Breakup dynamics in 6Li elastic scattering with four-body and three-body CDCC

We have investigated projectile breakup effects on 6Li + 209Bi elastic scattering near the Coulomb barrier with the four-body and three-body continuum-discretized coupled- channels methods. In the analysis, the elastic scattering is well described by the p + n + 4He + 209Bi four-body model. Four-body dynamics of the elastic scattering is precisely investigated and we then found that d (p + n) in 6Li may hardly break up during the scattering.


Introduction
In reactions of weakly-bound nuclei, projectile breakup processes are essential and we need to treat the effect properly in the calculation. The continuum-discretized coupled-channels method (CDCC) has been proposed as an accurate method of treating breakup effects [1,2]. Nowadays, CDCC has become a standard model known as three-body CDCC in which the total system is assumed to be a three-body system (two-body projectile + target). Furthermore, three-body CDCC has been extended to four-body CDCC in which the total system is assumed to be fourbody system (three-body projectile + target) [3]. Thus, CDCC is a powerful method to describe not only three-body scattering [4] but also four-body scattering [5]. The recent developments of CDCC are shown in Ref. [6,7,8]. 6 He + 209 Bi scattering near the Coulomb barrier was analyzed with three-body CDCC based on a 2 n + 4 He + 209 Bi three-body model [9]; that is to say a pair of extra neutrons in 6 He was treated as a single particle, dineutron ( 2 n). The three-body CDCC calculation, however, did not reproduce the angular distribution of the measured elastic cross section and overestimated the measured total reaction cross section by a factor of 2.5. This problem has been solved by four-body CDCC in which the total system is described as a n + n + 4 He + 209 Bi four-body system [3]. Also for 6 Li + 209 Bi scattering near the Coulomb barrier, three-body CDCC was applied [9]. In the three-body CDCC, a d + 4 He + 209 Bi three-body model was assumed. However, the calculation could not reproduce the data without normalization factors for the potential between 6 Li and 209 Bi. These studies [3,9] strongly suggest that 6 Li + 209 Bi scattering should also be treated with four-body CDCC.
In this work, we analyze 6 Li + 209 Bi elastic scattering at 29.9 and 32.8 MeV with four-body CDCC by assuming the p + n + 4 He + 209 Bi four-body model. This is the first application of four-body CDCC to 6 Li scattering. We deal with four-body dynamics explicitly and compare with the results of three-body CDCC. This poster presentation is mainly based on our recent work of Ref. [10].
for the total wave function Ψ, where E is a total energy of the system. In order to clarify the difference between four-body (p + n + 4 He + 209 Bi) and three-body (d + 4 He + 209 Bi) dynamics, we set two types of the model Hamiltonian. One is for four-body CDCC defined by where h denotes the internal Hamiltonian of 6 Li described by three-cluster model, R is the centerof-mass coordinate of 6 Li relative to 209 Bi, K R stands for the kinetic energy and U x describes the nuclear part of the optical potential between x and 209 Bi as a function of the relative coordinate R x . Since the Coulomb breakup effect is negligible for the 6 Li elastic scattering [9], we approximate the Coulomb part of p-209 Bi and 4 He-209 Bi interactions by e 2 Z Li Z Bi /R, where Z A is the atomic number of the nucleus A. The other is for three-body CDCC defined by where h denotes the internal Hamiltonian of 6 Li described by two-cluster model. In general, the total wave function Ψ is expanded in terms of the orthonormal set of eigenstates where ξ is the Jacobi coordinate in 6 Li, subscripts 0 and ε denote the ground state and the continuum state with the internal energy ε of 6 Li respectively. The expansion coefficient χ 0 (χ ε ) describes the relative motion between 209 Bi and 6 Li in its ground state (continuum state with ε). In CDCC, the continuum state is truncated at an upper limit ε max , and the continuum up to ε max is discretized into a finite number of discrete states Readers are directed to Ref. [10] for more details. In Eqs. (2) and (3), the optical potentials (U n , U p , U α , and U d ) are taken from Refs. [11,12,13]. The proton optical potential U p is assumed to be the same as U n , and parameters of U n are refitted to reproduce experimental data on n + 209 Bi elastic scattering [14], where only the central interaction is taken for simplicity. We confirmed that these optical potentials well reproduce the experimental data for each subsystem of 6 Li + 209 Bi as shown in Fig. 1 [(a) Figure 2 shows the angular distribution of elastic cross section for 6 Li + 209 Bi scattering at 29.9 MeV and at 32. 8 MeV. The dotted line shows the result of three-body CDCC calculation with the d-optical potential U OP d . This calculation is similar to the one in the previous study [9] and underestimates the measured cross section. The solid (dashed) line, meanwhile, stands for  The experimental data are taken from Refs. [14,12,13].

Results
the result of four-body CDCC calculation with (without) projectile breakup effects. In CDCC calculations without 6 Li-breakup, the model space is composed only of the 6 Li ground state (φ 0 ). The solid line reproduces the measured cross section but the dashed line does not, indicating the projectile breakup effects are thus significant. As just described, the present 6 Li scattering is well described by the p + n + 4 He + 209 Bi four-body model. In order to understand breakup dynamics in the 6 Li scattering, we investigate how d breakup affects the cross section in the framework of three-body CDCC. As a calculation in the limit of no d-breakup effect, the optical potential between d and 209 Bi (U OP d ) should be replaced by the single-folding potential (U SF d ) which is obtained by folding U n and U p with the ground-state deuteron density. Note that we use the same U n and U p as for four-body CDCC (see Eq. 2). In Fig. 2, the dot-dashed line shows the result of the three-body CDCC calculation with U SF d . The result well simulates that of four-body CDCC calculation (the solid line). This result suggests d breakup is suppressed in the 6 Li scattering.
Finally, we would like to mention about d + 209 Bi scattering at 12.8 MeV which is the same reaction as in Fig. 1(c). For this scattering, we can apply three-body CDCC in which the p + n + 209 Bi model is assumed and both Coulomb and nuclear breakup effects are taken into account. The results are as follows. The CDCC calculation with breakup effects reproduces the experimental data but the CDCC calculation without breakup (one-channel calculation with U SF d ) does not. d-breakup effect is thus quite important for "normal" d scattering. As already mentioned, the one-channel calculation with U OP d well reproduces the data [see Fig. 1 [15,16]. effect which is almost absent in d in 6 Li scattering. However, we need to discuss carefully whether the d breakup is always suppressed in the 6 Li scattering. Further analysis such as energy and target dependences of d breakup should be done.