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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.

2

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

3

• 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

4

Hamiltonian and variational equations for 4He

4

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)