Comprehensive analysis of large \alpha yields observed in {}^{6}Li induced reactions

Comprehensive analysis of large yields observed in Li induced reactions

Jin Lei jinlei@us.es Present address: Institute of Nuclear and Particle Physics, and Department of Physics and Astronomy, Ohio University, Athens, Ohio 45701, USA Departamento de FAMN, Universidad de Sevilla, Apartado 1065, 41080 Sevilla, Spain.    Antonio M. Moro moro@us.es Departamento de FAMN, Universidad de Sevilla, Apartado 1065, 41080 Sevilla, Spain.
July 14, 2019
Abstract
Background

Large yields have been reported over the years in reactions with Li and Li 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 inclusive breakup cross sections, pointing toward the dominance of non-elastic breakup (NEB) mechanisms.

Purpose

We aim to provide a systematic study of the inclusive cross sections observed in nuclear reactions induced by Li projectiles. In addition to estimating the total singles cross sections, it is our goal to evaluate angular and energy distributions of these particles and compare 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 the model proposed in the eighties by Ichimura, Austern and Vincent [Phys. Rev. C32, 432 (1982)], within the DWBA approximation.

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 Li projectiles. It provides also a natural explanation of the previously observed underestimation of the measured yields by CDCC calculations. The method used here can be extended to other weakly-bound projectiles, including halo nuclei.

pacs:
25.60.Dz,25.60.Gc,25.60.Bx,21.10.Gv,27.20.+n.

I Introduction

Reactions induced by the Li nucleus have been extensively studied giving rise to a large body of experimental data at present. Given its marked structure, with a separation energy of MeV (to be compared with the single nucleon separation energy of 5.39 MeV), one may anticipate that the breakup of this nucleus into and is a major reaction channel. In fact, experimental data show remarkably large yields of particles but, contrary to what naively expected, these yields are typically much larger than the corresponding yields. This suggests that the breakup of the Li is not a simple direct breakup mechanism.

From the theoretical point of view, a proper interpretation of these yields is still lacking. Continuum-discretized coupled-channels (CDCC) calculations, which treat the Li breakup as an inelastic excitation to the continuum, reproduce successfully the coincidence measurements Signorini et al. (2003) but they largely underestimate the inclusive cross 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 Li projectile impinging on a target , this corresponds to the processes of the form in which the two-projectile clusters survive after the collision and the target remains in the ground state.111If a three-body description of the Li is used, +p+n, the three-body breakup mode would be also part of the elastic breakup channel. Since we resort here to a two-body model of Li we include this channel in the NEB part.Thus, the underestimation of the inclusive yields 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 and , the projectile dissociation accompanied by target excitation, and the fusion of by , among others, that we will globally denote as non-elastic breakup (NEB) channels. An explicit account of these process 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 particles), 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 Lei and Moro (2015a, b); Moro and Lei (2016); Carlson et al. (2016); Potel et al. (2015a). In these models, the sum over all the possible final states through which the unobserved fragment may interact with the target is done in a formal way, making use of the Feshbach projection formalism Feshbach (1962) and closure.

In this work, we will show that inclusive singles cross sections from Li-induced reactions can be remarkably well reproduced using the inclusive breakup model proposed by Ichimura, Austern and Vincent (IAV) Ichimura et al. (1985). To our knowledge, this is the first study of this kind providing a systematic explanation of these data.

Although the IAV model provides a common formalism for the calculation of the elastic and non-elastic 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 continuum-discretized 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 cross sections in several Li-induced reactions comparing with the available data. In Sec. V the role of the transfer channels on the NEB cross section is discussed. In Sec. VI we investigate the systematic behaviour 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, Vincent (IAV) model

In this section we briefly summarize the model of Ichimura, Austern and Vincent (IAV), whose original derivation can be found in Ichimura et al. (1985); Austern et al. (1987), and has been also recently revisited by several authors Lei and Moro (2015a, b); Carlson et al. (2016); Potel et al. (2015a). 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,

(1)

where the projectile , composed of and , collides with a target , emitting fragments and any other fragments. Thus, denotes any final state of the system.

This process will be described with the effective Hamiltonian

(2)

where is the total kinetic energy operator, is the interaction binding the two clusters and in the initial composite nucleus , is the Hamiltonian of the target nucleus (with denoting its internal coordinates) and and are the fragment–target interactions. The relevant coordinates are depicted in Fig. 1.

Figure 1: (Color online) Coordinates used in the non-elastic 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 , the one that is assumed to be observed in the experiment, is described with a (complex) optical potential. Non-elastic processes arising from this interaction (e.g. target excitation), are included only effectively through . The particle is said to act as spectator. On the other hand, the interaction of the particle with the target retains the dependence of the target degrees of freedom ().

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 fragments:

(3)

where is the projectile-target relative velocity, is the density of states for the particle , is the imaginary part of the optical potential describing elastic scattering and is the so-called -channel wave function, which governs the evolution of after the projectile dissociation, when scatters with momentum and the target remains in the ground state. This function satisfies the following inhomogeneous differential equation

(4)

where , is the distorted-wave describing the scattering of in the final channel with respect to the + sub-system, and (with the optical potential in the final channel) is the post-form transition operator. The notation indicates integration over the coordinte only. This equation is to be solved with outgoing boundary conditions.

Austern et al. Austern et al. (1987) suggest approximating the three-body wave function appearing in the source term of Eq. (4), , by the CDCC one. Since the CDCC wave function is also a complicated object by itself, a simpler choice is to use the DWBA approximation, i.e., , where is a distorted wave describing elastic scattering and is the projectile ground state wave function.

The IAV model has been recently revisited by several groups Lei and Moro (2015a); Potel et al. (2015a); Carlson et al. (2016). All the calculations performed so far by these groups make use of the DWBA approximation for the incoming wave function. In Refs. Potel et al. (2015a); Carlson et al. (2016), the theory was applied to deuteron induced reactions of the form , and in Ref. Lei and Moro (2015a) the model was extended to Li projectiles, presenting a first application to the Bi(Li,) 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 limitations of the model.

Iii Extension of IAV model to

The sort of breakup cross section considered by Ichimura, Austern and Vincent can be regarded as transfer to continuum process populating states with positive relative energy (). In general, the inclusive cross section will contain also contributions coming from the population of states below the breakup threshold (). For example, in a (Li, ) reaction, the ’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 unpractical because of their large number and the uncertainties in their spin/parity assignments and spectroscopic factors. An alternative procedure was proposed by Udagawa and co-workers Udagawa et al. (1989). 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. Potel et al. (2015b), 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

(5)

where is the source term of the inhomogeneous Eq. (4) and is the Green’s function

(6)

where satisfies the equation

(7)

As usual, the solution of this equation is obtained from the regular () and irregular () solutions of the corresponding homogeneous equation. From these two solutions, can be expressed as

(8)

where is the lesser value of and and is the larger one. The normalization constant can be found by integrating Eq. (7) over an infinitesimal interval around

(9)

Where denotes a Wronskian, which is independent of the value of .

It is worth noting that the integral form of the channel wave function (5) can be also be used for positive 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 the channel function describes elastic scattering and the imaginary part is therefore associated with the flux leaving this channel in favor of non-elastic channels. For , the channel wave-function describes the motion of the particle 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 Mahoux and Sartor Mahaux and Sartor (1986, 1991) and recently reexamined by several groups (see e.g. Dickhoff et al. (2016)).

Iv Comparison with experimental data

In this section, we compare the formalism with existing Li inclusive breakup data on different targets. The Li nucleus is treated in a two-cluster model (+), with and 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 Austern et al. (1987), using the coupled-channels code FRESCO Thompson (1988). In this method, the breakup is treated as an inelastic excitation to the continuum states of the projectile. Although four-body CDCC calculations for Li scattering have become recently available Watanabe et al. (2012), we rely here on the more conventional di-cluster model. Thus, diagonal and off-diagonal coupling potentials are generated from the +target and +target interactions, evaluated at 2/3 and 1/3 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 fragment-target potentials Hirabayashi and Sakuragi (1991); Watanabe et al. (2012). This has been recently found to be a consequence of the shortcomings of the two-body description of the Li nucleus, which results in an effective suppression of the deuteron-target absorption Watanabe et al. (2012). In our previous work Lei and Moro (2015a), 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 potential, we use the potential model from Ref. Nishioka et al. (1984), which contains both central and spin-orbit terms, with the latter required to place correctly the resonances.

For the non-elastic breakup calculations, we rely also on a model, 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 Lei and Moro (2015a, b); Moro and Lei (2016).

iv.1 Pb (Li, )

Figure 2: (Color online) Angular distribution of particles produced in the reaction Li+Pb 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. Signorini et al. (2001); Kelly et al. (2000). See text for details.

First, the results for the reaction Pb(Li,), at several energies between 29 and 39 MeV are presented, comparing with the data from Refs. Signorini et al. (2001); Kelly et al. (2000). The nominal Coulomb barrier for this system is around 29.5 MeV Signorini et al. (2001). The CDCC calculations use the same structure model and bin discretization as in our previous calculations for Li+Bi Lei and Moro (2015a). The Pb and Pb optical potentials are taken from Refs. Han et al. (2006) and Barnett and Lilley (1974), respectively. To improve the reproduction of the elastic data, the surface term of the imaginary part of the Pb potential was removed. For the NEB calculations, the optical potential of Li+Pb is taken from Ref. Cook (1982).

Figure 2 shows the comparison of the calculated and experimental angular distributions of particles produced in this reaction at the measured incident energies. The squares and circles are the experimental data from Refs. Signorini et al. (2001) and Kelly et al. (2000), 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 Li+Bi case Lei and Moro (2015a), 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. Signorini et al. (2001), except for some overestimation for the lowest energies. Thus, our calculations clearly favour the data presented in Ref. Signorini et al. (2001) over those presented in Kelly et al. (2000).

From the results shown here and in Ref. Lei and Moro (2015a), it can be concluded that the nonelastic breakup process is the dominant emitting channel in the Li induced reactions on heavy targets. To investigate whether this conclusion is a general feature of Li induced reactions or it holds only for heavy targets we extend our analysis to lighter targets.

iv.2 Tb (Li, )

This reaction has been measured by Pradhan et al. Pradhan et al. (2013) at several energies between 23 MeV and 35 MeV.

In Ref. Pradhan et al. (2013), the following processes were invoked to explain the observed yields: (i) breakup of Li into and fragments where both fragments escape without being captured by the target, referred to in some works as non-capture breakup; (ii) particles resulting from capture by the target (deuteron incomplete fusion), following the breakup of Li into and or a deuteron transfer to the target; (iii) single-proton stripping from Li to produce the unbound He nucleus that decays into an particle and a neutron; (iv) single-neutron stripping from Li to produce Li, which will subsequently decay into an +p; and (v) single-neutron pickup from Li to produce Li, which breaks into an particle and a triton if Li is excited above its breakup threshold of 2.468 MeV. In Ref. Pradhan et al. (2013) these processes were treated separately, using several reaction formalisms and their sum reasonably reproduced the total particle cross sections, but not their angular distributions.

Within the inclusive breakup model adopted here, the processes discussed by Pradhan et al. Pradhan et al. (2013) can be re-defined as follows: process (i) can be divided into two parts. First, the non-capture breakup with the target remaining in its ground state, i.e., EBU. Second, the non-capture breakup accompanied by target excitation, which we call inelastic breakup and is part of our non-elastic 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 following the breakup of Li into and ; it can also happen that after the breakup of Li, the deuteron picks a neutron to become a tritium, contributing to the process (v). Processes (ii)-(v) as well as the inelastic breakup can be considered as nonelastic breakup and should be therefore accounted by the IAV formalism.

Figure 3: (Color online) Elastic scattering of Li+Sm at 22.1 MeV (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 Cook (1982). Experimental data are from Ref. Figueira et al. (2010) .

Elastic data for this reaction are not available. Thus, the CDCC calculation is tested against the data for the nearby system Li+Sm Figueira et al. (2010). The Sm and Sm optical potentials were taken from Refs. Huizenga and Igo (1962) and Han et al. (2006), respectively. The optical model calculation using the potential of Cook Cook (1982) (dashed lines) is also shown. It can be seen that the CDCC result is similar to the optical model calculation, particularly at MeV. At this energy, the calculations reproduce very well the elastic data. For the lower energy ( MeV), both calculations underestimate the data at backward angles. Note that, in contrast to the Li+Pb case, no apparent modification of the deuteron potential was required in this case.

Figure 4: (Color online) Angular distribution of particle production of the reaction Li+Tb 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. Pradhan et al. (2013)

Now the inclusive breakup cross sections Tb(Li,) are discussed. The EBU contribution was obtained from the CDCC calculations discussed in the previous paragraph. For the NEB calculation, the same optical potentials Tb were used. The Cook potential Cook (1982) was used to calculate the distorted wave of the incoming channel.

In Fig. 4 the calculated and experimental angular distributions of particles are compared for several incident energies of Li. 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 and magnitude of the data, except for a slight overestimation at some energies. Similarly to the heavy-target systems, i.e., Li+Bi Lei and Moro (2015a) and Li+Pb (Sec. IV.1), the NEB is found to account for most of the inclusive breakup cross section.

iv.3 Sn (Li, )

Figure 5: (Color online) Elastic scattering of Li+Sn at different incident energies. The solid and dashed lines are, respectively, the CDCC calculation and the optical model calculation with the optical potential from Pfeiffer et al. (1973). Experimental data are from Ref. Pfeiffer et al. (1973).

Inclusive breakup data for the Sn(Li, ) reaction are available in Ref. Pfeiffer et al. (1973) at energies between 18 and 24 MeV. The optical model parameterizations of Refs. Huizenga and Igo (1962) and Han et al. (2006) are used for the Sn and Sn systems. For the NEB calculations, the optical potential of Li+Sn is taken from Ref. Pfeiffer et al. (1973).

In Fig. 5 we compare the elastic data with the CDCC (solid lines) and optical model (dashed lines) calculations. Overall, both types of calculations reproduce well the data, with some discrepancy observed at 18 and 21 MeV.

Figure 6: (Color online) Angular distribution of particles produced in the reaction Li+Sn 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. Pfeiffer et al. (1973).

Figure 6 shows the comparison of the calculated and experimental angular distributions of particles 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 degrees. The summed EBU + NEB result (solid line) reproduces remarkably well the shape and magnitude of the data.

iv.4 Co (Li, )

Figure 7: (Color online) Elastic scattering of Li+Co at an incident energy of MeV. The solid and dashed lines are, respectively, the CDCC calculation and the optical model calculation with the optical potential from Cook (1982). Experimental data are from Ref. Souza et al. (2007).
Figure 8: (Color online) Angular distribution of particles produced in the reaction Li+Co at an incident energy of MeV. The dashed, dotted, and solid lines are, respectively, the EBU (CDCC), NEB(IAV model) and their sum. Experimental data are taken from Ref. Souza et al. (2009).
Figure 9: (Color online) Experimental and calculated inclusive energy spectra for 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. Souza et al. (2009).

Experimental data for the -production channel for the reaction Li+Co have been reported by Souza et al. Souza et al. (2009) at MeV, which is above the Coulomb barrier ( MeV).

Elastic data are available at the somewhat smaller energy MeV Souza et al. (2007) so we first compare these data with the optical model and CDCC calculations. For the former, we employed the global optical potential of Cook Cook (1982). For the CDCC calculations, the optical potentials for Co and Co were taken from Refs. Huizenga and Igo (1962) and Han et al. (2006), respectively. The results are shown in Fig. 7. It can be seen that both the CDCC and optical model calculations reproduce fairly well the data. We notice that no renormalization of the deuteron potential was required in this case.

The experimental and calculated angular distributions of inclusive particles are shown in Fig. 8. The NEB is seen to dominate the inclusive production. 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 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 particles in this reaction, such as the formation of a compound nucleus followed by evaporation. In fact, statistical model calculations performed in Ref. Souza et al. (2009) predicted a significant amount of particles coming from this channel. The evaluation of this contribution is beyond the scope of the present work.

The energy spectra for selected scattering angles are also available for this reaction. These are compared with our calculations in Fig. 9, with each panel corresponding to a given scattering angle, as indicated by the labels. Except at , the sum of EBU and NEB reproduces the peak of the energy distribution. However, the low-energy tail is clearly underestimated. At these energies, the main contribution of the inclusive production may arise from compound nucleus followed by evaporation and pre-equilibrium, which are not considered in the present calculations. We note that high energy particles stem from a deuteron transfer mechanism to the target and are well reproduced by our calculations.

iv.5 Ni (Li, )

Figure 10: (Color online) Elastic scattering of Li+Ni 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 Pfeiffer et al. (1973). Experimental data are from Ref. Pfeiffer et al. (1973).
Figure 11: (Color online) Angular distribution of particles produced in the reaction Li + Ni 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. Pfeiffer et al. (1973).

The production of the Li + Ni reaction at several incident energies between 12 MeV and 20 MeV was measured by Pfeiffer et al. Pfeiffer et al. (1973). 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 Li+Co case. For the OM calculations we use the global OM potential by Cook Cook (1982). Both calculations reproduce rather well the data, although the CDCC calculations slightly underestimates the data at large angles.

We present now the inclusive alpha cross sections. For the NEB calculation, the Li optical potential from Ref. Cook (1982) was used. Figure 11 shows the comparison of the calculated and experimental angular distributions of particles produced in this reaction, for several incident energies. Notice that the NEB (dotted lines) includes also the contribution coming from the transfer to target bound states. Again, the NEB part dominates the inclusive production. In general, the summed EBU + NEB cross section (solid lines) reproduces well the shape and magnitude of the data. At , and MeV some underestimation is observed, which might be associated with other -production channels, as pointed out in the Li+Co case.

From the results presented in the previous sections, we may conclude that the strong -production channel observed in Li experiments originates mostly from non-elastic breakup mechanisms. In all cases analyzed, the EBU mode turns out to account for a relatively small fraction of the total inclusive alpha cross section and its contribution is only important for the alpha particles emitted at small angles. For the lighter targets, we found also a indirect evidence of other alpha production mechanisms, such as fusion.

V Transfer content of the NEB cross section

Figure 12: (Color online) Top: NEB cross section as a function of the Pb relative energy in the c.m. frame for the reaction Li+Pb. The vertical dotted line indicates the energy of the Coulomb barrier for the +Pb reaction. The solid line is the reaction cross section for +Pb, arbitrarily normalized. Bottom: same as in top panel but for the Li+Ni system.
Figure 13: (Color online) Contour plots for the double differential cross section (upper panels) and the angle-integrated enery differential cross section as a function of the outgoing energy in the c.m. frame (lower panels) for the reactions: (a) Li+Pb, (b) Li+Tb, (c) Li+Sn and (d) Li+Co. The vertical lines indicate the breakup threshold for the +target system ().

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 a nd the charge of the target nucleus. For heavy targets, the transfer channel is 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 Pb(Li, X) NEB cross sections as a function of -Pb relative energy at three different incident energies, 29 MeV, 35 MeV and 39 MeV. The vertical dotted line indicates the nominal Coulomb barrier for the -Pb system. The black solid curve is the reaction cross section for the -Pb system, arbitrarily normalized to fit within the same scale. The bottom panel shows similar curves for the Li+Ni reaction at 12, 16 and 20 MeV. In both cases, it can be seen that the NEB is a Trojan Horse type process Baur (1986), which means that the Li projectile brings the deuteron inside the Coulomb barrier and let it interact with the target nucleus, giving a sizable cross sections for deuteron energies for which the reaction cross section has already become negligibly small. For the Pb target, due to the strong Coulomb repulsion, the NEB cross section becomes negligible at negative -Pb relative energies and this behavior is independent of the incoming Li energy. By contrary, for the Ni target, there is a low energy tail extending to negative deuteron energies (transfer).

We expect also some correlation between the -particle angular and energy distribution. This is shown in Fig. 13 in the form of contour plots of double differential cross sections and angle-integrated cross section as a function of the outgoing energy in the c.m. frame for the reactions (a) Li+Pb, (b) Li+Tb, (c) Li+Sn and (d) Li+Co. It can seen that the most energetic particles 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 (Co), the transfer channel becomes more relevant.

Vi Systematics of inclusive production

Figure 14: (Color online) Inclusive breakup cross sections involving Li projectile with several targets as a function of .
Figure 15: (Color online) Ratios of calculated EBU over TBU (= EBU + NEB) for different systems. See text for details.

Systematic studies of production yields in Li reactions show an interesting universal behaviour when plotted as a function of the incident energy scaled by the Coulomb barrier energy as reported for instance by Pakou et al. Pakou et al. (2003). In this section, we will investigate whether our calculations exhibit also this universal behaviour. For this study, we have considered the target systems Co, Sn, Tb, Pb, which have been analyzed in the preceding sections, and Bi, analyzed in Ref. Lei and Moro (2015a). The results are shown in Fig. 14, where we plot the calculated cross sections as a function of the reduced energy (), with the energy of the Coulomb barrier, estimated as , where () and () are the atomic number and atomic mass of the projectile (target), respectively, and fm. As expected, the breakup cross section drops quickly as the incident energy decreases below the barrier. This effect is enhanced for the Bi nucleus, possibly due to the larger Coulomb repulsion. Above the barrier, the medium-heavy and heavy targets the inclusive breakup cross sections show a similar trend, but not for the medium mass targets Ni and Co at larger energies. We recall however that, for these lighter systems, there might be additional contributions from other channels, such as compound nucleus following evaporation, which are not accounted for by the IAV formalism.

Figure 16: (Color online) Ratios of calculated TBU (= EBU + NEB) cross sections over the reaction cross section for the systems and energies analyzed in this work. See text for details.

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 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 Moro and Lei (2016). 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 production, 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 cross section. To address this question, we plot in Fig. 16 the ratio of the calculated TBU and reaction cross sections as a function of the reduced energy , for the systems studied in this work. Several interesting features emerge from this plot: (i) first, for all systems analyzed the ratio decreases smoothly as the incident energy increases; (ii) second, the heavier the target nucleus, the larger the ratio. For example, for the Pb and Bi target nuclei the ratio exceeds 80% at sub-Coulomb energies. Result (i) may be understood as a consequence of the competition with other channels which will open and increase their importance as the incident energy increases, such as other breakup modes not associated with the production of particles (e.g. H+He), target excitation not accompanied by projectile breakup, neutron pickup from the target, etc.

Vii Summary and conclusions

To summarize, we have performed a comprehensive analysis of inclusive breakup cross sections in Li-induced reactions with the aim of understanding the experimentally observed yields. For that, we have calculated separately the EBU and NEB contributions using the CDCC method (for the EBU part) and the closed-form model proposed by Ichimura, Austern and Vincent IAV theory (for the NEB part). For the latter model, we used the DWBA approximation, including finite-range effects and the remnant term of the transition operator.

Overall, the calculations show a very good agreement with the available data, providing a consistent and neat explanation of the large yields reported over the years for Li reactions, without the need of evaluating the individual channels contributing to the inclusive cross section. Furthermore, in all cases analyzed, the total breakup is largely dominated by the NEB part, with the EBU part representing only a small fraction of the total inclusive cross section. This explains why the CDCC calculations tend to largely underpredict the measured yields. The EBU becomes only dominant at very small angles, or at energies well below the Coulomb barrier. For the heavy target systems, the singles cross section accounts for a large fraction of the reaction cross section (above 80% at sub-Coulomb energies). For the lighter mass targets, we found that part of the yields corresponds to transfer to bound states of the residual nucleus. To account for this contribution, the IAV model has been conveniently extended, following the formalism developed by previous authors Udagawa et al. (1989); Potel et al. (2015b).

Finally, we have investigated whether our calculations support the observed universal trend of yields as a function of the reduced incident energy (). We find that the computed total breakup cross sections (EBU+NEB) exhibit this trend for the heavy targets, but significant deviations have been found for the light targets. This could indicate that the latter do not obey the universal behavior, but we cannot rule out that the deviations are due to the presence of additional production mechanisms, not included in our calculations. This problem deserves further investigation.

Acknowledgements.
We are grateful to Gregory Potel for his guidance in the extension of the IAV model to negative energies. This work has been partially supported by the Spanish Ministerio de Economía y Competitividad and FEDER funds under project FIS2014-53448-C2-1-P and by the European Union’s Horizon 2020 research and innovation program under grant agreement No. 654002.

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