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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. - PowerPoint PPT Presentation
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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.)