1. Cluster-Orbital Shell Model for neutron-lich nuclei Hiroshi MASUI Kitami Institute of Technology Collaborators: Kiyoshi KATO, Hokkaido Univ. Kiyomi IKEDA, RIKEN CEA/Saclay workshop “Importance of continuum coupling for nuclei close to the drip-lines” May 18-20, 2009, Saclay, France

2. Introduction Formalism of COSM Applications –O-isotopes, He-isotopes Comparison with GSM

3. Experimental situations and theoretical pictures Neutron separation energies R.m.s.radii Experiments Stable side Single-particle state Bound states (H.O. basis) Deeply bound Neutron-rich side Single-particle state Bound, continuum, Resonant states Weakly bound

4. 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

5. M-Scheme COSM 1. Hamiltonian and Interaction 2. Basis function 3. Stochastic variational approach Semi-microscopic approach Radial: Gaussian, Angular momentum: M-Scheme To reduce the basis size

6. Cluster-Orbital shell model (COSM) Y. Suzuki and K. Ikeda, PRC38(1998) Original: study of He-isotopes Shell-model Matrix elements (TBME) For many-particles COSM is suitable to describe systems: Weakly bound nucleons around a core Cluster-model Center of mass motion

7. 1. Hamiltonian and interactions Core part Valence part Treated by OCM A-body Hamiltonian Different size of the core gives different energy Decompose: core + valence parts Recoil: Semi-microscopic way: Anti-sym. Core and N: S. Saito PTPS 62(1977)11 Folding. direct + exchange Dynamics of the core H. M, K. Kato, and K. Ikeda, PRC73, 034318 (2006). “Cluster-Orbital Shell Model” (COSM) Y. Suzuki and K. Ikeda, PRC38(1988)

8. Interactions: semi-microscopic approach Core-N: M=0.58, B=H=0 N-N: M=0.58, B=H=0.07 N-N interaction : Volkov No.2 All interactions are based on the N-N interaction (basically) Parameters: LS-interaction: Phenomenological one 17 O: 5/2 +, 1/2 +, 3/2 + A. B. Volkov, NP74 (1965) 33 To reproduce 17 O(5/2 +,1/2 +,3/2 + ), 18 O (0 + )

9. 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

10. 3. Stochastic Variational Approach V. I. Kukulin and V. M. Krasnopol’sky, J. Phys. G3 (1977) “Refinement” procedure K. Varga, Y. Suzuki and R. G. Lovas, Nucl. Phys. A571 (1994) K. Varga and Y. Suzuki, Phys. Rev. C52(1995) Stochastic Variational procedure To reduce the basis size “exact” method 18 O ( 16 O+2n) : N=2100 Stochastic approach: N=138 H. M, K. Kato, and K. Ikeda, PRC73, 034318 (2006).

11. Application for the oxygen isotopes Same Hamiltonian with the (J-scheme) COSM work H. M, K. Kato, and K. Ikeda, PRC73, 034318 (2006). N-N: Volkov No.2 (M=0.58, B=H=0.07), adjusted to 18 O 0 + ground state Model space L max  5 L max  2 Valence nucleons N  4 N  10

12. S n for O-isotopes Exp. COSM (J-scheme) [1] COSM (M-scheme) : present [1] H. M, K. Kato, and K. Ikeda, PRC73, 034318 (2006).

13. J 2 -expectation values J 2 -value is almost good J=5/2 J=3/2 J=1/2 J=0

14. However, What is the key mechanism? N-N int.? Core-N int.? Others? The abrupt increase of R rms at 23 O can hardly be reproduced

15. Different NN-interactions Minnesota: u=1.0 Volkov No.2, M=0.58, B=H=0.25 Y. C. Tang, M. LeMere, and D. R. Thompson, Phys. Rep. 47 (1978)167. Different type of NN-int Weaker than the original so as to reproduce drip-line Case A Case B

16. B=H=0.25 B=H=0.07 Minnesota S n for O-isotopes “Case B” reproduce the dlip-line Case A Case B

17. B=H=0.25 B=H=0.07 Calculated R rms for O-isotopes Minnesota The abrupt increase of R rms is much more enhanced in “Case B” Case A Case B

18. B=H=0.07 Comparison with experments: R rms B=H=0.25 Minnesota However, the discrepancy is still large… Case A Case B

19. Components of the wave functions 22 O 23 O 24 O (d 5/2 ) 6 (s 1/2 ) 2 (d 5/2 ) 4 (s 1/2 )(d 5/2 ) 6 (s 1/2 )(…) (s 1/2 )(d 5/2 ) 4 (d 3/2 ) 2 (s 1/2 ) 2 (…) (s 1/2 ) 2 (d 5/2 ) 6 (s 1/2 ) 2 (…) (s 1/2 ) 2 (d 5/2 ) 4 (d 3/2 ) 2 22 O 23 O 24 O B=H=0.07 B=H=0.25 78.7% 15.9% 16.6% 95.0% 3.1% 3.2% 91.2% 2.1% 99.6% 97.0% 0.1% 99.9% 94.6% 4.3% 99.0% 98.5% 1.2% 99.8% S-wave component is enhanced at 23 O and 24 O Case B

20. Volkov: B=H=0.07 Volkov: B=H=0.25 Matter density of oxygen isotopes

21. Matter density of 24 O with Volkov B=H=0.25 R rms = 2.87 (fm) Exp: 3.19 (0.13)

22. He-isotopes Core-N: KKNN potential ( H. Kanada et al., PTP61(1979) ) N-N: Minnesota (u=1.0) ( T.C. Tang et al. PR47(1978) ) An effective 3-body force ( T. Myo et al. PRC63(2001) ) calc. Ref.1 Ref.2 4 He 1.48 1.57 1.49 6 He 2.48 2.48 2.30 2.46 8 He 2.66 2.52 2.46 2.67 [1] I. Tanihata et al., PRL55(1985) [2] G. D. Alkhazov et al. PRL78 (1997) R rmss H. M, K. Kato, K. Ikeda, PRC75 (2007)

23. Summary 1. M-scheme COSM approach Qualitative improvement of R rms By using Volkov No.2: over binding, R rms  A 1/3 2. Different NN-int (so as to reproduce the drip-line) Number of valence nucleons form 4 to 10 R rms is still not completely reproduced e.g. Three-body force, core-excitation (clustering),…

24. Comparison between COSM and GSM Collaboration with K. Kato, N. Michel, M. Ploszajczak

25. Im.k Re. k Bound states Anti-bound states (Virtual states) Resonant states Complex k-plane Continua

26. Cluster-orbital shell model (COSM) approachGamow shell model (GSM) approach Poles (bound, resonant, anti-bound states) Continua Single-particle states Many-particle states

27. 6 He Hamiltonian V 1, V 2:, Core-N int.. V 12c : Effective 3-body int. V nn: :NN int. “KKNN”  -n phase shift Minnesota potential Model space Maximum angular momentum Comparison Energy, pole-contribution, density

28. Calculation COSM N 20 20 20 20 20 20 20 20 20 20 20 Number of Gaussian functions for each core-N space Max. total basis size: 2310 Max. total basis size: 636 A) Full B) Reduced N=20 partial waves: s 1/2 p 3/2 p 1/2 d 5/2 d 3/2 f 7/2 f 5/2 g 92 g 7/2 h 11/2 h 9/2 L 0 1 2 3 4 5 N 8 20 20 8 8 8 8 5 5 2 2 partial waves: s 1/2 p 3/2 p 1/2 d 5/2 d 3/2 f 7/2 f 5/2 g 92 g 7/2 h 11/2 h 9/2 L 0 1 2 3 4 5

29. Calculation GSM k max = 3 (fm -1 ) Maximum momentum for continuum: Re. k Imag. k Re. k Imag. k Continuum Pole : 0p 3/2

30. 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. Components of the poles continua

31. Results Ground state energy: E(6He: 0 + ) GSMCOSM (B:Reduced)L max 1 2 3 4 5          

32. Ground state energy: E(6He: 0 + )

33. Results Ground state energy: E(6He: 0 + ) GSMCOSM (B:Reduced)L max 1 2 3 4 5 COSM (A: Full)               

34. Ground state energy: E(6He: 0 + ) More bound

35. Ground state energy: E(6He: 0 + )

36. Results Pole contribution: (0p 3/2 ) 2 COSM (A: Full) GSMCOSM (B:Reduced)L max 1 2 3 4 5               

37. Pole contribution: (0p 3/2 ) 2c Real partImaginary part

38. Density distribution for valence neutron:

39. Why do we have the difference?

40. Treatment (discretization) of the continuum GSM Re. k Imag. k COSM Gaussian basis function Non-discretized continuum (Fourier trans.) Discretized continuum

41. Non-discretized continuum To illustrate…

42. Summary COSM approach –J-Scheme and M-Scheme COSM have been performed. –Rrms of 24 O is not reproduced only by changing the NN- interaction/ Continuum coupling in COSM –COSM and GSM give almost the same feature for the coupling. However, the difference appears in the higher partial waves (pure continuum states). Discretization of the continuum is the key. (Same kind of discussion has been done in the CDCC approach.)