The art of modeling nuclear reactions with weakly bound nuclei: status and perspectives
Abstract
We give an overview of the theoretical description of nuclear reactions involving weakly-bound nuclei. Some of the more widespread reaction formalisms employed in the analysis of these reactions are briefly introduced, including various recent developments. We put special emphasis on the continuum-discretized coupled-channel (CDCC) method and its extensions to incorporate core and target excitations as well as its application to three-body projectiles. The role of the continuum for one-nucleon transfer reactions is also discussed. The problem of the evaluation of inclusive breakup cross sections is addressed within the Ichimura–Austern–Vincent (IAV)model.Other methods, such as those based on a semiclasical description of the scattering process, are also briefly introduced and some of their applications are discussed and a brief discussion on topics of current interest, such as nucleon-nucleon correlations, uncertainty evaluation and non-locality is presented.
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Eur. Phys. J. A (2025) 61:47 https://doi.org/10.1140/epja/s10050-025-01500-0 Review The art of modeling nuclear reactions with weakly bound nuclei: status and perspectives Antonio M. Moroa, Jesús Casal , Mario Gómez-Ramos Departamento de Física Atómica, Molecular y Nuclear, Facultad de Física, Universidad de Sevilla, Apartado 1065, 41080 Sevilla, Spain Received: 31 July 2024 / Accepted: 22 January 2025 © The Author(s) 2025 Communicated by Maria Borge Abstract We give an overview of the theoretical description of nuclear reactions involving weakly-bound nuclei. Some of the more widespread reaction formalisms employed in the analysis of these reactions are briefly introduced, including various recent developments. We put special emphasis on the continuum-discretized coupled-channel (CDCC) method and its extensions to incorporate core and target excitations as well as its application to three-body projectiles. The role of the continuum for one-nucleon transfer reactions is also discussed. The problem of the evaluation of inclusive breakup cross sections is addressed within theIchimura–Austern–Vincent(IAV)model.Other methods, such as those based on a semiclasical description of the scattering process, are also briefly introduced and some of their applications are discussed and a brief discussion on topics of current interest, such as nucleon-nucleon correlations, uncertainty evaluation and non-locality is presented. Contents 1 Introduction ...................... 2 The impact of weak-binding on the elastic cross section: optical model approach .............. 2.1 Evaluation of the polarization potential in a simple case: the adiabatic polarization potential ... 2.2 The phenomenological optical model ....... 3 Inclusion of breakup: the CDCC method ....... 3.1 Brief resumè of the CDCC method ........ 3.2 The role of closed channels ............ 3.3 Trivially equivalent polarization potential .... 3.4 Inclusion of core excitations ........... 3.5 Inclusion of target excitations ........... 3.6 Four-body CDCC ................. 3.7 Microscopic CDCC ................ ae-mail: [email protected] (corresponding author) 3.8 Interpretation of the CDCC wavefunction and cross sections ................... 3.8.1 Smoothingprocedurefortwo-bodyobservables .................... 3.8.2 Three-body observables .......... 3.8.3 Pseudostates versus bins .......... 4 Transfer reactions with weakly bound nuclei ..... The Johnson–Soper approximation .......... The Johnson–Tandy approximation .......... The CDCC-BA approximation ............. 4.1 Transfer reactions populating unbound systems . 4.2 Transfer reactions involving three-body projectiles 4.3 Simultaneous inclusion of projectile breakup and target excitation in transfer reactions .... 5 Inclusive breakup reactions .............. 5.1 The Ichimura–Austern–Vincent (IAV) model .. Application to three-body projectiles ......... 5.2 The Eikonal Hussein–McVoy formula (EHM) .. 5.3 Interpretation of inclusive breakup data ..... 6 Fusion involving weakly bound nuclei ........ 6.1 Computation of CF and ICF with CDCC .... 6.2 Evaluation of CF and ICF cross sections with the IAV model ................... 6.3 Application to surrogate reactions ........ 7 Semiclassical description of breakup and transfer reactions ........................ 7.1 The semiclassical formalism of Alder and Winther 7.2 Dynamic Coulomb polarization potential from the AW theory ................... 7.3 Semiclassical transfer-to-the-continuum model . 7.4 Dynamical Eikonal approximation ........ 8 Extraction the electric transition probabilities from Coulomb dissociation data ............... 8.1 Semiclassical analysis of Coulomb dissociation data 8.2 Quantum-mechanicaleffects:CDCCanalysisof Coulomb dissociation data ............ 0123456789().: V,-vol 123
47 Page 2 of 57 Eur. Phys. J. A (2025) 61:47 9 Study of nucleon–nucleon correlations from nucleon removal reactions ................... 10Uncertainty evaluation ................. 11Inclusion of non-local potentials ............ 12Conclusions and perspectives ............. References ......................... 1 Introduction Nuclear reactions are key tools to extract information about the structure of atomic nuclei, and the dynamical phenomena arising from nucleus-nucleus forces. When one of the colliding partners is weakly bound, new phenomena and mechanisms arise, requiring in some cases specific reaction frameworks tailored to the peculiarities of these systems and their interactions. Weakly-bound nuclei appear in the proximity of the neutron and proton driplines, a region where new exotic structures and phenomena are found. Promiment examples are halo nuclei, weakly bound nuclei composed of a compact core and one or two loosely bound nucleons with an unusually large matter radius, or Borromean systems, threebody systems with no bound binary subsystems, such as 9Be (α +α+n)or 6He (α +n+n). Although the field has experienced a great impulse in the last decades, the physics of nuclear scattering with weaklybound projectiles is not new and many of the problems that are being addressed were already recognized much earlier as, forexample,inthecontextoflow-energydeuteronscattering. The deuteron, while being a stable nucleus, displays many halo-like features, such as weak binding (Eb=2.22 MeV), large spatial extension (the proton-neutron separation is about 3.8 fm) and no bound excited states. To illustrate the role of the weak binding on the scattering observables, let us consider the scattering of lowenergy deuterons (a few MeV) by a heavy target nucleus, like 208Pb. Initially, when the deuteron is far apart from the target nucleus, it is in an internal state given by the ground-state of the proton-neutron Hamiltonian. As the deuteron approaches the target, it will feel its Coulomb repulsion. If the deuteron were a point-like particle, the effect of this interaction would be to distort the trajectory of the deuteron, without altering its internal structure. However, this is not the case. The Coulomb interaction acts on the proton, whose distance from the center-of-mass of the deuteron is ∼2 fm. Consequently, in addition to the monopole Coulomb potential (∝1/R)the deuteron will feel higher-order Coulomb multipoles arising from the expansion, valid for rR, given by [1] VC(r, R)=Zte2 | R+r/2|=Zte2 R−Zte2 2R2rcos(θ) +... (1) Fig. 1 Relevant coordinates for a deuteron+target scattering problem where Ris the coordinate from the target to the deuteron c.m. rthe proton-neutron relative coordinate and θis the angle between them (see Fig. 1). The main deviation from the point Coulomb interaction is caused by the dipole term which depends on the orientation of the proton-neutron relative coordinate with respect to the deuteron-target coordinate. Specifically, when the proton is closer to the target nucleus (cos(θ) < 0), the dipole potential will add a positive contribution, whereas when the proton is farther from the c.m. of the deuteron the contribution will be negative. Classically, we may think of this problem as an electric dipole moving in a slowly varying electric field, in which the tidal force exerted on the dipole will favour the configuration of Fig. 1. Quantum-mechanically, the perturbation induced by the dipole force will couple the deuteron ground state (positiveparity)with negative-paritystates.Sincethe ground-state is the only bound state, these states necessarily appear in the continuum. Therefore, the polarization effect will modify the state of the system, producing a new one in which the ground state is mixed with continuum states of negative parity. This has a twofold effect on the outcome of the scattering process. First, it will produce deviations of the elastic scattering with respect to the Rutherford formula. Second, the coupling with the positive-energy states will give rise to some dissociation probability, that is, the breakup of the deuteron into a proton and a neutron. In the next sections, we will discuss several methods to incorporate these two effects in the reaction formalisms. The discussed example, albeit describing a specific problem, exhibits some general features commonly found in the scattering of weakly bound nuclei. In particular, the coupling to the breakup channels will play a role, to a larger or smaller extent, in essentially all reaction observables. Therefore,reactionmodelsemployedto describe these observables will have to incorporate this effect. We enumerate some fingerprints of the weak binding on reaction observables: –Large interaction cross sections in nuclear collisions at high energies. Historically, the first evidence of the unusual properties of halo nuclei came from the pioneering experiments performed by Tanihata and co-workers at Berkeley using very energetic (800 MeV/nucleon) sec123
Eur. Phys. J. A (2025) 61:47 Page 3 of 57 47 ondary beams of radioactive species [2,3]. At these high energies, interaction cross sections are approximately proportional to the size of the colliding nuclei. It was found that some exotic isotopes of light nuclei (6He, 11Li, 14Be) presented much higher interaction cross sections than their neighbour isotopes, which was interpreted as an abnormally large radius. –Narrow momentum distributions of residues following fast nucleon removal. Momentum distributions of the residual nucleus following the removal of one or more nucleons of a energetic projectile colliding with a target nucleus are closely related to the momentum distribution of the removed nucleon(s) in the original projectile. Kobayashi et al. [4] found that the momentum distributions of 9Li following the fragmentation process 11Li +12C→9Li +X were abnormally narrow, which, according to the Heisenberg’s uncertainty principle, suggested a long tail in the density distribution of the 11Li nucleus. This result was later found in other weakly bound nuclei. –Abnormal elastic scattering cross sections. Elastic scattering is affected by the coupling to non-elastic processes. In particular, when coupling to breakup channels is important, elastic scattering cross sections are depleted with respect to the case of tightly bound nuclei. Some other key signatures are the departure of the elastic cross section from the Rutherford cross section at subCoulomb energies and the disappearance of the Fresnel peak at near-barrier energies in reactions induced by halo nuclei on heavy targets [5–8]. –Enhanced near-threshold breakup cross section in Coulomb dissociation experiments of neutron-halo nuclei. When a neutron-halo nucleus, composed of a charged core and one or two weakly-bound neutrons (11Be, 6He, 11Li,…) collides with a high-Ztarget nucleus, the projectile structure is heavily distorted due to the tidal force originated from the uneven action of the Coulomb interaction on the charged core and the neutrons. This produces a stretching which may eventually break up the loosely bound projectile. This gives rise to a large population of the continuum states close to the breakup threshold. –Complete fusion suppression Experiments with weakly bound light stable nuclei (such as 6,7Li and 9Be) have shown a systematic suppression of complete fusion (CF) cross sections (defined as capture of the complete charge of the projectile) of ∼20-30% compared to the case of tightly bound nuclei [9–15]. The effect has been attributed to the presence of strong competing channels, such as the breakup of the weakly bound projectile prior to reaching the fusion barrier, with the subsequent reduction of capture probability. This interpretation is supported by the presence of large αyields as well as target-like residues which are consistent with the capture of one of the fragment constituents of the projectile, a process which is usually termed as incomplete fusion (ICF). Aproper andquantitativeunderstandingofthese andother phenomena requires the use of an appropriate reaction theory. When dealing with weakly-bound systems, one expects acertaindecouplingbetweenthedegreesof freedom describing the relative motion between the weakly-bound nucleon (or cluster) from that of the internal excitations of the clusters themselves. Following this argument, one may be tempted to adopt an extreme model in which only the degree of freedom for the relative motion of the weakly-bound nucleon(s) or clusters are considered, while the others are simply ignored or frozen. This approach, which reminds of the separation between active nucleons and core nucleons in the shellmodel, has in fact become very useful to understand the main features of the structure of halo, and other weaklybound, nuclei and their dynamics. In some cases, such as the deuteron system or one-nucleon halos, such as 19Cor11Be, a two-body model may provide a reasonable starting point for the description of these nuclei. However, in some other systems, such as the Borromean nuclei 11Li, 6He or 9Be, it will be in general mandatory to resort to (at least) a threebody model for a meaningful description of their structure and reactions. One of the most successful models for describing reactions involving weakly-bound nuclei, which incorporates in a natural way the few-body structure of these systems as well as its breakup, is the Continuum Discretized CoupledChannels (CDCC) [16]. The model and its recent extensions will be discussed in Sect. 3. Other models, tailored to specific processes, will be discussed along this review. It is the purpose of this paper to review some of the theories developed and applied for describing nuclear reactions with weakly bound nuclei. As any review, the present one will be necessarily incomplete and possibly biased by the expertise and personal taste of the authors but with the hope that the reader will obtain, at least, a flavour of how reaction theory has helped the interpretation of experiments with these nuclear species. 2 The impact of weak-binding on the elastic cross section: optical model approach Elastic scattering provides a valuable source of information on the structure and dynamics of nuclei. As explained in the Introduction, it was long realised that the elastic scattering of deuterons, admittedly the simplest example of weakly bound nucleus, does not follow the expected behavior for tightly bound nuclei. 123
47 Page 4 of 57 Eur. Phys. J. A (2025) 61:47 As in the case of well-bound nuclei, the natural framework to study elastic scattering is the Optical Model (OM), which consists in solving a two-body Schrödinger equation with an effective nucleus-nucleus potential, referred to as the optical model potential (OMP) [17]. The form of this OMP can be formally derived from the microscopic nucleus-nucleus interaction. In his seminal work, Feshbach [18] showed that the OMP can be formally written as a sum of two terms, a bare potential, which is the expectation value of the nucleusnucleus potential in the ground state of the projectile+target system, and an additional term, the polarization potential, which accounts for the effect of non-elastic channels (inelastic scattering, transfer, breakup, fusion, …) on elastic scattering: Explicitly: V=V00 +V01 E(+)−HV† 0,(2) where the first and second terms correspond, respectively, to the bare and polarization potentials. Here, Hstands for the full, microscopic Hamiltonian, V0is an operator with non-vanishing matrix elements between the ground-state and other states of the system, V0=(V01,V02,...). This effective potential gives rise to the Schrödinger equation for the relative-motion wavefunction: T R+V−Eχ0( R)=0(3) where T Ris the kinetic energy operator and χ0the wavefunction for the relative motion between projectile and target in their ground states. The effect of weak binding of any of the colliding partners affects both the bare and polarization potentials. The ground-state of the projectile will exhibit an extended tail (as compared to the case of tightly-bound nuclei) and this will influence the form of V00. In addition, the weak binding will enhance certain couplings described by the V0operator, very prominently the breakup channels, and this will also modify the polarization term. In general, the evaluation of the Feshbach operator associated with the polarization potential is very involved and very often it is replaced by some phenomenological form. A few cases exist however in which an explicit evaluation is feasible, at least for specific nonelastic processes. One such example is the so-called adiabatic Coulomb polarization potential, briefly introduced in the next subsection. 2.1 Evaluation of the polarization potential in a simple case: the adiabatic polarization potential In the introductory section, we described the case of deuteron scattering by a heavy target nucleus. It was argued that the tidal force caused by the action of the Coulomb field on the proton produced a modification of the Coulomb-target interaction with respect to the point-Coulomb case [c.f. Eq. (1)]. Quantum-mechanically, the effect of this modified Coulomb interaction will be to induce coupling to deuteron breakup states. This is in fact one of the contributions included in the Feshbach polarization potential given by the second term of Eq. (3). This can be analytically evaluated in the so-called adiabatic limit, which assumes that the excitation energies are high enough so the characteristic time for a transition to a state is small compared to the characteristic time for the collision[1].Applyingsecond-orderperturbation theory,one gets the following expression for this adiabatic polarization potential [1,19]: Vpol(R)=− n=0 |n|Vdip|0|2 En−E0=−1 2α(ZTe)2 R4,(4) where Vdip is the second term of Eq. (1) and αis the dipole polarizability parameter [1], which measures the electric response of the nucleus (the deuteron in this case) to an external electric field. It is therefore a structure property of any nucleus. It is to be noted that the adiabiatic polarization potential given by Eq. (4) is purely real and, as such, does not describetheeffectofthedeuteronbreakupthatwouldremove flux from the elastic channel. This point will be revisited in Sec. 7.2, where another version of the polarization potential, which accounts for deuteron breakup, will be presented. The polarization potential (4) adds an attractive contribution to the Coulomb point-particle potential that, when used in the Schrödinder equation, will produce a small, but measurable deviation of the elastic cross section with respect to the well-known Rutherford formula. An accurate measurement of sub-Coulomb elastic scattering data can therefore be used to extract the dipole polarizability of the deuteron (and other polarizable systems). This idea was used by Rodning et al. [20] to infer the deuteron polarizability parameter from the analysis of sub-Coulomb deuteron elastic scattering on lead. To avoid the determination of absolute cross sections, in their analysis the authors of Ref. [20] introduced the adimensional quantity R(E)=σ(E=3MeV,θ 1=60◦) σ(E=3MeV,θ 2=150◦) σ(E,θ 2=150◦) σ(E,θ 1=60◦). (5) which is identically unity for pure Rutherford scattering. The deuteron polarizability will give rise to values of R(E)which areless than unity and the deviation increases with increasing scattering energy. In Fig. 2we show the measured values of R(E)from [20] along with two calculations. One in which only a nuclear potential is included, which plays the role of a bare part of Eq. (2) and a second calculation includ123
Eur. Phys. J. A (2025) 61:47 Page 5 of 57 47 Fig. 2 Experimental [20] values for the quantity R(E),definedin Eq. (5) for deuteron scattering on 208Pb at sub-Coulomb energies. The dashed line is a single-channel calculation including only the nuclear potential. The solid line is the single-channel calculation include also theeffect of the Coulomb dipole polarizability by means of the adiabatic polarization potential of Eq. (4). See text for details ing the effect of the deuteron poloarizability in the adiabatic approximation, Eq. (4), with α=0.70±0.05 fm3,thevalue extracted in [20] by comparison with the data. This constitutes a neat and beautiful example in which an important structure quantity can be inferred from reaction observables. 2.2 The phenomenological optical model Most practical applications of the OM rely on approximate forms of the Feshbach potential, consisting typically of a central potential (with possibly spin-orbit and tensor terms) whose radial parts are parametrized in terms of simple analytical forms, such as the popular Woods-Saxon potential. Because the Feshbach potential is complex, so is the phenomenological optical potential. In principle, the potential should also be non-local and angular-momentum and energy dependent. While the energy dependence is a common feature of phenomenological potentials, nonlocalilty andangular-momentumdependence is morerarelytakeninto account (see Sect. 11). In the presence of strong absorption, elastic scattering angular distributions between composite nuclei display some common features regardless of the interacting nuclei. At near-barrier energies, elastic distributions present two distinct regions. At small angles, where the Coulomb interaction dominates, the cross section remains close to the Fig. 3 Experimental data for 4,6He+208Pb compared with OM calculations, with Woods-Saxon forms with parameters given in Table 1. Experimental data from [22,23] Rutherford formula prediction, but oscillates around it. The amplitudes of the oscillations increase as the scattering angle increases but remain roughly 25% within the Rutherford prediction. This is the so-called “illuminated” region. After this maximum, the cross section drops rapidly, becoming much smaller than the Rutherford cross sections at large angles (the “shadow” region). An example of this behaviour can be seen in the elastic scattering data of αparticles on lead, shown in the left panel of Fig. 3. These features have been commonly interpreted by invoking classical models. The oscillatory behaviour in the illuminated region can be qualitatively interpreted within a Fresnel picture as an interference between distant pure Coulomb trajectories with closer trajectories affected by the nuclear attractive potential, while the shadow region can be understood as a consequence of the existence of a maximum angle in the classical deflection function [17]. A complete understanding of the observed pattern requires also the introduction of refractive effects which arise in nuclear scattering as a result of absorption [21]. All these effects are naturally accommodated within the OM. For example, the solid line in the left panel of Fig. 3 represents an OM fit of the data using a standard (complex) Woods-Saxon potential. Although the OM is not unique, one may find potentials with radii close to the sum of the geometricalradiioftheinteractingnucleianddiffusenessparameters close to those of the nuclear densities. In the case of reactions involving weakly bound nuclei, the observed elastic scattering angular distributions display notable deviations with respect to the aforementioned behaviour. The effect becomes more evident in the case of halo nuclei, such as 6He or 11Li. The oscillations in the illuminated region are damped (or even absent). The drop of the cross section with respect to the Rutherford prediction starts at smaller angles and hence the cross section exhibits a smoother angular dependence. An example is given by the 6He+208Pb data taken at Elab =22 MeV shown in the right panel of Fig. 3. 123
47 Page 6 of 57 Eur. Phys. J. A (2025) 61:47 Table 1 Woods–Saxon parameters for 4,6He+208Pb optical models. Reduced radii (rx) are converted into absolute (physical) radii as Rx= rx(A1/3 p+A1/3 t) System V0r0a0Wvriai [MeV] [fm] [fm] [MeV] [fm] [fm] 4He+208Pb 96.44 1.085 0.625 32 0.958 0.42 6He+208Pb 124.8 1.085 0.564 6.8 0.958 1.91 This behaviour can also be described within the OM but with potential parameters very different from those used in standard parametrizations extracted from well-bound nuclei. In particular, one needs very diffuse potentials with long-range absorptive tails. The effect of this long-range absorption is twofold: (i) it damps or suppresses the elastic Coulomb amplitude at small angles (i.e., at very large impact parameters), and (ii) it reduces the nuclear amplitude. The result is that the forward cross-section not only becomes smaller overall, but also loses its Coulomb-nuclear oscillations. In order to reproduce the elastic scattering data of weaklybound nuclei within the OM framework using standard Woods-Saxon potentials, a very long-range imaginary part is needed. Table 1lists the OMP parameters for 4He+208Pb and 6He+208Pb at the same incident energy (22 MeV). The radii of the real and imaginary parts are kept the same for both systems. To reproduce the data, the 6He OM requires a significantly larger imaginary diffuseness parameter. The result of the OM calculation for 6He is compared to the data for this reaction in the right panel of Fig. 3. This property is a clear indication of the presence of reaction channels dominated by long-range couplings. Natural candidates are the breakup of the projectile and neutron transfer (either one or two) from the projectile to the target nucleus. Both processes are of a peripheral nature, which explains the long-range imaginary potential. To identify the actual channels producing this effect one must go beyond the single-channel OM scheme and incorporate those channels into the reaction framework. In the case of the breakup of the projectile, this can be efficiently accomplished with the Continuum-DiscretizedCoupled-Channels(CDCC)method, which is described in the next sections. 3 Inclusion of breakup: the CDCC method 3.1 Brief resumè of the CDCC method The CDCC method was originally introduced by G. Rawitscher [24] and later refined by the Pittsburgh-Kyushu collaboration [16,25] to describe the effect of the breakup channels on the elastic scattering of deuterons. Denoting the reaction by a+A, with a=b+x(referred hereafter as the core and valence particles, respectively), the method assumes the effective three-body Hamiltonian H=Hproj +T R+UbA(rbA)+UxA(rxA), (6) with Hproj =Tr+Vbx the projectile internal Hamiltonian, Trand T Rare kinetic energy operators, Vbx the inter-cluster interaction and UbA and UxA are the core-target and valencetarget optical potentials (complex in general) describing the elastic scattering of the corresponding b+Aand x+Asubsystems, at the same energy per nucleon of the incident projectile. In the CDCC method the three-body wave function of the system is expanded in terms of the eigenstates of the Hamiltonian Hproj including both bound and unbound states. Since the latter form a continuum, a procedure of discretization is applied, consisting in representing this continuum by a finiteand discreteset ofsquare-integrablefunctions.Inactual calculations, this continuum must be truncated in excitation energy and limited to a finite number of partial waves associated with the relative co-ordinate r. Normalizable states representing the continuum should be obtained for each set of orbital and total angular momenta. Two main methods are used for this purpose: –The pseudo-state method, in which the b+xHamiltonian is diagonalized in a basis of square-integrable functions, such as Gaussians [26] or transformed harmonic oscillatorfunctions[27].Negativeeigenvaluescorrespondtothe boundstatesofthesystems,whereaspositiveeigenvalues are regarded as a finite representation of the continuum. –The binning method, in which normalizable states are obtained by constructing a wave packet (bin) by linear superpositionoftheactualcontinuumstatesoveracertain energy interval [16]. Wedescribethelattermethodinsomemoredetail.Assuming for simplicity a spinless core, these discretized functions are denoted as φn(r)=unjn(r) r[Y(ˆr)⊗χs]jnmjn,(7) 123
Eur. Phys. J. A (2025) 61:47 Page 7 of 57 47 where n≡ {[kn,kn+1]n,s,jn,mjn}specifies the n-th bin, with [kn,kn+1]the wavenumber interval of the bin, the valence-core orbital angular momentum, sthe valence spin, and j= +sthe total angular momentum. The symbol ⊗ denotes angular momentum coupling. The radial part of the bin is obtained as a linear combination (i.e., a wave packet) of scattering states as unjn(r)=2 πNnkn+1 kn wn(k)uk,j(r)dk,(8) where uk,j(r)is the scattering states for a continuum energy ε=¯ h2k2/2μand angular momentum quantum numbers , s,j,wn(k)is a weight function (for non-resonant continuum wn(k)is usually taken as eiδ, where δare the phase shifts [28] of the scattering states within the bin) and Nn is a normalization constant. The effect of this averaging is to damp the oscillations at large distances, making the bin wavefunction normalizable. Assuming a single bound state for simplicity, the CDCC wavefunction can be written ΨCDCC( R,r)=χ(+) 0( R)φ0(r)+ N n=1 χ(+) n( R)φn(r), (9) where the index n=0 denotes the ground state of the b+x system. This model wave function must verify the Schrödinger equation: [H−E]ΨCDCC( R,r)=0. This gives rise to a set of coupled differential equations for the unknowns χ(+) n( R) E−εn−T R−Unn( R)χ(+) n( R)= m=n Unm( R)χ(+) m( R), (10) where εn=φn|Hproj|φnand Unm( R)are the coupling potentials given by Unm( R)=drφ∗ n(r)[UbA +UxA]φm(r). (11) The inclusion of the breakup channels in expansion (9) will affect the elastic channel wavefunction through the coupling potentials U0n(n>0). The entrance flux, given by the norm of the plane wave associated with the entrance channel, will be distributed among the elastic and breakup channels and this will naturally reduce the elastic cross section with respect to the situation in which those breakup channels are omitted. This effect turns out to be essential for a correct descriptionoftheelasticcrosssections, as illustrated in Fig. 4 for the d+58Ni reaction at 80 MeV. The figure also depicts the discretization scheme employed in these calculations, which comprises =0,2 continuum states. In actual calculations, such as those performed by the popular coupled-channels code FRESCO [29], the total threeFig. 4 Left: Application of the CDCC method to d+58Ni elastic scattering at Ed=80 MeV. The solid line is the full CDCC calculation. The dashed line is the calculation omitting the breakup channels. The calculations were performed ignoring the internal spins of the proton and neutron and so =j. Right: Illustration of continuum discretization for the same problem Fig. 5 Relevant coordinates for the description of the scattering of a two-body composite nucleus by a target body wavefunction is expanded in states with good total angular momentum (JT), i.e., Ψ( R,r)= βi,JT,MT Cβi,JT,MTΨβi,JT,MT( R,r)(12) with Ψβi,JT,MT( R,r)= β χJT β,βi(R) RYL(ˆ R)⊗φn,J(r)JT,MT , (13) where, for the internal states, φn,J,M, we have included an additional subscript Jto write explicitly the projectile spin andhence nisnow anindextoenumerate stateswiththesame J. Also, in the former equation Ris the relative coordinate between the projectile center of mass and the target (assumed by now to be structureless), see Fig. 5. The index βgathers the quantum numbers compatible with a given total angular momentum JT,β≡{L,J,n}, where L(projectile-target orbital angular momentum) and Jboth couple to the total spin of the three-body system JT. The spin of the target is ignored for simplicity of notation. Each of these sets is called achannel. In particular, βidenotes the channels compatible with the initial state of the system (typically, the ground state of the projectile and target nuclei). 123
47 Page 8 of 57 Eur. Phys. J. A (2025) 61:47 The radial coefficients, χJT β,βi(R), from which the scattering observables are extracted, are calculated by inserting Eq. (13) in the Schrödinger equation, giving rise to a system of coupled differential equations: −¯ h2 2μ d2 dR2+¯ h2L(L+1) 2μR2+εn−EχJT β,βi(R) + β UJT β,β(R)χ JT β,βi(R)=0(14) where εnis the nominal energy of function φnand with the coupling potentials: UJT β,β(R)=β;JT|VbA( R,r)+VxA( R,r)|β;JT,(15) where ˆ R,r|β;JT=YL(ˆ R)⊗φn,J(r)JT .(16) The system of equations (14) is to be solved numerically followingthemethodsdescribedelsewhere (see,e.g.[30]and references therein) and subject to the asymptotic boundary conditions: χJT β:βi(Kβ,R)→i 2H(−) L(KβR)δβ,βi−SJT β,βiH(+) L(KβR) (17) →FL(KβR)δβ,βi+TJT β,βiH(+) L(KβR)(18) where Kβ=√2μ(E−n),SJT β,βiare the S-matrix elements, TJT β,βiare the T-matrix elements, H±) L(KR)are the ingoing (−) and outgoing (+) Coulomb functions and FL(KR)is the regular Coulomb function [17]. Because of the relation between the Coulomb functions, the two equations above are fully equivalent. In fact, the coefficients SJT β,βiand TJT β,βi are related by SJT β,βi=δβ,βi+2iTJT β,βi. Scattering amplitudes and differential cross sections are expressed in terms of these coefficients (see, e.g., [17,31]). ThestandardCDCCmethod isbased onastrictthree-body reaction model (b+x+A), and has proven rather successful in describing elastic and breakup cross sections of deuterons and other weakly bound two-body nuclei, such as 6,7Li and 11Be(seeFig. 6).However,it haslimitations. Theassumption of inert bodies is not always justified, since excitations of the projectile constituents (band x) and of the target (A)may take place along with projectile dissociation. Furthermore, the two-body picture may be inadequate for some nuclei as, for example, in the case of the Borromean systems (e.g. 6He, 11Li). Some extensions of the CDCC method to deal with these situations are outlined below. Fig. 6 Application of the CDCC method to 6Li+40Ca elastic scattering at 156 MeV. The solid and dashed lines are the CDCC calculations with and without inclusion of the 6Li (α+d) continuum. Experimental data are from Ref. [32]. Adapted from [27] 3.2 The role of closed channels In principle, expansion (9) may include internal states with excitation energies below (εi<E) or above (εi>E)the total energy, which are referred to as open and closed channels, respectively. Closed channels violate energy conservation and, as such, cannot contribute asymptotically to the scattering wavefunction. However, due to the couplings with other channels, they can affect the open channels and hence the cross sections. The solution of the CDCC equations (14) in the presence of closed channels is totally analogous to the case with open channels, except for the modification of the boundary condition of the radial functions associated with closed channels, which now reads χJT β:βi(Kβ,R)→Cβ,βiW−ηi,L+1/2(−2iKβR), (19) where W−ηi,L+1/2is a Whittaker function with ηthe Sommerfeld parameter, defined as η=ZpZte2/¯ hv. In general, the effect of closed channels is small at sufficiently high energies. However, at low energies, they can have a strong influence on the scattering observables. An example is shown in Fig. 7, corresponding to the reaction d+12 C→p+n+12 C at 12 MeV [33]. CDCC calculations with and without the inclusion of closed channels are compared with Faddeev calculations performed in the momentum-space formulation of Alt, Grassberger and Sandhas (FAGS) [34], which provide an essentially exact solution of the same Hamiltonian and therefore also account for the effect of closed channels. The upper and lower panels are for the angular and excitation energy distributions. The importance of closed channels is 123
Eur. Phys. J. A (2025) 61:47 Page 9 of 57 47 converged open channels only w/o odd partial waves FAGS T (deg) dV/d: (mb/sr) (a) FAGS converged open channels only H (MeV) dV/dH (mb/MeV) (b) Fig. 7 (a) Angular distribution and (b) breakup energy distribution of the elastic breakup cross section for 12C(d,pn)12Cat12MeV.The solid, dashed, and dash-dotted lines in each panel show the converged CDCC result, the result of the CDCC method calculated with only the open channels, and the result of the Faddeev-AGS (FAGS) theory taken from Ref. [35], respectively. The dotted line in (a) is the same as the solid line but omitting the odd partial waves between pand n. Taken from [33], with authorization from APS clearly seen. Only when closed channels are included, the CDCC calculation approaches the Faddeev solution. 3.3 Trivially equivalent polarization potential From a coupled-channel calculation (in particular, CDCC) onemayextractaneffectivepolarizationpotential,alsocalled trivial equivalent local polarization potential (TELP). For simplicity, we assume that there is only one entrance channel (βi=β0) so that the label βiin (20) can be omitted. Then the radial equation for the elastic channel becomes: −¯ h2 2μ d2 dR2+¯ h2L0(L0+1) 2μR2+UJT β0,β0(R)+ε0−EχJT β0(R) =− β UJT β0,β(R)χ JT β(R)=0(20) Fig. 8 AveragepolarizationpotentialUpol (solid lines), compared with the ground state diagonal monopole potential U00 (dashed) for the d+58Ni reaction at 80 MeV The right-hand side of this equation is then used to define the angular-momentum dependent polarization potential: Upol(R)=JTwJT(R)UTELP JT(R) JTwJT(R),(21) where UTELP JT(R)is the “trivial equivalent potential” for the total angular momentum JTdefined by UTELP JT(R)=1 χJT β0(R) β=β0 UJT β0,β(R)χ JT β(R), (22) and wJT,βi(R)are weight factors chosen as wJT(R)=(2JT+1)(1−|SJT β0,β0|2)|χJT β0(R)|2,(23) where SJT β0,β0are the elastic S-matrix elements for each JT value.Asinglechannelcalculation(elasticchannel)usingthe sum of “UJT β0,β0(R)+Upol(R)” should approximately reproduce the elastic scattering cross sections. InFig.9,weshowforillustrationthepolarizationpotential derived from the CDCC calculation for the reaction d+58Ni reaction at 80 MeV (c.f. Fig. 4). For comparison, we show the ground state diagonal monopole coupling potential (U00)In this case, the effect produced by the coupling to the breakup channels is mostly repulsive and absorptive. 3.4 Inclusion of core excitations Excitations of the projectile constituents (band xin our case) may take place along with projectile breakup. This mechanism is neglected in the standard formulation of the CDCC method. For example, for the scattering of halo nuclei, the core fragment bis assumed to be inert and, as such, the projectile states are described by pure single-particle or pure cluster states. Except for very specific cases (e.g. when bis an alpha particle), the structure of a composite system of the 123
47 Page 16 of 57 Eur. Phys. J. A (2025) 61:47 Fig. 20 8Li+12C(toppanel)and8Li+209Bi(bottompanel)elasticcross section,relativetoRutherford,computedwithinthemicroscopicCDCC approach for three-body projectiles. Dotted lines represent the calculations without breakup channels and the solid lines are the full calculations. Experimental data from Refs. [72,73]. Figure from Ref. [71], with authorization from Springer Figure19 shows an application of the method to the reaction 7Li+208Pb at near-barrier energies. Experimental data are compared with a one-channel calculation (only 7Li g.s.), two-channel calculation (ground plus first excited state) and several CDCC calculations including continuum states up to a certain projectile angular momentum. It is seen that a good description of the data is achieved when a sufficiently large number of continuum states are included. Other applications of the microscopic CDCC method include its extension for three-body projectiles including a single nucleon as one of the three particles, such as 8Li or 8Bas(4He+t+n)and (4He+3He+p), respectively [71]. Figure20 shows results for the elastic scattering of 7Li on 12C and 209Bi targets at low energies. The dashed line corresponds to the calculation including only the projectile ground state, while the solid line is the full microscopic CDCC calculations that exhibit a better agreement with the data. 3.8 Interpretation of the CDCC wavefunction and cross sections For a meaningful comparison with experimental data, it is important to properly understand which information and observables are (and which are not!) provided by the CDCC wavefunction. Of course, elastic scattering is the most direct outcome of the method and the one that can be compared with data more easily. By construction, the elastic scattering produced by the CDCC calculation incorporates the effect of the coupling to the inelastic and breakup channels included in the CDCC modelspace. According to our former notation, these breakup channels correspond to the three-body final states of the form b+x+Ain which the target and the projectile subsystems band xescape the interaction region and remain in their ground state. Morover, in addition to these channels explicitly included in the modelspace, CDCC includes effectively the effects of other channels associated with possible excitations of the target and/or of the fragments via the optical potentials. Regarding breakup observables, in principle, the CDCC wavefunction provides only the elastic breakup ones. The most immediate observables are the breakup angular distributions for specific bin intervals, i.e., dσn dΩc.m. =|f0,n(θ)|2.(35) When the binning method is employed, an approximate double differential cross section (with respect to the c.m. angle and relative energy) can be obtained by dividing each angular distribution by the bin width, as follows: d2σ dΩc.m.d≃1 Δn dσn dΩc.m. .(36) Regardingthenonelastic breakupcontributions,whichare effectively included in the fragment-target imaginary potentials, the CDCC method by itself does not provide detailed breakup cross sections involving excitations. However, one can define and compute a total absorption cross section that accounts for their overall contribution. For that, one first derives a reaction cross section in terms of the CDCC Smatrices as1: σreac =π K2 i JTπ Li 2JT+1 (2Ji p+1)(2Ji t+1)1−|SJT βi,βi|2, (37) and an integrated elastic breakup cross section σbu =π K2 i K Ki JTπ LLi 2JT+1 (2Ji p+1)(2Ji t+1)|SJT β,βi|2.(38) Then, one may define an absorption cross section as σabs =σreac−σbu. This absorption cross section will account for all nonelastic processes associated with the imaginary part of the fragment-target potentials, such as target excitation (i.e., inelastic scattering), complete fusion (i.e., capture 1See Ref. [74] for a different convention. 123
Eur. Phys. J. A (2025) 61:47 Page 17 of 57 47 Fig. 21 Comparison of one-channel (i.e., no continuum) calculations and full CDCC calculations for the elastic scattering of a8B+90Zr and b6He+208Pb at near-barrier energies. See text for details of the whole projectile) or incomplete fusion (i.e., capture of one of the projectile fragments). Very often, comparison with experimental data requires more detailed cross sections, such astheangularorenergydistributions of the fragment thatsurvives after the capture (absorption) of the other fragment. As noted above, these observables are not directly provided by the CDCC method, at least in its standard formulation. However, as will be shown in Sect. 3.8.2, the CDCC wavefunction can be used as an input for a suitable formalism capable of providing such observables. The absorption cross section plays an important role in understanding the CDCC results. To quantify the importance of the continuum on the elastic cross sections it is common to compare full CDCC results with no-continuum calculations, in which only the ground-state to ground-state coupling potential is retained. An example is shown in Fig. 21, which depicts the results for the 6He+208Pb reaction at 18 MeV and the 8B+90Zr reaction at 26.5 MeV. Dashed and solid lines denote the no-continuum and full CDCC calculations, respectively (in the 6He case, the improved dineutron model of Ref. [75] was used). Table 2lists the reaction, absorption, and (elastic) breakup cross sections from Table 2 Comparison of reaction, absorption and elastic breakup cross sections for 6He+208Pb at 18 MeV and 8B+90Zr at 26.5 MeV Reaction σreac σabs σbu (mb) (mb) (mb) 8B+90Zr Full 362 136 226 No cont 136 124 – 6He+208Pb Full 477 359 117 No cont 134 134 – the CDCC calculations. Clearly, inclusion of the continuum has a much larger effect on the elastic cross section for 6He. This is in apparent contradiction with the fact that the elastic breakup cross section is larger in the 8B case, as shown in the Table. The apparent inconsistency can be explained by looking at the absorption and reaction cross sections. In the 8B reaction, the inclusion of the continuum couplings produces only a small increase of the absorption cross section, and the reaction cross section is essentially increased by the elastic breakup cross section. By contrast, in the 6He reaction, the absorption cross section increases by almost a factor of 3 when adding the continuum couplings and this results also in an equivalent increase of the reaction cross section, hence the enhanced effect on the elastic cross section. One may anticipate that the increased absorption cross section for 6He is associated with processes in which the valence neutrons experience some kind of nonelastic interaction with the target, such as transfer to bound states. These processes will be discussed in a subsequent section. The different effect of the breakup channels on the elastic channel for neutron versus proton halo nuclei has been addressed in the literature by several authors [76–78]. 3.8.1 Smoothing procedure for two-body observables Owing to the discretization procedure inherent to the CDCC method, the breakup cross sections are given only for the discrete energies of the bins or pseudostates. In reality, breakup cross sections should be continuous functions of the relative energy between the fragments. Several procedures have been proposed to convert the discrete breakup energy distributions into continuous distributions for arbitrary values of the interfragment relative energy. We discuss here two such methods, one based on the S-matrices and the other on the scattering amplitudes. TheS-matricescalculatedinCDCCcorrespondtodiscrete valuesoftheenergy.Inthebinningmethod,anapproximation to this continuous S−matrix can be obtained dividing the discrete S−matrix by the square root of the bin width, i.e., SJT β,βi(k)≈1 √ΔEnˆ SJT β,βi(39) 123
47 Page 18 of 57 Eur. Phys. J. A (2025) 61:47 with β={L,J,n},βi={Li,J0,n0}and ΔEnis the width of the n-th bin. In the PS method, one could apply a similar procedure by assigning a width to each pseudostate. In fact, this approach was used, for example, in Ref. [79] to calculate the differential breakup cross section from the cross section to individual pseudostates, assuming that the width of the ith pseudostate is approximately given by ΔEn=(εn+1−εn−1)/2. A more accurate procedure, previously proposed in Ref. [80], is to obtain continuous S−matrix elements Sβ,βi(k), depending on the continuous variable k, as well as on the initial and final angular momenta, by an appropriate superposition of the discrete S−matrix elements ˆ Sβ,βiresulting from the solution of the coupled channel equations. Ignoring for simplicity intrinsic spins of the projectile constituents as [80,81] one gets SJT β,βi(k)≈ N n=1φ(−) k, |φ(N) n, ˆ SJT β,βi,(40) where φ(−) k, (r)and φ(N) n, (r)are the radial parts of the exact and pseudostate wavefunctions, respectively. The sum runs over theset of pseudostates included in the coupled–channels calculation. From the S-matrices, the double differential cross section for a b−xbreakup state with orbital angular momentum and wavenumber kis given by: d2σ(k) dk dΩc.m. = m=0 π K2 0 J,L (2J+1)mL −m|J0 ×|YL−m(Ωα)SJT β,βi(k)2 ,(41) As an application of this method, in Fig. 22 we plot the modulus of the breakup S−matrix elements for the reaction d+58Ni→p+n+58Ni for a total angular momentum J=17. Intrinsic spins of the proton and neutron were omitted, so the relevant channels are (, L)=(0,17), (2,15), (2,17) and (2,19). For the CDCC-Bin calculations, the continuum was divided into Ns=Nd=35 bins up to a maximum excitation energy of 70 MeV. For the CDCC-PS calculation, a transformed harmonic oscillator (THO) basis of N=50 states was necessary to obtain full convergence at high excitation energies, although N=30 gives already rather good results. After diagonalization of the internal Hamiltonian, only the eigenstates below 70 MeV were retained, reducing the actual size of the basis to only 24 for each partial wave (along with the ground state). In Fig. 22, the histogram represents the CDCC-Bin calculation, the filled circles correspond to the CDCC-PS calculation, in which each discrete S−matrix has been divided by the square root of the pseudostate width, and the line is the CDCC-PS calculation folded with the continuum wavefunctions, Eq. (40). It is clearly seen that both discretization methods are in almost perfect agreement. FurFig. 22 Modulus of the breakup Smatrix elements for the total angular momentum J=17 for the reaction d+58Ni at 80 MeV as a function of thep–nrelativemomentuminthe finalstate.ThehistogramistheCDCC calculation with binning method. The filled and open circles represent the CDCC calculations using the analytical THO pseudostate basis of Ref. [27]. The solid lines are obtained folding the discrete S-matrices with the true continuum wavefunctions. Adapted from Ref. [27] ther details about these calculations can be found in Ref. [27]. The connection between the discrete and continuous breakup cross sections can be also done at the level of the transition amplitudes themselves. In both CDCC and XCDCC, the solution of the coupled-channel equations provides some discrete breakup transition amplitudes, denoted Ti,J0,J M0,M(θi,Ki), connecting an initial state |J0M0with a three-body final state comprised of the target (assumed to be structureless), the valence particle and the core, at some discrete value of the final projectile-target c.m. momentum Ki={θi,Ki}. The first step of the formalism is to relate these discrete amplitudes with the actual breakup scattering amplitudes,withoutcontinuumdiscretization, that wedenote as TIs;J0 μσ;M0( k, K), where kis the projectile internal relative momentum. Formally, these breakup transition amplitudes can be written in integral form as: TIs;J0 μσ;M0( k, K)=φ(−) k;Iμ;sσei K· R|U|ΨJ0,M0( K0),(42) where U=UbA(r, R,ξ)+UxA(r, R)and φ(−) k;Iμ;sσare twobody exact scattering wave functions of the b+xsystem for a relative final momentum kand given core and valence spins. In order to relate the discrete and continuous amplitudes, one can approximate the exact wavefunction ΨJ0,M0in the equation above by its (X)CDCC counterpart and introduce the approximate completeness relation in the truncated discrete basis {φ(N) n,J,M;i=1,...,N}: 123
Eur. Phys. J. A (2025) 61:47 Page 19 of 57 47 TIs;J0 μσ;M0( k, K)≃ n,J,Mφ(−) k;Iμ;sσ|φ(N) n,J,Mφ(N) n,J,Mei K· R ×|U|ΨCDCC J0,M0( K0) = n,J,Mφ(−) k;Iμ;sσ|φ(N) n,J,MTn,J0,J M0,M( K), (43) where the transition matrix elements Tn,J0,J M0,M( K)are to be interpolated from the discrete ones Tn,J0,J M0,M(θi,Ki). Expressions for the overlaps between the final scattering states and the discrete states, φ(−) k;Iμ;sσ|φ(N) n,J,M, are given explicitly in Refs. [81] and [42] for bin and PS functions, respectively. The transition amplitudes of Eq. (43), TIs;J0 μσ;M0( k, K), contain the dynamics of the process in the coordinates describing the relative and center of mass motion of the core and the valence particle. From these amplitudes one can derive two-body observables for a fixed spin of the core, I, the solid angles describing the orientations of k(Ωk) and K(ΩK), as well as the relative energy between the valence and the core, Erel. These observables factorize into the transition matrix elements and a kinematical factor: d3σ(I) dΩkdΩKdErel =μbxkI (2π)5¯ h6 K K0 μ2 aA 2J0+1 × μ,σ,M0|TIs;J0 μσ;M0( k, K)|2,(44) where μbx and μaA are the valence-core and projectile-target reduced masses. 3.8.2 Three-body observables Within the CDCCandXCDCCreactionformalisms,breakup is treated as an excitation of the projectile to the continuum, so the theoretical cross sections are described in terms of the c.m. scattering angle of the projectile and the relative energy of the constituents, using two-body kinematics. For comparison with experimental data, it is useful to have also the cross sections in terms of the angle and energy of the projectile fragments, since these quantities are more directly connected with the actual measurements. In the case of the standard CDCC framework, fivefold fully exclusive cross sections have been derived and presented by several authors [81,82]. The method was generalized in Ref. [42] to the case of XCDCC. We briefly review the main formulas of the latter, noting that the case without core excitation is recovered when a single core state is considered. For simplicity, we ignore the target spin. Using the two-body transition amplitudes discussed in the previous subsection, the three-body observables, assuming the energy of the core is measured, are given by [81]: d3σ(I) dΩbdΩxdEb=2πμaA ¯ h2K0 1 2J0+1 × μ,σ,M0|TIs;J0 μσ;M0( k, K)|2ρ(Ωb,Ω x,Eb), (45) where the phase space term ρ(Ωb,Ω x,Eb), i.e., the number of states per unit core energy interval at solid angles Ωband Ωx, takes the form [83]: ρ(Ωb,Ω x,Eb)=mbmx¯ hkb¯ hkx (2π¯ h)6 ×mA mx+mA+mx( kb− Ktot)· kx/k2 x. (46) Here, the particle masses are given by mb(core), mx (valence), and mA(target) while ¯ h kband ¯ h kxare the core and valence particle momenta in the final state. The total momentum of the system corresponds to ¯ h Ktot and the connection with the momenta in Eq. (43) is made through: K= kb+ kx−ma Mtot Ktot; k=mb ma kx−mx ma kb(47) with ma=mb+mxand Mtot =mb+mx+mAthe total masses of the projectile and the three-body system, respectively. An application of this formalism is presented in Fig. 23, corresponding to the breakup of 11Be on a proton target at Ep=63.7 MeV/u. The top panel corresponds to the differential cross section with respect to the final n+10Be relative energy and the bottom panel to the breakup with respect to the final 10Be energy (integrated in the 10Be and neutron angles). These observables were computed by transforming the XCDCC breakup transition amplitudes by means of Eqs. (43)–(45), and integrating over the unmeasured variables. The XCDCC calculations were performed with a 11Be model including the 10Be ground (0+) and first excited (2+) states.Theformalismallowsto separateandquantifythecontribution of these two states of 10Be. Note that, due to energy conservation, in the relative energy distribution, the contribution coming from the 10Be(2+) state contributes only above the excitation energy of this state (Ex=3.367 MeV). The importance of the 10Be excitation during the reaction due to its interaction with the target nucleus is illustrated also in the top panel by means of a calculation in which the deformed part of the 10Be+p interaction is omitted (dot-dashed line). 3.8.3 Pseudostates versus bins As discussed in the preceding sections, discretization procedures employed in the CDCC method are either based on the 123
47 Page 20 of 57 Eur. Phys. J. A (2025) 61:47 Fig. 23 Differential breakup cross sections, with respect to the n-10Be relative energy (a) and with respect to the final 10Be energy (b), for the breakup of 11Be on protons at 63.7 MeV/nucleon. The contributions of the 10Be ground (0+) and first excited (2+) states are shown in each panel. In the top panel, the result of the calculation neglecting the 10Be excitation mechanism (labelled “no DCE”) is also shown. The vertical arrow denotes the threshold for 10Be(2+ 1)+n. Figure adapted from Ref. [42] pseudostate (PS) method or the binning method. Whereas reaction observables should be independent of the adopted discretization method (provided the calculations are fully converged), in practical applications one of the two methods might result more convenient than the other. The PS discretization method becomes particularly suitable when dealing with narrow resonances. In the case of bins,oneneeds fine discretization inordertoobtain a detailed description of the resonance region in the relative energy spectrum. By contrast, using PS’s one can obtain a detailed description of the resonance profile with a relatively small basis, with the aid of the convolution procedure discussed in Sect. 3.8.2. This is illustrated in Fig. 24, where we show the 10Be+n relative energy differential cross section correspondingtothe11Be+p →10Be+n+p reaction at an incident energy Fig. 24 Comparison of pseudostate (PS) and binning methods for the 11Be+p →10Be+n+p reaction at 63.7 MeV/nucleon. Both methods use the same structure model [84] and include continuum states with configuration s1/2,p1/2,p3/2,d3/2,d5/2. For the CDCC-PS calculation, the dominant individual contributions are shown separately of 63.7 MeV/nucleon. The 11Be bound and continuum states were generated with the structure model of Ref. [84], which doesnotaccount for10Beexcitations.Continuum wavess1/2, p1/2,p3/2,d3/2,d5/2were included. For the CDCC-Bin calculation, the continuum was truncated at Erel =12 MeV and divided into 12 bins for each partial wave, evenly spaced in momentum. The relative-energy differential cross section was obtained by dividing the cross section of each bin by the width of that bin [see Eq. (36)]. The CDCC-PS calculations employ a THO basis with N=30 (N=25) oscillator functions for =1,2(=0). After diagonalization of the 11Be Hamiltonian on this basis, only the eigenvalues below 12 MeV were retained. This results in 13 states for each partial wave (15 for =0). The calculated breakup scattering amplitudes were then convoluted using Eq. (43), from which the differential cross sections, Eq. (44), were then evaluated. Finally, the latter were integrated over the solid angles Ωk and ΩK. As seen in Fig. 24, the binning and PS methods yield, as expected, almost identical results but the PS method allows for a finer description of the resonance region with a comparable, or even smaller, number of basis functions. However, there are other situations where the binning procedure turns out to be more efficient than the PS method. One such situation is the case of reactions of highly polarizable weakly-boundnuclei(suchas neutron-halo nuclei) onhigh-Z targets. These reactions are characterized by the presence of strong long-range Coulomb couplings, which tend to emphasize low-lying excitation states and probe large separations. In this case, the binning method has been shown to provide more stable results with respect to the basis size [85]. 123
Eur. Phys. J. A (2025) 61:47 Page 21 of 57 47 Fig. 25 Relevant coordinates for a transfer reaction of the form A(d,p)B 4 Transfer reactions with weakly bound nuclei Transfer reactions are key spectroscopic tools for both stable and unstable nuclei. Angular distributions of outgoing fragments provide information on the angular momentum content of the populated states and the magnitude is closely related to the single-particle content of these states, quantified in terms of spectroscopic factors. The standard tool for analysingtransfer reactions,thedistorted-waveBornapproximation (DWBA) method, assumes that the transfer occurs in a single-step. Possible excitations of the colliding or outgoing nuclei are ignored or, at most, taken into account effectively through the choice of the effective interactions. These interactions are typically chosen so as to reproduce the corresponding elastic scattering cross sections in the incident and exit channels. To see how this method works in practice, let us consider asa particularcaseaprocessof theform A(d,p)B,schematically depicted in Fig. 25. Using the post-form representation, the transition matrix for this process can be written as [17] Tdp =χ(−) pΦB|Vpn +UpA −UpB|Ψ(+) d,(48) where Vpn,UpA are the proton-neutron and proton-target interactions, UpB is an auxiliary (and, in principle, arbitrary) potentialfor the p-Bsystem,ΦBis theinternalwavefunction of the residual nucleus Band Ψ(+) dis the total wave function correspondingtoanincidentdeuteronbeamofkineticenergy Edand binding energy εd. If target excitation is not considered explicitly, this total wavefunction can be approximated as Ψ(+) d(r, R,ξA)≈Ψ3b(r, R)dΦ(0) A(ξA)(49) where Φ(0) A(ξA)denotes the target ground-state wavefunction. Ignoringantisymmetrizationforsimplicity,thewavefunction ΦBfor a total angular momentum Jand projection M canbeexpanded in Astates using the usual parentage decomposition ΦJM B(rnA,ξ)= I,, j[ΦI A(ξ) ⊗φj(rnA)]JM,(50) where Iis the spin of A,and jthe orbital and total ( j= +s)angularmomentumofthevalenceparticleandφj(rnA) is a function describing the neutron-core relative motion. The normalization S, j=|φj(rnA)|2drnA (51) canberegardedasaspectroscopicfactorfortheconfiguration {, j}.2 If the interactions UpA and UpB are assumed to be independent of the target degrees of freedom (ξA), the integral in ξAcan be readily performed, transforming Eq. (48)into T3b dp =S, jχ(−) pφj|Vpn +UpA −UpB|Ψ3b,(+) d,(52) In DWBA, the three-body wavefunction Ψ3b,(+) dis further approximated by Ψ(+) d≈χ(+) d( R)ϕd(r). When the transfer reaction involves weakly-bound nuclei, including the deuteron discussed here, this choice is not well justified. Due to the presence of the short-range Vpn interaction, the evaluation of DWBA matrix element is mostly sensitive to small p-nseparations. These configurations do not only containcontributionscomingfromthe deuteron groundstate, but also from p-nunbound states. By contrast, the deuteron optical potential describing deuteron-target elastic scattering is not restricted to small p-ndistances. Therefore, the use of the deuteron optical potential in the (d,p)or (p,d)transfer amplitude is likely to lack important deuteron breakup components. This problem was recognized long ago and several solutions have been proposed. One of the first and most popular ones is the adiabatic method of Johnson and Soper [86]. The Johnson–Soper approximation In the Johnson–Soper (JS) approximation, this effect is approximately taken into account by means of the choice Ψ3b,(+) d( R,r)≃χ(+) JS ( R)φd(r), (53) where χ(+) JS ( R)is the solution of a two-body scattering problem, on the coordinate R, in which the interaction is given by a p−nzero-range approximation: UJS(R)=UpA(R)+UnA(R). (54) We see that, in this limit, the adiabatic theory of the transfer amplitude adopts a form akin to that found in DWBA, but this analogyisonlyformalbecausethefunctionχ(+) JS ( R)includes contributions from breakup and the potential UJS(R)may have little to do with the optical potential describing the deuteron elastic scattering, so χ(+) JS does not provide a good description of the elastic channel. Due to the adiabatic approximation, the JS theory is not expected to be accurate at low incident energies. 2Strictly, the functions φjand the spectroscopic factors Sjdepend in general on the core state Ibut, in the present case, this quantum number can be readily inferred from the , jvalues, so it is omitted for brevity. 123
47 Page 22 of 57 Eur. Phys. J. A (2025) 61:47 Fig. 26 Schematic representation of DWBA, ADWA and CDCC-BA approaches for a (d,p)transfer reaction The adiabatic approximation is equivalent to neglecting the excitation energy of the projectile states, [86], which amounts at setting Hbx →ε0in the three-body Hamiltonian (6). The adiabatic wave function takes into account the excitation tobreakup channels, assuming that these states are degenerate in energy with the projectile ground state, as illustrated in Fig. 26b. Therefore, the ADWA approach takes into account, approximately, the effect of deuteron break-up on thetransfercrosssection,withintheadiabaticapproximation. So, it should be well suited to describe deuteron scattering at highenergies,around100MeVpernucleon.Systematicstudies [87–89] have shown that ADWA is superior to standard DWBA for (d,p)scattering at these relatively high energies. The Johnson–Tandy approximation Although the zero-range adiabatic model of JS provides a systematic improvement over the conventional DWBA, there aresituationsinwhichtheformerfailstoreproducetheexperimentaldata[90,91].Modelswhichgobeyondthezero-range and adiabatic approximations are therefore needed. One of such models is the Weinberg expansion method of Johnson and Tandy (JT) [92]. The idea is to expand Ψ3b,(+) din terms of a set of functions which are complete within the range of Vpn. A convenient choice is the set of Weinberg states (also called Sturmians), given by Ψ3b,(+) d( R,r)= N i=0 φW i(r)χ W i( R), (55) where φW i(ξ) are the Weinberg states, which are solutions of the eigenvalue equation [Tr+αiVpn]φW i(r)=−εdφW i(r), (56) where εd=2.225 MeV is the deuteron binding energy and where αiare the eigenvalues, to be determined along with the eigenfunctions. Beyond the range of the potential all the Weinberg states decay exponentially, like the deuteron ground-state wave function. For i=0, α0=1 and so φW 0(r) is just proportional to the deuteron ground state. As iand αiincrease, they oscillate more and more rapidly at short distances. The Weinberg states form a complete set of functions of rwithin the range of the potential Vpn.Theyare Fig. 27 Comparison of the ADWA and DWBA methods for the 56Ni(p,d)57Ni (left) and 48Ca(d,p)49Ca (right) reactions at incident energies of 37 MeV and 10 MeV, respectively. The nucleon optical potentials are taken from the global CH89 [50] systematics while the Daehnick [95] global potential is used for the deuteron optical potential in the DWBA calculations. For the ADWA calculations, the prescriptionsofJohnson–Soper(JS)and Johnson-Tandy(JT) are shown.Quoted from Ref. [96] well suited to expand Ψ3b,(+) din this region, as is required by the amplitude in Eq. (48). They do not satisfy the usual orthonormality relation but the less conventional one φW i|Vpn|φW j=−δij.(57) IfweretaininEq.(55)onlytheleadingterm,Ψ3b,(+) d( R,r)≈ φW 0χW 0( R),one finds[92]thatχW 0verifiesthesingle-channel equation [T R+UJT( R)−Ed]χW 0( R)=0,(58) with Ed=E−εdand where the potential UJT is given by: UJT(R)=φW 0(r)|Vpn(UnA +UpA)|φW 0(r) φW 0(r)|Vpn|φW 0(r).(59) The bra and ket in this equation mean integration over r, with fixed R. Interestingly, in the zero-range limit, UJT(R) reduces to the JS potential, Eq. (54). Therefore, the zeroorderresultgivenbyEq.(59)canbe regardedasafinite-range version of the adiabatic (JS) potential. These two models are globallyreferredtoas Adiabatic Distorted WaveApproximation (ADWA). However, it is worth noting that the full Weinberg expansion makes no reference to the incident energy and, as such, does not involve the adiabatic approximation. This suggests that a stripping theory based on this Weinberg expansion can be used at low energies, where the adiabatic condition is not well satisfied. The inclusion of higher-order terms (i>0) in the Weinberg expansion has been investigated in Refs. [93,94]. In Fig. 27 we present a comparison between the DWBA and ADWA methods for the 56Ni(p,d)57Ni (left) and 48Ca (d,p)49Ca (right) reactions. For the ADWA calculations, the prescriptions of JS and JT are shown. It is clearly seen that the ADWA, in both its zero-range and finite-range forms, provides an improved description of the data as compared to the DWBA method. An appealing feature of the ADWA method is that its ingredients are completely determined by experiments. 123
Eur. Phys. J. A (2025) 61:47 Page 23 of 57 47 These ingredients are the proton-target and neutron-target optical potentials, evaluated at half the deuteron incident energy, as well as the well-known proton-neutron interaction. On the negative side, the ADWA approach does not consistently describe elastic scattering and nucleon transfer. Although, physically, elastic scattering, transfer and breakup should be closely related by flux conservation, this connection is not present in ADWA. Furthermore, the arguments leading to ADWA are strongly dependent on the assumption that the transfer process is governed by a short-range operator. Thus, it is not obvious that the approximations remain valid for other weakly bound systems, like 11Be. Even in the case of (d,p)scattering, the transfer matrix element is determined, in addition to the n-pinteraction, by the proton-target and proton-composite interactions which define the remnant term.Theroleoftheseterms,whichwouldhavecontributions of three-body configurations in which proton and neutron are not so close together, is not clear apriori. The latter problem can be avoided by using an alternative expression for the scattering amplitude. Following Goldberg and Watson [97], one can choose the auxiliary interaction UpB that appears in the remnant term of Eq. (52) to cancel exactly the core-target interaction, UpA. This results in an alternative, but still exact, scattering amplitude (the spectroscopic factor is omitted for simplicity), namely, T3bGW dp =˜ Φ(−) pB |Vpn|Ψ3b,(+) d,(60) where ˜ Φ(−)is a solution of the three-body equation: [Tr+T R+UpA +UnA −E]˜ Φ(−) pB =0.(61) In [98] Timofeyuk and Johnson use an adiabatic approximation for ˜ Φ(−) pB to produce a tractable expression that still includes recoil excitation and breakup effects, while keeping the matrix element constrained to the range of the Vpn interaction: ˜ Φ(−) pB ≃χ(−) pA (rpA,kα)φnA(rnA)e−iαkαrnA.(62) In this expression, kαis the relative momentum between Band pand α=mn/mB, and it was found to produce a good agreement between this calculation and experimental data for the 16O(d,p)17O and 10Be(d,p)11Be reactions. Nevertheless, given that this expression is better suited for the transfer of weakly bound and halo nuclei, the ADWA model has enjoyed more widespread use. The CDCC-BA approximation Another way of accounting for the breakup channels in transfer reactions is by insertion of the CDCC wavefunction, Eq. (9), in the transition amplitude of Eq. (52). The resultant amplitude is formally analogous to that found in the CCBA method [17], so we refer to it as CDCC-BA approxFig. 28 Comparison of the DWBA, ADWA and CDCC-BA methods for 58Ni(d,p)59Ni reaction at Ed=10 MeV (upper panel) and Ed= 56 MeV (lower panel). Adapted from Ref. [99] imation. This is expected to be a good approximation since the CDCC expansion is accurate for small p-nseparations, for which the matrix transition amplitude presents the largest magnitude. The resultant amplitude, however, will be clearly much more computationally demanding than that obtained with the DWBA or ADWA methods. In Fig. 28 we compare the performance of the DWBA, ADWA and CDCC-BA methods for the 58Ni(d,p)59Ni reaction at Ed=10 MeV (upper panel) and Ed=56 MeV (lower panel), taken from Ref. [99]. It becomes clear that the ADWA and CDCC methods provide a more reliable description of the data compared to the traditional DWBA approach. In [100], a systematic comparison between the CDCC-BA method and an adiabatic approximation (CDCC-AD) was performed for (d,p)reactions on various targets, showing good agreement between both, except in some cases, such as when the deuteron beam energy is small and similar to the binding energy of the nucleon in the target or when the binding energy of the nucleon is very small (Sn≃0.1MeV). It should be noted that in that work the adiabatic approximation is performed by setting the nominal energies of the continuum bins to that of the bound state, which is formally equivalent to the ADWA approach, although the calculation 123
47 Page 24 of 57 Eur. Phys. J. A (2025) 61:47 is rather different, so extensions of these results to standard ADWA calculations should be made with caution. The CDCC-BA approximation is reasonable as long as thecouplingtotherearrangement(transfer)channelsisweak, typically,whenthetransfercrosssectionismuchsmallerthan thereaction crosssection.Otherwise,othermethods thattreat transfer on equal footing to breakup, such as the CoupledReaction-Channel method (CRC) [17], are required. 4.1 Transfer reactions populating unbound systems So far, we have considered transfer reactions as a tool for investigating the bound states of a given nucleus. However, in a rearrangement process, the transferred particle can also populate unbound states of the final nucleus. This opens the possibility of studying and characterizing structures in the continuum, such as resonances or virtual states. In fact, due to the matching conditions for this type of processes [101], reactionsinducedby weakly-bound nuclei favour thepopulation of highly excited states of the residual nucleus, including those above the breakup threshold. As in the case of transfer to bound states, the simplest formalism to analyze these processes is the DWBA method. In this case, the bound wavefunction φ, j(r)appearing in the final state in Eq. (52) should be replaced by a positiveenergy wavefunction describing the state of the transferred particle (neutron in this case) with respect to the core target. In principle, for this purpose, one could use the suitable scattering state of the v+bsystem at the appropriate relative energy. However, this procedure tends to give numerical difficulties in evaluating the transfer amplitude due to the oscillatory behaviour of both the final distorted wave and the wavefunction φ, j(r). To avoid this problem, several alternative methods have been used. We enumerate here some of them: (i) The bound state approximation [102]. In the case of transfer to a resonant state, this method replaces the scattering state ϕ, j(r)by a weakly bound wavefunction with the same quantum numbers and j. In practice, this can be achieved by starting with the potential that generates a resonance at the desired energy and increase progressively the depth of the central potential until the state becomes bound. (ii) Huby and Mines [103] used a scattering state for φ, j(r)modulated by a convergence factor e−αr(with αa positive real number), which is intended to eliminate its contribution to the integral coming from large rvalues, and then extrapolate numerically to the limit α→0. (iii) Vincent and Fortune [104] put into question the validity of the bound state approximation arguing that, in general, the bound state and resonant form factors can Fig. 29 Radial part of the d3/2single-particle resonance wavefunction in 17OatEr=0.95 MeV compared with a slightly bound wavefunction (E=−0.1 MeV) and a bin wavefunction, centered at the nominal energy of the resonance and with a width of 0.5 MeV be very different and, even in those cases in which the fictitious form factor gives the correct shape, they can lead to very different absolute cross sections. They suggest using the actual scattering state, but choosing an integration contour along the complex plane in such a waythat the oscillatory integrandistransformedintoan exponential decay, thus palliating the slow convergence problem of the post-form transfer amplitude. (iv) In a real transfer experiment leading to positive-energy states, one does not have access to a definite final energy, but to a certain region of the continuum. That is to say, the extracted observables, such as energy differential cross sections, are integrated over some energy range which, at least, is of the order of the energy resolution of the experiment. This suggests a method of dealing with the unbound states consisting of discretizing the continuum states in energy bins, as in the CDCC approximation. In Fig. 29, we show as an example the radial part of a 3/2+resonance in 17O, described in terms of a d3/2neutron coupled to a zero-spin 16O core. The solid line is a scattering wavefunction evaluated at the nominal resonance energy (Erel =0.95 MeV). Note the oscillatory behaviour at large distances. The dotted line is a bin wavefunction, constructed by a superposition of scattering states, within the range of 0.5 MeV around the resonance energy. It is seen that, asymptotically, the oscillations are damped with respect to the original scattering states. Finally, the dot-dashed line is a bound state wavefunction, with a 1d3/2single-particle configuration, and a separation energy of 0.1 MeV. This wavefunction is very 123
Eur. Phys. J. A (2025) 61:47 Page 25 of 57 47 similar to the scattering state at short distances but decays exponentially at large distances. An advantage of the method (iv) is that it can be applied to situations in which one is interested in the description of the population of a range of continuum energies possibly covering both resonant and non-resonant states. For that, it is convenient to resort to the prior-form expression of the transition amplitude. Considering again the A(d,p)Breaction for simplicity, this amplitude reads Tprior if =Ψ(−) f( R,r)|VnA +UpA −UdA|φd(r)χ(+) dA ( R). (63) The function χ(+) dA is the distorted wave generated by the optical potential UdA and Ψ(−) f( R,r)is the exact threebody wave function for the final p+n+A state, with rand Rdenoting the n-Aand p-Brelative coordinates, respectively. The wave function Ψ(−) f( R,r)is the time-reversed of Ψ(+) f( R,r), which satisfies the three-body equation: [T R+Tr+Vpn +UpA +VnA −E]Ψ(+) f( R,r)=0, (64) with Ethe total energy of the system. To solve this equation, the wavefunction Ψ(−) f( R,r)can be expanded in n+A states with well-defined energy and angular momentum, as in the continuum-discretized coupled-channels (CDCC) method. An example is shown in Fig. 30, which corresponds to the differential cross section, as a function of the n-9Li relative energy, for the reaction 2H(9Li, p)10Li∗at 2.36 MeV/u (upper) and 11.1 MeV/u (lower panel), corresponding to the measurementsofISOLDE [105] andTRIUMF[106], respectively. The lines are the results of transfer-to-the-continuum calculations populating 10Li∗continuum states using the same 10Li structure in both cases. The figure shows the separate contribution of the s-wave (1−,2 −) and p-wave (1+,2 +) continuum states. The strength of the measured cross section close to zero energy is due to the presence of a virtual state in the n+9Li s-wave, whereas the peak around 0.4 MeV is due to a p1/2resonance. This is an example of how the use of transfer reactions can provide information on the continuum structure of weakly-bound or even unbound systems. For more details on the calculations, see Ref. [107]. 4.2 Transfer reactions involving three-body projectiles The formalism presented in the previous sections can be extended and applied to more complex systems. We discuss here the interesting case of the stripping reaction induced by a three-body Borromean system, in which one of the projectile fragments is transferred to the target, leaving a Fig. 30 Illustration of the transfer-to-the-continuum method, using a binning discretization, for the reaction 2H(9Li,p)10Li∗. Calculations are comparedwiththeexperimentaldatafromRefs. [105]and[106]. Figure taken from Ref. [107] Fig. 31 Diagram for a (p,d)or (p,pN)reaction induced by a threebody projectile in inverse kinematics. Taken from Ref. [108] two-body residual unbound system. To be more specific, we consider the case of a two-neutron halo projectile, such as 11Li, 6He impinging on a proton target. In these cases, given the unstable nature of the projectile, the reaction experiment must be performed in inverse kinematics. This process can be schematically represented as (see Fig. 31). (C+N1+N2) A +p→(C+N2) B +d,(65) 123
47 Page 32 of 57 Eur. Phys. J. A (2025) 61:47 Fig. 40 IAV calculations for the reaction 197Au(9Be,8Be*)198Au populating bound states of the residual nucleus. aNorm of the overlaps between the 9Be g.s. and the 8Be continuum pseudostates for different relative angular momenta L, as a function of the α-αrelative energy up to 20 MeV. Dashed lines are included as a guide. bDifferential cross section, at Ecm =36 MeV, as a function of the α-αrelative energy in the final 8Be∗system, for 0+states (solid line) and 2+states (dashed line). The symbols indicate the nominal energies of the g.s. (0+)andthefirst 2+ 1resonance of 8Be. cAngle-integrated total transfer cross section as a function of Ecm. The contribution from the 0+(dot-dashed line) and 2+(dotted) states of 8Be, together with their sum (blue solid), are compared to the experimental data of Ref. [129]. The IAV result rescaled by ab-initio Variational Monte Carlo (VMC) spectroscopic factors (dashed line) is also shown (see text). Adapted from Ref. [128] to z. The exponent in the second factor is Δk(z,b)≡−k 2EU(z,b), (87) which,whenintegratedalongtheentiretrajectory,gives the optical phase shift 2δ(b)=∞ −∞ Δkz,b dz=2∞ 0 Δkz,b dz, (88) that is related to the partial-wave optical S-matrix, i.e., S(b)=exp[2iδ(b)]. (ii) The Uapotential, distorting the incident wave, is taken as the sum of the corresponding fragment-target potentials: Ua=UbA +UxA (89) This particular choice has the virtue of taking into account breakup effects in the entrance channel. However, since each potential in Eq. (89) is evaluated in the corresponding fragment-target coordinate, the associated initial state wavefunction of the system would be a solution of a complicated three-body equation. (iii) The above complication vanishes thanks to the use of the eikonal approximation: the xand bfragments move with the same average velocity as the projectile and hence their momenta are given by kx=(mx/ma) ka, kb=(mb/ma) ka.(90) With the particular choice of Eq. (89) and the assumption in Eq. (90) one obtains the following result for the x-channel wavefunction: ϕEHM x(rx)=d3rbχ(−)∗ b(rb)χ(+) a(ra)φa(rbx) =ei kx·rxexp izx −∞ Δkxz,bx dz! ×d3rbeiq·rbSbA (bb)φa(rbx) (91) with q= kb− k b, the average momentum transferred in b−A elastic scattering. Inserting this expression into the general expression (78) (see details in [118,130]) one obtains for the double differential cross section: d2σ dEbdΩb EHM NEB =2 ¯ hva ρb(Eb)Ex kxd2 bx˜ φa,b(q, bx)2 ×1−|SxA(bx)|2.(92) 123
Eur. Phys. J. A (2025) 61:47 Page 33 of 57 47 where ˜ φa,b(q, bx)is given explicitly in Ref. [118] It should be noticed that the NEB depends only on the asymptotic properties, this is, the Smatrices, of the interaction of band xwith the target. There is no sensitivity on the wavefunctions in the interaction region. This is a result of the eikonal approximation, plus the particular choice of the distorted interaction, which included the imaginary potential WxA which ultimately generates the NEB. In many applications, one is interested in the total yield of fragment b, which is obtained upon integration of the previous formula over the angular and energy variables, resulting: σEHM NEB =2 va(2π)3Ex ¯ hkxd3rbd3rx|φa(rbx)|2 ×|SbA(bb)|21−|SxA(bx)|2.(93) This equation has an appealing and intuitive form: the integrand contains the product of the probabilities for the core being elastically scattered by the target, |SbA(bb)|2, times the probability of the valence particle being absorbed, (1−|SxA(bx)|)2. These probabilities are weighted by the projectile wave function squared and integrated over all possible impact parameters. Due to the Glauber approximation, Eq. (92) is expected to be accurate at high energies (above ∼100 MeV per nucleon). In fact, this formula has been extensively employed in the analysis of intermediateenergy knockout reactions (see e.g. [81,131,132] and references therein) mostly aimed at obtaining spectroscopic information of nucleon hole states. 5.3 Interpretation of inclusive breakup data In the previous sections, we have considered elastic breakup and nonelastic breakup (with the latter possibly including transfer and incomplete fusion) taking place in a breakup reaction. In actual inclusive breakup experiments these contributions will appear entangled in the data, although they produce some distinctive features. For example, the energy distribution of the observed fragment at a given scattering angle will exhibit a characteristic shape, as shown in Fig. 41 for a hypothetical A(d,pX)reaction. To understand this spectrum, it is important to recall that a given proton energy and angle will univocally determine the excitation energy of the residual n+Asystem. According to the proton energy (or, equivalently, the residual system excitation energy), we may distinguish the following regions: (i) The highest proton energies will be characterized by some narrow peaks corresponding to bound states of the n+Asystem. Depending on the experimental resolution, these peaks will appear separated or, instead, will merge with neighbouring peaks. Theoretically, the Fig. 41 Proton energy spectrum from a A(d,pX)inclusive breakup reaction (blue line). The vertical dotted line marks the neutron separation threshold in the A+nsystem. The low-energy peak arises from compound nucleus (CN) and pre-equilibrium (PE) processes followed by proton evaporation (magenta line) crosssectionfortheseisolatedboundstatescan be evaluated with the standard formalisms for transfer reactions, such as DWBA, CCBA, ADWA or CRC. (ii) As the excitation energy of the residual nucleus increases,sodoesthedensityofstatesandthelow-lying bound states will merge to form a quasi-continuum. Here, the treatment with the aforementioned methods becomes more troublesome because of the impossibility of disentangling unambiguously the contribution of each state from the data. When one is interested in energy averaged cross sections, the combination of the IAV model of Sect. 5.1 with dispersive models provides an appealing alternative to the traditional methods. (iii) At a certain excitation energy, the residual system will reach the neutron separation threshold (Ex=Sn). Just above this excitation energy, the proton spectrum will exhibit narrow peaks, corresponding to low-lying resonances, superimposed to a nonresonant continuum background. It is important to realize that the properties of the system just above the threshold (Ex>Sn) and just below it (Ex<Sn) states are qualitatively similar. In particular, the discrete energy levels extend above Sneven if those states are not strictly stationary. Therefore, the wavefunctions of the narrow resonances resemble very much those for the bound states below threshold and correspond to situations in which the system remains bound for a long time before decaying by barrier penetration. This is important from the reaction point of view because it indicates that the transfer cross section should evolve smoothly from negative to positive energies, as it actually happens in the distribution shown in Fig. 41 123
47 Page 34 of 57 Eur. Phys. J. A (2025) 61:47 (iv) As the excitation energy of the residual nucleus increases above the threshold, the narrow resonances disappear, the spectrum becomes continuous and structureless, giving rise to a bell-shaped bump. The most probableenergyoftheemergingproton can beobtained by assuming that the proton gets half the kinetic energy of the deuteron at the point of breakup, plus the Coulomb energy of the deuteron at that place, and half the internal energy of the deuteron (EB=−2.22 MeV) (see Ref. [133], p.509), i.e., Ep≃1 2Ed−Ze2 Rbu +Ze2 Rbu −1 2EB(94) where Edis the kinetic energy of the relative motion of the deuteron and the target nucleus A in the c.m. frame and Rbu the deuteron-target separation at the time of the deuteron breakup. The bump will contain contributions coming from both EBU and NEB components discussed in previous sections. The EBU can be conveniently evaluated with the DWBA or CDCC methods whereas, for the NEB part, the IAV model provides a very convenient framework. (v) Intheformercontributions,wehaveimplicitlyassumed that the observed particle (proton in this case) scatters elastically by the target nucleus. There will be situations in which this not the case; for example, when the projectile fuses completely with the target nucleus forming a compound nucleus that will eventually thermalize by emitting particles and gamma rays. Among these particles, there will be protons that will add up incoherently to the proton spectrum. These protons are typically emitted with low energy in the c.m. frame and will therefore contribute to the low energy part of the proton spectrum (see magenta line in Fig. 41). Somephenomenarelatedtothefusionofweakly-bound nuclei will be discussed in the next section. 6 Fusion involving weakly bound nuclei The previous sections have been devoted to the modeling of direct nuclear reactions. Fusion reactions involving weakly-bound nuclei display also distinctive features which require adequate reaction formalisms and considerations with respect to the case of well-bound nuclei. The topic of fusion with both stable and unstable nuclei has motivated manyworksandexcellent review papers havebeenpublished in recent years, so we refer the reader to these works for a detailed account of the present status of the description of fusion (e.g. [134–136]). We shall discuss two phenomena which are the focus of many theoretical and experimental efforts by several groups. One is the phenomenon of complete fusion suppression. Complete fusion is conveniently defined as the process in which the whole charge of the projectile and target nuclei merge, giving rise to an excited compound nucleus that will subsequently decay by particle and/or gamma emission. The other phenomenon is the large observed yields compatible with the partial fusion of the projectile (incomplete fusion, ICF). Experimental results and theoretical calculations indicate that these two phenomena are actually related since they appear simultaneously and so a plausible explanation of the CF suppression might be in fact the leak of flux going to the ICF channels. A variety of models have been proposed to evaluate the CF cross section, from the simple single-barrier penetration model to more sophisticated coupled-channels methods, in which collective excitations of the projectile and/or target nucleus are taken into account explicitly. These models are very successful at predicting the cross section for well bound nuclei, but tend to overestimate them for weakly bound projectiles at energies above the barrier. For example, for the light weakly bound nuclei 6,7,8Li, 9Be the experimental CF cross sections are found to be suppressed by ∼20-30% compared to the case of tightly bound nuclei [9–15]. Early analyses of these experiments tried to explain the phenomenon using coupled-channels calculations, including the coupling to low-lying excited states of the projectile and target [9,11,77,137,138]. Yet, these calculations systematically failed to reproduce the experimental suppression. This failure has been attributed to the omission in these calculations of the breakup of the projectile; a scenario was suggested in which the weakly bound projectile breaks up prior to reaching the fusion barrier, with the subsequent reduction of the complete fusion probability. This interpretation is supported by the presence of large αyields (in 6,7Li-induced reactions) as well as target-like residues which are consistent with the capture of one of the fragment constituents of the projectile, that is, ICF. To account for these observations, some authors have proposed a two-step scenario [11,139] consisting on the elastic dissociation of the projectile followed by the capture of one of the fragments by the target. However, calculations based on a three-dimensional classical dynamical model [139], which incorporates this two-step breakup-fusion mechanism, can only explain a small fraction of the observed CF suppression for 9Be [140] and 8Li [72] reactions. More encouraging results have been obtained with different methods based on the CDCC formalism, as described in the following subsections. 6.1 Computation of CF and ICF with CDCC Although the CDCC method was originally envisaged as a practical tool to evaluate the elastic and breakup observables, some works have been done to use this method to obtain complete and incomplete fusion cross sections. 123
Eur. Phys. J. A (2025) 61:47 Page 35 of 57 47 Fig. 42 Calculated total fusion cross sections for 11Be + 208Pb (full stars) compared with the experimental data from [143]for11Be + 209Bi (full squares) using the CDCC wavefunction. Adapted from Ref. [142], with permission from APS Someauthors[141,142]haveproposedtoidentifythetotal fusion with the amount of flux that leaves the coupled channels set due to a short-range imaginary potential iWF(R), while CF is identified with the absorption due to such potential, but restricted to bound states of the projectile only. Thus, total fusion is computed as σTF =π ¯ h2K0 JT (2JT+1)PJT,(95) where K0is the wavenumber of the incident channel and PJT is the complete fusion probability for total angular momentum JT PJT=− 8μ ¯ h2K0 β,βi ∞ 0|χJT β,βi(R)|2WF(R)dR.(96) whereχJT β,βiaretheradialsolutionsobtainedfromthecoupled equations (14). In the case of CF, the expression for the cross section is identical but the sum in βis restricted to those channels associated with bound states. The CF obtained in this way represents a lower limit of the physical CF cross section, since one has assumed no capture of all projectile fragments from breakup channels. In reality, these events should contribute to the CF, but cannot be distinguished in this model from the capture of only one projectile fragment. In this model, the incomplete fusion σICF is therefore defined as the absorption from breakup channels. Fig. 43 Schematicillustrationofthefourintegrationregionsemployed in the method of Hashimoto et al. [144] for evaluating complete and incompletefusionfromCDCCwavefunctionQuotedwithauthorization from Oxford University Press An application of this method is shown in Fig. 42, where the calculations for the total fusion of 11Be + 208Pb (full stars) are compared with the total fusion data for the nearby reaction 11Be + 209Bi from Ref. [143]. The agreement is reasonable around the Coulomb barrier, but the calculation underestimates the data by ∼41% for energies well above the Coulomb barrier. A limitation of this method is that it can only be applied to projectiles composed of a heavy charged fragment and a light uncharged one (such as 11Be), since it relies on the assumption that the center of mass of the projectile is close to that of the heavy fragment and far from the light one. Thus, it cannot be used for projectiles like 6,7Li that break up into two fragments of comparable masses. To overcome this difficulty, Hashimoto et al. [144] proposed an alternative approach based also on the CDCC method and applied it to the case of deuteron scattering. Their idea is to transform the CDCC wavefunction from its natural coordinates {r, R}to the coordinates {rp,rn}, |Ψ(r, R)|2drd R=|" Ψ(rp,rn)|2drpdrn,(97) and then associate the CF and ICF cross sections with the absorption taking place in different regions of the {rp,rn} space, as illustrated in Fig. 43. The distances rab pand rab n denote the absorption radii for the proton and neutron, such that for rp>rab p(rn>rab n) the proton (neutron) absorption becomes negligible. The CF is identified with the absorption taking place when both the proton and neutron are inside their respective absorption radii, i.e., σCF =2μ ¯ h2K0rp<rab p drprn<rab n drn|" Ψ(rp,rn)|2 ×{Wp(rp)+Wn(rn)},(98) 123
47 Page 36 of 57 Eur. Phys. J. A (2025) 61:47 Fig. 44 a) Comparison of σ(p) ICF (short-dashed line) and σ(n) ICF (dashdotted line) obtained in [144] with σ(p) STR (squares) and σ(n) STR (triangles) given by the Glauber model [145]b) Complete fusion cross sections calculated with the Glauber model (dots) and with the method of [144] (solid line). Quoted with authorization from Oxford University Press where K0is the incident wave number, and Eis the energy. Similarly, the ICF cross section is obtained from the absorption occurring in the region where one of the two fragments is inside its absorption radius while the other is outside it, i.e., σ(p) ICF =2μ ¯ h2K0rp<rab p drprn>rab n drn|" Ψ(rp,rn)|2Wp(rp), (99) σ(n) ICF =2μ ¯ h2K0rp>rab p drprn<rab n drn|" Ψ(rp,rn)|2Wn(rn). (100) The method was applied to the reaction d+7Li [144]. Although no comparison with experimental data was attempted,theauthorscomparedtheirresultswiththosecomputed with the Glauber calculations of Ref. [145]. This comparison is shown in Fig. 44 with the upper and lower panels corresponding respectively to the ICF and CF cross sections as a function of the deuteron incident energy. The agreement for the ICF is found to be very satisfactory for energies as low as Ed=10 MeV. Since the Glauber model is a high-energy approximation, the agreement at these relatively low energies is somewhat unexpected. By contrast, for the CF part (bottom panel), the Glauber method is found to give significantly smaller cross sections. A discussion of these results is provided in [145]. Another procedure to extract the CF and ICF cross section from the CDCC method was proposed by Parkar and coworkers [146]. The central idea of their method is to perform a series of CDCC calculations with different choices of the fragment-target potentials. In particular, to obtain the ICF crosssectionforthecaptureofagivenfragment,theyperform a CDCC calculation using a short-range imaginary potential for the interaction of that fragment and the target, and a real potential for the other fragment. Using this method, these authors have been able to obtain a reasonable account of the CF, ICF and TF cross sections of reactions induced by 6,7Li projectiles [146,147], although the short-range fusion potential needs to be adjusted for each reaction. More recently, the authors of Refs. [148,149]haveproposed an alternative method which requires a single CDCC calculation to compute the TF, CF and ICF cross sections. In this CDCC calculation, the fragment-target interactions are also modeled with optical potentials with a short-ranged imaginary part. For a two-body projectile, the total fusion cross section is computed as: σTF =2μ ¯ h2K0#Ψ(+)W(1)+W(2)Ψ(+)$,(101) where W(1,2)represent the imaginary parts of the fragmenttargetinteractions.ThisCDCCwavefunctioncanbesplitinto bound (ΨB) and continuum (ΨC) components: Ψ(+)( R,r)=ΨB( R,r)+ΨC( R,r), (102) where ΨBand ΨCare given by the channel expansions ΨB( R,r)= β∈B χβ( R)φβ(r)(103) ΨC( R,r)= γ∈C χγ( R)φγ(r), (104) where φβand φγare, respectively, the bound and unbound states of the projectile, and χβand χγare the corresponding wave function describing the projectile-target relative motion. Assumingthatmatrixelementsoftheimaginarypotentials connecting bound channels to bins are negligible, Eq. (101) can be put in the form σTF =σB TF +σC TF,(105) with σB TF =2μ ¯ h2K0 β,β∈B#χβW(1) ββ+W(2) ββχβ$(106) σC TF =2μ ¯ h2K0 γ,γ∈C#χγW(1) γγ+W(2) γγχγ$.(107) where W(i) αα=φαW(i)φα ,with α, αstanding for either β,βorγ,γ, arethematrixelementsoftheimaginarypotentials. 123
Eur. Phys. J. A (2025) 61:47 Page 37 of 57 47 Then, by performing an angular momentum expansion of the wave functions and the imaginary potentials, Eqs. (106) and (107) become σB TF =π K2 0 JT (2JT+1)PTF B(JT)(108) σC TF =π K2 0 JT (2JT+1)PTF C(JT), (109) with PTF B(JT)=P(1) B(JT)+P(2) B(JT)(110) PTF C(JT)=P(1) C(JT)+P(2) C(JT). (111) where P(i) B(JT)and P(i) C(JT)are the probabilities of absorption of fragment ciin bound channels and in the continuum, respectively, resulting from the contributions of W(i)to the TF cross section. In terms of these probabilities, the authors of Ref. [148, 149] introduce the ICF probabilities PICF1(JT)=P(1) C(JT)×1−P(2) C(JT)(112) PICF2(JT)=P(2) C(JT)×1−P(1) C(JT),(113) and the sequential complete fusion probability PSCF(JT)=P(1) C(JT)×P(2) C(JT). (114) In terms of the introduced probabilities, the following fusion cross sections are defined: – Direct complete fusion (CF): σDCF =σB TF,(115) whichdescribesthesimultaneouscaptureofthetwofragments. – Sequential complete fusion (SCF): σSCF =π K2 0 JT (2JT+1)PSCF(JT). (116) – ICF of fragment ci(ICFi) σICFi =π K2 0 JT (2JT+1)PICFi(JT). (117) In this formalism, the CF, ICF and TF cross sections are then given by σCF =σDCF +σSCF,(118) σICF =σICF1+σICF2,(119) σTF =σCF +σICF.(120) Fig. 45 CF (left) and ICF (right) cross sections for 7Li+209Bi. Experimental data from Refs. [11,12] are compared with the calculations of Ref. [149]. With permission of APS An application of this model is shown in Fig. 45, where CF (left panel) and ICF (right panel) data for the reaction 7Li+209Bi [11,12] are compared with the predictions of the model.Theupperandlowerpanelsdisplaythesameresultsin logarithmic and linear scale, respectively. For CF, the agreement with the data is very satisfactory, both above and below thebarrier(indicatedby thearrow).ForICF,theseparatecontributions for triton capture and αcapture are shown, with the former giving a much larger cross section. The calculations are found to reproduce well the data up to Ec.m.≈34 MeV, but overestimate them at higher energies. A detailed discussion of these results can be found in Ref. [149]. 6.2 Evaluation of CF and ICF cross sections with the IAV model As discussed in Sect. 5, the IAV model provides the total inclusivecross section corresponding to the detection of the b fragmentin reactions of the form A(a,b)X. This results from the fact that the imaginary part that appears in the expectation value of Eq. (78) accounts in principle for all processes in whichthe participantfragment xinteractsnonelasticallywith the target nucleus. This will include the ICF cross section, but also other NEB processes not associated with the formation of a compound nucleus of the x+Asystem, such as target excitation. The isolation of the ICF cross section from the total NEB cross section is indeed not a trivial problem. An intuitive approach consists in identifying the ICF with the absorption due to a short-ranged imaginary potential. A tentative application of this idea is shown in the bottom panel of Fig. 46, quoted from [150], corresponding to the reaction 7Li+209Bi at energies below and above the barrier. The symbols correspond to the ICF data of Dasgupta et al. [11,12]. For the calculations we show the individual contributions to the ICF cross section, namely, α-ICF (i.e., αabsorbed) and t-ICF (t-absorbed) as well as their 123
47 Page 38 of 57 Eur. Phys. J. A (2025) 61:47 Fig. 46 CF(top)andICF(bottom)crosssectionsfor7Li+209Bi.Experimental data from Dasgupta et al. [11,12] are compared with the calculations based on the IAV model. In the CF plot, we include also the reaction cross section (from a CDCC calculation) and the fusion computed with the barrier penetration model (BPM). Top panel adapted from [120]. Bottom panel quoted from [150] sum. To compute the α-ICF (t-ICF), the imaginary part of the α+209Bi (t+209Bi) system was replaced by a short-range imaginary potential of Woods-Saxon form and parameters W0=−50 MeV, ri=1.0fm,a=0.2 fm. The results are very similar to those reported in Ref. [149] and shown in Fig. 45, in which the authors made use of the absorption and survival probabilities extracted from the CDCC calculations. Further calculations are needed to elucidate the usefulness and applicability of the IAV model to evaluate ICF cross sections. The IAV model has also been used to infer CF cross sections of weakly bound nuclei [120,151]. The idea of this method is to decompose the reaction cross section as follows σR≈σCF +σinel +σEBU +σ(b) NEB +σ(x) NEB.(121) In this expression, σinel corresponds to the excitation of the projectile and/or target without dissociation (i.e., inelastic scattering). The terms σEBU and σ(b,x) NEB correspond to the elastic breakup (EBU) and nonelastic breakup (NEB) contributions already discussed in Sect. 5. In the latter, one distinguishes the cases in which either the fragment xor binteracts nonelastically with the target whereas the other scatters elastically. A successful determination of the CF cross section from the decomposition (121) requires that all other quantities involved in this formula can be evaluated accurately. The pure inelastic scattering cross sections (σinel)are standardly computed by means of coupled-channels calculations including low-lying collective excitations of the projectile and target. The EBU part can be accurately calculated with the continuum-discretized coupled-channels (CDCC) method.Finally,theNEBcontributionscan beevaluatedwith the IAV model, at least for projectiles with a developed twoboy structure, such as 6,7Li. An application of the method to the 7Li+209Bi reaction is shown in the top panel of Fig. 46, adapted from Ref. [120]. ThecirclesaretheCFdatafromRef.[12],thesolidgreenline is the reaction cross section obtained from a CDCC calculation and the solid red line is the calculated CF cross section inferred from Eq. (121) assuming a two-body model (α+t) for 7Li. For comparison, a single-channel barrier penetration model (BPM) calculation (dashed lined) is shown. As can be seen, the data are largely suppressed with respect to the BPM. In contrast, the CF extracted from Eq. (121) explains very well the data. As discussed in [151], the reduction with respect to the BPM is found to be mainly due to the competition due to the 209Bi(7Li, α)X channel, which includes, among others, the t-ICF channel (see the bottom panel). 6.3 Application to surrogate reactions Thegradual improvementinmodelsorientedto the computation of ICF cross sections has driven their applicability to the so-called surrogate method (SRM). This method provides an indirect way of evaluating compound-nucleus cross sections in reactions for which the direct measurement is difficult or even unpossible. An example is the extraction of neutroninduced cross sections of the form (n,χ), where χis a given decay channel product (γ, fission fragment, etc). Following the Bohr hypothesis, it is customarily assumed that in these reactions the formation and decay of a compound nucleus take place independently of each other. To obtain informationon thedecayof thecompoundnucleus (B∗)thatoccursin the reaction of interest (n+A→B∗→χ+C), one uses the alternative(surrogate)reactiond+D→b+B∗→b+χ+C that involves a projectile-target combination (d+D) that is experimentally more accessible. For example, one can use the stripping reaction d+A→p+B∗→p+χ+C,in which pand χare measured in coincidence. Compound nuclear reactions are adequately described in the Hauser–Feshbach formalism, which considers the conservation of angular momentum Jand parity π. The cross 123
Eur. Phys. J. A (2025) 61:47 Page 39 of 57 47 section for the “desired” reaction A(n,χ)Cis given by σ(n,χ)(En)= JT,π σCN(Eex,JT,π) GCN χ(Eex,JT,π), (122) where σCN JT,π (Eex,JT,π) is the cross section for the CN formation and GCN χ(Eex,JT,π) the branching ratio for the decay to channel χ. The objective of the surrogate method is to experimentally determine the decay probabilities GCN χ(Eex,JT,π), which are often difficult to calculate accurately. In the surrogate reaction, d+D→b+B∗→b+χ+C, the same CN nucleus B∗is formed and the decay product of interest (χ) is measured in coincidence with the outgoing particle b. The probability for this process can be written as: PS,χ (Eex)= JT,π FCN S(Eex,JT,π)GCN χ(Eex,JT,π), (123) where the subscript Sdenotes the specific surrogate reaction, FCN S(Eex,JT,π) is the probability of forming B∗in this surrogate reaction (with specific values of Eex,Jand π) and where GCN χ(Eex,JT,π) are the same branching ratios appearing in equation (122). The probability PS,χ (Eex)can be obtained experimentally as the ratio between the number of coincidences between the bparticle and the decay particle χ,NS,χ , and the total number of surrogate events, NS, i.e.: Pexp S,χ (Eex)=NS,χ NSχ ,(124) where χis the efficiency of detecting the exit-channel χfor the reactions in which bis detected. Ideally, if a reliable prediction of FCN S(Eex,JT,π) is possible, with an accurate determination of Pexp S,χ (Eex)for a range of energies and angles of b, it might be possible to extractthebranchingratiosGCN χ(Eex,JT,π)whichcanthen be used to calculate the desired cross section using Eq. (122). In practice, this approach is not always feasible due to the lack of some of this required information and the approach has relied on additional approximations. In particular, early applications made use of the so-called “Weisskopf-Ewing approximation”, which assumes that the branching ratios GCN χ(Eex,JT,π) are independent of the angular momentum and spin, giving rise to the simplified cross section: σ(n,χ)(En)=σCN(Eex)GCN χ(Eex), (125) where σCN(Eex)is the CN cross section summed over all possible JT,π values. Applying the same approximation to thesurrogatereaction,andusingJT,π FCN S(Eex,JT,π)= 1wehave PS,χ (Eex)=GCN χ(Eex), (126) allowing the determination of the desired cross section as σ(n,χ)(En)=σCN(Eex)PS,χ (Eex), (127) which avoids the need of the probabilities FCN S(Eex,JT,π). So, under the validity of theWeisskopf-Ewing approximation, the neutron (n,χ)cross section can be readily inferred from the measured probabilities for the surrogate reaction. Note also that the CN cross section must be estimated in some way, using, for example, an optical model calculation. In practice, the Weisskopf-Ewing approximation is rarely justified in most cases so one needs to resort to the more general expressions (122) and (123). The probabilities FCN S(Eex,JT,π)that appear in the latter can be estimated with the IAV model discussed in Sec. 5.1. This idea has been successfully applied to the 95Mo(n,γ) reaction [152]. The direct (n,γ) cross section for this reaction are compared in Fig. 47 with the cross section extracted from the surrogate reaction 95Mo(d,p) using the aforementioned formalism. As can be seen, they are in excellent agreement. As also shown in this figure, the result using the Weisskopf-Ewing approximation departs significantly from the direct measurement. 7 Semiclassical description of breakup and transfer reactions When the de Broglie wavelength of the projectile is small compared to some characteristic distance of the collision process one may describe its motion in terms of classical trajectories. This provides a more intuitive and usually mathematically simpler description of the reaction. This approximation cannot be applied to the internal motion of the nucleons inside the nucleus because their typical wavelength is of the same order as the size of the nucleus and therefore quantum effects are important (for example, for an energy of 30 MeV, v∼c/5 and so =¯ h/p≈1 fm). The methods in which the internal excitations are treated quantummechanically, while the projectile–target relative motion is treated classically, are called semiclassical methods. A large variety of such models have been proposed in the literature [84,155–159]. As examples, we discuss here the one developed by Alder and Winther and the semiclassical transfer-tothe-continuum model of Bonaccorso and Brink. 7.1 The semiclassical formalism of Alder and Winther In its simplest form, the theory of Alder and Winther [19] assumes that the projectile moves along a classical trajectory, which is weakly affected by the internal excitations of the colliding nuclei. This means that: Δ 1 and Δεn E1. 123
47 Page 40 of 57 Eur. Phys. J. A (2025) 61:47 Fig. 47 Application of the surrogate method to the 95Mo(n,γ)reaction. Top (panel (A)): Calculations of the cumulative probability of forming the CN through 95Mo(d,p)(FCN 95 Mo(d,p)). The shaded region, which spans excitation energies in the range from Ex=8.55 MeV to Ex=10.65 MeV, indicates the excitation energies over which the surrogate data are fit. The vertical dotted line represents the neutron evaporation threshold. Middle (panel (B)): histogram of the total contribution to CN formation over the shaded range in (A) as a function of angularmomentum,decomposed intopositiveandnegativeparities,and normalized to one over the integration region. Bottom: The solid blue region is the (n,γ)cross section obtained from the SRM, the red circles and black squares are the direct measurements [153,154]. The uncertainty resulting from experimental data and fitting error is indicated by the shaded band. The result obtained using the Weisskopf-Ewing approximation is also shown (gold diamonds). Quoted from Ref. [152], with permission from APS The projectile–target interaction is split as: V( R,ξ) = V0( R)+Vcoup( R,ξ), where V0( R)is independent of the internal coordinates and determines the classical trajectory R(t). The time evolution of the total wave function of the system verifies the Schrödinger equation i¯ hdΨ(ξ,θ,t) dt =V( R(θ, t), ξ) +Hp(ξ)Ψ(ξ,θ,t) (128) subject to the initial condition |Ψ(−∞)=|0. Inthespiritofthecoupled-channelsmethod,thetotalwave function is expanded in a basis of internal states of the projectile Hamiltonian [Hp(ξ) −εn]φn(ξ)=0: Ψ(ξ,θ,t)= n=0 cn(θ, t)e−iεnt/¯ hφn(ξ) (129) which, when inserted in Eq. (128), gives rise to a set of coupled equations for the expansion coefficients: i¯ hdcn(θ, t) dt = m e−i(εm−εn)t/¯ hVnm(θ, t)cm(θ, t)(130) with the initial condition cn(θ, −∞)=δn0. The timedependent coupling potentials Vnm(θ, t)are given by: Vnm(θ, t)=dξφ∗ n(ξ)V( R(θ, t), ξ)φm(ξ). (131) Once the coefficients are obtained, the excitation probability for a 0 →ntransition is given by: Pn(θ) =|cn(θ, ∞)|2, and the differential cross section by: dσ dΩ0→n=dσ dΩclas Pn(θ). where (dσ/dΩ)clas is the classical differential elastic cross section which, for a pure Coulomb case, coincides with the Rutherford cross section. Due to the conservation of the total probability (flux), one has n Pn(t)= n|cn(t)|2=1. When the couplings are weak, one may solve Eq. (130) perturbatively, assuming that c0≈1 and cn1forn>0. This gives the first-order solution cn(θ) ≡cn(θ, ∞)≃1 i¯ h∞ −∞ e−i(ε0−εn)t/¯ hVn0(θ, t)dt. (132) In the important case of pure Coulomb scattering, which was the case studied in detail by Alder and Winther [19], one finds analytical expressions for the excitation probabilities. Inparticular,thefirst-orderexcitationprobabilityfora0 →n transition, due to the electric Coulomb operator Eλ, results 123
Eur. Phys. J. A (2025) 61:47 Page 41 of 57 47 246810 E J [MeV] 10 100 1000 n E1 (E J ) 28 MeV/nucl. 70 MeV/nucl. 280 MeV/nucl. Pb target b min = 12.3 fm Fig. 48 Total number of virtual photons for the E1 multipolarity, corresponding to the scattering of a projectile by a 208Pb target at different incident energies and integrated for impact parameters b>12.3fm. Quoted from Ref. [160] dσ dΩ0→n=Zte2 ¯ hv2B(Eλ,0→n) e2a2λ−2 0 fλ(θ, ξ) (133) which is valid only for angles smaller than the grazing 3one (θ<θ gr)andwherea0ishalfthedistanceofclosestapproach in a classical head-on collision, ξ0→n=(εn−ε0) ¯ h a0 vis the adiabaticity parameter and fλ(θ, ξ) is an analytic function, depending on the kinematical conditions, but independent of the structure of the projectile. Forweaklyboundnuclei, the excitationwilltypicallypopulate unbound (continuum) states. The previous formula can be generalized to: dσ(Eλ) dΩdε=Zte2 ¯ hv21 e2a2λ−2 0 dB(Eλ) dε ×df λ(θ, ξ) dΩ(θ < θgr)(134) where df λ(θ, ξ)/dΩis also a well-defined analytic function and dB(Eλ)/dεis the electric reduced probability to the continuum states. It is common to express (134) in terms of the so-called number of equivalent photons, dσ(Eλ) dΩdε=16π3 9¯ hc dNEλ(Exθ) dΩ dB(Eλ) dε(135) where dNEλ(Exθ)/dΩis the number of equivalent photons per solid angle. This is typically referred to as Equivalent Photon Method (EPM). 3The grazing angle refers to the angle at which the projectile interacts with the surface of the target in such a way that the projectile barely “grazes” the surface of the target, rather than fully penetrating or colliding head-on. Fig. 49 Breakup of 8B→7Be+p on a 58Ni target at 26 MeV. Coupledchannel semiclassical calculations (labeled “present work”), using Coulomb+nuclear trajectories, are compared with CDCC calculations (labeled “Nunes and Thompson”), which include also nuclear and Coulomb couplings. Quoted from [130] with permission from APS Figure48,quotedfrom[160],showsthenumberofequivalent virtual photons, integrated for impact parameters beyond b=12.3 fm (note that b=a0cot (θ/2)), for the collision of a projectile with a Z=82 target, for three different incident energies. It can be seen that increasing the incident energy decreases the population of low-lying states and enhances the population of high-energy ones. The assumption of pure Coulomb trajectories can be relaxed, at the expense of losing some of the simplicity of the method. A compelling application is shown in Fig. 49 (taken from Ref. [130]) where semiclassical coupled-channels calculations, using trajectories modified by the nuclear interaction, are compared with CDCC calculations for the breakup angular distribution of the 8B+58Ni reaction at the nearbarrier energy of 26 MeV, finding a nice agreement between both. 7.2 Dynamic Coulomb polarization potential from the AW theory As discussed in Sect. 2, the Feshbach theory of nuclear reactions provides a formal expression for the nucleus-nucleus optical potential. This can be expressed as a sum of a bare potential (that is, the static part of the interaction due to the ground-state densities for the projectile and target) and a polarization potential. Under suitable approximations, it is possible to derive simple, analytical expressions for specific 123
47 Page 48 of 57 Eur. Phys. J. A (2025) 61:47 Fig. 59 Relative energy distribution of the fragments following the breakupof11Be on aleadtarget at69MeV/u,integrated inthescattering angle up to a maximum value of θmax =6◦(top) and 1.3◦(bottom). Full CDCC calculations, including both Coulomb and nuclear couplings are compared with CDCC calculations including only nuclear couplings Fig. 60 Relative energy distribution of the fragments following the breakup of 11Be on a lead target at 520 MeV/u. EPM and XCDCC calculations, based on the same structure model, are compared. The effect of the nuclear/Coulomb interference is illustrated is seen that (i) nuclear breakup is rather significant (consistently with the estimate of [179]) and, more importantly, (ii) the incoherent sum of the nuclear and Coulomb contributions overestimates the full XCDCC calculations with interference effects. From these calculations, it becomes apparent that the procedureofsummingincoherentlythenuclearandCoulomb Fig. 61 Comparison of the Coulomb dissociation data of 15Con208Pb at 68 MeV/u [180] with CDCC calculations using different 14C+n models for 15C. Figure from [181] contributions introduces some error in the extraction of the electric transition probabilities from Coulomb dissociation experiments. The error will become larger at lower incident energies. Ideally, one may also analyze directly the experimental cross sections with the CDCC method to extract the electric transition probabilities and other structure properties. The practical applicability of this idea is hindered by the fact that there is no simple connection between the reaction observables (cross sections) and the underlying B(Eλ) distributions. In particular, breakup cross sections are no longer proportional to the B(Eλ). One possibility is to compare the measured cross sections with a series of CDCC calculations assuming different structure models, with different electric strength distributions. ThisisexemplifiedinFig.61wherethedatacorrespondtothe relativeenergydistributionof 15Con208Pbat68MeV/umeasured at RIKEN [180] and the lines are CDCC calculations for different potential models of the 15C nucleus. From the comparison one may select the best structure model among those considered. Recently, an alternative to this procedure to extract B(Eλ) distributions from the comparison of CDCC calculations with breakup cross section data has been proposed [182]. The strategy can be summarized as follows: – Start with some trial structure model of the projectile. This will predict some distribution B0(E1;ε) depending on the continuum energy. – Introduce some small changes (perturbation)inthe model, by multiplying all Coulomb dipole matrix elements by arbitrary factors (1+2δ(εi)), where εiare the measured energies. This will modify the B(E1)at each measured εi Bmod(E1,ε i)≃B0(E1,ε i)(1+2δ(εi)),(155) 123
Eur. Phys. J. A (2025) 61:47 Page 49 of 57 47 012345 H (MeV) -0.2 0 0.2 0.4 0.6 0.8 1 dB/dH (e2 fm2/MeV) Starting 11Be model Extracted B(E1) from GSI data Extracted B(E1) from RIKEN data (Tc.m.< 1.3o) Extracted B(E1) from RIKEN data (Tc.m.< 6o) NCSMC: Calci et al. Fig. 62 Unfolded B(E1)distributionsextractedfrom theexperimental breakup cross sections measured in [179] (squares) and [178](circles) using the procedure based on the XCDCC method proposed in [182]. For comparison, the starting B(E1)distribution is shown by the black line and a NCSMC ab-initio calculation of Ref. [183] (red solid line) is also shown. Figure adapted from [182] – For small perturbations, the calculated breakup cross sections are modified simply as [182] σmod i≃σ0 i+δ(εi)σi.(156) where σ0 iand σiare constants which can be determined by performing several calculations for different δvalues. – By comparing the model calculations with the measured breakup data, one may infer the optimal value of δ(εi) for each excitation energy and, from them, the values of B(E1;εi)that best describe the data within the CDCC framework. An application of this method is shown in Fig. 62, adapted from[182].Thesolidblacklineisthe B(E1)distributionprovided by the starting (trial) B(E1)model used in a XCDCC calculation. Following the outlined procedure, this B(E1)is corrected for each excitation energy by comparing the calculated and measured cross sections. The procedure is applied separately to the data of Ref. [178], for the two angular cuts discussed above, as well as to the data of [179]. The corrected B(E1)distributions are displayed in this figure with the symbols. It is seen that the three of them agree reasonably well within the estimated errors. This is in contrast to the B(E1) extracted in the original works and shown in Fig. 58.Itis important to note that the latter are affected by the experimentalresolution,whilethoseshowninFig.62donotinclude the effect of the experimental energy resolution. Differences between these experimental resolutions in the two experiments are partially responsible for the differences observed in the B(E1)distributions reported in the original works. 9 Study of nucleon–nucleon correlations from nucleon removal reactions One particular point of interest for nucleon-Borromean systems of the form N + N + C is the study of the correlation between their two valence nucleons, since the nucleonnucleon interaction is essential to hold the Borromean system bound. As many Borromean systems also present nucleon halos, the study of the correlation between their valence nucleons permits the exploration of the nucleonnucleon interaction in low-density, isospin-asymmetric environments. A quantity related to the nucleon-nucleon correlation is the average opening angle between nucleons with respect to the core θNN. If both nucleons follow independent uncorrelated orbits, this angle will on average be 90◦ [184,185]. A smaller angle corresponds to a small average distance between nucleons compared to the distance of their center of mass to the core, in which is known as a “dineutron” (or diproton) configuration [186], which is associated with an attractive correlation between nucleons. Meanwhile, an angle larger than 90◦indicates a larger internucleon distance compared to their distance to the core, which corresponds to both nucleons being on opposite sides of the core, in the so-called “cigar” configuration, which corresponds to a repulsive nucleon-nucleon correlation. Note that three-body calculations show that both “dineutron” and “cigar” components appear in the wave function [58], and it is their relative strength which leads to a more dominant configuration. An experimental probe for θNNin the case of neutronBorromean systems is the dipole electric strength B(E1): assuming point-like particles, its expression reduces to: B(E1)=3 πZc A2 e2#r2 n+r2 n+2rnrncos θNN$ ≃6 πZc A2 e2#r2 n$(1+cos θNN), (157) where 'r2 n(is the mean square distance between neutron and core, Zcisthechargeofthecoreand Athemassofthenucleus [187], so an enhanced B(E1)indicates a smaller angle and dineutron correlations while a reduced B(E1)indicates the existence of a cigar-like configuration. B(E1)strengths have been measured for various Borromean nuclei, from which values for θNNcan be extracted. The values extracted in [185] are presented in Table 3, where the angles can be seen to suffer from significant uncertainties, due both to experimental uncertainties and the model dependence due to 'r2 n(. Another method of extracting the opening angle is the use of nucleon knockout reactions, where one of the valence nucleons is removed via a sudden reaction, such as nucleonknockout with 9Be or 12C targets or a (p,pN)reaction with protontargets, assuming the validity of the spectator approximationfortheremaining(C+N)system.Withinthisapprox123
47 Page 50 of 57 Eur. Phys. J. A (2025) 61:47 Table 3 B(E1) strength function and deduced average opening angle θNNfor various Borromean nuclei. Table adapted from [185] 6He 11Li 14Be 17Ne B(E1) (e2fm2) 1.20(20) 1.42(18) 1.69 1.56 [188][177][189][190] θNN(deg.) [185]83 +20 −10 66+22 −18 64+9 −10 110 Fig. 63 Opening angle and momenta measured in nucleon-removal reactions on Borromean systems. Figure adapted from [191] imation, in the projectile rest frame the Borromean nucleus starts at rest, so that when the nucleon is removed, the final momentum of the (C+N)system will be equal and opposite to that of the removed nucleon within the Borromean system (ky). Since the (C+N)system is unbound, the relative momentum between Cand the remaining nucleon can also be measured (kx), and the angle between them θkdetermined. A scheme of the quantities of interest is shown in Fig. 63. It should be noted that, as θkis an angle between the momenta associated with the nucleons instead of their positions, it can be viewed as a Fourier transform of the opening angle θNN. As such, a value of θklarger than 90◦is associated with a value of θNN lower than 90◦, and thus to a dineutron configuration, while the cigar configuration corresponds to θk<90◦and θNN >90◦. Various measurements of θkhave been performed for Borromean nuclei [192–195], finding for its average value θkthe following results (in degrees): 90 (6He), 90 (8He), 103.6+0.7 −0.9(11Li), 96.2+1.8 −1.6(14Be) [195]. It can be seen that, indeed, values for θk>90◦correspond to θNN<90◦, as shown in Table 3. It should also be noted that an asymmetric distribution in either θkor θNN (resulting in θ= 90◦) requires the interference of waves with opposite parities [196], which neatly explains the symmetry for 6,8He, since the valence neutrons find themselves in the negative-parity p3/2and p1/2waves, so their angular distribution must be symmetric. This also sets the average opening angle as a measure of the admixture of opposite parity components [197]. Asseenabove,forneutron-Borromeansystems,thedineutroncorrelationseemstodominateoverthecigardistribution. Recent experiments have tried to explore the spatial location of this dineutron correlation, as some nuclear matter calculations point to the low-density surface of Borromean systems asaregionwherethesecorrelationsshouldbefavoured[198]. Recent (p,pn)experiments on Borromean nuclei [191,197] have used the modulus of the momentum of the removed neutron (ky) as a proxy for the nuclear location (smaller momenta correspond to the nuclear halo and surface, while larger momenta correspond to the nuclear interior) and the average opening angle as a proxy for dineutron correlations. Through the use of an eikonal zero-range approximation for the (p,pn)nucleon-nucleon collision process, the cross section can be expressed as [199,200]: σ∝#φCn(kx,x)⊗eiky·y|S(y)φgs(x,y)$2 ,(158) where xand yare defined as in Sec. 4.2,φgs describes the bound state of the Borromean nucleus, φCn the final-state wave-function between the core and the remaining neutron, and S(y)is a S−matrix that includes the absorptive potential between target proton and core C. As such, the cross section can be interpreted as the square of the Fourier transform of the bound state, distorted by the absorption of the target proton and the final-state interaction between neutron and core. In [197], the maximum θkfor 11Li was found for small values of ky∼0.3fm −1, which was interpreted as the dineutron being located at the nuclear surface where the nuclear density is low, as predicted by previous models [198,201]. As the low nuclear density is a feature of the surface of all neutron-Borromean nuclei, it is expected that the dineutron also appears in the surface of other such systems. Indeed, in [191], the maximum of θkwas found for roughly the same value of kyalso for 14Be and 17B, as shown in Fig. 64, making a tantalizing case for the universality of dineutron correlations in the surface of neutron-Borromean nuclei, as predicted in [198], although experiments on more nuclei, such as 19Bor22C, are required in order to confirm this feature. In the figure the green bands correspond to systematic errors in experimental data, while statistical errors are indicated by the error bars. The theoretical calculations are able to match experimental results for 11Li and 17Bbut overestimate the maximum average angle for 14Be. This has been associated with the contribution of excited states of the 12Be beyond the 2+state included in the theoretical model [191]. 10 Uncertainty evaluation Giventheamountofingredientsrequired for the computation of reaction observables (bound-state wavefuncions, optical potentials, spectroscopic factors,...) there has been a push for the quantification of the uncertainties in the calculated observables due to the ambiguities of these ingredients (the Igo ambiguity [202] of the optical potentials is a classical exampleofsuchambiguities).While the classical responseto this demand was the comparison of calculations using differentingredients(forexample, optical potentials fromdifferent global parametrizations), this evaluation of the uncertainties is at most qualitative and highly dependent on the choice of 123
Eur. Phys. J. A (2025) 61:47 Page 51 of 57 47 Fig. 64 Average opening angle in momentum space θkas a function of ky. Green bands correspond to the systematic errors in the experimental measurement. In all nuclei the maximum average momentum is found at similar values of kyfor all considered nuclei. Figure adapted from Ref. [191] inputs. Recently, Bayesian methods have been proposed to quantify the uncertainties that optical potentials suffer when they are obtained from the fitting of elastic scattering data (as is usual for optical potentials) and to study the propagation of these uncertainties to observables such as (d,p) transfer angular or charge exchange (p,n)differential cross sections [203–210] or nucleon-knockout momentum distributions [211]. The determination of Bayesian posterior distributions and confidence intervals requires the evaluation of the reaction observables for a range of values of every considered parameter, which rapidly results in a humongous number of calculations, which are feasible in facilities with high computing power for light-computation models such as DWBA and ADWA, but become prohibitive for models which require heavier computations, such as CDCC or CRC.Assuch,methodsto accelerate these calculations using emulators [212,213] to predict and interpolate the result of these heavier calculations for the required parameter values are being explored. Through these methods, uncertainty analyses have been performed using CDCC [214], although admittedly only the uncertainties due to the valence-core interaction were studied, further proof of the computational magnitude of the task. In this regard, new global potential parametrizations with quantified uncertainties have recently been developed, with uncertainties originating either from the chiral interactions used to generate the potentials [215] or from fitting to extensive elastic scattering data [216], although it should be noted that in [216] the fit was not performedwithinaBayesianframework.Thelatterpotentialhas been used to study the uncertainty in the reduction factors for transfer and knockout reactions [217], which is still an open problem [218]. 11 Inclusion of non-local potentials It has long been known that nucleon-nucleus and nucleusnucleus optical potentials are non-local [219], both due to antisymmetrization and the effect of non-elastic channels (encoded in the second term of the Feshbach potential in Eq. (2)). The use of non-local potentials transforms the Schrödinger equation into an integro-differential equation, which is far more cumbersome to solve, so local potentials have traditionally been favoured in reaction calculations. In fact, for a long time only a few non-local parametrizations for optical potentials have been published [220–223] and non-locality is not explicitly considered in the majority of potential parametrizations [50,224]. In recent years there has been a development of dispersive optical potentials based on the nuclear self-energy, which is inherently non-local [225– 228], thus reigniting the interest in fully non-local potentials. For the study of elastic scattering, the neglect of nonlocality is not problematic, as one can always find a local equivalent potential which reproduces the phase-shifts and observables of the non-local one [229,230]. Analogously, reactions that are only sensitive to the asymptotics of the wave functions are usually well described by local potentials. However, non-locality reduces wavefunctions in the nuclear interior, in what is called the Perey effect [229,231], so non-locality should be taken into consideration in reactions that are sensitive to the nuclear interior, such as transfer reactions. This has usually been approximated through the multiplication of wavefunctions obtained from a local potential by the so-called Perey factor [229], which reduces the wavefunctions in the interior without altering their asymptotics. Although the Perey factor can be obtained for any non-locality shape, the prescriptions typically used assume a Gaussian dependence on non-locality [232] (usually referred to as Perey-Buck geometry). These approximations (local equivalent potential and Perey factor) have been compared to the computation of the wavefunction using the full nonlocal potential in [233]for(d,p)transfer reactions, finding a difference in spectroscopic factors of ∼10%, although these 123
47 Page 52 of 57 Eur. Phys. J. A (2025) 61:47 results did not consider non-locality in the outgoing deuteron wavefunction. The study of non-locality in (d,p)transfer reactions has received particular attention, due to the larger effect nonlocality has on the observables, the fact that there exist more reliable non-local nucleon-nucleus potentials than nucleusnucleus ones and the relative simplicity of their description, which allows for more manageable computations. As such, non-locality has been explicitly included in Faddeev calculations for (d,p)transfer reactions [234]aswellasinthe widespread ADWA calculations [235–237], with a special focus on the local-equivalent adiabatic deuteron potential, whose main feature was the energy shift in nucleon-nucleus potentials from the nominal value of half the deuteron energy due to the internal kinetic energy of the nucleons in the deuteron. Later implementations included non-locality in the deuteron channel fully [238–240], with the intriguing finding [239,240] that cross sections for the 26Al(d,p)27Al reaction were significantly dependent on the deuteron model, and in particular on its d-wave component, a dependence that was not observed in calculations with local potential. This dependence was later found to be an artifact due to the interplay of the ADWA formalism and the dwave component of the deuteron, as the inclusion of nonlocality in CDCC calculations (through the use of firstorder local equivalent potentials [241]) and Faddeev calculations [242] resulted in a reduced dependence on the deuteron model, showing the need for precise reaction theories to extract reliable information from experimental data. The CDCC formalism has been further extended to include non-locality through the consideration of the Perey effect [243] and the extension to 6Li→α+dbreakup reactions [244]. The effect of non-locality in other reactions, such as (d,p)Xsurrogate reactions [245] and eikonal descriptions of knockout reactions [246], has also been recently studied, as corresponds to the mounting interest in nonlocality arising from the new ab initio and dispersive potentials. 12 Conclusions and perspectives Thepresentreviewaimedtoprovideaglimpseintotheunique physics behind reactions involving weakly-bound nuclei and the formalisms and methods commonly used in the modeling and analysis of these reactions. We have shown the importance of breakup channels for weakly-bound systems and the methods developed to properly consider them, such as the CDCC method (Sec. 3), several semiclassical descriptions (Sec.7), or the ADWA and CDCC-BA formalisms for transfer reactions (Sec. 4), as well as their extension to consider collective excitations and/or 3-body projectiles, where the particular characteristics of Borromean systems have been exploited to explore nucleon-nucleon correlations (Sec. 9). As well, inclusive breakup (Sec. 5) and incomplete and complete fusion (Sec. 6) reactions with weakly-bound systems have been explored due to the unique phenomenon of complete fusion suppression found for weakly-bound species. Thanks to the increase in computational power, demanding methodssuchasCDCC andIAVbecomemoreaccessibleand useful in the analysis of experimental data. In addition, questionswhichrequirelargernumericalefforts,suchastheinclusion of non-locality in these sophisticated methods (Sec. 11) and the study of theoretical uncertainties (Sec. 10), are starting to be explored. Admittedly, the selection of reactions and methods presented in this work has been heavily curated by the expertise and preferences of the authors and many other methods for the study of reactions with weakly-bound nuclei exist and/or are currently being developed: – Some of the most promising advances are those related to few-body and ab-initio methods. The application of the Faddeev method to nuclear reactions, while potentiallyprovidinganexact,rigoroussolutionofagivenfewbody Hamiltonian, has only been possible in recent years thanks to the increasing computational capabilities, the inclusion of complex optical potentials and the development of techniques to deal with the long-range Coulomb couplings. The method has been very useful for benchmarking some of the methods discussed in this review, such as the CDCC and ADWA methods [35,247]. It has also proven very useful for analyzing nucleon knockout reactions [36,248,249]. – There has also been remarkable progress in the No-CoreShell-Model (NCSM) and its extension, the NCSM with Continuum (NCSMC) [250]. While originally developed to study nuclear structure, the method has been applied with great success to the calculation of scattering observables [183,251,252]. – Ab-initio methods provide not only alternative ways for evaluating scattering observables by themselves, but they also furnish very useful structure inputs for traditional reaction frameworks. For example, the Variational Monte Carlo (VMC) [253] and its variants, such as the Green’s function Monte Carlo and the Cluster Variational Monte Carlo [254], provide microscopic overlap functions which can be directly plugged into reaction formalisms for transfer [255] and knockout [256] reactions. – The expansion of experimental studies, reaching heavier exotic nuclei, has also shown the need to extend and upgrade the existing reaction models to incorporate the new features found in the structure and reactions of the newly discovered nuclei. For example, it has become clear that a proper description of newly discovered halo nuclei in the islands of inversion requires the simultane123
Eur. Phys. J. A (2025) 61:47 Page 53 of 57 47 ous inclusion of deformation, pairing and Pauli blocking. Reaction formalisms, such as DWBA or CDCC, must therefore be extended to accommodate such effects. Admittedly, the XCDCC method discussed in this review has used so far relatively simple particle-plus-rotor models, and hence it is mandatory to resort to more sophisticated models. – The challenge is even larger for the case of three-body systems which, in addition to the properties cited above, require specific formalisms to account for their threebody nature. For example, knockout experiments with 14Be have clear evidenced the need of including the excitations of the 12Be “core” [191,257]. Even beyond, models including fourand five-body systems may be necessary, as the recent discovery of a four-neutron correlated system in 8He attests [258]. All of these avenues for advancement show that, despite its numerous and significant successes, the study of nuclear reactions with weakly-bound nuclei remains a developing and active field, pushedby theoretical and technical advances and by new experimental measurements in ever more exotic nuclei. While we hope that this work has transmitted to the reader at least a glimpse of the current methods in this fascinating field, we also hold the paradoxical hope that new developments promptly render this work somewhat obsolete thanks to new and perhaps surprising advancements. Acknowledgements WearegratefultoAngelaBonaccorsoandJinLei fortheirfeedbackonthesemiclassicaltransferto thecontinuumcalculations.Thisworkhas been partially funded by theMinisteriodeCienciae Innovación, MCIN/AEI/10.13039/501100011033 under I+D+i project No. PID2020-114687GB-I00. M.G.-R. acknowledges financial support by MCIN/AEI/10.13039/501100011033 under grant IJC2020-043878I (also funded by “European Union NextGenerationEU/PRTR”). Funding Funding for open access publishing: Universidad de Sevilla/ CBUA Data Availability Statement This manuscript has no associated data. [Author’s comment: Data sharing not applicable to this article as no datasets were generated or analysed during the current study.] Code Availability Statement This manuscript has no associated code/software. [Author’s comment: Code/Software sharing not applicable to this article as no code/software was generated or analysed during the current study.] Open Access This article is licensed under a Creative Commons Attribution4.0InternationalLicense,whichpermitsuse, sharing,adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/. 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