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Tensor optimized shell model using bare interaction for light nuclei. 明 孝之  大阪工業大学       阪大 RCNP. 共同研究者 土岐 博   阪大 RCNP 池田 清美  理研. 1. 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. Tensor Optimized Shell Model (TOSM) Unitary Correlation Operator Method (UCOM)

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slide1

Tensor optimized shell model

using bare interaction

for light nuclei

明 孝之  大阪工業大学

      阪大RCNP

共同研究者

土岐 博  阪大RCNP

池田 清美  理研

1

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
Outline
    • 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
motivation for tensor force
Motivation for tensor force
  • Tensor force (Vtensor) plays a significant role in the nuclear structure.
    • In 4He,
    • ~ 80% (GFMC)

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)
  • Structures of light nuclei with bare interaction
    • tensor correlation + short-range correlation

3

tensor short range correlations
Tensor & Short-range correlations
  • Tensor correlation in TOSM (long and intermediate)
    • 2p2h mixing optimizing the particle states (radial & high-L)
  • Short-range correlation
    • Short-range repulsionin 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

4

4

property of the tensor force
Property of the tensor force

Long and intermediate ranges

5

tensor optimized shell model tosm
Tensor-optimized shell model (TOSM)

TM, Sugimoto, Kato, Toki, Ikeda

PTP117(2007)257

  • Tensor correlation in the shell model type approach.
  • Configuration mixingwithin 2p2h excitationswith high-L orbitTM 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 VtensorCPP-HF by Sugimoto et al,(NPA740) / Akaishi (NPA738)CPP-RMF by Ogawa et al.(PRC73), CPP-AMD by Dote et al.(PTP115)

4He

6

6

hamiltonian and variational equations in tosm
Hamiltonian and variational equations in TOSM

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

  • 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

7

4 he in tosm
4He in TOSM

vnn: G-matrix

Shrink

Length parameters

0 1 2 3 4 5 6

Lmax

Higher shell effect

good convergence

8

8

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

configuration of 4 he in tosm
Configuration of 4He in TOSM

4 Gaussians instead of HO

c.m. excitation = 0.6 MeV

  • 0 of pion nature.
  • deuteron correlation with (J,T)=(1,0)

Cf. R.Schiavilla et al. (GFMC)

PRL98(’07)132501

9

tensor short range correlations1
Tensor & Short-range correlations
  • Tensor correlation in TOSM (long and intermediate)
    • 2p2h mixing optimizing the particle states (radial & high-L)
  • Short-range correlation
    • Short-range repulsionin the bare NN force
    • Unitary Correlation Operator Method (UCOM)

S

D

TOSM+UCOM

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

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

10

10

unitary correlation operator method
Unitary Correlation Operator Method

TOSM

short-range correlator

Bare Hamiltonian

Shift operator depending on the relative distance r

2-body cluster expansion

of Hamiltonian

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

11

11

short range correlator c or c r
Short-range correlator : C (or Cr)

1E

Originalr2

3GeV repulsion

3E

Vc

1O

C

3O

AV8’ : Central+LS+Tensor

12

slide14

4He with AV8’in TOSM+UCOM

AV8’ :

Central+

LS+Tensor

exact

Kamada et al.

PRC64

(Jacobi)

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

14

extension of ucom s wave ucom
Extension of UCOM : S-wave UCOM

for only relative S-wave wave function

– minimal effect of UCOM

SD coupling

5 MeV gain

15

15

different effects of correlation function
Different effects of correlation function
  • S-wave

S

No Centrifugal Barrier

Short-range repulsion

D

  • D-wave

due to Centrifugal Barrier

16

16

saturation of 4 he in ucom
Saturation of 4He in UCOM

UCOM: short-range

+ tensor

T. Neff, H. Feldmeier

NPA713(2003)311

Short tensor

Energy

TOSM+UCOM

Long tensor

Benchmark cal.

Kamada et al. PRC64

17

slide18

4He in TOSM + S-wave UCOM

T

(exact)

Kamada et al.

PRC64 (Jacobi)

Remaining effect :

3-body cluster term

in UCOM

VLS

E

VC

VT

18

summary
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

19

slide20

Pion exchange interaction vs. Vtensor

Involve large

momentum

Delta interaction

Tensor operator

Yukawa interaction

- Vtensor produces the high momentum component.

20

characteristics of li isotopes
Characteristics of Li-isotopes

Halostructure

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

11Li

21

slide22

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.

22

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

TOSM

  • Orthogonality Condition Model (OCM) is applied.

23

11 li g s properties s 2n 0 31 mev
11Li G.S. properties (S2n=0.31 MeV)

Rm

Simon et al.

P(s2)

Tensor

+Pairing

Pairing correlation couples (0p)2 and (1s)2 for last 2n

2n correlation density in 11 li
2n correlation density in 11Li

s2=4%

s2=47%

9Li

9Li

n

n

Di-neutron type config.

Cigar type config.

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

25

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

short range correlator c or c r1
Short-range correlator : C (or Cr)

Hamiltonian in UCOM

2-body approximation in the cluster expansion of operator

ls splitting in 5 he with tensor corr
LS splitting in 5He with tensor corr.
  • 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)
  • Orthogonarity Condition Model (OCM) is applied.

27

6 he in coupled 4 he n n model
6He in coupled 4He+n+n model

System is solved based on RGM

TOSM

  • Orthogonality Condition Model (OCM) is applied.

29

tensor correlation in 6 he
Tensor correlation in 6He

Ground state

Excited state

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

30

30

slide31

6He results in coupled 4He+n+n model

Theory

With Tensor

complex scaling

for resonances

  • (0p3/2)2 can be described in Naive 4He+n+n model
  • (0p1/2)2 loses the energy

Tensor suppression in 0+2

31

31

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