Status of experimental knowledge on the unbound nucleus $^{13}$Be

The structure of the unbound nucleus $^{13}$Be is important to understanding the Borromean, two-neutron halo nucleus $^{14}$Be. 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. One focus of this paper is the comparison of new data from a $^{12}$Be(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 either $p$-wave 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.


INTRODUCTION
With just four protons, the particle-bound members of the beryllium chain of isotopes stretch from 7 Be (N/Z = 0.75) on the proton-rich side of stability, to 14 Be (N/Z = 2.5), the two-neutron halo on the neutron-rich side.Adding a single neutron to 7 Be results in the 8 Be system of two α particles, which is unbound by only 92 keV.Adding a second neutron produces 9 Be, 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 von Oertzen (1996Oertzen ( , 1997)); Seya et al. (1981), and core excitation Parfenova and Leclercq-Willain (2005).The molecular structure is closely connected to the Borromean nature of 9 Be; the three-body system of α − α − n is bound despite both of the two-body subsystems, α − n, and α − α being unbound.The delocalized neutron in 9 Be can be viewed as being exchanged between the α particles Freer et al. (2018).The neutron in the ground state of 9 Be is well understood as being in a π-type orbital, in analogy with atomic molecules, 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 Canavan and Freer (2020).
The cluster structure appears to weaken in 10 Be, as evidenced by the reduced size of its charge radius Nörtershäuser et al. (2009); Krieger et al. (2012).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 E x = 6.179MeV, which has a small γ branch to the more compact first 2 + state Freer et al. (2018).
Adding another neutron makes the archetypal one-neutron halo nucleus 11 Be.A 16% core-excited component in the ground state of 11 Be was required to reproduce the results from the 11 Be(p,d) 10 Be reaction Fortier et al. (1999).This is much less than for 9 Be, where the core excited component was calculated as around half of the ground state wave function Parfenova and Leclercq-Willain (2005).Additionally, dynamical core excitation needs to be included in calculations of both transfer Deltuva et al. (2016) and break up Moro and Crespo (2012) reactions.
The parity inversion in 11 Be, where the 1/2 + ground state dips 320 keV below the only other bound state with J π = 1/2 − , along with the large collectivity in 10 Be, led to questions about the robustness of the N = 8 shell closure at 12 Be.Using a three-body model with core excitation, Nunes et al were able to show an increased sphericity in the core 10 Be within 12 Be compared that seen in 11 Be Nunes et al. (2002).This in turn led to greater mixing between the 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 + 10 Be core severely restricts the formation of a halo in 12 Be Nunes (2005).Notably, 12 Be is not Borromean, as the n-10 Be system is bound but is still well described by three-body models.
The N = 10 isotope 14 Be presents the heaviest particle-stable beryllium isotope.The naive shell model would predict a d 5/2 dominated ground state wave function for 14 Be.However, with the level inversion seen in the other neutron-rich beryllium isotopes, some low-lying s 1/2 strength is expected.With two neutrons in the halo, and with significant ℓ > 0 components of the wave function, the halo of 14 Be is much more contained than that of 11 Be, despite being closer to the drip line.It would seem natural to study 14 Be in a three-body model, with 12 Be as a core and two valence neutrons Descouvemont (1995); Thompson and Zhukov (1996).Thompson and Zhukov found that adding an s-wave virtual state below the well know the energy, i.e. increasing the scattering length, of the virtual state resulted in the binding energy of 14 Be being too low.The three-body approach of Descouvemont found that only 66% of the ground state wave function of 14 Be could be described as 12 Be +n + n Descouvemont (1995).Labiche et al Labiche et al. (1999), using the model of Vinh Mau and Pacheco Vinh Mau and Pacheco (1996) found that assuming a 1 2 − ground state for 13 Be, consistent with the melting of the N = 8 shell closure seen in 11 Be and 10 Li could reproduce the measured properties of 14 Be.
Beyond the neutron drip line, 15 Be has been observed to decay to 12 Be through unbound states in 14 Be Spyrou et al. (2011).The last isotope to be observed is 16 Be, which is bound with respect to one neutron emission, but unbound to the emission of two neutrons Spyrou et al. (2012).The two neutrons from the decay were observed in a small emission angle.

THE UNBOUND NUCLEUS 13 Be
Theoretical studies of 13 Be have used the shell model Poppelier et al. (1985) or a potential model Fortune ( 2019), the Nilsson model Macchiavelli et al. (2018), microscopic cluster models Descouvemont (1994), antisymmetrized molecular dynamics Kanada-En'yo (2012), and relativistic mean field theory Ren et al. (1997).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 MeV 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, Bonaccorso (2012) and references therein.Casal et al. (2020), used a transfer to the continuum model including deformation in the 12 Be+n potential, following the prescription of Thompson et al. Thompson et al. (2004), to interpret the data from Corsi et al.Corsi et al. (2019).This work indicates a p-wave resonance at between 0.4 and 0.5 MeV above the threshold.
There have been many experiments on 13 Be since the first discovery of a resonance Aleksandrov et al. (1983) at 1.8 MeV above the neutron threshold.Some of the experimental results from the last four decades are collated 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 13 Be the fragments are 12 Be and a neutron.The early measurements, e.g.Aleksandrov et al. (1983); Ostrowski et al. (1992) Ribeiro et al. (2018).Marks et al. (2015) used a nucleon exchange reaction to populate unbound states in 13 Be from a 13 B 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.
Table 2. Previous studies of the low-lying structure of 13 Be, up to around 2.5 MeV above the neutron threshold.A 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 Ground State in this table.Strength 1b is around 0.5 -0.9 MeV above the neutron threshold.Strength 2 is the well-known d-wave resonance at around 2 MeV above the neutron threshold.Where the literature reports scattering length instead of energy this is quoted in parentheses. Author

RECENT TRANSFER REACTION MEASUREMENT
The most recent experiment, Kovoor (2022); Kovoor et al. (2023), used a 12 Be beam from ISAC-II, and the solid deuterium target and detector system IRIS Kanungo (2014) to perform a one-neutron transfer reaction.The advantages of using the (d,p) reaction at low energy (E beam = 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 Hooker et al. (2022).The initial analysis included the  2. The fit with the lowest χ 2 /NDF for the region below 1 MeV in resonance energy, for both the single and the mixed case, are the ones quoted in Table 2.The rest of this paper relates to the findings in Kovoor et al. (2023)   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 13 Be, even between recent measurements.The 5 2 + state is in a similar position in all the measurements, however, the width is much larger according to the works of Kondo et al. (2010) and Randisi et al. (2014).Similarly, Randisi et al. (2014) and Ribeiro et al. (2018) show a broad 1 2 + structure below 1 MeV, whereas Kondo et al. (2010) and Corsi et al. (2019)

et al. agree on an s-wave virtual state.
There is some consistency in the presence of a p-wave resonance around 0.5 MeV, with the exception of Randisi et al. (2014).This resonance is noticeably narrower (Γ = 0.11 MeV) in the work of Kovoor et al. (2023).
The mixed case in Kovoor et al. (2023) is not shown in Figure 1 as the ordering of the sand p-wave strength cannot be extracted from the data, only the relative intensities.The lowest χ 2 /NDF was found for a mixture of s − 70 +8 −6 % and p − 30 +6 −8 % wave strengths.It should be noticed that the χ 2 /NDF (3.87) for the s − 93 +2 −2 % and d − 7 +2 −2 % wave mixture below 1 MeV is similar to that for the sand p-wave mixture (3.32).Therefore, the initial conclusions of the study were that the near-threshold strength was either pure p-wave or a mixture, dominated by an s-wave strength, with a weaker resonance of either a por d-wave nature.
To make a more robust comparison between the recent invariant-mass measurements and the transfer reaction, the resonance parameters extracted from Kondo et al. (2010), Corsi et al. (2019), Randisi et al. (2014), andRibeiro et al. (2018) were used as inputs for the GEANT4 Agostinelli et al. (2003) 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 12 Be(d,p) reaction experiment was produced from the simulations using resonance parameters from Kondo et al. (2010) 2 resulted in a χ 2 /NDF of 2.02.The single p-wave below the 2 MeV 5 2 + 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 Casal et al. (2020).

SUMMARY
The beryllium chain of isotopes displays various clustering phenomena including molecular structures and one-and two-neutron halos in 11 Be and 14 Be respectively.The isotope 13 Be is an unbound sub-system of the Borromean nucleus 14 Be.Its structure has been investigated experimentally for forty years using both missing mass and invariant mass techniques.However, with the exception of the 2 MeV 5 2 + 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 either p-(39%) or d-wave (11%) resonance.Using the resonance parameters from either Corsi et al. (2019) or Ribeiro et al. (2018) produced better fits to the new data from the literature.The literature resonance parameters producing the poorer fits were those with a broader 5 ;von Oertzen et al. (1995);Belozyorov et al. (1998) mostly populated13 Be through multinucleon transfer reactions.An exception is the 12 Be(d,p) experiment performed byKorsheninnikov et al. (1995) 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 LaboratoryThoennessen et al. (2000) revealed any structure below the 5 2 + resonance at 2 MeV.Most of the more recent experiments used invariant mass techniques at energies between 40 AMeV and 400 AMeV.Of these, all except one employed nucleon removal, either a neutron from 14 Be Simon et al. (2007); Kondo et al. (2010); Aksyutina et al. (2013); Corsi et al. (2019) or a proton from 14 B Lecouey (2004); Randisi et al. (2014); including the comparisons with recent invariant mass measurements Kondo et al. (2010); Randisi et al. (2014); Ribeiro et al. (2018); Corsi et al. (2019).

Figure 1 .
Figure 1.Comparison of low-lying structure in 13 Be according to recent invariant mass measurements Kondo et al. (2010); Randisi et al. (2014); Ribeiro et al. (2018); Corsi et al. (2019) and a recent transfer reaction measurementKovoor et al. (2023).Only the case with a pure p-wave is shown forKovoor et al. (2023) as the position of the waves in the mixtures could not be resolved.Red, blue, and green lines depict s-, p-, and dwaves respectively.The red-dashed lines the threshold.The region shows the presence of a virtual state.

Figure 2 .
Figure 2. Data fitted with GEANT4 simulations with energy and widths obtained from (a) Kondo et al. (2010), (b) Corsi et al. (2019), (c) Randisi et al. (2014), (d)Ribeiro et al. (2018).The amplitudes of the states were used from the angular distributions.The global fit is shown as the red line and the background is denoted as black dots.The lowest-lying strength is shown as a blue-dashed line irrespective of its nature.The higher-lying states are depicted as solid brown and green dot-dashed lines.

Table 1 .
The energy (and J π assignments) of low-lying states in 13 Be according to a selection of theoretical studies.Strength 2 refers to any virtual states or resonances around 2 MeV above the neutron threshold.Energy above the neutron threshold (MeV)