Satoru SugimotoKyoto University

1. Introduction2. Charge- and parity-projected Hartree-Fock method (a mean field

type model) and its application to sub-closed shell oxygen isotopes

3.Shell model calculation to 15,16,17O4. Summary

Study of the Tensor　Correlation in Oxygen　Isotopesusing Mean-Field-Type and Shell Model Methods

Introduction The tensor force is important in nuclear struct

ure. There remain many open problems to be solve

d. How does the tensor correlation change in neutron-

rich nuclei? Shell evolution (Ostuka, PRL 95 232502 (2005)). The breakdown of the magic number in 11Li (Myo et al.)

The relation to the ls splitting in 5He (Myo et al. PTP 113 (2005) 763)

The correlation to be included

• In the simple HF calculation, 2p-2h correlations are hard to be treated.

• We need to include at least 2p-2h correlations to exploit the tensor correlation.beyond mean field modelbeyond mean field model

cf. Single-particle (H-F) correlation

Tj j jV j

0p1/2

0p3/2

0d3/2

0d5/2

1s1/2

VT

Proton Neutron

2p-2h correlation

0p1/2

0p3/2

0d3/2

0d5/2

1s1/2

Proton Neutron

0p1/2

0p3/2

0d3/2

0d5/2

1s1/2

VT

Proton Neutron0p-0h 2p-2h

Tj j jV j

VT

VT

VT

Active core

Charge- and parity-symmetry breaking mean field method

Tensor force is mediated by the pion. Pseudo scalar ()

To exploit the pseudo scalar character of the pion, we introduce parity-mixed single particle state. (over-shell correlation)

Isovector () To exploit the isovector character of the

pion, we introduce charge-mixed single particle state.

Projection Because the total wave function made from

such parity- and charge-mixed single particle states does not have good parity and a definite charge number. We need to perform the parity and charge projections.

Refs. Toki et al., Prog. Theor. Phys. 108 (2002) 903. Sugimoto et al., Nucl. Phys. A 740 (2004) 77; ; nucl-th/0607045. Ogawa et al., Prog. Thoer. Phys. 111 (2004) 75; Phys. Rev. C 73 (2006) 034301.

Results for 16O

MV1(VC)+G3RS(VT,VLS) By performing the parity and charge projection

the potential energy from the tensor force becomes sizable value.

　 xT E K V VT VLS

HF 1.0 -124.1 230.0 -354.1 0.0 -0.9

CPPHF 1.0 -127.1 237.1 -364.24 -11.7 -1.0

CPPHF 1.5 -127.6 253.9 -381.6 -38.3 -1.0

VT and VLS per particle

The potential energy from the tensor force has the same order in magnitude as that from the LS force.

The tensor potential energy decreases as neutron numbers.

14 16 18 20 22 24 26 28- 6

- 5

- 4

- 3

- 2

- 1

0

VT and V

LSV

T a

nd

VL

S (

MeV

)

Mass number

VT x

T=1.0

VLS

xT=1.0

VT x

T=1.5

VLS

xT=1.5

VLS

xT=1.0 HF

p1/2 d5/2

d3/2

s1/2

p3/2

Wave function （１６ O, ｘ T=1.5)

Opposite parity components mixed by the tensor force have narrow widths. It suggests that the tensor correlation needs high-momentum components.

s1/2 proton dominant

0 1 2 3 4 5

-0.4

-0.2

0.0

0.2

0.4

P(-): 19.5%P(): 27.4%

Wav

e fu

nctio

n (f

m-3

/2)

R (fm)

Proton (+) Proton (-) Neutron (+) Neutron (-)

Mixing of the opposite parity components in single-particle states

If a next j=1/2 orbit is occupied newly, the mixing probabilities of the j=1/2 orbit reduce by a blocking effect.

Mixing of the opposite-parity component may affect excitation spectra of nuclei.

Neutron xT=1.0

0

0.1

0.2

0.3

14 16 18 20 22 24 26 28A

Pro

babi

lity

0s1/ 2'0p3/ 2'0p1/ 2'0d5/ 2'1s1/ 2'0d3/ 2'

0p1/2 1s1/2

Shell model calculation

We perform the shell model calculation including 1p-1h and 2p-2h configurations to study the tensor correlation. inclusion of narrow-width single-particle

wave functions The shell model calculation can treat the

correlation which cannot be treated in a mean-field-type calculation.

cf. Myo et al. PTP 113 (2005) 763

Model space 16O: (0p-0h)+(1p-1h)+(2p+2h)

17O: (1p-0h)+(2p-1h)+(3p+2h)15O: (0p-1h)+(1p-2h)+(2p+3h)

Core (hole state) (0s1/2)4(0p3/2)6(0p1/2)4

Harmonic oscillator single-particle wave functions Particle state

Harmonic oscillator single-particle wave functions+Gaussian single-particle wave functions with narrow (half) width parameters

These are ortho-normalized by the G-S methodG 2 2

, ,( ) exp( ( / 2 )) ( )ljm a l a ljmN r a r Y

Effective interaction Central force: Volkov No. 1

A.B. Volkov, Nucl. Phys. 74 ( 1965 ) 33 Tensor: Furutani force

H. Furutani et al., Prog. Theor. Phys. Suppl. 68 ( 1980 ) 193

LS: G3RS . Tamagaki, Prog. Theor. Phys. 39 ( 1968 ) 91

No Coulomb force

16O

HO: (1s 0d)+(1p 0f)+(2s 1d 0g) NWG: bNW = bHO/2 = 1.8 fm

d-orbit: (1s 0d)+sNW+pNW+dNW

f-orbit: (1s 0d)+sNW+pNW+dNW+fNW

By including single-particle orbits with narrow width parameters the correlation energy from the tensor force becomes large.

　 E KE VC VT VLS P(CS) CM /

HO 2-

137.0 249.6 -

383.8 -2.6 -0.1 75% 1.50

NWG (d-orbit)

-136.

2 257.

5

-356.

9 -

32.9 -

3.9 74% 1.58

NWG (f-orbit)

-132.

5 266.

5

-337.

1 -

56.6 -

5.3 72% 1.58

17,15O (NWG (up to f-orbit))

KE+VC+VT≈0 ls-splitting nearly equals to VLS

E KE VC VT VLS P(NP) CM /

5/2+

-132.2

289.5

-357.4

-57.0 -7.3 72% 1.58

3/2+

-127.7

288.6

-356.7

-56.7 -2.8 73% 1.58

diff (5/2+ - 3/2+) -4.5 0.8 -0.6 -0.2 -

4.5 E KE VC VT VLS P(NP) CM /

1/2-

-114.8

250.2

-301.6

-55.7 -7.8 71% 1.58

3/2-

-110.3

247.3

-300.0

-54.1 -3.5 73% 1.58

diff (1/2- - 3/2-) -4.5 2.9 -1.5 -1.6 -

4.3 　

15O

17O

Magnetic Moment

Magnetic moments change a little in spite of the large correlation energy form the tensor force.

　5/2+ 3/2+

17O 17F 17O 17F

Schmidt value -1.91 4.79 1.15 0.13

HO 2 -1.87 4.75 1.17 0.11

NWG (f-orbit)

-1.89 4.77 1.15 0.13

　1/2- 3/2-

15N 15O 15N 15O

Schmidt value -0.26 0.64 3.79 -1.91

HO 2 -0.27 0.65 3.72 -1.85

NWG (f-orbit)

-0.25 0.63 3.74 -1.87

A=17

A=15

Summary We apply a mean-field model which treats the tensor

correlation by mixing parities and charges in single-particle states (the CPPHF method) to oxygen isotopes.

The opposite parity components induced by the tensor force is compact in size. (high-momentum component)

We perform the shell model calculation up to 2p-2h states to 15,16,17O.

The tensor correlation energy becomes large by including Gaussian single-particle wave functions with narrow widths.

The tensor correlation changes ls splitting and magnetic moments in 15,17O a little in spite of its large correlation energy.