Abstract
The structure of the unbound nucleus 13Be is important for understanding the Borromean, two-neutron halo nucleus 14Be. The experimental studies conducted over the last four decades are reviewed in the context of the beryllium chain of isotopes and some significant theoretical studies. The focus of this paper is the comparison of new data from a 12Be(d,p) reaction in inverse kinematics, which was analyzed using Geant4 simulations and a Bayesian fitting procedure, with previous measurements. Two possible scenarios to explain the strength below 1 MeV above the neutron separation energy were proposed in that study: a single p-wave resonance or a mixture of an s-wave virtual state with a weaker p- or d-wave resonance. Comparisons of recent invariant mass and the (d,p) experiments show good agreement between the transfer measurement and the two most recent high-energy nucleon removal measurements.
1 Introduction
With just four protons, the particle-bound members of the beryllium chain of isotopes stretch from 7Be (N/Z = 0.75) on the proton-rich side of stability to 14Be (N/Z = 2.5), the two-neutron halo on the neutron-rich side. Adding a single neutron to 7Be results in the 8Be system of two α particles which is unbound by only 92 keV. Adding a second neutron produces 9Be, the only beryllium isotope that is stable against β decay. This stability is a product of two phenomena that occur across the beryllium chain of isotopes, molecular structures [1]; [2]; [3], and core excitation [4]. The molecular structure is closely connected to the Borromean nature of 9Be; the three-body system of α − α − n is bound despite the two-body subsystems, α − n and α − α, being unbound. The delocalized neutron in 9Be can be viewed as being exchanged between α particles [5]. Analogous to atomic molecules, the neutron in the ground state of 9Be is well understood as being in a π-type orbital, whereas the first excited state may be better described with a neutron in the σ-type orbital. These molecular orbits are intimately related to the prolate nature of the two-α cluster structure [6].
The cluster structure appears to weaken in 10Be, as evidenced by the reduced size of its charge radius [7]; [8]. The two-α plus two-neutron structure is more apparent in excited states closer to the α and neutron separation energies, such as the isomeric second 0+ state at Ex = 6.179 MeV, which has a small γ branch to the more compact first 2+ state [5].
Adding another neutron makes the archetypal one-neutron halo nucleus 11Be. A 16% core-excited component in the ground state of 11Be was required to reproduce the results from the 11Be(p,d)10Be reaction [9]. This is less than that of 9Be, where the core-excited component was calculated as approximately half of the ground-state wave function [4]. Additionally, dynamical core excitation needs to be included in the calculations of both transfer [10] and breakup [11] reactions.
The parity inversion in 11Be, where the 1/2+ ground state decreases 320 keV below the only other bound state with Jπ = 1/2−, along with larger collectivity in 10Be, led to the questioning about the robustness of the N = 8 shell closure at 12Be. Using a three-body model with core excitation, Nunes et al. were able to show an increased sphericity in the core 10Be within 12Be compared to that observed in 11Be [12]. This in turn led to greater mixing between p- and sd-shell valence neutron states and a melting of the N = 8 shell closure. The coupling of a d-wave neutron with the excited 2+10Be core severely restricts the formation of a halo in 12Be [13]. Notably, 12Be is not Borromean, as the n-10Be system is bound but is still well described by three-body models.
The N = 10 isotope 14Be presents the heaviest particle-stable beryllium isotope. The naive shell model would predict a d5/2-dominated ground-state wave function for 14Be. However, with the level inversion seen in the other neutron-rich beryllium isotopes, some low-lying s1/2 strength is expected. With two neutrons in the halo and with significant ℓ > 0 components of the wave function, the halo of 14Be is much more contained than that of 11Be, despite being closer to the drip line. It would seem natural to study 14Be in a three-body model, with 12Be as the core and two valence neutrons [14]; [15]. Thompson and Zhukov found that adding an s-wave virtual state below the well-known state bound 13Be, in contradiction to its non-observance in fragmentation reactions. Reducing the energy, i.e., increasing the scattering length, of the virtual state resulted in the binding energy of 14Be being too low. The three-body approach of Descouvemont found that only 66% of the ground-state wave function of 14Be could be described as 12Be +n + n [14]. Labiche et al. [16], using the model of Vinh Mau and Pacheco [17], found that assuming a ground state for 13Be, consistent with the melting of the N = 8 shell closure observed in 11Be and 10Li, could reproduce the measured properties of 14Be.
Beyond the neutron drip line, 15Be has been observed to decay to 12Be through unbound states in 14Be [18]. The last isotope to be observed is 16Be, which is bound with respect to one neutron emission, but unbound to the emission of two neutrons [19]. The two neutrons from the decay were observed in a small emission angle.
2 The unbound nucleus 13Be
Theoretical studies of 13Be have used the shell model [20] or a potential model [21], the Nilsson model [22], microscopic cluster models [23], antisymmetrized molecular dynamics [24], and relativistic mean-field theory [25]. As shown in Table 1, with the exception of the earliest work, the calculations agree on the existence of a 5/2+ resonance between 2 and 2.5 MeV above the neutron threshold. However, the location of the ground state relative to the neutron threshold is disputed, as is the parity of the ground state. There have also been reaction theory studies of, and comparing to, experimental data, for example, the work of Bonaccorso [26] and references therein. Casal et al. [27] used a transfer to the continuum model including deformation in the 12Be + n potential, following the prescription in Thompson et al. [28], to interpret the data obtained from Corsi et al. [29]. This work indicates a p-wave resonance at between 0.4 and 0.5 MeV above the threshold.
TABLE 1
| Energy above the neutron threshold (MeV) | ||
|---|---|---|
| Author (year) | Ground state | Strength 2 |
| Poppelier et al. [20] | 1.16 a | 2.44 |
| Lenske Ostrowski et al. [30] | 0.9 | 2.3 and 2.45 |
| Descouvemont [23] v2 | −0.009 | 2.02 |
| Descouvemont [23] v4 | −0.038 | 2.05 |
| Fortune [21] | 0.86 | 2.11 |
Energy (and Jπ assignments) of low-lying states in 13Be according to a selection of theoretical studies.
Poppelier also calculates a state at 1.21 MeV.
Strength 2 refers to any virtual states or resonances approximately 2 MeV above the neutron threshold.
There have been many experiments on 13Be since the first discovery of a resonance [31] at 1.8 MeV above the neutron threshold. Some of the experimental results from the last four decades are shown in Table 2. The reactions used to probe the structure of halo nuclei can be broadly divided into missing mass and invariant mass techniques. Transfer reactions, where excitation energies are found from the reaction Q-values, fall into the missing mass category. Knockout, breakup, and Coulomb dissociation, where the final state is reconstructed from two or more fragments, represent invariant mass techniques. In the case of 13Be, the fragments are 12Be and a neutron. The early measurements, e.g., [31]; [30]; [32]; and [33], mostly populated 13Be through multinucleon transfer reactions. An exception is the 12Be(d,p) experiment performed in [34] at RIKEN. At a beam energy of 55 AMeV, the conditions are not well matched to observe low ℓ transfer, and the carbon in the target largely masked any structure below 2 MeV. Indeed, none of the experiments before the fragmentation experiment at the National Superconducting Cyclotron Laboratory [35] revealed any structure below the resonance at 2 MeV.
TABLE 2
| Author (year) | Reaction | Energy above the threshold (MeV) or (as) and Jπ | ||
|---|---|---|---|---|
| Ground state (strength 1a) | Strength 1b | Strength 2 | ||
| Aleksandrov et al. [31] | 14C + 7Li | — | — | 1.8 |
| Ostrowski et al. [30] | 13C + 14C | — | ||
| Korsheninnikov et al. [34] | 12Be + d | — | — | 2.0 |
| von Oertzen et al. [32] | 13C + 14C | — | — | |
| Belozyorov et al. [33] | 14C + 11B | — | 0.80 | 2.02 |
| Thoennessen et al. [35] | 9Be + 18O | |||
| Lecouey [36] | 14B + C | — | ||
| Simon et al. [37] | 14Be + C | — | ||
| Kondo et al. [38] | 14Be + p | |||
| Aksyutina et al. [39] | 14Be + p | — | ||
| Randisi et al. [40] | 14,15B + natC | |||
| Marks et al. [41] | 13B + 9Be | — | ||
| Ribeiro et al. [42] | 14B + CH2 | |||
| Corsi et al. [29] | 14Be + p | |||
| Kovoor et al. [43] single | 12Be + solidD | — | ||
| Kovoor et al. [43] mix | 12Be + solidD | anda | ||
Previous studies of the low-lying structure of 13Be, up to approximately 2.5 MeV above the neutron threshold.
Order undetermined (see text).
Strength is defined as a single resonance, or a virtual state, or a mixture of resonances, or a virtual state and a resonance. All experiments found some strength below 0.5 MeV above the neutron threshold, and that is labeled as the ground state in this table. Strength 1b is approximately 0.5–0.9 MeV above the neutron threshold. Strength 2 is the well-known d-wave resonance at approximately 2 MeV above the neutron threshold, where the literature reports scattering length instead of energy which is quoted in parentheses.
Most of the more recent experiments used invariant mass techniques at energies between 40 and 400 AMeV. Of these, all except one employed nucleon removal, either a neutron from 14Be [37]; [38]; [39]; and [29] or a proton from 14B [36]; [40]; and [42]. In [41], a nucleon exchange reaction was used to populate unbound states in 13Be from a 13B beam. Spectra from invariant mass methods contain information relating to the initial state of the target (the beam in an inverse kinematics reaction) and reflect final state interactions between outgoing fragments. There can also be complications relating to the reaction mechanism, which can be diffractive or absorptive.
3 Recent transfer reaction measurement
The most recent experiment, [44]; [43], used a 12Be beam from ISAC-II and the solid deuterium target and detector system IRIS [45] to perform a one-neutron transfer reaction. The advantages of using the (d,p) reaction at low energy (Ebeam = 9.5 AMeV) are that the energy and angular momentum-matching conditions are ideal for populating low-ℓ, near-threshold states. The non-Gaussian experimental response caused by increasingly poor resolution for lower-energy protons exiting the solid deuterium target necessitated an analysis technique that included both simulation and Bayesian fitting methods, as demonstrated in [46]. The initial analysis included the resonance at approximately 2 MeV and a single s-, p-, or d-wave virtual state or resonance closer to the threshold, referenced as “single” in Table 2. Additionally, mixes of two out of s-, p-, and d-wave strengths were included to allow for two resonances or a resonance and a virtual state below the 2 MeV d-wave resonance, referenced as “mixed” in Table 2. The fits with the lowest χ2/NDF for the region below 1 MeV in resonance energy, for both the single and mixed cases, are the ones shown in Table 2. The rest of this paper relates to the findings in [43], including the comparisons with recent invariant mass measurements [38]; [40]; [42]; [29].
A comparison of the transfer measurement (for the case assuming a single state below 2 MeV) with some recent invariant mass measurement results is shown in Figure 1. Discrepancies clearly persist in the low-lying structure of 13Be, even between recent measurements. The state is in a similar position in all the measurements; however, the width is much larger according to the works in [38] and [40]. Similarly, in [40] and [42], a broad structure below 1 MeV was shown, whereas [38] and [29] agree on an s-wave virtual state. There is some consistency in the presence of a p-wave resonance of approximately 0.5 MeV, with the exception of [40]. This resonance is noticeably narrower (Γ = 0.11 MeV) in the work in [43].
FIGURE 1
The mixed case in [43] is not shown in Figure 1 as the ordering of the s- and p-wave strength cannot be extracted from the data, only the relative intensities. The lowest χ2/NDF was found for a mixture of and wave strengths. It should be noticed that the χ2/NDF (3.87) for the and wave mixture below 1 MeV is similar to that for the s- and p-wave mixture (3.32). Therefore, the initial conclusion of the study was that the near-threshold strength was either a pure p wave or a mixture dominated by an s-wave strength, with a weaker resonance of either a p- or d-wave nature.
To make a more robust comparison between the recent invariant-mass measurements and the transfer reaction, the resonance parameters extracted from [38], [29], [40], and [42] were used as inputs for the Geant4 [47] simulation, as shown in Figure 2. The centroid energies and widths from the analyses were used, and the relative intensities of the resonances and virtual states were fitted as free parameters. A relatively poor fit to the data from the 12Be(d,p) reaction experiment was produced from the simulations using resonance parameters from [38] (χ2/NDF = 5.51). Using the parameters from [40] provided a better fit (χ2/NDF = 3.47), and those from [39] (not shown) provided a fit with χ2/NDF = 3.18. The parameters that provided the best fit from the literature were those from [29] (χ2/NDF = 2.62), closely followed by those from [42] (χ2/NDF = 2.76). These can all be compared to the Bayesian fit of the data with a single p-wave resonance along with the well-known d-wave resonance of approximately 2 MeV. This fit allowed the locations and widths of the two resonances to vary along with the intensities. The parameters shown in Table 2 resulted in a χ2/NDF of 2.02. The single p wave below the 2 MeV resonance is the scenario that best agrees with these data. This dominance of p-wave strength near the particle threshold is in agreement with the results of [27].
FIGURE 2
4 Summary
The beryllium chain of isotopes displays various clustering phenomena including molecular structures and one- and two-neutron halos in 11Be and 14Be, respectively. The isotope 13Be is an unbound subsystem of the Borromean nucleus 14Be. Its structure has been investigated experimentally for 40 years using both missing mass and invariant mass techniques. However, with the exception of the 2 MeV resonance, the low-lying structure is still disputed. A new single-neutron transfer reaction experiment has brought new data and a new analysis technique involving Geant4 simulations and a Bayesian fitting routine. The best fit of the Q-value data was obtained with a narrow 0.55 MeV p-wave resonance and the d-wave resonance located at 2.22 MeV. Adding either a virtual state or a second resonance below 1 MeV produced somewhat poorer fits (χ2/NDF = 3.0–3.4) that were dominated by the s-wave contribution (61%–89%) with a small p-wave (39%) or d-wave (11%) resonance. Using the resonance parameters from either [29] or [42] produced better fits to the new data from the literature. The literature resonance parameters producing poorer fits were those with a broader resonance [38]; [40].
Statements
Author contributions
KJ was the spokesperson for the new 12Be + 2H experiment and wrote the first draft of this manuscript. JK analyzed the data and produced the figures from the new 12Be + 2H experiment. RK was the co-spokesperson for the new 12Be + 2H experiment and was instrumental in the design and running of the experiment. All authors contributed to the article and approved the submitted version.
Funding
This research was supported by the U.S. Department of Energy, the Office of Science, the Office of Nuclear Physics under contract Nos DE-FG02-96ER40963 (UTK). The authors are grateful for the support from the NSERC, Canada Foundation for Innovation and Nova Scotia Research and Innovation Trust, and RCNP, the grant-in-aid program of the Japanese government. TRIUMF is supported by a contribution from the National Research Council, Canada.
Acknowledgments
The authors would like to acknowledge the collaborators on the TRIUMF 12Be + 2H experiment and the beam delivery team for providing the 12Be beam, and DE-AC05-00OR22725 (ORNL), and the U. S. National Science Foundation under Award Numbers PHY-1404218 (Rutgers) and PHY-2011890 (Notre Dame). This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) grant nos 2020R1A2C1005981 and 2016R1A5A1013277. This work was partially supported by STFC grant no. ST/L005743/1 (Surrey).
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
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Summary
Keywords
direct reactions, clustering, beryllium, 13Be, 12Be
Citation
Jones KL, Kovoor J and Kanungo R (2023) Status of experimental knowledge on the unbound nucleus 13Be. Front. Phys. 11:1242668. doi: 10.3389/fphy.2023.1242668
Received
19 June 2023
Accepted
28 August 2023
Published
15 September 2023
Volume
11 - 2023
Edited by
Valdir Guimaraes, University of São Paulo, Brazil
Reviewed by
Zilong Chang, Indiana University, United States
Magda Cicerchia, University of Padua, Italy
Updates
Copyright
© 2023 Jones, Kovoor and Kanungo.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: K. L. Jones, kgrzywac@utk.edu
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.