Introduction

With the advent of radioactive ion beam technology and facilities, the nuclear landscape has been extended to a large map, encompassing Z=118 and reaching the proton and neutron drip line [1-3]. Contrary to the proton-rich side of the nuclear chart, experimental exploration of neutron-rich nuclei appears to be quite limited because of their short lifetimes and tiny cross sections [4-7]. Among these dripline nuclei, the heaviest carbon isotope is experimentally determined to be ²²C, which is of particular interest in both nuclear physics and Efimov physics in connection with its Borromean nature [8-11]. In detail, approximately two decades ago, ²²C was predicted to be an ideal s-wave two-neutron halo nucleus with a dominant configuration [12, 13], which was then confirmed in the measurement of the reaction cross section, regardless of the unexpectedly large matter radius [14]. In addition, this dripline nucleus ²²C plays a crucial role in other exotic structural properties, such as the possible new magicity at N=16 and shape decoupling [15-20]. On the other hand, ²²C is, as mentioned before, a typical Borromean nucleus with a bound three-body (²⁰C+n+n) system, while no pair of its components, namely ²⁰C+n or n+n, is found. However, there are still large uncertainties and puzzles in its nuclear quantities, such as the separation energy and matter radius [2, 21-23].

In fact, since the astonishing experiment of the reaction of ²²C on a liquid hydrogen target in Ref. [14], extensive efforts have been made to clarify the ambiguity regarding the bulk properties of this intriguing nucleus [9, 21, 22]. For example, the interaction cross sections (σ_I) of ²²C on a carbon target were accurately measured at 235 MeV/nucleon, resulting in a σ_I of 1.280±0.023 b [9]. Within a four-body Glauber reaction model, the root-mean-squared matter radius of ²²C was deduced to be 3.44±0.08 fm, which is obviously smaller than the previous extraction. Several other experiments have also been performed to understand the ²²C nuclear structure, such as neutron removal reactions from carbon isotopes and the reconstruction of ²⁰C+n decay-energy spectrum [8, 21, 22]. As another typical quantity for the dripline nuclei, the binding energy or the (two) neutron separation energy of ²²C is still uncertain. To date, there seems to be only one direct mass measurement of ²²C, leading to an upper limit of S_2n≤ 320 keV [23]. According to a recent atomic mass evaluation (AME)20, the value of S_2n was determined to be 35±20 keV [2]. From a theoretical perspective, there are different choices for predicting separation energies within the shell model context and three-body approach [12, 24-26]. In addition, extensive efforts have been devoted to this exotic nucleus ²²C and its isotopes via the ab initio methods [27-29]. However, a debate or discussion seemingly exists regarding the knowledge of ²²C.

It is well expected and accepted that there is consistency between the neutron separation energy and matter radius for neutron-rich nuclei [12, 24, 30, 31]. In other words, such a correlation can allow us to at least place the constraint on the structural property of ²²C, although its absolute value of separation energy and matter radius may not be reached at present. Based on the extracted matter radii from the reaction measurements, the two-neutron separation energy of ²²C can be obtained via the empirical formula of the relationship between the energy and radius [23, 24, 32]. For example, under such a procedure, the S_2n of ²²C would be approximately 10 keV based on the large radius reported decades ago [24]. Recently, the reaction cross sections of neutron-rich carbon isotopes on ¹²C were systematically measured over a wide range of incident energies from 30 to 950 MeV/nucleon [33]. The matter distributions and radii of the carbon isotopes are then determined by using the finite-range Glauber model with Coulomb correction, including the exotic nucleus ²²C. The simultaneously obtained charge radii were in excellent agreement with those directly extracted from the charge-change cross sections [33, 34]. Keeping these in mind, it is of particular significance to determine what one can obtain through the constraint of these new data with high accuracy. Against this background, the three-body Faddeev method is employed here to further understand the binding mechanism of ²²C via the two-neutron separation energy, with the condition of reproducing its matter radius. The details of the theoretical approach and the parameterization choice are presented in the next section, and the specific results and related discussions on the nucleon density distributions are provided in Sect. III. Finally, a summary is presented in the last section.

Theoretical approach

The Faddeev equations provide a rigorous mathematical framework for three-body systems, whose essence lies in decomposing the total wave function into three Faddeev components , each corresponding to and describing the pair-wise interaction between particles [35]. For instance, in the ²²C(n+n+²⁰C) system, Ψ₁ may correspond to the interaction between the two neutrons, whereas Ψ₂ and Ψ₃ correspond to neutron-core interactions. Each component corresponds to a different Jacobi coordinate system (as illustrated in Fig. 1). The equation is as follows: (1)The left-hand side represents the independent dynamics of the subsystem, including the kinetic energy Ti, core Hamiltonian , and two-body interaction potential Vi (encompassing both nuclear and Coulomb forces). The right-hand side represents how the superposition of the other two components feeds back into the evolution of the current component through the potential energy terms. This reflects the three-body coupling, the root cause of complexity in three-body problems.

Fig. 1Full-screen View

(Color online) Three sets of Jacobi coordinates in the Faddeev formalism for ²²C(n+n+²⁰C) three-body system

Through this decomposition, the Faddeev equations transform the three-body problem into a coupled system of integro-differential equations, thereby avoiding the complexities of directly solving the high-dimensional Schrödinger equation. However, even with this simplified form, the numerical solution presents significant challenges owing to the multidimensional coordinate systems. To further reduce the computational dimensionality and achieve a unified description of the dynamic behavior across different Jacobi coordinates, the introduction of hyperspherical coordinates is essential.

By adopting the hyperspherical coordinate system, which includes the hyperradius ρ and hyperangles θi, the two-dimensional system of partial differential equations is transformed into a set of coupled, one-dimensional equations. The transformation of the Jacobi coordinates into hyperspherical coordinates takes the following form [36]: (2)The hyperradius ρ characterizes the global scale of the three-body system, being invariant under translational and rotational transformations as well as permutations of particle pairs (i, j), while correlating with the size of the nuclear core. In contrast, the hyperangle θi describes the relative configuration between the particles, which exhibits radial dependence and is correlated with the relative magnitudes of the two Jacobi coordinates. The wave function is expanded within the hyperspherical coordinates as [36, 37] (3) denotes the hyperangular component, described by the Jacobi polynomial ; is the normalization coefficient and Ki represents the hyperangular momentum directly related to the order of the corresponding Jacobi polynomial through . represents the hyperradial component, which is expanded in terms of Laguerre polynomial basis functions.

Upon introducing the hyperspherical expansion into the Faddeev coupled equations, one obtains the following set of simultaneous linear equations: (4)Within the Faddeev equations, the matrix H requires several types of matrix elements: (5) denotes the central interaction. The spin-orbit (SO) coupling is described by the operator and radial form factor . The tensor operator and radial form factor account for the multipole-deformed potential. The standard tensor interaction is given by operator with radial dependence , whereas the spin-spin interaction is represented by and its form factor .

The central potential depends solely on the interparticle distance, with its operator form expressed as and is diagonal in the angular momentum and spin quantum numbers. Within a fixed Jacobi coordinate system, the matrix elements are given by (6) represents the hyperspherical basis function. The spin-orbit potential takes the form , where its matrix elements incorporate scalar products of the orbital angular momentum with particle spins s_j or s_k.

numerical results and discussions

In this section, we perform detailed calculations of the ground-state properties of the ²²C nucleus using a three-body model based on the Faddeev equations. As a typical neutron-dripline nucleus, the experimentally deduced root-mean-square (r.m.s.) radius of ²²C, fm [33], serves as a key constraint in this study. Based on this, the parameters of the Woods-Saxon potential for the neutron-core interaction were determined using an inverse optimization approach.

In developing an interaction model n+²⁰C, critical assumptions are implemented due to the absence of experimental data on core excitations: the ²⁰C core is treated as a rigid structure with spin-parity Jπ = 0⁺, enforcing Pauli blocking of the core orbitals (1s_1/2)², (1p_3/2)⁴,(1p_1/2)² and (1d_5/2)⁶ [38]; valence neutrons predominantly occupy the (2s_1/2) orbital with very weak binding [21].

The core-neutron interaction is described using a Woods-Saxon potential under the aforementioned assumptions, as shown in the following Eq. (7) [12, 37, 39], which takes the first two terms from Eq. (5). (7)The interaction potential parameters are determined by reproducing the experimentally deduced r.m.s. matter radii. Ref. [33] reports the experimental r.m.s. matter radii of ²²C as fm. Achieving convergence within the experimental constraints necessitates reducing the radius parameter to r₀ = 1.13A^1/3 fm (versus the conventional 1.25A^1/3 fm used in Ref. [12, 37]) – motivated by the monotonic increase of r.m.s. matter radii with r₀. When setting r₀=1.13A^1/3 fm, the minimum achievable r.m.s. matter radii falls below the experimental lower limit, thereby encompassing the experimental range. The core-neutron potential strength is incrementally adjusted until precisely matches the experimental value of 3.173 fm. The maximum attainable r.m.s. matter radii through variation remains below the critical experimental values (3.296 fm and 3.419 fm). Consequently, a systematic adjustment of r₀ was implemented to translate the theoretical radius distribution, ensuring full coverage of the experimental value range. Specifically, distinct r₀ values are sequentially fixed to ensure that the adjustable range of the r.m.s. matter radii, achieved by tuning the neutron-core potential, encompasses each target experimental value.

From Table 1, it can be observed that also depends on the three-body interaction potential. Entries 2 and 4 demonstrate that identical r.m.s. matter radii values can correspond to multiple distinct sets of neutron-core and three-body interaction potentials. The analysis of entries 3 and 5 reveals that the neutron-core interaction potential plays a predominant role, whereas the three-body interaction potential also contributes significantly. The unique determination of this parameter set requires the incorporation of an additional constraint, namely, the two-neutron separation energy S_2n. A comparison of entries 3 and 5 reveals that within a constrained range of neutron-core potentials, S_2n exhibits a stronger correlation with the three-body interaction potential. Consequently, S_2n provides an approximate constraint for the three-body potential, enabling the subsequent precise calibration of .

The two-neutron separation energy S_2n, obtained as an outcome of the calculation constrained by the matter radius, was then compared with theoretical and experimental values for validation. Ref. [9] reports MeV, but calculations considering only two-body potentials exhibit significant deviations from this value. This discrepancy is mitigated by introducing a Gaussian-form three-body interaction in Eq. (8), which emulates the effects of core deformation or core excitation [35, 40, 41]. (8)

When identical three-body potential parameters , r_3b=14 are applied to parameter sets 1, 4, and 5, an inverse correlation is observed: stronger neutron-core interaction potentials yield larger two-neutron separation energies S_2n. These results exhibited significant deviations from the theoretical values. The deviation is reduced when the three-body potential strength is increased from the baseline of parameter set 5. This adjustment brings the calculated two-neutron separation energy S_2n into closer agreement with the theoretical value of MeV reported in Ref. [9]. The rationale for avoiding universally larger three-body potentials lies in Set1’s extremely shallow S_2n binding. Further enhancement induces a transition from the bound states to the continuum. Consequently, each parameter set requires a customized three-body potential strength.

At this stage, the input parameters, including two-body and three-body potentials have been calibrated by fixing the r.m.s. matter radii and benchmarking against the theoretical values of the two-neutron separation energy. These parameters are listed in Table 1, with Sets 1, 2, and 3 selected as the primary configurations. Across these sets, only three parameters were varied, while all others remained fixed.

The hyperangular momentum cutoff K_max=20 was employed throughout calculations to guarantee numerical convergence. The choice of K_max is briefly discussed in Ref. [42], larger values of K_max yield better convergence behavior and enable higher accuracy of the numerical solution. However, the influence of K_max on the results was significantly smaller than that of the two-body potential. Moreover, our study of the nucleus ²²C did not involve long-range interactions. Therefore, choosing K_max=20 is adequate for the calculations.

The neutron-neutron interaction is described using the Gogny-Pires-Tourreil (GPT) potential [43], with the central, tensor, and spin-orbit terms included while omitting the spin-spin contribution. This potential provides good fits to the low-energy properties of nucleon-nucleon scattering.

Based on the parameter sets (Sets 1, 2, and 3) listed in Table 1, the orbital energy levels, spatial configurations, and density distributions were systematically calculated. The bound states prohibited by the Pauli principle, such as the 1s_1/2 and 1p_3/2 orbitals, are eliminated through supersymmetric transformations, restricting the valence neutrons to the allowed 1d_5/2 orbital. Furthermore, adjusting the s-wave and p-wave potential strengths along with the spin-orbit coupling strength revealed that the (1s_1/2), (1p_3/2), and (1p_1/2) states depend mainly on s-wave and p-wave contributions, which are not listed explicitly.

Table 1 lists the key single-particle orbital energies (units: MeV), where the orbital energy 1d_5/2 ranges from -0.323 to -0.72 MeV, indicating that the orbital is in a weakly bound state. The deeply bound 1s_1/2 orbital (-18.306 to -19.889 MeV) has a compact wave function and is Pauli-blocked for the valence neutrons. It should be noted that our research results do not rely on these energy values but rather on the single-particle wave functions. The different potential energies selected in this study produced almost identical single-particle wave functions for each occupied orbital, and all potential energy sets were configured such that the energy of the 2s_1/2 single-particle state approached zero, consistent with the prescription in Ref. [12].

Because there is no centrifugal potential barrier for neutrons in the 2s_1/2 orbital in the average potential field, the extremely weak neutron binding energy leads to significant tunneling effects. This significant tunneling implies the radial expansion of the wave function of the s_1/2 orbital. This energy level structure is consistent with the typical characteristics of the neutron drip-line nuclei. Additionally, owing to the expansion of the wave function, the dynamic coupling effects with other strongly bound orbitals are weakened beyond the static effects of the average potential field, indicating the presence of a pure configuration and an unperturbed core. If there are two neutrons in the 2s_1/2 orbital, their interaction is the only source of additional binding energy. Because the s-wave potential energy may be further weakened, the ground state energy of ²²C is considered to be the minimum value in this analysis.

We used the two- and three-body interaction models in Table 1 and solved the Faddeev equations to obtain the energy values and r.m.s. matter radii of the ground and excited states (relative to the n+²⁰C threshold) of ²²C. The calculation results are presented in Table 2. All energy values are given relative to the ²⁰C+n+n threshold, and the values are similar.

Z=6, N=14 closed shell cores are referred to as closed cores. The neutron part of the ²⁰C ground state not only contains the 1d_5/2 closed-shell configuration but also other configurations such as 2s_1/2. In the ground state, the double neutron separation energy S_2n=-E=0.265–0.656 MeV. The parameters were set based on the experimental values in Ref. [33], S_2n falls within the experimental range and is consistent with other theoretical values, S_2n=0.423 ± 1.140 MeV from Ref. [44] and S_2n=-0.140 ± 0.460 MeV from Ref. [23] within the error range, confirming the reliability of the theoretical model.

In addition, the ²²C ground-state correlation core is introduced, featuring two valence neutrons occupying the halo s-orbital that must satisfy the orthogonality condition with the ²⁰C core s-orbital. The excited state has a spin-parity of Jπ = 0⁺. Notably, the excited state energy E is positive, indicating that the employed two-body and three-body potentials cannot bind the neutrons, which is consistent with the experimental phenomenon that there is no bound state in the ²¹C nucleus [45].

To analyze the three-body configuration of ²¹C, the average distance parameter of the valence neutrons was calculated (Table 3). In the ground state, the average distance between two neutrons is fm, and the distance from the nucleus to the center of mass of the pair of neutrons is fm. In the excited state, these values extend to fm and fm, which correspond to the size of a stable atomic nucleus with mass number . Therefore, the excited state of ²²C can be described as a giant halo state. The key finding was that the ratio remained constant. (9)The identical proportions obtained in both states indicate that these two states have similar geometric structures, with the main difference being the spatial discrete scaling factor.

In addition, the correlation density distribution between the ground and excited states of ²²C was analyzed. These values were calculated using the Jacobi coordinate system, with ²⁰C as the bystander particle. The formula for calculating the spatial correlation density distribution is as follows [41]. (10)The spatial distributions of the two valence neutrons in the ground and excited states are shown in Figs. 2 and 3, respectively. The spatial distribution function peaks at fm (Fig. 2) in the ground state, corresponding to the maximum probability density. This configuration is consistent with a compact three-body configuration, whereas the main peak of the maximum probability density for the excited state is at (9.5, 4.5) fm, with a secondary peak at (12, 5.5) fm, which originates from the contribution of the (1d)² orbital (Fig. 3). This double-peak structure reflects the orbital rearrangement effect in the excited state: some neutrons transition from the s-wave to the d-wave, causing the spatial configuration to differentiate into two modes: compact and extended.

Fig. 2Full-screen View

(Color online) Spatial distribution of two valence neutrons in the ground state

Fig. 3Full-screen View

(Color online) Spatial distribution of two valence neutrons in the excited state

Orbital occupancy analysis reveals the s-wave dominant characteristics of ²²C. In the ground state, (2s_1/2)² accounts for 97.77%, whereas the d-wave component accounts for only 2.23%; in the excited state, (2s_1/2)² decreases to 72.90%, whereas (1d_5/2)² increases to 27.10%. This change coincides with the secondary peak position of the density distribution, indicating the existence of s-d orbital mixing in the excited state. This may be because the d-wave component acquires significant weight and core polarization, and the extended halo structure enhances the coupling between the core and valence neutrons, inducing orbital mixing. It is worth noting that despite the increase in d-wave components, s-waves still dominate (>70%); thus, the excited state is essentially the same as the ground state, with an s-wave halo nucleus [12], but exhibiting more complex multi-orbit coupling effects.

The Efimov effect is a universal quantum phenomenon in three-body systems, in which an effective long-range attraction emerges from short-range two-body interactions near resonance, leading to a series of weakly bound states exhibiting discrete scale invariance [46]. Given the significantly larger r.m.s. radii and more diffuse spatial distribution of the excited state, it can be characterized as an Efimov state in the halo nucleus [47, 48]. Since ²²C has been confirmed to be a double-neutron halo nucleus, Ref. [49] pointed out that the Efimov effect is most likely to be observed in double neutron halo nuclei. The study also revealed an interesting geometric similarity between the ground and excited state configurations—in Efimov physics, two consecutive Efimov states can be related through discrete spatial scaling factors [50]. The identical ratio relationship between r_nn and r_c,nn in the ground and excited states of ²²C [Eq. (9)] indicates that the nuclear configurations of the two states have highly similar geometric shapes, indicating that discrete scaling symmetry exists. The features we discovered in ²²C are highly consistent with the theoretical explanation of the Efimov states [51].

Conclusion

In this study, we systematically investigated the structural properties of the neutron-rich nucleus ²²C using a three-body Faddeev approach constrained by the experimentally determined matter radius. Our calculations employed the Woods-Saxon potential for the neutron-core interaction and introduced a Gaussian-form three-body potential to account for core deformation effects. By fine-tuning the potential parameters, the experimental matter radius ( fm) was successfully reproduced, while the derived two-neutron separation energy S_2n was in agreement with both theoretical and experimental values, confirming the weakly bound nature of ²²C.

The ground state of ²²C exhibits a dominant (2s_1/2)² configuration with a weakly bound 1d_5/2 orbital, yielding S_2n = 0.265–0.656 MeV, consistent with previous results. The excited state displays an extended s-d hybrid structure (s-wave >70%) and a double-peak spatial distribution. The constant ratio between states suggests a discrete scaling symmetry, supporting ²²C as an Efimov candidate in Borromean nuclei.

Our results highlight the interplay between the neutron and core interaction and three-body forces in shaping the properties of ²²C. The consistency between our predictions and experimental data validates the three-body approach as a powerful tool for studying neutron-rich nuclei. Future work could explore the dynamical effects of core excitation and the role of higher-order interactions in refining the description of ²²C and similar dripline nuclei. These insights contribute to a deeper understanding of exotic nuclear structures and their connections to universal few-body phenomena.