11

明　孝之　　大阪工業大学　　　　　　阪大 RCNP

Tensor optimized shell model using bare interaction

for light nuclei

共同研究者 土岐　博 　 阪大 RCNP 池田　清美　　理研

RCNP 研究会「少数粒子系物理の現状と今後の展望」 @RCNP 2008.12.23-25

Outline

2

Tensor Optimized Shell Model (TOSM)

Unitary Correlation Operator Method (UCOM)

TOSM + UCOM with bare interaction

• Application of TOSM to Li isotopes

• Halo formation of 11Li

• TM, K.Kato, H.Toki, K.Ikeda, 　　　　　　　PRC76(2007)024305

• TM, K.Kato, K.Ikeda, PRC76(2007)054309• TM, Sugimoto, Kato, Toki, Ikeda, PTP117(2007)257• TM. Y.Kikuchi, K.Kato, H.Toki, K.Ikeda, PTP119(2008)561• TM, H. Toki, K. Ikeda, Submited to PTP

3

Motivation for tensor force

tensor centralV V

• Structures of light nuclei with bare interaction

tensor correlation + short-range correlation

• Tensor force (Vtensor) plays a significant role in the nuclear structure.

– In 4He,

– ~ 80% (GFMC)V

VNN

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

• We would like to understand the role of Vtensor in the nuclear structure by describing tensor correlation explicitly.

model wave function (shell model and cluster model) He, Li isotopes (LS splitting, halo formation, level inversion)

4

Tensor & Short-range correlations

4

• Tensor correlation in TOSM (long and intermediate)

–

– 2p2h mixing optimizing the particle states (radial & high-L)

12 2 1 2 2 0ˆ( ),[ , ] 2S Y L Sr

• Short-range correlation– Short-range repulsion

in the bare NN force

– Unitary Correlation

Operator Method (UCOM)

S

D

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

T. Neff, H. Feldmeier, NPA713(2003)311

5

Property of the tensor force

5

Long and intermediate ranges

• Centrifugal potential (1GeV@0.5fm) pushes away the L=2 wave function.

6

Tensor-optimized shell model (TOSM)

6

• Tensor correlation in the shell model type approach.

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

• Length parameters such as are determined independently and variationally.

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

{ }b 0 0 1/2, ,...s pb b

4He

TM, Sugimoto, Kato, Toki, IkedaPTP117(2007)257

7

Hamiltonian and variational equations in TOSM

0k

H E

C

7

1

,A A

i G iji i j

H t T v

: central+tensor+LS+Coulombijv

k kk

C : shell model type configurationk

0H

0,H E

b

• 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 radius of 4He

TM, Sugimoto, Kato, Toki, Ikeda, PTP117(’07)257

8

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

8

Length parameters

good convergence Higher shell effect 16 Lmax

Shrink

0 1 2 3 4 5 6

vnn: G-matrix

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

9

Configuration of 4He in TOSM

(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

P[D] 9.69

Energy (MeV) 28.0

51.0tensorV

• 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

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

10

Tensor & Short-range correlations

10

• Tensor correlation in TOSM (long and intermediate)

–

– 2p2h mixing optimizing the particle states (radial & high-L)

12 2 1 2 2 0ˆ( ),[ , ] 2S Y L Sr

• Short-range correlation– Short-range repulsion

in the bare NN force

– Unitary Correlation

Operator Method (UCOM)

S

D

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

T. Neff, H. Feldmeier NPA713(2003)311

TOSM+UCOM

11

Unitary Correlation Operator Method

11H. 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

(

( )(

)( ))

R rR r

s

s r

† 1 (Unitary trans.)C C

rp p p

Bare HamiltonianShift operator depending on the relative distance r

( ) ( )R r r s r

TOSM

2-body cluster expansionof Hamiltonian

12

Short-range correlator : C (or Cr)

: ( ) for Hamiltonian C r R r †

†

( )C r C R r

C lC l

AV8’ : Central+LS+Tensor

3GeV repulsion Originalr2

C

†C sC s

Vc

1E

3E

1O3O

13

4He in UCOM (Afnan-Tang, Vc only)

13

C

14

4He with AV8’ in TOSM+UCOM

Kamada et al.PRC64 (Jacobi)

AV8’ : Central+LS+Tensor

exact

• Gaussian expansion for particle states (6 Gaussians)• Two-body cluster expansion of Hamiltonian

15

Extension of UCOM : S-wave UCOM

, ( )SsC R r

15

for only relative S-wave wave function

– minimal effect of UCOM†

†

†

central central

LS LS

rel relrel reltensor tensor

for s-wave

for s-wave

N

ˆ( ) ( )

( ) ( )

( ) ( )

o change

S D S S D

s s

s s

s s

C TC T T

C V r C V r

C V r C V r

V r V rC

SD coupling

tensorV 5 MeV gain

16

Different effects of correlation function

16

S

D

rel 0 at 0S r

rel 0 at 0D r due to Centrifugal Barrier

• D-wave

• S-wave

No Centrifugal Barrier

Short-range repulsion

17

Saturation of 4He in UCOM

17

T. Neff, H. Feldmeier

NPA713(2003)311Short tensor

Long tensor

TOSM+UCOM

Benchmark cal. Kamada et al. PRC64

UCOM: short-range + tensor

corr. uncorr.rC C

En

erg

y

18

4He in TOSM + S-wave UCOM

T

VT

VLS

VC

E

(exact)Kamada et al.PRC64 (Jacobi)

Remaining effect :

3-body cluster term

in UCOM

19

Summary• Tensor and short-range correlations

– Tensor-optimized shell model (TOSM)

• He & Li isotopes (LS splitting, Halo formation)

– Unitary Correlation Operator Method (UCOM)

• Extended UCOM : S-wave UCOM

• In TOSM+UCOM, we can study the nuclear structure starting from the bare interaction.

– Spectroscopy of light nuclei (p-shell, sd-shell)

19

2020

Pion exchange interaction vs. Vtensor

Delta interaction

Yukawa interaction

Involve largemomentum

2 2 2

1 2 1 2 122 2 2 2 2 2

2 2 2 2

1 2 122 2 2 2 2 2

ˆ ˆ3( )( ) ( )

( )

q q qq q S

m q m q m q

m q m qS

m q m q m q

12 1 2 1 2ˆ ˆ3( )( ) ( )S q q

Tensor operator

- Vtensor produces the high momentum component.

21

Characteristics of Li-isotopes

21

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

• Mechanism is unclear

Halo structure

11Li

2222Pairing-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.

23

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

9 11 9

1

( Li) ( Li) ( Li) ( ) 0N

j i ii

H E nn

A

23

11 9( Li) ( Li) nnH H H 11 9

1

( Li) ( Li) ( )N

i ii

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 2 neutrons with Gaussian expa( ) nsion me: od th nn A{

TOSM

24

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

2525

2n correlation density in 11Li

nDi-neutron type config.

9Li

Cigar type config.

n

s2=4% s2=47%

K. Hagino and H. Sagawa, PRC72(2005)044321

9Li

H.Esbensen and G.F.Bertsch, NPA542(1992)310

26

Short-range correlator : C (or Cr)

: ( ) for hamiltonian,

( ) for relative wave function

C r R r

r R r

† 1 1

( ) ( )r rC p C p

R r R r

†C lC l

† ( )C r C R r †12 12C S C S †C sC s

corr. uncorr.( ) ( )r C r

†( ) i cm iji i j

H T V t T v r C HC H T V

2, ij

i j

T T T T u

( ( ))iji j

V v R r

2-body approximation in the cluster expansion of operator

( ( ))R R r r

Hamiltonian in UCOM

†C tC t

27

LS splitting in 5He with tensor corr.

5 4( He) ( He) relH H H 5 4

1

( He) ( He) ( )

N

i ii

nA

27

44H

1

( He) ( ) ( ) ( )N

ij rel e n ij j ii

H T V n E n

4Orthogonarity to the Pauli-forbidden states 0 : { ; He}i

• Orthogonarity Condition Model (OCM) is applied.

•T. Terasawa,PTP22(’59)

•S. Nagata, T. Sasakawa, T. Sawada R. Tamagaki, 　　 PTP22(’59)

• K. Ando, H. Bando PTP66(’81)

• TM, K.Kato, K.Ikeda PTP113(’05)

28

Phase shifts of 4He-n scattering

28

29

6He in coupled 4He+n+n model• System is solved based on RGM

6 4( He) ( He) nnH H H 6 4

1

( He) ( He) ( )N

i ii

nn

A

4 6 4

1

( He) ( He) ( He) ( ) 0N

j i ii

H E nn

A

4 shell model type configu( He) rat: on ii

• Orthogonality Condition Model (OCM) is applied.

41 2 1 2 12

1

( He) ( ) ( ) ( )N

ij c c ij j ii

H T T V V V nn E nn

4 44( He) : Hamilt ( He onian for ) Heij i jH H

4 Orthogonality to the Pauli0 -, forbidden { He} : st t s a ei

1 2 2 neutrons with Gaussian expa( ) nsion me: od th nn A{

TOSM

30

Tensor correlation in 6He

30

Ground state Excited state

TM, K. Kato, K. Ikeda, J. Phys. G31 (2005) S1681

3131

Theory

With Tensor

6He results 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

complex scaling for resonances