1. Faddeev Calculation for Neutron-Rich Nuclei Eizo Uzu (Tokyo Univ. of Science) Collaborators Masahiro Yamaguchi (RCNP) Hiroyuki Kamada (Kyusyu Inst. Tech.) Yasuro Koike (Hosei Univ., CNS) Focus on He isotopes

2. (extraction) TUNL Nuclear Data Evaluation Project http://www.tunl.duke.edu/nucldata/ (unit: MeV) unboundbound

3. He isotopes (even nuclei) 6 He nn 3-cluster model 4 He nn 6 He 8 He Borromean-nuclei unbound

4. (extraction) TUNL Nuclear Data Evaluation Project http://www.tunl.duke.edu/nucldata/ (unit: MeV) Borromean

5. He isotopes (even nuclei) 6 He nn 3-cluster model 4 He nn 8 He nn 6 He 8 He 10 He Borromean-nuclei series Faddeev equations

6. 6 He 6 Li A. Eskandarian and I. R. Afnan, PRC46, 1992 (2344). T = 1: Faddeev eqs. Yamaguchi-type potentials 1 S 0 for n - n S 1/2, P 1/2, and P 3/2 for 4 He- n

7. JJ Eskandarian and AfnanOurs E  E  0+0+ -0.560.00-0.5650.000 2+2+ +0.950.30+0.9470.287 T = 1 (in MeV) Checking on 6 Li

8. 6 He 6 Li A. Eskandarian and I. R. Afnan, PRC46, 1992 (2344). T =1: Faddeev eqs. Yamaguchi-type potentials 1 S 0 for n - n S 1/2, P 1/2, and P 3/2 for 4 He- n Apply to 8 He and 10 He with our extention

9. (extraction) TUNL Nuclear Data Evaluation Project http://www.tunl.duke.edu/nucldata/ (unit: MeV)

10. Odd Nuclei

13. (extraction) TUNL Nuclear Data Evaluation Project http://www.tunl.duke.edu/nucldata/ (unit: MeV)

14. Even Nuclei

16. Interesting Nature “He” isotopes are getting stable with increasing the number of neutrons … ? Remembering the three-nucleon forces… NNN NNN  The binding energy of 3 H becomes deeper

17. (extraction) TUNL Nuclear Data Evaluation Project http://www.tunl.duke.edu/nucldata/ (unit: MeV) Core excitation is important!

18. 6 He, 8 He, 10 He … Borromean-nuclei series 4 He, 6 He, 8 He … Core: Ground states 6 He *, 8 He * … Core: Excited states 5 He, 7 He, 9 He … resonances Faddeev eqs. employ Yamaguchi-type potentials for n - n and (core)- n subsystems Brief Summary

19. 8 He, 10 He … Borromean-nuclei series 6 He, 8 He … Core: Ground states 6 He *, 8 He * … Core: Excited states Potentials between (core)- n nn 8 He 8 He * nn 6 He 6 He * couple 7 He 9 He

20. 7 He, 9 He … resonances employ Yamaguchi-type potentials for n - n and (core)- n systems Fix the parameters to reproduce energy levels and decay widths for 7 He and 9 He Potentials between (core)- n nn 8 He 8 He * nn 6 He 6 He * couple 7 He 9 He

21. for the study of 10 He Potentials between 8 He- n nn 8 He 8 He * 12 Separable Potential Yamaguchi-type form factor c i : mixing parameter (state probability) b i : “width” parameter ` i : angular momentum

22. parameters: c 1, c 2,  1,  2 Potentials between 8 He- n 8 He 9 He 10 He Fit! (fit to 5 He data) c 1, c 2  1,  2 = 1.5166 fm  1 fixed determined uniquly 8 He 8 He *

23. Ground (0 + ) State of 10 He J p of 9 He J p of 9 He * 3/2  1/2  1/2 + 1/2  1/2 + 1/2  3/2  3/2 + 1/2  1/2  E (MeV)  (MeV)  0.187 0.591  0.534  1.862 0.306 0.782 0.131 0.034 0.657 exp. 1.070.3

24. parameters: c 1, c 2,  1,  2 Potentials between 6 He- n 6 He 7 He 8 He Fit! (fit to 5 He data) c 1, c 2  1,  2 = 1.5166 fm  1 fixed determined uniquly 6 He 6 He *

25. Ground (0 + ) State of 8 He J  of 7 HeJ  of 7 He * E (MeV) 3/2   1.317 3/2  1/2   2.175 3/2  5/2   1.318 exp.  2.140

26. State Probability 8 He 6 He n n + n n + 66.2% 33.8% 6 He * n n n n 6 He 66.2% 33.8% 0+0+ 2+2+ 6 He nn nn nn

27. Summary Study the He isotopes (Borromean nuclei) –within the 3-cluster model … core + 2 n The three-body Faddeev eqs. Yamaguchi-type potentials –parameters are fit to the smaller next odd nuclei –taking into account the core excitation states of 9 He(1/2  ) and 9 He * (1/2 + ) (for 10 He) probability of 6 He(66.2%) and 6 He * (33.8%) in 8 He

28. Outlook For improvement the potentials … Response function of the Coulomb break-up reactions (e.g. 8 He + Pb) –compare directly with the experimental data titi E R(E)R(E)

29. Response Function (preliminary) only include 0 + state of ( 6 He + 2 n ) 01234 0.01 0.02 0.00 c.m. energy (MeV) R ( E ) (MeV -1 ) our prediction 7 He g.s. 01234 0.01 0.02 0.00 c.m. energy (MeV) simulation (integrate ±0.3MeV) for 8 He + Pb reaction 7 He 1st 8 He 1st 8 He 2nd

30. Summary Study the He isotopes (Borromean nuclei) –within the 3-cluster model … core + 2 n The three-body Faddeev eqs. Yamaguchi-type potentials –parameters are fit to the smaller next odd nuclei –taking into account the core excitation states of 9 He(1/2  ) and 9 He * (1/2 + ) (for 10 He) probability of 6 He(66.2%) and 6 He * (33.8%) in 8 He

31. Reaction Low energy reaction of (core + 2 n ) + (heavy ion) e.g. ( 8 He + Pb) 6 He nn Pb 6 He nn Coulomb interaction nn

32. Cross Section response function Coulomb scattering : W.F. of the bound state: W.F of the final (continuum) state : Momentum shift op. Pb n 6 He ( ) n projection op.

33. Brief Summary (Faddeev-like eqs.) 6 He nn v1v1 v2v2 v3v3 Faddeev Eqs.

34. Brief Summary titi fitting potential parameters to peaks of the response function E 6 He nn v1v1 v2v2 R(E)R(E) v3v3

35. Response Function 6 He nn v1v1 v2v2 v3v3

36. Three-Cluster System 6 He nn nn nn channel 1channel 2channel 3 12 3 12 3 1 2 3 t i : T-matrix for two-body subsystem (Faddeev theory)

37. State Probability (preliminary) 8 He 6 He n n + n n + 66.2% 33.8% 6 He * n n n n 6 He 66.2% 33.8% 0+0+ 2+2+ 10 He 8 He (0 + ) 8 He * (2 + ) n n n n 48.8% 51.2%

38. (extraction) TUNL Nuclear Data Evaluation Project http://www.tunl.duke.edu/nucldata/ (unit: MeV)

39. Application : He isotopes Binding Energy :: 6 He < 8 He Resonance Levels :: 5 He ¼ 7 He ¼ 9 He Decay Width :: 5 He > 7 He ¼ 9 He Is nucleus stable with increasing neutron?

40. (extraction) TUNL Nuclear Data Evaluation Project http://www.tunl.duke.edu/nucldata/ not well established (unit: MeV)

41. 8 He- n 8 He nn 8 He * coupling channel 1 channel 2 8 He nn 9 He 10 He 6 He+2 n (extraction) TUNL Nuclear Data Evaluation Project http://www.tunl.duke.edu/nucldata/ unit: MeV

42. Neutron-rich Nuclei core nn Focus on “He” isotopes Faddeev Equations Exact treatment of the break-up thresholds (or boundary conditions) 3-cluster model

43. Yamaguchi-type Pot. Separable Potential Yamaguchi-type form factor c i : mixing parameter (state probability) l: coupling constant b i : “width” parameter ` i : angular momentum titi E R(E)R(E) (for the first attempt)

44. Content Motivation Potential Model Calculations