Comprehensive analysis of large α yields observed in 6Li-induced reactions

PHYSICAL REVIEW C 95, 044605 (2017) Comprehensive analysis of large αyields observed in 6Li-induced reactions Jin Lei*and Antonio M. Moro† Departamento de FAMN, Universidad de Sevilla, Apartado 1065, 41080 Sevilla, Spain (Received 10 January 2017; published 6 April 2017) Background: Large αyields have been reported over the years in reactions with 6Li and 7Li projectiles. Previous theoretical analyses have shown that the elastic breakup (EBU) mechanism (i.e., projectile breakup leaving the target in its ground state) is able to account only for a small fraction of the total α-inclusive breakup cross sections, pointing toward the dominance of nonelastic breakup (NEB) mechanisms. Purpose: We aim to provide a systematic study of the α-inclusive cross sections observed in nuclear reactions induced by 6Li projectiles. In addition to estimating the total αsingles’ cross sections, it is our goal to evaluate angular and energy distributions of these αparticles and compare them with experimental data, when available. Method: We compute separately the EBU and NEB components of the inclusive breakup cross sections. For the former, we use the continuum-discretized coupled-channels (CDCC) method, which treats this mechanism to all orders. For the NEB part, we employ the model proposed by Ichimura et al. [Phys. Rev. C 32,431 (1985)], within the distorted-wave Born approximation (DWBA). Results: Overall, the sum of the computed EBU and NEB cross sections is found to reproduce very well the measured singles’ cross sections. In all cases analyzed, we find that the inclusive breakup cross section is largely dominated by the NEB component. Conclusions: The presented method provides a global and systematic description of inclusive breakup reactions induced by 6Li projectiles. It provides also a natural explanation of the previously observed underestimation of the measured αyields by CDCC calculations. The method used here can be extended to other weakly bound projectiles, including halo nuclei. DOI: 10.1103/PhysRevC.95.044605 I. INTRODUCTION Reactions induced by the 6Li nucleus have been extensively studied, giving rise to a large body of experimental data. Given its marked α+dstructure, with a separation energy of 1.474 MeV (to be compared with the single-nucleon separation energy of 5.39 MeV), one may anticipate that the breakup of this nucleus into αand dis a major reaction channel. In fact, experimental data show remarkably large yields of αparticles but, contrary to expectations, these yields are typically much larger than the corresponding dyields. This suggests that the breakup of the 6Li is not a simple direct breakup mechanism. From the theoretical point of view, a proper interpretation of these αyields is still lacking. Continuum-discretized coupled-channels (CDCC) calculations, which treat the 6Li breakup as an inelastic excitation to the continuum, reproduce successfully the coincidence α+dmeasurements [1]butthey largely underestimate the inclusive αcross sections. It is worthwhile recalling that the CDCC method provides only the so-called elastic breakup (EBU) component of the total breakup cross section. For the reaction of a 6Li projectile impinging on a target A, this corresponds to the processes of the form 6Li +A→α+d+Ag.s.in which the two-projectile clusters survive after the collision and the target remains in *Present address: Institute of Nuclear and Particle Physics, and Department of Physics and Astronomy, Ohio University, Athens, Ohio 45701, USA; [email protected] †[email protected] the ground state.1Thus, the underestimation of the inclusive αyields by the CDCC calculations means that there other mechanisms contributing to the inclusive breakup cross section other than the EBU. These include the exchange of nucleons between dand A, the projectile dissociation accompanied by target excitation, and the fusion of dby A, among others, that we will globally denote as nonelastic breakup (NEB) channels. An explicit account of these processes is very challenging due to the huge number of accessible final states and the variety of competing different mechanisms. When one is only interested in the evaluation of the singles’ cross section (for example, the energy or angular distribution of αparticles), rather than on the separate contributing mechanisms, one may resort to the inclusive breakup models proposed in the 1980s and recently reexamined by several groups [2–6]. In these models, the sum over all the possible final states through which the unobserved fragment dmay interact with the target is done in a formal way, making use of the Feshbach projection formalism [7] and closure. In this work, we will show that inclusive αsingles cross sections from 6Li-induced reactions can be remarkably well reproduced using the inclusive breakup model proposed by Ichimura, Austern, Vincent model (IAV) [8]. To our knowledge, this is the first study of this kind providing a systematic explanation of these data. 1If a three-body description of the 6Li is used, α+p+n, the threebody breakup mode 6Li +A→α+p+n+Ag.s.would be also part of the elastic breakup channel. Since we resort here to a two-body model of 6Li we include this channel in the NEB part. 2469-9985/2017/95(4)/044605(11) 044605-1 ©2017 American Physical Society JIN LEI AND ANTONIO M. MORO PHYSICAL REVIEW C 95, 044605 (2017) Although the IAV model provides a common formalism for the calculation of the elastic and nonelastic breakup components of the inclusive breakup cross section, in our analysis we will employ this model only for the NEB part, whereas for the EBU part we will use the continuumdiscretized coupled-channels (CDCC) method, which treats breakup to all orders. The paper is organized as follows. In Sec. II we give a short overview of the IAV theory, highlighting only its main formulas. In Sec. III the extension of the formalism to negative deuteron energies (bound states) is discussed. In Sec. IV,the formalism is applied to describe the αcross sections in several 6Li-induced reactions, comparing with the available data. In Sec. Vthe role of the transfer channels on the NEB cross section is discussed. In Sec. VI we investigate the systematic behavior of the inclusive cross section with respect to the incident energy and for all analyzed targets. Finally, in Sec. VII we summarize the main results of this work. II. THE ICHIMURA, AUSTERN, AND VINCENT (IAV) MODEL In this section, we briefly summarize the model of Ichimura, Austern, and Vincent (IAV), whose original derivation can be found in Refs. [8,9] and has been also recently revisited by several authors [2,3,5,6]. We outline here the main results of this model and refer the reader to these references for further details on their derivations. We write the process under study in the form a(=b+x)+A→b+B∗,(1) where the projectile a, composed of band x, collides with a target A, emitting bfragments and any other fragments. Thus, B∗denotes any final state of the x+Asystem. This process will be described with the effective Hamiltonian H=K+Vbx +UbA(rb)+HA(ξ)+VxA(ξ,rx),(2) where Kis the total kinetic energy operator, Vbx is the interaction binding the two clusters band xin the initial composite nucleus 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. FIG. 1. Coordinates used in the nonelastic breakup calculations. 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 observed in the experiment, is described with a (complex) optical potential. Nonelastic processes arising from this interaction (e.g., target excitation) are included only effectively through UbA. The particle bis said to act as spectator.On the other hand, the interaction of the particle xwith the target retains the dependence of the target degrees of freedom (ξ). Starting from Hamiltonian (2) IAV derived the following expression for the double differential cross section for the NEB with respect to the angle and energy of the bfragments: d2σ dEbdbNEB =− 2 ¯hva ρb(Eb)ψ(0) x( kb,rx)Wxψ(0) x( kb,rx), (3) where vais the projectile-target relative velocity, ρb(Eb)= kbμb/((2π)3¯h2) is the density of states for the particle b,Wx is the imaginary part of the optical potential describing x+A elastic scattering, and ψ(0) x( kb,rx)istheso-calledx-channel wave function, which governs the evolution of xafter the projectile dissociation, when bscatters with momentum  kband the target remains in the ground state. This function satisfies the following inhomogeneous differential equation (Ex−Kx−UxA)ψ(0) x( kb,rx)=(χ(−) b( kb,rb)|Vpost|3b,(4) where Ex=E−Eb,χ(−) bis the distorted-wave describing the scattering of bin the final channel with respect to the x+Asubsystem, and Vpost ≡Vbx +UbA −Ub(with Ubthe optical potential in the final channel) is the post form transition operator. The notation (|| indicates integration over the rb coordinate only. This equation is to be solved with outgoing boundary conditions. Austern et al. [9] suggest approximating the three-body wave function appearing in the source term of Eq. (4), 3b, by the CDCC one. Since the CDCC wave function is also a complicated object by itself, a simpler choice is to use the distorted-wave Born approximation (DWBA), i.e., ψ3b x≈ χ(+) a(ra)φa(rbx), where χ(+) ais a distorted wave describing a+Aelastic scattering and φais the projectile ground-state wave function. The IAV model has been recently revisited by several groups [2,5,6]. All the calculations performed so far by these groups make use of the DWBA approximation for the incoming wave function. In Refs. [5,6], the theory was applied to deuteron-induced reactions of the form A(d,pX), and in Ref. [2] the model was extended to 6Li projectiles, presenting a first application to the 209Bi(6Li,αX) reaction. In general, the agreement with the data has been found to be very encouraging, although further comparisons with experimental data are advisable to better assess the validity and understand the limitations of the model. III. EXTENSION OF IAV MODEL TO Ex<0 The sort of breakup cross section considered by IAV can be regarded as transfer to continuum process populating x+A states with positive relative energy (Ex>0). In general, the inclusive cross section will contain also contributions 044605-2 COMPREHENSIVE ANALYSIS OF LARGE αYIELDS . . . PHYSICAL REVIEW C 95, 044605 (2017) coming from the population of states below the breakup x+A threshold (Ex<0). For example, in a (6Li, αX) reaction, the α’s emitted at the higher energies will actually correspond to deuteron transfer to bound states of the target nucleus. One would like to have a common framework to describe transfer to continuum states as well as to bound states. The explicit inclusion of all possible final bound states is impractical because of their large number and the uncertainties in their spin-parity assignments and spectroscopic factors. An alternative procedure was proposed by Udagawa and coworkers [10]. The key idea is to extend the complex potential to negative energies. Then, the bound states of the system are simulated by the eigenstates in this complex potential. The imaginary part will be associated with the spreading width of the single-particle states, which accounts for the fragmentation of these states into more complicated configurations due to the residual interactions. The method has been recently reexamined by Potel et al. [11], who have provided an efficient implementation of this idea. Here, we closely follow their formulation. For that, we first rewrite Eq. (4) in integral form ψ(0) x( kb,rx)=∞ 0 Gx(rx, r x)ρ( kb, r x)d3r x,(5) where ρ( kb, r x)=(χ(−) b( kb,rb)|Vpost|3bis the source term of the inhomogeneous Eq. (4) and Gx(rx, r x) is the Green’s function Gx(rx, r x)=1 rxr x lxmx glx(rx,r x)Ymx∗ lx(ˆ r x)Ymx lx(ˆ rx),(6) where glx(rx,r x) satisfies the equation (Ex−Kx−UxA)glx(rx,r x)=δ(rx−r x).(7) As usual, the solution of this equation is obtained from the regular [flx(rx)] and irregular [h(+) lx(rx)] solutions of the corresponding homogeneous equation. From these two solutions, glx(rx,r x) can be expressed as glx(rx,r x)=Nlxflx(r<)h(+) lx(r>),(8) where r<is the lesser value of rxand r xand r>is the larger one. The normalization constant Nlxcan be found by integrating Eq. (7) over an infinitesimal interval around r x 2μx ¯h2=r x+δ r x−δ drx d2 dr2 x glx(rx,r x) =d drx glx(rx,r x) r x+δ r x−δ =Nlxflx(r x)d drx h(+) lx(r x+δ) −h(+) lx(r x)d drx flx(r x−δ) δ→0 −−→ NlxWflx(r x),h(+) lx(r x),(9) where Wdenotes a Wronskian, which is independent of the value of r x. It is worth noting that the integral form of the x-channel wave function (5) can be also be used for positive x−A energies. Proceeding in this way, the application of the IAV formalism to positive and negative energies is formally analogous. Despite this formal similitude, the interpretation of the channel function and of the underlying imaginary part of the potential is somewhat different in both regions. For Ex>0 the channel function ψ(0) xdescribes x−Aelastic scattering and the imaginary part is therefore associated with the flux leaving this channel in favor of nonelastic channels. For Ex<0, the channel wave function describes the motion of the xparticle in a bound single-particle configuration state of the residual nucleus, and the imaginary part is connected with the spreading width of this configuration, which accounts for the fragmentation of these states into more complicated configurations. The connection between both regimes becomes more transparent within a dispersive formulation of the optical potential, as suggested long ago by Mahaux and Sartor [12,13] and recently reexamined by several groups (see, e.g., Ref. [14]). IV. COMPARISON WITH EXPERIMENTAL DATA In this section, we compare the formalism with existing 6Li inclusive breakup data on different targets. The 6Li nucleus is treated in a two-cluster model (α+d), with αand d playing the roles of spectator and participant in the IAV model, respectively. The elastic breakup (EBU) contribution of the inclusive breakup cross section is evaluated with the CDCC method [9], using the coupled-channels code FRESCO [15]. In this method, the breakup is treated as an inelastic excitation to the continuum states of the projectile. Although four-body CDCC calculations for 6Li scattering have become recently available [16], we rely here on the more conventional α+d di-cluster model. Thus, diagonal and off-diagonal coupling potentials are generated from the d+target and α+target interactions, evaluated at 1/3 and 2/3 of the projectile incident energy, respectively. In order to reproduce correctly the elastic scattering data, CDCC calculations based on this two-body model typically require some renormalization of the fragmenttarget potentials [16,17]. This has been recently found to be a consequence of the shortcomings of the two-body description of the 6Li nucleus, which results in an effective suppression of the deuteron-target absorption [16]. In our previous work [2], we found that this effect could be well simulated by removing the surface part of the deuteron-target optical potential. In the calculations presented in this work, we also allow for such kind of modification, in order to reproduce correctly the elastic scattering data. For the α+dpotential, we use the potential model from Ref. [18], which contains both central and spin-orbit terms, with the latter required to place correctly the =2 resonances. For the nonelastic breakup calculations, we rely also on a α+dmodel, but the spin of the deuteron is ignored, since our current implementation of the IAV model ignores the intrinsic spin of the fragments. This approximation was also used in our previous works [2–4]. 044605-3 JIN LEI AND ANTONIO M. MORO PHYSICAL REVIEW C 95, 044605 (2017) 0 50 100 150 0 20 40 60 dσ/dΩ (mb/sr) 29 MeV C.Signorini G.R.Kelly NEB EBU TBU 0 50 100 150 0 20 40 60 33 MeV 0 50 100 150 θlab(deg) 0 20 40 60 80 100 dσ/dΩ (mb/sr) 0 50 100 150 θlab(deg) 0 50 100 150 35 MeV 39 MeV FIG. 2. Angular distribution of αparticles produced in the reaction 6Li +208Pb at the incident energies indicated by the labels. The dotted, dashed, and solid lines correspond to the NEB (IAV model), EBU (CDCC), and their sum (TBU), respectively. Experimental data are from Refs. [19,20]. See text for details. A. 208Pb (6Li, αX) First, the results for the reaction 208Pb(6Li,αX), at several energies between 29 and 39 MeV are presented, compared with the data from Refs. [19,20]. The nominal Coulomb barrier for this system is around 29.5 MeV [19]. The CDCC calculations use the same structure model and bin discretization as in our previous calculations for 6Li +209Bi [2]. The d−208Pb and α−208Pb optical potentials are taken from Refs. [21,22], respectively. To improve the reproduction of the elastic data, the surface term of the imaginary part of the d+208Pb potential was removed. For the NEB calculations, the optical potential of 6Li +208Pb is taken from Ref. [23]. Figure 2shows the comparison of the calculated and experimental angular distributions of αparticles produced in this reaction at the measured incident energies. The squares and circles are the experimental data from Refs. [19,20], respectively. It is evident that there is an appreciable difference between the two sets of data. The dashed and dotted lines are the EBU (CDCC) and NEB (IAV model) results. As in the 6Li +209Bi case [2], the NEB is found to account for most of the inclusive breakup cross section. The sum EBU+NEB (TBU) reproduces reasonably well the magnitude and shape of the data of Ref. [19], except for some overestimation for the lowest energies. Thus, our calculations clearly favor the data presented in Ref. [19] over those presented in Ref. [20]. From the results shown here and in Ref. [2], it can be concluded that the nonelastic breakup process is the dominant α-emitting channel in 6Li induced reactions on heavy targets. To investigate whether this conclusion is a general feature of 6Li reactions or it holds only for heavy targets, we extend our analysis to lighter targets. 050 100 150 0.3 0.6 0.9 1.2 dσ/dσR Figueira et al. OM CDCC 050 100 150 θ c.m. (deg) 0 0.3 0.6 0.9 1.2 dσ/dσR 22.1 MeV 35.1 MeV FIG. 3. Elastic scattering of 6Li +144Sm at 22.1 (top) and 35.1 MeV (bottom). The solid and dashed lines are, respectively, the CDCC calculation and the optical model calculation with the optical potential from Ref. [23]. Experimental data are from Ref. [25]. B. 159Tb (6Li, αX) This reaction has been measured by Pradhan et al. [24]at several energies between 23 and 35 MeV. In Ref. [24], the following processes were invoked to explain the observed αyields: (i) breakup of 6Li into αand dwhere both fragments escape without being captured by the target, referred to in some works as noncapture breakup; (ii) αparticles resulting from dcapture by the target (deuteron incomplete fusion), following the breakup of 6Li into αand d or a deuteron transfer to the target; (iii) single-proton stripping from 6Li to produce the unbound 5He nucleus that decays into an αparticle and a neutron; (iv) single-neutron stripping from 6Li to produce 5Li, which will subsequently decay into an α+p; and (v) single-neutron pickup from 6Li to produce 7Li, which breaks into an αparticle and a triton if 7Li is excited above its breakup threshold of 2.468 MeV. In Ref. [24], these processes were treated separately using several reaction formalisms and their sum reasonably reproduced the total α-particle cross sections, but not their angular distributions. Within the inclusive breakup model adopted here, the processes discussed by Pradhan et al. [24] can be redefined as follows: Process (i) can be divided into two parts. First, the noncapture breakup with the target remaining in its ground state, i.e., EBU. Second, the noncapture breakup accompanied by target excitation, which we call inelastic breakup and is part of our nonelastic breakup cross section; processes (ii)–(iv) may be also embedded in the NEB part, in which the deuteron is absorbed by the target or it breaks up into p+nfollowing the breakup of 6Li into αand d; it can also happen that after the breakup of 6Li, the deuteron picks a neutron to become a tritium, contributing to the process (v). Processes 044605-4 COMPREHENSIVE ANALYSIS OF LARGE αYIELDS . . . PHYSICAL REVIEW C 95, 044605 (2017) (ii)–(v) as well as the inelastic breakup can be considered as nonelastic breakup and should be therefore accounted by the IAV formalism. Elastic data for this reaction are not available. Thus, the CDCC calculation is tested against the data for the nearby system 6Li +144Sm [25]. The α+144Sm and d+144Sm optical potentials were taken from Refs. [26] and [21], respectively. The results are shown in Fig. 3. The optical model calculation using the potential of Cook [23] (dashed lines) is also shown. It can be seen that the CDCC result is similar to the optical model calculation, particularly at E=35.1 MeV. At this energy, the calculations reproduce very well the elastic data. For the lower energy (E=22.1MeV), both calculations underestimate the data at backward angles. Note that, in contrast to the 6Li +208Pb case, no apparent modification of the deuteron potential was required in this case. Now the inclusive breakup cross sections 159Tb(6Li,αX) are discussed. The EBU contribution was obtained from the CDCC calculations discussed in the previous paragraph. For the NEB calculation, the same optical potentials α/d +159Tb were used. The Cook potential [23] was used to calculate the distorted wave of the incoming channel. In Fig. 4the calculated and experimental angular distributions of αparticles are compared for several incident energies of 6Li. The dashed and dotted lines are the EBU (CDCC) and NEB (IAV model) results. The summed EBU +NEB cross sections (solid lines) reproduce fairly well the shape 0 50 100 150 0 10 20 30 dσ/dΩ (mb/sr) EBU (CDCC) NEB (FR-DWBA) TBU 0 50 100 150 0 10 20 30 40 0 50 100 150 0 20 40 60 dσ/dΩ (mb/sr) 0 50 100 150 θlab (deg) 0 50 100 0 50 100 150 θlab (deg) 0 50 100 150 dσ/dΩ (mb/sr) 23 MeV 25 MeV 27 MeV 30 MeV 35 MeV 159Tb(6Li,αX) FIG. 4. Angular distribution of αparticle production of the reaction 6Li +159Tb at the incident energies indicated by the labels. The dashed, dotted, and solid lines are EBU calculated with CDCC, NEB calculated with finite-range DWBA, and their sum (TBU), respectively. The experimental data are taken from Ref. [24]. 0 50 100 150 0.6 0.8 1 dσ/dσR CDCC OM 0 50 100 150 0.4 0.6 0.8 1 0 50 100 150 0 0.5 1 dσ/dσR 0 50 100 150 0 0.5 1 0 50 100 150 θlab (deg) 0 0.5 1 dσ/dσR 0 50 100 150 θlab (deg) 0 0.5 1 18 MeV 19 MeV 20 MeV 21 MeV 22.5 MeV 24 MeV FIG. 5. Elastic scattering of 6Li +118Sn 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. [27]. Experimental data are from Ref. [27]. and magnitude of the data, except for a slight overestimation at some energies. Similarly to the heavy-target systems, i.e., 6Li +209Bi [2] and 6Li +208Pb (Sec. IV A), the NEB is found to account for most of the inclusive breakup cross section. C. 118Sn (6Li, αX) Inclusive breakup data for the 118Sn(6Li, αX) reaction are available in Ref. [27] at energies between 18 and 24 MeV. The optical model parametrizations of Refs. [26] and [21] are used for the α-118Sn and d-118Sn systems. For the NEB calculations, the optical potential of 6Li +118Sn is taken from Ref. [27]. In Fig. 5we compare the elastic data with the CDCC (solid lines) and optical model (dashed lines) calculations. Overall, both types of calculations reproduce the data well, with small discrepancies observed at some of the energies. Figure 6shows the comparison of the calculated and experimental angular distributions of αparticles produced in this reaction, for several incident energies. Again, the NEB part (dotted lines) accounts for most of the inclusive breakup cross section and the EBU (dashed lines) becomes the dominant breakup mode for angles smaller than ∼50 deg. The summed EBU +NEB result (solid line) reproduces remarkably well the shape and magnitude of the data. 044605-5 JIN LEI AND ANTONIO M. MORO PHYSICAL REVIEW C 95, 044605 (2017) 0 50 100 150 0 4 8 12 16 dσ/dΩ (mb/sr) EBU NEB TBU 0 50 100 150 0 10 20 0 50 100 150 0 10 20 30 dσ/dΩ (mb/sr) 0 50 100 150 0 10 20 30 0 50 100 150 θ lab (deg) 0 15 30 45 dσ/dΩ (mb/sr) 0 50 100 150 θ lab (deg) 0 20 40 60 18 MeV 19 MeV 20 MeV 21 MeV 22.5 MeV 24 MeV FIG. 6. Angular distribution of αparticles produced in the reaction 6Li +118Sn at the incident energies indicated by the labels. The dotted, dashed, and solid lines correspond to the NEB (IAV model), EBU (CDCC), and their sum (TBU), respectively. Experimental data are from Ref. [27]. D. 59Co (6Li, αX) Experimental data for the α-production channel for the reaction 6Li +59Co have been reported by Souza et al. [29] at Elab =21.5 MeV, which is above the Coulomb barrier (VB=12 MeV). Elastic data are available at the somewhat smaller energy Elab =18 MeV [28] so we first compare these data with the optical model and CDCC calculations. For the former, we employed the global optical potential of Cook [23]. For the CDCC calculations, the optical potentials for α+59Co and d+59Co were taken from Refs. [26] and [21], respectively. The results are shown in Fig. 7. It can be seen that both the CDCC and optical model calculations reproduce the data fairly well. We notice that no renormalization of the deuteron potential was required in this case. The experimental and calculated angular distributions of inclusive αparticles are shown in Fig. 8. The NEB is seen to dominate the inclusive αproduction. It should be noticed that, in this case, the NEB part includes also the transfer populating bound states of the target, which was obtained using the formalism discussed in Sec. III. A more detailed discussion of this contribution is left for Sec. V. The total cross section, TBU =EBU +NEB, reproduces well the 0 306090 120 150 θc.m. (deg) 0.5 1 dσ/dσR Souza et al. CDCC OM 6Li+59Co @ 18 MeV FIG. 7. Elastic scattering of 6Li +59Co at an incident energy of 18 MeV. The solid and dashed lines are, respectively, the CDCC calculation and the optical model calculation with the optical potential fromRef.[23]. Experimental data are from Ref. [28]. shape of the experimental data, although the magnitude is underestimated by ∼30% at the maximum. This might indicate the presence of other relevant mechanisms leading to the production of αparticles in this reaction, such as the formation of a compound nucleus followed by αevaporation. In fact, statistical model calculations performed in Ref. [29] predicted a significant amount of αparticles coming from this channel. The evaluation of this contribution is, however, beyond the scope of the present work. The energy spectra for selected αscattering angles are also available for this reaction. These are compared with our calculations in Fig. 9, with each panel corresponding to a given αscattering angle, as indicated by the labels. Except at θlab = 15◦, the sum of EBU and NEB reproduces the peak of the αenergy distribution. However, the low-energy tail is clearly underestimated. At these energies, the main contribution of the inclusive αproduction may arise from compound nucleus 0 306090 120 150 θlab (deg) 0 50 100 150 dσ/dΩ (mb/sr) Souza et al. EBU NEB TBU 59Co(6Li,αX) @ 21.5 MeV FIG. 8. Angular distribution of αparticles produced in the reaction 6Li +59Co at an incident energy of 21.5 MeV. The dashed, dotted, and solid lines are, respectively, the EBU (CDCC), NEB(IAV model), and their sum. Experimental data are taken from Ref. [29]. 044605-6 COMPREHENSIVE ANALYSIS OF LARGE αYIELDS . . . PHYSICAL REVIEW C 95, 044605 (2017) 0 102030 0 10 20 d2 σ/dEdΩ (mb/(sr MeV)) EBU NEB TBU 0 102030 0 10 20 0 102030 0 15 30 d2 σ/dEdΩ (mb/(sr MeV)) 0 102030 0 15 30 0 102030 0 10 20 d2 σ/dEdΩ (mb/(sr MeV)) 0 102030 Eα lab (MeV) 0 5 10 0 102030 Eα lab (MeV) 0 2 4 6 d2 σ/dEdΩ (mb/(sr/MeV)) θα=15o 59Co(6Li,αX)@21.5MeV θα=35oθα=45o θα=55oθα=65o θα=75o θα=25o FIG. 9. Experimental and calculated inclusive αenergy spectra for Elab =21.5 MeV, at selected scattering angles. The dashed, dotted, and solid lines are respectively the EBU (CDCC), NEB (IAV model), and their sum. Experimental data are taken from Ref. [29]. followed by evaporation and pre-equilibrium, which are not considered in the present calculations. We note that highenergy αparticles stem from a deuteron transfer mechanism to the target and are well reproduced by our calculations. E. 58Ni (6Li, αX) The αproduction of the 6Li +58Ni reaction at several incident energies between 12 and 20 MeV was measured by Pfeiffer et al. [27]. Elastic scattering data, which were also measured, are compared with CDCC and OM calculations in Fig. 10 (note that the angles and cross sections are referred to the laboratory frame, as in the original reference). For the former, we use the same optical potentials as in the nearby 6Li +59Co case. For the OM calculations we use the global OM potential by Cook [23]. Both calculations reproduce rather well the data, although the CDCC calculations slightly underestimates the data at large angles. We present now the inclusive αcross sections. For the NEB calculation, the 6Li optical potential from Ref. [23]was used. Figure 11 shows the comparison of the calculated and experimental angular distributions of αparticles produced in this reaction, for several incident energies. Notice that the NEB (dotted lines) includes also the contribution coming from the 0 50 100 150 0.5 1 dσ/dσR Pfeiffer et al. CDCC OM 0 50 100 150 0 0.5 1 0 50 100 150 0 0.5 1 dσ/dσR 0 50 100 150 θlab (deg) 0 0.5 1 0 50 100 150 θlab (deg) 0 0.5 1 dσ/dσR 12 MeV 14 MeV 16 MeV 18 MeV 20 MeV 6Li+58Ni FIG. 10. Elastic scattering of 6Li +58Ni at several energies indicated by the labels. The solid and dashed lines are, respectively, the CDCC calculation and the optical model calculation with the optical potential from Ref. [27]. Experimental data are from Ref. [27]. transfer to target bound states. Again, the NEB part dominates the inclusive αproduction. In general, the summed EBU + NEB cross section (solid lines) reproduces well the shape and magnitude of the data. At 16, 18, and 20 MeV, some underestimation is observed, which might be associated with other α-production channels, as pointed out in the 6Li +59Co case. From the results presented in the previous sections, we may conclude that the strong α-production channel observed in 6Li experiments originates mostly from nonelastic breakup mechanisms. In all cases analyzed, the EBU mode turns out to account for a relatively small fraction of the total inclusive αcross section and its contribution is only important for the αparticles emitted at small angles. For the lighter targets, we found also a indirect evidence of other αproduction mechanisms, such as fusion. V. TRANSFER CONTENT OF THE NEB CROSS SECTION The relative importance of the transfer to bound states within the NEB cross section will depend on several parameters, such as the projectile incident energy and the charge of the target nucleus. For heavy targets, the transfer channel is 044605-7 JIN LEI AND ANTONIO M. MORO PHYSICAL REVIEW C 95, 044605 (2017) 0 50 100 150 0 5 10 15 dσ/dΩ (mb/sr) EBU NEB TBU 0 50 100 150 0 10 20 30 0 50 100 150 0 20 40 60 dσ/dΩ (mb/sr) 0 50 100 150 θlab (deg) 0 20 40 60 80 0 50 100 150 θlab (deg) 0 50 100 150 dσ/dΩ (mb/sr) 12 MeV 14 MeV 16 MeV 18 MeV 20 MeV 58Ni(6Li,αX) FIG. 11. Angular distribution of αparticles produced in the reaction 6Li +58Ni at the incident energies indicated by the labels. The dashed, dotted, and solid lines are, respectively, the EBU, NEB, and their sum (TBU). Experimental data are from Ref. [27]. suppressed due to the strong Coulomb interaction between the deuteron and the target, whereas for light targets this channel is expected to play a more important role. This is illustrated in Fig. 12 for two such cases; the upper panel displays the calculated 208Pb(6Li, αX) NEB cross sections as a function of d-208Pb relative energy at three different incident energies, 29, 35, and 39 MeV. The vertical dotted line indicates the nominal Coulomb barrier for the d-208Pb system. The black solid curve is the reaction cross section for the d-208Pb system, arbitrarily normalized to fit within the same scale. The bottom panel shows similar curves for the 6Li +58Ni reaction at 12, 16, and 20 MeV. In both cases, it can be seen that the NEB is a Trojan horse type process [30], which means that the 6Li projectile brings the deuteron inside the Coulomb barrier and lets it interact with the target nucleus, giving sizable cross sections for deuteron energies for which the reaction cross section has already become negligibly small. For the 208Pb target, due to the strong Coulomb repulsion, the NEB cross section becomes negligible at negative d-208Pb relative energies and this behavior is independent of the incoming 6Li energy. In contrast, for the 58Ni target, there FIG. 12. Top: NEB cross section as a function of the d-208Pb relative energy in the c.m. frame for the reaction 6Li +208Pb. The vertical dotted line indicates the energy of the Coulomb barrier for the d+208Pb reaction. The solid line is the reaction cross section for d+208Pb, arbitrarily normalized. Bottom: same as in top panel but for the 6Li +58Ni system. is a low-energy tail extending to negative deuteron energies (transfer). We expect also some correlation between the α-particle angular and energy distribution. This is shown in Fig. 13 in the form of contour plots of double differential cross sections and FIG. 13. Contour plots for the double differential cross section (upper panels) and the angle-integrated energy differential cross section as a function of the outgoing αenergy in the c.m. frame (lower panels) for the reactions (a) 6Li +208Pb, (b) 6Li +159Tb, (c) 6Li +118Sn, and (d) 6Li +59Co. The vertical lines indicate the breakup threshold for the d+target system (Ex=0). 044605-8 COMPREHENSIVE ANALYSIS OF LARGE αYIELDS . . . PHYSICAL REVIEW C 95, 044605 (2017) 0.8 1 1.2 1.4 1.6 Ec.m./VB 10 100 1000 σα TBU 6Li+58Ni 6Li+59Co 6Li+118Sn 6Li+159Tb 6Li+208Pb 6Li+209Bi FIG. 14. Inclusive breakup αcross sections involving 6Li projectile with several targets as a function of Ec.m./VB. angle-integrated cross section as a function of the outgoing αenergy in the c.m. frame for the reactions (a) 6Li +208Pb, (b) 6Li +159Tb, (c) 6Li +118Sn, and (d) 6Li +59Co. It can seen that the most energetic αparticles are preferably emitted at forward angles, whereas those with lower energies contribute to both forward and backward angles. Moreover, when the charge of the target is small (59Co), the transfer channel becomes more relevant. VI. SYSTEMATICS OF INCLUSIVE αPRODUCTION Systematic studies of αproduction yields in 6Li reactions show an interesting universal behavior when plotted as a function of the incident energy scaled by the Coulomb barrier energy as reported for instance by Pakou et al. [31]. In this section, we will investigate whether our calculations exhibit also this universal behavior. For this study, we have considered the target systems 59Co, 118Sn, 159Tb, and 208Pb, which have been analyzed in the preceding sections, and 209Bi, analyzed in Ref. [2]. The results are shown in Fig. 14, where we plot the calculated σTBU αcross sections as a function of the reduced energy (Ec.m./VB), with VBthe energy of the Coulomb barrier, estimated as VB=ZpZte2/[rB(A1/3 p+A1/3 t)], where Zp(Zt) and Ap(At) are the atomic number and atomic mass of the projectile (target), respectively, and rB=1.44 fm. As expected, the breakup cross section drops quickly as the incident energy decreases below the barrier. This effect is enhanced for the 209Bi nucleus, possibly due to the larger Coulomb repulsion. Above the barrier, the inclusive breakup cross sections show a similar trend for the medium-heavy and heavy targets, but not for the medium mass targets 58Ni and 59Co at larger energies. We recall, however, that for these lighter systems, there might be additional contributions from other channels, such compound nucleus followed by evaporation, which are not accounted for by the IAV formalism. We have also studied the relative importance of EBU versus NEB as a function of the incident energy. For that, we display in Fig. 15 the ratio of EBU over TBU (=EBU +NEB) for the 0.8 1 1.2 1.4 1.6 Ec.m./VB 0 0.1 0.2 0.3 0.4 σEBU / σTBU 6Li+58Ni 6Li+59Co 6Li+118Sn 6Li+159Tb 6Li+208Pb 6Li+209Bi FIG. 15. Ratios of calculated EBU over TBU (=EBU +NEB) for different systems. See text for details. analyzed systems. It is seen that, for incident energies below the Coulomb barrier, the elastic breakup cross section becomes comparatively more important as the energy decreases. This can be attributed to the fact that, below the barrier, the breakup takes place at large projectile-target separations, and the deuteron absorption (responsible for the NEB part) will be less important [4]. By contrast, for energies above the Coulomb barrier, the ratio shows an almost constant behavior. It can also be seen that while for the heavy mass targets elastic breakup plays an important role in the inclusive αproduction, especially below the Coulomb barrier, for the medium mass targets elastic breakup is less important and the nonelastic breakup is dominant. Another relevant question regards the fraction of the reaction cross section that is exhausted by the αcross section. To address this question, we plot in Fig. 16 the ratio of the calculated TBU and reaction cross sections as a function 0.8 1 1.2 1.4 1.6 Ec.m./VB 0 0.2 0.4 0.6 0.8 1 σα TBU/σR 6Li+58Ni 6Li+59Co 6Li+118Sn 6Li+159Tb 6Li+208Pb 6Li+209Bi FIG. 16. Ratios of calculated TBU (=EBU +NEB) αcross sections over the reaction cross section for the systems and energies analyzed in this work. See the text for details. 044605-9