Reexamining closed-form formulae for inclusive breakup: Application to deuteron- and 6 Li-induced reactions

PHYSICAL REVIEW C 92, 044616 (2015) Reexamining closed-form formulae for inclusive breakup: Application to deuteron- and 6Li-induced reactions Jin Lei*and A. M. Moro† Departamento de FAMN, Universidad de Sevilla, Apartado 1065, 41080 Sevilla, Spain (Received 13 June 2015; published 28 October 2015) The problem of the calculation of inclusive breakup cross sections in nuclear reactions is reexamined. For that purpose, the post-form theory proposed by Ichimura, Austern, and Vincent [Phys. Rev. C 32,431 (1985)] is revisited, and an alternative derivation of the nonelastic breakup part of the inclusive breakup is presented, making use of the coupled-channels optical theorem. Using the distorted-wave Born approximation (DWBA) version of this model, several applications to deuteron and 6Li reactions are presented and compared with available data. The validity of the zero-range approximation of the DWBA formula is also investigated by comparing zero-range with full finite-range calculations. DOI: 10.1103/PhysRevC.92.044616 PACS number(s): 24.10.Eq,25.70.Mn,25.45.−z I. INTRODUCTION The breakup of a nucleus into two or more fragments is an important mechanism occurring in nuclear collisions, particularly when one of the colliding nuclei is weakly bound. The analysis of this kind of processes has provided useful information on the structure of the broken nucleus, such as binding energies, spectroscopic factors, and angular momentum (e.g., [1,2]), and has contributed to the understanding of the dynamics of the reactions among composite systems. In the simplest scenario, in which the projectile is broken up into two fragments, these processes can be schematically represented as a+A→b+x+A, where a=b+x.From the theoretical point of view, this problem is difficult to treat because one has to deal with three-body final states. When the state of the three outgoing fragments (b,x, and A)is fully determined, the reaction is said to be exclusive.If, in addition, the three particles are emitted in their ground state, the corresponding cross section is referred to as elastic breakup (EBU). In this case, the reaction can be treated as an effective three-body problem interacting via some effective two-body interactions. Although the rigorous formal solution of this problem is given by the Faddeev formalism [3,4], the difficulty of solving these equations has led to the development of simpler approaches, such as the distorted-wave Born approximation (DWBA) [5], the continuum-discretized coupled-channels (CDCC) method [6], and a variety of semiclassical approaches [7–10]. A more complicated situation occurs when the final state of one or more fragments is not specified. In this case, thereactionissaidtobeinclusive with respect to this unobserved particle(s). This is the case of reactions of the form a+A→b+B∗, where B∗is any possible configuration of the x+Asystem. This includes the breakup processes in which xis elastically scattered by A, which corresponds to the EBU defined above, but also breakup accompanied by target excitation, particle(s) exchange between xand A,xtransfer to *[email protected] †[email protected] A, and the fusion of xby A, which are globally referred to as nonelastic breakup (NEB). The total inclusive breakup (TBU) is therefore the sum of EBU and NEB components, i.e., TBU = EBU +NEB. Measured observables usually correspond to single- or double-differential cross sections with respect to the angle and/or energy of band hence include both EBU and NEB contributions. The evaluation of NEB cross sections are needed, for example, in the calculation of total fusion cross sections in reactions induced by weakly bound projectiles (e.g., 6Li, 7Li, 9Be). A significant fraction of the total fusion cross section comes from incomplete fusion (ICF), in which only part of the projectile fuses with the target, the other fragment surviving after the collision [11]. Although many theoretical efforts have been made to develop suitable models to calculated ICF cross sections [12–14], the unambiguous calculation of CF and ICF within a fully quantum mechanical model remains a challenging problem [15,16]. Because the ICF is part of the inclusive breakup, the study of inclusive breakup reactions may lead, in turn, to a better understanding of ICF. A related problem is that of the indirect determination of neutron-induced cross sections on short-lived nuclei, from a surrogate reaction which gives rise to the same compound nucleus [17]. This is the case, for example, of the process A(n,f ) (where fis a fission fragment), for which the surrogate reaction A(d,pf) may be used. To extract the cross section for the former, one needs to know the fraction of protons produced in the surrogate reaction, which are accompanied by the formation of a n+Acompound nucleus. Therefore, the applicability of the method requires the separation of the EBU component (which does not lead to compound-nucleus formation) from the NEB (which contains the absorption cross section). The calculation of inclusive breakup observables is more involved than that for the exclusive ones because they require the inclusion of all the possible processes through which the particle xcan interact with the target A. Given the large number of accessible states, this procedure is impractical in most cases. As an alternative to this approach, one may try to replace the physical final states with a set of representative states (also named doorway states). These can be taken, for 0556-2813/2015/92(4)/044616(14) 044616-1 ©2015 American Physical Society JIN LEI AND A. M. MORO PHYSICAL REVIEW C 92, 044616 (2015) example, as the eigenstates of the x+AHamiltonian in a mean-field potential. As long as the basis used to describe these final states is complete, one may argue that the sum over these representative states should provide results close to those obtained if the sum were done over the true physical states. This procedure, referred to in some works as transfer to the continuum method, has been used recently with success to describe some inclusive breakup reactions of weakly bound projectiles at Coulomb barrier energies, such as 208Pb(8Li, 7Li) [18], 208Pb(6He, α)[19], and 120Sn(6He, α)[20]. However, despite this relative success, this method is based on a heuristic approach rather than on a rigorous formal theory. Lacking this formal justification, it is not clear how these doorway states should be chosen and how the final calculated cross sections depend on this choice. Another drawback of this approach is that it does not make it possible to separate the contributions coming from EBU and NEB. At intermediate energies (above ∼100 MeV/u), the problem can be greatly simplified using the adiabatic (fast collision) and eikonal (forward scattering) approximations, which makes it possible to obtain closed formulas of the inclusive process in terms of the absorption and survival probabilities of the unobserved particle as a function of the impact parameter. This approach has been used extensively in the analysis of nucleon removal (knockout) experiments at intermediate energies, in which typically the removed particle is not observed and only the momentum distributions of the residual core is measured (see, e.g., Refs. [21,22], and references therein). These models, however, cannot be applied to low incident energies (a few MeV/u) and when the energy or momentum transfer is large. The problem of the calculation of inclusive breakup cross sections is nevertheless not new. This problem was studied in detail by some groups since the late 1970s, and several theories were proposed and applied. The aim of these theories was to derive closed-form formulas, in which the sum over final states of the x+Asystem is formally reduced to some expectation value of the imaginary part of the x+Aoptical potential. In the pioneering works by Baur and coworkers [23–25], the sum is done making use of unitarity and a surface approximation of the form factors of excited states of the residual nucleus. These two approximations were avoided in later works by Udagawa and Tamura [26,27], who used a prior-form DWBA formalism, and by Austern and Vincent [28], who used the post-form DWBA. The latter was refined by Kasano and Ichimura [29], who found a formal separation between the EBU and the NEB contributions. These results were carefully reviewed by Ichimura, Austern, and Vincent [30] and the model was subsequently referred to as the IAV formalism. Later on, Austern et al. reformulated this theory within a more complete three-body model [6]. It is worth noting that the prior-form model of Udagawa and Tamura (UT), on one side, and the post-form DWBA model of Austern and Vincent (AV), on the other side, although formally similar, give different predictions for the NEB part. This led to a long-standing dispute between these two groups, which was finally settled in the IAV work [30], where it was demonstrated that a proper derivation of the prior-form formula gives rise to additional terms not considered by UT. Although the comparison of these theories with experimental data showed very encouraging results, they have apparently fallen into disuse. Moreover, some of these theories, such as the three-body model of Austern, has never been tested, to our knowledge, probably owing to the computational limitations at that time. This is in contrast to the experimental situation, in which inclusive breakup measurements are used for many applications, with both stable and unstable beams. Therefore, it seems timely to reexamine these theories and study their applicability to problems of current interest. The revival and increasing interest on this problem is evidenced by two recent theoretical works on this subject [31,32]. Both of them use the IAV model in DWBA. In Ref. [31], the authors use the zero-range post-form of this model, whereas in Ref. [32] the finite-range prior-form version of the model is used instead. Both of them apply the method to deuteron-induced reactions, with encouraging results. In this paper, we revisit also the IAV model, with special emphasis on the calculation of the NEB part, for which we provide a new derivation. We have implemented the DWBA version of this model both in zero range and in exact finite range. To assess the validity of this theory, we have performed calculations for several reactions induced by deuterons and, for the first time, the method is applied to 6Li scattering. In both cases, we compare with available data. The paper is organized as follows. In Sec. II we give a short overview of the theory, including a new derivation of the NEB formula within the IAV model. In Sec. III, the formalism is applied to several inclusive reactions induced by deuterons and 6Li. Finally, in Sec. IV we summarize the main results of this work and outline some future developments. II. THE ICHIMURA, AUSTERN, VINCENT MODEL In this section we briefly review the model of Ichimura, Austern, and Vincent [6,30]. The final formula obtained in this model has been derived in different ways. Here, we closely follow the early derivation done by AV [28] because it provides an interesting physical insight. We write the process under study as a(=b+x)+A→b+B∗.(1) This process is described with the Hamiltonian H=K+Vbx +UbA(rbA)+HA(ξ)+VxA(ξ,rx),(2) where Kis the total kinetic energy operator, Vbx is the interaction binding the two clusters band xin the projectile a,HA(ξ) is the Hamiltonian of the target nucleus (with ξdenoting its internal coordinates), and VxA and UbA are the fragment-target interactions. The relevant coordinates are depicted in Fig. 1. Note that the coordinate rbconnects particle bwith the center of mass (c.m.) of the x+Asystem. In writing the Hamiltonian of the system in the form (2)we make a clear distinction between the two cluster constituents; the interaction of the fragment b, the one that is assumed to be detected in the experiment, is described with an optical potential. Nonelastic processes arising from this interaction (e.g., target excitation) are included only effectively through UbA. Particle bis said to act as spectator. However, the 044616-2 REEXAMINING CLOSED-FORM FORMULAE FOR . . . PHYSICAL REVIEW C 92, 044616 (2015) FIG. 1. Coordinates used in the breakup reaction. interaction of the particle xwith the target retains the dependence of the target degrees of freedom (ξ). Within the assumed three-body model, and using the postform representation, the total wave function of the system can be written in integral form as (ξ,rx,rb)=[E+−Kb−UbB −HB]−1 ×Vpost(ξ,rx,rb),(3) where E+=E+i,→0, UbB is an auxiliary (and, in principle, arbitrary) potential between band the composite B,Vpost ≡Vbx +UbA −UbB, and HBis the Hamiltonian of the x+Apair, given by HB(ξ,rx)=HA(ξ)+Kx+VxA(ξ,rx).(4) The eigenstates of the target Hamiltonian are denoted as φc A(ξ), i.e., [HA(ξA)−Ec A]φc A(ξ)=0, with c=0 corresponding to the target ground state, for which we assume E0 A=0. We consider now a specific final state of the detected particle b, characterized by a given final momentum of this fragment ( kb). The motion of bis described by a distorted wave with momentum  kb, obtained as a solution of the single-channel equation [Kb+U† bB −Eb]χ(−) b( kb,rb)=0.(5) The wave function describing the motion of xafter the breakup, denoted as Zx(ξ,rx), can be obtained projecting the total wave function [Eq. (3)] onto this particular state of the bparticle; i.e.,1 Zx( kb,ξ,rx)≡(χ(−) b| =[E+−Eb−HB]−1(χ(−) b|Vpost|,(6) where the round bracket denotes integration over rbonly. The last equation can be also written in differential form as [E+−Eb−HB]Zx( kb,ξ,rx)=(χ(−) b|Vpost|.(7) 1Note that this function also depends on  ka, which indicates the direction of the incident beam. Because this direction is fixed, this dependence is omitted for simplicity of the notation. The source term of this equation involves the exact and hence unknown wave function , which, in actual calculations, must be approximated by some calculable form. For example, in DWBA, one assumes the factorized form (ξ,rx,rb)≈φ0 A(ξ)φa(rbx)χ(+) a( ka,ra),(8) where φa(rbx) is the projectile ground-state wave function and χ(+) a( ka,ra) is a distorted wave describing the a+Amotion in the incident channel. In practice, the latter is commonly replaced with the solution of some optical potential describing a+Aelastic scattering. Austern et al. [6] proposed also the three-body approximation (ξ,rx,rb)≈φ0 A(ξ)3b(rx,rb),(9) where 3b is a three-body wave function for the three fragments (x+b+A) and contains, in addition to the b+x ground state, contributions from b+xinelastic scattering and breakup. It is worth noting that, either in the approximation (8)orin (9), the three-body wave function does not contain explicitly excited states of A. Thus, in the IAV model, the NEB can be viewed as a two-step process in which the first step is the dissociation of the projectile, leaving the target in the ground state, while the second step is the absorption of x or the excitation of A. A possible procedure to solve Eq. (7) is to expand the function Zxin a complete set of x+Astates, i.e., Zx( kb,ξ,rx)= c ψc x( kb,rx)φc A(ξ),(10) where ψc x( kb,rx) describes the x−Arelative motion when the target is in the state c. The expansion (10) can be inserted into Eq. (7), giving rise to a set of coupled equations for the unknown functions ψc x( kb,rx). This approach will be, in general, unpractical because the expansion (10) involves a very large number of final states. If one is not interested in the description of the transition to specific x+Astates, but rather in their summed contribution, one can proceed as follows. Following Feshbach, the Zx( kb,ξ,rx) function is decomposed as Zx( kb,ξ,rx)=PZx+QZx,(11) where Pis the projector operator onto the target ground state and Q=1−P.FromEq.(10) we see that PZx= ψ0 x( kb,rx)φ0 A(ξ). The function ψ0 x( kb,rx), which describes the x+Arelative motion when the target is in the ground state, verifies the equation (E+ x−Kx−Ux)ψ0 x( kb,rx)=ρ( kb,rx),(12) where Ex=E−Eb,ρ( kb,rx)≡(χ(−) b|Vbx|is the so-called source term, and Uxis the formal optical model potential describing x+Aelastic scattering. Explicitly, Ux=φ0 AVxA +VxAQ[E+−Eb−HQQ]−1VxAφ0 A,(13) where HQQ ≡QHBQ. The formal potential Uxis a complicated nonlocal, angular-, and energy-dependent object. However, as done in two-body scattering problems, it can be approximated by some energy-averaged (possibly local) 044616-3 JIN LEI AND A. M. MORO PHYSICAL REVIEW C 92, 044616 (2015) potential or by some phenomenological representation (denoted Uxhereafter) with parameters adjusted to describe x+A elastic scattering. Note that Eq. (12) is formally analogous to the inhomogeneous equation appearing in DWBA and coupled-channel Born approximation (CCBA) calculations between bound states, as formulated in the source term method of Ascuitto and Glendenning [33] and used by several coupled-channels codes [34]. A. Separation of elastic and nonelastic breakup In their original paper, AV provide only the total inclusive cross section. Later, Kasano and Ichimura [29] showed that this expression can be formally decomposed into two pieces, corresponding to the EBU and NEB contributions. Here we present an alternative derivation of these formulas, which exploits the aforementioned analogy of Eq. (12) with that found in the DWBA and CCBA formalisms. This equation is to be solved with purely outgoing boundary conditions (because there are no incoming waves in the x−Achannel); that is, ψ0 x( kb,rx)→f( kb,ˆ rx)eikxrx rx .(14) The function f( kb,ˆ rx) depends on the angular part of rx, in addition to the direction of  kb. Asymptotically, when rxis large, the position vector rxbecomes parallel to the momentum  kxand we may write f( kb,ˆ rx)→f( kb, kx). We therefore recognize f( kb, kx) as the scattering amplitude for the EBU process, and its square is proportional to the differential cross section for the detection of the xparticle in the direction of  kxand the bparticle in the direction  kb. To obtain this amplitude, one can proceed in two different ways. One possibility is to integrate the differential equation (12) and, at a sufficiently large distance (beyond the range of the short-ranged potentials), equate the solution to the asymptotic form (14), from which the scattering amplitude can be obtained. A second approach to solve Eq. (12)isto use integral (i.e., Green’s function) techniques. This gives a closed-form expression for the scattering amplitude, f( kb, kx)=− μx 2π2χ(−) xχ(−) b|Vbx|3b,(15) where μxis the reduced mass of the x+Asystem and the distorted wave χ(−) x( kx,rx) is a solution of the homogeneous part of Eq. (12), i.e., [Kx+U† x−Ex]χ(−) x( kx,rx)=0,(16) whose solution consists of a plane wave of momentum  kxplus an ingoing spherical wave. The corresponding differential cross section, for a final differential volume in momentum space, is given by2 2Note that the factor (2π)4of Ref. [35] is replaced here with a (2π)−5factor, consistent with our definition of the amplitude for the plane waves as ei kr. (cf., for instance, Eq. (5.36) of Ref. [35]) dσ =(2π)−5 vid kxd kbd kAδ(Ef−Ei)δ( Pf− Pi)|Tfi|2, (17) where Tfi is the usual transition amplitude (or Tmatrix), which is related to the scattering amplitude by f=−(μx/2π2)Tfi. In the c.m. frame,  Pi=0. Also, the target momentum ( kA)is not measured, so we can integrate over it, making use of the momentum-conserving δfunction, dσ =(2π)−5 vid kxd kbδ(Ef−Ei)|Tfi|2.(18) The element d kbis conveniently expressed in terms of energy and solid angle elements using d kb=(2π)3ρb(Eb)dbdEb, where ρb(Eb)=kbμb/[(2π)32] is the density of states.3 UsingthisinEq.(18), dσ =(2π)−2 viδ(Ef−Ei)|Tfi|2ρb(Eb)dbdEbd kx.(19) The double-differential cross section with respect to the energy and the scattering angle of bis therefore given by d2σ dbdEbEBU =(2π)−2 vi ρb(Eb)δ(Ef−Ei)|Tfi|2d kx, (20) which coincides with the result of Austern et al. (Eq. (8.15) of Ref. [6]), noting that d kx→(2π)3 kx. Although it is not the purpose of the present work, we note also that the previous expression can be used to compute the fully exclusive cross section, with respect to the angles and energies of band x. For that, we use again d kx=(2π)3ρx(Ex)dxdExand use the energy-conserving δ function, resulting in d2σ dbdEbdxEBU =2π vi ρb(Eb)ρx(Ex)|Tfi|2.(21) To obtain the expression for the NEB component, we make use of the coupled-channels optical theorem recently formulated by Cotanch [36]. This work generalizes the wellknown optical theorem to the multichannel case. If χiis the channel wave function and Withe diagonal imaginary part for this channel, the contribution to the absorption in this particular channel is given by [36] σi abs =− 2 vel χi|Wi|χi,(22) where vel is the projectile-target relative velocity in the incident (elastic) channel. We may use this result to calculate the NEB contribution by noting that the latter is nothing but the absorption occurring 3These expressions result from N(k)d kb=ρb(Eb)dbdEb,where N(k) is the number of states in the differential volume d kb,which is determined from  k| k=δ( k− k)/N(k). In our case,  k| k= (2π)3δ( k− k), and hence N(k)=(2π)−3. 044616-4 REEXAMINING CLOSED-FORM FORMULAE FOR . . . PHYSICAL REVIEW C 92, 044616 (2015) in the x+Achannel. The channel wave function is given by ψ0 x( kb,rx), which is a solution of Eq. (12). Because Eq. (12) corresponds to a definite energy and direction of the bparticle, we consider the differential cross section corresponding to a range of the outgoing momenta of b, d2σ=−2 viψ0 xWxψ0 xN(kb)d kb,(23) with Wx≡Im[Ux]. Transforming the element of momentum into energy and solid angle elements, we get the doubledifferential cross section d2σ dEbdbNEB =−2 vi ρb(Eb)ψ0 xWxψ0 x.(24) This result was obtained, by different arguments, by Kasano and Ichimura [29]. A similar result was also obtained by Hussein and McVoy [37]. The alternative derivation presented here, based upon the generalized optical theorem, provides a clear interpretation of this term, as the flux leaving the x+Achannel following the breakup of the projectile into b+x. To recapitulate, in the IAV model, the breakup can be viewed as a two-step process. The first step corresponds to the dissociation of the projectile (a) into the fragments band x, leaving the target in the ground state. The subsequent motion of the participant particle (x) is described by the function ψ0 x( kb,rx), which is the solution of the inhomogeneous Eq. (12). This particle can then be scattered elastically by the target or can interact nonelastically (for example, excite the target or fuse with it). The former corresponds to the EBU part of the inclusive breakup cross section, whereas these nonelastic processes, corresponding to the second step in this two-step picture, yield the NEB contribution. Quantitatively, this contribution is obtained as the expectation value of Im[Ux] in the state ψ0 x( kb,rx)[Eq.(24)]. Note that, because this function depends on the final state of the spectator particle (b), the NEB expression (24) yields the absorption for each final state of b. B. Practical implementation of the IAV model The IAV formula for NEB breakup, Eq. (24), has a deceptively simple form. The function ψ0 xmust be first calculated from the inhomogeneous Eq. (12), whose source term contains the three-body wave function 3b, which is a complicated object by itself. Furthermore, this equation must be solved for each outgoing energy and angle of bcovering the range of interest. For these reasons, practical implementations of this theory have resorted to additional approximations. Standardly, all these applications rely on the DWBA approximation of the incident channel [that is, Eq. (8)], rather than on a three-body model [Eq. (9)]. Even at the DWBA level, Eq. (12) is not trivial. Usually, a partial wave decomposition of the scattering waves appearing in Eq. (12) will be used and this means that a large number of angular momenta for the a+A,x+A, and b+Bdistorted waves will be required for convergence of the calculated cross sections. In addition, the right-hand side of this equation contains nonlocal kernels (similar to those appearing in DWBA calculations between bound states, but involving a larger number of angular momenta). Consequently, in addition to the DWBA approximation, most of the existing calculations of this kind have been done in the zero-range (ZR) approximation. To assess the validity of this approximation, we have performed calculations with both the ZR and exact finiterange (FR) formulas. The detailed formulas for the NEB cross sections in these two approximations are given in the Appendixes. Other difficulties arising in solving Eq. (12)arethe well-known convergence problems of the post-form DWBA formula when applied to breakup reactions. This is because χ(−) b, being a scattering state, will be infinitely oscillatory and the operator in the matrix element Vbx and the initial state (ψain DWBA) depend on the rbx coordinate and hence there is no natural cutoff in the rbintegration. As a consequence, the source term has infinite range. To overcome this problem, Huby and Mines [38] and Vincent [39] multiply the source term by an exponential convergence factor that damps the contribution of the integral at large distances. Alternatively, following Vincent and Fortune [40], one may use the integration in the complex plane. Here we adopt a different procedure. Following Thompson [41], we consider energy bins for the bdistorted waves. For this, the scattering states are first expanded in partial waves (see Appendix A), and the radial coefficients, Rb(rb,kb), are then averaged over small energy or momentum intervals, i.e., ¯ Rbrb,ki b=Nki b+kb/2 ki b−kb/2 dkbRb(rb,kb),(25) where kbis the bin width, ki bis the central momentum of the bin, and Nis a normalization constant. The resulting bin wave function is square integrable and thus leads to convergent results when it is used in the source term of Eq. (12). The formulas discussed in this section are applied to specific cases in the following section. III. CALCULATIONS In this section, we present calculations for several reactions induced by deuterons and 6Li projectiles and compare the calculated inclusive cross sections with experimental data to assess the validity of the theory. In all cases, we compute the separate contributions for the elastic (EBU) and nonelastic (NEB) breakup cross sections. For the former, we use the CDCC formalism, using the coupled-channels code FRESCO [34]. This makes it possible to treat the EBU to all orders and should be equivalent to the post-form three-body model of Austern et al. For the NEB part, we use the DWBA version of Eq. (24). We have also tested the accuracy of the ZR approximation in the NEB formula by comparing ZR with FR calculations. A. Application to (d,pX) There is a large body of exclusive and inclusive breakup data for deuteron-induced reactions. We have considered the inclusive (d,pX) data for the reactions d+93Nb at 044616-5 JIN LEI AND A. M. MORO PHYSICAL REVIEW C 92, 044616 (2015) Ed=25.2 MeV from Ref. [42] and d+58Ni at 80 and 100 MeV from Refs. [43,44]. The data for d+93Nb were already analyzed in Ref. [42], using the so-called surface approximation, in Ref. [29], using the ZR version of the post-form DWBA formula discussed here, and in Ref. [32], using the prior form of the DWBA IAV model. These calculations give a reasonable account of the experimental data. In the CDCC calculations [6] the deuteron breakup is treated as inelastic excitations to the p-ncontinuum. This continuum is truncated at a maximum excitation energy and discretized in energy bins. For the present case, the p-nstates were included for =0–4 partial waves and up to a maximum excitation energy of 20 MeV. For the p-ninteraction, we considered the simple Gaussian form of Ref. [6]. The proton-target and neutron-target interactions were adopted from the global parametrization of Koning and Delaroche (KD) [45], omitting the spin-orbit term, and evaluated at half of the deuteron incident energy. In the CDCC method, the breakup cross sections are calculated in terms of the c.m. scattering angle and excitation energy of the p-n system. Therefore, to compare with the proton inclusive data, these breakup cross sections must be converted to the proton energy and scattering angle, making use of the appropriate kinematical transformation. This was done with the formalism and codes developed in Ref. [46]. For the NEB calculations, we use also the KD parametrization for the proton-target and neutron-target interactions, but evaluated at the corresponding proton (Ep) and neutron (En) energies. In DWBA, one needs also the incoming channel optical potential (d+93Nb), which was taken from Ref. [47]. For the ZR-DWBA calculations we used the ZR constant D0=125 MeV fm3/2and included the FR correction factor (see, e.g., Refs. [48,49] and Appendix A). In Fig. 2(a) we compare the experimental [42] and calculated inclusive double-differential cross section, d2σ/ dEpdp, corresponding to a proton energy of Ep=14 MeV. The dotted line is the EBU calculation (CDCC), which is found to underestimate the data at all angles. The thin solid line is the FR-DWBA calculation for the NEB part (see Appendix B). The thick solid line is the sum of the EBU and NEB contributions. Except at very large angles, it is found to explain satisfactorily the data. It is seen that, except for the smallest angles, the inclusive breakup cross section is largely dominated by the NEB contribution. Our results are consistent with those reported in Refs. [29,42]. In Fig. 2(b),we compare several approximations for the numerical evaluation of the NEB cross section. The dotted line is the ZR-DWBA calculation, including, nevertheless, the FR correction (rx) (see Appendix A). The dashed line is the FR-DWBA calculation, omitting the remnant term in the transition operator (i.e., using Vpost ≈Vpn). Finally, the solid line is the full FR-DWBA calculation. We find that the ZR calculation (with FR correction) provides a very accurate result in the present reaction, thus supporting the validity of this approximation in this case. Further, we see that the nonremnant term has a very small effect and can be also safely ignored in the FR calculation. 0 306090120150180 0.01 0.1 1 10 100 dσ/dΩpdEp(mb/sr MeV) Pampus et al. EBU (CDCC) NEB (FR-DWBA) Total 0 306090120150180 0 10 20 30 dσ/dΩpdEp(mb/sr MeV) NEB(ZR-DWBA) NEB(FR-DWBA without remnant) NEB(FR-DWBA) 0 306090120150180 θc.m.(deg) 0 10 20 30 40 dσ/dΩpdEp(mb/sr MeV) Δk=0.06 fm-1 Δk=0.04 fm-1 Δk=0.02 fm-1 Δk=0.015 fm-1 93Nb(d,pX) @ Ed=25.5 MeV (Ep=14 MeV) (a) (b) (c) FIG. 2. (Color online) (a) Experimental and calculated doubledifferential cross section, as a function of the proton scattering angle, for the protons emitted in the 93Nb(d,pX) reaction with an energy of 14 MeV and a deuteron incident energy of Ed=25.5MeV.The dotted, thin solid, and thick solid lines are the EBU (CDCC), the NEB (FR-DWBA), and their incoherent sum, respectively. Experimental data are from Ref. [42]. (b) NEB angular distribution calculated with ZR-DWBA (dotted line), FR-DWBA without remnant (dashed line), and full FR-DWBA (solid line). (c) Convergence of the NEB calculation with respect to the bin width, kb, used for the bdistorted waves. See text for details. To obtain meaningful results, the calculated observables must converge as the bin width kbis progressively decreased [cf. Eq. (25)]. This is verified in Fig. 2(c) for the present case, where we show the calculated NEB angular distribution for different values of kb. Although the rate of convergence was found to be small, it is seen that for kb≈0.02 fm−1the calculations are well converged for the full angular range. A similar convergence study was done in the other calculations presented below. We present now the results for the 58Ni(d,pX) reaction at 80 and 100 MeV and compare with the data from Refs. [43,44]. These data have been also analyzed in Refs. [50–52], using the CDCC method for the EBU part, and the semiclassical Glauber approach for the NEB part. In our CDCC calculations, 044616-6 REEXAMINING CLOSED-FORM FORMULAE FOR . . . PHYSICAL REVIEW C 92, 044616 (2015) 0 20406080 100 101 102 103 dσ/dEp (mb/MeV) J.R. Wu et al. EBU (CDCC) NEB (FR-DWBA) PE+CN Total 0 20406080 Ep (MeV) 0 10 20 30 dσ/dEp (mb/MeV) NEB (ZR-DWBA) NEB (FR-DWBA without remnant) NEB (FR-DWBA) 58Ni(d,pX) @ Ed=80 MeV (a) (b) FIG. 3. (Color online) (a) Experimental and calculated angleintegrated proton differential cross section, as a function of the outgoing proton energy in the LAB frame, for the 58Ni(d,pX) reaction at Ed=80 MeV. The dotted and thin solid lines are the EBU and NEB contributions, calculated with CDCC and FR-DWBA, respectively. The dot-dashed line is the contribution coming from preequilibrium and compound nucleus [50]. The thick solid line is the incoherent sum of the three contributions. Experimental data are from Ref. [43]. (b) Nonelastic breakup calculated with ZR-DWBA (dotted), nonremnant FR-DWBA (dashed), and full FR-DWBA (solid) formulas. the proton-target and neutron-target interactions are obtained again from the KD parametrization, and we employed the same p-ninteraction used in the d+93Nb calculations. For the p-ncontinuum we considered the partial waves =0–6, and excitation energies up to 50 and 90 MeV for the data at Ed=80 MeV ad Ed=100 MeV, respectively. For the NEB calculations, the d+58 Ni potential was taken from Ref. [47]. In Fig. 3, we present the angle-integrated energy differential cross section at Ed=80 MeV (dσ/dEp). In Fig. 3(a),the dotted and thin solid lines correspond to the EBU (CDCC) and NEB (FR-DWBA) calculations. It is seen that the NEB contribution is much larger than the EBU part. Both distributions show a bell-shaped behavior, with a maximum around half of the deuteron energy. However, it is observed that the sum of these two contributions cannot explain the experimental yield at small proton energies. As shown in Ref. [50], these lowenergy protons come mainly from compound nucleus followed by evaporation and preequilibrium. Because these processes are not accounted for by the present formalism, in this work we have adopted the estimate done in Ref. [50] [dot-dashed line in Fig. 3(a)]. The total inclusive cross section, including this contribution (thick solid line) reproduces reasonably well the shape and magnitude of the data. Note that protons with energies larger than ∼74 MeV correspond to bound states of the neutron-target system and they are associated with a stripping mechanism. This contribution could be accommodated in the present formalism solving Eq. (12)for Ex<0 and with boundary conditions appropriate for bound states instead of outgoing boundary conditions. Further, for high-lying bound excited states, where the density of levels will be very high, one may use the ideas of Udagawa and co-workers of extending the complex potential to negative energies to describe the spreading of single-particle states [53,54]. However, these extensions go beyond the scope of the present work. In Fig. 3(b), we compare different approximations for the transition amplitude used in the NEB calculation, namely, ZRDWBA (dotted line), FR-DWBA with no remnant (dashed line), and full FR-DWBA (solid line). As in the previous case, the ZR-DWBA and FR-DWBA calculations agree very well for proton energies around and above the maximum, although some small differences are visible. The effect of the remnant term is again found to be very small. We finally present the results for the d+58Ni reaction at 100 MeV. This is shown in Fig. 4, where the top panel contains the experimental and calculated proton angular distributions for protons detected at 50 MeV in the laboratory frame, and the bottom panel shows the energy distribution for the protons scattered at 8◦in the laboratory frame. Again, it is seen that the 0 20406080100 Ep (MeV) 0 20 40 60 80 100 d2σ/dEpdΩp (mb/sr MeV) D. Ridikas et al. EBU (CDCC) NEB (FR-DWBA) PE+CN Total 0 20406080 θlab (deg) 0.01 0.1 1 10 100 1000 d2σ/dEpdΩp (mb/sr MeV) EBU (CDCC) NEB (FR-DWBA) Total 58Ni(d,pX) @ Ed=100 MeV θp=8ο Ep=50 MeV (a) (b) FIG. 4. (Color online) Double-differential cross section of protons emitted in the 58Ni(d,pX) reaction at Ed=100 MeV in the laboratory frame. (a) Proton angular distribution for a fixed proton energy of Ep=50 MeV. (b) Energy distribution for protons emitted at a laboratory angle of 8 ◦(arrow in top panel). The meanings of the lines are the same as in Fig. 3and are also indicated by the labels. Experimental data are from Ref. [44]. 044616-7 JIN LEI AND A. M. MORO PHYSICAL REVIEW C 92, 044616 (2015) inclusive breakup is dominated by the NEB contribution in the full angular range, particularly at large scattering angles. As in the 80-MeV case, both the EBU and the NEB contributions exhibit bell-shaped distributions, with a maximum around ≈Ed/2. Note that the low-energy region is dominated by the protons coming from compound nucleus and preequilibrium. Except for some underestimation of the cross section at the maximum, the agreement between the theory and the data is rather satisfactory. B. Application to (6Li, αX) As a second example, we consider the αproduction following the breakup of the weakly bound nucleus 6Li. The understanding of the large αyields observed in reactions with 6Li has been subject of many studies [57–65]. These works have shown (see, e.g., Refs. [62,64]) that the total exclusive cross sections (α+dand α+p) are much smaller than the total αproduction cross section. Consequently, the αinclusive cross sections are largely underestimated by CDCC calculations. Furthermore, some of these works have shown that the total fusion cross section of these reactions is significantly enhanced owing to partial fusion of the projectile, usually referred to as ICF [66]. The calculation of ICF cross sections from a purely quantum mechanical framework is still a challenging problem [66,67]. Because the ICF is part of the NEB cross section, the inclusive breakup model considered in this work might provide useful starting point to tackle this problem. However, one has to bear in mind that the NEB cross section will contain, in addition to ICF contributions, other contributions, such as breakup accompanied by target excitation, without absorption of any of the fragments. These contributions should be subtracted from the total NEB cross section to extract the ICF part. Work in this direction is in progress and the results will be presented elsewhere. Here we focus on the calculation of the total inclusive cross sections. For that purpose, we have considered the 6Li +209Bi reaction at several bombarding energies between 10 and 50 MeV, for which experimental data exist [57]. The nominal Coulomb barrier for this system is around 30.1 MeV [11], so these data cover energies below and above the barrier. The 6Li nucleus is treated in a two-cluster model (α+d). CDCC calculations based on this model have been performed for many 6Li-induced reactions. To reproduce the elastic data, these calculations usually require a reduction of the imaginary part of the fragment-target interactions [68–70]. In contrast, four-body CDCC calculations, based on a more realistic three-body model of 6Li (α+p+n), are able to describe the elastic data for 6Li +209Bi without any readjustment of these potentials [71], thus suggesting that the need for a reduced absorption is related to the limitations of this two-body model for 6Li. Because the inclusive formulas considered in this work are based on a two-body model of the projectile, we perform our calculations with the α+dmodel and allow for the same kind of renormalization prescribed in previous works. For that, we first study the elastic scattering within the CDCC framework. These calculations include s-wave 0.9 1 1.1 dσ/dσR OMP CDCC 0.9 1 1.1 0.6 0.8 1 1.2 dσ/dσR 0 0.5 1 0 0.4 0.8 1.2 dσ/dσR 0 0.5 1 0 0.5 1 dσ/dσR 0 0.5 1 0 50 100 150 θc.m. (deg) 0 0.5 1 dσ/dσR 0 50 100 150 θc.m. (deg) 0 0.5 1 24 MeV 26 MeV 28 MeV 30 MeV 32 MeV 34 MeV 36 MeV 38 MeV 40 MeV 50 MeV FIG. 5. (Color online) Elastic scattering of 6Li +209Bi at different incident energies. The solid and dashed lines are, respectively, the CDCC calculation and the optical model calculation with the optical potential from Ref. [55]. The experimental data are from Ref. [56]. (Jπ=1+), p-wave (Jπ=0−,1−,2−), and d-wave (Jπ= 1+,2+,3+) continuum states. For the dwave, we make a finer division of bins to describe the 6Li resonant states at 2.186 MeV (Jπ=3+), 4.31 MeV (Jπ=2+), and 5.7 MeV (Jπ=1+). For the α+dground state we used a Woods-Saxon well with V0=78.46 MeV, r0=1.15 fm, and a=0.7fm[72]. We used a second Woods-Saxon well to describe the p- and d-wave states with parameters V0=80.0MeV,r0=1.15 fm, and a=0.7 fm and supplemented with a spin-orbit term, with the usual Woods- Saxon derivative form, and parameters Vso =2.5MeV,rso = 1.15 fm, and aso =0.7 fm to place the d-wave resonances correctly. The d-209Bi and α-209Bi optical potentials are taken from Refs. [73,74], respectively. Consistently with previous works, we find that these calculations tend to underestimate the elastic data. We found that, by removing the surface part of the d-209Bi imaginary potential, a good description of the experimental elastic angular distributions is achieved. This is shown in Fig. 5by solid lines. For comparison, we have included the optical model calculation using the potential of Cook [55] (dashed lines). We note that this reduction of the imaginary potential is consistent with the conclusions of 044616-8 REEXAMINING CLOSED-FORM FORMULAE FOR . . . PHYSICAL REVIEW C 92, 044616 (2015) 0 5 10 dσ/dΩ (mb/sr) EBU NEB (FR) Total 0 5 10 15 0 10 20 dσ/dΩ (mb/sr) 0 20 40 0 20 40 dσ/dΩ (mb/sr) 0 20 40 60 80 0 30 60 90 dσ/dΩ (mb/sr) 0 40 80 120 050 100 150 θlab (deg) 0 60 120 180 dσ/dΩ (mb/sr) 050 100 150 θlab (deg) 0 120 240 360 24 MeV 26 MeV 28 MeV 30 MeV 32 MeV 34 MeV 36 MeV 38 MeV 40 MeV 50 MeV FIG. 6. (Color online) Angular distribution of αparticles produced in the reaction 6Li +209Bi at the incident energies indicated by the labels. The dotted, dashed, and solid lines correspond to the EBU (CDCC), NEB (FR-DWBA), and their sum, respectively. Experimental data are from Ref. [57]. Ref. [71], which points toward an effective suppression of the deuteron breakup in 6Li scattering, compared to the free deuteron scattering. We discuss now the inclusive breakup cross sections (6Li,αX). The EBU contribution was obtained from the CDCC calculations discussed above. For the NEB calculations, we used Eq. (24), in both the ZR and the FR-DWBA approximations. We adopt the same optical potential of α/d + 209Bi as used in the CDCC calculations. For simplicity, the deuteron and target spins are ignored (note that, in the CDCC calculations, the inclusion of the deuteron spin is important to place correctly the =2 resonances). The distorted waves for the incoming channel are calculated with the optical potential of Cook quoted above. In Fig. 6, we compare the calculated and experimental angular distributions of αparticles, for several incident energies of 6Li. The dotted and dashed lines are the EBU (CDCC) and NEB (FR-DWBA) results. Except for the lowest energies, the NEB is found to account for most of the inclusive breakup cross section, in agreement with previous findings [62,64]. The summed EBU +NEB cross sections (thick solid lines) reproduce fairly well the shape and magnitude of the data, both above and below the barrier. These results give confidence in the possibility of extending the formulation of the IAV theory to situations in which the unobserved particle is a composite system. At the most forward angles (where the αyield is nevertheless small) the EBU is found to be larger than the NEB part. Using a semiclassical picture, this can be understood by noting that these small angles will correspond to distant trajectories. However, according to Eq. (24), the NEB is only effective for distances within the range of the deuteron-target imaginary potential, and, hence, it will be very small for these distant trajectories. It is worth noting, however, that the separation between EBU and NEB parts in the (6Li,αX) case is less clear than in the (d,pX) case. In the present model, the NEB is associated with the absorption produced by the d+target imaginary potential. If an empirical deuteron-target potential is used, part of this absorption will be attributable to the breakup of the deuteron into p+n. However, in a more realistic description of 6Li in terms of α+p+n, the breakup of 6Li into α+p+n(leaving the target in the ground state) would actually correspond to EBU. Despite this ambiguity, we believe that the sum of the two contributions, that is, the TBU cross section, can be reasonably well estimated by the present model, as supported by the comparison with the data. We study now the validity of the ZR approximation in the present reaction. This is shown in Fig. 7, where 0 50 100 150 0 1 2 3 4 5 dσ/dΩ (mb/sr) NEB (ZR-DWBA) NEB (FR-DWBA; no remnant) NEB (FR-DWBA) 0 50 100 150 θlab(deg) 0 40 80 120 dσ/dΩ (mb/sr) (a) E=24 MeV (b) E=38 MeV 209Bi(6Li,αX) FIG. 7. (Color online) Angular distribution of αparticles produced by NEB in the reaction 6Li +209Bi at the incident energies of (a) 24 MeV and (b) 38 MeV. The dotted, dashed, and solid lines are the ZR-DWBA, FR-DWBA without remnant term, and full FR-DWBA calculations, respectively. 044616-9