Proton-and neutron-rich breakups near zero binding energy

An analysis of the elastic scattering and breakup cross sections in the 17F/17O+208Pb reactions in the zero binding energy limit, is presented. The main motivation was to investigate whether finite reaction observables can be obtained for a neutron-rich system in this limit, where it can be regarded as an open quantum system. It is first found that as the binding energy tends to zero, the ground-state wave function of the 17O ( 16O +n) system falls asymptotically as ∼1/rℓb (where ℓb=2 , is the ground-state orbital angular momentum). For both projectiles, finite elastic scattering and breakup cross sections are obtained in the zero binding energy limit, where they also become insensitive to the variation of the binding energy. For the 17F projectile, this is due to the core-proton Coulomb barrier, whereas for the 17O projectile, it is due to the ∼1/rℓb asymptotic behavior of the ground-state wave function in this limit. In conclusion, finite reaction observables in the breakup of an open quantum system can be obtained for ℓb≥2 .


Introduction
The physics of halos and other loosely-bound systems has attracted an extensive attention over the past few decades (see for instance [1][2][3][4][5][6][7], for some of the latest developments in this field). Given their weak ground-state binding energy, halo nuclei are mainly characterized by the extension of the matter density to the peripheral region (well outside the core nucleus radius), and a ground-state that is strongly coupled to the continuum, particularly for a neutron-halo system where there is no core-neutron Coulomb barrier. Therefore, given a strong correlation between the low binding energy and the extension of the density matter, if the ground-state binding energy tends to zero, the density matter can be expected extend to infinity, leading to an open quantum system (for example, see [8][9][10][11], for more discussion on open quantum systems). In this case, the ground-state wave function may no longer fulfil the natural condition of square-integrability, and a closed gap between the ground-state and the continuum. The bound-state can then be, to some extent, regarded as embedded in the continuum, probably 'leading' to the well-known bound-state in the continuum (BIC) phenomenon. Born out of quantum mechanical curiosity [12], this phenomenon has become a topic of great interest, with practical applications in various fields, as exemplified by [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]. With the current state-of-art experimental equipment, extremely loosely-bound systems are no longer out of sight as evidenced by [33]. In some studies, the spectroscopy of newly observed systems is determined through the analysis of the spectroscopic factor as function of the ground-state binding energy, where this energy ranges from 0 to around 1 MeV (see for example [3]). This makes the study of the breakup process in the zero binding energy limit of utmost importance.
Due to the Coulomb barrier between two charged fragments, the proton-halo ground-state matter density is not expected to extend to infinity, thus preventing this system from reaching the state of an open quantum system, as shown for example in [34], for the  + B Be p 8 7 system. In this reference, it is obtained that the ground-state wave function converges to a finite function as the binding energy approaches zero, where it remains square-integrable. This agrees with [35], where a general asymptotic expression of the proton-halo wave function is derived in the zero binding energy limit. Studying the breakup process in the zero binding energy limit [34], the breakup cross section is reported to remain finite and become independent of the binding energy, similar to the ground-state wave function. The continuum-continuum couplings effect, which is otherwise known to increase as the binding energy decreases, is also found to be independent of the ground-state binding energy in this limit. Notice that apart from a dissociation of the projectile into its constituent fragments, the breakup process is regarded as an excitation (transition) of the projectile from the bound-state to the continuum. It could be interesting to investigate whether a finite breakup cross section in the zero binding energy limit is associated with the valence nucleon being embedded in the continuum. Although this could be a formidable task, it might shed some light on whether a BIC can be associated with a finite breakup cross section in the zero binding energy limit.
Although the conclusion reached in [34], can be expected to extend to any system with two charged fragments, it is not the case for neutron-rich systems. For a neutron-halo system, where there is no Coulomb barrier, the extension of the matter density to infinity in the zero binding energy limit, means that finite reaction observables may not be obtained. For example, for an s-wave ground-state, the wave function falls asymptotically as ), where C, is the normalization coefficient and k b the ground-state wave number. This implies that in the zero binding energy limit (k b →0), the wave function will approach C. However, for a non-s-wave ground-state, the asymptotic expression of the wave function obtained in [35], suggests that it remains square-integrable even for k b →0. In this case, finite reaction observables may be expected even for a neutron-halo system in the zero binding energy limit.
In n), respectively [36]. The two main objectives of this study are: (1) using the expression of the non-s-wave ground-state wave function in the asymptotic region from [35], to investigate whether finite elastic scattering and breakup cross sections can be obtained for a neutron-rich system in the zero binding energy limit, and (2) to extend the conclusion of [34] to other proton-halo systems. This study will also serve to further emphasize the peripheral nature of the breakup process (which is well documented as exemplified by [37][38][39][40][41]), and the crucial importance of the ground-state orbital angular momentum in the breakup of open quantum systems. To this end, the same procedure as in [34] is adopted, where apart from the experimental ground-state binding energies, four values, arbitrary chosen in the interval - ], are considered. For practical reason, the value −0.01 keV which obtained by scaling S p and S n by about 6×10 4 and 4×10 5 respectively, can be regarded as approaching zero compared to the experimental values. Among other factors, the choice of these reactions is primarily motivated by the nonzero orbital angular momentum in the ground-states of the projectile nuclei. The elastic scattering and breakup cross sections are obtained through a numerical solution of the coupled differential equations, derived from the Continuum-Discretized Coupled-Channel (CDCC) formalism [42,43]. This formalism is uniquely designed to handle breakup reactions induced by weakly-bound projectiles, due to its accurate inclusion of the continuumcontinuum couplings in the coupling matrix elements.
The detail of the numerical calculations is given in section 2, the results are presented and discussed in section 3, and the conclusions are reported in section 4.

Numerical calculations
The detail of the CDCC formalism (which is not repeated in the present work), can be obtained in [42,43]. This section outlines the description of the projectiles and the numerical parameters used to generate the projectiles' ground-state and continuum wave functions as well as the ones used to solve the CDCC coupled differential equations (CDCC model space).  ,

Projectile description
), m j is the z-projection of j] that describes the relative motion of the core nucleus and the valence nucleon, is , is radial wave function (with k, the wave number, and it is related to the energy ε in the continuum through m e =  k 2 cv 2 , where μ cv , is the core-nucleon reduced mass), r, the core-nucleon radial coordinate, W Y r ( ) ℓ , the usual spherical Harmonics, with q j W º , r r r ( ), the solid angle in the direction of the vector r, expressed in spherical coordinates. The wave function F a k r , jm ( ), is an eigenfunction of the following internal Hamiltonian where ∇ 2 , is the usual nabla operator, and V cv (r), the core-nucleon interacting potential, which in the present case is given by with V 0 and V so , being the depths of the central and spin-orbit coupling terms of the nuclear component, R 0 , the radius, a, the diffuseness, and V C (r), the Coulomb component (which is considered here to be a Coulomb pointsphere potential), where V C (r)=0, for + O n 16 system. In this equation, the function f r R a , , In the asymptotic region (r?R 0 ) where nuclear forces are nonexistent, the radial bound-state wave function f k r , From this equation, one can deduce that in the S p →0 limit, which shows that in this limit, the proton-halo system has a square-integrable wave function, as also observed in [34], for the + Be , is the spherical Hankel function [45]. In the zero binding energy limit, one obtains [35] f  - [35]. With this proportionality, the wave function falls asymptotically as~r 1 b ℓ . Consequently, the wave function also satisfies equation (7), depending on the value of ℓ b , in particular for  2 b ℓ . This reveals the crucial importance of the orbital angular momentum ℓ b for the neutron-rich system as S n →0. For an s-wave bound-state in which case the wave function does not satisfy equation (7). Owing to the peripheral nature of the breakup process, one can expect a finite breakup cross section for the + O n 16 as S n →0, due to the ground-state orbital angular momentum ℓ b . This assertion is what the present study intends to highlight.
In the CDCC formalism, the projectile-target wave function is expanded on the projectile bound and discretized bin states. To this end, once the relative momentum k has been truncated by k max , and the interval [0: , the pure scattering wave functions are transformed into square-integrable bin wave functions, using for example the binning technique as follows [46] ò ) . The substitution of the discretized projectile-target wave function into the Schrödinger equation, yields a finite set of coupled differential equations, which are solved numerically.

Numerical parameters
To numerically solve the two-body Schrödinger equation and obtain the projectiles' different states, I used the following parameters of the V cv (r) potential, V 0 =−56.7 MeV, = V 25.14 MeV fm 16 ), R 0 =3.023 fm, and a=0.6415 fm, taken from [44]. This potential reproduces the ground and first-excited bound-states as well as the resonance energies. The same parameters were also used for the non-resonant continuum states. The depth V 0 was adjusted to obtained the other groundstate binding energies considered, i.e., S p,n =−100 keV, −10 keV, −0.1 keV, and −0.01 keV. The different core-target and nucleon-target optical potential parameters are listed in table 1. The parameters of the neutrontarget potential were obtained from the global parametrization of [47], whereas the ones of the proton-target potential were taken from [48]. The latter were adopted as they provide a better fit of the + F Pb 17 208 elastic scattering experimental data. The CDCC model space parameters used in the numerical solution of the coupled differential equations, are summarized in table 2. In this table, max ℓ , is the maximum core-nucleon orbital angular momentum, λ max , the maximum order in the potential multipole expansion, k max , the maximum relative momentum, r max , the maximum matching radius for bin integration, Δr, the integration step size associated with r max , R max , the maximum matching radius in the numerical integration of the coupled differential equations, ΔR, the integration step size associated with R max , and L max , the maximum orbital angular momentum of the projectile-target relative center-of-mass motion. The interval [0: k max ] was discretized into momentum bins of widths, D =  ground-state wave function falls asymptotically to zero in accordance with equation (6). Also, for S p −10 keV, the wave function converges to a finite function as it becomes insensitive to the variation of S p , and it is more extended to the asymptotic region compared to S p −100 keV. Given the convergence of the ground-state wave function, finite breakup cross sections can be anticipated for S p −10 keV. · , r so =1.076 fm and a so =0.59 fm.  Table 2. Parameters used in the numerical solution of the CDCC coupled differential equations. Considering the + O n 16 system in panel (b), it is observed that for S n −100 keV, the wave function exhibits the natural square-integrable property as it quickly decays to zero in the asymptotic region, similar to panel (a). However, for S n −10 keV, the wave function falls asymptotically according equation (9), i.e., it appears to slowly decay as~r 1 b ℓ (being largely extended to infinity compared to the case of S n −100 keV). It is also noticed that the wave function becomes insensitive to the variation of S n , for S n −10 keV.
A careful observation of this figure shows that at r=200 fm, the wave function approaches -10 fm 4 1 2 . This value may be lower enough to ensure a square-integrable wave function even as S n →0. For ℓ b =0,1, it can be expected to be much larger, meaning that the wave function might no longer be square-integrable. Hence, for s-wave neutron-halos such as Be 11 , C 15,19 , where ℓ b =0, finite reaction observables might not be expected in the zero binding energy limit. To show that the O 16 +n ground-state wave function does in fact fall asymptotically as r 1 b ℓ in the zero binding energy limit, I compare in figure 2, the ground-state wave function for S n =−0.01 keV, with the function = f r r It is seen in this figure that indeed, the wave function is identical to f (r) for r10 fm.
In order to assess the implication of a large extension of the ground-state wave function to the peripheral region, I analyze the function f f = I r k r r k r , , , which reflects the radial behavior of the dipole electric response function. According to the first-order perturbation theory [50,51]   n ground-state wave function is remarkably clear. It is noticed that while the function I(r) falls rapidly to zero for S n −100 keV, similar to panel (c), for S n −10 keV, it becomes oscillatory in the asymptotic region, due to the slow decay of the ground-state wave function in this energy region. If one were to consider pure scattering wave functions, there would be a convergence issue in the breakup calculations in the zero binding energy limit.
However, in the CDCC formalism, the pure scattering wave functions are transformed into squareintegrable bin wave functions as shown by equation (11). To display how this transformation affects the function I(r) in figure 1(d), the function f j = I r k r r k r , ,  is plotted in figure 3. To obtain the bin wave functions j k r , has no convergence issues. This highlights the uniqueness of the CDCC method to handle breakup calculations in the zero binding energy limit.

Elastic scattering and breakup cross sections
The elastic scattering cross sections, which were calculated at E lab =170 MeV for both projectiles, are plotted in reaction). Inspecting panel (a), one sees that the elastic scattering cross section is not sensitive to the variation of the ground-state wave function, despite of the fact that for S p −10 keV, the ground-state wave function is relatively more extended to the asymptotic  reaction were taken from [52].
region compared to the case of S p −100 keV, as seen in figure 1(a). A good agreement with the experimental data from [52] is also observed. It follows that the ground-state binding energy below the experimental value has no effect on the + F Pb 17 208 elastic scattering cross section, such that any energy in this range provides a better fit of the experimental data. In other words, the ground-state binding energy that provides a better fit of the experimental elastic scattering cross section data for a core-proton system is not unique. If the calculations of [34] were compared with the experimental data, a similar conclusion would have been reached.
Considering the + O Pb 17 208 reaction in panel (b), it follows that the scattering cross section decreases as the binding energy decreases from S n =−4144 keV to −10 keV, while it also becomes insensitive to the variation of this energy for S n −10 keV. This lack of sensitivity can be attributed to the fact that the ground-state wave function falls asymptotically as~r 1 b ℓ in the S n →0 limit, leading to the convergence of the radial integral of the elastic scattering matrix elements. In conclusion, a finite elastic scattering cross section for a neutron-rich projectile is obtained in the zero binding energy limit, for ℓ b =2. This result is quite important for a better understanding of the breakup dynamics of an open quantum system. Another observation in figure 4(b), is the persistence of the Coulomb-nuclear interference peak (CNIP) around 30°, similar to panel (a), which is not completely suppressed even as S n →0. For the + Be Pb 11 208 reaction which also involves a neutron-halo projectile, the CNIP is found to be completely suppressed at an incident well above the Coulomb barrier [53]. Among other factors, the persistence of the CNIP in this case can be mainly attributed to the centrifugal barrier in the ground-state of the + O n 16 system, which is absent in the + Be n 10 system (see for instance [7] for more discussion).
The angular distributions breakup cross sections for both + F Pb reaction [panel (a)], one observes, unlike in the case of elastic scattering, a substantial increase of the breakup cross section from S p =−601 keV to S p =−10 keV. This is attributed to a relatively large extension of the ground-state wave function as reported in figure 1(a). As anticipated from figure 1, finite breakup cross sections are obtained for S p −10 keV, where they are also insensitive to the variation of S p , in agreement with conclusion drawn in [34]. A further look at this figure, one sees that the theoretical calculations overestimate the experimental data (obtained from [52]) at lower angles. This is also the case in [48,52], where the same reaction is analyzed. It could be that more breakup dynamics such as projectile and target excitations, among others, are needed in order to successfully describe the data. Turning to the + O Pb 17 208 reaction [panel (b)], a substantial increase of the breakup cross section from S n =−4144 keV to S n =−10 keV is noticed. Finite breakup cross sections are as well obtained for S n −10 keV, where they also become insensitive to the variation of S n . This is ascribed to the fact that the ground-state wave function falls asymptotically as~r 1 b ℓ . In conclusion, a finite breakup cross section of a neutron-rich projectile in the zero binding energy limit is obtained ℓ b =2. This amounts to saying that finite reaction observables for an open quantum system can be obtained, provided the ground-state orbital angular momentum is non-zero. This highlights the crucial importance of the ground-state orbital angular momentum in the zero binding energy limit for an open quantum system. To ensure that the convergence of the breakup cross section in the zero binding energy is exclusively related to the convergence of of the ground-state wave function, I show in figure 6, the breakup cross sections in the absence of couplings among continuum states (continuum-continuum couplings), i.e., only couplings to and from the ground-state are taken into account. These couplings are known to have a large effect on the breakup cross section. Observing this figure, it can be noticed that the breakup cross sections for both systems remain finite for S p,n −10 keV, which points to a negligible effect of the continuum-continuum couplings.

Conclusions
In this paper, I have studied the breakup of proton-and neutron-rich systems in the zero binding energy limit, considering F 17 and O 17 nuclei, which are modeled as  + F O p 17 16 and  + F O 17 16 n. The main objective was to investigate whether finite elastic scattering and breakup cross sections can be obtained for a neutron-rich system in the zero binding energy limit. To this end, the experimental binding energy values were artificially reduced down to S p, n =−0.01 keV. For practical reason, this value was then considered to be in the zero energy limit compared to S p =−0.601 MeV and S n =−4.144 MeV. It is first shown that the + O p 16 system has a square-integrable ground-state wave function in the zero binding energy limit. In the same limit, the + O n 16 ground-state wave function falls asymptotically as~r 1 b ℓ (where ℓ b =2, is the orbital angular momentum in the ground-state). Therefore, the square-integrability of the neutron-rich ground-state wave function the zero binding energy limit strongly depends on ℓ b . In this binding energy limit, finite + F Pb reaction, it is due to the~r 1 b ℓ asymptotic behavior of the + O n 16 ground-state wave function.
In conclusion, the results obtained in [34], can be generalized to other proton-rich systems. Finite reaction observables in the breakup of an open quantum system can be obtained, provided ℓ b 2. It follows that for swave neutron-halos such as Be 11 , C 15,19 where ℓ b =0, finite reaction observables in the S n →0 limit may not be expected, in particular for the elastic scattering cross section. These results provide an important step into the breakup dynamics of an open quantum system, and highlight the crucial role of the orbital angular momentum.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files). [panel (b)] reactions as function of the c.m. angle θ, for different ground-state separation energies. These results were obtained in the absence of couplings among continuum states in the coupling matrix elements (no continuum-continuum couplings.