11

Role of tensor force in light nuclei based on

the tensor optimized shell model

Hiroshi TOKI RCNP, Osaka Univ. 　Manuel Valverde RCNP, Osaka Univ.Atsushi UMEYA RIKEN Kiyomi IKEDA RIKEN

　　　　　 Takayuki MYO 　明 孝之 Osaka Institute of Technology

International conference on the structure of baryons "BARYONS’10” @ Osaka Univ. , 2010.12

Outline

2

Tensor correlation in light nuclei

• Tensor Optimized Shell Model (TOSM)to describe tensor correlation

• Unitary Correlation Operator Method (UCOM)for short-range correlation

• TOSM+UCOM with bare nuclear force

• Application of TOSM to Li isotopes

• halo formation in 11Li

• Tensor force (Vtensor) plays a

significant role in the nuclear structure.

– In 4He, Vtensor ~ Vcentral

– ~ 80% (GFMC)

3

Importance of tensor force

R.B. Wiringa, S.C. Pieper, J. Carlson, V.R. Pandharipande, PRC62(2001)

• We would like to understand role of Vtensor in the nuclear

structure by describing tensor correlation explicitly.

tensor & short range correlations from VNN (bare)

He, Li isotopes (LS splitting, halo formation, level inversion)

V

VNN

S

D

Energy -2.24 MeV

Kinetic 19.88

Central -4.46

Tensor -16.64

LS -1.02

P(L=2) 5.77%

Radius 1.96 fmVcentral

Vtensor

AV8’

Deuteron & tensor force

Rm(s)=2.00 fm

Rm(d)=1.22 fmd-wave is “spatially compact”

r

AV8’

TM, Sugimoto, Kato, Toki, Ikeda

PTP117(2007)257

5

Tensor-optimized shell model (TOSM)

5

4He

• Tensor correlation in the shell model type approach.

• Configuration mixingwithin 2p2h excitationswith high-L orbits.　 TM et al., PTP113(2005) TM et al., PTP117(2007) T.Terasawa, PTP22(’59))

• Length parameters {b} such as b0s, b0p , … are optimized

independently (or superposed by many Gaussian bases).

– Describe high momentum component from Vtensor (Shrinkage)HF by Sugimoto et al,(NPA740) / Akaishi (NPA738)RMF by Ogawa et al.(PRC73), AMD by Dote et al.(PTP115)

Configurations in TOSM

proton neutron

Gaussian expansion

nlj

particle states

hole states (harmonic oscillator basis) c.m. excitation is

excluded by Lawson’s methodApplication to Hypernuclei by Umeya

coupling

C0 C1 C2 C3

7

4He in TOSM

Orbit bparticle/bhole

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

7

Length parameters

good convergence Higher shell effect 16 Lmax

0 1 2 3 4 5 6

vnn: G-matrix

Cf. K. Shimizu, M. Ichimura and A. Arima, NPA226(1973)282.( ) in q V

shrink

8

Unitary Correlation Operator Method

8H. Feldmeier, T. Neff, R. Roth, J. Schnack, NPA632(1998)61

corr. uncorr.C

12

( ) ( )exp( ), r ij ij rij iji j

p s r s r pC i g g

short-range correlator

† H E C HC H E

† 1 (Unitary trans.)C C

rp p p

Bare HamiltonianShift operator depending on the relative distance r

† 12( ) ( ) ( )C rC r s r s r s r

TOSM

2-body cluster expansion

9

Short-range correlator : C (or Cr)

3GeV repulsion Originalr2

C

Vc

1E

3E

1O3O

s(r)

[fm]

We further introducepartial-wave dependencein “s(r)” of UCOM

S-wave UCOM

Shift function Transformed VNN (AV8’)

10

T

VT

VLS

VC

E

(exact)Kamada et al.PRC64 (Jacobi)

• Gaussian expansion with 9 Gaussians

• variational calculation

TM, H. Toki, K. IkedaPTP121(2009)511

4He in TOSM + S-wave UCOM

good convergence

11

Configurations of 4He in TOSM + short-range UCOM

(0s1/2)4 83.0 %

(0s1/2)−2JT(p1/2)2

JT JT=10 2.6

JT=01 0.1

(0s1/2)−210(1s1/2)(d3/2)10 2.3

(0s1/2)−210(p3/2)(f5/2)10 1.9

Radius [fm] 1.54

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• 0 of pion nature.

• deuteron correlation with (J,T)=(1,0)

Cf. R.Schiavilla et al. (GFMC) PRL98(’07)132501

• 4He contains p1/2 of “pn”-pair.

5,6He in TOSM+UCOM

• Argonne V8’ with no Coulomb force.

• Convergence for particle states.– Lmax ~10

– 6~8 Gaussian for radial component

• Lawson method to eliminate CM excitation.

• Bound state approximation for resonances.

Cf) GFMC : K.M. Nollett et al. , PRL99 (2007) 022502 NCSM : S. Quaglioni, P. Navrátil, PRL101 (2008) 092501 SVM : Y.Suzuki, W.Horiuchi, K. Arai, NPA823 (2009) 1

4,5,6He with TOSM+UCOM

• Difference from 4He in MeV

AV8’

Preliminary

5He : Hamiltonian component

• Difference from 4He in MeV

5He 3/2 1/2

T 24.1 17.9

Central 9.0 7.0

Tensor 5.6 1.1

LS 2.6 1.0

E 6.9 10.8 Diff.=3.9 [MeV]

=18.4 MeV (hole)

bhole=1.5 fm for 4,5He

Tensor correlation & Splitting in 5He

Vtensor~exact

in 4He

Enhancement of Vtensor

LS splitting in 5He with tensor correlation

• T. Terasawa, A. Arima, PTP23 (’60) 87, 115.• S. Nagata, T.Sasakawa, T.Sawada, R.Tamagaki, PTP22(’59)• K. Ando, H. Bando PTP66 (’81) 227• TM, K.Kato, K.Ikeda PTP113 (’05) 763 (+n OCM)

30% of the observed splitting from Pauli-blocking d-wave spliiting is weaker than p-wave splitting

Pauli-Blocking

Phase shifts of 4He-n scattering

18

Characteristics of Li-isotopes

Breaking of magicity N=8• 10-11Li, 11-12Be• 11Li … (1s)2 ~ 50%. (Expt by Simon et al.,PRL83)

• Mechanism is unclear

11LiTanihata et al., PRL55(1985)2676. PLB206(1998)592.

Halo structure

19

9Li in TOSM• Tensor-optimized shell model

TM et al., PTP121(2009), PTP117(2007).

• 0s+0p+1s0d within 2p2h excitations, G-matrix (Akaishi)

• Length parameters b0s, b0p , … are determined independently and variationally (or Gaussian expansion).

– Describe high momentum component from Vtensor cf. CPP-HF by Sugimoto et al.,NPA740 / Akaishi NPA738

Energy surface for b-parameter in 9Li

Vtensor is optimized with shrunk HO basis

cf. 1st order (residual interaction): T. Otsuka et al. PRL95(2005)232502.

TOSM

pairing correlation

nn

Dominant part of the tensor correlation

pn

21Pairing-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.

22

11Li in coupled 9Li+n+n model• System is solved based on RGM

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 configu( Li) rat: on ii

• Orthogonality Condition Model (OCM) is applied.

91 2 1 2 12

1

( Li) ( ) ( ) ( )N

ij c c ij j ii

H T T V V V nn E nn

9 99( Li) : Hamilt ( Li onian for ) Liij i jH H

9 Orthogonality to the Pauli0 -, forbidden { Li} : st t s a ei

1 2 hybrid-TVmodel with Gaussi( an expansion) : nn A{

TOSM

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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

(pn)(nn)

24

Coulomb breakup strength of 11Li

E1 strength by using the Green’s function method

+Complex scaling method+Equivalent photon method (TM et al., PRC63(’01))

• Expt: T. Nakamura et al. , PRL96,252502(2006) • Energy resolution with 　　　 =0.17 MeV.E

11 9Li (G.S.) Li n n

No three-body resonance

T.Myo, K.Kato, H.Toki, K.IkedaPRC76(2007)024305

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Summary

• TOSM+UCOM with bare nuclear force.

• Vtensor enhances LS splitting energy.

• Coexistence of tensor and pairing correlations, explicitly

• Halo formation in 11Li

Review Di-neutron clustering and deuteron-like tensor

correlation in nuclear structure focusing on 11Li

K. Ikeda, T. Myo, K. Kato and H. Toki Springer, Lecture Notes in Physics 818 (2010)

"Clusters in Nuclei" Vol.1, 165-221.