Introduction

Clustering is a fundamental phenomenon in the nucleus [1-8]. Cluster models are widely used and have been shown to be effective in describing the characteristics of light nuclei [9, 10]. With the cluster model, the study of diproton and dineutron correlations is crucial for understanding nucleon-nucleon interactions and the underlying nuclear structure, providing information on pairing mechanisms and the behavior of nucleons in short-range interactions, which are essential for understanding phenomena such as nuclear stability and reaction dynamics [11-14].

⁶He, the lightest Borromean halo nucleus, together with its mirror nucleus ⁶Be, the lightest two-proton emitter, has attracted extensive studies on the dinucleon correlations in their decaying modes and structures [15-21]. Previous studies have described the structure of ⁶He [22-24], a more precise description can be found in [25], which addresses both six-body correlations and clustering in the ⁶He ground state using the no-core shell model with continuum (NCSMC), where the “dineutron” configuration is shown to prevail over the “cigar” structure. Recently, the $2_1^{+}$ state of ⁶He was suggested to exhibit a dineutron correlation based on simulations of its decay mode [16]. Regarding its mirrored nucleus, ⁶Be stands out as the lightest two-proton emitter featuring a distinct structure of α and two protons [26, 27]. This characteristic, in accordance with Goldansky’s framework, establishes a robust benchmark for conducting comprehensive investigations of two-proton decay and diproton correlations within nuclear structure [28, 29].

Dineutron and diproton correlations have been intensively discussed through 2n and 2p emissions from unbound nuclei in connection with recent experiments [30-33]. In the case of ⁶He, experiments have investigated the decay mode of its $2_1^{+}$ resonant state via the ⁶He breakup reaction by ¹²C at 240 MeV/nucleon, revealing the coexistence of dineutron decay and democratic decay, which suggests the possible existence of a dineutron structure in the $2_1^{+}$ state of ⁶He [34]. For ⁶Be, experiments using a high-resolution array to detect its α+p+p three-body decay have provided precise three-body correlation data that agree well with theoretical models, thereby validating the theoretical approach over a wide range of energies [13, 28, 35-38].

The main objective of this study was to investigate the correlations between diprotons and dineutrons in ⁶He and ⁶Be in several low-lying states of ⁶He and ⁶Be at the structural level using a microscopic nuclear model. We employed the Generator Coordinate Method (GCM) with Brink wave functions [39, 40] as a robust framework to model and analyze these correlations.

By calculating the Two-Cluster Overlap Amplitude (TCOA), we aim to quantify the spatial distribution and correlation strength of nucleon pairs, providing insights into the nucleon-nucleon interactions within these nuclei [41, 42]. This approach enables detailed examination of the structural and correlation properties of ⁶He and ⁶Be, contributing to a deeper understanding of nucleon correlations in light nuclear systems [43, 44].

Theoretical framework

In the present GCM calculations, the total wave function of ⁶He (⁶Be) can be written as the superposition of angular-momentum-projected and parity-projected Brink wave functions ${\text{Ψ}}_M^{J\pi }={\displaystyle \sum _{i\mathrm{,}K}}{c}_{i\mathrm{,}K}{\widehat{P}}_{MK}^J{\widehat{P}}^{\pi }{\text{Φ}}^{\text{B}}\left({\left\{R\right\}}_i\right)\mathrm{,}$ (1) where ${\widehat{P}}_{MK}^J$ and ${\widehat{P}}^{\pi }$ are the angular-momentum and parity projectors, respectively. The index i indicates a specified set of generator coordinates {R₁, R₂, R₃}. An illustration of ⁶He is shown in Fig. 1. The Brink wave function is fully antisymmetrized and the wave function of the k-th nucleon is defined as a Gaussian wave packet ${\varphi }_k\left(R_j\right)=\frac{1}{{\left(\pi b^2\right)}^{3/4}}\exp \left[-\frac{1}{2b^2}{\left(r_k-R_j\right)}^2\right]{\chi }_k{\tau }_k\mathrm{.}$ (2) In the present calculation, the oscillator parameter for the single-particle wave functions was set to $b=1.46\text{fm}$, which is the same as that used in Refs. [45, 46]. The Hamiltonian of the system includes kinetic, central N-N, spin-orbit, and Coulomb parts $\begin{array}{ll}\widehat{H}=\hfill & -\frac{{\hslash }^2}{2m}{\displaystyle \sum _i}{\nabla }_i^2-T_{\text{c}\text{.m}\text{.}}+{\displaystyle \sum _{i<j}}{\widehat{V}}_{ij}^{\text{NN}}+{\displaystyle \sum _{i<j}}{\widehat{V}}_{ij}^{\text{LS}}+{\displaystyle \sum _{i<j}}{\widehat{V}}_{ij}^{\text{C}}\mathrm{.}\hfill \end{array}$ (3) The Volkov No.2 potential [47] was taken as the central N-N potential ${\widehat{V}}_{ij}^{\text{NN}}={\displaystyle \sum _{n=1}^2}{v}_n{e}^{-\frac{r_{ij}^2}{a_n^2}}{\left(W+B{\widehat{P}}_{\sigma }-H{\widehat{P}}_{\tau }-M{\widehat{P}}_{\sigma }{\widehat{P}}_{\tau }\right)}_{ij}$ (4) with $a_1=1.01\text{fm}$, $a_2=1.8\text{fm}$, $v_1=61.14\text{MeV}$, $v_2=-60.65\text{MeV}$, $W=1-M$, M=0.6 and B=H=0.125. The G3RS potential [48, 49] is used for the spin-orbit term ${\widehat{V}}_{ij}^{\text{LS}}=v_0\left(e^{-d_1{r}_{ij}^2}-e^{-d_2{r}_{ij}^2}\right)\widehat{P}\left({}^{3}O\right)L\cdot S\mathrm{,}$ (5) where $\widehat{P}\left({}^{3}O\right)$ is the projection operator onto a triplet odd state, strength $v_0=2000\text{MeV}$, and parameters d₁ and d₂ are set to $5.0{\text{fm}}^{-2}$ and $2.778{\text{fm}}^{-2}$, respectively. The coefficients { $c_{i\mathrm{,}K}$} in Eq. (1) are determined by solving the Hill-Wheeler equation as follows:

Fig. 1Full-screen View

(Color online) Schematic diagram of α+n+n clustering structure of the Brink wave function of ⁶He

TCOA was introduced as an extension of the RWA method to quantitatively analyze the spatial distribution and correlation strength of nucleon pairs [50]. This approach has been successfully applied to study core + N + N + N structures [51], providing a detailed description of clustering dynamics.

To illustrate the three-cluster structure, the TCOA [52] of ⁶He is defined as: $\begin{array}{l}{\mathcal{Y}}_{l_1{l}_{23}L}^{J^{\pi }}\left(a_{\alpha -\text{nn}}\mathrm{,}{a}_{\text{nn}}\right)=\sqrt{\frac{6!}{4!1!1!}}\\ \langle \frac{\delta \left(r_1-a_{\alpha -\text{nn}}\right)\delta \left(r_{23}-a_{\text{nn}}\right)}{r_1^2{r}_{23}^2}\\ {\left[{\left[Y_{l_1}\left({\widehat{r}}_1\right)\otimes Y_{l_{23}}\left({\widehat{r}}_{23}\right)\right]}_L\otimes \left[{\text{Φ}}_{\alpha }\otimes {\text{Φ}}_n\otimes {\text{Φ}}_n\right]\right]}_0{}_{JM}|{\text{Ψ}}_M^{J^{\pi }}\rangle \end{array}$ (6) where a_α-nn and a_nn represent the distances from the center of mass(c.o.m.) of the two neutrons to the α cluster, and the distance between the two neutrons, respectively, as shown in Fig. 1. l₁ and l₂₃ correspond to the orbital angular momenta associated with distances a_α-nn and a_nn, respectively, while L denotes the total angular momentum obtained from their coupling. The reference wave function for the α cluster is denoted by ${\text{Φ}}_{\alpha }$.

To characterize the relative motion between the α cluster and the two neutrons, we introduce the relative-motion coordinates r₁ and r₂₃, which are defined as $r_1=X_1-\frac{X_2+X_3}{2}\mathrm{,}{r}_{23}=X_2-X_3\mathrm{,}$ (7) where $X_i$ is the c.o.m. of the physical coordinates of the α and neutrons. (Similarly, the structure of ⁶Be is analogous, with two valence neutrons replaced by two valence protons.)

The TCOA provides the spatial distribution of valence nucleons in terms of the distance between the two valence nucleons, $a_{\text{NN}}$, and the distance between their center and the α-core nucleus, $a_{\alpha -\text{NN}}$. It should be noted that the other degrees of freedom are integrated into in Eq. 6. The description provided by the TCOA can be viewed as the averaged isosceles triangle configuration. This allows us to estimate the opening angle $\theta =2\text{arctan}\left(a_{\text{NN}}/2a_{\alpha -\text{NN}}\right)$, of the two nucleons with respect to the core. The opening angle $\theta$ is a key measure for dinucleon correlations; for instance, $\theta {=90}^{\circ }$ corresponds to two non-correlated nucleons.

Results and discussion

By superposing 600 distinct three-body spatial configurations, $\alpha +n+n$ and $\alpha +p+p$, we obtained clustering wave functions for both ⁶He and ⁶Be. The resulting positive-parity low-lying state energy spectra are shown in Fig. 2, exhibiting an overall shift compared with the experimental data. The calculated positive-parity low-lying energy spectra (Fig. 2) exhibited qualitative consistency with the experimentally observed 0⁺ and 2⁺ states. For example, the calculated excitation energy of the $2_1^{+}$ state for ⁶He agrees well with the experimental observations—corresponding to very narrow resonances - showing only a small deviation of approximately 0.3 MeV. Moreover, the mirror symmetry breaking for ⁶He and ⁶Be in the ground state energy was also well reproduced. This consistency indicates that mirror symmetry breaking caused by isospin effects and Coulomb interactions, as well as the spatial extension of the valence nucleons, is effectively described. In the following section, we focus on the detailed spatial distribution of the valence nucleons relative to the α-core nucleus.

Fig. 2Full-screen View

The calculated energy spectra of ⁶He and ⁶Be compared with the corresponding experimental values. The gray dashed lines represent thresholds

Based on the definition of the TCOA discussed above, this framework effectively characterizes critical three-body cluster correlations, with specific emphasis on the dineutron correlation in ⁶He and the diproton correlation in ⁶Be. Figures 3 and 4 present the TCOA distributions for three-cluster systems in ⁶He and ⁶Be, where the orbital angular momenta quantum numbers l₁ = l₂₃ = 0 and l₁ = l₂₃ = 1 were chosen because these specific combinations exhibited the most pronounced TCOA distribution amplitudes. In a purely non-correlated scenario, the distributions would exhibit equal weights on both sides of the dashed lines in the figure, which divide two distinct regions in the hyperspherical description of three-body nuclei [53]. For the ground states of ⁶He and ⁶Be, two distinct peaks were observed: a dinucleon-like peak in the region $a_{\alpha -\text{NN}}>a_{\text{NN}}/2$ and a cigar-like peak in the region $a_{\alpha -\text{NN}}>a_{\text{NN}}/2$. These peaks arise from the two valence nucleons that predominantly occupy the p-shell. The TCOA further indicates that a dinucleon-like configuration is favored, as evidenced by its higher maximum TCOA and asymmetric distribution. For example, in ⁶He, the dineutron-like peak is characterized by ${\mathcal{Y}}_{l_1{l}_{23}L}^{J^{\pi }}\left(\left\{a_{\alpha -\text{nn}}\mathrm{,}{a}_{\text{nn}}\right\}=\left\{2.1\text{fm}\mathrm{,3}\text{fm}\right\}\right)=0.32$, which is consistent with the ab initio results in Ref. [25], whereas the cigar-like peak is given by ${\mathcal{Y}}_{l_1{l}_{23}L}^{J^{\pi }}\left(\left\{a_{\alpha -\text{nn}}\mathrm{,}{a}_{\text{nn}}\right\}=\left\{4.4\text{fm}\mathrm{,1.1}\text{fm}\right\}\right)=-0.26$.

Fig. 3Full-screen View

(Color online) TCOA calculation for the ground states of ⁶Be and ⁶He. The internal orbital angular momenta are l₁=0 and l₂₃=0. The dashed line represents $a_{\alpha -\text{NN}}=a_{\text{NN}}/2$

Fig. 4Full-screen View

(Color online) TCOA calculation for the 2⁺ states of ⁶Be and ⁶He. The internal orbital angular momenta are l₁=1 and l₂₃=1. The dashed line represents $a_{\alpha -\text{NN}}=a_{\text{NN}}/2$

The conclusion of the favored dineutron correlation in the ground state of ⁶He is consistent with a recent experimental work [15], which extracted $\text{B(E1)}$ values to infer an average opening angle of the two valence neutrons of approximately 56°, supporting the presence of a dineutron correlation. The TCOA for ⁶Be exhibits similar behavior, with a maximum characterized by ${\mathcal{Y}}_{l_1{l}_{23}L}^{J^{\pi }}\left(\left\{a_{\alpha -\text{pp}}\mathrm{,}{a}_{\text{pp}}\right\}=\left\{2.1\text{fm}\mathrm{,3}\text{fm}\right\}\right)=0.28$, and another peak given by ${\mathcal{Y}}_{l_1{l}_{23}L}^{J^{\pi }}\left(\left\{a_{\alpha -\text{pp}}\mathrm{,}{a}_{\text{pp}}\right\}=\left\{4.5\text{fm}\mathrm{,1.1}\text{fm}\right\}\right)=-0.23$. However, the peak amplitudes are suppressed owing to the Coulomb repulsion. Additionally, a halo(-like) nature was revealed in the TCOA, as indicated by the diffuse distribution. For ⁶He and ⁶Be, the TCOA extends up to $a_{\alpha -\text{NN}}\approx 10\text{fm}$, whereas heavier ¹⁰Be exhibits a more compact distribution using a 2n overlap function [54].

Compared to the ground states, the $0_2^{+}$ states of both ⁶He and ⁶Be, there are mainly three main peak regions: dinucleon-like and cigar-like retained, while a stronger third peak emerged in the acute opening angle region. The three peaks might be due to siginificant d-wave occupation in the $0_2^{+}$ states; for example, the ground state in ¹⁸O nucleus in [55]. Compared with the ground state, this state is more diffuse and exhibits a gas-like feature, which indicates a more complex correlation structure.

For the first excited state $2_1^{+}$ in both ⁶He and ⁶Be, as shown in Fig. 4, we found that there was only one peak, predominantly distributed in the dinucleon correlation region, that is, $a_{\text{NN}}>2a_{\alpha -\text{NN}}\left({\theta }_{\text{NN}}=2\text{arctan}\left(a_{\text{NN}}/2a_{\alpha -\text{NN}}\right){<90}^{\circ }\right)$. The maximum value is characterized by ${\mathcal{Y}}_{l_1{l}_{23}L}^{J^{\pi }}\left(\left\{a_{\alpha -\text{nn}}\mathrm{,}{a}_{\text{nn}}\right\}=\left\{3.8\text{fm}\mathrm{,2.1}\text{fm}\right\}\right)=0.16$ for ⁶He, corresponding to an opening angle of ${\theta }_{\text{NN}}{=85}^{\circ }$. This result is consistent with a recent three-body model calculation using the complex-scaling method, which shows a peak in the opening angle density profile at approximately 60°~80°. For ⁶Be, the peak is characterized by ${\mathcal{Y}}_{l_1{l}_{23}L}^{J^{\pi }}\left(\left\{a_{\alpha -\text{pp}}\mathrm{,}{a}_{\text{pp}}\right\}=\left\{4.0\text{fm}\mathrm{,2.2}\text{fm}\right\}\right)=0.13$, with a noticeably smaller amplitude. This difference represents the observed mirror symmetry breaking in the dinucleon correlations, where the effects of the Coulomb interaction result in a weaker diproton correlation.

The TCOA distributions of the $2_2^{+}$ states in both ⁶Be and ⁶He exhibited remarkably similar gas-like characteristics, manifested as diffuse patterns without distinct peaks. Notably, both nuclei showed a single dominant peak in their respective distributions, with suppressed two-nucleon correlations compared to their $2_1^{+}$ states. For ⁶Be, this corresponds to a weakened diproton correlation in the $2_2^{+}$ state, wheras for ⁶He the analogous suppression occurs in the dineutron component.

Summary

In this study, we investigated the diproton and dineutron correlations in the ground and low-lying 2⁺ states of ⁶Be and ⁶He using the TCOA method within the GCM framework. Our calculations reveal that both ⁶Be and ⁶He exhibit pronounced diproton and dineutron correlations in their ground states, characterized by a cigar-like spatial configuration with a localized nucleon pair. The TCOA distributions for the $2_1^{+}$ states also show a single-peak structure, which is indicative of dinucleon correlations consistent with previous descriptions.

The present theoretical framework, combining the GCM with TCOA analysis, has proven effective in providing a detailed description of nucleon-nucleon correlations and clustering behavior in light nuclear systems, offering insights into the structural evolution of mirror nuclei across different excitation energies.