Study of Weakly Bound Nuclei with an Extended Cluster-Orbital Shell Model Hiroshi MASUI Kitami Institute of Technology, Kitami, Japan K. Kato Hokkaido University, Sapporo, Japan K. Ikeda RIKEN, Wako, Japan 18 th International IUPAP Conference on Few-Body Problems in Physics “FB18” August 21-26, 2006, Santos, Sao-Paulo, BRAZIL

2. New aspects for the halo structure An extended Cluster-orbital shell model 1.A model to describe weakly bound, “many-nucleon” systems Introduction Gamow shell-model picture

A. Ozawa, from From experiments: Widening of R rms near the drip-lines

Abrupt changes happen near the neutron drip-line O-isotopes Separation Energy R rms

Difference from typical halo nuclei: 6 He, 11 Be, 11 Li Core + Multi-valence neutrons(?)Core+n (+2n) Large S n values of 23 O and 24 O ( 2.7MeV and 3.7MeV ) 6 He : 4 He+2n (S n : 0.98MeV) 11 Li : 9 Li+2n (S n : 0.33MeV) 11 Be: 10 Be+n (S n : 0.50MeV) 23 O : 22 O+n (S n : 2.7MeV) 24 O : 22 O+2n (S n : 3.7MeV) Weak-bound neutrons (Relatively) Strong-bound neutrons 16 O 22 O

From experiments: part 1 RIKEN (R. Kanungo et al., PLB512(2001) ) Reaction cross-section deduced by the Glauber model 22 O alone ＜ 22 O in 23 O 22 O の R rms “Core” is soft enough 22 O is not appropriate to be considered as a Core

23 O ground state : 5/2 + (Lowest config. :1/2 + ) (0d 5/2 ) 6 is no good picture of 22 O = Not a “inert” core From experiments: part 2 RIKEN ( R. Kanungo et al., PRL88(2002) ) Momentum distribution fitted by the Glauber model Gives the best fit d 5/2 s 1/2 d 5/2 s 1/2 J  5/2 + J   1/2 +

From experiments: part 3 23 O-ground state is 1/2 + GSI (D. Cortina-Gil et al., PRL93(2004) ) Analysis using the Eikonal model Still this picture is true d 5/2 J   1/2 +

What we need is a model to describe weakly bound, “many-nucleon” systems An extended Cluster-Orbital Shell Model

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

− Neo Cluster-Orbital Shell-Model − We extend the model space 2. Dynamics of the total system Microscopic treatment of the core and valence nucleons Structure of the core Interaction between the core and a valence nucleon H.M, K. Kato and K. Ikeda, PRC73(2006), Description of weakly bound systems Gaussian basis function Stochastically chosened basis sets A sort of full-space calculation

Basis function for valence nucleons in COSM i-th basis function Gaussian Shell model: Single-particle states COSM: Non-orthogonal 1. Description of weakly bound systems

Anti-symmetrized wave function C.F.P.-like coefficients

“exact” method SVM-like approach V. I. Kukulin and V. M. Krasnopol’sky, J. Phys. G3 (1977) K. Varga and Y. Suzuki, Phys. Rev. C52(1995) H. Nemura, Y. Akaishi and Y. Suzuki, Phys. Rev. Lett. 89(2002) “Refinement” procedure 18 O ( 16 O+2n) : N=2000 Stochastic approach: N=138

2. Dynamics of the total system Size-parameter of the core: b 0p 3/2 0p 1/2 0s 1/2 h.o. config. We change core-size parameter b

16 O+XN systems

Microscopic Core-N interaction NN-int. : Volkov No.2 17 O (M k =0.58, H k =B k =0.07) directexchange Pauli (OCM)

16 O+XN systems Energies are almost reproduced

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

R rms radius

Dynamics of the core T. Ando, K. Ikeda, and A. Tohsaki-Suzuki, PTP64 (1980). Additional 3-body force Energy of 16 O-coreCore-N potential Described by the same core-size parameter b

Different minima of b b: 18 Ne case is larger O 18 Ne fixed-b changed 2.81 ± ±0.08 Exp. Energy of the total system corevalence

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

What is the difference? Change of Core - N interaction: Effect for the S-wave potential is different This could be a key to solve the structure of 23 O and 24 O If d5/2 is closed in 22 O, s-wave becomes dominant in 23 O Core+nCore+p 0d 5/2 1s 1/2

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 He He [1] I. Tanihata et al., PRL55(1985) [2] G. D. Alkhazov et al. PRL78 (1997) R rmss Tail part of wave function

2. Comparison with GSM “Gamow Shell Model (GSM)” Single-particle states Bound states (h.o. base) Pole (bound and resonant ) + Continuum R. Id Betan, et al., PRC67(2003) N. Michel, et al., PRC67 (2003) G. Hagen, et al., PRC71 (2005) “Gamow” state

Progresses N. Michel, W. Nazarewicz, M. Ploszajczak, J. Okolowicz G. Hagen, M. Hjorth-Jensen, J. S. Vaagen R. Id Betan, R. J. Liotta, N. Sandulescu, T. Vertse He-, O-isotopes (Core+Xn), Li-isotopes (Core+Xn+p) Effective interaction, Lee-Suzuki transformation Many-body resonance, Virtual states

Preparation for a comparison 1. Completeness relation 2. Expansion of the wave function Solved by CSM Single-particleCOSM

18 O Even though the NN-int. and model space are different, pole and continuum contributions are the same [21] N. Michel et al., PRC67 (2003) [26] G. Hagen et al., PRC71 (2005) “SN” : N-particles in continuum

6 He “COSM” S. Aoyama et al. PTP93 (1995) V-base “ECM” T-base Correlation of n-n T-base is important

Poles and Continua of 6 He 0p 1/2 : 0p 3/2 : Almost the same Different [21] N. Michel et al., PRC67 (2003) [26] G. Hagen et al., PRC71 (2005) “SM” approaches:

Even though angular momenta In the basis set increase Contributions of the sum of p 3/2 and p 1/2 do not change

Details of poles and continua p 3/2 p 1/2 Almost the same Changes drastically!!

Summary 1. An extended COSM (Neo-COSM) Energies, R rms are reasonably reproduced Dynamics of the core is a key to study multi-valence nucleon sytems 2. Comparison to GSM Stable nuclei: Weakly bound nuclei: Same as GSM Different from GSM Correlations of poles and continua are included at a maximum Useful method to study stable and unstable nuclei within the same footing