Tensor optimized shell model
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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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明 孝之  大阪工業大学       阪大 RCNP

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Rcnp

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

    • Centrifugal potential ([email protected]) pushes away the L=2 wave function.

    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


    4 he in ucom afnan tang v c only

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

    C

    13

    13


    Rcnp

    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


    Rcnp

    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


    Rcnp

    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


    Rcnp

    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


    Phase shifts of 4 he n scattering

    Phase shifts of 4He-n scattering

    28


    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


    Rcnp

    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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