Cluster-Orbital Shell Model for neutron-lich nuclei

Hiroshi MASUIKitami 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

Introduction

• Formalism of COSM

• Applications– O-isotopes, He-isotopes

• Comparison with GSM

Experimental situations and theoretical pictures

Neutron separation energies

R.m.s.radii

ExperimentsStable side

Single-particle state

Bound states(H.O. basis)

Deeply bound

Neutron-rich side

Single-particle state

Bound, continuum,Resonant states

Weakly bound

Wave function to describe the weakly bound systems

cm (m )

m

Shell-model-like approach

(m ) (m )

Basis function:

Our COSM approach

(m ) A' G(r1,r2,,rN )(m ) JM (m ) Basis function:

1 m m

b,r,a

dk (k) (k)L

Completeness relation:

•Continuum shell model•Gamow shell model

G(r1,r2,,rN )(m )

exp( (1(m )r1

2 2(m )r2

2 N(m )rN

2))

Linear combination of Gaussian:

Long tail of halo w.f.

Cluster-orbital shell model

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

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

1. Hamiltonian and interactions

Core part Valence part

ˆ h i ˆ t 'i ˆ V i

Treated by OCM

ˆ H ˆ t ii

A

ˆ v iji j

A

ˆ T GA-body Hamiltonian

ˆ H ˆ t ik

C

ˆ v kl

kl

C

ˆ t 'i ˆ v ik

k

C

i

V

( ˆ v ij ˆ T ij )i j

V

ˆ V i Cˆ v ik' C V

k

c

Vd Vex ˆ VLS

Different size of the core gives different energy

Decompose: core + valence parts

ˆ T ij 1

i j

Recoil:

Semi-microscopic way:

Anti-sym. Core and N:

S. Saito PTPS 62(1977)11

Folding. direct + exchangeDynamics 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)

Interactions: semi-microscopic approach

ˆ H ˆ t ik

C

ˆ v kl

kl

C

ˆ t 'i ˆ v ik

k

C

i

V

( ˆ v ij ˆ T ij )i j

V

•Core-N: M=0.58, B=H=0

•N-N: M=0.58, B=H=0.07

N-N interaction :

ˆ v ij Volkov No.2

All interactions are based on the N-N interaction (basically)

ˆ V i Cˆ v ik' C V

k

c

Vd Vex ˆ VLS

Parameters:

LS-interaction:Phenomenological one

17O: 5/2+, 1/2+, 3/2+

A. B. Volkov, NP74 (1965) 33

To reproduce 17O(5/2+,1/2+,3/2+), 18O (0+)

2. Basis function

V(m ) A' G(r1,r2,,rN )(m ) MTz

(m )

cm V(m )

m

Radial part:Gaussian

Angular momentum part:Z-component

“M-Scheme”Basis function

G(r1,r2,,rN )(m )

exp( (1(m )r1

2 2(m )r2

2 N(m )rN

2))

Shell-model

Nmax H.O.basis:

Gamow S.M.:

kmax

Non-Orthogonal

Each coordinate is spanned from the c.m. of the core,and is expressed by Gaussian with a different width parameter

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)

Em( i) Em

( i1)

m1

mmax

Stochastic Variational procedure

To reduce the basis size

“exact” method 18O (16O+2n) : N=2100Stochastic approach: N=138

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

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 18O 0+ ground state

•Model space

Lmax 5 　 Lmax 2

•Valence nucleons

N 4 　 N 10

Sn for O-isotopes

Exp.

COSM (J-scheme) [1]

COSM (M-scheme) : present

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

J2-expectation values

J2-value is almost good

J=5/2

J=3/2

J=1/2

J=0

However,

What is the key mechanism? N-N int.? Core-N int.? Others?

The abrupt increase of Rrms at 23O can hardly be reproduced

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

B=H=0.25

B=H=0.07

Minnesota

Sn for O-isotopes

“Case B” reproduce the dlip-line

Case ACase B

B=H=0.25

B=H=0.07

Calculated Rrms for O-isotopes

Minnesota

The abrupt increase of Rrms is much more enhanced in “Case B”

Case A

Case B

B=H=0.07

Comparison with experments: Rrms

B=H=0.25Minnesota

However, the discrepancy is still large…

Case A

Case B

Components of the wave functions

22O 23O 24O

(d5/2)6

(s1/2)2(d5/2)4

(s1/2)(d5/2)6

(s1/2)(…)

(s1/2)(d5/2)4(d3/2)2

(s1/2)2 (…)

(s1/2)2(d5/2)6

(s1/2)2 (…)

(s1/2)2(d5/2)4 (d3/2)2

22O

23O

24O

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 23O and 24O

Case B

Volkov: B=H=0.07 Volkov: B=H=0.25

Matter density of oxygen isotopes

Matter density of 24O with Volkov B=H=0.25

Rrms = 2.87 (fm)

Exp: 3.19 (0.13)

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

H. M, K. Kato, K. Ikeda, PRC75 (2007)

Summary

1. M-scheme COSM approach

Qualitative improvement of Rrms

By using Volkov No.2: over binding, Rrms A1/3

2. Different NN-int (so as to reproduce the drip-line)

Number of valence nucleons form 4 to 10

Rrms is still not completely reproduced

e.g. Three-body force, core-excitation (clustering),…

Comparison betweenCOSM and GSM

Collaboration with K. Kato, N. Michel, M. Ploszajczak

Im.k

Re. k

Bound states

Anti-bound states(Virtual states)

Resonant states

Complex k-plane

Continua

G(r1,,rN )JMCluster-orbital shell model (COSM) approach

Gamow shell model (GSM) approach

•Poles (bound, resonant, anti-bound states)

•Continua

Single-particle states

Many-particle states

2/1 jL

6HeHamiltonian

•V1, V2:, Core-N int..

• V12c : Effective 3-body int. ))(exp( 22

21

01212 rrVV cc

• Vnn: :NN int.

“KKNN” -n phase shift

cnncccc VVVVttH 122121

Minnesota potential

Model space Maximum angular momentum

5)2/1(max jL

Comparison

Energy, pole-contribution, density

Calculation

COSMJMrgrgCJMrrGC

m

mmm

m

mm )()(),( 2

)(1

)()(21

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: s1/2 　 p3/2 p1/2 d5/2 d3/2 f7/2 f5/2 g92 g7/2 h11/2 h9/2

L 0 1 2 3 4 5

N 8 20 20 8 8 8 8 5 5 2 2

partial waves: s1/2 　 p3/2 p1/2 d5/2 d3/2 f7/2 f5/2 g92 g7/2 h11/2 h9/2

L 0 1 2 3 4 5

)2102/2120(2/)1( NNN lj

Calculation

GSM JM;

kmax= 3 (fm-1)

Maximum momentumfor continuum:

Re. k

Imag. k

Re. k

Imag. k

•Continuum•Pole : 0p3/2

Cn2 G(r1,,rN )

2

Shell model COSM

Preparation of s.p. completeness relation: Diagonalize the s.p. Hamiltonian by using complex scaling method (CSM)

rre i ( ppe i )

1b,r

ki ki

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

ResultsGround state energy: E(6He: 0+)

GSMCOSM (B:Reduced)Lmax

1

2

3

4

5

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

ResultsGround state energy: E(6He: 0+)

GSMCOSM (B:Reduced)Lmax

1

2

3

4

5

COSM (A: Full)

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

More bound

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

ResultsPole contribution: (0p3/2)

2

COSM (A: Full)GSMCOSM (B:Reduced)Lmax

1

2

3

4

5

Pole contribution: (0p3/2)2c

Real part Imaginary part

Density distribution for valence neutron:

Why do we have the difference?

Why do we have the difference?

Treatment (discretization) of the continuum

•GSM Re. k

Imag. k i

irb

kk,

1

•COSM

)4/exp(')exp( 22m

rikm akedkNraN

Gaussian basis function

Non-discretized continuum

(Fourier trans.)

)( ikk Discretized continuum

Discretized continuum Non-discretized continuum

)4/exp(' 2 akedkN rikm

mk

To illustrate…

Summary

• COSM approach– J-Scheme and M-Scheme COSM have been performed.– Rrms of 24O is not reproduced only by changing the NN-inter

action/

• Continuum coupling in COSM– COSM and GSM give almost the same feature for the coupli

ng. 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.)