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Study of Weakly Bound Nuclei with an Extended Cluster-Orbital Shell Model

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18th International IUPAP Conference on Few-Body Problems in Physics “FB18”

August 21-26, 2006, Santos, Sao-Paulo, BRAZIL

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

Introduction

A model to describe weakly bound,

“many-nucleon” systems

An extended Cluster-orbital shell model

2. New aspects for the halo structure

Gamow shell-model picture

From experiments: Widening of Rrms near the drip-lines

A. Ozawa, from [email protected]

Abrupt changes happen near the neutron drip-line

O-isotopes

Rrms

Separation Energy

16O

22O

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

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)

(Relatively) Strong-bound neutrons

Weak-bound neutrons

Core+n (+2n)

Core + Multi-valence neutrons(?)

From experiments: part 1

RIKEN (R. Kanungo et al., PLB512(2001) )

Reaction cross-section deduced by the Glauber model

22OのRrms

22O alone ＜ 22O in 23O

“Core” is soft enough

22O is not appropriate to be considered as a Core

From experiments: part 2

RIKEN ( R. Kanungo et al., PRL88(2002) )

Momentum distribution fitted by the Glauber model

Gives the best fit

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

s1/2

s1/2

J=5/2+

Jp = 1/2+

d5/2

d5/2

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

From experiments: part 3

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

Analysis using the Eikonal model

23O-ground state is 1/2+

Jp = 1/2+

d5/2

Still this picture is true

a model to describe weakly bound,

“many-nucleon” systems

An extended Cluster-Orbital Shell Model

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

- Gaussian basis function

- Stochastically chosened basis sets

- Structure of the core

- Interaction between the core and a valence nucleon

We extend the model space

−Neo Cluster-Orbital Shell-Model−

H.M, K. Kato and K. Ikeda, PRC73(2006), 034318

1. Description of weakly bound systems

A sort of full-space calculation

2. Dynamics of the total system

Microscopic treatment of the core and valence nucleons

Single-particle states

Shell model:

COSM:

1. Description of weakly bound systems

Basis function for valence nucleons in COSM

i-th basis function

Gaussian

Non-orthogonal

Anti-symmetrized wave function

C.F.P.-like coefficients

SVM-like approach

V. I. Kukulin and V. M. Krasnopol’sky, J. Phys. G3 (1977)

K. Varga and Y. Suzuki, Phys. Rev. C52(1995)

“exact” method

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

Stochastic approach: N=138

“Refinement” procedure

H. Nemura, Y. Akaishi and Y. Suzuki, Phys. Rev. Lett. 89(2002)

0p1/2

0p3/2

0s1/2

h.o. config.

Size-parameter of the core: b

2. Dynamics of the total system

We change core-size parameter b

- Microscopic Core-N interaction

NN-int. : Volkov No.2

(Mk=0.58, Hk=Bk=0.07)

17O

Pauli (OCM)

direct

exchange

Energies are almost reproduced

Calculated levels of O-isotopes

18O

19O

20O

Order of levels: good

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

Rrms radius

Additional 3-body force

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

Described by the same core-size parameter b

Energy of 16O-core

Core-N potential

Different minima of b

b: 18Ne case is larger

fixed-b

Exp.

changed

18O

2.64

2.61 ±0.08

2.65

2.66

2.81 ±0.14

18Ne

2.68

Energy of the total system

core

valence

Inclusion of the dynamics of the core:

Rrms are improved

What is the difference?

Core+n

Core+p

Change of Core - N interaction:

Effect for the S-wave potential is different

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

This could be a key to solve the structure of 23O and 24O

1s1/2

0d5/2

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) )

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)

Tail part of wave function

“Gamow Shell Model (GSM)”

R. Id Betan, et al., PRC67(2003)

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

G. Hagen, et al., PRC71 (2005)

Single-particle states

Bound states (h.o. base)

Pole (bound and resonant ) + Continuum

“Gamow” state

- R. Id Betan, R. J. Liotta, N. Sandulescu, T. Vertse

Many-body resonance, Virtual states

- N. Michel, W. Nazarewicz, M. Ploszajczak, J. Okolowicz

He-, O-isotopes (Core+Xn), Li-isotopes (Core+Xn+p)

- G. Hagen, M. Hjorth-Jensen, J. S. Vaagen

Effective interaction, Lee-Suzuki transformation

1. Completeness relation

Solved by CSM

2. Expansion of the wave function

Single-particle

COSM

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

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

“SN” : N-particles in continuum

Even though the NN-int. and model space are different,

pole and continuum contributions are the same

“ECM”

T-base

S. Aoyama et al. PTP93 (1995)

“COSM”

V-base

Correlation of n-n

T-base is important

Poles and Continua of 6He

“SM” approaches:

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

0p3/2 :

Almost the same

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

0p1/2 :

Different

Even though angular momenta

In 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!!

2. Comparison to GSM

Same as GSM

Stable nuclei:

Weakly bound nuclei:

Different from GSM

1. An extended COSM (Neo-COSM)

- Energies, Rrms are reasonably reproduced
- Dynamics of the core is a key to study
- multi-valence nucleon sytems

Useful method to study stable and unstable nuclei

within the same footing

Correlations of poles and continua are included at a maximum