Resonating group method

The RGM was formulated as early as 1937 to study scattering between light nuclei microscopically. In this method, the nucleons are separated into several groups, as the precursor of the concept of “cluster,” whereas the exchange effect between identical nucleons from different groups is taken as if the nucleon is resonating between each group. Since the 1960s, intensive research has been conducted using RGM to analyze the clustering structures of nuclei. Considering a two-cluster system $A=C_1+C_2$, the RGM wave function is defined as [18] ${\text{Ψ}}^{\text{RGM}}=\mathcal{A}\left\{{\varphi }_{C_1}{\varphi }_{C_2}\chi \left(\rho \right)\right\}\mathrm{,}$ (2) where ${\varphi }_{C_1}$ and ${\varphi }_{C_2}$ denote the internal wave functions of clusters C₁ and C₂, and $\chi \left(\rho \right)$ is the inter-cluster relative wave function, depending on the relative coordinates between the COM of the two clusters ρ. The antisymmetrization operator $\mathcal{A}$ is applied to the dynamic coordinates of the nucleons between clusters C₁ and C₂. The relative wave function $\chi \left(\rho \right)$ is obtained as the solution to the equation of motion as follows: ${\displaystyle \int }\left[H\left(\rho \mathrm{,}{\rho }^{\prime }\right)-EN\left(\rho \mathrm{,}{\rho }^{\prime }\right)\right]\chi \left({\rho }^{\prime }\right)d{\rho }^{\prime }=\mathrm{0,}$ (3) where $H\left(\rho \mathrm{,}{\rho }^{\prime }\right)$ and $N\left(\rho \mathrm{,}{\rho }^{\prime }\right)$ represent the RGM kernels of Hamiltonian and normalization operators, respectively.

Generator coordinate method and Brink wave function

The actual calculation using the RGM is tedious and requires solving both an integro-differential equation and an analysis of the Hamiltonian and norm kernels. In addition, extension to three or more cluster systems within the RGM framework is much more cumbersome. The proposal of the generator coordinate method (GCM), which is essentially equivalent to the RGM [18, 38], makes performing the calculation and extending the framework to multi-cluster systems much easier. The GCM wave function of the nucleus can be expressed as [18] ${\text{Ψ}}^{\text{GCM}}={\displaystyle \int }d\alpha f(\alpha )\text{Φ}(\alpha )\mathrm{,}$ (4) where $d\alpha =d{\alpha }_{1}d{\alpha }_2\cdots$, $\text{Φ}(\alpha )$ is the generating wave function specified by the generator coordinates ${\alpha }_1\mathrm{,}{\alpha }_2\mathrm{,}\cdots$, which serve as variation parameters rather than physical coordinates. The weight function $f(\alpha )$ is determined by solving the Hill– Wheeler equation ${\displaystyle \int }\left[H\left(\alpha \mathrm{,}{\alpha }^{\prime }\right)-EN\left(\alpha \mathrm{,}{\alpha }^{\prime }\right)\right]f\left({\alpha }^{\prime }\right)d{\alpha }^{\prime }=\mathrm{0,}$ (5) where the Hamiltonian and norm kernels are defined as $\begin{array}{l}H\left(\alpha \mathrm{,}{\alpha }^{\prime }\right)=\langle \text{Φ}(\alpha )\left|\mathcal{H}\right|\text{Φ}\left({\alpha }^{\prime }\right)\rangle \\ N\left(\alpha \mathrm{,}{\alpha }^{\prime }\right)=\langle \text{Φ}(\alpha )|\text{Φ}\left({\alpha }^{\prime }\right)\rangle \mathrm{.}\end{array}$ (6) The GCM is extensively used in the analyses of clustering structures in nuclei, combined with the Brink wave function. The Brink wave function serves as the basis wave function that is superposed to obtain the wave function of the total system. The Brink wave function is defined as a fully antisymmetrized many-body wave function consisting of several cluster wave functions characterized by various generator coordinates. For a nucleus including A nucleons with a clustering configuration of $C_1+C_2+\cdots +C_N$, the Brink wave function is given by [39] ${\text{Φ}}^{\text{B}}\left(R_1\mathrm{,}\cdots \mathrm{,}{R}_N\right)=\mathcal{A}\left\{{\text{Φ}}_{C_1}\left(R_1\right)\cdots {\text{Φ}}_{C_N}\left(R_N\right)\right\}\mathrm{,}$ (7) where Cj (j=1, …, N) denotes the jth cluster and its mass number. The wave function of cluster Cj is defined as ${\text{Φ}}_{C_j}\left(R_j\right)=\mathcal{A}\left\{{\varphi }_1\left(R_j\right)\cdots {\varphi }_{C_j}\left(R_j\right)\right\}\mathrm{,}$ (8) and R_j denotes the generator coordinates. The single-particle wave function is expressed in the Gaussian form as ${\varphi }_k\left(R_j\right)=\frac{1}{{\left(\pi b^2\right)}^{3/4}}\exp \left[-\frac{1}{2b^2}{\left(r_k-R_j\right)}^2\right]{\chi }_k{\tau }_k$ (9) where the spins and isospins are fixed as $|\uparrow \rangle$ or $|\downarrow \rangle$. b denotes the harmonic-oscillator parameter of single nucleons.

To describe a realistic nuclear system, the GCM–Brink wave function is defined as the superposition of the angular-momentum and parity-projected Brink wave functions: ${\text{Ψ}}_M^{J\pi }={\displaystyle \sum _{i\mathrm{,}K}{c}_{i\mathrm{,}K}}{\widehat{P}}_{MK}^{J\pi }{\text{Φ}}^{\text{B}}\left({\left\{R\right\}}_i\right)\mathrm{,}$ (10) with the projectors defined as $\begin{array}{c}{\widehat{P}}_{MK}^J=\frac{2J+1}{8{\pi }^2}{\displaystyle \int }d\text{Ω}{D}_{MK}^{J*}\left(\text{Ω}\right)\widehat{R}\left(\text{Ω}\right)\\ {\widehat{P}}^{\pi }=\frac{1+\pi {\widehat{P}}_r}{2}\mathrm{,}\pi =\pm ,\end{array}$ (11) and the index i specifies each cluster configuration indicated by a set of generator coordinates $\left\{R\right\}=\left\{R_1\mathrm{,}\cdots \mathrm{,}{R}_N\right\}$. The coefficients $\left\{c_{i\mathrm{,}K}\right\}$ are then determined by solving the Hill–Wheeler equation.

In the Brink cluster model, the existence of clusters is assumed a priori. By contrast, more flexible theoretical methods for studying nuclear clustering, such as antisymmetrized molecular dynamics (AMD) [40-42] and fermionic molecular dynamics (FMD) [43-45], treat all nucleons independently without assuming any clustering structure. The Brink wave function can be considered as a special case of AMD or FMD wave functions, with a fixed harmonic oscillator parameter b and frozen degrees of freedom for the nucleons in clusters.

Orthogonality condition model

In RGM and GCM, the forbidden states, resulting from the Pauli exclusion principle between fermions, are eliminated when solving the equations of motion due to antisymmetrization in the nuclear systems. As a semi-microscopic method, the orthogonality condition model (OCM) [16-18] reduces computation cost by artificially removing the forbidden states before solving the equation of motion. Considering the semi-microscopic approximation of the RGM, the equation of motion can be expressed as $(\mathcal{H}-E\mathcal{N})\chi =\mathrm{0,}$ (12) where $\mathcal{H}$ and $\mathcal{N}$ are the Hamiltonian and normalization operators, respectively. The RGM equation can be rewritten as $\text{Λ}\left({\mathcal{N}}^{-1/2}\mathcal{H}{\mathcal{N}}^{-1/2}-E\right)\text{Λ}\left({\mathcal{N}}^{1/2}\chi \right)=\mathrm{0,}$ (13) where $\text{Λ}\equiv 1-{\displaystyle \sum {}_F}|{\chi }_F\rangle \langle {\chi }_F|$ and χF denote forbidden states. In OCM, we assume that the nonlocality effect of the RGM can be approximated using the orthogonality operator Λ with respect to the Pauli-forbidden space. Accordingly, before the orthogonality operation, ${\mathcal{N}}^{-1/2}\mathcal{H}{\mathcal{N}}^{-1/2}$ is first approximated by the Hamiltonian without the exchange part (e.g., with an effective localized potential $V^{\text{eff}}$). ${\mathcal{N}}^{-1/2}\mathcal{H}{\mathcal{N}}^{-1/2}\approx -\frac{{\hslash }^2}{2\mu }{\nabla }_r^2+V^{\text{eff}}\left(r\right)\mathrm{.}$ (14) In this case, we obtain the OCM equation $\text{Λ}\left[-\frac{{\hslash }^2}{2\mu }{\nabla }_r^2+V^{\text{eff}}\left(r\right)-E\right]\text{Λ}\left({\mathcal{N}}^{1/2}\chi \right)=\mathrm{0,}$ (15) which is much easier to solve, even for heavier nuclei.

Tohsaki–Horiuchi–Schuck–Röpke wave function

In 2001, the THSR wave function [19] was originally proposed to describe the α-condensation of the Hoyle state, defined as $\begin{array}{l}{\text{Ψ}}_{3\alpha }^{\text{THSR}}\left(B\right)\\ =N\left(B\right)\mathcal{A}\left\{\exp \left[-\frac{2}{B^2}{\displaystyle \sum _{k=1}^3{\left(X_k-X_{\text{cm}}\right)}^2}\right]{\displaystyle \prod _{i=1}^3}\varphi \left({\alpha }_i\right)\right\}\end{array}$ (16) where $B={\left(b^2+2R_0^2\right)}^{1/2}$ and X_i denotes the coordinates of the COM of the ith α cluster, whose internal wave function is $\varphi \left({\alpha }_i\right)$. It can be seen that in the THSR wave function, the three α clusters occupy the same lowest 0S harmonic-oscillator orbit characterized by a width parameter B. When the B parameter is as large as the size of the whole nucleus, the α clusters can be considered to be moving freely in the nucleus, occupying the lowest 0S orbit. This behavior of α clusters is associated with the concept of Bose–Einstein condensation (BEC) of bosons. However, when B has the same value as the width parameter of the free α particle $B=b$, the THSR wave function is reduced to the Brink wave function.

An important feature of the clustering revealed by the THSR wave function is the BEC nature of the Hoyle state. It was found that by varying the total energy, the Hoyle state could be obtained using the THSR wave function with a rather large B value, which means that the Hoyle state can be well interpreted as a 3-α condensate state, where the α clusters all move in the 0S orbit within a relatively large volume, consistent with the large radius of the Hoyle state [19]. More interestingly, a subsequent study showed that the THSR wave function of the Hoyle state is nearly equivalent to the 3α cluster model wave functions obtained from the RGM or GCM [46]: ${\left|\langle {\text{Ψ}}_{3\alpha }^{\text{THSR}}|{\text{Ψ}}_{3\alpha }^{\text{RGM/GCM}}\rangle \right|}^2\approx 100\%\mathrm{.}$ (17) Starting from the nonlocalized character of cluster motion, Ref. [22, 47, 48] proposed a container picture in which the size parameters $\left\{B_i\right\}$ of the cluster relative-motion wave functions are considered as true dynamical quantities for describing the correlations between clusters, and the cluster correlations are also considered. In addition, by addressing the separation of the center-of-mass problem and considering different cluster correlations, Ref. [49] proposed a new trial wave function: $\begin{array}{c}{\text{Ψ}}^{\text{new}}={\widehat{L}}_{N-1}\left(\beta \right){\widehat{G}}_N\left({\beta }_0\right)\widehat{D}\left(Z\right){\text{Φ}}_0\left(r\right)\\ ={\displaystyle \int }{d}^3{\tilde{T}}_1\cdots d^3{\tilde{T}}_{N-1}\exp \left[-{\displaystyle \sum _{i=1}^{N-1}\frac{{\tilde{T}}_i^2}{{\beta }_i^2}}\right]\\ {\displaystyle \int }{d}^3{R}_1\cdots d^3{R}_N\exp \left[-{\displaystyle \sum _{i=1}^N\frac{C_i{\left(R_i-Z_i-T_i\right)}^2}{{\beta }_0^2-2b_i^2}}\right]\\ {\text{Φ}}_0\left(r-R\right)\\ =n_0\exp \left[-\frac{A}{{\beta }_0^2}{X}_{\text{cm}}^2\right]\\ \mathcal{A}\left\{{\displaystyle \prod _{i=1}^{N-1}}\exp \left[-\frac{{\left({\xi }_i-S_i\right)}^2}{2B_i^2}\right]{\displaystyle \prod _{i=1}^N}{\varphi }_i^{\text{int}}\left(b_i\right)\right\},\end{array}$ (18) where $\widehat{D}\left(Z\right)$ shifts the nucleons to the positions of the corresponding clusters, ${\widehat{G}}_N\left({\beta }_0\right)$ performs an integral transformation to separate the COM from the wave function, and ${\widehat{L}}_{N-1}\left(\beta \right)$ describes the cluster correlations in the form of the container picture. For further details, please refer to Ref. [49]. The physical meaning of the quantities in Eq. (18) are clear. X_cm is the COM coordinate of the entire nucleus. ${\xi }_i$ and S_i are the Jacobi coordinates of the cluster COM coordinate X_i and generator coordinate Z_i, respectively. The internal wave function of the ith cluster ${\varphi }_i^{\text{int}}\left(b_i\right)$ depends on the width variable bi and includes the spin and isospin parts. By applying the integral, we can determine the width of the Gaussian relative wave function as follows: $B_k^2=\frac{1}{2}\left[{\displaystyle \sum _{i=1}^{k+1}{C}_i}/\left(C_{k+1}{\displaystyle \sum _{i=1}^k{C}_i}\right)\right]{\beta }_0^2+\frac{1}{2}{\beta }_k^2\mathrm{.}$ (19) In the future, by utilizing this wave function, we can achieve a more realistic description of various cluster states in light nuclei.

Calculation methods of RWA

Calculation methods for the RWA were established many years ago [13]. In RGM, the calculation for the RWA is straightforward, where the RGM-type wave function is written as $\begin{array}{c}{\text{Ψ}}^{\text{RGM}}=\sqrt{\frac{C_1!C_2!}{A!}}\\ \times \mathcal{A}\left\{{\chi }_l\left(r\right){\left[Y_l\left(\widehat{r}\right)\otimes {\left[{\text{Φ}}_{C_1}^{j_1{\pi }_1}\otimes {\text{Φ}}_{C_2}^{j_2{\pi }_2}\right]}_{j_{12}}\right]}_{JM}\right\}.\end{array}$ (34) The relative-motion wave function χl(r) can be expanded using the radial harmonic oscillator functions $R_{nl}\left(r\mathrm{,}{\nu }^{\prime }\right)$ ${\chi }_l\left(r\right)={\displaystyle \sum _n{e}_n}{R}_{nl}\left(r\mathrm{,}{\nu }^{\prime }\right)\mathrm{,}$ (35) in which $e_n={\displaystyle \int }{R}_{nl}\left(r\mathrm{,}{\nu }^{\prime }\right){\chi }_l\left(r\right)r^{2}dr\mathrm{.}$ (36) and ${\nu }^{\prime }=\left(C_1{C}_2/A\right)\nu$, where $\nu =1/2b^2$ denotes the width. The RWA can then be calculated by $y_c^{J\pi }\left(r\right)={\displaystyle \sum _n{\mu }_{nl}}{e}_n{R}_{nl}\left(r\mathrm{,}{\nu }^{\prime }\right)$ (37) where μnl denotes the eigenvalues of the RGM norm kernel: $\begin{array}{l}{\mu }_{nl}=\langle R_{nl}\left(r\right){\left[Y_l\left(\widehat{r}\right){\left[{\text{Φ}}_{C_1}^{j_1{\pi }_1}{\text{Φ}}_{C_2}^{j_2{\pi }_2}\right]}_{j_{12}}\right]}_J|\\ \mathcal{A}\left\{R_{nl}\left(r\right){\left[Y_l\left(\widehat{r}\right){\left[{\text{Φ}}_{C_1}^{j_1{\pi }_1}{\text{Φ}}_{C_2}^{j_2{\pi }_2}\right]}_{j_{12}}\right]}_J\right\}\rangle \mathrm{.}\end{array}$ (38) Within the GCM–Brink framework, the coefficient enl can be calculated as [13] $\begin{array}{c}{e}_{nl}={(-)}^{\left(n-l\right)/2}\sqrt{\frac{2l+1}{(n-l)!!(n+l+1)!!}}\\ {\displaystyle \sum _{pq}\frac{{\left(\nu S_p^2\right)}^{n/2}}{\sqrt{n!}}}{e}^{-\nu S_p^2/2}{B}_{pq}^{-1}\langle {\text{Φ}}_{j_1{\pi }_1{j}_2{\pi }_2{j}_{12}l}^{J\pi }\left(S_q\right)|{\text{Ψ}}_{MA}^{J\pi }\rangle \mathrm{,}\end{array}$ (39) where $B_{pq}=\langle {\text{Φ}}_{j_1{\pi }_1{j}_2{\pi }_2{j}_{12}l}^{J\pi }\left(S_p\right)|{\text{Φ}}_{j_1{\pi }_1{j}_2{\pi }_2{j}_{12}l}^{J\pi }\left(S_q\right)\rangle \mathrm{,}$ (40) ${\text{Φ}}_{j_1{\pi }_1{j}_2{\pi }_2{j}_{12}l}^{J\pi }\left(S_p\right)$ is the Brink wave function projected onto the angular momenta and parities of the nucleus and the two clusters involved, and Sp is the discretized inter-cluster distance.

The traditional method for calculating the RWA using the GCM-Brink wave function requires significant computational resources. Recently, Chiba and Kimura [53] proposed a Laplace expansion method to calculate the RWA within the GCM/AMD framework. Through the Laplace expansion, the AMD wave function of the A-nucleon system, which is defined as the determinant of an $A\times A$ matrix $B_{A\times A}$, can be split into two AMD wave functions of clusters C₁ and C₂ ($A=C_1+C_2$): $\begin{array}{l}{\text{Φ}}_A^{\text{AMD}}=\sqrt{\frac{C_1!C_2!}{A!}}{\displaystyle \sum _{1\le i_1<\cdots <i_{C_1}\le A}P\left(i_1\mathrm{,}\cdots \mathrm{,}{i}_{C_1}\right)}\\ {\text{Φ}}_{C_1}^{\text{AMD}}\left(i_1\mathrm{,}\cdots \mathrm{,}{i}_{C_1}\right){\text{Φ}}_{C_2}^{\text{AMD}}\left(i_{C_1+1}\mathrm{,}\cdots \mathrm{,}{i}_A\right)\end{array}$ (41) with a phase factor $P\left(i_1\mathrm{,}\cdots \mathrm{,}{i}_{C_1}\right)$ defined as $P\left(i_1\mathrm{,}\cdots \mathrm{,}{i}_{C_1}\right)={(-)}^{C_1\left(C_1+1\right)/2+{\displaystyle {\sum }_{s=1}^{C_1}{i}_s}}\mathrm{.}$ (42) The cluster wave function ${\text{Φ}}_{C_1}^{\text{AMD}}\left(i_1\mathrm{,}\cdots \mathrm{,}{i}_{C_1}\right)$ is defined as the determinant of the matrix composed of the 1, …, C₁th rows and the $i_1\mathrm{,}\cdots \mathrm{,}{i}_{C_1}$th columns of the matrix $B_{A\times A}$, and the determinant defining ${\text{Φ}}_{C_2}^{\text{AMD}}\left(i_{C_1+1}\mathrm{,}\cdots \mathrm{,}{i}_A\right)$ consists of the elements that remain after removing the $\mathrm{1,}\cdots \mathrm{,}{C}_1\text{th}$ rows and the $i_1\mathrm{,}\cdots \mathrm{,}{i}_{C_1}$th columns. The RWA can then be calculated as the sum of the coupling results of the three kernels with angular momentum and parity: $\begin{array}{c}{y}_c^{J\pi }\left(a\right)=\frac{1}{\sqrt{{\mathcal{N}}_K^{J\pi }}}{\displaystyle \sum _{1\le i_1<\cdots <i_{C_1}\le A}P\left(i_1\mathrm{,}\cdots \mathrm{,}{i}_{C_1}\right)}\\ \times {\left[{\chi }_l\left(a;i_1\mathrm{,}\cdots \mathrm{,}{i}_A\right)\otimes {\left[N^{j_1{\pi }_1}\left(i_1\mathrm{,}\cdots \mathrm{,}{i}_{C_1}\right)\otimes N^{j_2{\pi }_2}\left(i_{C_1+1}\mathrm{,}\cdots \mathrm{,}{i}_A\right)\right]}_{j_{12}}\right]}_{JK}\mathrm{,}\end{array}$ (43) in which $\begin{array}{c}{\chi }_{l{m}_l}\left(a;i_1\mathrm{,}\cdots \mathrm{,}{i}_A\right)=\langle \frac{\delta (r-a)}{r^2}{Y}_{l{m}_l}\left(\widehat{r}\right)|\chi \left(r;i_1\mathrm{,}\cdots \mathrm{,}{i}_A\right)\rangle \\ N_{m_1}^{j_1{\pi }_1}\left(i_1\mathrm{,}\cdots \mathrm{,}{i}_{C_1}\right)=\langle {\text{Φ}}_{m_1{C}_1}^{j_1{\pi }_1}|{\text{Φ}}_{C_1}^{\text{int}}\left(i_1\mathrm{,}\cdots \mathrm{,}{i}_{C_1}\right)\rangle \\ N_{m_2}^{j_2{\pi }_2}\left(i_{C_1+1}\mathrm{,}\cdots \mathrm{,}{i}_A\right)=\langle {\text{Φ}}_{m_2{C}_2}^{j_2{\pi }_2}|{\text{Φ}}_{C_2}^{\text{int}}\left(i_{C_1+1}\mathrm{,}\cdots \mathrm{,}{i}_A\right)\rangle .\end{array}$ (44) Note that the application of the Laplace expansion method to a symmetric clustering structure is time-consuming because the number of possible combinations for Laplace expansion increases when the system is heavy and the cluster mass number C₁ is close to C₂. In this case, traditional calculation methods can be applied more efficiently.

Asymptotic behavior

For cluster states in self-conjugated nuclei, such as α+α, as shown in Fig. 2, the RWA exhibits distinct features in different regions, namely suppressed inner oscillation, enhanced surface peak, and damping of the outer tail. The inner oscillation and enhanced peak are closely related to the antisymmetrization effect between the clusters. Due to the fermionic nature of the nucleons in the α clusters, the formation probability of the α+α structure at a small distance is suppressed, and forbidden states appear at nodal distances. However, the tail part of the RWA, where the distance between clusters is sufficiently large that the antisymmetrization effect between clusters becomes very weak, is mainly determined by the separation energy as well as the centrifugal and Coulomb barriers. Consequently, the asymptotic behavior of the RWA should be well defined, and is important for examining the delocalization of α clusters in weakly bound cluster states [54].

Fig. 2Full-screen View

(Color online) RWA and approximated RWA of the α+α channel in $0_1^{+}$ and $2_1^{+}$ states of ⁸Be. The figure is taken from Ref. [24]

In addition, when nuclear reactions are analyzed, the asymptotic behavior of the RWA determines the angular distributions of the nucleon or cluster removal cross-sections [27]. For bound systems and narrow resonances, the tail part of RWA decreases as $a{y}_c^{J\pi }\left(a\right)\to C_c^{J\pi }{W}_{-\eta \mathrm{,}l+1/2}\left(2\kappa a\right)\mathrm{,}$ (45) where $W_{-\eta \mathrm{,}l+1/2}\left(2\kappa a\right)$ is the Whittaker function, $\eta =Z_1{Z}_2{e}^2\mu /{\hslash }^2\kappa$ is the Sommerfeld parameter, and $\kappa =\sqrt{-2\mu E}/\hslash$ is the wave number. $C_c^{J\pi }$ is the asymptotic normalization coefficient, which plays an important role in reaction analyses.

The significance of the asymptotic behavior of RWA is further demonstrated by the consistency between the tail parts of the relative wave functions with distinctive definitions. It should be noted that the RWA is not normalized to unity but to $S_c^2$ as shown in Eq. (21). If we define $u_l\left(r\right)={\displaystyle \sum _n{e}_n}\sqrt{{\mu }_{nl}}{R}_{nl}\left(r\right)\mathrm{,}$ (46) it can be seen that, for the cluster model wave function normalized to unity (e.g., $\langle \text{Ψ}|\text{Ψ}\rangle =1$), we have ${\displaystyle \int }{\left|u_l\left(r\right)\right|}^2{r}^2\text{d}r=1.$ (47) The difference between the relative wave functions yl(r), ul(r), and χl(r) is in the treatment of the antisymmetrization effect between clusters [24]. Based on Eq. (34), we can see that χl(r) is the cluster relative wave function before antisymmetrization and, as a result, contains nonphysical forbidden states whose norm kernel has an eigenvalue of μnl=0. By contrast, in the antisymmetrized relative wave functions yl(r) and ul(r), the forbidden-state components are eliminated by the coefficients μnl and $\sqrt{{\mu }_{nl}}$, respectively, whereas the treatments of the partially allowed states are different, and thus exhibit different normalization results. These distinctions can be inferred more explicitly by comparing Eqs. (37), (46), and (35). Notably, although the wave functions χl(r), ul(r), and yl(r) are different in the region in which the antisymmetrization effect plays a major role (i.e., the inner region with a small distance between clusters), they exhibit the same asymptotic behavior in the region with a large r, where the antisymmetrization effect is weak and can be ignored, i.e., for large r: ${\chi }_l\left(r\right)\approx y_l\left(r\right)\approx u_l\left(r\right)\mathrm{.}$ (48) Using this important feature, Kanada-En'yo et al. [24] showed that, for two-body cluster channels, the tail part of the RWA can be well approximated using the overlap of the cluster-model wave function with the single-Brink wave function: $\left|a{y}_l\left(a\right)\right|\approx \frac{1}{\sqrt{2}}{\left(\frac{2{\nu }^{\prime }}{\pi }\right)}^{1/4}\left|\langle {\text{Ψ}}_M^{J\pi }|{\text{Φ}}^{\text{B}}\left(S=a\right)\rangle \right|\equiv a{y}^{\text{app}}\left(a\right)\mathrm{,}$ (49) where S denotes the intercluster distance of the Brink wave function. In Fig. 2, we show both the approximated RWA and exact RWA of the $0_1^{+}$ and $2_1^{+}$ states of ⁸Be. The approximated RWA describes the tail part well, although the inner oscillation is absent. As previously mentioned, when analyzing the decay characteristics, we need to examine only the RWA value for a relatively larger channel radius a. In this case, the approximation of RWA using a single-Brink overlap provides a practical calculation method that effectively reduces computational cost.

Testing the abilities of different trial wave functions is also interesting for describing the asymptotic behavior of the cluster relative motion. Based on the equivalence of the three types of relative wave function in the tail region, Kanada-En'yo [54] further calculated the relative wave functions obtained from the Brink, spherical THSR (sTHSR), and deformed THSR (dTHSR) wave functions as well as a function with Yukawa tail (YT), and compared them with the exact solution obtained by GCM wave function. The relative wave functions of these trail wave functions can be expressed as various types of Gaussians. For the Brink and sTHSR wave functions, the intercluster wave function can be adopted as a shifted spherical Gaussian (ssG) ${\chi }^{\text{ssG}}\left(r\right)=\exp \left[-\frac{{\left(r-S\right)}^2}{{\sigma }^2}\right]$ (50) where the partial wave expansion is ${\chi }_l^{\text{ssG}}\left(S\mathrm{,}\sigma ;r\right)\propto i_l\left(\frac{2Sr}{{\sigma }^2}\right)\exp \left(-\frac{r^2+S^2}{{\sigma }^2}\right)\mathrm{,}$ (51) where il(r) is the regular modified spherical Bessel function. This function is controlled by two parameters, S and σ. When σ is fixed at $\sigma =1/\sqrt{{\nu }^{\prime }}=\sqrt{A/C_1{C}_2}b$, the relative wave function corresponds to the Brink wave function, whereas when S→0, the limit is equal to that of the sTHSR wave function. For the dTHSR wave function, the relative wave function is described by a deformed Gaussian (dG) function around the origin. If we consider the axially symmetric case as an example, then $\begin{array}{c}{\chi }^{\text{dG}}\left({\sigma }_{\perp }\mathrm{,}{\sigma }_z;r\right)\propto \exp \left(-\frac{x^2}{{\sigma }_{\perp }^2}-\frac{y^2}{{\sigma }_{\perp }^2}-\frac{z^2}{{\sigma }_z^2}\right)\\ =\exp \left(-\frac{r^2}{{\sigma }_{\perp }^2}+\frac{r^2}{\text{Δ}}\cos {\theta }^2\right)\end{array}$ (52) with $\frac{1}{\text{Δ}}\equiv \frac{1}{{\sigma }_{\perp }^2}-\frac{1}{{\sigma }_z^2}\mathrm{.}$ (53) For an even-l wave, the partial wave function can be calculated as $\begin{array}{c}{\chi }_l^{\text{dG}}\left({\sigma }_{\perp }\mathrm{,}{\sigma }_z;r\right)\propto 2\sqrt{(2l+1)\pi }\exp \left(-\frac{r^2}{{\sigma }_{\perp }^2}\right)\\ \times {\displaystyle {\int }_0^1}{P}_l\left(t\right)\exp \left(\frac{r^2}{\text{Δ}}{t}^2\right)\text{d}t\mathrm{,}\end{array}$ (54) and for an odd-l wave, we have $\begin{array}{c}{\chi }_l^{\text{dG}}\left({\sigma }_{\perp }\mathrm{,}{\sigma }_z;r\right)\propto 2\sqrt{(2l+1)\pi }r\exp \left(-\frac{r^2}{{\sigma }_{\perp }^2}\right)\\ \times {\displaystyle {\int }_0^1}{P}_l\left(t\right)t\exp \left(\frac{r^2}{\text{Δ}}{t}^2\right)\text{d}t\mathrm{,}\end{array}$ (55) where $P_l\left(t\right)$ is the Legendre polynomial.

To determine the accuracy of the trial wave functions in describing the asymptotic behavior of the cluster relative motions, Fig. 3 shows the relative wave function $r{u}_l\left(r\right)$ obtained by precise GCM solutions and trial wave functions that include the Brink, spherical and deformed THSR, and Yukawa-type wave functions calculated by Kanada-En'yo [54]. For comparison, the results of the SM wave function are also shown. The results show that the SM and Brink wave functions can provide the correct number of oscillation nodes in the inner region. However, both decrease too quickly in the tail region to provide a reasonable asymptotic feature. In addition, the SM wave function exhibits an enhanced peak that is narrower and more inward than that of the precise wave function, whereas the Brink wave function shows a more outward peak. Notably, the Brink wave function reproduces the amplitudes of the inner oscillation part better and exhibits a longer tail than the SM wave function. However, the tail of the Brink relative wave function deviates significantly from the exact function because the Brink wave function is a localized model wave function. Introducing the nonlocalization character enables the sTHSR wave function to describe the relative wave function better than the Brink and SM wave functions, although minor deviations appear in the tail part. Surprisingly, the dTHSR and YT wave functions can generate very precise results for the cluster relative wave function, suggesting that they can serve as efficient trial wave functions in describing the relative motion between clusters.

Fig. 3Full-screen View

(Color online) α-α cluster relative wave function obtained by GCM wave function of ⁸Be(0⁺), compared with that obtained by the shell-model wave function, Brink wave function, spherical THSR wave function, deformed THSR wave function, and Yukawa-tail function. The figure is taken from Ref. [54]

Two-body overlap amplitude

The RWA is essentially a one-body overlap amplitude that depends on a single intercluster distance parameter. To observe the correlations between clusters or nucleons more clearly and to understand the much more complex three-body cluster motion, we can extend the analysis of overlap amplitudes to three-body channels. This extension was recently applied to the analysis of $\text{core}+N+N$ structures [55-57]. Here, we provide a more general formula for calculating the two-body overlap amplitude by iteratively applying the Laplace expansion method [53].

The two-body overlap amplitude is defined as $\begin{array}{l}{\mathcal{Y}}_c^{J\pi }\left(a_1\mathrm{,}{a}_2\right)=\sqrt{\frac{A!}{C_1!C_2!C_3!}}\\ \times \langle \frac{\delta \left(r_1-a_1\right)\delta \left(r_2-a_2\right)}{r_1^2{r}_2^2}\\ {\left[{\left[Y_{l_1}\left({\widehat{r}}_1\right)\otimes Y_{l_2}\left({\widehat{r}}_2\right)\right]}_L\otimes {\left[{\text{Φ}}_{C_1}^{j_1{\pi }_1}\otimes {\left[{\text{Φ}}_{C_2}^{j_2{\pi }_2}\otimes {\text{Φ}}_{C_3}^{j_3{\pi }_3}\right]}_{j_{23}}\right]}_{j_{123}}\right]}_{JM}|{\text{Ψ}}_M^{J\pi }\rangle ,\end{array}$ (56) in which $c=\left\{j_1{\pi }_1{j}_2{\pi }_2{j}_3{\pi }_3{j}_{23}{j}_{123}{l}_1{l}_{2}L\right\}$ according to the three-body coupling channel of ${\left[{\left[C_1(j_1)\otimes {\left[C_2\left(j_2\right)\otimes C_3\left(j_3\right)\right]}_{j_{23}}\right]}_{j_{123}}\otimes {\left[l_1\otimes l_2\right]}_L\right]}_J$ and the parity relation of $\pi ={(-)}^{l_1+l_2}{\pi }_1{\pi }_2{\pi }_3$. The relative motion coordinates r₁ and r₂ are defined as follows. $\begin{array}{l}{r}_1=X_2-X_3\\ r_2=X_1-\frac{C_2{X}_2+C_3{X}_3}{C_2+C_3}\mathrm{,}\end{array}$ (57) where X_i is the COM of the physical coordinates of cluster Ci. By applying the Laplace expansion method twice to the AMD/Brink wave function, which is defined as a Slater determinant, we obtain $\begin{array}{l}{\text{Φ}}_A^{\text{AMD}}=\sqrt{\frac{A!}{C_1!C_2!C_3!}}\\ \begin{array}{ll}{\displaystyle \sum _{\begin{array}{c}1\le i_1<\cdots <i_{C_1}\le A\\ 1\le j_1<\cdots <j_{C_2}\le C_2+C_3\end{array}}}\hfill & P_A\left(i_1\mathrm{,}\cdots \mathrm{,}{i}_{C_1}\right)P_{C_2+C_3}\left(j_1\mathrm{,}\cdots \mathrm{,}{j}_{C_2}\right)\hfill \end{array}\\ \times {\text{Φ}}_{C_1}^{\text{AMD}}\left(i_1\mathrm{,}\cdots \mathrm{,}{i}_{C_1}\right){\text{Φ}}_{C_2}^{\text{AMD}}\left(j_1\mathrm{,}\cdots \mathrm{,}{j}_{C_2}\right){\text{Φ}}_{C_3}^{\text{AMD}}\left(j_{C_2+1}\mathrm{,}\cdots \mathrm{,}{j}_{C_2+C_3}\right)\end{array}$ (58) with the phase factor $\begin{array}{c}{P}_A\left(i_1\mathrm{,}\cdots \mathrm{,}{i}_{C_1}\right)={(-)}^{\frac{C_1\left(C_1+1\right)}{2}+{\displaystyle {\sum }_s{i}_s}}\\ P_{C_2+C_3}\left(j_1\mathrm{,}\cdots \mathrm{,}{j}_{C_2}\right)={(-)}^{\frac{C_2\left(C_2+1\right)}{2}+{\displaystyle {\sum }_s{j}_s}}\mathrm{.}\end{array}$ (59) Note that js $(s=\mathrm{1,}\cdots \mathrm{,}{C}_2+C_3)$ corresponds to the index in the matrix composed of the remaining elements after removing the rows and columns constituting ${\text{Φ}}_{C_1}^{\text{AMD}}$ from the matrix $B_{A\times A}$. Following the analogous procedure in the two-body Laplace expansion method calculation, the product of the three AMD wave functions can be rewritten as $\begin{array}{l}{\text{Φ}}_{C_1}^{\text{AMD}}{\text{Φ}}_{C_2}^{\text{AMD}}{\text{Φ}}_{C_3}^{\text{AMD}}\\ ={\text{Φ}}_{C_1}^{\text{cm}}{\text{Φ}}_{C_2}^{\text{cm}}{\text{Φ}}_{C_3}^{\text{cm}}{\text{Φ}}_{C_1}^{\text{int}}{\text{Φ}}_{C_2}^{\text{int}}{\text{Φ}}_{C_3}^{\text{int}}\\ ={\left(\frac{C_1{C}_2{C}_3}{{\left(\pi b^2\right)}^3}\right)}^{3/4}\exp \left\{-\frac{1}{2b^2}\left[C_1{\left(X_1-Z_1\right)}^2+C_2{\left(X_2-Z_2\right)}^2+C_3{\left(X_3-Z_3\right)}^2\right]\right\}{\text{Φ}}_{C_1}^{\text{int}}{\text{Φ}}_{C_2}^{\text{int}}{\text{Φ}}_{C_3}^{\text{int}}\\ ={\left(\frac{C_1{C}_2{C}_3}{{\left(\pi b^2\right)}^3}\right)}^{3/4}\exp \left\{-\frac{1}{2b^2}\left[A{X}_G^2+\frac{C_2{C}_3}{C_2+C_3}{\left(r_1-S_1\right)}^2+\frac{C_1\left(C_2+C_3\right)}{C_1+C_2+C_3}{\left(r_2-S_2\right)}^2\right]\right\}{\text{Φ}}_{C_1}^{\text{int}}{\text{Φ}}_{C_2}^{\text{int}}{\text{Φ}}_{C_3}^{\text{int}}\\ ={\text{Φ}}_A^{\text{cm}}{\chi }_{23}\left(r_1\right){\chi }_{123}\left(r_2\right){\text{Φ}}_{C_1}^{\text{int}}{\text{Φ}}_{C_2}^{\text{int}}{\text{Φ}}_{C_3}^{\text{int}}\mathrm{,}\end{array}$ (60) where Z_i is the COM of the generator coordinates of the nucleons in cluster Ci, and S_i is the corresponding Jacobi coordinate. ${\text{Φ}}_A^{\text{cm}}={\left(\frac{A}{\pi b^2}\right)}^{3/4}\exp \left[-\frac{A}{2b^2}{X}_G^2\right]$ (61) is the COM wave function of the nucleus, and $\begin{array}{c}{\chi }_{23}\left(r_1\right)={\left(\frac{C_2{C}_3}{C_2+C_3}\frac{1}{\pi b^2}\right)}^{3/4}\\ \exp \left[-\frac{C_2{C}_3}{C_2+C_3}\frac{1}{2b^2}{\left(r_1-S_1\right)}^2\right]\\ {\chi }_{123}\left(r_2\right)={\left(\frac{C_1\left(C_2+C_3\right)}{A}\frac{1}{\pi b^2}\right)}^{3/4}\\ \exp \left[-\frac{C_1\left(C_2+C_3\right)}{A}\frac{1}{2b^2}{\left(r_2-S_2\right)}^2\right]\end{array}$ (62) are the relative wave functions between C₂ and C₃ and between C₁ and the COM of $C_2+C_3$, respectively. ${\text{Φ}}_{C_1}^{\text{int}}$, ${\text{Φ}}_{C_2}^{\text{int}}$, and ${\text{Φ}}_{C_3}^{\text{int}}$ are the intrinsic wave functions of C₁, C₂, and C₃. Then, the two-body overlap amplitude can be calculated by $\begin{array}{c}{\mathcal{Y}}_c^{J\pi }\left(a_1\mathrm{,}{a}_2\right)\\ \sqrt{\frac{A!}{C_1!C_2!C_3!}}\langle \frac{\delta \left(r_1-a_1\right)\delta \left(r_2-a_2)\right)}{r_1^2{r}_2^2}\\ {\left[{\left[Y_{l_1}\left({\widehat{r}}_1\right)\otimes Y_{l_2}\left({\widehat{r}}_2\right)\right]}_L\otimes {\left[{\text{Φ}}_{C_1}^{j_1{\pi }_1}\otimes {\left[{\text{Φ}}_{C_2}^{j_2{\pi }_2}\otimes {\text{Φ}}_{C_3}^{j_3{\pi }_3}\right]}_{j_{23}}\right]}_{j_{123}}\right]}_{JM}|{\text{Ψ}}_M^{J\pi }\rangle \\ =\sqrt{\frac{A!}{C_1!C_2!C_3!}}\langle \frac{\delta \left(r_1-a_1\right)\delta \left(r_2-a_2\right)}{r_1^2{r}_2^2}{P}_{KM}^{J\pi }\\ {\left[{\left[Y_{l_1}\left({\widehat{r}}_1\right)\otimes Y_{l_2}\left({\widehat{r}}_2\right)\right]}_L\otimes {\left[{\text{Φ}}_{C_1}^{j_1{\pi }_1}\otimes {\left[{\text{Φ}}_{C_2}^{j_2{\pi }_2}\otimes {\text{Φ}}_{C_3}^{j_3{\pi }_3}\right]}_{j_{23}}\right]}_{j_{123}}\right]}_{JM}|{\text{Ψ}}_A^{\text{int}}\rangle \\ =\sqrt{\frac{A!}{C_1!C_2!C_3!}}\langle \frac{\delta \left(r_1-a_1\right)\delta \left(r_2-a_2\right)}{r_1^2{r}_2^2}\\ {\left[{\left[Y_{l_1}\left({\widehat{r}}_1\right)\otimes Y_{l_2}\left({\widehat{r}}_2\right)\right]}_L\otimes {\left[{\text{Φ}}_{C_1}^{j_1{\pi }_1}\otimes {\left[{\text{Φ}}_{C_2}^{j_2{\pi }_2}\otimes {\text{Φ}}_{C_3}^{j_3{\pi }_3}\right]}_{j_{23}}\right]}_{j_{123}}\right]}_{JK}|{\text{Ψ}}_A^{\text{int}}\rangle \\ =\frac{1}{\sqrt{{\mathcal{N}}_K^{J\pi }}}{\displaystyle \sum _{\begin{array}{l}1\le i_1<\cdots <i_{C_1}\le A\\ 1\le j_1<\cdots <j_{C_2}\le C_2+C_3\end{array}}{P}_A\left(i_1\mathrm{,}\cdots \mathrm{,}{i}_{C_1}\right)P_{C_2+C_3}\left(j_1\mathrm{,}\cdots \mathrm{,}{j}_{C_2}\right)}\\ \times {\left[{\left[{\chi }_{l_1}^{23}\otimes {\chi }_{l_2}^{123}\right]}_L\otimes {\left[N^{j_1{\pi }_1}\left(i_1\mathrm{,}\cdots \mathrm{,}{i}_{C_1}\right)\otimes {\left[N^{j_2{\pi }_2}\left(j_1\mathrm{,}\cdots \mathrm{,}{j}_{C_2}\right)\otimes N^{j_3{\pi }_3}\left(j_{C_2+1}\mathrm{,}\cdots \mathrm{,}{j}_{C_2+C_3}\right)\right]}_{j_{23}}\right]}_{j_{123}}\right]}_{JK}\end{array}$ (63) with the overlap kernels defined as $\begin{array}{c}{\chi }_{l_1{m}_{l1}}^{23}\left(a_1;j_1\mathrm{,}\cdots \mathrm{,}{j}_{C_2+C_3}\right)\\ =\langle \frac{\delta \left(r_1-a_1\right)}{r_1^2}{Y}_{l_1{m}_{l1}}\left({\widehat{r}}_1\right)|{\chi }^{23}\left(r_1;j_1\mathrm{,}\cdots \mathrm{,}{j}_{C_2+C_3}\right)\rangle \\ {\chi }_{l_2{m}_{l2}}^{123}\left(a_2;i_1\mathrm{,}\cdots \mathrm{,}{i}_{C_1}\mathrm{,}{j}_1\mathrm{,}\cdots \mathrm{,}{j}_{C_2+C_3}\right)\\ =\langle \frac{\delta \left(r_2-a_2\right)}{r_2^2}{Y}_{l_2{m}_{l2}}\left({\widehat{r}}_2\right)|{\chi }^{123}\left(r_2;i_1\mathrm{,}\cdots \mathrm{,}{i}_{C_1}\mathrm{,}{j}_1\mathrm{,}\cdots \mathrm{,}{j}_{C_2+C_3}\right)\rangle \\ N_{m_1}^{j_1{\pi }_1}\left(i_1\mathrm{,}\cdots \mathrm{,}{i}_{C_1}\right)\\ =\langle {\text{Ψ}}_{m_1{C}_1}^{j_1{\pi }_1}|{\text{Φ}}_{C_1}^{\text{int}}\left(i_1\mathrm{,}\cdots \mathrm{,}{i}_{C_1}\right)\rangle \\ N_{m_2}^{j_2{\pi }_2}\left(j_1\mathrm{,}\cdots \mathrm{,}{j}_{C_2}\right)\\ =\langle {\text{Ψ}}_{m_2{C}_2}^{j_2{\pi }_2}|{\text{Φ}}_{C_2}^{\text{int}}\left(j_1\mathrm{,}\cdots \mathrm{,}{j}_{C_2}\right)\rangle \\ N_{m_3}^{j_3{\pi }_3}\left(j_{C_2+1}\mathrm{,}\cdots \mathrm{,}{j}_{C_2+C_3}\right)\\ =\langle {\text{Ψ}}_{m_3{C}_3}^{j_3{\pi }_3}|{\text{Φ}}_{C_3}^{\text{int}}\left(j_{C_2+1}\mathrm{,}\cdots \mathrm{,}{j}_{C_2+C_3}\right)\rangle .\end{array}$ (64)