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Three-body approaches for inclusive breakup reactions
Nuclear Physics A 787 (2007) 463c–470c
Three-body approaches for inclusive breakup reactions∗ A. M. Moro,a F. M. Nunes,b D. Escrigc and J. G´omez-Camachoa a
Universidad de Sevilla, P.O. Box 1065, E-41080, Seville, Spain
b
NSCL and Department of Physics and Astronomy, Michigan State University, U.S.A.
c
Instituto de Estructura de la Materia, CSIC, E-28006 Madrid, Spain
The formal problem of three-body breakup in nuclear reactions is addressed. Two alternative treatments, namely, the direct breakup and transfer to the continuum approaches, are discussed and compared. Calculations for the reactions 8 B+ 58 Ni at 26 MeV and 6 He + 208 Pb at 22 MeV are presented and compared with existing data. 1. Introduction An important reaction mechanism that occurs when a loosely bound projectile collides with a target is the breakup of the projectile into two or more fragments. In the case of a projectile consisting of two weakly bound fragments, scattered by a stable target, this process gives rise to three fragments. The formal treatment of these reactions involves a complicated three-body problem. In the case in which the internal structure of the constituents can be safely ignored, such as in deuteron-proton or deuteron-neutron reactions at intermediate energies this problem has been successfully solved by solving the Faddeev equations [1,2]. In the Faddeev formalism, the total wavefunction of the three-body system is expanded into three components. Each component contains the bound states as well as continuum states for a specific binary system. Therefore, each rearrangement channel is confined to a particular Faddeev component. By contrast, three-body breakup will be distributed among the three Faddeev components. Notice that, since each Jacobi representation provides a complete basis, the Faddeev wavefunction uses an overcomplete basis. The difficulties inherent to the solution of the Faddeev equations has prevented its application to realistic heavy ion reactions. In practical calculations, it is customary to select a particular representation, depending on the problem under study. For instance, in single-particle excitations populating bound or narrow resonances of the projectile, the natural representation will be that based on the eigenstates of the projectile. On the other hand, for a process in which part of the projectile is transferred to a bound or resonant state of the target, the obvious representation is that containing the fragmenttarget states. The choice is less obvious in the case of non-resonant breakup. According to the way in which the breakup process is modelled, current breakup reaction theories ∗ Work partially supported by National Science Foundation (USA) under grant PHY-0456656, and by the Junta de Andaluca (Spain).
0375-9474/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2006.12.069
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can be divided into two main categories. On one hand, some approaches, such as the conventional Continuum-Discretized Coupled Channels method [3], treat this process as an inelastic excitation of the projectile to the continuum, that we will refer hereafter as the direct breakup picture (BU). Bound as well as unbound states are represented by the eigenstates of the two-body Hamiltonian of the projectile. The Schr¨odinger equation is then projected onto these states, giving rise to a system of coupled equations. To make the number of equations finite, the two-body continuum is truncated in two ways: by retaining a given number of partial waves between the two fragments, and by discretizing the continuum into a set of energy intervals, or bins, up to a maximum excitation energy. In practical calculations, the number of partial waves, the maximum excitation energy, and the bin widths are chosen in order to obtain convergence of the observables considered. On the other hand, the breakup process can be also considered as a transfer of a piece of the projectile to the target continuum. This procedure has been referred to in the literature as transfer to the continuum (TC) method. Some authors have applied this method within a semiclassical picture [4]. Also, it has been applied as a generalisation of the Distorted Wave Born Approximation (DWBA) [5]. A natural question that arises from the considerations above is to what extent the BU and TC representations provide similar results for the breakup observables. Furthermore, even if this is the case, it is also important to discern which representation will be more appropriate in a particular reaction. It is the objective of this contribution to shed some light into these two questions. For this purpose, we have performed calculations for several systems using the BU and TC approaches. In this work, the results for the 8 B+58 Ni and 6 He +208 Pb reactions will be presented and discussed.
58
Ni + p
7
Be + p
8
(a)
8
58 B + Ni
(b)
58 B + Ni
7
58
Be + Ni
Figure 1. Schematic representation of the couplings included in the BU (a) and TC calculations (b) in the 8 B+ 58 Ni reaction.
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2. Calculations 2.1. The 8 B+58 Ni reaction In this section we consider the breakup reaction 8 B+58 Ni → 7 Be + p + 58 Ni at Elab = 25.8 MeV, which is close to the Coulomb barrier. This reaction was measured at the Notre Dame facility. Details of this experiment can be found in [6]. A detailed analysis of this reaction within the BU and TC approaches was already presented in Ref. [7]. Here, we briefly summarise the main results. To compare the BU and TC methods, we considered the angular and energy distribution of the scattered 7 Be fragments. The BU calculation (see Fig. 1a) was performed according to the standard CDCC approach, following closely the work of Ref. [8]. For each partial wave, the 8 B two-body continuum (7 Be+p) was discretized into energy bins, up to a maximum excitation energy of max = 8 MeV. To achieve convergence, we had to include s, p, d, f and g waves for the 7 Be+p relative motion. Bound state–continuum as well as continuum–continuum couplings were included. Within the TC, the breakup is treated as a proton transfer to unbound states of the p+58 Ni system, as illustrated in Fig. 1b. As in the CDCC calculation, these unbound states were represented by energy bins. The p+58 Ni continuum was truncated up to a maximum excitation energy of 10 MeV, above the breakup threshold. In this case, the convergence rate for the forward scattering angles was slower, forcing us to include up to 17 waves to achieve a reasonable convergence of this observable. Further details of these calculations, including the choice of the binary interactions between the fragments, can be found in Refs. [7] and [8]. In Fig. 2 we present the TC and BU calculations for the observables associated with the 7 Be fragments. The left panel of this figure shows the differential cross section of the scattered 7 Be products as a function of the scattering angle, in the laboratory frame. For each angle, this distribution is integrated with respect to the energy of 7 Be. The circles correspond to the data of Guimar˜aes et al. [6], the thick solid line is the BU calculation and the thick dashed-line is the full TC calculation. It can be noticed that both calculations reproduce satisfactorily well the data. To illustrate the slow convergence rate of the TC calculation we have included also a calculation with 6 partial waves. The latter, represented by the thin dashed line, clearly fails to reproduce the strong peak at forward angles. In the right panel of Fig. 2, the energy distribution of the 7 Be fragments, integrated from 0 to 180◦ , is represented for the BU (solid line) and TC (dashed line) calculations. Again, a very reasonable agreement, both in shape and absolute normalisation, is found between both approaches. ¿From this comparison, we can conclude that, at least for this reaction, the BU and TC schemes predict essentially the same breakup observables. This means that the two representations describe to a large extent the same piece of the three-body continuum. We may also conclude that the 7 Be observables are dominated by pure three-body breakup, in contrast to what occurs in breakup reactions proceeding through two-body resonances. ¿From this analysis, it becomes also apparent that the BU approach is more suitable to describe this reaction, in the sense that a reduced model space is required for convergence. As an additional argument for this conclusion, we have estimated the energy of the
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TC: lf = 0-17
100
40
dσ/dE (mb/MeV)
dσbu/dΩ (mb/sr)
Direct Breakup: li=0-3 TC: lf = 0-6
50
30
20
10
0 0
50 7
θlab ( Be) (degrees)
100
0
10
15 20 7 Elab ( Be) (MeV)
25
Figure 2. Angular (left panel) and energy distribution (right panel) of the measured 7 Be fragments produced in 8 B+58 Ni → 7 Be + p + 58 Ni reaction at Elab = 26 MeV. The solid line is the BU calculation, performed within the CDCC approach. The thick dashed line, is the TC calculation. The thin solid line in the left panel is corresponds to a TC calculation in a reduced model space. The arrow is the expected energy of 7 Be fragments detected at θlab = 10◦ in the BU picture.
outgoing 7 Be ejectile assuming an extreme direct breakup picture, in which the projectile breaks up leaving the proton and the 7 Be fragments with zero relative energy. Then, the energy of 7 Be in the laboratory frame (E7 ) is related to the energy of the 8 B∗ system (E8 ) as E7 87 E8 . For a scattering angle θlab = 20◦ , this yields the value E7 = 22.1 MeV, which is indicated by the arrow in the right panel of Fig. 2. This value is close to the maximum of the energy distribution, thus supporting the fact that the energy of the 7 Be fragments is consistent with a direct breakup process, rather than a transfer-like process. 2.2. The 6 He +208 Pb reaction We now consider the 6 He +208 Pb reaction, recently measured at energies around the Coulomb barrier at the Louvain-la-Neuve facility (Belgium). This case represents a more complicated problem from both the experimental and theoretical sides, since the 6 He nucleus will fragment into three fragments, 4 He+n+n. However, to simplify the analysis, we will treat the 6 He nucleus within the so called di-neutron model, in which the two valence neutrons are assumed to constitute a fully correlated neutron pair. The 6 He ground state is obtained by solving the Schr¨odinger equation with an effective 2n-α interaction, and assuming a 2S single particle state, with unit spectroscopic factor. Recent calculations for the 6 He +208 Pb [9] and 6 He +209 Bi [10] reactions show that this simple di-neutron model fails to reproduce the elastic data. This is mainly due to the fact that this model largely overestimates the couplings between the ground and the continuum states. This deficiency of the model is very important for the Pb and Bi targets, since these heavy
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targets produce very intense dipole couplings on the projectile. In Ref. [9], this problem was solved by arbitrary reducing the strength of these couplings by 50%. Here, we use an alternative procedure in which the two-neutron separation energy S2n is replaced by an effective value. To determine this value, we compare the density probability derived from the nn-α relative wavefunction, with that calculated in a realistic three-body model of 6 He. We found that, using the effective separation energy S2n 1.5 MeV, the nn-α relative wavefunction from the di-neutron model simulates reasonably well the three-body calculation. Besides, the B(E1) and B(E2) strength distributions, are also well reproduced. Details of this method will be published elsewhere.
208
Pb + 2n
4
He + 2n
6
(a)
6
208
He + Pb
(b)
208
He + Pb 4
210
He + Pb
Figure 3. Schematic representation of the couplings included in the BU (a) and TC calculations (b) in the 6 He + 208 Pb reaction.
In the BU calculations the 6 He continuum was discretized into energy bins, up to a maximum excitation energy of 8 MeV above the two-neutron breakup threshold. For the α+2n relative motion, the partial waves s, p and d were included. Couplings between the bound and continuum states, as well as among the continuum states, were included. These couplings are schematically depicted in Fig. 3a. In the TC approach, the α particles are assumed to arise from two-neutron transfer to bound and unbound states of the 208 Pb target, as represented schematically in Fig. 3b. The transfer amplitude was evaluated within the Distorted Wave Born Approximation (DWBA). For each 2n-208 Pb partial wave, the bound and unbound spectrum of twoneutron states in the 210 Pb nucleus was represented by a uniform distribution of states, evenly spaced in steps of 1 MeV. Unbound states were represented by a distribution of energy bins of width 1 MeV, and up to a maximum excitation energy of 8 MeV above the 210 Pb→208 Pb+2n breakup threshold. In order to achieve convergence of the angular differential cross section within the angular range of interest, we had to include up to 6 partial waves for the 2n+208 Pb relative motion. In Fig. 4 we present the BU and TC calculations with the experimental data. In the left panel, the distribution of the α products is represented as a function of the scattering angle, in the laboratory frame. The solid and dashed lines correspond to the
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Figure 4. Angular (left panel) and energy (right panel) distribution of the 4 He fragments produced in the 6 He +208 Pb → 4 He + 2n + 208 Pb reaction at Elab = 22 MeV. The circles (left panel) and the the histogram (right panel) represent the experimental data. In both plots, the solid and dashed line correspond to the TC and BU calculations. The arrow (right panel) is the expected energy of α particles detected at θlab = 150◦ assuming a direct breakup picture.
TC and BU calculations, respectively. For clarity, in the right plot, the BU calculation has been multiplied by a factor of 5. The former reproduces reasonably well the trend of the data, although the absolute normalisation is underestimated by about 40%. This underestimation could be due to the contribution of other channels (such as the one neutron transfer), but also it could indicate the degree of inaccuracy of our simplified structure model. From this figure, we see also that the BU calculation completely fails to explain the data. The BU distribution peaks at forward angles (about 30◦ ) and then falls quickly at backward angles, predicting a too small cross section as compared to the data. The right panel of Fig. 4 displays the energy distribution of the α fragments, integrated within the angular range of the experiment (θlab = 132 − 165◦ ). It can be seen that the TC calculation is in fairly good agreement with the data. In particular, the position of the peak is very well reproduced. The BU calculation which, as noted before, largely underestimates the data, fails also to reproduce the shape of the spectrum. It is interesting to note that the maximum of the experimental distribution corresponds to a energy of the α particles about 18 MeV. For comparison, the expected energy of these fragments, estimated in a direct breakup picture at θlab = 150◦ , is about Eα 12.5 MeV, significantly smaller than the observed value. Consequently, the α particles are post-accelerated after the scattering process. This suggests that the transferred neutrons lose part of their kinetic energy. This interpretation is consistent with a reaction picture in which the incoming 6 He particles are stretched mainly due to the tidal forces exerted by the target. This effect is very important in the case of the Pb target, due to the strong repulsive effect of the Coulomb field on the α core. The halo neutrons, by contrast, will easily penetrate the Coulomb barrier and dissociate from the projectile, due to the small separation energy.
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80
20 8
58
7
58
6
60
dσ/dε (mb/sr)
dσ/dε (mb/sr)
B+ Ni → Be+p+ Ni
40 20 0 0
2
4
εp-Ni (MeV)
6
8
208
He+
4
Pb → He + 2n +
208
Pb
15 10 5 0
-4
-2
0
2
ε2n-Pb (MeV)
4
6
8
Figure 5. Left panel: calculated energy distribution of the measured 7 Be fragments produced in the reaction 8 B+ 58 Ni , and integrated in the whole angular range, as a function of the relative energy between the protons and target. Right panel: analogous distribution for the 4 He fragments produced in the 6 He +208 Pb at Elab = 22 MeV, as a function of the 2n-208 Pb relative energy, and integrated in the angular range θlab = 132◦ − 165◦ . Both calculations are performed using the TC approach. The arrows indicate the expected position of the peak in an extreme direct breakup picture.
This favours that the neutrons are captured by the target, or left in a continuum state of low energy. The α particles will be then post-accelerated, carrying most of the available kinetic energy. By contrast, in the 8 B+58 Ni case, the Coulomb repulsion will tend to repel both the 7 Be core and the valence proton. The calculations here presented clearly show that the BU approach is more efficient to describe the 8 B+58 Ni breakup, while in the 6 He + 208 Pb reaction the TC method provides a more suitable representation. To have a better understanding of this result, in Fig. 5 we show the breakup cross section, calculated in the TC approach, as a function of the relative energy between the transferred particle and the target. Negative energies in this scale represent transfer to bound states. In each plot, the arrow is the estimated relative energy assuming an extreme direct breakup picture, as explained above. In the 8 B+58 Ni case (left panel), the maximum of the distribution occurs at a energy which is fully consistent with the prediction of the extreme direct breakup model. By contrast, in the 6 He +208 Pb case (right panel), the average relative energy between the neutrons and the target is considerably smaller than the direct breakup estimate. This explains why in the 8 B+58 Ni reaction a good agreement is found between the BU and TC calculations, while in the 6 He +208 Pb case they predict very different results. 3. Summary and conclusions In this contribution, we have compared two different approaches to study breakup reactions leading to three fragments in the final state. In the BU approach, three-body continuum is represented in terms of a basis of the internal states of the projectile and the breakup process is treated as an inelastic-like process. By contrast, the TC method is
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based on a target representation of the continuum, and the breakup process is interpreted as the transfer of part of the projectile to the target continuum. In the 8 B+ 58 Ni → 7 Be+p+58 Ni reaction at Elab = 26 MeV energies, both descriptions predict similar results for the breakup observables, although the BU approach turns out to be more efficient, in the sense that less partial waves are required to describe these observables. This agreement suggests that, at least in the observables analysed, the process is dominated by three-body breakup, and there is no strong correlation between the final binary systems. This is probably favoured by the mutual repulsion between the three (charged) fragments. By contrast, in the 6 He + 208 Pb reaction at Elab = 22 MeV the TC and BU calculations predict very different results. At backward angles, the measured α products are in much better agreement with the TC calculations, while the BU calculations largely underestimates the data. We attribute this result to the fact that the α particles emitted at backward angles arise from two-neutron transfer to slightly unbound states of the target, which is the region of the three-body continuum which is better described by the TC scheme. Then, in this reaction the transferred neutrons are highly correlated in the final state to the target, and these configurations are hardly described in a basis representation based on the projectile. Although further calculations should be required to achieve more definite conclusions, from the present analysis we may conclude that the choice of the most appropriate representation depends on the reaction under study, but also on the kinematical conditions of the experiment. Furthermore it is expected that the incident energy has an important effect on the correlations between the fragments and hence, it can influence the adequacy of each representation. A detailed comparison between these two approaches and the more general Faddeev method is also underway for the p+11 Be → p+n+10 Be reaction. We believe that this study will provide a deeper insight on the relationship between these methods. REFERENCES 1. L. D. Faddeev, JETP 39 (1960) 1459. 2. C. J. Joachain, Quantum collision theory, North-Holland, 1987. 3. N. Austern, Y. Iseri, M. Kamimura, M. Kawai, G. Rawitscher, M. Yahiro, Phys. Rep. 154 (1987) 125. 4. A. Bonaccorso, D. M. Brink, Phys. Rev. C 38 (1998) 1776. 5. R. Chatterjee, P. Banerjee, R. Shyam, Nucl. Phys. A 675 (2000) 477. 6. V. Guimar˜aes et al, Phys. Rev. Lett. 84 (9) (2000) 1862–1865. 7. A. M. Moro, F. M. Nunes, Nucl. Phys. A 767 (2005) 138–154. 8. J. A. Tostevin, F. M. Nunes, I. J. Thompson, Phys. Rev. C 63 (2001) 024617. 9. K. Rusek, I. Martel, J. G´omez-Camacho, A. M. Moro, R. Raabe, Phys. Rev. C 72 (2005) 037603. 10. T. Matsumoto, T. Egami, K. Ogata, Y. Iseri, M. Kamimura, M. Yahiro, Phys. Rev. C 73 (2006) 051602.
Three-body approaches for inclusive breakup reactions∗ A. M. Moro,a F. M. Nunes,b D. Escrigc and J. G´omez-Camachoa a
Universidad de Sevilla, P.O. Box 1065, E-41080, Seville, Spain
b
NSCL and Department of Physics and Astronomy, Michigan State University, U.S.A.
c
Instituto de Estructura de la Materia, CSIC, E-28006 Madrid, Spain
The formal problem of three-body breakup in nuclear reactions is addressed. Two alternative treatments, namely, the direct breakup and transfer to the continuum approaches, are discussed and compared. Calculations for the reactions 8 B+ 58 Ni at 26 MeV and 6 He + 208 Pb at 22 MeV are presented and compared with existing data. 1. Introduction An important reaction mechanism that occurs when a loosely bound projectile collides with a target is the breakup of the projectile into two or more fragments. In the case of a projectile consisting of two weakly bound fragments, scattered by a stable target, this process gives rise to three fragments. The formal treatment of these reactions involves a complicated three-body problem. In the case in which the internal structure of the constituents can be safely ignored, such as in deuteron-proton or deuteron-neutron reactions at intermediate energies this problem has been successfully solved by solving the Faddeev equations [1,2]. In the Faddeev formalism, the total wavefunction of the three-body system is expanded into three components. Each component contains the bound states as well as continuum states for a specific binary system. Therefore, each rearrangement channel is confined to a particular Faddeev component. By contrast, three-body breakup will be distributed among the three Faddeev components. Notice that, since each Jacobi representation provides a complete basis, the Faddeev wavefunction uses an overcomplete basis. The difficulties inherent to the solution of the Faddeev equations has prevented its application to realistic heavy ion reactions. In practical calculations, it is customary to select a particular representation, depending on the problem under study. For instance, in single-particle excitations populating bound or narrow resonances of the projectile, the natural representation will be that based on the eigenstates of the projectile. On the other hand, for a process in which part of the projectile is transferred to a bound or resonant state of the target, the obvious representation is that containing the fragmenttarget states. The choice is less obvious in the case of non-resonant breakup. According to the way in which the breakup process is modelled, current breakup reaction theories ∗ Work partially supported by National Science Foundation (USA) under grant PHY-0456656, and by the Junta de Andaluca (Spain).
0375-9474/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2006.12.069
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can be divided into two main categories. On one hand, some approaches, such as the conventional Continuum-Discretized Coupled Channels method [3], treat this process as an inelastic excitation of the projectile to the continuum, that we will refer hereafter as the direct breakup picture (BU). Bound as well as unbound states are represented by the eigenstates of the two-body Hamiltonian of the projectile. The Schr¨odinger equation is then projected onto these states, giving rise to a system of coupled equations. To make the number of equations finite, the two-body continuum is truncated in two ways: by retaining a given number of partial waves between the two fragments, and by discretizing the continuum into a set of energy intervals, or bins, up to a maximum excitation energy. In practical calculations, the number of partial waves, the maximum excitation energy, and the bin widths are chosen in order to obtain convergence of the observables considered. On the other hand, the breakup process can be also considered as a transfer of a piece of the projectile to the target continuum. This procedure has been referred to in the literature as transfer to the continuum (TC) method. Some authors have applied this method within a semiclassical picture [4]. Also, it has been applied as a generalisation of the Distorted Wave Born Approximation (DWBA) [5]. A natural question that arises from the considerations above is to what extent the BU and TC representations provide similar results for the breakup observables. Furthermore, even if this is the case, it is also important to discern which representation will be more appropriate in a particular reaction. It is the objective of this contribution to shed some light into these two questions. For this purpose, we have performed calculations for several systems using the BU and TC approaches. In this work, the results for the 8 B+58 Ni and 6 He +208 Pb reactions will be presented and discussed.
58
Ni + p
7
Be + p
8
(a)
8
58 B + Ni
(b)
58 B + Ni
7
58
Be + Ni
Figure 1. Schematic representation of the couplings included in the BU (a) and TC calculations (b) in the 8 B+ 58 Ni reaction.
A.M. Moro et al. / Nuclear Physics A 787 (2007) 463c–470c
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2. Calculations 2.1. The 8 B+58 Ni reaction In this section we consider the breakup reaction 8 B+58 Ni → 7 Be + p + 58 Ni at Elab = 25.8 MeV, which is close to the Coulomb barrier. This reaction was measured at the Notre Dame facility. Details of this experiment can be found in [6]. A detailed analysis of this reaction within the BU and TC approaches was already presented in Ref. [7]. Here, we briefly summarise the main results. To compare the BU and TC methods, we considered the angular and energy distribution of the scattered 7 Be fragments. The BU calculation (see Fig. 1a) was performed according to the standard CDCC approach, following closely the work of Ref. [8]. For each partial wave, the 8 B two-body continuum (7 Be+p) was discretized into energy bins, up to a maximum excitation energy of max = 8 MeV. To achieve convergence, we had to include s, p, d, f and g waves for the 7 Be+p relative motion. Bound state–continuum as well as continuum–continuum couplings were included. Within the TC, the breakup is treated as a proton transfer to unbound states of the p+58 Ni system, as illustrated in Fig. 1b. As in the CDCC calculation, these unbound states were represented by energy bins. The p+58 Ni continuum was truncated up to a maximum excitation energy of 10 MeV, above the breakup threshold. In this case, the convergence rate for the forward scattering angles was slower, forcing us to include up to 17 waves to achieve a reasonable convergence of this observable. Further details of these calculations, including the choice of the binary interactions between the fragments, can be found in Refs. [7] and [8]. In Fig. 2 we present the TC and BU calculations for the observables associated with the 7 Be fragments. The left panel of this figure shows the differential cross section of the scattered 7 Be products as a function of the scattering angle, in the laboratory frame. For each angle, this distribution is integrated with respect to the energy of 7 Be. The circles correspond to the data of Guimar˜aes et al. [6], the thick solid line is the BU calculation and the thick dashed-line is the full TC calculation. It can be noticed that both calculations reproduce satisfactorily well the data. To illustrate the slow convergence rate of the TC calculation we have included also a calculation with 6 partial waves. The latter, represented by the thin dashed line, clearly fails to reproduce the strong peak at forward angles. In the right panel of Fig. 2, the energy distribution of the 7 Be fragments, integrated from 0 to 180◦ , is represented for the BU (solid line) and TC (dashed line) calculations. Again, a very reasonable agreement, both in shape and absolute normalisation, is found between both approaches. ¿From this comparison, we can conclude that, at least for this reaction, the BU and TC schemes predict essentially the same breakup observables. This means that the two representations describe to a large extent the same piece of the three-body continuum. We may also conclude that the 7 Be observables are dominated by pure three-body breakup, in contrast to what occurs in breakup reactions proceeding through two-body resonances. ¿From this analysis, it becomes also apparent that the BU approach is more suitable to describe this reaction, in the sense that a reduced model space is required for convergence. As an additional argument for this conclusion, we have estimated the energy of the
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TC: lf = 0-17
100
40
dσ/dE (mb/MeV)
dσbu/dΩ (mb/sr)
Direct Breakup: li=0-3 TC: lf = 0-6
50
30
20
10
0 0
50 7
θlab ( Be) (degrees)
100
0
10
15 20 7 Elab ( Be) (MeV)
25
Figure 2. Angular (left panel) and energy distribution (right panel) of the measured 7 Be fragments produced in 8 B+58 Ni → 7 Be + p + 58 Ni reaction at Elab = 26 MeV. The solid line is the BU calculation, performed within the CDCC approach. The thick dashed line, is the TC calculation. The thin solid line in the left panel is corresponds to a TC calculation in a reduced model space. The arrow is the expected energy of 7 Be fragments detected at θlab = 10◦ in the BU picture.
outgoing 7 Be ejectile assuming an extreme direct breakup picture, in which the projectile breaks up leaving the proton and the 7 Be fragments with zero relative energy. Then, the energy of 7 Be in the laboratory frame (E7 ) is related to the energy of the 8 B∗ system (E8 ) as E7 87 E8 . For a scattering angle θlab = 20◦ , this yields the value E7 = 22.1 MeV, which is indicated by the arrow in the right panel of Fig. 2. This value is close to the maximum of the energy distribution, thus supporting the fact that the energy of the 7 Be fragments is consistent with a direct breakup process, rather than a transfer-like process. 2.2. The 6 He +208 Pb reaction We now consider the 6 He +208 Pb reaction, recently measured at energies around the Coulomb barrier at the Louvain-la-Neuve facility (Belgium). This case represents a more complicated problem from both the experimental and theoretical sides, since the 6 He nucleus will fragment into three fragments, 4 He+n+n. However, to simplify the analysis, we will treat the 6 He nucleus within the so called di-neutron model, in which the two valence neutrons are assumed to constitute a fully correlated neutron pair. The 6 He ground state is obtained by solving the Schr¨odinger equation with an effective 2n-α interaction, and assuming a 2S single particle state, with unit spectroscopic factor. Recent calculations for the 6 He +208 Pb [9] and 6 He +209 Bi [10] reactions show that this simple di-neutron model fails to reproduce the elastic data. This is mainly due to the fact that this model largely overestimates the couplings between the ground and the continuum states. This deficiency of the model is very important for the Pb and Bi targets, since these heavy
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targets produce very intense dipole couplings on the projectile. In Ref. [9], this problem was solved by arbitrary reducing the strength of these couplings by 50%. Here, we use an alternative procedure in which the two-neutron separation energy S2n is replaced by an effective value. To determine this value, we compare the density probability derived from the nn-α relative wavefunction, with that calculated in a realistic three-body model of 6 He. We found that, using the effective separation energy S2n 1.5 MeV, the nn-α relative wavefunction from the di-neutron model simulates reasonably well the three-body calculation. Besides, the B(E1) and B(E2) strength distributions, are also well reproduced. Details of this method will be published elsewhere.
208
Pb + 2n
4
He + 2n
6
(a)
6
208
He + Pb
(b)
208
He + Pb 4
210
He + Pb
Figure 3. Schematic representation of the couplings included in the BU (a) and TC calculations (b) in the 6 He + 208 Pb reaction.
In the BU calculations the 6 He continuum was discretized into energy bins, up to a maximum excitation energy of 8 MeV above the two-neutron breakup threshold. For the α+2n relative motion, the partial waves s, p and d were included. Couplings between the bound and continuum states, as well as among the continuum states, were included. These couplings are schematically depicted in Fig. 3a. In the TC approach, the α particles are assumed to arise from two-neutron transfer to bound and unbound states of the 208 Pb target, as represented schematically in Fig. 3b. The transfer amplitude was evaluated within the Distorted Wave Born Approximation (DWBA). For each 2n-208 Pb partial wave, the bound and unbound spectrum of twoneutron states in the 210 Pb nucleus was represented by a uniform distribution of states, evenly spaced in steps of 1 MeV. Unbound states were represented by a distribution of energy bins of width 1 MeV, and up to a maximum excitation energy of 8 MeV above the 210 Pb→208 Pb+2n breakup threshold. In order to achieve convergence of the angular differential cross section within the angular range of interest, we had to include up to 6 partial waves for the 2n+208 Pb relative motion. In Fig. 4 we present the BU and TC calculations with the experimental data. In the left panel, the distribution of the α products is represented as a function of the scattering angle, in the laboratory frame. The solid and dashed lines correspond to the
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Figure 4. Angular (left panel) and energy (right panel) distribution of the 4 He fragments produced in the 6 He +208 Pb → 4 He + 2n + 208 Pb reaction at Elab = 22 MeV. The circles (left panel) and the the histogram (right panel) represent the experimental data. In both plots, the solid and dashed line correspond to the TC and BU calculations. The arrow (right panel) is the expected energy of α particles detected at θlab = 150◦ assuming a direct breakup picture.
TC and BU calculations, respectively. For clarity, in the right plot, the BU calculation has been multiplied by a factor of 5. The former reproduces reasonably well the trend of the data, although the absolute normalisation is underestimated by about 40%. This underestimation could be due to the contribution of other channels (such as the one neutron transfer), but also it could indicate the degree of inaccuracy of our simplified structure model. From this figure, we see also that the BU calculation completely fails to explain the data. The BU distribution peaks at forward angles (about 30◦ ) and then falls quickly at backward angles, predicting a too small cross section as compared to the data. The right panel of Fig. 4 displays the energy distribution of the α fragments, integrated within the angular range of the experiment (θlab = 132 − 165◦ ). It can be seen that the TC calculation is in fairly good agreement with the data. In particular, the position of the peak is very well reproduced. The BU calculation which, as noted before, largely underestimates the data, fails also to reproduce the shape of the spectrum. It is interesting to note that the maximum of the experimental distribution corresponds to a energy of the α particles about 18 MeV. For comparison, the expected energy of these fragments, estimated in a direct breakup picture at θlab = 150◦ , is about Eα 12.5 MeV, significantly smaller than the observed value. Consequently, the α particles are post-accelerated after the scattering process. This suggests that the transferred neutrons lose part of their kinetic energy. This interpretation is consistent with a reaction picture in which the incoming 6 He particles are stretched mainly due to the tidal forces exerted by the target. This effect is very important in the case of the Pb target, due to the strong repulsive effect of the Coulomb field on the α core. The halo neutrons, by contrast, will easily penetrate the Coulomb barrier and dissociate from the projectile, due to the small separation energy.
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80
20 8
58
7
58
6
60
dσ/dε (mb/sr)
dσ/dε (mb/sr)
B+ Ni → Be+p+ Ni
40 20 0 0
2
4
εp-Ni (MeV)
6
8
208
He+
4
Pb → He + 2n +
208
Pb
15 10 5 0
-4
-2
0
2
ε2n-Pb (MeV)
4
6
8
Figure 5. Left panel: calculated energy distribution of the measured 7 Be fragments produced in the reaction 8 B+ 58 Ni , and integrated in the whole angular range, as a function of the relative energy between the protons and target. Right panel: analogous distribution for the 4 He fragments produced in the 6 He +208 Pb at Elab = 22 MeV, as a function of the 2n-208 Pb relative energy, and integrated in the angular range θlab = 132◦ − 165◦ . Both calculations are performed using the TC approach. The arrows indicate the expected position of the peak in an extreme direct breakup picture.
This favours that the neutrons are captured by the target, or left in a continuum state of low energy. The α particles will be then post-accelerated, carrying most of the available kinetic energy. By contrast, in the 8 B+58 Ni case, the Coulomb repulsion will tend to repel both the 7 Be core and the valence proton. The calculations here presented clearly show that the BU approach is more efficient to describe the 8 B+58 Ni breakup, while in the 6 He + 208 Pb reaction the TC method provides a more suitable representation. To have a better understanding of this result, in Fig. 5 we show the breakup cross section, calculated in the TC approach, as a function of the relative energy between the transferred particle and the target. Negative energies in this scale represent transfer to bound states. In each plot, the arrow is the estimated relative energy assuming an extreme direct breakup picture, as explained above. In the 8 B+58 Ni case (left panel), the maximum of the distribution occurs at a energy which is fully consistent with the prediction of the extreme direct breakup model. By contrast, in the 6 He +208 Pb case (right panel), the average relative energy between the neutrons and the target is considerably smaller than the direct breakup estimate. This explains why in the 8 B+58 Ni reaction a good agreement is found between the BU and TC calculations, while in the 6 He +208 Pb case they predict very different results. 3. Summary and conclusions In this contribution, we have compared two different approaches to study breakup reactions leading to three fragments in the final state. In the BU approach, three-body continuum is represented in terms of a basis of the internal states of the projectile and the breakup process is treated as an inelastic-like process. By contrast, the TC method is
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based on a target representation of the continuum, and the breakup process is interpreted as the transfer of part of the projectile to the target continuum. In the 8 B+ 58 Ni → 7 Be+p+58 Ni reaction at Elab = 26 MeV energies, both descriptions predict similar results for the breakup observables, although the BU approach turns out to be more efficient, in the sense that less partial waves are required to describe these observables. This agreement suggests that, at least in the observables analysed, the process is dominated by three-body breakup, and there is no strong correlation between the final binary systems. This is probably favoured by the mutual repulsion between the three (charged) fragments. By contrast, in the 6 He + 208 Pb reaction at Elab = 22 MeV the TC and BU calculations predict very different results. At backward angles, the measured α products are in much better agreement with the TC calculations, while the BU calculations largely underestimates the data. We attribute this result to the fact that the α particles emitted at backward angles arise from two-neutron transfer to slightly unbound states of the target, which is the region of the three-body continuum which is better described by the TC scheme. Then, in this reaction the transferred neutrons are highly correlated in the final state to the target, and these configurations are hardly described in a basis representation based on the projectile. Although further calculations should be required to achieve more definite conclusions, from the present analysis we may conclude that the choice of the most appropriate representation depends on the reaction under study, but also on the kinematical conditions of the experiment. Furthermore it is expected that the incident energy has an important effect on the correlations between the fragments and hence, it can influence the adequacy of each representation. A detailed comparison between these two approaches and the more general Faddeev method is also underway for the p+11 Be → p+n+10 Be reaction. We believe that this study will provide a deeper insight on the relationship between these methods. REFERENCES 1. L. D. Faddeev, JETP 39 (1960) 1459. 2. C. J. Joachain, Quantum collision theory, North-Holland, 1987. 3. N. Austern, Y. Iseri, M. Kamimura, M. Kawai, G. Rawitscher, M. Yahiro, Phys. Rep. 154 (1987) 125. 4. A. Bonaccorso, D. M. Brink, Phys. Rev. C 38 (1998) 1776. 5. R. Chatterjee, P. Banerjee, R. Shyam, Nucl. Phys. A 675 (2000) 477. 6. V. Guimar˜aes et al, Phys. Rev. Lett. 84 (9) (2000) 1862–1865. 7. A. M. Moro, F. M. Nunes, Nucl. Phys. A 767 (2005) 138–154. 8. J. A. Tostevin, F. M. Nunes, I. J. Thompson, Phys. Rev. C 63 (2001) 024617. 9. K. Rusek, I. Martel, J. G´omez-Camacho, A. M. Moro, R. Raabe, Phys. Rev. C 72 (2005) 037603. 10. T. Matsumoto, T. Egami, K. Ogata, Y. Iseri, M. Kamimura, M. Yahiro, Phys. Rev. C 73 (2006) 051602.