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Role of Tensor Correlation on the Spin-Orbit Splitting in Neutron Halo Nuclei
In collaboration with Satoru Sugimoto (Kyoto), Kiyoshi Kato (Hokkaido) Hiroshi Toki (RCNP) and Kiyomi Ikeda (RIKEN)
Takayuki Myo RCNP, Osaka Univ.1. 4He with tensor-optimized shell model.
2. Tensor correlation in 5,6He.
3. 11Li with tensor and pairing correlations for halo formation.
The 17th International Spin Physics Symposium, SPIN2006@Kyoto Univ.
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Motivation
• Spectroscopy of neutron-rich nuclei , 5,6He, 10,11Li
We would like to understand the roles of Vtensor on the nuclear structure by describing tensor correlation (TC) explicitly in the model space.
• Tensor force (Vtensor) plays a significant role in the nuclear structure.
– In 4He,
– GFMC shows ~ 70-80% of two-body attraction energy.
• Physical effect of Vtensor is not easy to understand in a transparent manner.
πVtensor centralV V
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• We start the study of TC from 4He in the shell model type approach.
• Configuration mixing within 2p2h excitations
TM et al., PTP113(2005)TM et al., nucl-th/0607059T.Terasawa, PTP22(’59))
• Length parameters such as are determined independently and variationally.
– Describe high momentum component from Vtensor ( cf. CPPHF by Sugimoto et al,(NPA740) / Akaishi (NPA738))
{ }b 0 0 1/ 2 0 3/ 2, , ,...s p pb b b
Tensor-optimized shell model for 4He
4He
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Hamiltonian and variational equations for 4He
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1
( He) ,A A
i G iji i j
H t T v
: central+tensor+LS+Coulombijv
4( He) k kk
C : shell model type configurationk
0H
0,H E
b
0
k
H E
C
• Effective interaction : Akaishi force (NPA738)
– G-matrix from AV8’ with kQ=2.8 fm-1
– Long and intermediate ranges of Vtensor survive.
– Adjust Vcentral to reproduce B.E. and radii of 4He
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4He with 0s+0p space
(0s)4 87.5%
(0s)2JT(0p1/2)2
JT JT=10 7.7
JT=01 1.3
(0s)2(0p3/2)2 3.4
(0s)2(0p1/2)(0p3/2) 0.06
P(D) 6.9
Rm 1.48 fm
• Narrow 0p-orbit
• = 0- coupling of 0s1/2-0p1/2
• (J,T)=(1,0), p-n pair
Energy surface
J
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4He with adding high-L orbits
Orbit bnlj/b0s
0p1/2 0.65
0p3/2 0.58
1s1/2 0.63
0d3/2 0.58
0d5/2 0.53
0f5/2 0.66
0f7/2 0.55
Length parameters
Solutions shows a good convergence Higher shell effect 16
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4He with 0s+p+1s+d+f+g
(0s1/2)4 85.0 %
(0s1/2)2JT(0p1/2)2
JT JT=10 5.0
JT=01 0.3
(0s1/2)210(1s1/2)(0d3/2)10 2.4
(0s1/2)210(0p3/2)(0f5/2)10 2.0
(0s1/2)210(0p1/2)(0p3/2)10 0.9
P[D] 9.6
Energy (MeV) - 28.0
- 51.0tensorV
Three cases are mixed.
• 0- of pion nature.
• Deuteron correlation with (J,T)=(1,0)
4 Gaussians instead of HO
central
71.2 MeV
V 48.6 MeV
T
c.m. excitation = 0.6 MeV
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LS splitting in 5He with tensor correlation
5 4( He) ( He) relH H H
[Ref] T. Terasawa, PTP22(’59), S. Nagata, T. Sasakawa, T. Sawada and R. Tamagaki, PTP22(’59), K. Ando and H. Bando PTP66(’81)
5 4
1
( He) ( He) ( )
N
i ii
nA
44H
1
( He) ( ) ( ) ( )N
ij rel e n i ij j ii
h T V n E n
4He
Projection operator to remove the Pauli-forbid d: en states
i
• Orthogonarity Condition Model (OCM) is applied.
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Phase shifts of 4He-n scattering
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Tensor correlation in 6He
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6He in coupled 4He+n+n model
• (0p3/2)2 can be described in Naive 4He+n+n model
• (0p1/2)2 loses the energy Tensor suppression in 0+2
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Characteristics of 11Li• S2n = 0.31 MeV
• Rm = 3.12±0.16 / 3.53±0.06 fm (9Li: 2.32±0.02 fm)
Breaking of magic number N=8• Simon et al.(exp,PRL83) (1s)2~50%.• Mechanism is unclear
Halo structure Borromean systemNo bound state in any two-body subsystem
0.31 MeV
9Li+n+n
10Li(*)+n
11Li
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11Li with coupled 9Li+n+n model• System is solved based on the RGM equation
11 9( Li) ( Li) nnH H H 11 9
1
( Li) ( Li) ( )N
i ii
nn
A
9 11 9
1
( Li) ( Li) ( Li) ( ) 0N
j i ii
H E nn
A
9 shell model type configuration (0s( Li) : +0p+sd)i
• Orthogonarity Condition Model (OCM) is applied.
91 2 1 2 12 1, 2,
1
( Li) ( ) ( ) ( )N
ij c c i i ij j ii
h T T V V V nn E nn
9 99 Internal( Li) Hamiltonian for( Li) : Liij i jh H
9L
9i
Projection operator to remove the Pauli-forbidden states
from the relative motion between Li
n
:
-
i
Up to 2p2h
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Pairing-blocking : K.Kato,T.Yamada,K.Ikeda,PTP101(‘99)119, Masui,S.Aoyama,TM,K.Kato,K.Ikeda,NPA673('00)207. TM,S.Aoyama,K.Kato,K.Ikeda,PTP108('02)133, H.Sagawa,B.A.Brown,H.Esbensen,PLB309('93)1.
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9Li with tensor and pairing correlations
• Configuration mixing with H.O. basis function(TM, K.Kato, K. Ikeda, PTP113(2005), nucl-th/0607059).
• 0s+0p+1s0d within 2p2h excitations.
• Length parameters such as are determined independently and variationally.
{ }b 0 0 1/ 2 0 3/ 2, , ,...s p pb b b
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Properties of local minimum in 9Li
(a) :
Tensor correlation with (0s1/2)–2(0p1/2)2 excitation of p-n pair (T=0).
Vtensor is optimized with spatially shrunk H.O. basis.
Pairing correlationof p-shell neutrons
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Superposition of two minima in 9Li
0p0h 78.5 %
(0p3/2)-201(0p1/2)2
01 8.8
(0s1/2)-2JT(0p1/2)2
JT JT=10 6.8
JT=01 0.2
(0s1/2)210(1s1/2)(0d3/2)10 1.9
(0s1/2)-210(0d3/2)2
10 1.2
Energy (MeV) - 44.3
- 31.9
Rm (fm) 2.31
tensorV
Pairing correlation of n-n pair
(exp. : - 45.3 MeV)
(exp. : 2.32±0.02)
Tensor correlation of p-n pair
0– coupling of 0s1/2–0p1/2
•pion nature of Vtensor
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Hamiltonian for 11Li
•V9Li-n : folding potential
•Same strength for s- and p-waves
•Adjust to reproduce
S2n=0.31 MeV
•Vn-n : Argonne potential (AV8’)
[Ref] TM, S. Aoyama, K. Kato, K. Ikeda, PTP108(2002)
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11Li G.S. properties (S2n=0.31 MeV)
Tensor+Pairing
Simon et al.
P(s2)
Rm
E(s2)-E(p2) 2.1 1.4 0.5 -0.1 [MeV]
Pairing correlation couples (0p)2 and (1s)2 for last 2n
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G
Tensor+Pairing
Charge Radii of 11Li
2 11 2 2ch proton neutro
2C
2 11p
22 11 2 9
-2p n
n
p
( Li)
2( Li) ( Li)
1
Li)
1
( RN
R
R
R
R R
R
Z
RC-2n
RC-2n 4.67 3.74 5.36 5.69 [fm]
InertCore
Pairing Tensor Expt. (Sanchez et al., PRL96(2006))
9Li
11LiCharge Radii
9Li
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Coulomb breakup strength of 11Li
No three-bodyresonance
E1 strength by using the Green’s function method
+Complex scaling method+Equivalent photon method (TM et al., PRC63(’01))
•Energy resolution is considered with =0.17 MeV.
•Expt: T. Nakamura et al. , PRL96,252502(2006)
E
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Summary
• We investigate the explicit effects of the tensor correlation in 4,5,6He and 9,10,11Li.
• 4He with tensor-optimized shell model– Spatial shrinkage of particle states
– Specific 2p2h excitations with 0- coupling 0s1/2(h)-0p1/2(p)
• 5He and 6He– Pauli-blocking effect on 0p1/2 orbit
• 11Li– Breaking of magic number, halo formation
– Charge radii, Coulomb breakup strength
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Expt: Th. Stammbach, and R. L.Walter, NPA180(’72)No Tensor: S. Aoyama, S. Mukai, K. Kato and K. Ikeda, PTP93.(’95)
microscopic 4He[(0s)4]-n int. Kanada et al. ,RGM, PTP61(’79).
KKNNP P
C C LS LSV V V P V V P
KKNN-TP P
C CP
C C LLS LS Sa a aV V V P V V P
1.48, 0.60, 0.37PC C LSa a a Tensor-optimized 4He,
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2n correlation density in 11Li Talk by Y. Kikuchi (9/21PM)
Cf. H.Esbensen and G.F.Bertsch, NPA542(1992)310
9Li
n n
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Gaussian expansion for 4He
• We determin variationally.
• Solutions converge with 4 Gaussians
• Gaussian with narrow lengths are important
2' '
q,q'
( ) with HO ( )( )q q q qlj a a lj lj : withGa lussi engtan h q qlj b
qa
0/ 0.5 0.7q sb b
Gaussian expansion of particle states instead of HO for the quantitative description of the radial shrinkage
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Hamiltonian and variational equations for 9Li
9
1
( Li) ,A A
i G iji i j
H t T v
: central+tensor+LS+Coulombijv
9( Li) k kk
C : shell model type configurationk
0H
0,H E
b
0
k
H E
C
• Effective interaction : Akaishi force (NPA738)
– G-matrix from AV8’ with kQ=2.8 fm-1
– Long and intermediate ranges of Vtensor survive.
– Adjust Vcentral to reproduce B.E. and radii of 9Li
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Energy surface of 9Li for length parameters of HO
• (a) shows b0p1/2 ~ b0s x 0.5
• (b) shows b0p1/2 ~ b0p3/2
two minima, (a), (b)with a common b0p3/2 value.
Pairing correlation of neutronswith (0p3/2)-2(0p1/2)2 excitation.
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Correlations in the final states of 11Li breakup
(PW)