The Study of Nuclear Structures wThe Study of Nuclear Structures with the Brueckner-AMDith the Brueckner-AMD

Tomoaki Togashi and Kiyoshi Kato

Department of Physics, Hokkaido University

INPC2007, Tokyo

June 6, 2007

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PurposesPurposes

- We develop a new ab initio calculation framework based on the antisymmetrized molecular dynamics (AMD) and the Brueckner theory.

Brueckner-AMD; the Brueckner theory

+ Antisymmetrized molecular dynamics

(AMD)

T.Togashi and K.Kato; Prog. Theor. Phys. 117 (2007) 189

},{),( 2121 AA ΑZZZ

Zrr ii ))(exp()( 2

jiijB iij

jij CCB i

iiC

1~

　

}~~{ˆ~~~ ˆ ~ Gt

Gtt

QVVG ˆ

)ˆˆ(ˆˆˆ

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Framework of the Brueckner-AMD

1. AMD wave function

2. Single particle orbits are constructed with the AMD-HF

3. G-matrix is calculated with the single particle orbits

( B-matrix ) ( diagonalization )

single particle orbit*

( Slater determinant )

( Bethe-Goldstone equation )

( Gaussian wave packet )

i

i

i

i

Z

H

dt

Zd

Z

H

dt

Zd

4. The frictional 　 cooling method

t t t

The state at the energy minimum

self-consistent

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H.Bando, Y.Yamamoto , S.Nagata , PTP 44 (1970) 646

A.Dote, H.Horiuchi, PTP 103 (2000) 91A.Dote, Y.Kanada-En’yo, H.Horiuchi, PRC56 (1997) 1844

*The single particle orbits should be Hartree-Fock hamiltonian eigen states, however, in the case that 　 those state are adopted as the single particle orbits, the results have scarcely been changed until now.

Brueckner-AMD Results

Intrinsic Density (8Be)

ρ(r)

Ｘ

ＹIntrinsic Density (12C)

ρ(r)

Ｘ

Ｙ

ρ(r)Intrinsic Density ( ４

He)

*In the case of 8Be and 12C, the Gaussian width parameter is the same value as the case of 4He. Ｘ

Ｙ

Av8’; P.R.Wringa and S.C.Pieper, PRL89 (2002), 182501.

Parity- & Jπ- projection in the Brueckner-AMD

Parity-projected state :

Space inversion operator　 parity

⇒ 　 the liner combination of two Slater determinants

Ex) Parity Projection

The G-matrix between the different Slater determinants is necessary. ba

ba G

ˆ

Bethe-Goldstone Equation

)())(1()( ijijGe

Qij

ijF̂

))()()()(( ijijGijijv

Correlation function

Model (AMD) wave function

)(/)(ˆ ijijFij

bl

bk

bkl

aij

aj

ai

bl

bk

aj

ai FvvFG )()( ˆˆˆˆ

2

1ˆ

The G-matrix is constructed with the correlation functions.

⇒

Y. Akaishi, H. Bando, and S. Nagata, 　 PTP. Suppl. No.52 (1972), 339.

Results of 4He with variation after projection (VAP)

Argonne v8’（ｎｏ　Ｃｏｕｌｏｍｂ　ｆｏｒｃｅ）

A. Variation (cooling) with no projection

Binding Energy (4He) ： -22.5(MeV)

B. Variation after projection (VAP): parity+

Binding Energy (4He: +) ： -23.6(MeV) +

Projection after variation (PAV): J=0

Binding Energy (4He: 0+) ： -24.7(MeV)

-25.9(MeV)benchmark calculations† :†Ref: H. Kamada et al. , PRC64 (2001) 044001

The result is comparable with that of the benchmark calculations

ρ(r)Ｙ4He (Parity +)

Ｘ

Results of Be isotopes with VAP (Parity)

8Be (Parity +)

Argonne v8’（ｎｏ　Ｃｏｕｌｏｍｂ　ｆｏｒｃｅ）

9Be (Parity -)

10Be (Parity +)

Binding energy: -37.2(MeV) Binding energy: -36.5(MeV)

Binding energy: -39.6(MeV)

*The Gaussian width parameter is the optimized value of 8Be in the case with no projection.

Ｘ

ＸＸ

ρ(r) ρ(r)

ρ(r)

Ｙ Ｙ

Ｙ

（ matter ）

（ proton ）

（ neutron ）

Intrinsic density of 9Be (Parity -)

Argonne v8’（ｎｏ　Ｃｏｕｌｏｍｂ　ｆｏｒｃｅ）

Binding energy: -36.5(MeV)

Ｘ

Ｘ

Ｘ

Ｙ

Ｙ

Ｙ

ρ(r)

ρ(r)

ρ(r)

*The Gaussian width parameter is the optimized value of 8Be in the case with no projection.

π-orbit

Intrinsic density of 9Be (Parity +)

（ matter ）

（ proton ）

（ neutron ）

Ｙ

Ｘ

Ｘ

Ｘ

Ｙ

Ｙ

ρ(r)

ρ(r)

ρ(r)

Argonne v8’（ｎｏ　Ｃｏｕｌｏｍｂ　ｆｏｒｃｅ）

Binding energy: -34.0(MeV)

*The Gaussian width parameter is the optimized value of 8Be in the case with no projection.

σ-orbit

Intrinsic density of 10Be (Parity +)

（ matter ）

（ proton ）

（ neutron ）

Argonne v8’（ｎｏ　Ｃｏｕｌｏｍｂ　ｆ

ｏｒｃｅ）Binding energy: -39.6(MeV)

*The Gaussian width parameter is the optimized value of 8Be in the case with no projection.

Ｘ

Ｙ ρ(r)

Ｘ

Ｘ

Ｙ

Ｙ

ρ(r)

ρ(r)

Intrinsic density of 10C (Parity +)

（ matter ）

（ proton ）

（ neutron ）

ρ(r)

ρ(r)

ρ(r)

Ｘ

Ｙ

Ｘ

Ｘ

Ｙ

Ｙ

Argonne v8’（ｎｏ　Ｃｏｕｌｏｍｂ　

ｆｏｒｃｅ）Binding energy: -39.6(MeV)

*The Gaussian width parameter is the optimized value of 8Be in the case with no projection.

Summary & Future works

Summary

・ We constructed the framework of AMD with realistic interactions 　 based on the Brueckner theory.

・ Furthermore, we proposed the projection method in the Brueckner-AMD 　 with the correlation functions based on the Bethe-Goldstone equation.・ With the Brueckner-AMD and the variation(cooling) after projection, 　　　　 we obtained the reasonable results for light nuclei starting with the realistic 　 interaction.

Future works

・ Systematic calculations in light nuclei

・ Calculations with other realistic interactions: 　 Argonne v18, CD-Bonn, ・・・・ Three-body force

http://wwwndc.tokai-sc.jaea.go.jp/CN04/CN001.html

: Calculated

: Calculating

Molecular structures will appear close to the respective cluster threshold.

Nuclear Cluster Structures IKEDA Diagram

Threshold rules

Threshold Physics

Unstable nuclei Ground states of drip-line nuclei are observed near the thresholds.

It is desired to understand exotic properties of drip-line nuclei and various kinds of cluster structures in light nuclei from more basic points of view, namely with a realistic nuclear force and a wide model space.

Intrinsic density of 9B (Parity -)

（ matter ）

（ proton ）

（ neutron ）

Argonne v8’（ｎｏ　Ｃｏｕｌｏｍｂ　ｆｏｒｃｅ）Binding energy: -36.5(MeV)

Ｘ

Ｙ ρ(r)

ρ(r)

ρ(r)

Ｘ

Ｘ

Ｙ

Ｙ

*The Gaussian width parameter is the optimized value of 8Be in the case with no projection.

( preliminary)

Intrinsic density of 7Li (Parity -)

（ matter ）

（ proton ）

（ neutron ）

Argonne v8’（ｎｏ　Ｃｏｕｌｏｍ

ｂ　ｆｏｒｃｅ）Binding energy: -25.1(MeV)

Ｘ

Ｙ ρ(r)

ρ(r)

ρ(r)

Ｘ

Ｘ

Ｙ

Ｙ

*The Gaussian width parameter is the optimized value of 8Be in the case with no projection.