Microscopic cluster study of the ${}^{10}{\rm Be}$ and ${}^{11}{\rm Be}$ nuclei

Nuclear clustering is a well-established phenomenon. In particular, the $\alpha$ particle, due to its large binding energy, is a typical cluster in light nuclei. About 50 years ago, the seminal paper of Ikeda and his collaborators ITH68 showed that $\alpha$-clustering is expected near the $\alpha$ threshold of $4N$ nuclei. This conjecture leads to the famous Ikeda diagram and was remarkably confirmed by theory and by experiment. It was even extended to $N\neq Z$ nuclei (see recent reviews in Refs. HIK12 ; DD12 ; KSK16 ; KH18 . For example, $\alpha+^{14}$C cluster states are known in ¹⁸O for a long time DB85 .

Neutron-rich nuclei, in particular in the low-mass region of the nuclear chart, require specific attention. If $\alpha$ clustering is expected, the role of the external neutrons should be addressed by specific methods. A typical example is ⁶He, which requires three-body models to accurately describe the halo neutrons. Beryllium isotopes are particularly interesting: ⁸Be is the archetype of $\alpha$-cluster nuclei. Although unstable, the ground state is well known to have a marked $\alpha+\alpha$ cluster structure. Going to heavier Be isotopes require multicluster approaches, where the $\alpha+\alpha$ structure persists, but where the additional neutrons play a role. Multicluster descriptions have been proposed in the past within the Generator Coordinate Method De02 , the Antisymmetrized Molecular Dynamics KH01 ; Ka02 , the molecular model IO00 ; IOI00 ; IHO02 ; It06 ; IIM08 ; IIS08 or the Resonating Group Method Ar04 ; OAS00 . An experimental review of $Z=2-4$ neutron-rich isotopes can be found in Ref. Fo18 .

In the present work, we aim to investigate the ${}^{10}{\rm Be}$ and ${}^{11}{\rm Be}$ isotopes within the $\alpha+\alpha+n+n$ and $\alpha+\alpha+n+n+n$ Generator Coordinate Method (GCM). A previous study on ⁹Be DI18 within this microscopic approach shows that the $\alpha+\alpha+n$ model is able to reproduce many ⁹Be properties. The main issue in ${}^{10}{\rm Be}$ and ${}^{11}{\rm Be}$ is a large number of independent coordinates. In other words, accurate bases require large numbers of functions. This problem can be efficiently addressed by the Stochastic Variational Method (SVM), where a random choice of the basis sets is performed, which permits optimizing the basis KK77 ; VS95 . A recent application of the SVM to ⁶Li, considered as a six-body system, has been performed in Ref. SH19 .

The paper is organized as follows. In sect. II, we briefly present the microscopic model, and provide some detail about our use of the SVM. Sections III and IV are devoted to the ${}^{10}{\rm Be}$ and ${}^{11}{\rm Be}$ nuclei, respectively. Concluding remarks and outlook are presented in sect. V.

The energy convergence is illustrated in Fig. 2 for several $J\pi$ values. We use the Minnesota parameter $u=0.973$ which reproduces the experimental binding energy with respect to the $\alpha+\alpha+n+n$ threshold ($-8.64$ MeV). Experimental states are shown on the right part of the figure. The calculation predicts a $2^{+}$ excitation energy in excellent agreement with the experiment, although a $2^{+}_{2}$ state is found below the experimental energy. The convergence is reasonable with about 400 basis functions. We find $0^{+}_{2}$ and $1^{-}$ states whose energies differ by $1-2$ MeV from the experiment. The $0^{+}_{2}$ state is known to have an $\alpha+^{6}$He cluster structure FBD99 , and its theoretical energy might be slightly improved by constraining the random selection to such configurations. For the negative-parity states, a different $u$ value would permit to reproduce more precisely the experimental energy.

The ground-state proton and neutron densities of the ${}^{10}{\rm Be}$ ground state are presented in Fig. 3. The transition density between an initial state $i$ and a final state $f$ is defined as

where $\boldsymbol{t}_{i}$ is the isposin of nucleon $i$, and the signs ”+” and ”-” correspond to the neutron and proton densities, respectively. These densities are expanded in multipoles Ka81 as

and are computed as explained in Ref. DI18 . The monopole ($\lambda=0$) densities of the ground state are normalized such that

where $Z=4$ and $N=6$ for the ${}^{10}{\rm Be}$ nucleus.

As expected, the neutron density extends to larger distances, owing to the presence of the external neutrons. From these densities, we obtain the rms radii presented in Table 1, and compared with the experimental data of Ref. THH85 . The proton radius $\sqrt{<r^{2}>_{p}}$ is an excellent agreement with experiment, but the matter radius $\sqrt{<r^{2}>}$ is slightly larger. Notice that the experimental values are partly model dependent.

With the GCM wave functions, we can also compute the E2 transition probability. Our value is lower than experiment, which suggests that an effective charge is necessary. This is not surprising in neutron-rich nuclei, where polarization effects usually require an effective charge which simulates neutron effects.

To have a deeper insight on the ${}^{10}{\rm Be}$ structure, we have investigated energy curves, where one of the generator coordinate is fixed. The energy curves are presented in Fig. 4 for $R_{\alpha\alpha},R_{\alpha n}$ and $R_{nn}$. Figure 4(a) shows that the minimum of the energy is obtained for $R_{\alpha\alpha}\approx 3$ fm, which is lower than in ⁹Be ($\approx 4$ fm) but still significant. The $\alpha$-cluster structure is stronger for $J=1^{-}$, in agreement with the $\alpha+^{6}$He configuration suggested in Ref. FBD99 . For $R_{\alpha n}$, the minimum is found near $R_{\alpha n}\approx 2$ fm. For $R_{nn}$, however, the energy is minimum near $R_{nn}\approx 3$ fm. This result stresses the importance of a 4-body model for ${}^{10}{\rm Be}$. A simple dineutron approximation for the external neutrons would not provide accurate wave functions.

The ${}^{11}{\rm Be}$ nucleus has attracted much interest over the last decades, owing to the well-known parity inversion TU60 and to the low binding energy of the ground state. This property makes ${}^{11}{\rm Be}$ an ideal example of a one-neutron halo nucleus. Many microscopic studies have been devoted to ${}^{11}{\rm Be}$: the GCM De97 ; De02 , the AMD KHD01 and, more recently the No Core Shell Model CNR16 where it is shown an explicit treatment of the ${}^{10}{\rm Be}+n$ cluster structure is necessary to reproduce the large $B(E1)$ transition probability between the $1/2^{+}$ ground state and the $1/2^{-}$ first excited state. In most models, however, the parity inversion cannot be reproduced with a common interaction. A parity-dependent interaction must be adopted.

In the present work, we aim to investigate the ${}^{11}{\rm Be}$ structure in the framework of a multicluster approach. An improvement with respect to Ref. De02 is the use of a more efficient method to select the optimal basis, and therefore to get more precise properties of ${}^{11}{\rm Be}$. As mentioned in the introduction, the GCM description of ${}^{11}{\rm Be}$ involves 9 generator coordinates, and the use of the SVM turns out to be quite useful to keep the basis within reasonable sizes.

We first illustrate the energy convergence of various states in Fig. 5. The admixture parameter $u$ has been adjusted to the experimental neutron separation energy. With $u=1.066$ for positive parity and $u=0.893$ for negative parity, we reproduce the energy of the $1/2^{+}$ and $1/2^{-}$ states ($-0.50$ MeV and $-0.18$ MeV, respectively, with respect to the ${}^{10}{\rm Be}+n$ threshold). Reproducing the experimental binding energies is crucial for the asymptotic part of the wave functions. Figure 5 shows that a fair convergence can be achieved with about $600-700$ basis functions. A similar number of basis functions has been employed for the 6-nucleon description of ⁶Li SH19 . The model not only provides the ground state and the first excited state, but a realistic description of low-lying resonances is also obtained. For these resonances, the energies are in reasonable agreement with the experiment.

Figure 6 presents the ${}^{11}{\rm Be}$ proton and neutron densities for the $1/2^{+}$ and $1/2^{-}$ states. For both states, the proton density is rather peaked near the origin. In contrast, the neutron densities extend to large distances. This is a well-known effect, due to the weak binding energy of the last neutron. The rms radii, obtained from the densities, are displayed in Table 2. The proton and matter radii of the ground state are smaller than the experimental values, a result consistent with the previous study of Ref. De02 . Most likely, other configurations are necessary to improve the comparison with experiment. The $B(E1)$ value is also underestimated by the GCM. This was already observed in previous multicluster calculations KHD01 ; De02 , and in the NCSM CNR16 . The authors of Ref. CNR16 suggest that an explicit account of ${}^{10}{\rm Be}+n$ configurations is necessary to reproduce the large experimental value.

In Fig. 7, we analyse various energy curves of ${}^{11}{\rm Be}$. In each case, a generator coordinate is kept fixed, and the SVM is applied to the eight remaining generator coordinates. Although these curves cannot be strictly considered as potentials, they provide a valuable insight into the structure of ${}^{11}{\rm Be}$. The minimum for $R_{\alpha\alpha}$ is found near $R_{\alpha\alpha}\approx 2$ fm, i.e. a value smaller than in ${}^{10}{\rm Be}$. This result confirms that the $\alpha+\alpha$ distance decreases when the number of external nucleon increases De02 . For $R_{\alpha n}$, the minimum is rather flat up to $R_{\alpha n}\approx 2$ fm. Large values are therefore unlikely. The $R_{nn}$ dependence is quite interesting: it shows a weak variation of the total energy. Consequently, it is important to include several configurations covering a wide interval. This conclusion holds for all considered states. The dependence on $R_{\alpha n2}$, i.e. on the distance between the $\alpha+\alpha$ c.m. and the third neutron, presents a flat minimum around $R_{\alpha n}\approx$ 2 fm.

The main goal of this paper is to investigate the ${}^{10}{\rm Be}$ and ${}^{11}{\rm Be}$ nuclei within a microscopic multicluster model. The only adjustable parameter is the admixture parameter $u$, involved in the Minnesota interaction, and fitted on the binding energy of the ground states. A challenge with many-body approaches is to cope with a large number of generator coordinates or, in other words, with a large number of degrees of freedom. We have confirmed that the SVM provides an excellent framework to address this issue. Although computer times are still quite long, they remain within reasonable limits on modern computers.

The multicluster model is based on two $\alpha$ clusters, and 2 or 3 surrounding neutrons. It provides an excellent description of the low-energy spectrum of both nuclei. In particular, ${}^{11}{\rm Be}$ is nicely reproduced, not only for bound states but also for resonances. The stability of the energies with the number of basis functions (see Fig. 5) shows that we also have a fair description of the continuum.

We have used the GCM wave functions to compute various properties. In particular, the ${}^{10}{\rm Be}$ and ${}^{11}{\rm Be}$ densities could be used to determine folding potentials. In ${}^{10}{\rm Be}$, the rms radii are in good agreement with the experiment. A small effective charge should be introduced to improve the agreement for the $B(E2)$ value. The rms radii in ${}^{11}{\rm Be}$, however, are somewhat underestimated, and the $B(E1)$ value is much smaller than the experiment. This is not surprising in a multicluster model De02 ; KHD01 , and even in the NCSM CNR16 . As suggested in Ref. CNR16 , couplings to ${}^{10}{\rm Be}+n$ configurations should be introduced explicitly.

Finally, we have analysed the structure of ${}^{10}{\rm Be}$ and ${}^{11}{\rm Be}$ with the energy curves, where one of the generator coordinates is fixed. This approach provides a qualitative overview of the nucleus. The $\alpha+\alpha$ clustering decreases from ${}^{10}{\rm Be}$ to ${}^{11}{\rm Be}$. It should likely disappear for heavier Be isotopes, such as ¹⁴Be.

P. D. acknowledges the hospitality of the Yukawa Institute for Theoretical Physics, where most of this work was performed. This work was supported by the Fonds de la Recherche Scientifique - FNRS under Grant Numbers 4.45.10.08 and J.0049.19. Computational resources have been provided by the Consortium des Équipements de Calcul Intensif (CÉCI), funded by the Fonds de la Recherche Scientifique de Belgique (F.R.S.-FNRS) under Grant No. 2.5020.11 and by the Walloon Region. P. D. is Directeur de Recherches of F.R.S.-FNRS, Belgium.