- Email: [email protected]
Hindrance of fusion induced by 6He
Nuclear Physics A 787 (2007) 225c–230c
Hindrance of fusion induced by 6He E.Cremaa P. R .S. Gomes
b
and L. C. Chamona
a
Departamento de F´ısica Nuclear, Instituto de F´ısica da Universidade de S˜ao Paulo, Caixa Postal 66318, 05315-970, S˜ao Paulo, SP, Brazil.
b
Instituto de F´ısica, Universidade Federal Fluminense, Niter´ oi, R.J., 24210-340, Brasil
We analyze fusion cross sections induced by the 6 He halo projectile in the framework of the double folding S˜ao Paulo potential, which has been shown to be a reliable potential for the analysis of fusion cross sections induced by stable weakly bound nuclei. The results show that the fusion cross sections of 6 He + 209 Bi, 238 U are suppressed at energies above the barrier, when compared with coupled channel calculations that do not take into account the projectile breakup, whereas they are in agreement with the calculations at sub-barrier energies. 1. Introduction In nucleus-nucleus reactions at energies around the Coulomb barrier, the bare nuclear interaction between the nuclei has been widely represented by a frozen potential with a Woods-Saxon shape. As nuclear reactions calculations are strongly dependent on the choice of this primary interaction, comparison between data and theoretical calculations are also strongly potential-dependent. As one has not direct access to the true bare potential, some procedures have been developed in order to constrain the very important choice of the bare potential to be used. The first procedure is the analysis of the elastic scattering data within the context of an optical potential description, where a real potential is connected to an imaginary potential, which is added to take into account all the flux deviated from the elastic channel. The optical potential that gives the best fit to the elastic scattering data is then used to describe other reaction channels, even in coupled channel calculations with inelastic and transfer reactions included in the coupling matrix. This kind of calculation suffers from several inconsistencies: 1) as the imaginary and real parts of the optical potential are not completely independent, the real part will carry information of the complete reaction mechanism, and can not be used as bare potential; 2) the elastic scattering at forward angles is governed by the external part of the potential and can not test the more inner region of the interaction where fusion takes place; 3) as coupled channel calculation (CCC) intents to understand the role of each reaction channel in the global reaction mechanism, it is senseless to perform a CCC with an imaginary potential that had already accounted for these channels. Another procedure that has been used to obtain the bare potential is the fit of the high-energy fusion data with the single barrier penetration model (BPM). However this 0375-9474/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2006.12.036
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E. Crema et al. / Nuclear Physics A 787 (2007) 225c–230c
method also has uncertainties. The most important is the evidence of the existence of channels that hinder the fusion cross section at energies near and above the Coulomb barrier. Another problem is the deep inelastic process that hinders the fusion cross section at energies well above the barrier. Hence, the bare potential extracted with this method can be contaminated by different processes. In principle, the barrier distribution measurement is the best way to constrain the choice of the bare potential, once its the height and the shape should be reproduced by the calculations. Indeed, vast experimental work is demanded to obtain the barrier distribution, which is not possible to be done with low-intensity radioactive beams. Of course, the true bare potential could be thought to be deduced from a complete coupled channel calculation where all reaction channels are explicitly included in the coupling matrix. However, this complete calculation is not possible for practical reasons. Then, reducing the model space in the CCC, the resulting potential will be the bare potential added to a polarization-potential contribution arising from the the channels not included in the calculation. This situation becomes even more complicated when a weakly bound nucleus is involved in the reaction, since the breakup channel might couple strongly with other degrees of freedom. In order to overcome the difficulties discussed above, we have shown recently [1] that the parameter-free S˜ao Paulo potential (SPP) [2,3] is a reliable bare interaction for studying fusion of systems involving stable weakly-bound nuclei in situations in which experimental barrier distributions cannot be obtained. As it is deduced from more fundamental grounds, the SPP does not contain the limitations discussed above and does not contain any adjustable parameter. In this paper we propose to adopt the SPP as the bare potential in the study of fusion reactions induced by the radioactive weakly-bound nucleus 6 He. This paper is organized as follows. In Section 2, we describe briefly the SPP, already discussed in several papers [4]. In Section 3, we present the main results of our analysis of fusion induced by stable weakly-bound projectiles, 6,7 Li + 209 Bi and 9 Be + 208 Pb, demonstrating that the SPP is a reliable potential for the study of these systems. The extension of this analysis to the fusion of 6 He + 238 U and 6 He + 209 Bi systems is presented in Section 4. Conclusions are drawn in Section 5. 2. Brief Description of S˜ ao Paulo Potential. The SPP is a model for the interaction between two nuclei, based on the effects of the Pauli non-locality [3,4]. Apart from theoretical considerations, the model is strongly supported on experimental evidences. Within this model the nuclear interaction is connected with the folding potential through [2]: 2 2 V (R, E) = V (R) e−4v /c , (1) SP
F
where c is the speed of light, v is the local relative velocity between the two nuclei, v2 (R, E) =
2 [E − VC (R) − VN (R, E)] , μ
(2)
and VC is the Coulomb potential. With the aim of providing a parameter-free description of the nuclear interaction, the SPP model includes an extensive systematic of nuclear
E. Crema et al. / Nuclear Physics A 787 (2007) 225c–230c
227c
densities [2]. The two-parameter Fermi (2pF) distribution is assumed to be a good approximation to describe the densities. The radii of the 2pF distributions are well represented by R0 = 1.31A1/3 − 0.84 fm, where A is the number of nucleons of the nucleus. The values obtained for the diffuseness of the matter distributions are similar throughout the periodic table and present small variations around the average value a = 0.56 fm. Within the context of this realistic systematic, the SPP does not contain any adjustable parameter. This is, in fact, an essential feature, because the lack of adjustable parameters characterizes the model as a powerful tool to make predictions for quite different systems and energies. Finally, it should be stressed that the energy-dependent term in Eq. (1) can be ignored for bombarding energies around the Coulomb barrier, since the relative velocity between the interacting nuclei is close to zero in this energy regime. Therefore, all theoretical predictions obtained in this paper are basically due to the double-folding potential, within the context of the systematics for the nuclear densities.
3. Testing the SPP with the Fusion in the
6,7
Li+209 Bi and 9 Be+208 Pb Systems
A good test for the ability of the SPP to describe the bare nucleus-nucleus interaction is to compare its predictions with the most reliable analyses in the literature, where the bare potentials have been inferred from barrier distribution measurements associated with coupled channel calculations. Since we are interested in studying the fusion induced by 6 He, which is an unstable weakly-bound nucleus, a good starting point is the analysis of the fusion induced by 6,7 Li and 9 Be, which are also weakly-bound nuclei and have some similarities with 6 He. The fusion excitation functions for 6,7 Li+209 Bi and 9 Be+208 Pb systems were measured and their fusion barrier distributions deduced by the Dasgupta et al [5–7]. Employing the SPP as bare interaction, in a previous paper [1] we performed successful coupled channel analyses of these systems, as it will be briefly shown here. The aim of that work was not to improve Dasgupta’s calculations by including the breakup channel, but rather to show that the SPP predictions could replace the barrier distributions measurements for those systems. Therefore, we coupled exactly the same channels (with the same nuclear parameters) as in ref. [6], and the same special version of the CCFULL code [15]. For 208 Pb, the 3− and 5− collective states and double octupole phonon states were coupled. For 9 Be, couplings to the 5/2− and 7/2− states in the ground state rotational band were included. Also, following the same procedure of ref. [6], the breakup channel was not included in our calculations. Fig. 1 shows the prediction of the SPP for the 9Be+208 Pb system. The fusion excitation function and the fusion barrier distribution are very well predicted by our calculation if we use the same suppression factor that Dasgupta et al used in order to take into account the breakup channel not included in the coupling matrix. Similar results were found for the other two systems: 6,7 Li+209 Bi. Finally, we would like to stress once more that our calculations do not contain any freeparameter and that we did not use the available fusion barrier distributions. Hence, we believe that we have shown that our method can be a good starting point for the analysis of systems involving radioactive projectiles, for which it is very difficult to measure their barrier distributions.
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E. Crema et al. / Nuclear Physics A 787 (2007) 225c–230c
Figure 1. (a) Fusion cross section data and (b) experimental barrier distribution for the Be+208 Pb system, from ref. [7]. The dotted and dashed lines correspond to theoretical results obtained with SPP within BPM and coupled channel calculations, respectively. The solid line represents the CCC result multiplied by the same suppression factor used in ref [7]. 9
4. Fusion Induced by 6 He with
238
U and
209
Bi targets
The success of the SPP to predict fusion cross sections for stable weakly bound nuclei was due to its realistic double-folding potential calculations. The basis for these calculations is a good knowledge of the matter distributions of the interacting nuclei. As the ground state density of 6 He has been determined from several experimental and theoretical studies [2,8–12], we are able to calculate the SPP for the 6 He+238 U and 6 He+209 Bi systems, for which the fusion excitation functions were already measured [13,14]. The present conclusions about the role of the 6 He on the fusion reaction mechanism are contradictory. Kolata et al [14] claim that there is a large sub-barrier fusion enhancement for the 6 He + 209 Bi system, whereas Raabe et al [13] sustain that there is no enhancement in the sub-barrier fusion of the 238 U system. Could these opposite conclusions be due to the target structure differences? In this paper we will answer this question. As in the previous section, the aim of the present work is not to improve the coupled
E. Crema et al. / Nuclear Physics A 787 (2007) 225c–230c
229c
channel calculations performed in ref. [13] and ref. [14]. Instead, in order to evidence the effect of the potential choice on the conclusions, we coupled exactly the same channels as it was done in ref [13] and [14]. The following excited states were coupled: a) for the 6 He projectile, the resonance state at the continuum at E* = 1.8 MeV, with β2 = 0.90 and r0 = 1.06 fm; b) for the 238 U target, E*(2+ , 4+ , 6+ , 8+ ) = 0.049, 0.148, 0.307, 0.518 MeV, respectively, and the E*(3− ) = 0.7319 MeV, with the deformation parameters β2 = 0.37 and β3 = 0.1213, with r0 = 1.06f m; and c) for the 209 Bi target, E*(5/2+ ) = 2.62 MeV and E*(7/2+ ) = 3.09 MeV, with β3 = 0.153 and β5 = 0.110, with r0 = 1.06f m. We used the FRESCO code [16] in our CC calculations, with the SPP interaction supplied numerically. In order to evidence the sensibility of our results with the spread of the Fermi density parameters around their corresponding average values, calculations were performed with the strength of SPP changed by ±10%. The results are shown in figures 2a and 2b, for the 238 U and 209 Bi targets, respectively, where the broad lines represent the regions covered by the variation of ±10% in the intensity of the bare potential.
Figure 2. Fusion excitation functions for (a) 6 He+209 Bi and (b) 6 He+238 U. The broad lines are the results of coupled channel calculations with S˜ao Paulo Potential as bare interaction. See text for details. The most striking result that comes from the analysis of these two figures is the very similar behavior of the two systems, both above and below the Coulomb barrier. For both systems, no sub-barrier fusion enhancement is observed and a hindrance of about 15% is found at above barrier energies. It should be stressed that similar fusion suppression factors were found for fusion induced by the stable weakly-bound nuclei, 6,7 Li and 9 Be [1]. This hindrance of the fusion cross section is believed to be due to the effect of the breakup
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E. Crema et al. / Nuclear Physics A 787 (2007) 225c–230c
process, which produces a strong coupling between the elastic channel and the continuum states. So, we have established a very coherent and systematic behavior for the fusion induced by weakly-bound nuclei. For the 6 He+238 U system, our results are coincident with those obtained from Raabe’s analysis [13], which also used a bare potential calculated by a double-folding procedure (M3Y). On the other hand, our results for the 6 He+209 Bi system are in opposition to the Kolata’s analysis [14], when the bare potential was deduced from a BPM fit to the highest energies data. 5. Conclusion In summary, we have used the trustworthy SPP to analyze fusion induced by the halo radioactive projectile 6 He. We have found a similar behavior as the one for stable weakly bound 6 Li, 7 Li and 9 Be projectiles: at energies above the barrier there is fusion cross section suppression, of the order of 15%, probably due to the effect of the breakup that produces a strong coupling between the elastic channel and the continuum states. At subbarrier energies, there is no remarkable effect of the breakup on the fusion cross section, probably due to competition between break-up and other effects that may enhance fusion, as it occurs for tightly bound projectiles. This work was partially supported by Financiadora de Estudos e Projetos (FINEP), Funda¸ca˜o de Amparo a` Pesquisa do Estado de S˜ao Paulo (FAPESP), Funda¸ca˜o de Amparo a` Pesquisa do Estado do Rio de Janeiro (FAPERJ), and Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq). REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
E. Crema, P.R.S. Gomes and L.C. Chamon, Phys. Rev. C 72 (2005) 034610. L.C. Chamon et al., Phys. Rev. C 66 (2002) 014610. L.C. Chamon et al., Phys. Rev. Lett. 79 (1997) 5218. L.R. Gasques et al., Phys. Rev. C 69 (2004) 034603 and references therein M. Dasgupta et al; Phys. Rev. C 66 (2002) 041602(R) M. Dasgupta et al; Phys. Rev. C 70 (2004) 024606 M. Dasgupta et al; Phys. Rev. Lett 82 (1999) 1395 L. R. Gasques et al, Phys. Rev. C67 (2003) 024602. L. R. Gasques et al, Phys. Rev. C67 (2003) 067603. M. V. Zukhov et al, Nucl. Phys. A552 (1993) 353. J. S. Al-Khalili et al, Phys. Rev. C54 (1996) 1843. G. D. Alkazov et al, Phys. Rev. Lett. 78 (1997) 2313. R. Raabe et al, Nature 431 (2004) 823 J.J. Kolata et al, Phys. Rev. Lett 81 (1998) 4580 ; Phys. Rev. C 57 (1998), R6 K. Hagino (private communication) I. Thompson, Comput. Phys. Rep. 7 (1988),167
Hindrance of fusion induced by 6He E.Cremaa P. R .S. Gomes
b
and L. C. Chamona
a
Departamento de F´ısica Nuclear, Instituto de F´ısica da Universidade de S˜ao Paulo, Caixa Postal 66318, 05315-970, S˜ao Paulo, SP, Brazil.
b
Instituto de F´ısica, Universidade Federal Fluminense, Niter´ oi, R.J., 24210-340, Brasil
We analyze fusion cross sections induced by the 6 He halo projectile in the framework of the double folding S˜ao Paulo potential, which has been shown to be a reliable potential for the analysis of fusion cross sections induced by stable weakly bound nuclei. The results show that the fusion cross sections of 6 He + 209 Bi, 238 U are suppressed at energies above the barrier, when compared with coupled channel calculations that do not take into account the projectile breakup, whereas they are in agreement with the calculations at sub-barrier energies. 1. Introduction In nucleus-nucleus reactions at energies around the Coulomb barrier, the bare nuclear interaction between the nuclei has been widely represented by a frozen potential with a Woods-Saxon shape. As nuclear reactions calculations are strongly dependent on the choice of this primary interaction, comparison between data and theoretical calculations are also strongly potential-dependent. As one has not direct access to the true bare potential, some procedures have been developed in order to constrain the very important choice of the bare potential to be used. The first procedure is the analysis of the elastic scattering data within the context of an optical potential description, where a real potential is connected to an imaginary potential, which is added to take into account all the flux deviated from the elastic channel. The optical potential that gives the best fit to the elastic scattering data is then used to describe other reaction channels, even in coupled channel calculations with inelastic and transfer reactions included in the coupling matrix. This kind of calculation suffers from several inconsistencies: 1) as the imaginary and real parts of the optical potential are not completely independent, the real part will carry information of the complete reaction mechanism, and can not be used as bare potential; 2) the elastic scattering at forward angles is governed by the external part of the potential and can not test the more inner region of the interaction where fusion takes place; 3) as coupled channel calculation (CCC) intents to understand the role of each reaction channel in the global reaction mechanism, it is senseless to perform a CCC with an imaginary potential that had already accounted for these channels. Another procedure that has been used to obtain the bare potential is the fit of the high-energy fusion data with the single barrier penetration model (BPM). However this 0375-9474/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2006.12.036
226c
E. Crema et al. / Nuclear Physics A 787 (2007) 225c–230c
method also has uncertainties. The most important is the evidence of the existence of channels that hinder the fusion cross section at energies near and above the Coulomb barrier. Another problem is the deep inelastic process that hinders the fusion cross section at energies well above the barrier. Hence, the bare potential extracted with this method can be contaminated by different processes. In principle, the barrier distribution measurement is the best way to constrain the choice of the bare potential, once its the height and the shape should be reproduced by the calculations. Indeed, vast experimental work is demanded to obtain the barrier distribution, which is not possible to be done with low-intensity radioactive beams. Of course, the true bare potential could be thought to be deduced from a complete coupled channel calculation where all reaction channels are explicitly included in the coupling matrix. However, this complete calculation is not possible for practical reasons. Then, reducing the model space in the CCC, the resulting potential will be the bare potential added to a polarization-potential contribution arising from the the channels not included in the calculation. This situation becomes even more complicated when a weakly bound nucleus is involved in the reaction, since the breakup channel might couple strongly with other degrees of freedom. In order to overcome the difficulties discussed above, we have shown recently [1] that the parameter-free S˜ao Paulo potential (SPP) [2,3] is a reliable bare interaction for studying fusion of systems involving stable weakly-bound nuclei in situations in which experimental barrier distributions cannot be obtained. As it is deduced from more fundamental grounds, the SPP does not contain the limitations discussed above and does not contain any adjustable parameter. In this paper we propose to adopt the SPP as the bare potential in the study of fusion reactions induced by the radioactive weakly-bound nucleus 6 He. This paper is organized as follows. In Section 2, we describe briefly the SPP, already discussed in several papers [4]. In Section 3, we present the main results of our analysis of fusion induced by stable weakly-bound projectiles, 6,7 Li + 209 Bi and 9 Be + 208 Pb, demonstrating that the SPP is a reliable potential for the study of these systems. The extension of this analysis to the fusion of 6 He + 238 U and 6 He + 209 Bi systems is presented in Section 4. Conclusions are drawn in Section 5. 2. Brief Description of S˜ ao Paulo Potential. The SPP is a model for the interaction between two nuclei, based on the effects of the Pauli non-locality [3,4]. Apart from theoretical considerations, the model is strongly supported on experimental evidences. Within this model the nuclear interaction is connected with the folding potential through [2]: 2 2 V (R, E) = V (R) e−4v /c , (1) SP
F
where c is the speed of light, v is the local relative velocity between the two nuclei, v2 (R, E) =
2 [E − VC (R) − VN (R, E)] , μ
(2)
and VC is the Coulomb potential. With the aim of providing a parameter-free description of the nuclear interaction, the SPP model includes an extensive systematic of nuclear
E. Crema et al. / Nuclear Physics A 787 (2007) 225c–230c
227c
densities [2]. The two-parameter Fermi (2pF) distribution is assumed to be a good approximation to describe the densities. The radii of the 2pF distributions are well represented by R0 = 1.31A1/3 − 0.84 fm, where A is the number of nucleons of the nucleus. The values obtained for the diffuseness of the matter distributions are similar throughout the periodic table and present small variations around the average value a = 0.56 fm. Within the context of this realistic systematic, the SPP does not contain any adjustable parameter. This is, in fact, an essential feature, because the lack of adjustable parameters characterizes the model as a powerful tool to make predictions for quite different systems and energies. Finally, it should be stressed that the energy-dependent term in Eq. (1) can be ignored for bombarding energies around the Coulomb barrier, since the relative velocity between the interacting nuclei is close to zero in this energy regime. Therefore, all theoretical predictions obtained in this paper are basically due to the double-folding potential, within the context of the systematics for the nuclear densities.
3. Testing the SPP with the Fusion in the
6,7
Li+209 Bi and 9 Be+208 Pb Systems
A good test for the ability of the SPP to describe the bare nucleus-nucleus interaction is to compare its predictions with the most reliable analyses in the literature, where the bare potentials have been inferred from barrier distribution measurements associated with coupled channel calculations. Since we are interested in studying the fusion induced by 6 He, which is an unstable weakly-bound nucleus, a good starting point is the analysis of the fusion induced by 6,7 Li and 9 Be, which are also weakly-bound nuclei and have some similarities with 6 He. The fusion excitation functions for 6,7 Li+209 Bi and 9 Be+208 Pb systems were measured and their fusion barrier distributions deduced by the Dasgupta et al [5–7]. Employing the SPP as bare interaction, in a previous paper [1] we performed successful coupled channel analyses of these systems, as it will be briefly shown here. The aim of that work was not to improve Dasgupta’s calculations by including the breakup channel, but rather to show that the SPP predictions could replace the barrier distributions measurements for those systems. Therefore, we coupled exactly the same channels (with the same nuclear parameters) as in ref. [6], and the same special version of the CCFULL code [15]. For 208 Pb, the 3− and 5− collective states and double octupole phonon states were coupled. For 9 Be, couplings to the 5/2− and 7/2− states in the ground state rotational band were included. Also, following the same procedure of ref. [6], the breakup channel was not included in our calculations. Fig. 1 shows the prediction of the SPP for the 9Be+208 Pb system. The fusion excitation function and the fusion barrier distribution are very well predicted by our calculation if we use the same suppression factor that Dasgupta et al used in order to take into account the breakup channel not included in the coupling matrix. Similar results were found for the other two systems: 6,7 Li+209 Bi. Finally, we would like to stress once more that our calculations do not contain any freeparameter and that we did not use the available fusion barrier distributions. Hence, we believe that we have shown that our method can be a good starting point for the analysis of systems involving radioactive projectiles, for which it is very difficult to measure their barrier distributions.
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E. Crema et al. / Nuclear Physics A 787 (2007) 225c–230c
Figure 1. (a) Fusion cross section data and (b) experimental barrier distribution for the Be+208 Pb system, from ref. [7]. The dotted and dashed lines correspond to theoretical results obtained with SPP within BPM and coupled channel calculations, respectively. The solid line represents the CCC result multiplied by the same suppression factor used in ref [7]. 9
4. Fusion Induced by 6 He with
238
U and
209
Bi targets
The success of the SPP to predict fusion cross sections for stable weakly bound nuclei was due to its realistic double-folding potential calculations. The basis for these calculations is a good knowledge of the matter distributions of the interacting nuclei. As the ground state density of 6 He has been determined from several experimental and theoretical studies [2,8–12], we are able to calculate the SPP for the 6 He+238 U and 6 He+209 Bi systems, for which the fusion excitation functions were already measured [13,14]. The present conclusions about the role of the 6 He on the fusion reaction mechanism are contradictory. Kolata et al [14] claim that there is a large sub-barrier fusion enhancement for the 6 He + 209 Bi system, whereas Raabe et al [13] sustain that there is no enhancement in the sub-barrier fusion of the 238 U system. Could these opposite conclusions be due to the target structure differences? In this paper we will answer this question. As in the previous section, the aim of the present work is not to improve the coupled
E. Crema et al. / Nuclear Physics A 787 (2007) 225c–230c
229c
channel calculations performed in ref. [13] and ref. [14]. Instead, in order to evidence the effect of the potential choice on the conclusions, we coupled exactly the same channels as it was done in ref [13] and [14]. The following excited states were coupled: a) for the 6 He projectile, the resonance state at the continuum at E* = 1.8 MeV, with β2 = 0.90 and r0 = 1.06 fm; b) for the 238 U target, E*(2+ , 4+ , 6+ , 8+ ) = 0.049, 0.148, 0.307, 0.518 MeV, respectively, and the E*(3− ) = 0.7319 MeV, with the deformation parameters β2 = 0.37 and β3 = 0.1213, with r0 = 1.06f m; and c) for the 209 Bi target, E*(5/2+ ) = 2.62 MeV and E*(7/2+ ) = 3.09 MeV, with β3 = 0.153 and β5 = 0.110, with r0 = 1.06f m. We used the FRESCO code [16] in our CC calculations, with the SPP interaction supplied numerically. In order to evidence the sensibility of our results with the spread of the Fermi density parameters around their corresponding average values, calculations were performed with the strength of SPP changed by ±10%. The results are shown in figures 2a and 2b, for the 238 U and 209 Bi targets, respectively, where the broad lines represent the regions covered by the variation of ±10% in the intensity of the bare potential.
Figure 2. Fusion excitation functions for (a) 6 He+209 Bi and (b) 6 He+238 U. The broad lines are the results of coupled channel calculations with S˜ao Paulo Potential as bare interaction. See text for details. The most striking result that comes from the analysis of these two figures is the very similar behavior of the two systems, both above and below the Coulomb barrier. For both systems, no sub-barrier fusion enhancement is observed and a hindrance of about 15% is found at above barrier energies. It should be stressed that similar fusion suppression factors were found for fusion induced by the stable weakly-bound nuclei, 6,7 Li and 9 Be [1]. This hindrance of the fusion cross section is believed to be due to the effect of the breakup
230c
E. Crema et al. / Nuclear Physics A 787 (2007) 225c–230c
process, which produces a strong coupling between the elastic channel and the continuum states. So, we have established a very coherent and systematic behavior for the fusion induced by weakly-bound nuclei. For the 6 He+238 U system, our results are coincident with those obtained from Raabe’s analysis [13], which also used a bare potential calculated by a double-folding procedure (M3Y). On the other hand, our results for the 6 He+209 Bi system are in opposition to the Kolata’s analysis [14], when the bare potential was deduced from a BPM fit to the highest energies data. 5. Conclusion In summary, we have used the trustworthy SPP to analyze fusion induced by the halo radioactive projectile 6 He. We have found a similar behavior as the one for stable weakly bound 6 Li, 7 Li and 9 Be projectiles: at energies above the barrier there is fusion cross section suppression, of the order of 15%, probably due to the effect of the breakup that produces a strong coupling between the elastic channel and the continuum states. At subbarrier energies, there is no remarkable effect of the breakup on the fusion cross section, probably due to competition between break-up and other effects that may enhance fusion, as it occurs for tightly bound projectiles. This work was partially supported by Financiadora de Estudos e Projetos (FINEP), Funda¸ca˜o de Amparo a` Pesquisa do Estado de S˜ao Paulo (FAPESP), Funda¸ca˜o de Amparo a` Pesquisa do Estado do Rio de Janeiro (FAPERJ), and Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq). REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
E. Crema, P.R.S. Gomes and L.C. Chamon, Phys. Rev. C 72 (2005) 034610. L.C. Chamon et al., Phys. Rev. C 66 (2002) 014610. L.C. Chamon et al., Phys. Rev. Lett. 79 (1997) 5218. L.R. Gasques et al., Phys. Rev. C 69 (2004) 034603 and references therein M. Dasgupta et al; Phys. Rev. C 66 (2002) 041602(R) M. Dasgupta et al; Phys. Rev. C 70 (2004) 024606 M. Dasgupta et al; Phys. Rev. Lett 82 (1999) 1395 L. R. Gasques et al, Phys. Rev. C67 (2003) 024602. L. R. Gasques et al, Phys. Rev. C67 (2003) 067603. M. V. Zukhov et al, Nucl. Phys. A552 (1993) 353. J. S. Al-Khalili et al, Phys. Rev. C54 (1996) 1843. G. D. Alkazov et al, Phys. Rev. Lett. 78 (1997) 2313. R. Raabe et al, Nature 431 (2004) 823 J.J. Kolata et al, Phys. Rev. Lett 81 (1998) 4580 ; Phys. Rev. C 57 (1998), R6 K. Hagino (private communication) I. Thompson, Comput. Phys. Rep. 7 (1988),167