Ab Initio Study on the Halo Structure in ¹¹Be

Abstract

We present an ab initio study on the one-neutron halo nucleus ¹¹Be using nuclear lattice effective field theory with high-fidelity chiral interactions at N3LO. By employing the wavefunction matching method to mitigate the sign problem and the pinhole algorithm to sample many-body correlations, we successfully reproduce the ground-state parity inversion and the extended matter radius characteristic of the halo structure. We analyze the intrinsic density distributions and geometric shapes of ¹¹Be in comparison with the core nucleus ¹⁰Be. Our results reveal a prominent two-cluster structure in both nuclei and the occupation of the $\sigma$ molecular orbital by the valence neutron in ¹¹Be. It enhances the prolate deformation as well as the diffuse neutron tail, distinct from the $\pi$-orbital occupation observed in the ¹⁰Be ground state.

1. Introduction

Radioactive ion beams have opened new avenues for the study of nuclear structures and dynamics far away from the stability line. Exotic phenomena such as unusually large matter distributions, later known as halo structures, were discovered in light neutron-rich nuclei [1]. The ¹¹Be nucleus is one of the earliest identified and most extensively studied one-neutron halo nuclei [2,3], serving as a benchmark system for understanding the properties of loosely bound quantum systems.

A striking feature of ¹¹Be is the “parity inversion” of its ground state. According to the standard shell model, the seventh neutron should occupy the $1p_{1/2}$ orbital, resulting in a $1/2^{-}$ ground state. However, experiments established long ago that the ground state of ¹¹Be has spin-parity $1/2^{+}$ [4,5], while the $1/2^{-}$ state lies at an excitation energy of 0.32 MeV. This inversion indicates the breakdown of the N = 8 magicity and the lowering of the $2s_{1/2}$ orbital relative to the $1p_{1/2}$ orbital. The weak binding of the s-wave neutron leads to a spatially extended density distribution, forming the characteristic halo structure.

Experimentally, the structure of ¹¹Be has been investigated using various probes. Transfer reactions, such as $(d, p)$ and $(p, d)$, have been used to extract spectroscopic factors for the halo neutron. Schmitt et al. [6] reported spectroscopic factors of 0.71(5) for the $1/2^{+}$ ground state and 0.62(4) for the $1/2^{-}$ excited state. Breakup and knockout reactions have also confirmed the dominant s-wave component of the ground state and provided information on the core excitation [7,8,9]. Electromagnetic dissociation measurements have extracted the $B\left(E1\right)$ strength and the root mean square distance of the halo neutron [10,11,12]. Elastic scattering and breakup data have also been analyzed to constrain the core excitation and reaction mechanisms [13].

Theoretically, reproducing the parity inversion and halo structure of ¹¹Be has been a challenge. Early studies highlighted the importance of a dynamic coupling between the core and the loosely bound neutron [14]. Approaches incorporating particle–vibration coupling [15,16] or core deformation [17] have been successful in explaining the level order. Microscopic cluster models, such as antisymmetrized molecular dynamics, have also reproduced the anomalous parity and suggested a molecular orbit picture for the valence neutron [18,19]. In the realm of ab initio calculations, the no-core shell model initially faced difficulties in reproducing the parity inversion with realistic two-nucleon forces, suggesting a need for three-nucleon interactions or larger model spaces [20]. More recent ab initio studies using chiral interactions have emphasized the role of continuum effects and specific features of the three-nucleon force in correctly describing the spectrum [21]. Unified descriptions using density functional theory linking parity inversion, halo, and clustering continue to be a focus of recent work [22].

Recently, a systematic ab initio study was performed on beryllium isotopes from ⁷Be to ¹²Be [23] using nuclear lattice effective field theory (NLEFT) [24,25,26]. This comprehensive investigation employed high-fidelity chiral interactions up to next-to-next-to-next-to-leading order (N3LO), made possible by the wavefunction matching (WFM) technique [27], which mitigates the Monte Carlo sign problem. It successfully reproduced the ground-state energies, radii, and electromagnetic properties of the beryllium chain, including the correct ground state of ¹¹Be.

This proceeding will present a more detailed analysis of the ¹¹Be halo structure as a supplement to the previous study [23]. Specifically, we investigate the density distributions and underlying geometric structure of the ground and excited states using the pinhole algorithm.

2. Nuclear Lattice Effective Field Theory

In this work, we employ nuclear lattice effective field theory [24,25] to investigate the structure of ¹¹Be. The high-fidelity χEFT Hamiltonian at N3LO employed in this work follows Ref. [27]:

$$H=K+V_{\mathrm{OPE}}^{{\mathsf{\Lambda }}_{\pi }=300}+V_{\mathrm{Cou}.}+V_{2\mathrm{N}}^{Q^4}+W_{2\mathrm{N}}^{Q^4}+V_{2\mathrm{N},\mathrm{WFM}}^{Q^4}+W_{2\mathrm{N},\mathrm{WFM}}^{Q^4}+V_{3\mathrm{N}}^{Q^3},$$

(1)

where K is the kinetic energy; $V_{\mathrm{OPE}}^{{\mathsf{\Lambda }}_{\pi }=300}$ is the one-pion exchange (OPE) interaction with cutoff ${\mathsf{\Lambda }}_{\pi }=300$ MeV; $V_{\mathrm{Cou}.}$ is the Coulomb interaction; $V_{2\mathrm{N}}^{Q^4}$ are the two-nucleon (2N) contact interactions at $Q^4$ chiral expansion order; $W_{2\mathrm{N}}^{Q^4}$ denotes the corresponding Galilean invariance-restoring (GIR) terms; $V_{2\mathrm{N},\mathrm{WFM}}^{Q^4}$ and $W_{2\mathrm{N},\mathrm{WFM}}^{Q^4}$ are the 2N wave function matching (WFM) interactions and GIR terms; and $V_{3\mathrm{N}}^{Q^3}$ are the three-nucleon (3N) contact interactions at $Q^3$ chiral expansion order. The details on the interaction are presented in Refs. [27,28,29].

To address the Monte Carlo sign problem inherent in realistic nuclear forces, we utilize the wavefunction matching (WFM) method [27]. This method introduces a simple Hamiltonian $H^S$ with an SU(4)-symmetric contact interaction and a smeared OPE potential. A unitary transformation is performed on the full Hamiltonian H to $H^{\prime }$ so that the short-distance wave functions of $H^{\prime }$ matches those of $H^S$, while the difference $H^{\prime }-H^S$ is treated in first-order perturbation theory.

The ground-state wave function of the simple Hamiltonian is obtained using projection Monte Carlo simulations with auxiliary fields. The trial wave function $|{\mathsf{\Psi }}_0^S\rangle$ was chosen as a single Slater determinant of harmonic oscillator wave functions. To analyze the spatial correlations and halo structure, we use the pinhole algorithm [30], which samples nucleon coordinates according to the following amplitude:

$$Z=\langle {\mathsf{\Psi }}_0^S\left|M^{L_t/2}\rho ({\mathbf{n}}_1,\dots ,{\mathbf{n}}_A)M^{L_t/2}\right|{\mathsf{\Psi }}_0^S\rangle .$$

(2)

where $M=:exp(-a_t{H}_S):$ is the transfer matrix operator of the simple Hamiltonian, $a_t$ is the Euclidean time projection step, and $L_t$ is the total number of time slices. The many-body density operator $\rho ({\mathbf{n}}_1,\dots ,{\mathbf{n}}_A)$ is defined as

$$\rho ({\mathbf{n}}_1,\dots ,{\mathbf{n}}_A)=:a^{†}\left({\mathbf{n}}_1\right)a\left({\mathbf{n}}_1\right)\dots a^{†}\left({\mathbf{n}}_A\right)a\left({\mathbf{n}}_A\right) : .$$

(3)

Observables are then calculated by combining the unperturbed contributions from the sampled configurations with the perturbative corrections from the N3LO Hamiltonian [27]. See the Supplemental Materials of Refs. [30,31] for more details of the pinhole algorithm and its perturbative corrections. This framework allows for the precise determination of radii and intrinsic density distributions essential for studying halo nuclei.

3. Results and Discussion

We adopted a lattice size of $L=10$ with lattice spacing $a=1.32$ fm and Euclidean time lattice spacing of $a_t=0.001$ MeV ⁻¹. The low-lying spectrum from ⁷Be to ¹²Be was calculated in Ref. [23], and the energies and radii of ¹⁰Be and ¹¹Be are summarized in

Table 1. Ground-state energies and radii of ¹⁰Be and ¹¹Be obtained through NLEFT calculation with N3LO interaction [23], in comparison with experimental data [32,33,34,35].

Nuclear State	Energy (MeV)		Charge Radius (fm)		Matter Radius (fm)
	NLEFT	Exp.	NLEFT	Exp.	NLEFT	Exp.
10Be	$-63.7\left(3\right)$	$-65.0$	2.51(4)	2.36(2)	2.53(2)	2.30(2)
11Be	$-65.6\left(3\right)$	$-65.5$	2.54(4)	2.46(2)	2.86(1)	2.91(5)

. The experimental values are also listed for comparison [32,33,34,35]. The binding energy of ¹⁰Be calculated via NLEFT is 7.2 MeV with respect to the $\alpha$-$\alpha$-neutron–neutron threshold, and 6.1 MeV with respect to the ⁹Be-neutron threshold. The binding energy of the last neutron in ¹¹Be calculated via NLEFT is 1.9 MeV with respect to the ¹⁰Be-neutron threshold, closer to the 500 keV binding observed in nature. To achieve a more accurate description of the binding energies, more efforts are needed to improve the accuracy of the ab initio calculations, such as through the fine tuning of three-body interactions [36].

The ground-state parity inversion of ¹¹Be can be reproduced through NLEFT calculations with the N3LO interaction, and the overall agreement with the experimental data is good. The radii of ¹⁰Be are about 10% larger than the experimental values, while the radii of ¹¹Be agree better. A sudden increase in the matter radius of ¹¹Be is observed in the halo structure.

In Ref. [23], the A-nucleons were grouped into two clusters composed of two protons and two neutrons that are the closest to each other according to their spatial coordinates ${\mathbf{r}}_i$ sampled by the pinhole algorithm. The extra neutrons (two in ¹⁰Be and three in ¹¹Be) are considered to be the valence neutrons. By randomly rotating one of the clusters along the z-axis, the intrinsic 3D density distribution $\rho \left(\mathbf{r}\right)$ is obtained. The results of the ¹⁰Be and ¹¹Be ground states are shown in Figure 1a,b, respectively. Their difference (¹¹Be-¹⁰Be) is shown in Figure 1c.

One can see a clear two-cluster structure in the density distribution. Due to the different occupations of the $\pi$ and $\sigma$ orbitals, the two nuclei show quite different shape patterns. In the ¹⁰Be ground state, the valence neutrons occupy the $\pi$ orbital surrounding and between the two clusters [37,38,39,40,41]. The extra neutron in ¹¹Be occupies the $\sigma$ orbital [38]. As shown in Figure 1c, its center part has a higher density, whereas at the top/bottom of the clusters, the density is lower. Furthermore, the halo structure can also be seen in the difference in density distributions.

To see the details of the density distribution and halo character more closely, we plotted the radial density distributions of the ¹⁰Be and ¹¹Be ground states calculated by NLEFT in Figure 2. From left to right are the total densities, densities of the two clusters, and densities of the valence neutrons. The dashed line in Figure 2a,c indicates the expected asymptotic density distribution proportional to $e^{-2\kappa r}/r^2$, with $\kappa =\sqrt{2mE}/\hslash \approx 0.303$ fm⁻¹ calculated for $\left|E\right|=1.9$ MeV, and the shadow band indicating the error propagated from the binding energy uncertainty. One can see at a large radial distance that the density of ¹¹Be is much more extended, showing the halo structure. By comparing panels (a) and (c), one can see that the density at such a large radial distance mainly comes from the valence neutrons. Due to the occupation of the $\sigma$ orbital, the density of ¹¹Be near the center is larger, where both the $\sigma$-orbital valence neutron and the clusters contribute. Due to the more prolate shape of ¹¹Be, its cluster radial density distribution is also slightly more extended, as seen in panel (b).

In Figure 3, we plot the angular distribution of two clusters (cluster1–center–cluster2) in the ¹⁰Be and ¹¹Be ground states calculated by NLEFT. This distribution is due to the valence neutrons, as in the case of ⁸Be, the angle between two clusters will always be 180°. The integration of the probability over the whole range gives one. More valence neutrons (three in ¹¹Be compared to two in ¹⁰Be) lead to a wider distribution of this angle.

In Figure 4, we plot the probability distribution of the angle between each nucleon, with the z-axis in the ¹⁰Be and ¹¹Be ground states calculated by NLEFT in the intrinsic frame shown in Figure 1. The distribution is symmetric to 90° and therefore only half is plotted. Integration over the whole range gives the particle numbers: 8 for panel (a), and 2 (¹⁰Be) and 3 (¹¹Be) for panel (b). The dashed line in Figure 4b indicates the expected angular distribution for a homogeneous sphere, $p\left(\theta \right)={\displaystyle \frac{\pi }{360}}sin\theta$ (normalized to 1). For nucleons in the clusters, the distribution is highest around 0° and gradually decreases to 90°. This is because we aligned the clusters along the z-axis. For valence neutrons, the distribution is much higher around 90° compared to nucleons in the clusters, as they mostly move around the neck of the two clusters. For nucleons in the $\pi$-orbital in ¹⁰Be, while a higher probability is observed towards 90 degrees, a certain probability can also be seen at small angles. This is related to how the one-body density is constructed from the fully correlated many-nucleon distributions. After grouping the closest two protons and two neutrons, one cluster is rotated to the z-axis. With this procedure, the valence neutron has nonvanishing probability to locate near the z-axis. In panel (b), the difference between ¹⁰Be and ¹¹Be is also plotted. It can be seen that the extra valence neutron in ¹¹Be leads to a higher probability around 0° near the top of the cluster region. Overall, this extra valence neutron is distributed rather smoothly in all angles, showing both the center part of the $\sigma$ orbital and the s-wave character of a halo.

4. Conclusions and Perspectives

In this work, we investigated the structure of the paradigmatic halo nucleus ¹¹Be using ab initio NLEFT calculations with high-precision N3LO chiral interactions. By leveraging the wavefunction matching method, we overcame the sign problem and obtained a good description of the binding energies and radii for both ¹⁰Be and ¹¹Be, properly reproducing the well-known parity inversion in the ¹¹Be ground state.

Using the pinhole algorithm, we computed the intrinsic 3D density distributions and radial profiles, providing a microscopic view of the internal structure. We observed well-developed cluster structures in both isotopes. A detailed comparison of the angular and radial distributions between ¹⁰Be and ¹¹Be demonstrates that the halo formation in ¹¹Be is driven by the valence neutron occupying a $\sigma$-type molecular orbital. This configuration results in a more prolate deformation and a significant extension of the neutron density at large distances compared to the ¹⁰Be core, where valence neutrons occupy $\pi$-type orbitals.

Future studies should apply this framework to explore other exotic systems near the driplines. The ability to successfully describe the clustering and halo correlations using this lattice EFT approach opens new avenues for understanding the emergence of nuclear molecular structures from first principles.