Post on 03-Jan-2016
description
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
P
P
Pn
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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
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jij CCB i
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)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
P n
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Pn
nn nn
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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.