- Email: [email protected]
Fusion reaction of halo nuclei : A real-time wave-packet method for three-body tunneling dynamics
Nuclear Physics A 787 (2007) 267c–274c
Fusion reaction of halo nuclei : A real-time wave-packet method for three-body tunneling dynamics M. Itoa ∗ , K. Yabanab , T. Nakatsukasab , M. Uedac a
RIKEN, Hirosawa 2-1, Wako, Saitama 351-0198, Japan
b
Institute of Physics and Center for Computational Sciences, University of Tsukuba, Tsukuba 305-8571, Japan c
Akita National College of Technology, Akita 011-8511, Japan
Fusion reactions of halo nuclei are investigated theoretically with a three-body model. The time-dependent wave-packet method is used to solve the three-body Schr¨odinger equation. Dynamics of the halo nucleons are visualized in the coordinate space. Significant differences between the neutron-halo and the proton-halo systems are discussed in connection to the Coulomb breakup process. The fusion cross sections are calculated for 11 Be - 209 Bi and 6 He - 238 U. Our calculation indicates that the fusion cross section is slightly hindered by the presence of weakly bound neutrons. The present results nicely agree with recent measurements. 1. Introduction Physics at the drip-line is one of current subjects in nuclear physics. In particular, nuclear reactions involving halo nuclei have been attracting many researchers since the discovery of halo structure in the secondary beam experiment [1]. In the studies on reaction mechanism of halo nuclei, developments of the reaction theories capable of describing coupling to the continuum and transfer channels are quite important because of its weakly binding properties. For reactions in medium and high incident energies, the Glauber and the eikonal theories have been successful [2–5]. This is due to the fact that the dynamics can be well separated into fast and slow motions. That is, the relative motion between projectile and target is much faster than the intrinsic motion of halo neutrons in the projectile. In the low energy collision, however, it becomes difficult to separate the intrinsic motion and the relative one. In spite of many experimental and theoretical efforts in the last decade, reaction mechanism of halo nuclei is still an open question at low incident energy. The sub-barrier fusion of halo nuclei is one of the interesting subjects at low energy reaction. In early stages, fusion probability was expected to be strongly enhanced due to the spatially extended density of the halo neutron and the coupling to soft modes inherent to the halo structure [6–8]. These early studies utilized a specific reaction model. Fur∗
Corresponding author, Email: [email protected] ; TEL: +81 48-467-9326 ; FAX: +81 48-462-4462
0375-9474/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2006.12.042
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thermore, recent coupled-channel calculations using a three-body model, again, indicate a substantial enhancement in the fusion probability at sub-barrier energies [9,10]. On the other hand, experimental results show a rather confusing status. In early measurements, an enhancement of the fusion cross section at sub-barrier energies was reported for reactions using a two-neutron-halo nucleus 6 He [11,12]. Recently, however, Ref. [13] has reported that the enhancement previously obtained in Ref. [12] is not due to the fusion but to the transfer-induced fission. In case of a one-neutron-halo nucleus, 11 Be, the experimental results did not present an evident conclusion either [14]. A recent comparison between 10 Be and 11 Be fusion cross sections suggests no enhancement for 11 Be [15]. Therefore, we may say, at least, that the present experimental status indicates no evidence for a fusion enhancement predicted by the previous studies [6–10]. To elucidate reaction mechanism of the halo nuclei at low energies, we have developed a time-dependent wave packet method for fully quantum calculations using a three-body model [16–21]. Our results have agreed with the fusion cross sections observed in recent experiments [13,14]. In fact, the calculation has even suggested a slight hindrance of the fusion probability because of the presence of the halo neutron. In the present report, we will present our extended analysis with the time-dependent wave packet method [20,21]. First, we will show the time evolutions of the wave packet for the three body systems with the proton and neutron halo structures. In a comparison of the proton with neutron halos, a distinctive difference is observed not only in the wave packet dynamics but also in the fusion probability. Next, we will also show the calculation for fusion cross sections of 11 Be - 209 Bi and 6 He - 238 U. Results are compared with recent experimental data [13,15]. 2. Formulation In our three-body model, the projectile is described as a weakly bound two-body system of the core and the halo nucleon(s). The projectile (P ), which is composed of the core (C) plus nucleon(s) (n), and the target (T ) constitute the three-body model. For 11 Be, we assume a 10 Be+n structure in which the halo neutron occupies a 2s orbital in the projectile ground state. For 6 He, a di-neutron model of α + 2n is assumed. The time-dependent Schr¨odinger equation for the three-body model is given as
∂ h ¯2 h ¯2 2 i¯ h Ψ(R, r, t) = − ∇2R − ∇ + VnC (r) + VCT (RCT ) + VnT (rnT ) Ψ(R, r, t), ∂t 2μ 2m r
(1)
where we denote the relative n-C coordinate as r and the relative P -T coordinate as R. The reduced masses of n-C and P -T motions are m and μ, respectively. The Hamiltonian in Eq. (1) includes interaction potentials among constituents. The nC potential, VnC (r), is a real potential. This potential should produce a halo structure of the projectile. The core-target potential, VCT (RCT ), consists of the nuclear and Coulomb potentials. The nuclear potential is complex and its short-ranged imaginary part is inside the Coulomb barrier between the core and the target nuclei. The n-T potential, VnT (rnT ), is taken to be real. We treat the constituents as spin-less point particles, and ignore the non-central forces. The nuclear potentials of VnC , VCT , and VnT are taken to be of a Woods-Saxon shape. The VnC provides the 2s bound orbital at −0.6 MeV for 11 Be, and
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−0.97 MeV for 6 He, which are close to the experimental separation energies of one and two neutrons, respectively. We define the fusion cross section in terms of the loss of flux caused by the imaginary potential in VCT . Once the core and the target nuclei are in the range of the imaginary VCT inside the Coulomb barrier, the total flux decreases by the absorption, no matter whether the neutron is captured by the target or not. Therefore, this includes not only complete but also the incomplete fusion in which the charged core (10 Be for 11 Be and 4 He for 6 He) and the target fuse while the neutron escapes. The wave-packet calculation is performed with the partial wave expansion. Here, we use the partial wave expansion in the body-fixed frame [22,23]. Although this is equivalent to the one in the space-fixed frame, the body-fixed frame is computationally superior for states at finite total angular momentum, J = 0. We adopt the model space of the radial coordinates, 0 < R < 50 fm and 0 < r < 60 fm, and discretize them with steps (ΔR, Δr) = (0.2, 1.0) fm, (ΔR, Δr) = (0.3, 0.6) fm for 11 Be and 6 He, respectively. The maximum n-C relative angular momentum is taken as lmax = 70 to obtain converged results. In the present calculation, we adopt the no-Coriolis approximation to calculate the cross sections. The discrete-variable representation is used for the second differential operator [24]. The time evolution of the wave packet is calculated employing the fourth-order Taylor-expansion method. The initial wave packet in the incident channel is a spatially localized Gaussian wave packet multiplied with an incoming wave for the P-T relative motion while the n-C wave function is a bound orbital corresponding to the halo structure. The details are given in Ref. [21]. 3. Results 3.1. Wave packet dynamics Now, let us show the typical time evolution of the wave packet solution in Eq. (1). We show the density distribution of the wave packet ρ(R, r, t) = |Ψ(R, r, t)|2 which depends on two relative vectors, R and r. In the central collision (J = 0), it depends only on the magnitude of individual vectors (R, r) and a relative angle θ between R and r and hence, ρJ=0 (R, r, t) = ρJ=0 (R, r, θ, t). In this report, we visualize the two dimensional density which is obtained by integrating one of the degree of freedom. In Fig.1, we show the time evolution of the density distribution ρ(R, r, t) which is obtained by integrating θ degree of freedom. In Fig. 1, the left-most panel shows the initial wave packet and it starts to move after the time evolution is switched on. The wave packet spreads over a wide spatial region along the direction of r, which indicates the breakup component of the valence neutron. In the right-most two panels, furthermore, we can clearly see some enhancements in the amplitude along the diagonal line R ∼ r. This represents the transfer component, because r is almost equal to R when the valence neutron is transferred from the projectile to the target. Therefore, we can clearly see both the breakup and the transfer components of the valence neutron in the wave packet dynamics. To visualize the dynamics of halo nucleon during the reaction from a different viewpoint, we show the density distribution of the wave packet integrated over R, ρ(r, θ, t). In Fig. 2
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Figure 1. Time-dependent density distribution of the three-body system, ρ(R, r, t), for the central collision, J = 0. The ordinate and abscissa denote the R and r, respectively. the density ρ(r, θ, t) is plotted in the two-dimensional x-y plane with x = r cos θ and y = r sin θ. The direction of the x axis is taken to be parallel to the relative coordinate R and hence, the target collides with the projectile from the right side of the figure in the time evolution. Since we assume the 2s-orbital for the relative motion between the core and the neutron, the density ρ(r, θ, t) is spherical in the initial state shown in the left-most panel in Fig. 2. As the projectile and the target begin to collide, the distribution flows to the positive x-direction. This means that the halo neutron escapes toward the forward (target side) direction .
Figure 2. Time-dependent density distribution of the halo neutron, ρ(r, θ, t), for the central collision. The ordinate and the abscissa denote the x = r cos θ and the y = r sin θ, respectively.
To obtain deeper understanding on the breakup mechanism, we artificially construct the 11 Be nucleus with the “proton halo”, 11 Be=10 Li+p. We modified the VnC so as to reproduce the proton binding energy of −0.6 MeV and performed the similar calculation of the time evolution. In Fig. 3, we show the time evolution of the density distribution ρ(r, θ, t) plotted in the x-y plane. In this figure, we can clearly see different behavior from the neutron halo case. The proton is mainly scattered to the opposite direction to the target. The breakup towards backward direction is a dominant process in the proton halo case. These behaviors may be understood as the Coulomb breakup process. We first consider the neutron halo case. At low incident energy, the core nucleus is scattered to the backward direction by the Coulomb repulsion between the core and the target nuclei. On the other hand, the halo neutron is scattered only by the attractive VnT interaction. Therefore,
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the projectile is dissociated by the VCT Coulomb field and the halo neutron goes to the forward direction.
Figure 3. The same as Fig. 2 but for the case of the proton halo. See text for details.
In contrast to the neutron halo case, the halo proton as well as the core nucleus are scattered to the backward direction by the target Coulomb field. Since the charge-mass ratio of the proton is larger than that of the core nucleus., the acceleration of the proton is stronger than that of the core. Therefore, in the center-of-mass frame of the projectile, the proton is scattered to the backward direction. 3.2. Fusion probability and lmax convergence A single wave packet solution contains information for a certain range of energy. We extract the fusion probability for a fixed incident energy by the energy projection procedure shown in Refs. [17–20]. The calculated fusion probabilities for the central collision are shown in Fig. 4. The fusion probability (solid line) is suppressed by the presence of the halo neutron in comparison with that without the halo neutron (dotted line). On the other hand, for the proton-halo case, we find the strong enhancement in the fusion probability (dashed line). In order to understand the difference between the neutron halo and the proton one, we should look carefully at the time evolution of the density distributions shown in Figs. 2 and 3. When the core nucleus is decelerated by the target Coulomb field, the halo neutron keeps their incident velocity, leaving the core nucleus. This process corresponds to the Coulomb breakup of the projectile in which the neutron is dominantly scattered to the forward direction as shown in Fig. 2. After the breakup, the fusion takes place between two charged particles, the core and the target nuclei. That is, the halo neutrons behave like a spectator. Based on this picture, we propose a mechanism that may explain the reason for the slight suppression of the fusion probability. The incident energy of the projectile is initially carried by the core and the halo neutron. After the Coulomb breakup, since the dissociated neutrons, as a spectator, keep their shared energy, the energy of the core nucleus becomes smaller than the total incident energy of the projectile. This practical decrease in the incident energy makes the fusion probability suppressed for a neutron-halo nucleus in comparison with the nucleus without halo neutrons. For the proton-halo case, the halo proton is scattered to the backward direction, as is seen in Fig. 3. The charge number of the core nucleus is smaller than that of the projectile
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after the breakup. Therefore, the Coulomb barrier height becomes lower between the core and the target nuclei and hence, the core can easily penetrate the Coulomb barrier. We thus conclude that the essential features of the low-energy fusion reactions of halo nuclei may be determined by the Coulomb breakup process.
1.2
1.2 Proton halo
Fusion Probability
Fusion Probability
lmax = 4 0.8 Two Body
0.4
Neutron halo
0.8 lmax = 10 0.4
10
Be +
209
Bi
lmax = 30, 50
J=0 0 32
36 40 Ec.m. ( MeV )
44
Figure 4. Fusion probability as a function of the incident energy (J = 0). Solid, dashed, and dotted lines indicate, respectively, those for neutron-halo, protonhalo, and the core nuclei.
0 30
35 40 Ec.m. ( MeV )
45
Figure 5. Fusion probability of 11 Be - 209 Bi with different lmax J = 0. For reference, the fusion probability without a halo neutron is plotted by a thin-solid curve.
Our results of the neutron-halo nuclei differ from those shown in Refs. [9,10] that report the strong enhancement of the sub-barrier fusion probability. This is simply because the model space in Refs. [9,10] is too small and not fully converged. In Fig. 5, we show the fusion probability for different cutoff values of lmax . If we adopt a small lmax value, a strong enhancement of the sub-barrier fusion probability is obtained in our three-body calculations. This is consistent to results of the CDCC calculation [9,10]. However, increasing lmax , the fusion probability gradually decreases, and eventually becomes smaller than the values without a halo neutron (thin solid line). The result of lmax = 30 cannot be distinguished from that of lmax = 50. To get a converged result, therefore, one needs to adopt lmax ≥ 30. 3.3. Fusion cross section Extending the calculations to finite J up to J = 30, we achieve the total fusion cross section. We show calculated fusion cross sections of 10,11 Be - 209 Bi in Fig. 6. The VCT for 10 Be - 209 Bi is constructed so that the fusion cross section of this system is reproduced by a simple potential picture. Adding a halo neutron, the fusion cross section decreases slightly for an entire energy region. This is because the suppression of the fusion probability observed in the J = 0 also appears in finite J. The measurement indicates that the fusion cross sections of 11 Be do not differ from those
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of 10 Be within experimental errors. At this stage, it is difficult to judge whether or not there is a slight suppression in the experimental fusion cross section. In our calculation, a part of the halo neutron proceeds to the forward direction keeping the incident velocity. Therefore, the observation of these forward neutrons in the incomplete fusion process may support our picture. In Ref.[15], the forward neutron emission accompanying the fusion is discussed.
10
Be +
209
4
He +
Bi
Fusion cross section ( mb )
Fusion cross section ( mb )
3
10
2
10
11
Be +
209
10
U
2 6
10
Bi
238
He +
238
U
0
1
10
36
40 44 Ec.m. ( MeV )
48
Figure 6. Fusion cross section of 11 Be - 209 Bi (solid line) and 10 Be - 209 Bi (dotted line). Measured cross sections [15] are also shown: 11 Be by filled squares and 10 Be by open circles.
12
16
20 24 Ec.m. ( MeV )
28
32
Figure 7. The same as Fig. 6 but for 6 He - 238 U. Experimental data are taken from Ref. [13], shown by open circles (4 He) and by filled circles (6 He). Two data points with large error bars at Ec.m. ≤ 18 MeV indicate the data for 6 He.
We next consider the fusion cross section of 6 He - 238 U. In Fig. 7, we show the calculated fusion cross sections of 6 He and 4 He on 238 U. The experimental data in Ref.[13] indicate, again, that there is no difference between 6 He and 4 He fusion cross section within a statistical error. Our calculation indicates that the cross section for 6 He is suppressed from the one for 4 He at energies above the barrier. The magnitude of suppression in 6 He is larger than that in 11 Be at high energies. On the other hand, at sub-barrier energies, the calculated fusion cross sections of 4 He and 6 He are almost identical to each other. We again need to wait for measurements with high statistics to make a definite conclusion whether our calculation agrees with the experimental data. 4. Summary In summary, we calculated the fusion cross section for nuclei with halo structure. The reaction is described as the three-body model, and the Schr¨odinger equation is solved exactly with the time-dependent wave-packet method. The reaction process was investigated for the proton halo system as well as the neutron halo one and we pointed out a distinctive difference between them. From the visualization of the wave packet, we found
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that the forward and backward scatterings were dominant process for the neutron and proton halos, respectively. Furthermore, there appears a large difference in the fusion probabilities : the suppression for the neutron halo and the enhancement for the proton halo. These difference was discussed in connection to the Coulomb breakup induced by the independent scattering of the core and halo nucleon. We then compared calculated cross sections with measurements for 11,10 Be - 209 Bi and 6,4 He - 238 U collisions. Recent measurements of these systems suggest that there is no evidence for an enhancement in the fusion cross section of halo nuclei. Our calculation indicates that the fusion cross section is even suppressed by adding a halo neutron. Further measurements with increased statistics are desirable to obtain definite conclusion whether the fusion cross section is increased or suppressed by the presence of the halo neutron. This work is supported in part by the Grant-in-Aid for Scientific Research in Japan (Nos. 17540231, 17740160, and 18740129). The numerical calculations were performed at SIPC, University of Tsukuba, at YITP, Kyoto University, and at RCNP, Osaka University. REFERENCES 1. I. Tanihata et al., Phys. Rev. Lett. 55, 2676 (1985). 2. Y. Suzuki, R.G. Lovas, K. Yabana, and K. Varga, Structure and Reactions of Light Exotic Nuclei, Taylor and Francis, 2003. 3. G.F. Bertsch, B.A. Brown, and H. Sagawa, Phys. Rev. C39, 1154 (1989). 4. Y. Ogawa, K. Yabana, and Y. Suzuki, Nucl. Phys. A543, 722 (1992). 5. K. Yabana, Y. Ogawa, and Y. Suzuki, Nucl. Phys. A539, 295 (1992);Phys. Rev. C45, 2909 (1992). 6. N. Takigawa and H. Sagawa, Phys. Lett. B265, 23 (1991). 7. M.S. Hussein et al., Phys. Rev. C46, 377 (1992). 8. N. Takigawa, M. Kuratani, and H. Sagawa, Phys. Rev. C 47, R2470 (1993). 9. K. Hagino, A. Vitturi, C.H. Dasso, and S.M. Lenzi, Phys. Rev. C61, 037602 (2000). 10. A. Diaz-Torres and I.J. Thompson, Phys. Rev. C65, 024606 (2002). 11. J.J. Kolata et al., Phys. Rev. Lett. 81, 4580 (1998). 12. M. Trotta et al., Phys. Rev. Lett. 84, 2342 (2000). 13. R. Raabe et al., Nature 431, (2004) 823. 14. C. Signorini et al., Eur. Phys. J. A 2, 227 (1998); 5, 7 (1999). 15. C. Signorini et al., Nucl. Phys. A735, 329 (2004). 16. K. Yabana and Y. Suzuki, Nucl. Phys. A588, 99c (1995). 17. K. Yabana, Prog. Theor. Phys. 97, 437 (1997). 18. K. Yabana, M. Ueda, T. Nakatsukasa, Nucl. Phys. A722, 261c (2003). 19. K. Yabana, M. Ito, M. Kobayashi, M. Ueda, T. Nakatsukasa, Nucl. Phys. A738, 303 (2004). 20. T. Nakatsukasa, K. Yabana, M. Ito, M. Kobayashi, M. Ueda, Prog. Theor. Phys. Suppl. 154, 85 (2004). 21. M. Ito, K. Yabana, T. Nakatsukasa, M. Ueda, Phys. Lett. B637, 53 (2006). 22. T.T. Pack, J. Chem. Phys. 60, 633 (1974). 23. B. Imanishi and W. von Oertzen, Phys. Rep. 155, (1987) 29. 24. D.T. Colbert and W.H. Miller, J. Chem. Phys. 96, 1982 (1992).
Fusion reaction of halo nuclei : A real-time wave-packet method for three-body tunneling dynamics M. Itoa ∗ , K. Yabanab , T. Nakatsukasab , M. Uedac a
RIKEN, Hirosawa 2-1, Wako, Saitama 351-0198, Japan
b
Institute of Physics and Center for Computational Sciences, University of Tsukuba, Tsukuba 305-8571, Japan c
Akita National College of Technology, Akita 011-8511, Japan
Fusion reactions of halo nuclei are investigated theoretically with a three-body model. The time-dependent wave-packet method is used to solve the three-body Schr¨odinger equation. Dynamics of the halo nucleons are visualized in the coordinate space. Significant differences between the neutron-halo and the proton-halo systems are discussed in connection to the Coulomb breakup process. The fusion cross sections are calculated for 11 Be - 209 Bi and 6 He - 238 U. Our calculation indicates that the fusion cross section is slightly hindered by the presence of weakly bound neutrons. The present results nicely agree with recent measurements. 1. Introduction Physics at the drip-line is one of current subjects in nuclear physics. In particular, nuclear reactions involving halo nuclei have been attracting many researchers since the discovery of halo structure in the secondary beam experiment [1]. In the studies on reaction mechanism of halo nuclei, developments of the reaction theories capable of describing coupling to the continuum and transfer channels are quite important because of its weakly binding properties. For reactions in medium and high incident energies, the Glauber and the eikonal theories have been successful [2–5]. This is due to the fact that the dynamics can be well separated into fast and slow motions. That is, the relative motion between projectile and target is much faster than the intrinsic motion of halo neutrons in the projectile. In the low energy collision, however, it becomes difficult to separate the intrinsic motion and the relative one. In spite of many experimental and theoretical efforts in the last decade, reaction mechanism of halo nuclei is still an open question at low incident energy. The sub-barrier fusion of halo nuclei is one of the interesting subjects at low energy reaction. In early stages, fusion probability was expected to be strongly enhanced due to the spatially extended density of the halo neutron and the coupling to soft modes inherent to the halo structure [6–8]. These early studies utilized a specific reaction model. Fur∗
Corresponding author, Email: [email protected] ; TEL: +81 48-467-9326 ; FAX: +81 48-462-4462
0375-9474/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2006.12.042
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M. Ito et al. / Nuclear Physics A 787 (2007) 267c–274c
thermore, recent coupled-channel calculations using a three-body model, again, indicate a substantial enhancement in the fusion probability at sub-barrier energies [9,10]. On the other hand, experimental results show a rather confusing status. In early measurements, an enhancement of the fusion cross section at sub-barrier energies was reported for reactions using a two-neutron-halo nucleus 6 He [11,12]. Recently, however, Ref. [13] has reported that the enhancement previously obtained in Ref. [12] is not due to the fusion but to the transfer-induced fission. In case of a one-neutron-halo nucleus, 11 Be, the experimental results did not present an evident conclusion either [14]. A recent comparison between 10 Be and 11 Be fusion cross sections suggests no enhancement for 11 Be [15]. Therefore, we may say, at least, that the present experimental status indicates no evidence for a fusion enhancement predicted by the previous studies [6–10]. To elucidate reaction mechanism of the halo nuclei at low energies, we have developed a time-dependent wave packet method for fully quantum calculations using a three-body model [16–21]. Our results have agreed with the fusion cross sections observed in recent experiments [13,14]. In fact, the calculation has even suggested a slight hindrance of the fusion probability because of the presence of the halo neutron. In the present report, we will present our extended analysis with the time-dependent wave packet method [20,21]. First, we will show the time evolutions of the wave packet for the three body systems with the proton and neutron halo structures. In a comparison of the proton with neutron halos, a distinctive difference is observed not only in the wave packet dynamics but also in the fusion probability. Next, we will also show the calculation for fusion cross sections of 11 Be - 209 Bi and 6 He - 238 U. Results are compared with recent experimental data [13,15]. 2. Formulation In our three-body model, the projectile is described as a weakly bound two-body system of the core and the halo nucleon(s). The projectile (P ), which is composed of the core (C) plus nucleon(s) (n), and the target (T ) constitute the three-body model. For 11 Be, we assume a 10 Be+n structure in which the halo neutron occupies a 2s orbital in the projectile ground state. For 6 He, a di-neutron model of α + 2n is assumed. The time-dependent Schr¨odinger equation for the three-body model is given as
∂ h ¯2 h ¯2 2 i¯ h Ψ(R, r, t) = − ∇2R − ∇ + VnC (r) + VCT (RCT ) + VnT (rnT ) Ψ(R, r, t), ∂t 2μ 2m r
(1)
where we denote the relative n-C coordinate as r and the relative P -T coordinate as R. The reduced masses of n-C and P -T motions are m and μ, respectively. The Hamiltonian in Eq. (1) includes interaction potentials among constituents. The nC potential, VnC (r), is a real potential. This potential should produce a halo structure of the projectile. The core-target potential, VCT (RCT ), consists of the nuclear and Coulomb potentials. The nuclear potential is complex and its short-ranged imaginary part is inside the Coulomb barrier between the core and the target nuclei. The n-T potential, VnT (rnT ), is taken to be real. We treat the constituents as spin-less point particles, and ignore the non-central forces. The nuclear potentials of VnC , VCT , and VnT are taken to be of a Woods-Saxon shape. The VnC provides the 2s bound orbital at −0.6 MeV for 11 Be, and
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−0.97 MeV for 6 He, which are close to the experimental separation energies of one and two neutrons, respectively. We define the fusion cross section in terms of the loss of flux caused by the imaginary potential in VCT . Once the core and the target nuclei are in the range of the imaginary VCT inside the Coulomb barrier, the total flux decreases by the absorption, no matter whether the neutron is captured by the target or not. Therefore, this includes not only complete but also the incomplete fusion in which the charged core (10 Be for 11 Be and 4 He for 6 He) and the target fuse while the neutron escapes. The wave-packet calculation is performed with the partial wave expansion. Here, we use the partial wave expansion in the body-fixed frame [22,23]. Although this is equivalent to the one in the space-fixed frame, the body-fixed frame is computationally superior for states at finite total angular momentum, J = 0. We adopt the model space of the radial coordinates, 0 < R < 50 fm and 0 < r < 60 fm, and discretize them with steps (ΔR, Δr) = (0.2, 1.0) fm, (ΔR, Δr) = (0.3, 0.6) fm for 11 Be and 6 He, respectively. The maximum n-C relative angular momentum is taken as lmax = 70 to obtain converged results. In the present calculation, we adopt the no-Coriolis approximation to calculate the cross sections. The discrete-variable representation is used for the second differential operator [24]. The time evolution of the wave packet is calculated employing the fourth-order Taylor-expansion method. The initial wave packet in the incident channel is a spatially localized Gaussian wave packet multiplied with an incoming wave for the P-T relative motion while the n-C wave function is a bound orbital corresponding to the halo structure. The details are given in Ref. [21]. 3. Results 3.1. Wave packet dynamics Now, let us show the typical time evolution of the wave packet solution in Eq. (1). We show the density distribution of the wave packet ρ(R, r, t) = |Ψ(R, r, t)|2 which depends on two relative vectors, R and r. In the central collision (J = 0), it depends only on the magnitude of individual vectors (R, r) and a relative angle θ between R and r and hence, ρJ=0 (R, r, t) = ρJ=0 (R, r, θ, t). In this report, we visualize the two dimensional density which is obtained by integrating one of the degree of freedom. In Fig.1, we show the time evolution of the density distribution ρ(R, r, t) which is obtained by integrating θ degree of freedom. In Fig. 1, the left-most panel shows the initial wave packet and it starts to move after the time evolution is switched on. The wave packet spreads over a wide spatial region along the direction of r, which indicates the breakup component of the valence neutron. In the right-most two panels, furthermore, we can clearly see some enhancements in the amplitude along the diagonal line R ∼ r. This represents the transfer component, because r is almost equal to R when the valence neutron is transferred from the projectile to the target. Therefore, we can clearly see both the breakup and the transfer components of the valence neutron in the wave packet dynamics. To visualize the dynamics of halo nucleon during the reaction from a different viewpoint, we show the density distribution of the wave packet integrated over R, ρ(r, θ, t). In Fig. 2
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Figure 1. Time-dependent density distribution of the three-body system, ρ(R, r, t), for the central collision, J = 0. The ordinate and abscissa denote the R and r, respectively. the density ρ(r, θ, t) is plotted in the two-dimensional x-y plane with x = r cos θ and y = r sin θ. The direction of the x axis is taken to be parallel to the relative coordinate R and hence, the target collides with the projectile from the right side of the figure in the time evolution. Since we assume the 2s-orbital for the relative motion between the core and the neutron, the density ρ(r, θ, t) is spherical in the initial state shown in the left-most panel in Fig. 2. As the projectile and the target begin to collide, the distribution flows to the positive x-direction. This means that the halo neutron escapes toward the forward (target side) direction .
Figure 2. Time-dependent density distribution of the halo neutron, ρ(r, θ, t), for the central collision. The ordinate and the abscissa denote the x = r cos θ and the y = r sin θ, respectively.
To obtain deeper understanding on the breakup mechanism, we artificially construct the 11 Be nucleus with the “proton halo”, 11 Be=10 Li+p. We modified the VnC so as to reproduce the proton binding energy of −0.6 MeV and performed the similar calculation of the time evolution. In Fig. 3, we show the time evolution of the density distribution ρ(r, θ, t) plotted in the x-y plane. In this figure, we can clearly see different behavior from the neutron halo case. The proton is mainly scattered to the opposite direction to the target. The breakup towards backward direction is a dominant process in the proton halo case. These behaviors may be understood as the Coulomb breakup process. We first consider the neutron halo case. At low incident energy, the core nucleus is scattered to the backward direction by the Coulomb repulsion between the core and the target nuclei. On the other hand, the halo neutron is scattered only by the attractive VnT interaction. Therefore,
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the projectile is dissociated by the VCT Coulomb field and the halo neutron goes to the forward direction.
Figure 3. The same as Fig. 2 but for the case of the proton halo. See text for details.
In contrast to the neutron halo case, the halo proton as well as the core nucleus are scattered to the backward direction by the target Coulomb field. Since the charge-mass ratio of the proton is larger than that of the core nucleus., the acceleration of the proton is stronger than that of the core. Therefore, in the center-of-mass frame of the projectile, the proton is scattered to the backward direction. 3.2. Fusion probability and lmax convergence A single wave packet solution contains information for a certain range of energy. We extract the fusion probability for a fixed incident energy by the energy projection procedure shown in Refs. [17–20]. The calculated fusion probabilities for the central collision are shown in Fig. 4. The fusion probability (solid line) is suppressed by the presence of the halo neutron in comparison with that without the halo neutron (dotted line). On the other hand, for the proton-halo case, we find the strong enhancement in the fusion probability (dashed line). In order to understand the difference between the neutron halo and the proton one, we should look carefully at the time evolution of the density distributions shown in Figs. 2 and 3. When the core nucleus is decelerated by the target Coulomb field, the halo neutron keeps their incident velocity, leaving the core nucleus. This process corresponds to the Coulomb breakup of the projectile in which the neutron is dominantly scattered to the forward direction as shown in Fig. 2. After the breakup, the fusion takes place between two charged particles, the core and the target nuclei. That is, the halo neutrons behave like a spectator. Based on this picture, we propose a mechanism that may explain the reason for the slight suppression of the fusion probability. The incident energy of the projectile is initially carried by the core and the halo neutron. After the Coulomb breakup, since the dissociated neutrons, as a spectator, keep their shared energy, the energy of the core nucleus becomes smaller than the total incident energy of the projectile. This practical decrease in the incident energy makes the fusion probability suppressed for a neutron-halo nucleus in comparison with the nucleus without halo neutrons. For the proton-halo case, the halo proton is scattered to the backward direction, as is seen in Fig. 3. The charge number of the core nucleus is smaller than that of the projectile
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after the breakup. Therefore, the Coulomb barrier height becomes lower between the core and the target nuclei and hence, the core can easily penetrate the Coulomb barrier. We thus conclude that the essential features of the low-energy fusion reactions of halo nuclei may be determined by the Coulomb breakup process.
1.2
1.2 Proton halo
Fusion Probability
Fusion Probability
lmax = 4 0.8 Two Body
0.4
Neutron halo
0.8 lmax = 10 0.4
10
Be +
209
Bi
lmax = 30, 50
J=0 0 32
36 40 Ec.m. ( MeV )
44
Figure 4. Fusion probability as a function of the incident energy (J = 0). Solid, dashed, and dotted lines indicate, respectively, those for neutron-halo, protonhalo, and the core nuclei.
0 30
35 40 Ec.m. ( MeV )
45
Figure 5. Fusion probability of 11 Be - 209 Bi with different lmax J = 0. For reference, the fusion probability without a halo neutron is plotted by a thin-solid curve.
Our results of the neutron-halo nuclei differ from those shown in Refs. [9,10] that report the strong enhancement of the sub-barrier fusion probability. This is simply because the model space in Refs. [9,10] is too small and not fully converged. In Fig. 5, we show the fusion probability for different cutoff values of lmax . If we adopt a small lmax value, a strong enhancement of the sub-barrier fusion probability is obtained in our three-body calculations. This is consistent to results of the CDCC calculation [9,10]. However, increasing lmax , the fusion probability gradually decreases, and eventually becomes smaller than the values without a halo neutron (thin solid line). The result of lmax = 30 cannot be distinguished from that of lmax = 50. To get a converged result, therefore, one needs to adopt lmax ≥ 30. 3.3. Fusion cross section Extending the calculations to finite J up to J = 30, we achieve the total fusion cross section. We show calculated fusion cross sections of 10,11 Be - 209 Bi in Fig. 6. The VCT for 10 Be - 209 Bi is constructed so that the fusion cross section of this system is reproduced by a simple potential picture. Adding a halo neutron, the fusion cross section decreases slightly for an entire energy region. This is because the suppression of the fusion probability observed in the J = 0 also appears in finite J. The measurement indicates that the fusion cross sections of 11 Be do not differ from those
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of 10 Be within experimental errors. At this stage, it is difficult to judge whether or not there is a slight suppression in the experimental fusion cross section. In our calculation, a part of the halo neutron proceeds to the forward direction keeping the incident velocity. Therefore, the observation of these forward neutrons in the incomplete fusion process may support our picture. In Ref.[15], the forward neutron emission accompanying the fusion is discussed.
10
Be +
209
4
He +
Bi
Fusion cross section ( mb )
Fusion cross section ( mb )
3
10
2
10
11
Be +
209
10
U
2 6
10
Bi
238
He +
238
U
0
1
10
36
40 44 Ec.m. ( MeV )
48
Figure 6. Fusion cross section of 11 Be - 209 Bi (solid line) and 10 Be - 209 Bi (dotted line). Measured cross sections [15] are also shown: 11 Be by filled squares and 10 Be by open circles.
12
16
20 24 Ec.m. ( MeV )
28
32
Figure 7. The same as Fig. 6 but for 6 He - 238 U. Experimental data are taken from Ref. [13], shown by open circles (4 He) and by filled circles (6 He). Two data points with large error bars at Ec.m. ≤ 18 MeV indicate the data for 6 He.
We next consider the fusion cross section of 6 He - 238 U. In Fig. 7, we show the calculated fusion cross sections of 6 He and 4 He on 238 U. The experimental data in Ref.[13] indicate, again, that there is no difference between 6 He and 4 He fusion cross section within a statistical error. Our calculation indicates that the cross section for 6 He is suppressed from the one for 4 He at energies above the barrier. The magnitude of suppression in 6 He is larger than that in 11 Be at high energies. On the other hand, at sub-barrier energies, the calculated fusion cross sections of 4 He and 6 He are almost identical to each other. We again need to wait for measurements with high statistics to make a definite conclusion whether our calculation agrees with the experimental data. 4. Summary In summary, we calculated the fusion cross section for nuclei with halo structure. The reaction is described as the three-body model, and the Schr¨odinger equation is solved exactly with the time-dependent wave-packet method. The reaction process was investigated for the proton halo system as well as the neutron halo one and we pointed out a distinctive difference between them. From the visualization of the wave packet, we found
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that the forward and backward scatterings were dominant process for the neutron and proton halos, respectively. Furthermore, there appears a large difference in the fusion probabilities : the suppression for the neutron halo and the enhancement for the proton halo. These difference was discussed in connection to the Coulomb breakup induced by the independent scattering of the core and halo nucleon. We then compared calculated cross sections with measurements for 11,10 Be - 209 Bi and 6,4 He - 238 U collisions. Recent measurements of these systems suggest that there is no evidence for an enhancement in the fusion cross section of halo nuclei. Our calculation indicates that the fusion cross section is even suppressed by adding a halo neutron. Further measurements with increased statistics are desirable to obtain definite conclusion whether the fusion cross section is increased or suppressed by the presence of the halo neutron. This work is supported in part by the Grant-in-Aid for Scientific Research in Japan (Nos. 17540231, 17740160, and 18740129). The numerical calculations were performed at SIPC, University of Tsukuba, at YITP, Kyoto University, and at RCNP, Osaka University. REFERENCES 1. I. Tanihata et al., Phys. Rev. Lett. 55, 2676 (1985). 2. Y. Suzuki, R.G. Lovas, K. Yabana, and K. Varga, Structure and Reactions of Light Exotic Nuclei, Taylor and Francis, 2003. 3. G.F. Bertsch, B.A. Brown, and H. Sagawa, Phys. Rev. C39, 1154 (1989). 4. Y. Ogawa, K. Yabana, and Y. Suzuki, Nucl. Phys. A543, 722 (1992). 5. K. Yabana, Y. Ogawa, and Y. Suzuki, Nucl. Phys. A539, 295 (1992);Phys. Rev. C45, 2909 (1992). 6. N. Takigawa and H. Sagawa, Phys. Lett. B265, 23 (1991). 7. M.S. Hussein et al., Phys. Rev. C46, 377 (1992). 8. N. Takigawa, M. Kuratani, and H. Sagawa, Phys. Rev. C 47, R2470 (1993). 9. K. Hagino, A. Vitturi, C.H. Dasso, and S.M. Lenzi, Phys. Rev. C61, 037602 (2000). 10. A. Diaz-Torres and I.J. Thompson, Phys. Rev. C65, 024606 (2002). 11. J.J. Kolata et al., Phys. Rev. Lett. 81, 4580 (1998). 12. M. Trotta et al., Phys. Rev. Lett. 84, 2342 (2000). 13. R. Raabe et al., Nature 431, (2004) 823. 14. C. Signorini et al., Eur. Phys. J. A 2, 227 (1998); 5, 7 (1999). 15. C. Signorini et al., Nucl. Phys. A735, 329 (2004). 16. K. Yabana and Y. Suzuki, Nucl. Phys. A588, 99c (1995). 17. K. Yabana, Prog. Theor. Phys. 97, 437 (1997). 18. K. Yabana, M. Ueda, T. Nakatsukasa, Nucl. Phys. A722, 261c (2003). 19. K. Yabana, M. Ito, M. Kobayashi, M. Ueda, T. Nakatsukasa, Nucl. Phys. A738, 303 (2004). 20. T. Nakatsukasa, K. Yabana, M. Ito, M. Kobayashi, M. Ueda, Prog. Theor. Phys. Suppl. 154, 85 (2004). 21. M. Ito, K. Yabana, T. Nakatsukasa, M. Ueda, Phys. Lett. B637, 53 (2006). 22. T.T. Pack, J. Chem. Phys. 60, 633 (1974). 23. B. Imanishi and W. von Oertzen, Phys. Rep. 155, (1987) 29. 24. D.T. Colbert and W.H. Miller, J. Chem. Phys. 96, 1982 (1992).