反対称化分子動力学でテンソル力を取り扱う試み

－更に前進するには？－

反対称化分子動力学でテンソル力を取り扱う試み

－更に前進するには？－A. Dote (KEK), Y. Kanada-En’yo （ KEK ） ,

H. Horiuchi (Kyoto univ.), Y. Akaishi (KEK), K. Ikeda (RIKEN)

1. Introduction

2. Requests for AMD’s wave function to treat the tensor force

3. Further requests found in the study of 4He•Use of wave packets with different width-

parameters•Importance of angular momentum projection

4. Single particle levels in 4He

5. Summary and future plan RCNP 研究会「核力と核構造」‘04.03.22 　 at 　 RCNP

でもその前に…

Introduction

Why tensor force ？1, large contribution in nuclear forces ex) deuteron : bound by the tensor force

In microscopic models, tensor force is incorporated into central force.

2, creation and annihilation of clustering structure

Tensor force works well on nuclear surface

more work in more developed clustering structure?

3, Development of effective interaction for AMD

AMD, Hartree-Fock : without tensor forceRelativistic Mean-Field approach : no πmeson

How in finite nuclei? Affection to nuclear structure?

•　 Unified effective interaction for light to heavy nuclei.

4

6MV1, Gogny

Tensor force

A

A

•　 reduction of calculation costs

IntroductionPrevious study of the tensor force† says…

• In the nuclear matter, the 3E effective interaction containing the tensor force is weakened. Saturation property

The tensor force might be weakened in heavy nuclei rather than in light nuclei, and inside of nuclei rather than near nuclear surface.

• In the perturbation theory, 0 0

T T

QV VE H

The incorporation of tensor force into central force is sensitive to the starting energy.

????

Tensor force might favor such a structure that the ratio of the surface is large, namely well-developed clustering structure.

The effective central force changes, corresponding to the nuclear structure.

If the nuclear structure changes, the starting energy changes also. Therefore, the effective central force changes.

Cluster

ShellTensor force

Following the tensor force, each of nuclei chooses shell- or clustering-structure,

which are qualitatively different from each other.

Our scenario

† Y.Akaishi, H.Bando, S.Nagata, PTP 52 (1972) 339

Requests for AMD’s wave function to treat the tensor force

ii. Superposition of wave packets with spin

tensor force : Strong correlation between spin and spacer

2

exp ii

ii iC

Z

r

iii. Changeability of isospin wave func. 2

expii i i

i

C

Zr

tensor force : 1 21 1 2 12 2

1

2z z

Change the charge of a single particle wave function

i. Parity-violated mean field P

Ex.) Furutani potential

DeuteronVt -1 MeV -10 MeV -21 MeVL2 0.01 0.5 0.5

4HeVt -1 MeV -3.3 MeV -22.4 MeVL2 0.01 0.06 0.4

ii

ii

iii

iii

i

i

tensor force : r

Change the parity of a single particle wave function

0

5

10

15

20

25

30

B.E. [MeV]

0 1 2 3

Further requests found in the study of 4He

Akaishi potentialBased on Tamagaki potential, The repulsive core of the central force is treated with G-matrix method. Tensor force : bare interaction.

Interaction ： Tensor force is not incorporated into central force.

1. Wave packets with different width-parameters

• Gain the binding energy without shrinkage

•ν for S-wave ≠ ν for P-wave

2. Angular momentum projection

Because of very narrow wave packets, only a little mixture of J≠0 components makes the kinetic energy increase.

3. J projection after J constraint variation

10 MeV

17 MeV25 MeV

28 MeV

0 1 2 3

Wave packets with different width-parameters

One nucleon wave function 　：　 superposition of Gaussian wave packets with different ν’s

2

expi

i ii

ii

iC

Z

r

＋ ＋～ ＋　・・・

ν

Kinetic energy

Tensor force

Radius

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5

r [fm]

Akaishi_sAkaishi_pEnyo_s'Enyo_p'

Why different ν’s ?

sEnyof

pEnyof

2

2

( ) exp[ ]

( ) exp[ ]

sEnyo s

pEnyo p

f r A r

f r B r r

0.32s vs

0.9p

Tensor force vs Swave× Pwave

- 60

- 40

- 20

0

20

40

60

80

0 1 2 3 4

fm

Tensor*r2S*P*r2

Tensor force vs Swave× Pwave

- 60

- 40

- 20

0

20

40

60

80

0 1 2 3 4

fm

Tensor*r2S*P*r2

Tensor max

S×P max

-30

-25

-20

-15

-10

-5

01 1.2 1.4 1.6 1.8

Rrms [fm]B.E

. [M

eV]

Test for different ν’s

(0.9, 0.9, 0.9)

(0.5, 0.5, 0.5)

(0.2, 0.2, 0.2)

(0.2, 0.5, 0.9)(0.2, 0.5, 0.9)

By using the wave packets with different ν’s, the binding energy can be gained without shrinkage.

•4He•3 wave packets•Vc×1.0 ／ Vt×3.0•Frictional cooling•spin/isospin free

•4He•3 wave packets•Vc×1.0 ／ Vt×3.0•Frictional cooling•spin/isospin free

Result of 4He •4 wave packets•Vc×1.0 ／ Vt×2.0•Frictional cooling•spin/isospin free

•4 wave packets•Vc×1.0 ／ Vt×2.0•Frictional cooling•spin/isospin free0. 4 wave packets with common ν’s － ν=0.6 －

B.E. Rrms Kin Vc Vt Ve J 2 L2 S2

- 10.8 1.23 94.1 - 39.4 - 66.4 0.9 0.32 1.50 1.31

1. 4 wave packets with different ν’s － ν=0.3 ～ 1.5 geometric ratio －

B.E. Rrms Kin Vc Vt Ve J 2 L2 S2

- 17.4 1.32 79.0 - 41.9 - 55.4 0.9 0.23 0.80 0.68

2. Angular momentum projection onto J=0

B.E. Rrms Kin Vc Vt Ve J 2 L2 S2

- 25.3 1.32 73.4 - 43.5 - 56.2 0.9 0.00 0.57 0.57

- 15.7 1.23 91.7 - 40.3 - 68.0 0.9 0.00 1.27 1.27

Importance of angular momentum projection 　

– details of energy gain–

〇 in case of ν = 0.3 ～ 1.5

Contributions from various J components to the kinetic energy

J J z Kin. overlap Kin× overlapMixed Mixed 79.04 - - - - - -

0 0 73.43 0.955 70.091 1 199.75 0.007 1.381 0 - - - 0.000 0.001 - 1 201.24 0.007 1.352 2 238.19 0.005 1.232 1 172.96 0.002 0.312 0 - - - 0.000 0.002 - 1 177.75 0.002 0.302 - 2 238.38 0.005 1.24… … … … …

B.E. Kin. Vc Vt VeBefore - 17.40 79.03 - 41.90 - 55.40 0.90After - 25.32 73.43 - 43.47 - 56.17 0.89Δ 7.92 5.60 1.57 0.77 0.01

Because of very narrow wave packets, J≠0 components have large kinetic energy.Only 1 ％ mixture of a J≠0 component increases the kinetic energy by 1 MeV.

-30

-25

-20

-15

-10

-5

00 0.2 0.4 0.6 0.8 1

<J 2> before J projectionB.E

. [M

eV]

BeforeAfter

J projection after J constraint variation •4He•4 wave packets•ν= 0.3 ～ 1.5•Vc×1.0 ／ Vt×2.0•Frictional cooling•spin/isospin free•J projection (VBP)

•4He•4 wave packets•ν= 0.3 ～ 1.5•Vc×1.0 ／ Vt×2.0•Frictional cooling•spin/isospin free•J projection (VBP)

B.E. Rrms Kin Vc Vt Ve J 2 L2 S2No const. - 25.3 1.32 73.4 - 43.5 - 56.2 0.9 0.00 0.57 0.57J 2 const. - 28.6 1.35 70.4 - 43.5 - 56.3 0.9 0.00 0.50 0.51

No constraintB.E. = -25.3 MeVRrms = 1.32 fm

J2 constraintB.E. = -28.6 MeVRrms = 1.35 fm

Single particle levels in 4He

What happens in the 4He obtained by the AMD calculation?

See single particle levels!

Extract the single particle levels from the intrinsic state with AMD+HF method.

40 0He J Tz PP P P

Single Slater determinant

1/ 2 1/ 20 0s p ?

Diagonalize the overlarp matrix

How to extract single particle levels (AMD+HF)

Bij i j

21

ˆ ˆA a

h f t f f f v f f

1i i

i

fB c

c c

p p pi

i

piph g f g g

1. Prepare orthonormal base.

2. Mimicking Hartree-Fock, construct a single particle Hamiltonian.

3. Diagonalizing h, get single particle levels.

Single particle energy

Single particle state

i : one nucleon wave function in AMD wave function

Single particle levelsS.P.E J 2 J z2 L2 P(+) P(proton)

-4.04 1.16 0.33 0.83 81% 61%

-4.52 1.17 0.31 0.85 81% 59%

-8.28 1.09 0.32 0.68 85% 30%

-8.78 1.06 0.31 0.65 86% 45%

• 2 groups Lower: neutron-like, P(-)=15% Upper: proton-like, P(-)=19% P-stateP-state

• High L?

• High J? If 0s1/2+0p1/2, J2=0.75. But J2=1.16.

If S(80%)+P(20%), L2=0.4. But L2=0.83.7% D-state?

Summary To treat the tensor force in AMD framework, following points are needed:

1, superposition of wave packets with spin, 2, changiability of charge wave function.

Other points are found to be important, by the study of 4He: 3, wave packets with different ν’s gain the B.E. without shrinkage. ν for S-wave is rather different from that for P-wave. 4, J projection Because of very narrow wave packets, J≠0 components have large kinetic energy. 4‘, J projection after J constraint variation leads to better solution.

Result of 4He

Akaishi potential, Vt×2.0.

B.E. Rrms L2 Kin Vc Vt Ve- 28.6 1.35 0.50 70.4 - 43.5 - 56.3 0.9

Summary and Future plan We have investigated the single particle levels in 4He.

Characteristics of our S.P. levels: 1, two groups (2+2) 2, 15 ～ 20% negative parity component (P-state) is mixed. 3, including some components except 0s1/2 + 0p1/2 higher L, higher J state.

• More detailed analysis of single particle levels.

How is each component?: proton-parity +, proton-parity -, neutron-parity +, neutron-parity -

• Vt×2 ↓

Treat the short range part by tensor correlator method (Neff & Feldmeier)or

Cut the high momentum component by G-matrix method (Akaishi-san)