Coupled-Channel analyses of three-body and four-body breakup reactions

1. Coupled-Channel analyses of three-body and four-body breakup reactions Takuma Matsumoto (RIKEN Nishina Center) T. Egami1, K. Ogata1, Y. Iseri2, M. Yahiro1 and M. Kamimura1 (1Kyushu University, 2Chiba-Keizai College) Unbound Nuclei Workshop 3-5 November 2008

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

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

4. Region of Interest • In a simplified picture, light neutron rich nuclei can be described by a few-body model n n core n Three-body System (Four-body reaction system) core 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 Discretized states Continuum discretization • 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 Breakup threshold Bound

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

7. Three-Body model of 6He 6He is a typical example of a three-body projectile and many theoretical calculations have been done. 1 2 n 3 n 4He 4 Four-body CDCC calculation of 6He breakup reactions 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).

8. Ip=0+ Ip=1- Ip=2+ Excitation energy of 6He [MeV] Gaussian Expansion Method • 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 • 6He : n + n + 4He (three-body model) n n n n n n • Hamiltonian • Vnn : Bonn-A • Van : KKNN int. 4He 4He 4He Channel 1 Channel 2 Channel 3

9. Elastic Scattering of 6He

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 6He

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

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

15. Non-resonance Component Add non resonance component

16. Breakup to continuum of 6He

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

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

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

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

21. Differential Cross Section Four-Body Breakup k K n n c P t Quintupledifferential 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. Ip = 1- ground state E1 Transition Strength • Discretized B(E1) strength • Smoothing procedure

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

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

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

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

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. Momentum-bin method • Four-body breakup (Three-body projectile) • Three-body continuum of the projectile cannot be calculated easily. 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 Pseudo-State Bound 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)