Binding-energy asymmetry in absorption explored through CDCC extended for complex potentials

Physics Letters B 832 (2022) 137252 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Binding-energy asymmetry in absorption explored through CDCC extended for complex potentials M. Gómez-Ramos a,∗, J. Gómez-Camacho a,b, A.M. Moro a,c aDepartamento de Física Atómica, Molecular y Nuclear, Facultad de Física, Universidad de Sevilla, Apartado 1065, E-41080 Sevilla, Spain bCentro Nacional de Aceleradores (U. Sevilla, J. Andalucía, CSIC), Tomás Alva Edison, 7, 41092 Sevilla, Spain cInstituto Interuniversitario Carlos I de Física Teórica y Computacional (iC1), Apdo. 1065, E-41080 Sevilla, Spain a r t i c l e i n f o a b s t r a c t Article history: Received 3 March 2022 Received in revised form 24 May 2022 Accepted 2 June 2022 Available online 14 June 2022 Editor: J.-P. Blaizot Keywords: Continuum-discretized coupled channels Optical model Direct reactions Three-body model In this work, we present an extension of the Continuum-Discretized Coupled-Channel formalism to include the effects of absorption and excitation of the core through its interaction with the removed particle in the description of nuclear breakup reactions. This extension is performed via the inclusion of a complex energy-dependent interaction between core and removed particle and the use of a binormal basis to ensure orthogonality. The formalism is applied to neutron breakup reactions with a 12C target at 70 MeV per nucleon for the loosely-bound 11Be nucleus and the more deeply-bound 41Ca nucleus, finding a moderate reduction in the cross section for the weakly-bound case and a strong reduction for the more deeply-bound case. Possible implications for the interpretation of intermediate-energy knockout reactions are discussed. ©2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 1. Introduction Nucleon-removal reactions have a long and successful history in the study of the single-particle properties of nuclei [1–4]. In particular, nuclear breakup (or elastic breakup) reactions have been used extensively, specially in the study of loosely-bound and halo nuclei [5–9]. In these breakup reactions, a nucleon is removed from a nucleus avia interaction with a target A, leaving the target Ain its ground state, as well as a residual core band nucleon x(a =b +x), all of them detected in what is called the elastic breakup channel. The usual description for these reactions relies in a single-particle model of the nucleus, where the interaction between core band removed nucleon xis assumed to be well described by an effective real interaction Vbx which reproduces the properties of the bound system aand in some cases the low-energy continuum of the b −x system. The eigenstates of this potential are then taken as a good description of the continuum of the b −xsystem and are used as the basis for Distorted Wave Born Approximation (DWBA) [10]or Continuum-Discretized Coupled-Channel (CDCC) [11] calculations to describe the reaction observables. However, in breakup reactions core band nucleon xmay end up in states with significant relative energies, where new channels beyond the single-particle excitation open, in particular the excitation and possible disinte- *Corresponding author. E-mail address: [email protected] (M. Gómez-Ramos). gration of b. The mere interaction between band xcan populate these open channels, thus reducing the cross section to the elastic breakup channel. These open channels cannot be considered in the description of breakup reactions described above but can nevertheless play a significant role, as has been shown for the breakup of 11Be [12,13]. Generally the exclusion of these nonelastic channels restricts the description of breakup observables to low relative energies between the two fragments and to low binding energies of the removed nucleon, where their effect can be neglected. The exclusion of these open channels is particularly questionable when removing deeply-bound nucleons, where the large energy transfer enhances their population and subsequent decay of core nucleus b[14]. It is particularly in the removal of deeply-bound nucleons where a current open problem exists, in which theoretical predictions of nucleon-knockout cross sections severely overestimate experimental data [2,15], while for weaklybound nucleons, the agreement is much better. We therefore find timely to explore the effects of non-elastic channels in breakup reactions, which will be included through the use of an effective complex energy-dependent interaction, in an approach similar to the widely-used optical model [10] and to the Ichimura-AusternVincent (IAV) description of non-elastic breakup [16]. It should be remarked that in Faddeev/AGS [17,18] calculations, the effects of complex potentials have been explored [19]but to our knowledge this study has not been extended to CDCC nor has it delved into the effects of non-orthogonality which naturally appear when considering complex potentials. The approach considered in this work https://doi.org/10.1016/j.physletb.2022.137252 0370-2693/©2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. M. Gómez-Ramos, J. Gómez-Camacho and A.M. Moro Physics Letters B 832 (2022) 137252 Fig. 1. Relevant coordinates for the description of the A(a,bx)Areaction. also opens CDCC to the use of state-of-the-art microscopic optical potentials [20–22], which are generally complex, non-local and energy-dependent, though their study will be left for a later publication. This work is structured as follows: in Section 2the formalism for CDCC with complex potentials is presented. In Section 3, results are presented for the 12C(11Be,10 Be +n)12C and 12C(41Ca,40 Ca + n)12Creactions. Finally, in Section 4, the results of this work are summarized and prospective lines of investigation are presented. 2. Theoretical framework 2.1. Effective three-body Hamiltonian We start our derivation with the following Hamiltonian (see Fig. 1for the relevant coordinates): H=TR+Tr+Hb(ξ) +Vbx(r,ξ)+UxA(rx)+UbA(rb), (1) where Hb(ξ) is the core Hamiltonian, which depends explicitly on the core degrees of freedom ξwhich couple strongly to those of the valence particle through Vbx. Its eigenstates will be denoted φ(c) b(ξ) (with c=0denoting the ground state). The interactions UxA and UbA are assumed to be complex, as in standard in CDCC calculations, and represented by some optical potentials describing the respective fragment-target elastic scattering. By contrast, the potential Vbx(r, ξ) is assumed to be real and depending on the considered core degrees of freedom (the xparticle is assumed to be structureless for simplicity). Note that the use of complex UxA and UbA interactions assumes that a projection on the target ground state has been performed and hence the target internal coordinates do not appear explicitly in the Hamiltonian above. The complex interaction UbA implicitly takes into account the excitation of the core b. However, the relevant core degrees of freedom that come into play in the interaction with the target UbA should not significantly interfere with the degrees of freedom ξthat are excited due to the interaction with the valence particle Vbx. Therefore, the interaction UbA can be assumed to be independent of ξ. A similar approximation was taken in the IAV description of inelastic breakup [16]. The three-body scattering wavefunction consistent with the Hamiltonian (1)can be expressed in integral form as (+)(ξ, r,R)=χaφa+G3BVpriorχaφa(2) where G3B=1 E+−TR−UxA −UbA −Hbx (3) with E+=E+i, →0(Ebeing the center-of-mass energy of the three-body system). Vprior =UxA +UbA −UaA is the transition potential in the prior representation of the matrix element, as considered, for example, in transfer and inelastic breakup [10,16], and φais the projectile ground state wavefunction. The distorted wave χais a solution of the (arbitrary) auxiliary potential UaA and Hbx is the projectile Hamiltonian Hbx(ξ, r)=Hb(ξ) +Tr+Vbx(ξ, r). (4) The three-body wavefunction (2) depends, in addition to the internal and relative coordinates rand R, on the core degrees of freedom ξ. In the standard CDCC formulation, the effects of the excitations of core bare ignored. Practical solutions of this threebody problem with core degrees of freedom have been recently carried out in a generalized CDCC method which includes these core excited components explicitly in the adopted model space [23,24]. In the present work we adopt a different approach in which these core excited components are not treated explicitly, but embedded in the effective two-body interactions. In other words, we seek for an approximate solution of the projected wavefunction (+)(r, R) ≡φ(0) b|(+). As we will see, the presence of these core excitations generates modifications in the effective Hamiltonian which render the choice of the Vbx interaction more complicated than usually assumed in practical implementations. To see this in an approximate way, we assume that, in the projected model space, the ground state is well described by the product wavefunction φa(ξ, r) ≃ϕ0(r)φ(0) b(ξ) (where ϕ0(r)describes the b +x relative motion in the projectile ground state). With this choice, the projected three-body scattering wavefunction results (+)(R,r)=φ(0) b|(1+G3BVprior)|χaφ(0) bϕ0 =1+φ(0) b|G3B|φ(0) bVpriorχaϕ0 =1+1 E+−TR−UxA−UbA−Tr−Ubx Vpriorχaϕ0 (5) where, in the last line, we have used the definition of the formal optical model Green’s function: Gopt(z)=φ(0) b|1 z−Hbx |φ(0) b= 1 z−Tr−Ubx (6) where Ubx is the formal b +xelastic scattering optical potential. The last line of equation (5) indicates that the formal elimination of the core degrees of freedom necessarily leads to an effective interaction (Ubx) which, in general, will be complex and energy dependent. Note that, even at excitation energies below the first b excited state, the Ubx interaction will contain polarization effects which induce deviations from the bare b +xinteraction [12,13]. 2.2. CDCC equations in a binormal basis It is therefore required to extend CDCC calculations to consider complex and energy-dependent potentials for the core-valence interaction. As a starting point, let us briefly re-derive the standard CDCC equations [11], starting from the Schrödinger equation: (TR+Tr+Vbx +UbA +UxA −E)(R,r)=0.(7) Now we introduce the CDCC expansion for the wavefunction of the system : = bγJπ χJπ bγ(R)ϕJπ bγ(r)+ γJπdkχJπ γ(k,R)ϕJπ γ(k,r) ≃ bγJπ χJπ bγ(R)ϕJπ bγ(r)+ nγJπ χJπ nγ(R)ϕJπ nγ(r)≡ i χi(R)ϕi(r), (8) 2 M. Gómez-Ramos, J. Gómez-Camacho and A.M. Moro Physics Letters B 832 (2022) 137252 where ϕJπ bγ(r)are the bound eigenstates of potential Vbx and ϕJπ γ(k, r)its eigenfunctions in the continuum as a function of k, the modulus of the relative momentum between band x. Both sets of wavefunctions have a defined total angular momentum and parity Jπand γ={(lsx)jIbM}, the other quantum numbers required to describe the spin-angular configuration (the orbital angular momentum l, the spin of xs x, their sum j, the spin of bI band the total magnetic angular momentum Min this case). The continuum wave functions are approximated by a discrete set of functions ϕJπ nγ [11], built to be orthogonal. The following derivations do not distinguish between bound ϕJπ bγ(r)and discretized continuum states ϕJπ nγso we will use index ito denote both bound and discretized continuum states. χiis the coefficient of state ϕiand depends on R. As the wavefunctions are eigenvalues of the real potential Vbx, they are orthogonal so one can obtain a set of equations for χiby multiplying to the left by ϕi|:  j(TR−Ei)ϕi|ϕj+ϕi|UbA +UxA|ϕjχj(R)=  j(TR−Ei)δij +Uijχj(R)=0,(9) where the bracket denotes integration over r, and we have used (Tr+Vbx)ϕi(r) =iϕi(r)with Ei=E−iand introduced the coupling potentials Uij =ϕi|UbA +UxA|ϕj. In order to derive these expressions one fundamental assumption is the orthogonality of states ϕiand ϕj. However, if one were to consider a complex or energy-dependent potential, its eigenstates would no longer constitute an orthogonal basis, as is well known. However, even in this case it is possible to derive analogous expressions through the use of a binormal or biorthogonal basis ˜ ϕi, which is defined as [25]: ˜ ϕi|ϕj=ϕi|˜ ϕj=δij.(10) In this case, one needs only to multiply Eq. (9)by ˜ ϕito obtain the set of equations for χj:  j(TR−Ei)δij +˜ ϕiUbA +UxA ϕjχj(R)=0.(11) Therefore, one can solve the Schrödinger equation for a complex energy-dependent potential simply by modifying the coupling potentials ˜ Uij =˜ ϕiUbA +UxA ϕj, which however requires the knowledge of ˜ ϕi, the binormal states to the eigenstates of the potential. To compute the binormal states, we use a procedure which is similar to the discretized version of the expressions by McKellar and McKay [26]. The states ϕiused in our CDCC calculations include the bound ground state ϕ0, which is an eigenstate of the nucleon-core Hamiltonian with a real potential Vbx, plus a set of bins of continuum states ϕ(−) iof the complex potential Ubx (Vbx and Ubx can be different since we allow for energy dependence). These bins are, as per usual, constructed from continuum states which are asymptotically given as a unit-amplitude outgoing wave plus incoming waves ϕ(−) i∼h(1) (kr) −Ch(2) (kr), with h(1), h(2)being the outgoing and incoming Hankel functions. It is convenient to obtain ϕ(−) ias the complex conjugate of the solution ϕ(+) iof the conjugate potential U∗ bx, which verifies ϕ(+) i∼h(2) (kr) −Sh(1) (kr), with unit-amplitude incoming wave plus outgoing waves [10]. In order to obtain the binormal states ˜ ϕito the states ϕidescribed above, we start with the observation that, if the potentials were energy independent, the binormal states to ϕ(−) iwould precisely be the states ϕ(+)∗ i, the conjugates to the solutions of the same complex potential Ubx, but with unit-amplitude-incomingwave boundary conditions [27]. This observation gives us a first approach for the binormal states. ˜ ϕ(−) i≃ϕ(+)∗ i.(12) Calculations using this approximation will be labeled as “Complex non-orthogonal” or “Complex NO” in the following. Note that, in this approach, the binormal for the bound state would be the same bound state. Due to energy dependence, this approximation does not produce states that are strictly binormal. However, the states ϕ(+)∗ ican be taken as a first approximation to the binormal states, which should be valid when the energy dependence of the potential is small. The approximation ˜ ϕ(−) i∼ϕ(+)∗ ialso gives some insight on the effect of the complex potential in breakup reactions. In particular, for reactions at sufficiently high energies, the breakup process can be seen as one step from the bound state ϕ0to the continuum state ϕi, led by the coupling potential Ui0=ϕi|UbA +UxA|ϕ0for a real Uxb interaction. The inclusion of an absorptive complex potential in Uxb results (within this approximation) in a coupling potential ˜ Ui0∼ϕ(+)∗ i|UbA +UxA|ϕ0. Now, it is well known that the effects of absorption reduce the wavefunction in the nuclear interior, the region explored by Ui0, since ϕ0is bound. As such, one can expect  ˜ Ui0<|Ui0|, resulting in a smaller cross section, which is the intuitive effect of absorption. The proper calculation of the binormal states ˜ ϕito ensure orthogonality can be performed starting from the states ϕ(+)∗ iand finding the adequate linear combination that fulfills: ˜ ϕ(−) i= jA−1ji ϕ(+)∗ j(13) Aij =ϕ(−) i|ϕ(+)∗ j.(14) Note that states of different angular momentum and parity Jπare already orthogonal so the orthogonalization procedure only needs to consider bins with the same Jπ. Trivially, when the potential is real and energy-independent ϕ(−) i=ϕ(+)∗ iand ϕ(−) j|ϕ(−) i=δij, so the binormal ˜ ϕ(−) iis the same ϕ(−) i. 3. Results We will first consider as a test case the breakup reaction 12C(11Be,n10Be)12Cat 70 MeV per nucleon, for which experimental data exist [7]. For the real part of the interaction between neutron and 10Be, we have considered the potential from Capel et al. [28]. There is unfortunately no data on low-energy reaction cross sections for the n+10Be system. As such we have opted to fit the reaction cross sections taken from the compilation of [29]for p, n+9Be, rescaled by a factor (10/9)2/3to account for the different sizes of the nuclei. It should be noted that for 9Be non-elastic channels (in this case dissociation in n +2α) already open at a relative energy between nand 9Be of 1.6 MeV while for 10Be the first non-elastic channel (excitation to the 2+level) opens at a relative energy between nand 10Be, En10Be, of 3.4 MeV. As such, this procedure likely overestimates the imaginary potential for En10Be in this range. For the imaginary part of the n+10Be potential, we have considered the parametrization (for En10Be >0): W(En10Be,r)=− W0(En10Be) 1+exp (r−R)/a0 W0(En10Be)=(a(En10Be −Eb)+b)E4 n10Be E4 n10Be +E4 b ,(15) 3 M. Gómez-Ramos, J. Gómez-Camacho and A.M. Moro Physics Letters B 832 (2022) 137252 Fig. 2. n+10Be reaction cross section computed with the parametrization considered in this work. Experimental data have been adapted from those in [29](see text for details). The top panel shows the reaction cross section while the bottom panel shows the depth of the potential as a function of energy. Fig. 3. Quadrupole electric transition probability for the nucleus 11Be. The top and bottom panels correspond to Jπ=5/2and Jπ=3/2, respectively. The red solid line corresponds to a calculation with a complex Un10Be potential and bins, the black solid line to a real Un10Be potential and bins and the green thin line to a real potential and exact continuum wavefunctions. which presents the sharp increase seen in the data while being analytical. The results of the fit are shown in Fig. 2, where the top panel shows the reaction cross section and the bottom panel the depth of the potential, both as a function of En10Be. The parameters giving this fit are as follows: R =4.33 fm, a0=0.8fm, Eb=3 MeV, a =0.0143 and b =2MeV. Despite the aforementioned overestimation of the imaginary potential, we consider this description of the n-10Be interaction to be realistic enough for the purposes of this work. With this parametrization of the interaction we are able to explore the continuum of 11Be. The calculations were performed in general using 60 bins up to an energy of 30 MeV for each partial wave. States of negative energy except for the ground state have not been considered. We find results to be converged with this number of bins and verified the orthogonality between the bins and their binormal states, finding non-orthogonality in the norm which amounted to less than 10−8between any pair of states. We first focus on the states with L =2 between nand 10Be. We present in Fig. 3the B(E2)quadrupole reduced transition probability for the 5/2+and 3/2+states as a function of En10Be. In the case of a complex energy-dependent potential the formula for the Fig. 4. Differential breakup cross section for the reaction 12C(11Be, n +10 Be)12Cat 70 MeV per nucleon. The top panel corresponds to final states with Jπ=5/2+ and the bottom panel to states with Jπ=3/2+. The red solid line corresponds to calculations with a complex Un10Be potential and the black solid line corresponds to a real Un10Be potential. The red dot-dashed line corresponds to DWBA calculations with the complex potential while the red dotted line corresponds to a full CDCC calculation using Eq. (12)(see text for details). B(E2)changes in the intuitive way: B(E2) ∝|˜ ϕk|O(E2)|ϕgs|2. In the figure we present the calculation of the B(E2)with a real potential (the one presented with W(r) =0) considering exact continuum wavefunctions in the green thin line and continuum bins in the black solid line. For the complex potential, we can only compute the binormal states for the bins. Hence only the results for bins are presented in the red solid line. In general, the results are very similar for real and complex potentials, as can be expected, since the imaginary part is very small for energies En10Be <Eb. Only for the narrow resonance we see a significant discrepancy between the results with bins and whose with exact continuum wavefunctions, due to the abruptness of the behavior of the B(E2), which makes it difficult to describe with bins of reasonable width. However, for the very narrow 5/2+resonance we see a small dip when comparing the results with bins for real and complex potentials. This effect is enhanced when studying the cross section as a function of energy for the 12C(11Be,10 Be+n)12Creaction at 70 MeV/A, which is presented for the 5/2+and 3/2+states in Fig. 4. The breakup cross section has been computed using the CDCC formalism, with similar inputs to those in [30], in particular the optical potentials: Between 10Be and 12C a folding potential with the CEG07 interaction [31]was considered, while between nand 12C the Köning-Delaroche interaction (KD) [32]was used. In Fig. 4, the black and red solid lines correspond to real and complex potentials respectively. Here it can be seen that for Jπ=5/2+the effect of the complex interaction is strongest in the resonance, 4 M. Gómez-Ramos, J. Gómez-Camacho and A.M. Moro Physics Letters B 832 (2022) 137252 Fig. 5. Breakup cross section for the 12C(11Be, n +10 Be)12Creaction at 70 MeV per nucleon as a function of n−10Be energy. The solid red line corresponds to calculations with a complex Un10Be potential and the solid black line corresponds to a real Un10Be potential. Experimental data are taken from [7]. whose peak is reduced by a factor of 40%, while the tail above the resonance presents a reduction of ∼10% and the overall cross section is reduced from 23.8 mb to 19.9 mb. For jπ=3/2+, the effect is quite more noticeable, with a clear reduction of the cross section starting around Eb=3MeV, reaching 40% at 5-6 MeV and slowly reducing to 20% at 10 MeV, with the cross section reduced from 16.1 mb to 13.7 mb. The increased effect of the complex potential in the 5/2+resonance can be understood, as a resonant state is enhanced in the nuclear interior, where the absorption due to the imaginary part of the potential is largest. This, in turn, also explains the reduced effect in the B(E2), which is more sensitive to larger n+10Be distances than the (mostly nuclear) 12C(11Be,10 Be +n)12Creaction. The larger reduction found for the 3/2+states has a similar explanation, as the 3/2+continuum for this potential presents a broad resonant-like structure that spans the interval between 3 and 8 MeV and is thus more sensitive to absorption, although perhaps not to the extent of the narrow 5/2+resonance. In fact, for the other partial waves considered (sand pwaves), which do not present a resonant behavior, the effect of absorption is smaller, of less than 10%. The effects of the complex potential on the cross section should not be understood only as a consequence of absorption for the final channel, as the orthogonalization of the continuum states and the more standard coupled channel effects can also play a role. In order to clarify their relevance in Fig. 4two additional calculations are presented: In the red dot-dashed line, a DWBA calculation is performed to study the effects of coupled-channels, which are found to be minor in this case, with a 4-5% change in the total cross section, although in the 5/2+case at ∼4MeV the changes reach ∼10%. The red dotted line in Fig. 4corresponds to the complex nonorthogonal approximation where, instead of the binormal states ˜ ϕ(−) i, the functions ϕ(+)∗ iwere used. These states are not orthogonal and should formally not be used, as they introduce spurious contributions due to non-orthogonality. However, in this case they are found to give identical results to the binormal states, which can be related to the rather small energy dependence of the complex potential and the fact that the bound state does not correspond to these two waves. As a final observable for this reaction, in Fig. 5the full breakup cross section (including s-, pand d-wave breakup, as in [33,34]) for θcm =0 −12◦is presented for the real (black solid line) and complex (red solid line) potential. Due to computational limitations the discretization of the continuum for each partial wave corresponds to 30 bins instead of 60. Experimental data from [7] are shown for reference. Although in this work we do not seek agreement with the experimental data, it is worth mentioning that this reaction was shown in [12]to have an important contribution from the dynamic excitation of the 10Be core due to its interaction with the target. This contribution is not included in the present models. Therefore, the calculation with the real potential underestimates the data, and more so the one with the complex potential, which produces a smaller cross section due to the nucleon-core absorption. This disagreement could be improved through the explicit inclusion of core excitation in the reaction model together with a complex potential. This, however, is beyond the objectives of this work. For the full cross section, the effects of absorption follow the trend shown in the previous results, showing no effects for small energies except for the resonance, which is significantly reduced, and then a small reduction in the cross section appearing around E∼2.5 −3MeV and remaining for larger energies. The integrated cross sections are 80 mb for the real potential and 73 mb for the complex one, so the effect of the imaginary part of the potential amounts to a moderate reduction in cross section of ∼9%. The overestimation of the imaginary part of the n−10Be interaction at low energies mentioned previously may produce too large a difference between the two calculations in the region of the resonance, so the reduction factor could be even smaller. The modesty of the effects of the complex potential can be ascribed to three factors. Firstly, the strength of the imaginary potential is quite small. As seen in Fig. 2, it barely reaches 2MeV for energies below 20 MeV. Secondly, the nucleus considered, 11Be, is weakly bound and exhibits a halo structure so the breakup reaction explores large n−10Be distances, where the effect of the imaginary potential is weaker. Thirdly, the breakup of a weakly-bound nucleus preferably populates states with smaller momentum and energy, where the lack of open channels which could absorb flux results in a smaller imaginary potential at these low energies, and thus in a smaller effect on the observables. It is therefore interesting to explore a more deeply-bound nucleus, to assess how the effects of absorption between nucleon and core evolve with binding energy. We have therefore chosen to study the reaction 12C(41Ca, n +40 Ca)12Cat 70 MeV/A, which is analogous to the previous one save for the studied nucleus. For the optical potentials in this reaction, we have kept the KD interaction for neutron-12C, and used the São Paulo interaction [35]with the standard normalization factors 1 and 0.78 for the real and imaginary parts of the 12C-40Ca interaction. For the interaction between neutron and 40Ca we used the KD global potential, which reproduces the reaction cross section between neutron and 40Ca for a wide range of energies as well as the binding energy of the valence neutron in 41Ca, at which the interaction becomes real. The calculation considered for the continuum all partial waves with L =0 −3 between nand 40Ca. Each partial wave was described with 25 bins up to a n-40Ca relative energy (En40Ca) of 30 MeV. The ground state was the only negative-energy state considered, as other bound states are not reproduced by the KD interaction. The cross section as a function of energy is presented in Fig. 6, where the red solid line corresponds to the full complex energydependent interaction between neutron and 40Ca while the black solid line corresponds to the real energy-independent interaction, which is taken as the KD interaction at the (negative) neutron separation energy, thus producing the same bound state. As can be seen in the figure the effects of the complex interaction are in this case significantly larger, reducing the cross section by ∼50%. The distribution for the real energy-independent potential presents a resonant-like structure with Jπ=5/2+at En40Ca ∼2 MeV and another very broad structure with Jπ=7/2−at En40Ca ∼6-7 MeV, 5 M. Gómez-Ramos, J. Gómez-Camacho and A.M. Moro Physics Letters B 832 (2022) 137252 Fig. 6. Energy differential breakup cross section for the 12C(41Ca, n +40 Ca)12Creaction at 70 MeV per nucleon as a function of the relative energy between nand 40Ca. The solid red line corresponds to calculations with a complex Un40Ca potential and the solid black line corresponds to a real Un40Ca potential. The dot-dashed red line corresponds to a DWBA calculation with the complex potential while the dotted red line corresponds to the full CDCC calculation using Eq. (12)(see text for details). both of which are significantly reduced by the complex energydependent potential. As in Fig. 4, we present in the red dot-dashed line the results of the DWBA calculation, finding also in this case that higher-order effects are minor, which can be expected, as the energy of the reaction is similar. However, we find a very strong effect when considering the complex non-orthogonal approximation (i.e., using ϕ(+)∗ iinstead of ˜ ϕ(−) i), producing a cross section that is much larger than even that for a real potential. Further exploration shows that the 7/2−wave presents the largest effect for this calculation. This large spurious contribution originates in the non-orthogonality between the continuum states and the bound state (which has also Jπ=7/2−). This shows that it is fundamental to ensure the orthogonality of states, specially for the partial wave associated to the bound state. The factors that explained the modesty of the effects of the complex potential in the weakly-bound 11Be can be used to understand the larger effect on 41Ca. The larger binding energy in 41Ca extends the cross section to larger n-40Ca energies, where the effect of the imaginary potential is larger. In addition, the imaginary potential is larger in the KD interaction, with a surface term that is present even at zero energy, which explains the strong effect even at low energies. This reduction of the cross section, which is found to be heavily dependent on the binding energy of the removed nucleon, bears a tantalizing resemblance to that found in nucleon-knockout reactions at intermediate energies, where experimental cross sections show a significant reduction when compared to theoretical calculations when removing the deeply-bound species in an asymmetric nucleus, but barely any reduction when removing the weakly-bound species [2,15]. This reduction is a currently heavilydebated topic, since a similar trend was not found for transfer [36] and quasifree proton-induced removal reactions [37–39]. A recent overview on this topic can be found in [40]. It should be noted that the kind of effects explored in this work, that is, the absorption of the valence particle due to the interaction with the core (which may lead to a destruction of the core) are not considered in the usual treatment of high-energy nucleonknockout reactions, which is usually described in the eikonal sudden approximation [41]in which, during the collision, the core and valence nucleon are assumed to move independently and not interact. On the other hand, both in proton-induced reactions, with the usual description with the Distorted-Wave Impulse Approximation (DWIA) [3,42–44], and in transfer reactions, with the widelyused DWBA, this interaction and possible absorption is considered, at least in an approximate way. The method presented in this work has been developed for the so-called exclusive or diffractive breakup reactions, in which the valence and core particles survive and are detected. However, in nucleon-knockout reactions, specially for deeply-bound nucleons, the main contributor to the cross section is the stripping reaction, in which the valence particle is absorbed and only the core is detected. This requires a description beyond CDCC, such as the aforementioned eikonal sudden approximation or the quantum IAV model [16,45]. Extensions of these models to include core-valence absorption are, by the results of this work, promising avenues to solve the open problem of the reduction factors in nucleonremoval cross sections and their exploration is currently underway. 4. Summary and conclusions In this work, we have presented a method for extending the CDCC formalism to include absorption and excitation effects between core and valence particle via the use of complex energydependent potentials, which require a binormal basis for discretized continuum states to ensure orthogonality. The method has been applied to study the breakup cross sections for the reactions 12C(11Be,10 Be +n)12C and 12C(41Ca,40 Ca +n)12Cat 70 MeV/A, finding a moderate reduction of the cross section for the breakup of 11Be and a much larger reduction for 41Ca, which is associated to the larger binding energy of the latter. This dependence of the reduction of the cross section on the binding-energy of the removed species is very similar to the open problem of the reduction factors in nucleon-knockout reactions and offers an explanation for these reduction factors in the absorption due to the interaction between valence nucleon and residual core. Further investigation is required on this topic, in particular the extension of these effects to stripping cross sections. The method presented in this work permits the inclusion of open channels in the study of breakup reactions, whose effects are ignored in their usual description. Therefore, it allows for a more realistic treatment of these reactions, in particular in the removal of more deeply-bound nucleons, for which these effects are more important. The effects of absorption on resonant states are found to be particularly intense, for resonances at moderate and high excitation energies, where open channels exist. So, they should be considered for a proper understanding of the relation of the structure properties of these resonances with the measured cross sections. Given that the present method only modifies the coupling potentials, its inclusion in standard nuclear reaction codes such as fresco [46]is straightforward. The results of this work also open up the study of breakup reactions with microscopic optical potentials, which up until now were precluded from use in CDCC calculations due to them being complex and energy-dependent. Therefore, the presented formalism serves as a connection between state-of-the-art descriptions of nuclear reactions and nuclear structure. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This work is partially supported by the I+D+i project PID2020114687GB-I00 funded by MCIN/AEI/10.13039/501100011033, by the grant Group FQM-160 funded by the Consejería de Economía, 6 M. Gómez-Ramos, J. Gómez-Camacho and A.M. Moro Physics Letters B 832 (2022) 137252 Conocimiento, Empresas y Universidad, Junta de Andalucía (Spain), and by project P20_01247, funded by the Consejería de Economía, Conocimiento, Empresas y Universidad, Junta de Andalucía (Spain) with funds from “ERDF A way of making Europe”. M.G.-R. is supported by the Junta de Andalucía under grant number DOC-01006 in the plan PAIDI 2020, which receives funds from the European Social Fund. References [1] M.H. Macfarlane, J.B. French, Stripping reactions and the structure of light and intermediate nuclei, Rev. Mod. Phys. 32 (1960) 567–691, https://doi .org /10 . 1103 /RevModPhys .32 .567, https://link.aps .org /doi /10 .1103 /RevModPhys .32 .567. [2] A. Gade, P. Adrich, D. Bazin, M.D. Bowen, B.A. Brown, C.M. Campbell, J.M. Cook, T. Glasmacher, P.G. Hansen, K. Hosier, S. McDaniel, D. McGlinchery, A. Obertelli, K. Siwek, L.A. Riley, J.A. Tostevin, D. Weisshaar, Reduction of spectroscopic strength: weakly-bound and strongly-bound single-particle states studied using one-nucleon knockout reactions, Phys. Rev. C 77 (2008) 044306, https://doi . org /10 .1103 /PhysRevC .77.044306, https://link.aps .org /doi /10 .1103 /PhysRevC .77. 044306. [3] G. Jacob, T.A.J. Maris, Quasi-free scattering and nuclear structure, Rev. Mod. Phys. 38 (1966) 121–142, https://doi .org /10 .1103 /RevModPhys .38 .121, https:// link.aps .org /doi /10 .1103 /RevModPhys .38 .121. [4] L. Lapikás, Quasi-elastic electron scattering off nuclei, Nucl. Phys. A 553 (1993) 297–308, https://doi .org /10 .1016 /0375 -9474(93 )90630 -G, https://www. sciencedirect .com /science /article /pii /037594749390630G. [5] R. Chatterjee, R. Shyam, Breakup reactions of light and medium mass neutron drip line nuclei, Prog. Part. Nucl. Phys. 103 (2018) 67–108, https://doi . org /10 .1016 /j .ppnp .2018 .06 .001, https://www.sciencedirect .com /science /article / pii /S014664101830053X. [6] V. Pesudo, M.J.G. Borge, A.M. Moro, J.A. Lay, E. Nácher, J. Gómez-Camacho, O. Tengblad, L. Acosta, M. Alcorta, M.A.G. Alvarez, C. Andreoiu, P.C. Bender, R. Braid, M. Cubero, A. Di Pietro, J.P. Fernández-García, P. Figuera, M. Fisichella, B.R. Fulton, A.B. Garnsworthy, G. Hackman, U. Hager, O.S. Kirsebom, K. Kuhn, M. Lattuada, G. Marquínez-Durán, I. Martel, D. Miller, M. Moukaddam, P.D. O’Malley, A. Perea, M.M. Rajabali, A.M. Sánchez-Benítez, F. Sarazin, V. Scuderi, C.E. Svensson, C. Unsworth, Z.M. Wang, Scattering of the halo nucleus 11Be on 197Au at energies around the Coulomb barrier, Phys. Rev. Lett. 118 (2017) 152502, https://doi .org /10 .1103 /PhysRevLett .118 .152502, https:// link.aps .org /doi /10 .1103 /PhysRevLett .118 .152502. [7] N. Fukuda, T. Nakamura, N. Aoi, N. Imai, M. Ishihara, T. Kobayashi, H. Iwasaki, T. Kubo, A. Mengoni, M. Notani, H. Otsu, H. Sakurai, S. Shimoura, T. Teranishi, Y.X. Watanabe, K. Yoneda, Coulomb and nuclear breakup of a halo nucleus 11Be, Phys. Rev. C 70 (2004) 054606, https://doi .org /10 .1103 /PhysRevC .70 .054606, https://link.aps .org /doi /10 .1103 /PhysRevC .70 .054606. [8] T. Nakamura, N. Fukuda, T. Kobayashi, N. Aoi, H. Iwasaki, T. Kubo, A. Mengoni, M. Notani, H. Otsu, H. Sakurai, S. Shimoura, T. Teranishi, Y.X. Watanabe, K. Yoneda, M. Ishihara, Coulomb dissociation of 19Cand its halo structure, Phys. Rev. Lett. 83 (1999) 1112–1115, https://doi .org /10 .1103 /PhysRevLett .83 .1112, https://link.aps .org /doi /10 .1103 /PhysRevLett .83 .1112. [9] T. Nakamura, A.M. Vinodkumar, T. Sugimoto, N. Aoi, H. Baba, D. Bazin, N. Fukuda, T. Gomi, H. Hasegawa, N. Imai, M. Ishihara, T. Kobayashi, Y. Kondo, T. Kubo, M. Miura, T. Motobayashi, H. Otsu, A. Saito, H. Sakurai, S. Shimoura, K. Watanabe, Y.X. Watanabe, T. Yakushiji, Y. Yanagisawa, K. Yoneda, Observation of strong low-lying e1strength in the two-neutron halo nucleus 11Li, Phys. Rev. Lett. 96 (2006) 252502, https://doi .org /10 .1103 /PhysRevLett .96 .252502, https:// link.aps .org /doi /10 .1103 /PhysRevLett .96 .252502. [10] G.R. Satchler, Direct Nuclear Reactions, Clarendon Press, Oxford (UK), 1983. [11] N. Austern, Y. Iseri, M. Kamimura, M. Kawai, G. Rawitscher, M. Yahiro, Continuum-discretized coupled-channels calculations for three-body models of deuteron-nucleus reactions, Phys. Rep. 154 (3) (1987) 125–204, https://doi .org /10 .1016 /0370 -1573(87 )90094 -9, https://www.sciencedirect . com /science /article /pii /0370157387900949. [12] A.M. Moro, J.A. Lay, Interplay between valence and core excitation mechanisms in the breakup of halo nuclei, Phys. Rev. Lett. 109 (2012) 232502, https://doi .org /10 .1103 /PhysRevLett .109 .232502, https://link.aps .org / doi /10 .1103 /PhysRevLett .109 .232502. [13] P. Capel, D. Phillips, H.-W. Hammer, Simulating core excitation in breakup reactions of halo nuclei using an effective three-body force, Phys. Lett. B (2021) 136847, https://doi .org /10 .1016 /j .physletb .2021.136847. [14] C. Louchart, A. Obertelli, A. Boudard, F. Flavigny, Nucleon removal from unstable nuclei investigated via intranuclear cascade, Phys. Rev. C 83 (2011) 011601, https://doi .org /10 .1103 /PhysRevC .83 .011601, https://link.aps .org /doi / 10 .1103 /PhysRevC .83 .011601. [15] J.A. Tostevin, A. Gade, Updated systematics of intermediate-energy singlenucleon removal cross sections, Phys. Rev. C 103 (2021) 054610, https:// doi .org /10 .1103 /PhysRevC .103 .054610, https://link.aps .org /doi /10 .1103 / PhysRevC .103 .054610. [16] M. Ichimura, N. Austern, C.M. Vincent, Equivalence of post and prior sum rules for inclusive breakup reactions, Phys. Rev. C 32 (1985) 431–439, https://doi .org / 10 .1103 /PhysRevC .32 .431, https://link.aps .org /doi /10 .1103 /PhysRevC .32 .431. [17] L.D. Faddeev, Scattering theory for a three-particle system, Zh. Eksp. Teor. Fiz. 39 (1960) 1459, Sov. Phys. JETP 12 (1961) 1014. [18] E. Alt, P. Grassberger, W. Sandhas, Reduction of the three-particle collision problem to multi-channel two-particle Lippmann-Schwinger equations, Nucl. Phys. B 2(2) (1967) 167–180, https://doi .org /10 .1016 /0550 -3213(67 )90016 -8, http://www.sciencedirect .com /science /article /pii /0550321367900168. [19] E. Cravo, R. Crespo, A. Deltuva, Valence neutron-core interaction effects on the breakup of halo nuclei, Phys. Rev. C 87 (2013) 034612, https://doi .org /10 .1103 / PhysRevC .87.034612, https://link.aps .org /doi /10 .1103 /PhysRevC .87.034612. [20] W. Dickhoff, R. Charity, Recent developments for the optical model of nuclei, Prog. Part. Nucl. Phys. 105 (2019) 252–299, https://doi .org /10 .1016 /j .ppnp .2018 . 11.002, https://www.sciencedirect .com /science /article /pii /S0146641018300875. [21] A. Idini, C. Barbieri, P. Navrátil, Ab initio optical potentials and nucleon scattering on medium mass nuclei, Phys. Rev. Lett. 123 (2019) 092501, https://doi .org / 10 .1103 /PhysRevLett .123 .092501, https://link.aps .org /doi /10 .1103 /PhysRevLett . 123 .092501. [22] J. Rotureau, P. Danielewicz, G. Hagen, G.R. Jansen, F.M. Nunes, Microscopic optical potentials for calcium isotopes, Phys. Rev. C 98 (2018) 044625, https://doi . org /10 .1103 /PhysRevC .98 .044625, https://link.aps .org /doi /10 .1103 /PhysRevC .98 . 044625. [23] N.C. Summers, et al., Extended continuum discretized coupled channels method: core excitation in the breakup of exotic nuclei, Phys. Rev. C 74 (2006) 014606, https://doi .org /10 .1103 /PhysRevC .74 .014606. [24] R. de Diego, J.M. Arias, J.A. Lay, A.M. Moro, Continuum-discretized coupledchannels calculations with core excitation, Phys. Rev. C 89 (2014) 064609, https://doi .org /10 .1103 /PhysRevC .89 .064609. [25] R.G. Newton, Scattering Theory of Waves and Particles, Dover, 1982. [26] B.H. McKellar, C.M. McKay, Formal scattering theory for energy-dependent potentials, Aust. J. Phys. 36 (1983) 607. [27] D.C. Brody, Biorthogonal quantum mechanics, J. Phys. A, Math. Theor. 47 (3) (2013) 035305, https://doi .org /10 .1088 /1751 -8113 /47 /3 /035305. [28] P. Capel, G. Goldstein, D. Baye, Time-dependent analysis of the breakup of 11Be on 12Cat 67 MeV/nucleon, Phys. Rev. C 70 (2004) 064605, https://doi . org /10 .1103 /PhysRevC .70 .064605, https://link.aps .org /doi /10 .1103 /PhysRevC .70 . 064605. [29] A. Bonaccorso, R.J. Charity, Optical potential for the n-9Be reaction, Phys. Rev. C 89 (2014) 024619, https://doi .org /10 .1103 /PhysRevC .89 .024619, https://link. aps .org /doi /10 .1103 /PhysRevC .89 .024619. [30] A. Moro, J. Lay, J. Gómez Camacho, Determining B(E1) distributions of weakly bound nuclei from breakup cross sections using continuum discretized coupled channels calculations. Application to 11Be, Phys. Lett. B 811 (2020) 135959, https://doi .org /10 .1016 /j .physletb .2020 .135959, https://www. sciencedirect .com /science /article /pii /S0370269320307620. [31] T. Furumoto, W. Horiuchi, M. Takashina, Y. Yamamoto, Y. Sakuragi, Global optical potential for nucleus-nucleus systems from 50 MeV/u to 400 MeV/u, Phys. Rev. C 85 (2012) 044607, https://doi .org /10 .1103 /PhysRevC .85 .044607, https:// link.aps .org /doi /10 .1103 /PhysRevC .85 .044607. [32] A. Koning, J. Delaroche, Local and global nucleon optical models from 1 keV to 200 MeV, Nucl. Phys. A 713 (3) (2003) 231–310, https:// doi .org /10 .1016 /S0375 -9474(02 )01321 -0, https://www.sciencedirect .com / science /article /pii /S0375947402013210. [33] J.A. Lay, A.M. Moro, J.M. Arias, J. Gómez-Camacho, Exploring continuum structures with a pseudo-state basis, Phys. Rev. C 82 (2010) 024605, https://doi . org /10 .1103 /PhysRevC .82 .024605, https://link.aps .org /doi /10 .1103 /PhysRevC .82 . 024605. [34] D.J. Howell, J.A. Tostevin, J.S. Al-Khalili, Coupled channels calculations of 11Be breakup, J. Phys. G, Nucl. Part. Phys. 31 (10) (2005) S1881–S1884, https://doi . org /10 .1088 /0954 -3899 /31 /10 /093. [35] L.C. Chamon, D. Pereira, M.S. Hussein, M.A. Cândido Ribeiro, D. Galetti, Nonlocal description of the nucleus-nucleus interaction, Phys. Rev. Lett. 79 (1997) 5218–5221, https://doi .org /10 .1103 /PhysRevLett .79 .5218, https://link. aps .org /doi /10 .1103 /PhysRevLett .79 .5218. [36] F. Flavigny, et al., Limited asymmetry dependence of correlations from single nucleon transfer, Phys. Rev. Lett. 110 (2013) 122503, https://doi .org / 10 .1103 /PhysRevLett .110 .122503, https://link.aps .org /doi /10 .1103 /PhysRevLett . 110 .122503. [37] L. Atar, et al., Quasifree (p,2p) reactions on oxygen isotopes: observation of isospin independence of the reduced single-particle strength, Phys. Rev. Lett. 120 (2018) 052501, https://doi .org /10 .1103 /PhysRevLett .120 .052501, https:// link.aps .org /doi /10 .1103 /PhysRevLett .120 .052501. [38] M. Holl, et al., Quasi-free neutron and proton knockout reactions from light nuclei in a wide neutron-to-proton asymmetry range, Phys. Lett. B 795 (2019) 682–688, https://doi .org /10 .1016 /j .physletb .2019 .06 .069, https://www. sciencedirect .com /science /article /pii /S0370269319304666. [39] M. Gómez-Ramos, A. Moro, Binding-energy independence of reduced spectroscopic strengths derived from (p, 2p)and (p, pn)reactions with nitrogen and oxygen isotopes, Phys. Lett. B 785 (2018) 511–516, https://doi .org /10 . 7 M. Gómez-Ramos, J. Gómez-Camacho and A.M. Moro Physics Letters B 832 (2022) 137252 1016 /j .physletb .2018 .08 .058, https://www.sciencedirect .com /science /article /pii / S0370269318306749. [40] T. Aumann, C. Barbieri, D. Bazin, C. Bertulani, A. Bonaccorso, W. Dickhoff, A. Gade, M. Gómez-Ramos, B. Kay, A. Moro, T. Nakamura, A. Obertelli, K. Ogata, S. Paschalis, T. Uesaka, Quenching of single-particle strength from direct reactions with stable and rare-isotope beams, Prog. Part. Nucl. Phys. 118 (2021) 103847, https://doi .org /10 .1016 /j .ppnp .2021.103847, https://www.sciencedirect . com /science /article /pii /S0146641021000016. [41] P. Hansen, J. Tostevin, Direct reactions with exotic nuclei, Annu. Rev. Nucl. Part. Sci. 53 (1) (2003) 219–261, https://doi .org /10 .1146 /annurev.nucl .53 .041002 . 110406. [42] G. Jacob, T.A.J. Maris, Quasi-free scattering and nuclear structure. II, Rev. Mod. Phys. 45 (1973) 6–21, https://doi .org /10 .1103 /RevModPhys .45 .6, https:// link.aps .org /doi /10 .1103 /RevModPhys .45 .6. [43] K. Ogata, K. Yoshida, K. Minomo, Asymmetry of the parallel momentum distribution of (p, pn) reaction residues, Phys. Rev. C 92 (2015) 034616, https://doi . org /10 .1103 /PhysRevC .92 .034616, https://link.aps .org /doi /10 .1103 /PhysRevC .92 . 034616. [44] T. Aumann, C.A. Bertulani, J. Ryckebusch, Quasifree (p,2p) and (p,pn) reactions with unstable nuclei, Phys. Rev. C 88 (2013) 064610, https://doi .org /10 .1103 / PhysRevC .88 .064610, https://link.aps .org /doi /10 .1103 /PhysRevC .88 .064610. [45] J. Lei, A.M. Moro, Reexamining closed-form formulae for inclusive breakup: application to deuteronand 6Li-induced reactions, Phys. Rev. C 92 (2015) 044616, https://doi .org /10 .1103 /PhysRevC .92 .044616, https://link.aps .org /doi / 10 .1103 /PhysRevC .92 .044616. [46] I.J. Thompson, Coupled reaction channels calculations in nuclear physics, Comput. Phys. Rep. 7(4) (1988) 167–212, https://doi .org /10 .1016 / 0167 -7977(88 )90005 -6, https://www.sciencedirect .com /science /article /pii / 0167797788900056. 8