Description of nuclear structures in Description of nuclear structures in light nuclei with Brueckner-AMDlight nuclei with Brueckner-AMD

Tomoaki Togashi

Division of Physics, Hokkaido University

The 19th International IUPAP Conference on Few-Body Problems in Physics

@Bonn, 31.Aug. - 5.Sep. 2009P

P

Pｎ

ｎｎｎｎ

Yuhei Yamamoto

Kiyoshi Katō

One of the recent remarkable developments in theoretical nuclear physics; ab initio calculations based on the realistic

nuclear force 　　　　

For heavier nuclei, Green’s function Monte Carlo (GFMC) no-core shell model (NCSM) coupled-cluster (CC) method unitary-model operator approach (UMOA) Fermionic molecular dynamics (FMD) + unitary correlation operator method (UCOM) tensor optimized shell model (TOSM) and so on…

R.B.Wiringa et al., PRC62 (2000) 014001

Introduction (1)

We can discuss cluster structures based on the realistic nuclear force.

3,4-body systems can be solved strictly by Faddeev, Gauss Expansion Method, and so on…

Introduction (2)

Advantages of AMD ：

－ We can treat alpha-nuclei and non-alpha nuclei

－ The wave function is written in a Slater determinant form

－ Both states of shell model and cluster model can be described

－ no assumption for configurations

Recently, the studies with the Antisymmetrized Molecular Dynamics (AMD) have been developed.

Y.Kanada-En’yo and H.Horiuchi,

PTP Supple 142 (2001), 205

AMD has succeeded in understanding and predicting many properties of both stable and unstable nuclei.

However, the AMD calculations have been performed with

phenomenological interactions.

Our strategy

We develop the new AMD framework based on realistic nuclear forces.

We apply the Brueckner theory to AMD and calculate G -matrix in AMD starting from realistic nuclear forces.

Brueckner-AMD; the Brueckner theory + AMD

T.Togashi and K.Katō; Prog. Theor. Phys. 117 (2007) 189.T.Togashi, T.Murakami and K.Katō; Prog. Theor. Phys. 121

(2009) 299.

In this framework, we introduce no phenomenological corrections.

AMD wave function

))(exp()( 2

i

i

Zrr

},,det{!

121 A

A

iiii r )(

neutronor proton ,or ii

…

Cooling method

i

i

i

i

Z

H

dt

Zd

Z

H

dt

Zd

Energy Variation of A-body problem

A-body wave function

(Slater determinant)

single-particle wave functions

A.Ono, H.Horiuchi, T.Maruyama,

A.Ohnishi, PTP 87 (1992) 1185

Nucleon-nucleon (NN) correlations cannot be described sufficiently.

We calculate G-matrix in AMD.

The initial configuration is chosen randomly.

　}~~{ˆ~~~ ˆ ~ Gth

The single-particle orbits in AMD AMD + Hartree-Fock methodA.Dote, Y.Kanada-En’yo, H.Horiuchi,

PRC56 (1997) 1844

A

iiiC

~~

Bethe-Goldstone equation

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

Gtt

QVVG ˆ

)ˆˆ(ˆˆˆ

21

single particle energy(s.p.e.)

self-consistent

)(ˆ cmT

starting energy :

Brueckner-AMD

single-particle w.f. of AMD

the eigenstate of Hartree-Fock Hamiltonian

~~~~1QQ -operator :

occupied states

Cooling methodThe configuration is changed.

G–matrices are changed because of dependence on s.p.e.

)()ˆ1()( ijGe

Qij

)(ˆ)(ˆ ijGijV

correlation function

)(ˆ ijFij

Correlation functions in Brueckner-AMD Bethe-Goldstone

Equation

Model (AMD) pair wave function

Solution of the Bethe-Goldstone equation

rdenominatoenergy :e

)(ˆˆ)(

)(ˆ)()(ˆ)(

ijFVij

ijVijijGij

ij

G -matrix element

Correlation function

)(/)(ˆ ijijFij

Correlation functions are determined for every pair.

Decomposition of G -matrix

G-matrix can be decomposed into central, LS, tensor by correlation functions.

S.Nagata, H.Bando, Y.Akaishi, PTPS 65 (1979) 10

correlation functionλ: rank of operators

G-matrix with operators can be defined as .

In the practical calculations, we use this potential with operators.

Application to light nucleiInteractio

ns We use Argonne v8’ (AV8’) as a realistic nuclear interaction.( cutting off electromagnetic interaction)

Projection

Spin (J ) : projection after variation (PAV)

)operator projectionSpin :(ΦΦ JKM

JKM

JKM PP

Parity eigenstate obtained with VAP

We present the Brueckner-AMD (B-AMD) results of some light nuclei (4He, 8Be, 7Li, 9Be, and 12C).

AV8’; P.R.Wringa and S.C.Pieper, PRL89 (2002), 182501.

: variation after projection (VAP)

Parity

)(

⇒ We calculate the binding energy of the ground state and the energy levels of the excited states.

K

JKMK

JM C

ΦΦ (K -mixing)

Argonne v8'

[B.E.: -44.0 (MeV)] [B.E.: -56.5 (MeV)]

4He

binding energy(0+)-24.6(MeV)

Few-body cal† -25.9(MeV)

EXP

-28.3(MeV)

[fm-3]

0.1

0.080.06

0.04

0.02

ρ/24

-4 4Z

Y

Result of 4He

†H. Kamada et al. , PRC64 (2001) 044001

Result of 4He & 8Be level scheme

α+α cluster

[B.E.: -29.6 (MeV)] [B.E.: -39.2 (MeV)]

Argonne v8'

7Li level scheme

[B.E.: -41.9 (MeV)] [B.E.: -58.2 (MeV)]

Argonne v8'

9Be (parity -) level scheme

[-41.9 (MeV)] [-58.2 (MeV)]

Argonne v8'

9Be level scheme

α+α+n picture

[-70.4 (MeV)] [-92.2 (MeV)]

Argonne v8'

12C level scheme

3α-like

3α cluster

Higher 0+ state of 12C

P.Navratil et.al., PRL84 (2000) 5728

The shell-model approaches have difficulty in describing the higher 0+ state of 12C.

Huge shell-model model space is needed.

On the other hand, cluster models have been succeeded in describing the 0+

2 state of 12C.Ex) E.Uegaki et.al., PTP 62 (1979) 1621

Cluster models indicate that a number of spatial configurations are needed to describe the 0+

2 state of 12C.

We perform the energy variation with the orthogonality to the lowest state.

Description of higher 0+ state of 12C

Y.Kanada-En’yo, PTP117 (2007) 655

.).(.).(

)(.).(.).()(

sgsg

ZsgsgZ

Intrinsic state for the excited state

:)(,.).( Zsg parity-projected states

In order to describe the higher 0+ state, it is necessary to superpose the intrinsic configurations different from 0+ ground state.

We perform the energy variation where the inertial axes are fitted in order to prevent us from obtaining the solution of rotated ground state.

In order to create various spatial configurations, we perform the above variation under r.m.s.-radius and deformation parameter β constraint.

[B.E.: -92.2 (MeV)] [B.E.: -73.7 (MeV)]

12C level schemeArgonne

v8'Argonne

v8'

All states contribute coherently.

Summary & Proceeding works

Summary• We developed the framework of AMD with realistic interactions based on the Brueckner theory.

Proceeding works

• We applied Brueckner-AMD to some light nuclei and succeeded to describe not only ground states but also excited states with cluster structures which shell model approaches cannot describe.

• The way how to introduce genuine 3-body forces

• Roles of tensor force for 8Be (discussed by Yuhei Yomamoto)

• In the Brueckner-AMD, we can construct the G-matrix and correlation functions in AMD self-consistently. That means this method has no phenomenological corrections.

Appendix

1E

1O 3O

G -matrix (central)

renormalization of tensor force

3E d=1.0 [fm]

3O LS 3O tensor

3E LS 3E tensor

G -matrix (spin-orbit, tensor)

reduction due to short range correlations

d=1.0 [fm]

Superposition of Slater determinants in Brueckner-AMD

The G -matrix between the different configurations in bra and ket states is necessary.

ba

ba G

ˆ

bl

bk

bij

aij

aj

ai

bl

bk

bkl

aij

aj

ai

bl

bk

aj

ai

GG

FvvFG

)()(

)()(

ˆˆ2

1

ˆˆˆˆ2

1ˆ

The G -matrix is calculated with the correlation functions. Correlation functions derived from bra and ket

states

T.Togashi, T.Murakami and K.Katō; Prog. Theor. Phys. 121 (2009) 299.

However, the G –matrix can be determined for only a single

configuration.

Features of G -matrix in Brueckner-AMD

①. The G-matrix can be solved in AMD framework because the single-particle orbits in AMD can be defined and applied to the Brueckner theory.

②. The G-matrix in Brueckner-AMD is changed with the structures through the starting-energy dependence.

Gtt

QVVG ˆ

)ˆˆ(ˆˆˆ

21

　

}~~{ˆ~~~ ˆ ~ Gt

~~~~1Q

)(ˆ cmT

starting energy:

The G-matrix in Brueckner-AMD is different from the case of nuclear matter and considers the property of finite nuclei.

Approximated β parameter

jiij rrQ : the moment of the inertia matrix

23

1222

32

4

153Tr

zyxrQ

ii

23

2222

3 22 153Tr

zyxrQQ

ii

Approximated β parameter

)13(5

2 D

2(TrQ)

TrQQD

[B.E.: -54.5 (MeV)] [B.E.: -76.3 (MeV)]

Argonne v8'

11B (parity-) level scheme

The inversion of 3/2- and 1/2- occurs.

What’s the origin of spin-orbit splitting ?

C.Forssen et.al., PRC71 (2005) 044312

No-core shell model calculations without genuine 3-body forces also indicate the inversion between 3/2- and 1/2- in 11B.

3-body forces may be essential to reproduce the energy levels of some nuclei.