Four- and three-body channel coupling effects on 6Li elastic scattering with CDCC

We investigate breakup dynamics in 6Li elastic scattering on heavy targets (T = 209Bi or 208Pb) near the Coulomb barrier energy. Since the subsystem of 6Li has a bound state as deuteron (n + p = d), a four-body channel (6Li + T → n + p + α + T) and a three-body channel (6Li + T → d + α + T) get entangled during scattering as a breakup channel. Both channels are precisely treated with the four-body version of the continuum-discretized coupled-channels method (four-body CDCC). Four-body CDCC well reproduces measured elastic cross sections with no adjustable parameter. We then estimate the channel coupling effects by dividing the breakup channel into fourand three-body channels. It is found that 6Li breakup is mainly induced by a three-body channel.


Introduction
Projectile breakup is essential in reactions of weakly-bound nuclei and appears as a strong coupling effect between elastic and breakup channels.The continuum-discretized coupled-channels method (CDCC) was proposed for treating various kinds of channels including breakup (continuum) channels [1][2][3].The coupling effect was first confirmed in deuteron scattering and later verified in halo nuclei near the drip-line in which a two-body projectile + target (T) three-body problem was assumed.Nowadays, CDCC is widely applied to describe three-body dynamics.
Our interest is now going to four-body dynamics in scattering of three-body projectiles.CDCC for three-and four-body scattering are now called three-and four-body CDCC, respectively.It is interesting to consider the difference of four-body scattering between 6 He and 6 Li as typical examples.Since 6 He is a Borromean nucleus in which no binary subsystem is bound, only a four-body channel ( 6 He + T → n + n + α + T) exists as a breakup channel.This property makes four-body dynamics relatively simpler and this is the reason why four-body CDCC was first applied to 6 He scattering [4,5].On the other hand, 6 Li has a bound state in the n + p subsystem.Therefore, not only a four-body channel ( 6 Li + T → n + p + α + T) but also a three-body channel ( 6 Li + T → d + α + T) exists in its breakup channel.This situation makes it more difficult to understand four-body dynamics of 6 Li scattering.≈ 30 MeV) was first analyzed with three-body CDCC based on the d+α + 209 Bi three-body model [6].However, the calculation could not reproduce the measured elastic cross section without a normalization factor 0.8 to d-209 Bi and α-209 Bi optical potentials.This problem was solved by four-body CDCC based on the n + p + α + 209 Bi fourbody model [7].The calculation well describes the experimental data with no adjustable parameter.As an interesting result, it was reported that d breakup in 6 Li is strongly suppressed during the elastic scattering.In this work, we focus on four-body dynamics of 6 Li elastic scattering from the point of view of four-and three-body channel coupling effects.

Theoretical framework and Model Hamiltonian
We recapitulate four-body CDCC based on the n + p + α + T four-body model; see Ref. [3,7] for the detail.The scattering state Ψ with the total energy E is governed by the four-body Schrödinger equation with the model Hamiltonian H 4 : where K R stands for the kinetic energy operator with respect to the relative coordinate R between 6 Li and T, and U x (x = n, p, α) represents the optical potential between x and T. Since Coulomb-breakup effects are negligibly small for the present elastic scattering [6,7], the Coulomb part of U p and U α is then approximated into e 2 Z Li Z T /R, where Z A is the atomic number of nucleus A. h npα denotes the internal Hamiltonian of 6 Li in the three-cluster model.In CDCC, Eq. ( 1) is solved in the model space P spanned by the ground and discretized continuum states of 6 Li: where Φ γ represents the γ-th eigenstate with eigenenergy ε γ ; note that the γ = 0 and γ = 1-N correspond to the ground and excited states in P, respectively.The Φ γ are obtained by diagonalizing h npα with the Gaussian basis functions [8].
In order to disentangle breakup dynamics, we divide the CDCC model space P in the following way.P can be decomposed into the ground-state part P 0 and the breakup-state part P * as P = P 0 + P * for For later discussion, P * is further divided into a subspace P npα dominated by npα configurations and a subspace P dα by dα configurations.The subspaces are defined as follows.The probability of dα configurations in the breakup state Φ γ is obtained by the overlap between Φ γ and the d ground state We then define a breakup state with Γ (dα) γ > 0.5 (Γ (dα) γ ≤ 0.5) as a dαdominant (npα-dominant) state.The subspace P dα (P npα ) is a model space spanned by dα-dominant (npα-dominant) breakup states.Consequently, the model space P of CDCC calculations is expressed as P = P 0 + P dα + P npα .

Results
We only show the results of 6 Li+ 208 Pb elastic scattering at 39 MeV; see Ref. [7] for the analysis of 6 Li+ 209 Bi scattering and the detail of model setting.As for U n , we take the potential of Koning and Delaroche [9] determined from the measured elastic cross section of n + 209 Bi.For simplicity, the spin-orbit interaction is neglected and U p is assumed to have the same geometry as U n .The potential U α is taken from Ref. [10] determined from measured differential cross sections of α + 209 Bi scattering at 19-22 MeV.The angular distribution of elastic cross sections for 6 Li + 208 Pb scattering at 39 MeV are plotted in Fig 1 .The one-channel (1ch) calculation with no breakup (dotted line) underestimates the experimental data, whereas the four-body CDCC calculation (solid line) reproduces the experimental data.The enhancement from the dotted to solid lines is induced by channel coupling effects, indicating 6 Li breakup effects are quite important.Thus, 6 Li scattering on heavy targets are well described by four-body CDCC with no adjustable parameter.
Next, we consider four-and three-body channel coupling effects on the present 6 Li scattering.For this purpose, the subspaces P dα and P npα are switched off from the full calculation, respectively.Figure 2 shows the angular distribution for 6 Li + 208 Pb scattering at 39 MeV, and the solid and dotted lines are the same as in Fig. 1.When the model space is limited to P 0 + P npα , the dot-dashed line is obtained [see Fig. 2(a)] and close to the result of 1ch calculation (dotted line).On the other hand, when the model space is limited to P 0 +P dα , we have the dashed line [see Fig. 2(b)] that well simulates the full calculation (solid line).Note that the number of dα-dominant states in P is as few as about one-eighth of that of npα-dominant states.The result indicates that the coupling between P 0 and P dα is dominant.In other words, 6 Li breakup is mainly induced by the d + α breakup, and the d-breakup is suppressed in 6 Li scattering.

Summary
We have analyzed the four-body dynamics of 6 Li elastic scattering.The elastic scattering of 6 Li + 208 Pb at 39 MeV is well described by four-body CDCC based on the n + p + α + 208 Pb model.The breakup channels are approximately divided into four-and three-body channels and the coupling effects are estimated. 6Li breakup is mainly caused by a strong transition between the elastic (P 0 ) 21 st International Conference on Few-Body Problems in Physics 06007-p.3 and three-body channel (P dα ), suggesting d-breakups effect are suppressed in 6 Li scattering.We will investigate what causes the suppression, and the energy and target dependence in the forthcoming paper.

Figure 1 .
Figure 1.(Color online) Elastic cross sections (normalized by the Rutherford cross section) for6 Li + 208 Pb scattering at 39 MeV.The solid line represents the result of four-body CDCC calculation with full channel coupling, whereas the dotted line shows the result of 1ch calculation with no channel coupling.The experimental data is taken from Ref.[11].

Figure 2 .
Figure 2. (Color online) Elastic cross sections (normalized by the Rutherford cross section) for 6 Li + 208 Pb scattering at 39 MeV.The solid (dotted) line represents the result of full (1ch) calculation.The dot-dashed line in the panel (a) shows the calculation in which the model space is limited to P 0 + P npα , whereas the dashed line in the panel (b) denotes the one with P 0 + P dα .