cage filled with the active gas, which serves as both a tracking medium and a reaction target. This figure was taken from Ref. [214]

The resonant scattering experiment conducted by Fritsch et al. in 2016 on the ¹⁰Be + ⁴He system [103] is a notable example. The trajectories of the initial and final state particles were recorded in an AT-TPC, as depicted in Fig. 29(a). The angular correlation between the two final particles can be used to distinguish the ground and excited 2⁺ states of the ¹⁰Be fragment, as shown in Fig. 29(b). Of course, it is still difficult to see the decay to the ～6 MeV states in ¹⁰Be and, thus, insufficient to sort out the decay paths related to various molecular configurations, as demonstrated by experimental works using advanced silicon-strip detectors [49, 50]. The choice of the active gas as both the target material and detection medium presents another limitation.

Fig. 29Full-screen View

(Color online) (a) Trajectories of ¹⁰Ben + ⁴He resonant scattering. (b) Kinematical correlation of the angle θ_lab between ¹⁰Be and ⁴He. This figure was taken from Ref. [103]

New detection techniques have been introduced by combining the precise tracking TPC (MicroMegas type) with surrounding Si-CsI(Tl) telescope detectors, as shown in Fig. 30 [217]. This combined detector system enhances the capabilities of resonant scattering experiments by providing an improved tracking accuracy, a better energy resolution, and an excellent PID capability. A series of works have reported the use of this equipment, including the level scheme in ⁹C [218], α clustering in ¹⁸Ne [219], and various cluster configurations in ¹³N [188].

Fig. 30Full-screen View

(Color online) Schematic of combined TPC-telescopes. The TPC is encircled by an array of 58 Si-CsI(Tl) telescope detectors and covers a total solid angle of approximately 3π. This figure was taken from Ref. [217]

Despite the numerous advantages of using AT-TPC in resonant scattering experiments, such as its high efficiency, tracking capability, and low detection threshold, the choice of the working gas (typically ⁴He) remains a significant limitation. One solution might be to incorporate a thin solid target inside the TPC, as has already been realized in previous fission measurements [220], as depicted in Fig. 31. Here, the TPC plays the role of a detector, not the active target. The primary advantages would come from having a lower detection threshold and larger detection solid angle, compared with those of silicon detectors. These factors are crucial, especially for low-energy experiments.

Fig. 31Full-screen View

(Color online) Schematic of a TPC-Si-CsI(Tl) detector with an internal solid target. The TPC is equipped with GEM/MicroMegas technology. The concept displayed in this figure was taken from Ref. [220, 217]

Using knockout reactions to investigate cluster composition in nuclear ground states

According to gRDF calculations, α clusters can be formed at the low-density surface regions of heavy nuclei, but, in this case, there is a close interplay between the α-cluster formation and development of the neutron skin[62]. This interplay leads to a reduction in the neutron-skin thickness, compared with the theoretical predictions, without considering the existence of α clusters, which will further impact the experimental constraint on the density-dependence parameter L of the symmetry energy. However, the formation of α clusters is gradually hindered as the neutron skin becomes thicker. This hindrance manifests itself as a monotonic decrease in the strength of the α-cluster formation along the isotopic chain and thus can be directly examined experimentally. Hence, it is crucial to carry out new experiments to find direct experimental evidence for the existence of preformed α particles (clusters) at the surfaces of heavy nuclei and further examine the systematics of α-clustering strength along an isotopic chain into a neutron-rich region accompanied by a steady increase of the neutron skin.

When unbound, α clusters can spontaneously come out of the mother nucleus to take advantage of the quantum tunneling effect (penetration of the Coulomb barrier). This phenomenon lies at the heart of α decay from heavy nuclei and experimental studies of α-cluster structures in the excited states of light nuclei. However, α clusters are generally bound in the ground states of nuclei, particularly in neutron-rich nuclei with well-developed neutron skins, and cannot be emitted spontaneously. Therefore, to probe α clustering in these nuclei, one needs a new reaction probe that takes the α clusters out of the mother nucleus with minimized impact from the residual.

The quasi-free (p, pα) reaction is the method of choice. Fig. 32(a) shows a schematic of the (p, pα) reaction in normal kinematics. Generally, a high-energy (several hundred MeV) proton beam is sent onto the target of interest. At the instant of the reaction, a certain amount of kinetic energy is transferred from the incident proton to the preformed α cluster, which gets knocked out of the target nucleus. When the transferred energy is significantly large, compared with the α separation energy, the (p, pα) reaction can be considered as quasi-free. Thus, the α cluster is knocked out without being disturbed by the residual, which can simply be considered as a spectator. Hence, the corresponding reaction cross-section provides a good measure of the strength of the preformed α cluster in the mother nucleus. Moreover, similar quasi-free nucleon knockout reactions, such as (e, ep), (p, 2p), and (p, pn) have been well-established as experimental probes to investigate the single-particle structures of nuclei [221, 222]. Furthermore, the (p, pα) reaction was widely used in the 1970s and 1980s to study the cluster structures in light stable nuclei, such as ⁹Be and ¹²C [223-225]. However, further extending such (p, pα) experiments into a heavier mass region was largely constrained by the then available experimental equipment and theoretical tools with which to connect the experimental results to the internal cluster structure. During the past decade, important advances have been made in both experimental techniques and theoretical tools, making the (p, pα) reaction a sensitive probe for clustering in the ground states of nuclei (e.g., Refs. [226, 227]).

Fig. 32Full-screen View

(Color online) Schematic of the quasi-free (p, pα) reaction in normal (a) and inverse (b) kinematics

In 2021, Tanaka et al. performed a quasi-free (p, pα) experiment at the Research Center for Nuclear Physics (RCNP) of Osaka University to measure the α-clustering strength and isotopic dependence of ^{112,116,120,124}Sn [61]. As illustrated in Fig. 33, the experiment was performed with the 392 MeV proton beam at the WS beam line. The scattered protons and α particles after the (p, pα) reaction were detected in coincidence using the Grand Raiden and LAS spectrometers. The experimental setup was designed according to the kinematics of the proton scattering off a preformed α particle and optimized to achieve the detection of low-energy α particles (down to ～50 MeV) and a high signal-to-noise ratio. For all four tin isotopes, the MM spectrum [Fig. 34 (top panel)] shows a clear peak located at the known α-separation energy, which is simply determined by the mass, as expected for the quasi-free knockout of the preformed α clusters. Thus, these results provide direct evidence for the formation of α clusters in these tin isotopes. In addition, the observed momentum distribution of the α particles further reaffirms that the formation of the α particle indeed occurs in the low-density surface region of heavy nuclei, as predicted by theoretical calculations.

Fig. 33Full-screen View

(Color online) Schematic of the experimental setup of the (p, pα) reaction on heavy nuclei at RCNP. This figure was adapted from Ref. [61]

Fig. 34Full-screen View

(Color online) MM spectra for the α-knockout reaction ¹¹²Sn(p, pα) (upper panel), and comparison of the experimental σ_{p, pα} cross-section with the theoretical prediction (lower panel). This figure was adapted from Ref. [61]

The MM spectrum is fitted using a combination of the Gaussian for the ground-state peak and the simulated line shape of the continuum background. For each tin isotope, the (p, pα) cross section σ_{p, pα} is then deduced from the integral of the ground-state peak. As shown in the bottom panel of Fig. 34, σ_{p, pα} gradually decreases as the mass number increases, with an approximately twofold decrease from ¹¹² Sn to ¹²⁴Sn. The observed isotopic systematics of σ_{p, pα} are well-reproduced by theoretical calculations that consider the radial density distributions of the α clusters of the gRDF prediction and reaction mechanism. Further analytical results confirmed that the observed decline in σ_{p, pα} was predominantly caused by the decrease in the α-clustering strength, whereas the effect of the reaction mechanism was minor. Thus, these results support the tight interplay between the surface α-clustering and Δr_np in heavy nuclei and thereby lead to a reduction of Δr_np, in comparison to theoretical calculations that do not consider the α-clustering effect as predicted by the gRDF calculations [62]. A linear correlation between Δr_np and the slope parameter L has been predicted via mean-field model calculations [228] and is generally used to constrain L. Many projects worldwide are ongoing to measure the neutron skin thicknesses of heavy nuclei, such as ²⁰⁸Pb, with sufficient precision. These results suggest the necessity of considering the effects of nuclear clustering when constraining EOS parameters from the neutron skin thickness [63].

The recent increase in the availability of secondary beams of radioactive isotopes provides new opportunities to investigate α cluster structures in the ground states of unstable nuclei [17, 229, 230]. Recently, Li et al. conducted the (p, pα) reaction in inverse kinematics (Fig. 32(b)) on the neutron-rich unstable nucleus ¹⁰Be [231]. A well-developed α-2 n-α molecular-like cluster structure, that is, a dumbbell-like arrangement of two α cores surrounded by two valence neutrons occupying the π orbit, was predicted for the ground state of ¹⁰Be by microscopic cluster models, THSR [226], and AMD [232]. The experiment was performed at the Radioactive Isotope Beam Factory (RIBF) of the RIKEN Nishina Center, and the experimental setup is illustrated in Fig. 35. From the measured angle and energy of the recoil proton and α particle, the excitation energy of the residual ⁶He was reconstructed. This reconstruction is equivalent to the α-particle separation energy spectrum usually used in (p, pα) reactions in normal kinematics, such as [61]. In the experiment, the two reaction channels leading to different final states of the residual, namely, ⁶He¹⁰Be(p, pα)⁶He(g.s.), ¹⁰Be(p, pα)⁶He(e.x.), were observed. Fig. 36 shows that the experimental triple differential cross section (TDX) for the ¹⁰Be(p, pα)⁶He(g.s.) channel is nicely reproduced by the distorted-wave impulse approximation (DWIA) calculations that incorporate the microscopic α-cluster w.f. provided by THSR and AMD calculations. This experiment provides strong evidence for the molecular-like cluster structure of the ground state of ¹⁰Be, as predicted by THSR and AMD.

Fig. 35Full-screen View

(Color online) Experimental setup of the (p, pα) reaction in inverse kinematics on the neutron-rich unstable nucleus ¹⁰Be at the RIBF of the RIKEN Nishina Center. This figure was adapted from Ref. [231]

Fig. 36Full-screen View

(Color online) The triple differential cross-section (TDX) for ¹⁰Be (p, pα) ⁶He(g.s.), compared with theoretical calculations. The angles of the proton (θ_p) and alpha (θ_α) are chosen according to the recoilless condition of the residual. The inset shows the density distribution of the protons and valence neutrons predicted by THSR, which exhibits a well-developed α-2 n-α cluster structure. This figure was adapted from Ref. [231]