1. Hiroshi MASUI Kitami Institute of Technology RCNP 研究会 「 核子・ハイペロン多体系におけるクラスター現象 」, KGU 関内, 26-27 Sep. 2013, 横浜 Collaborators:K. KatoHokkaido Univ. K. IkedaRIKEN

2. What we have done 1. Core-size and total system 2. Continuum shell model 3. M-scheme 4. Coupled-channel model H. M, K. Kato and K. Ikeda, PRC75, (2007) 034316. H. M, K. Kato and K. Ikeda, PRC73, (2006) 034318. H. M, K. Kato and K. Ikeda, EPJA42, (2009) 535. H. M, K. Kato and K. Ikeda, NPA895, (2012) 1.

3. What we have done 1. Core-size and total system 2. Continuum shell model 3. M-scheme 4. Coupled-channel model H. M, K. Kato and K. Ikeda, PRC75, (2007) 034316. H. M, K. Kato and K. Ikeda, PRC73, (2006) 034318. H. M, K. Kato and K. Ikeda, EPJA42, (2009) 535. H. M, K. Kato and K. Ikeda, NPA895, (2012) 1.

4. Matter radius of nuclei near the drip-lines An “abrupt” change of the radius due to the weakly bound neutron or proton A. Ozawa 2001

5. 8 He 6 He 11 Be 23 O C-iso. F-iso. O-iso. 11 Li 24 O 19 C 26 F Relation for the S n and (R rms /A 1/3 ) Large S n Large R rms

6. Difference from typical halo nuclei: 6 He, 11 Be, 11 Li Core + Xn Core+n (+2n) Large S n values of 23 O and 24 O ( 2.7MeV and 3.7MeV ) 6 He : 0.98MeV 11 Li : 0.38MeV 11 Be: 0.50MeV 23 O : 2.7MeV 24 O : 3.7MeV Weakly-bound neutrons Strongly-bound neutrons 22 O SnSn SnSn

7. 2. Basis function Radial part: Gaussian Angular momentum part: Z-component “M-Scheme” Basis function Shell-model H.O.basis: Gamow S.M.: Non-Orthogonal Each coordinate is spanned from the c.m. of the core, and is expressed by Gaussian with a different width parameter

8. Energies of 16 O+XN systems Energies are almost reproduced

9. 18 O 19 O 20 O Calculated levels of O-isotopes Order of levels: good GSM : N. Michel, et al., PRC67 (2003)

10. Dynamics of the core T. Ando, K. Ikeda, and A. Tohsaki-Suzuki, PTP64 (1980). Additional 3-body force Energy of 16 O-core for the h.o. w.f. with effective NN-int.. Volkov No.2 Energy and R rms of 16 O

11. Change the core size Core-N Hamiltonian: Microscopic core-N potential:

12. Dynamics of the core Energy of 16 O-coreCore-N part Optimum value of b is different in isotopes and isotones

13. R rms are improved Inclusion of the dynamics of the core:

14. What we have done 1. Core-size and total system 2. Continuum shell model 3. M-scheme 4. Coupled-channel model H. M, K. Kato and K. Ikeda, PRC75, (2007) 034316. H. M, K. Kato and K. Ikeda, PRC73, (2006) 034318. H. M, K. Kato and K. Ikeda, EPJA42, (2009) 535. H. M, K. Kato and K. Ikeda, NPA895, (2012) 1.

15. Wave function to describe the weakly bound systems Shell-model-like approach Basis function: Our COSM approach Basis function: Completeness relation: Continuum shell model Gamow shell model Linear combination of Gaussian: Long tail of halo w.f. Cluster-orbital shell model

16. Shell model COSM Preparation of s.p. completeness relation: Diagonalize the s.p. Hamiltonian by using complex scaling method (CSM) CSM: Resonant poles No explicit path for continua Comparison between the COSM w.f. and GSM w.f.. Re. E Imag. E  (Products of s.p.w.f.)(Gaussian w.f.) H. M, K. Kato and K. Ikeda, PRC75, (2007) 034316. Check the component of the continua

17. Im.k Re. k Bound states Anti-bound states (Virtual states) Resonant statesComplex momentum plane Gamow Shell Model

18. [26] G. Hagen et al., PRC71 (2005) L  5 L=1 (p 3/2, p 1/2 ) Comparison between COSM and GSM H. M, K. Kato and K. Ikeda, PRC75, (2007) 034316. 6 He (0 + )

19. (p 1/2 ) 2 (p 3/2 ) 2 Poles: (0p 3/2 ) 2  1.2 Poles: (0p 1/2 ) 2  1.45 Total: C[(p 1/2 ) 2 ]  0.04 Total: C[(p 3/2 ) 2 ]  0.9 Pole-Cont.  −0.2 Cont.-Cont. < −0.1 Cont.-Cont.  1.25 Pole-Cont.  −2.66 Large cancellation occurs in the p 1/2 case Total For L  5 case

20. What we have done 1. Core-size and total system 2. Continuum shell model 3. M-scheme 4. Coupled-channel model H. M, K. Kato and K. Ikeda, PRC75, (2007) 034316. H. M, K. Kato and K. Ikeda, PRC73, (2006) 034318. H. M, K. Kato and K. Ikeda, EPJA42, (2009) 535. H. M, K. Kato and K. Ikeda, NPA895, (2012) 1.

21. To reproduce the drip-line at 24 O  ab initio calc. + Realistic force  Effect of the thee-body interaction T. Otsuka et al, Phys. Rev. Lett. 105, 032501 (2010) G. Hagen et al., Phys. Rev. C 80, 021306(R) (2009)

22. Ab initio calc. + Realistic force G. Hagen et al., Phys. Rev. C 80, 021306(R) (2009) Coupled-cluster (2-body) + N3LO int.  -dependence: lack of many-body int.

23. How about the radius? h  ~ 27 MeV b ~ 1.24 (fm) Very small radius G. Hagen et al., Phys. Rev. C 80, 021306(R) (2009) Coupled-cluster (2-body) + N3LO int.

24. Our approaches  Role of many valence neutrons 16 O+Xn model m-scheme COSM + Gaussian basis  Role of last one- or two-neutrons “Core” + n or “Core”+2n model A simplified model approach

25. Wave function: Radial part Product of Gaussian with polynomial Spin-isospin part Total M and M T are fixed Coordinate system is defined from the center of mass of the core nucleus We check the expectation value of the total J as

26. Expectation value of J 2 J=0 J=1/2 J=5/2 J=3/2

27. To treat the Pauli-forbidden states (PFS) Basis function Conditions (Normalized) (Orthogonalized) (r 2k x Gaussian) Solve them Pauli Forbidden states (PFS) Gauss. with poly. (orthogonal to PFS)

28. B=H=0.25 B=H=0.07 S n for O-isotopes Attraction <

29. Change the core size with A 1/6 B=H=0.07 B=H=0.25 b: 1.723 (fm) b~A 1/6 Fixed-core Expanded-core

30. Comparison with other approaches [3] G. Hagen et al., PRC 80 (2009) [2] B. Ab-Ibrahim et al., JPSJ 78 (2009) [1] H. Nakada, NPA764 (2006) □: [2] ■: [1] ▲: [3] ○: fixed-b ●: m-COSM with b〜A 1/6 0.25fm C.C.M. (Hagen et al [3]) 0.31fm Exp. [4] Fixed-b b〜A 1/6 [4] A. Ozawa et al, NPA693 (2001)

31. What we have done 1. Core-size and total system 2. Continuum shell model 3. M-scheme 4. Coupled-channel model H. M, K. Kato and K. Ikeda, PRC75, (2007) 034316. H. M, K. Kato and K. Ikeda, PRC73, (2006) 034318. H. M, K. Kato and K. Ikeda, EPJA42, (2009) 535. H. M, K. Kato and K. Ikeda, NPA895, (2012) 1.

32. Radius : large 0p-0h Inclusion of the core excitation a coupled-channel picture for 16 O “Mean-field-like“ core 2p-2h ex. Radius : small (Submitted to NPA)

33. Inclusion of the core excitation TOSM in 9 Li T. Myo, K. Kato, H. Toki and K. Ikeda, PRC76(2007) 2. Some config. are suppressed due to the Pauli-blocking 1. Different size for each orbit

34. 0d 5/2 1s 1/2 0d 3/2 0d 5/2 1s 1/2 0d 3/2 [2p2h excitations] [XpXh excitations] 0d 5/2 1s 1/2 0d 3/2 …

35. A coupled-channel approach 1ch b = 1.1A 1/6 (fm) Coupling term Energy difference of the core ΔE = 2.5, 5, 7.5, 10 (MeV) Fit: E, R rms Hamiltonian: (0p0h) 1s 1/2 is allowed 16 O 17 O ( 16 O+n) 18 O 19 O ( 18 O+n) 20 O 21 O ( 20 O+n) 22 O 23 O ( 22 O+n) 2ch b = 1.5(fm) : fixed (2p2h) 1s 1/2 is forbidden Δ 12 = Δ 21 =  2MeV

36. 0d 5/2 1s 1/2 0d 3/2 16 O-core (2p2h) 17 O (5/2 + ) 0d 5/2 1s 1/2 0d 3/2 18 O-core (0p0h) 19 O (5/2 + ) 0d 5/2 1s 1/2 0d 3/2 16 O-core (0p0h) 0d 5/2 1s 1/2 0d 3/2 18 O-core (2p2h)

37. 0d 5/2 1s 1/2 0d 3/2 20 O-core (0p0h) 21 O (5/2 + ) 0d 5/2 1s 1/2 0d 3/2 22 O-core (2p2h) 23 O (1/2 + ) Pauli-forbidden 0d 5/2 1s 1/2 0d 3/2 20 O-core (2p2h) 0d 5/2 1s 1/2 0d 3/2 22 O-core (0p0h)

38. Fitted Results for a coupled-channel core+n model approach 16 O+n 18 O+n 20 O+n 22 O+nS n are almost reproduced Sn (MeV)

39. Results for a coupled-channel core+n model approach R rms (fm) R. Kanungo et al. PRC84 (2011) A. Ozawa et al. NPA693 (2001) 16 O+n 18 O+n 20 O+n 22 O+n Mean-field-like core Fixed core

40. Summary M-scheme COSM approach Oxygen isotopes Rrms 23 O and 24 O [An attempt to reproduce the Rrms] Many valence nucleons Semi-microscopic int., OCM, resonance (CSM) 16 O(fixed size) +Xn : failed to reproduce the experiment Modification of the core is important (Shrunk core) + (Broad core) Coupled-channel picture is a possible way

41. What we want to do 1. Systematics 2. More precise core-configuration 3. Coupled-channel and deformed core 4. GSM-like treatment Core + XN 0p0h, 2p2h, 4p4h,…. Valence-core competition S.P. basis function