Study of Weakly Bound Nuclei with an Extended

Cluster-Orbital Shell Model

Hiroshi MASUI

Kitami Institute of Technology, Kitami, Japan

K. KatoHokkaido University, Sapporo, Japan

K. IkedaRIKEN, Wako, Japan

18th 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 UEC-Workshop@RIKEN

From experiments: Widening of Rrms near the drip-lines

Abrupt changes happen near the neutron drip-line

O-isotopes

Separation Energy Rrms

Difference from typical halo nuclei: 6He, 11Be, 11Li

Core + Multi-valence neutrons(?)Core+n (+2n)

Large Sn values of 23O and 24O ( 2.7MeV and 3.7MeV )

6He : 4He+2n (Sn: 0.98MeV)11Li : 9Li+2n (Sn: 0.33MeV)11Be: 10Be+n (Sn: 0.50MeV)

23O : 22O+n (Sn: 2.7MeV)24O : 22O+2n (Sn: 3.7MeV)

Weak-bound neutrons (Relatively) Strong-bound neutrons

16O22O

From experiments: part 1 RIKEN (R. Kanungo et al., PLB512(2001) )

Reaction cross-section deduced by the Glauber model

22O alone ＜ 22O in 23O

22O の Rrms

“Core” is soft enough

22O is not appropriate to be considered as a Core

23O ground state : 5/2+ (Lowest config. :1/2+)

(0d5/2)6 is no good picture of 22O = Not a “inert” core

From experiments: part 2RIKEN ( R. Kanungo et al., PRL88(2002) )

Momentum distribution fitted by the Glauber model

Gives the best fit

d5/2

s1/2

d5/2

s1/2

J5/2+ J1/2+

From experiments: part 3

23O-ground state is 1/2+

GSI (D. Cortina-Gil et al., PRL93(2004) )

Analysis using the Eikonal model

Still this picture is true

d5/2J1/2+

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), 034318

1. 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

k l klShell model:

k Single-particle states

COSM: kllk Non-orthogonal

1. Description of weakly bound systems

Anti-symmetrized wave function

C.F.P.-like coefficients

“exact” method

SVM-like approachV. 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

18O (16O+2n) : N=2000Stochastic approach: N=138

2. Dynamics of the total system

Size-parameter of the core: b

0p3/2

0p1/2

0s1/2

h.o. config.

We change core-size parameter b

16O+XN systems

•Microscopic Core-N interaction

NN-int. : Volkov No.2

17O

(Mk=0.58, Hk=Bk=0.07)

direct exchange Pauli (OCM)

16O+XN systems

Energies are almost reproduced

18O 19O

20O

Calculated levels of O-isotopes

Order of levels: good

GSM : N. Michel, et al., PRC67 (2003)

Rrms radius

Dynamics of the core

T. Ando, K. Ikeda, and A. Tohsaki-Suzuki, PTP64 (1980).

Additional 3-body force

Energy of 16O-core Core-N potential

Described by the same core-size parameter b

Different minima of b

b: 18Ne case is larger

2.64

2.66

18O

18Ne

fixed-b

2.65

2.68

changed

2.81 ±0.14

2.61 ±0.08

Exp.

Energy of the total system

core valence

Rrms 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 23O and 24O

If d5/2 is closed in 22O, s-wave becomes dominant in 23O

Core+n Core+p

0d5/2

1s1/2

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.24He 1.48 1.57 1.49 6He 2.48 2.48 2.30 2.468He 2.66 2.52 2.46 2.67

[1] I. Tanihata et al., PRL55(1985)[2] G. D. Alkhazov et al. PRL78 (1997)

Rrmss

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

1b,r,a

dk L

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 comparison1. Completeness relation

2. Expansion of the wave function

Solved by CSM

Single-particle COSM

18O

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

6He

“COSM”

S. Aoyama et al. PTP93 (1995)

V-base

“ECM”

T-base

Correlation of n-n

T-base is important

Poles and Continua of 6He

0p1/2 :

0p3/2 : Almost the same

Different

[21] N. Michel et al., PRC67 (2003)

[26] G. Hagen et al., PRC71 (2005)

“SM” approaches:

Even though angular momentaIn the basis set increase

Contributions of the sum of p3/2 and p1/2 do not change

Details of poles and continua

p3/2 p1/2

Almost the same Changes drastically!!

21)( SSPolelA j

Summary1. An extended COSM (Neo-COSM)

•Energies, Rrms 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 nucleiwithin the same footing