Role of tensor force in He and Li isotopes with

tensor optimized shell model

Hiroshi TOKI RCNP, Osaka Univ. 　Kiyomi IKEDA RIKENAtsushi UMEYA RIKEN

Takayuki MYO 　Osaka Institute of Technology

The Fifth Asia-Pacific Conference on Few-Body Problems in Physics @ Seoul, Korea, 2011.8.22-26

Purpose & Outline

2

• We would like to understand role of Vtensor

in the nuclear structure by describing strong tensor correlation explicitly.

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

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

• TOSM+UCOM to He & Li isotopes with Vbare

TM, K. Kato, H. Toki, K. Ikeda, PRC76(2007)024305TM, H. Toki, K. Ikeda, PTP121(2009)511TM, A. Umeya, H. Toki, K. Ikeda, PRC(2011) in press.

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 properties & tensor force

Rm(s)=2.00 fm

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

r

AV8’

TM, Sugimoto, Kato, Toki, Ikeda PTP117(2007)257

4

Tensor-optimized shell model (TOSM)

4

4He• Configuration mixing

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

• Length parameters such as b0s, b0p , … are optimized

independently (or superposed by many Gaussian bases).

– Describe high momentum component from Vtensor

– Spatial shrinkage of relative D-wave component as seen in deuteron HF by Sugimoto et al,(NPA740) / Akaishi (NPA738) RMF by Ogawa et al.(PRC73), AMD by Dote et al.(PTP115)

Hamiltonian and variational equations in TOSM

0,k

H E

C

1

,A A

i G iji i j

H t T v

k k

k

C : shell model type configurationk

0n

H E

b

c.m. excitation is excluded by Lawson’s method

(0p0h+1p1h+2p2h)

TOSM code : p-shell region

1, ,( ) ( )

N

nlj lj n lj nC

r r

2/1/2,

ˆ( ) ( ) ( ),nr b

zl

nlj n l l j tN b r e Y r r

Particle state : Gaussian expansion for each orbit

0,lj

H E

C

Gaussian basis function

Configurations of 5He in TOSM

proton neutron

Gaussian expansion

nlj

particle states

hole states (harmonic oscillator basis)

c.m. excitation isexcluded by Lawson’s method

Application to Hypernuclei by Umeya

C0 C1 C2 C3

Tomorrow

7

Unitary Correlation Operator Method

7H. 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 Hamiltonian Shift operator depending on the relative distance

† 12( ) ( ) ( )C rC r s r s r s r

TOSM

2-body cluster expansion

Amount of shift, variationally determined.

(short-range part)

8

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

4-8He with TOSM+UCOM

• Difference from 4He in MeV

• No VNNN

• No continuum

~6 MeV in 8He using GFMC

~7 MeV in 8He using Cluster model (PLB691(2010)150 , TM et al.)

4p4h in TOSM

4-8He with TOSM+UCOM

• Excitation energies in MeV

• No VNNN

• No continuum • Excitation energy spectra are reproduced well

5-9Li with TOSM+UCOM

• Excitation energies in MeV

• No VNNN

• No continuum

Preliminary results

• Excitation energy spectra are reproduced well

1212

Matter radius of He isotopes

I. Tanihata et al., PLB289(‘92)261 G. D. Alkhazov et al., PRL78(‘97)2313O. A. Kiselev et al., EPJA 25, Suppl. 1(‘05)215. P. Mueller et al., PRL99(2007)252501

TOSM

Expt

TM, R. Ando, K. KatoPLB691(2010)150

Halo Skin

Cluster model

13

Configurations of 4He

(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

13

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

• 4He contains p1/2 of “pn-pair”.

• 0 of pion nature.

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

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Tensor correlation in 6He

14

Ground state Excited state

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

Tensor correlation is suppressed due to Pauli-Blocking

val.-n

halo state (0+)

6He : Hamiltonian component

• Difference from 4He in MeV

6He 0+1 2+

1 0+2

n2 config (p3/2)2 (p3/2)2 (p1/2)2

Kin. 53.0 52.4 34.3

Central 27.8 22.9 14.1

Tensor 12.0 12.8 0.2

LS 4.0 5.0 2.1

=18.4 MeV (hole)

bhole=1.5 fm

same trend in 5,7,8He

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Summary

• TOSM+UCOM with bare nuclear force.

• 4He contains “pn-pair” of p1/2 than p3/2.

• He isotopes with p3/2 has large contributions

of Vtensor & Kinetic energy than those with p1/2.

• Vtensor enhances LS splitting energy.

– TM, A. Umeya, H. Toki, K. Ikeda, PRC(2011) in press.

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.

4-8He in TOSM+UCOM

• Argonne V8’ w/o 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.

TM, H. Toki, K. Ikeda, PTP121(2009)511TM, A. Umeya, H. Toki, K. Ikeda, PRC, in press.

5He : Hamiltonian component

• Difference from 4He in MeV

5He 3/2 1/2

n config. p3/2 p1/2

Kin. 24.1 17.9

Central 9.0 7.0

Tensor 5.6 1.1

LS 2.6 1.0

=18.4 MeV (hole)

bhole=1.5 fm

LS splitting in 5He & 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 splitting is weaker than p-wave splitting

Pauli-Blocking

val.-n

Tensor correlation & Splitting in 5He

Vtensor~exact

in 4He

Enhancement of Vtensor

4-8He with TOSM

• No VNNN

• No continuum

• Difference from 4He in MeV

Minnesota force (Central+LS)

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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’)

• Tensor force (Vtensor) plays a significant

role in the nuclear structure.

– In 4He, Vtensor ~ Vcentral

(AV18, GFMC)

– Main origin : -exchange in pn-pair

23

Importance of tensor force

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

structure by describing tensor correlation explicitly.

tensor correlation + short range correlation model wave functions : shell model, cluster model (TOSM: Tensor Optimized Shell Model) He, Li isotopes (LS splitting, halo formation, level inversion)

7He : Hamiltonian component

• Difference from 4He in MeV

7He 3/21 3/2

2 3/23

n3 config (p3/2)3 (p3/2)2(p1/2) (p3/2)(p1/2)2

Kin. 82.9 74.1 65.8

Central 39.1 33.6 27.0

Tensor 19.9 16.0 10.0

LS 8.6 2.8 1.0

bhole=1.5 fm

8He : Hamiltonian component

• Difference from 4He in MeV

8He 0+1 0+

2 2+1

n4 config (p3/2)4 (p3/2) 2(p1/2)2 (p3/2) 3 (p3/2)

Kin. 114.2 101.0 109.6

Central 59.2 51.6 56.2

Tensor 25.2 17.1 23.4

LS 11.6 2.55 7.1

bhole=1.5 fm