1. Takuma Matsumoto (Kyushu Univ.) K. Minomo, K. Ogata a, M. Yahiro, and K. Kato b (Kyushu Univ, a RCNP, b Hokkaido Univ) Description for Breakup Reactions of Three-body Projectiles

2. 1 2 4 3 1 2 3 Introduction  The Continuum-Discretized Coupled-Channels method (CDCC)  Developed by Kyushu group about 20 years ago M. Kamimura, M.Yahiro, Y. Iseri, Y. Sakuragi, H. Kameyama, and M. Kawai, Prog. Theor. Phys. Suppl. 89, 1 (1986)  Three-body breakup reactions  Four-body breakup reactions Four-body CDCC method

3. 6 He Target 4 He n n The CDCC equation of four-body systems is the same as that of three-body systems.  6 He projectile : n + n + 4 He (three-body model) S 2n ~ 1 MeV Three-body bound and discretized continuum states Four-Body Breakup Reactions

4. V nn : D. Gogny, et al., PLB32, 591 (1970), V na : KKNN interaction Channel 1Channel 2Channel 3 n n n n n n 4 He  Gaussian Expansion Method : E. Hiyama et al., Prog. Part. Nucl. Phys. 51, 223 An accurate method of solving few-body problems. A variational method with Gaussian basis functions Take all the sets of Jacobi coordinates I  =0 + I  =1 - I  =2 + Excitation energy of 6 He [MeV] Ground and breakup states of 6 He

5. 6 He+ 12 C scattering at 18 MeV 6 He+ 209 Bi scattering at 22.5 MeV Nuclear Breakup Nuclear & Coulomb Breakup For elastic scattering, CDCC well reproduces the experimental data. T.M. Hiyama, Ogata, Iseri, Kamimura, Chiba, and Yahiro, Phys. Rev. C70, 061601 (2004). T.M. Egami, Ogata, Iseri, Kamimura, and Yahiro, Phys. Rev. C73, 051602 (2006). Elastic Cross Section

6. In CDCC breakup cross sections are discrete. E* (MeV)  (mb) 6 He+ 12 C scattering at 240 MeV/nucl. How to calculate the continuum breakup cross section EXP. PRC59, 1252(1999), T. Aumann et al. CDCCEXP Breakup Cross Section

7. Continuous breakup T-matrix element Smoothing factor : Discrete T-matrix element Smoothing factor for 3-body system

8. Smoothing factor for 4-body system Smoothing factor : Three-body continuum wave function  Difficult to solve  Not good convergency  M.Rodriguez-Gallardo, J. M. Arias, J. Gomez-Camacho,A. M. Moro, I. J. Thompson, and J. A. Tostevin,PRC80, 051601(R) (2009).  T. Egami, T.M., K. Ogata, M.Yahiro, PTP121, 780(2009)  T.M., T. Egami, K. Ogata, M.Yahiro, PTP121, 885(2009) New description of continuum breakup cross section with Complex-scaling method (CSM). T.M., K. Kato, and M. Yahiro, PRC82, 051602 (2010).

9. Complex-scaling operator: Coordinate: Momentum: k Im[k] Re[k] } Bound states Resonance Continuum Useful for searching many-body resonances Green’s function with Complex-Scaling Method (CDCS Green’s function) S. Aoyama, T. Myo, K. Kato, and K. Ikeda, Prog. Theor. Phys. 116, 1 (2006) Complex-Scaling Method

10. New Smoothing Procedure with CSM Final state of the projectile Response function T-matrix calculated by CDCC Green’s function with Complex-Scaling Method (CDCS Green’s function) T.M., K. Kato, and M. Yahiro, PRC82, 051602 (2010).

11. New description of differential breakup cross section Differential Breakup Cross Section

12. Convergence System : 6 He + 12 C scattering @ 40 MeV/A 1.Convergence of T-matrix elements calculated by CDCC 2.Convergence of Green’s function in calculating continuum cross sections. The number of basesGaussian range max Set I1010 fm Set II1520 fm Set III2050 fm We should confirm the convergence with extending the model space

13. Convergence of T-matrix (2 + ) 2 + (set II) 2 + (set I) The T-matrix calculated with set I gives good convergence

14. Convergence of Green’s Function 1-1- 0+0+ 2+2+ Dashed : set I Solid : set II Mark : set III Dashed : set I Solid : set II Mark : set III The result with set II gives good convergence for Green’s function

15. 6 He + 12 C and 208 Pb scattering at 240 MeV/A T. Aumann et al, PRC59, 1252(1999). Microscopic optical potential (Double folding model with Melbourne g-matrix)  n - 12 C and 4 He - 12 C potentials  n – 208 Pb and 4 He – 208 Pb potentials V nA V cA V nA n n 4 He 12 C, 208 Pb

16. 6 He+ 12 C scattering @ 240 MeV/nucl. Exp. data from PRC59, 1252 (1999), T. Aumann et al. Underestimation → Inelastic breakup effect ~ 20% Nuclear Breakup is dominant Breakup to 3 - continuum is negligible

17. 6 He+ 208 Pb scattering @ 240 MeV/nucl. Exp. data from PRC59, 1252 (1999), T. Aumann et al. Underestimation → Inelastic breakup effect Overestimation ??? Coulomb Breakup is dominant

18.  In order to obtain continuous breakup cross sections for four-body breakup, we propose a new smoothing method with the complex scaling method.  The convergence of breakup cross sections is confirmed with extending the model space.  The new smoothing method is applied to analyses for 6 He breakup reactions on 12 C and 208 Pb at 240 MeV/A.  In a future work, we will analyse a four-body breakup reaction of 6 He, 11 Li, 14 Be with the new smoothing method. Summary