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Cluster-Orbital Shell Model for neutron-lich nuclei

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

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Cluster-Orbital Shell Model for neutron-lich nuclei

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

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

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

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

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

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

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

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

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

  10. 3. Stochastic Variational Approach To reduce the basis size V. I. Kukulin and V. M. Krasnopol’sky, J. Phys. G3 (1977) K. Varga, Y. Suzuki and R. G. Lovas, Nucl. Phys. A571 (1994) K. Varga and Y. Suzuki, Phys. Rev. C52(1995) “Refinement” procedure Stochastic Variational procedure “exact” method 18O (16O+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 N-N: Volkov No.2 (M=0.58, B=H=0.07), adjusted to 18O 0+ ground state H. M, K. Kato, and K. Ikeda, PRC73, 034318 (2006). • Model space Lmax 5 Lmax 2 • Valence nucleons N  4 N  10

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

  13. J2-expectation values J=5/2 J=3/2 J=1/2 J=0 J2-value is almost good

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

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

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

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

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

  19. Components of the wave functions Case B B=H=0.25 B=H=0.07 (d5/2)6 78.7% 95.0% 22O (s1/2)2(d5/2)4 15.9% 3.1% (s1/2)2 (…) 3.2% 16.6% 24O 23O 22O 91.2% 97.0% (s1/2)(d5/2)6 23O 0.1% 2.1% (s1/2)(d5/2)4(d3/2)2 (s1/2)(…) 99.9% 99.6% 94.6% 98.5% (s1/2)2(d5/2)6 24O 4.3% (s1/2)2(d5/2)4 (d3/2)2 1.2% (s1/2)2 (…) 99.0% 99.8% S-wave component is enhanced at 23O and 24O

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

  21. Matter density of 24O with Volkov B=H=0.25 Rrms = 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) ) Rrmss calc. Ref.1 Ref.2 4He 1.48 1.57 1.49 6He 2.48 2.48 2.30 2.46 8He 2.66 2.52 2.46 2.67 [1] I. Tanihata et al., PRL55(1985) [2] G. D. Alkhazov et al. PRL78 (1997) H. M, K. Kato, K. Ikeda, PRC75 (2007)

  23. Summary 1. M-scheme COSM approach Number of valence nucleons form 4 to 10 By using Volkov No.2: over binding, Rrms A1/3 2. Different NN-int (so as to reproduce the drip-line) Qualitative improvement of Rrms Rrms is still not completely reproduced e.g. Three-body force, core-excitation (clustering),…

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

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

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

  27. 6He Hamiltonian • V1, V2:, Core-N int.. “KKNN” a-n phase shift • Vnn: :NN int. Minnesota potential • V12c : Effective 3-body int. Model space Maximum angular momentum Comparison Energy, pole-contribution, density

  28. Calculation COSM Number of Gaussian functions for each core-N space N=20 A) Full L 0 1 2 3 4 5 partial waves: s1/2p3/2 p1/2 d5/2 d3/2 f7/2 f5/2 g92 g7/2 h11/2 h9/2 N 20 20 20 20 20 20 20 20 20 20 20 Max. total basis size: 2310 B) Reduced L 0 1 2 3 4 5 partial waves: s1/2p3/2 p1/2 d5/2 d3/2 f7/2 f5/2 g92 g7/2 h11/2 h9/2 N 8 20 20 8 8 8 8 5 5 2 2 Max. total basis size: 636

  29. Re. k Imag. k Re. k Imag. k Calculation GSM • Continuum • Pole : 0p3/2 Maximum momentum for continuum: kmax= 3 (fm-1)

  30. Re. E  Imag. E Components of the poles continua H. M, K. Kato and K. Ikeda, PRC75, (2007) 034316. Comparison between the COSM w.f. and GSM w.f.. COSM Shell model (Gaussian w.f.) (Products of s.p.w.f.) 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

  31. Results Ground state energy: E(6He: 0+) Lmax COSM (B:Reduced) GSM -0.1998 -0.1959 1 -0.8057 -0.8006 2 -0.9218 3 -0.9153 4 -0.9597 -0.9582 5 -0.9738 -0.9755

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

  33. COSM (A: Full) -0.1992 -0.8202 -0.9518 -1.0131 -1.0568 Results Ground state energy: E(6He: 0+) Lmax COSM (B:Reduced) GSM -0.1998 -0.1959 1 -0.8057 -0.8006 2 -0.9218 3 -0.9153 4 -0.9597 -0.9582 5 -0.9738 -0.9755

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

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

  36. Results Pole contribution: (0p3/2)2 Lmax COSM (B:Reduced) GSM COSM (A: Full) (1.5569, -0.4619) (1.5478, -0.4225) (1.550,-0.42133) 1 (1.2519, -0.6898) (1.2561, -0.6737) (1.2643, -0.6684) 2 3 (1.2126, -0.7029) (1.2115, -0.6881) (1.2247,-0.6812) 4 (1.2046, -0.7051) (1.1947, -0.6919) (1.2127,-0.6840) 5 (1.2031, -0.7054) (1.1822, -0.6942) (1.2066, -0.6865)

  37. Pole contribution: (0p3/2)2c Real part Imaginary part

  38. Density distribution for valence neutron:

  39. Why do we have the difference?

  40. Re. k Imag. k Why do we have the difference? Treatment (discretization) of the continuum • GSM Discretized continuum • COSM Gaussian basis function (Fourier trans.) Non-discretized continuum

  41. To illustrate… Discretized continuum Non-discretized continuum

  42. Summary • COSM approach • J-Scheme and M-Scheme COSM have been performed. • Rrms of 24O 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.)

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