1. Coupled-Channel analyses of three-body and four-body breakup reactions Takuma Matsumoto (RIKEN Nishina Center) T. Egami 1, K. Ogata 1, Y. Iseri 2, M. Yahiro 1 and M. Kamimura 1 ( 1 Kyushu University, 2 Chiba-Keizai College) Unbound Nuclei Workshop 3-5 November 2008

2. Introduction Nuclear and Coulomb Target Unstable Nuclei  The unstable nuclear structure can be efficiently investigated via the breakup reactions.  Elastic cross section  Breakup cross section  Momentum distribution of emitted particles  An accurate method of treating breakup processes is highly desirable. Structure information

3. Breakup Reactions Three-body Breakup Reaction Four-body Breakup Reaction The projectile breaks up into 2 particles. Projectile (2-body) + target (1-body) 3-body breakup reaction The projectile breaks up into 3 particles. Projectile (3-body) + target (1-body) 4-body breakup reaction Ex.) d, 6 Li, 11 Be, 8 B, 15 C, etc.. Ex.) 6 He, 11 Li, 14 Be, etc.. One-neutron halo Two-neutron halo 1 2 3 2 1 4 3

4. Region of Interest  In a simplified picture, light neutron rich nuclei can be described by a few-body model core n n n Three-body System (Four-body reaction system) Two-body System (Three-body reaction system)

5. The CDCC Method  The Continuum-Discretized Coupled-Channels method (CDCC)  Developed by Kyushu group about 20 years ago M. Kamimura, Yahiro, Iseri, Sakuragi, Kameyama and Kawai, PTP Suppl.89, 1 (1986).  Fully-quantum mechanical method  Successful for nuclear and Coulomb breakup reactions  Three-body and Four-body reaction systems  Essence of CDCC Continuum Bound Breakup threshold discretization Discretized states  Breakup continuum states are described by a finite number of discretized states  A set of eigenstates forms a complete set within a finite model space

6. Discretization of Continuum Momentum-bin method Pseudo-state method Wave functions of discretized continuum states are obtained by diagonalizing the model Hamiltonian with basis functions Cannot apply to four-body breakup reactions Can apply to four-body breakup reactions

7. Three-Body model of 6He 6 He is a typical example of a three-body projectile and many theoretical calculations have been done. 4 He n n 1 2 3 4 1. T. M,, E. Hiyama, T. Egami, K. Ogata, Y. Iseri, M. Kamimura, and M. Yahiro, Phys. Rev. C70, 061601 (2004) and C73, 051602 (2006). 2. M. Rodríguez-Gallardo, J. M. Arias, J. Gómez-Camacho, R. C. Johnson, A. M. Moro, I. J. Thompson, and J. A. Tostevin, Phys. Rev. C77, 064609 (2008). Four-body CDCC calculation of 6 He breakup reactions

8. Gaussian Expansion Method  6 He : n + n + 4 He (three-body model) Channel 1Channel 2Channel 3 n n n n n n 4 He  Hamiltonian  V nn : Bonn-A  V  n : KKNN int.  Gaussian Expansion Method : E. Hiyama et al., Prog. Part. Nucl. Phys. 51, 223 A variational method with Gaussian basis functions An accurate method of solving few-body problems. Take all the sets of Jacobi coordinates I  =0 + I  =1 - I  =2 + Excitation energy of 6 He [MeV]

9. Elastic Scattering of 6 He

10. Nuclear Breakup of 6He  E >> Coulomb barrier : Negligible of Coulomb breakup effects  Elastic cross section

11. Coulomb Breakup of 6He  E ~ Coulomb barrier : Coulomb breakup effects are to be significant  Elastic cross section

12. Inelastic Scattering of 6 He

13. Inelastic scattering to 2 + 2+ resonance state 0+0+ 2+2+ 4 He n n n n ground state 2 + resonance state p 6 He+p@40 MeV/nucl. 2 + resonance state is shown as a discretized state. A. Lagoyannis et al., Phys. Lett. B 518, 27 (2001)

14. Elastic and Inelastic of 6 He Elastic cross section Inelastic cross section Exp data: A. Lagoyannis et al., Phys. Lett. B 518, 27 (2001) 6 He+p@40MeV/A 6 He(g.s.) -> 6 He(2 + ) For the inelastic cross section, the calculation underestimates the data. without breakup effects with breakup effects

15. Non-resonance Component Add non resonance component

16. Breakup to continuum of 6 He

17. Breakup Cross Section g.s → 2 + cont. g.s → 1 - cont. g.s → 0 + cont. Nuclear and Coulomb Breakup Nuclear Breakup 6 He+ 12 C @ 229.8 MeV resonance  nn   [MeV]  BU [mb] Discrete S matrix Continuum S matrix

18. Smoothing Procedure Discrete T matrix → Continuum T matrix Discrete T matrix Smoothing factor Approximately complete set Continuum T matrix

19. Triple Differential Cross Section Three-Body Breakup c n t k P Triple differential cross section Momentum distribution of c

20. Momentum Distribution 18 C + p → 17 C + n + p p 18 C 17 C n Exp. Data: Kondo et al.

21. Differential Cross Section Four-Body Breakup c n t n k K P Quintuple differential cross section Three-body smoothing function

22. Calculation of Smoothing Factor Schrodinger Eq. Lippmann-Schwinger Eq. Smoothing function : T. Egami (Kyushu Univ.) Complex-Scaled Solution of the Lippmann-Schwinger Eq.

23. E1 Transition Strength ground state I  = 1 -  Discretized B(E1) strength  Smoothing procedure

24. T. Aumann et al., Phys. Rev. C59, 1252(99). A. Cobis et al., Phys. Rev. Lett. 79, 2411(97). D. V. Danilin et al., Nucl. Phys. A632, 383(98). Calculated by Egami B(E1) strength of 6 He: 0 + → 1 - J. Wang et al., Phys. Rev. C65,034036(02).

25. Summary  We propose a fully quantum mechanical method called four-body CDCC, which can describe four-body nuclear and Coulomb breakup reactions.  We applied four-body CDCC to analyses of 6 He nuclear and Coulomb breakup reactions, and found that four-body CDCC can reproduce the experimental data.  Four-body CDCC is indispensable to analyse various four-body breakup reactions in which both nuclear and Coulomb breakup processes are to be significant  In the future work, we are developing a new method of calculation of energy distribution of breakup cross section and momentum distribution of emitted particle.

27. 6 He+p scattering @ 40 MeV/A 4 He n n p 6 He+p four-body system p+ 4 He scattering at 40 MeV

28. 6 He+p @ 25, 70 MeV/A 6 He+p@25MeV/A 6 He+p@70MeV/A

29. The Number of Coupling Channels Nuclear BU: E max = 25 MeV @ 230 MeV, 10 MeV @18MeV 0 + : 60 channels @ 230 MeV, 32 channels @18MeV 2 + : 85 channels @ 230 MeV, 42 channels @18MeV Nuclear & Coulomb BU: E max = 8 MeV 0 + : 43 channels 1 - : 48 channels 2 + : 61 channels

30. Continuum-Discretized Coupled-Channels Review : M. Kamimura, M. Yahiro, Y. Iseri, Y. Sakuragi, H. Kameyama and M. Kawai, PTP Suppl. 89, 1 (1986)  Three-body breakup (Two-body projectile)  Two-body continuum of the projectile can be calculated easily.  Four-body breakup (Three-body projectile)  Three-body continuum of the projectile cannot be calculated easily. Momentum-bin method We proposed a new approach of the discretization method

31. The Pseudo-State method Wave functions of discretized continuum states are obtained by diagonalizing the model Hamiltonian with basis functions  Variational method of Rayleigh-Ritz  Eigen equation Bound State Pseudo-State As the basis function, we propose to employ the Gaussian basis function. (Gaussian Expansion Method : E. Hiyama et al. Prog. Part. Nucl. Phys. 51, 223 )