Twonucleon emitters within a pseudostate method: The case of Be and Be.
Abstract
 Background

Since the first experimental observation, twonucleon radioactivity has gained renewed attention over the past fifteen years. The Be system is the lightest twoproton groundstate emitter, while Be has been recently proposed to be the first twoneutron groundstate emitter ever observed. A proper understanding of their properties and decay modes requires a reasonable description of the threebody continuum.
 Purpose

Study the groundstate properties of Be and Be within a general threebody model and investigate their nucleonnucleon correlations in the continuum.
 Method

The pseudostate (PS) method in hyperspherical coordinates, using the analytical transformed harmonic oscillator (THO) basis for threebody systems, is used to construct the Be and Be groundstate wave functions. These resonances are approximated as a stable PS around the known twonucleon separation energy. Effective core potentials, constrained by the available experimental information on the binary subsystems Li and Be, are employed in the calculations.
 Results

The ground state of Be is found to present a strong dineutron configuration, with the valence neutrons occupying mostly an state relative to the core. The results are consistent with previous matrix calculations for the actual continuum. The case of Be shows a clear symmetry with respect to its mirror partner, the twoneutron halo He: The diproton configuration is dominant, and the valence protons occupy an orbit.
 Conclusions

The PS method is found to be a suitable tool in describing the properties of unbound ground states. For both Be and Be, the results are consistent with previous theoretical studies and confirm the dominant dinucleon configuration. This favors the picture of a correlated twonucleon emission.
I Introduction
Exotic nuclei far from stability give rise to unusual properties and decay modes Pfützner et al. (2012). In the past few decades, the advances in radioactive beam physics has enabled the study and characterization of nuclear systems close to the neutron and proton driplines. Large efforts have been devoted to understanding the properties of twoneutron halo nuclei Zhukov et al. (1993); Tanihata et al. (2013). These are Borromean systems, in which all binary subsystems do not form bound states. Theoretical investigations within models indicate that the correlations between the valence neutrons play a fundamental role in shaping the properties of twoneutron halo nuclei Zhukov et al. (1993); Nielsen et al. (2001); Kikuchi et al. (2016).
The evolution of these correlations beyond the driplines has implications for twonucleon radioactivity. First proposed for twoproton decays in the sixties Goldansky (1960), this topic gained renewed attention after the first experimental observation of the correlated emission from the ground state of Fe Giovinazzo et al. (2002); Pfützner et al. (2002). Since then, other examples of twoproton emitters have been confirmed, e.g. Zn Blank et al. (2005), Mg Mukha et al. (2007) or Be Grigorenko et al. (2009). More recently, the case of twoneutron emission has also been observed from Be Spyrou et al. (2012), O Kohley et al. (2013) and O Jones et al. (2015).
The decay paths for twonucleon emitters can follow different mechanisms. If there is a narrow state available in the intermediate nucleus, i.e., below the ground state of the parent system, the process is expected to proceed sequentially. On the contrary, if this sequential decay is not energetically possible and the parent nucleus present a strong correlation between the two nucleons prior to emission, a simultaneous “dinucleon” decay takes place (see Ref. Lovell et al. (2017) and references therein). When these extreme pictures do not apply, the concept of true threebody “democratic” Pfützner et al. (2012) decay is introduced. The boundaries between these two and threebody dynamics are still not clear, specially in the decay of excited states, although there have been recent developments Egorova et al. (2012). In this context, threebody models are a natural choice to study the nucleonnucleon correlation and decay modes.
Very exotic beryllium isotopes offer a good opportunity to study twonucleon correlations. On the protonrich side, Be is known to be the lighest twoproton emitter in its original sense Goldansky (1960): The intermediate Li states are not accessible for sequential decay from the ground state of Be Grigorenko et al. (2009); Egorova et al. (2012). On the neutronrich side, the case of Be was claimed to be the first experimental observation of a groundstate decay showing a clear signature of correlated dineutron emission Spyrou et al. (2012). Threebody models in terms of and have been recently used to analyze the structure of these unbound systems Grigorenko et al. (2009); Egorova et al. (2012); Lovell et al. (2017); Oishi et al. (2017). This requires a proper description of continuum states. The threebody continuum problem for systems comprising a single charged particle can be solved, for instance, using the hyperspherical matrix theory Lovell et al. (2017). The extension for systems involving the Coulomb interaction is not an easy task, as the asymptotic behavior for these systems is not known in general. To deal with this problem, very involved procedures are needed ÁlvarezRodríguez et al. (2008); Nguyen et al. (2012); Ishikawa (2013), not free from uncertainties.
An alternative is the socalled pseudostate (PS) method Tolstikhin et al. (1997), which consists in diagonalizing the Hamiltonian in a complete set of squareintegrable functions. This provides the bound states of the system, and also a discrete representation of the continuum. In this context, a variety of bases have been proposed for twobody Hazi and Taylor (1970); Matsumoto et al. (2003); RodríguezGallardo et al. (2004); Moro et al. (2009) and threebody systems Descouvemont et al. (2003); Matsumoto et al. (2004); RodríguezGallardo et al. (2005); Casal et al. (2013). Lately, the PS method in hyperspherical coordinates Zhukov et al. (1993); Nielsen et al. (2001) has been successfully applied to describe the structure properties and reaction dynamics of threebody nuclei (e.g. Casal et al. (2014); Descouvemont et al. (2015); Casal et al. (2015, 2016, 2017); GómezRamos et al. (2017)). This approach, involving a standard eigenvalue problem, is computationally simpler than the calculation of actual continuum states.
It is the purpose of this work to study the groundstate properties of Be and Be by means of the PS method in a threebody () scheme. Calculations are constrained by the experimental information on the binary subsystems Be () Snyder et al. (2013) and Li () Tilley et al. (2002), and the known twonucleon separation energies in Be Spyrou et al. (2012), Be Tilley et al. (2002). The validity of the discretization is assessed by comparing with previous theoretical studies, and the results are analyzed in terms of twonucleon correlations.
The paper is structured as follows. In Sec. II, the threebody formalism used in this work is presented. Results for Be and Be are shown in Sec. III, where the reliability of the theoretical approach is discussed. Finally, Sec. IV summarizes the main conclusions and outlines possible further applications.
Ii Hyperspherical Harmonics (HH) Formalism
Threebody systems can be described using Jacobi coordinates , where the label or 3 indicates one of the three coordinate choices in Fig. 1. In these sets, the variable is proportional to the relative coordinate between two particles and is proportional to the distance from the center of mass of the subsystem to the third particle. The scaling factors between physical distances and Jacobi coordinates are given by Zhukov et al. (1993)
(1) 
and
(2) 
where is the total mass number and are in a cyclic order. It is then clear that the Jacobi set corresponds to the system where particles are related by the coordinate. From Jacobi coordinates, the hyperspherical coordinates can be introduced. Here, the hyperradius and the hyperangle are given by
(3)  
(4) 
and are the twodimensional angular variables related to . Note that, while the hyperangle depends on , the hyperradius does not.
In the hyperspherical harmonic (HH) formalism, the eigenstates of the system are expanded in hyperspherical coordinates as
(5) 
where is introduced for the angular dependence. For simplicity, the label has been omitted, assuming a fixed Jacobi set. Here, is a set of quantum numbers referred to as channel, where is the hypermomentum, and are the orbital angular momenta associated with the Jacobi coordinates and , respectively, is the total orbital angular momentum (), is the spin of the particles related by the coordinate , and results from the coupling . By denoting by the spin of the third particle, which is assumed to be fixed, the total angular momentum is . The angular functions in Eq. (5), , are states of good total angular momentum, which are expanded as Zhukov et al. (1993)
(6) 
where are the hyperspherical harmonics. These are the analytical eigenfunctions of the hypermomentum operator , given by
(7) 
(8) 
with a Jacobi polynomial of order and a normalization constant. With the above definition, the threebody Schrödinger equation leads to a set of coupled hyperradial equations
(9) 
where are the coupling potentials defined as
(10) 
In this work, the system given by Eq. (9) is replaced by a standard eigenvalue problem by using the pseudostate (PS) method Tolstikhin et al. (1997), Here, as in Refs. Casal et al. (2013, 2014, 2015, 2016), the analytical transformed harmonic oscillator (THO) basis is used. The radial functions are expanded as
(11) 
where denotes the hyperradial excitation and are just the diagonalization coefficients. Therefore, the wave functions (5) involve infinite sums over and . However, calculations are typically truncated at maximum hypermomentum and hyperradial excitations in each channel. These parameters have to be large enough to provide converged results.
The THO basis functions in Eq. (11) are obtained from the harmonic oscillator (HO) functions using a local scale transformation, , satisfying the relationship
(12) 
This transformation keeps the simplicity of the HO functions, but converts their Gaussian asymptotic behavior into an exponential one. This provides a suitable representation of bound and resonant states to calculate structure and scattering observables. For this purpose. the analytical form proposed by Karataglidis et al. Karataglidis et al. (2005) can be used,
(13) 
Note that the THO hyperradial wave functions depend, in general, on all the quantum numbers included in a channel , although the HO hyperradial wave functions only depend on the hypermomentum . The preceding transformation depends on parameters and . The most interesting feature of the analytical THO method is that the ratio governs the asymptotic behavior of the basis functions and controls the density of PSs as a function of the energy. This allows us to select an optimal basis depending on the system or observable under study Casal et al. (2013). In order to study the properties of a single threebody resonance, the Hamiltonian can be diagonalized using a THO basis with a small hyperradial extension. This gives a representation of the continuum characterized by a low level density, so that the resonant behavior can be associated with a single PS. Examples of this approach have been previously reported, for instance, to study the properties of the 2 resonance in He RodríguezGallardo et al. (2005); Casal et al. (2013) or the 5/2 resonance in Be Descouvemont et al. (2015). Here, the spatial distribution of the valence nucleons in unbound states is analyzed in terms of the groundstate probability written in the Jacobi1 set,
(14) 
where the wave function has been transformed back to Jacobi coordinates, and, after scaling to the relative distances and , it satisfies the normalization relationship
(15) 
Iii Application to exotic beryllium isotopes
The only stable beryllium isotope, Be, is already a weakly bound system Tilley et al. (2004). Exotic isotopes form bound states from Be to Be (with the exception of the unbound systems Be and Be). Beyond the driplines, Be and Be ground states have been observed as resonances characterized by twonucleon separation energies MeV Spyrou et al. (2012) and MeV Tilley et al. (2002). Their widths are 0.8 and 0.092 MeV, respectively, although a much narrower state, 0.17 MeV, was found for Be in recent calculations Lovell et al. (2017). The discrepancy was atributed to the effect of the experimental resolution. The properties of the relevant binary subsystems Be and Li have also been measured. The ground state of Be is a state 1.8(1) MeV above the neutron separation threshold and has a width of 0.58(20) MeV Snyder et al. (2013). On the other hand, the state in Li is unbound with respect to the proton emission by 1.96(5) MeV, and its accepted width is 1.5 MeV Tilley et al. (2002). Therefore, the sequential twonucleon emission from the ground state of Be (Be) is (mostly) unaccessible, as shown in Fig. 2. This favors a simultaneous twonucleon emission, either in the form of a “dinucleon” or in a true threebody (democratic) decay Pfützner et al. (2012).
Be Lovell et al. (2017)  He  

26.18  48.00  
30.50  42.35  
42.73  21.50  
10.00  40.00  
33.77  40.00  
0.65  0.70  
0.65  0.35  
3.02  2.00  
3.02  1.50  
2.00 
Threebody descriptions require, as input, a nucleonnucleon interaction and realistic potentials. For the former, in this work the GPT nucleonnucleon potential Gogny et al. (1970) is employed, including central, spinorbit and tensor terms. This potential, although simpler than the robust Reid93 Stoks et al. (1994) or AV18 Wiringa et al. (1995) interactions, reproduces observables up to 300 MeV. This makes it suitable for threebody calculations RodríguezGallardo et al. (2005); Casal et al. (2013); Lovell et al. (2017). The and potentials are adjusted to reproduce the position of the Be and Li ground states, respectively. These are dependent WoodsSaxon potentials with central and spinorbit terms, whose parameters are given in Table 1. Note that, in this work, the interaction is the same used in Ref. Lovell et al. (2017), while the potential is essentially the one used in Refs. RodríguezGallardo et al. (2005); Casal et al. (2013) for the case but including also the Coulomb repulsion. The later is a shallow potential, in the sense that the Pauli state has been removed by introducing a repulsive term. However, the potential gives rise to , and bound states which represent the neutronoccupied orbitals of the core. These states have to be projected out for the threebody calculations, and this can be achieved, as in Ref. Lovell et al. (2017), by using a supersymmetric transformation Baye (1987).
The core phase shifts corresponding to the potentials given in Table 1 are shown in the upper panels of Figs. 3 and 4 for Be(5/2) and Li(3/2) states, respectively. In the lower panels, the position of the twobody resonances can be associated with the maximum of the overlaps between Be (Li) continuum states and the threebody ground state of Be (Be). Details about these threebody calculations are given in the following sections. In these figures, vertical lines represent the experimental position of the resonances to which the interactions have beed adjusted. For completeness, in Fig. 4, the phase shifts for He scattering as well as the corresponding overlaps are shown together with those for He. These have been obtained by just switching off the Coulomb interaction in the binary potential. It is clear that both systems, He and Li, can be described using the same core potential, and this enables the description of the mirror nuclei Be and He using the same threebody Hamiltonian except for the Coulomb part. Details are presented in section III.2.
iii.1 configuration in Be
The 0 states in Be () are computed in the Jacobi1 set, where the two valence neutrons outside a Be core are related by the coordinate. Since threebody models are an approximation to the full manybody picture, realistic binary interactions alone are typically insufficient to reproduce the known spectra RodríguezGallardo et al. (2005); de Diego et al. (2010); Thompson et al. (2004); Casal et al. (2014). It is then customary to include also a simple hyperradial threebody force, whose parameters can be fixed to reproduce the (known) threebody energies without distorting the structure of the states. In this work, as in Ref. Casal et al. (2016), a Gaussian form is adopted,
(16) 
Using fm and MeV, a lowlying resonance around the twoneutron separation energy of Be is obtained. Note that this threebody force, with different geometry and parameters, was also employed in the previous Be threebody calculation of Ref. Lovell et al. (2017).
In this work, the threebody continuum problem is solved approximately within the threebody PS method using the THO basis. The parameters of the analytical transformation defining the basis control the level density after diagonalization Casal et al. (2013). Following the stabilization method by Hazi and Taylor Hazi and Taylor (1970), stable eigenstates close to resonance energies provide a good approximation of the inner part of the exact scattering wave function. The stability can be checked by changing the parameter of the transformation and keeping fixed the oscillator length Lay et al. (2012). The Be spectra obtained within different THO bases are shown in Fig. 5. In this calculations, fm, and the threebody problem is solved by truncating the basis expansion with and . From Fig. 5, it is clear that a state around 1.3 MeV shows a rather stable pattern and, for large values, is well isolated from the rest of discretized continuum states. With fm, the state has a variational minimum and can be used to study the groundstate properties. This PS approximation to analyze resonance properties of threebody systems was previously reported, for instance, for the 2 resonance in He RodríguezGallardo et al. (2005); Casal et al. (2013) or the 5/2 resonance in Be Descouvemont et al. (2015).
The stability of the calculations is further clarified in Fig. 6, where the convergence of the groundstate energy as a function of the maximum hypermomentum and the number of hyperradial excitations is presented. This corresponds to the lowest PS obtained using fm, which is taken as an approximation of the resonance groundstate wave function. With , corresponding to 136 channels in the wave function expansion (5), the resonance around 1.3 MeV is fully converged. It is also clear that hyperradial basis functions for each channel are sufficient to achieve convergence of the ground state.
The spatial distribution of the valence neutrons in Be can be studied from the probability function defined by Eq. (14). This is shown in Fig. 7 as a function of and . The maximum at fm and fm corresponds to the dineutron configuration, while the other smaller peaks are typically associated with the triangle and cigarlike arrangements. From Fig. 7, it is clear that the dineutron component dominates the ground state of Be. This state is governed by relative components between the valence neutrons, which amount for 75% of the total norm. The Be partial wave content of this state is 81%. A similar behavior was previously reported for the twoproton configuration in Ne Casal et al. (2016). The present calculations confirm the strong dineutron configuration in the Be, which favors the picture of a correlated twoneutron emission. These findings agree with the experimental interpretation in Ref. Spyrou et al. (2012) of Be as a groundstate dineutron emitter and are also consistent with the previous theoretical work Lovell et al. (2017).
Note that, in Ref. Lovell et al. (2017), the actual continuum was obtained within the matrix approach. In the present work, this problem has been approximated by solving a simple eigenvalue problem, which provides discrete PSs as a representation of the continuum. Results using the same threebody Hamiltonian are fully consistent, which supports the reliability of the PS method to study the properties of unbound threebody states. This can be achieved due to the versatility of the THO basis, whose analytical parameters enable the identification and analysis of single resonances. Note that, from the computational point of view, using the PS method is much less demanding than solving the actual continuum problem, such as in matrix calculations. Moreover, the present approach is general and can be easily applied to systems comprising any number of charged particles, for which the exact computation of scattering states is a wellknown open problem. An application in this line will be presented in the following subsection for the case of Be.
While the properties of the binary subsystem play a relevant role in shaping the properties of the compound system Be, the dominant dineutron configuration can be associated to the effect of the neutronneutron interaction Lovell et al. (2017). This can be studied by diagonalizing the threebody Hamiltonian without the potential. The groundstate probability corresponding to this unphysical solution is depicted in Fig. 8, where the same interaction is employed. In this calculation, the threebody force has been adjusted to recover the same twoneutron separation energy. The fundamental difference with respect to the physical ground state in Fig. 7 is the absence of a dominant dineutron configuration. Here, the dineutron and cigarlike contributions carry almost the same strength. In this case, the  relative components are reduced to 50%, while these valence neutrons occupy an almost pure orbit with respect to the core, i.e., 98%. This illustrates that the strong dineutron character of the Be ground state is driven by the interaction, and it is again consistent with the conclusions drawn in Ref. Lovell et al. (2017).
iii.2 configuration in Be
The threebody Hamiltonian for the system includes now, in addition to the binary nuclear potentials, the pairwise Coulomb repulsion between the three interacting particles. Nevertheless, from the point of view of the PS method, the Be case is totally analogous to the one of Be discussed in the preceding subsection. Moreover, its mirror partner He can be described using the same method by just switching off the Coulomb part of the binary interactions. This enables a comparative study between both systems in a threebody scheme.
The ground states of He and Be are generated in a THO basis with the same parameters used for Be: fm and fm. As in the previous case, this choice gives a stable PS in the continuum carrying the resonant properties of the Be groundstate. The position of the states is again adjusted using the threebody force introduced in Eq. (16) with fm. The depths to reproduce the experimental twoneutron separation energy in He, 0.975 MeV Brodeur et al. (2012), as well as the energy of the unbound Be are and 2.5 MeV, respectively. In Fig. 9, the convergence of the groundstate energy for both systems is shown as a function of , which determines the size of the model space. It is clear that, within the PS approximation, the convergence of the bound state in He is achieved much faster than that of unbound systems. As in the previous case, the basis is set to hyperradial excitations, which was found to be sufficient to provide converged results.
As in the previous example, the correlation between valence nucleons can be studied by plotting the groundstate probabilities. This is shown in Figs. 10 and 11 for He and Be, respectively. The twoneutron halo in He presents the typical dineutron configuration Zhukov et al. (1993) around fm and fm. This is a clear signal of the strong correlations in the halo. The situation for Be is found to be analogous, with the absolute maximum corresponding to two protons close to each other at some distance apart from the He core. The distribution is similar to the initial twoproton density presented in Ref. Oishi et al. (2017), where the decay from Be is described as the time evolution of the valence protons in the spherical mean field generated by the core. The spatial distribution for Be is more diffuse than that for He, as it corresponds to an unbound system under the influence of the Coulomb repulsion between the three bodies. The wave function contains 86% (83%) of relative components between the two valence protons (neutrons) in Be (He), and the nucleoncore content is close to 90%. The present calculations confirm the strong diproton configuration in Be, in clear symmetry with the twoneutron halo of He. These results favor the picture of a correlated twoproton emission from the ground state of Be.
Iv Summary and conclusions
Threebody calculations for the unbound Be () and Be () systems have been carried out to study the correlations between the valence nucleons, in relation with twonucleon radioactivity. Their ground states have been generated within the PS method using the analytical THO basis. This enables the identification of single resonances as discrete eigenstates in the continuum, which are stable with respect to the choice of the basis parameters. The models incorporate the GPT interaction, realistic corenucleon potentials adjusted to reproduce the known resonance energies of the binary subsystems Be() and Li(), and also a phenomenological threebody force to adjust the position of the threebody states to the known experimental energies.
The groundstate probability distribution for Be presents a strong dineutron configuration, consistent with recent experimental observations. The present approach agrees with the conclusions from matrix calculations of actual scattering states. This supports the reliability of the PS method to study the groundstate properties of unbound threebody systems. The method is computationally less demanding and can be applied in general to systems comprising several charged particles. In this line, the ground state of Be shows a dominant diproton component, in clear symmetry with the twoneutron halo of its mirror partner He. For both Be and Be, the present results favor the picture of a correlated twonucleon emission.
Possible applications of the PS method to study nucleonnucleon correlations in unbound systems include the description of exotic oxygen isotopes such as O, O or O, the later being the mirror partner of the twoneutron halo Li. The decay from excited states of dripline nuclei, e.g., the resonances in He or Ne, and the influence of these correlations for reaction observables, could also be studied. Work along these lines is ongoing.
Acknowledgements.
I am sincerely grateful to M. RodríguezGallardo and J. M. Arias for their constructive comments about the suitability of the present calculations. This work has been partially supported by the Spanish Ministerio de Economía y Competitividad under Projects No. FIS201453448C21P and No. FIS201451941P, and by the European Union Horizon 2020 research and innovation program under Grant Agreement No. 654002.References
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