The coexistence of cluster and shell model structures with AMD+GCM

Tadahiro Suhara (YITP) Yoshiko Kanada-En’yo (Kyoto U.)

Introduction

Shell-model structure

Nucleons make an one-center mean field and move in this field independently.

Cluster structure

The cluster means a spatially localized subsystem composed of strongly correlated nucleons.

Very contrasting at correlation.

In light nuclei, cluster and shell model structures often coexist.

Cluster and shell-model structures

The coexistence of cluster and shell-model structures

12C

Shell-model 0 MeV

Energy

7.4 MeV

3 alpha clusters develop and various structure appears

α condensed state

0+2

3 α threshold

equilateral-triangular

～ E. Uegaki, et al. Prog. Theor. Phys. 57, 1262 (1977) M. Kamimura, et al. J. Phys. Soc. Jpn. 44 (1978), 225. A. Tohsaki, et al. Phys. Rev. Lett. 87, 192501 (2001) T. Neff, et al. Nuc. Phys. A738, 357 (2004) Y. Kanada-En’yo, Prog. Theor. Phys. 117, 655 (2007) etc

linear-chainlike (Obtuse-angle triangular) 0+

3

3-1

Ab initio calculation No Core Shell Model 3α Cluster Model

E. Uegaki, et al. Prog. Theor. Phys. 57, 1262 (1977). P. Navratil, et al. Phys. Rev. C 68, 034305 (2003). J. Phys. G: Nucl. Part. Phys. 36 083101 (2009).

No Core Shell Model still have not been able to reproduce cluster developed states.

N.itagaki, et al. Phys. Rev. C 70, 054307 (2004).

It is important to include the α breaking (Shell-model structure).

Shell model structure

12C

3α 2α+2p+2n α+4p+4n

~6MeV 0+1

Breaking of cluster (Shell model structure)

・Propose a method which can describe various cluster and shell-model structures. β-γ constraint AMD + GCM ・To check the applicability, we applied this method to 12C and 10Be. ・Importance of the triaxiality γ. T.S. and Y. kanada-En’yo, Prog. Theor. Phys. 123, 303 (2010).

Purpose of our study

Framework

],,,det[ A21AMD ϕϕϕ =Φ

)()( iii ξZ χφϕ =)n p()(

])(exp[)(

or

2

×

=

−−∝

↓

↑

i

ii

ii

ξξ

χ

ννφ

ξ

ZrZ

, iiZ ξZ=

spatial

spin and isospin

Set of variational parameters

a wave function of A-body system

directionspin : packets aveGaussian w ofcenter :

　i

i

ξZ

β-γ constraint AMD + GCM

AMD（Antisymmetrized Molecular Dynamics）

det

iZ

Cluster Shell model

Cluster structure Shell-model structure

det det

β-γ constraint AMD + GCM

AMD can describe both of cluster structures and shell-model structures.

Model space of AMD

Constraints The quadrupole deformation (β, γ)

( )><+><+><=

><−><=

><−><−><=

2222

2

22

2

222

35

35sin

235cos

zyxR

RyxR

yxz

πγβ

πγβ

β γ 12C

β-γ constraint AMD + GCM

α α α

α α α

Parity and angular momentum projections

)()(8

122

1

2 ΩΩΩ+

=

±=

∫ ∗

±

RDdJP

PP

JMK

JMK π

In this study, we performed the variation after the parity projection (VAP). After the variation, we project the obtained wave function onto the total-angular-momentum eigenstates (PAV).

β-γ constraint AMD + GCM

( ) ( )∑∑ ±± Φ=ΦK i

iiJ

MKiinJn PKf γβγβ ,,,

GCM (Generator Coordinate Method) J± state

Effective interaction Hamiltonian

The central force : The Volkov No.2

The LS force : The LS part of the G3RS

∑∑∑∑<<<

+++−=ji

ijji

ijji

iji

i vvvTtH CoulombLScentralCM

eff

ijijij

ij Xar

var

vv ]))(exp[])(exp[( 2

22

2

11

central −+−=

]fm[01.1],MeV[14.61],fm[80.1],MeV[65.60

)125.0,6.0,4.0(

2211 ===−=

====−−+=

avavHBMWPMPHPBPWX ij τστσ

SL ⋅==−+−= )1()1(]))(exp[])(exp[( 2

22

2

11

LS TPSPar

uar

uv ijijij

]fm[600.0],MeV[1600],fm[447.0],MeV[16002

1)1(,2

1)1(

2211 =−===

+==

+==

auau

PTPPSP τσ

Structures in 12C

Calculated results of 12C

Shell-model

+ energy surface

0+ energy surface

Calculated results of 12C

Shell-model

Structures on the β-γ plane

3α cluster structures

Equilateral triangle

Obtuse-angle triangle

Linear-chain

Energy levels in 12C

unit: e2fm4

unit: fm

Calculated results of 12C

Our results reproduce the experimental values (B(E2) and rms radii) well.

Structures of 0+ states 0+

1: Shell-model-like

0+2: Various 3α cluster

configurations

0+3: Linear-chainlike

GCM amplitude

The present results is consistent with the earlier works. ⇒β-γ constraint AMD+GCM is effective for describing various cluster and shell structures.

Calculated results of 12C

The triaxial basis has an essential role in the description of excited states.

Importance of triaxiality Comparison with the axial symmetric GCM

Structures in 10Be

Calculated results of 10Be + energy surface

0+ energy surface

2α+2n(π3/2)2

2α+2n(π3/2+px)2

2α+2n (π3/2+px)2 α+6He 2α+2n(σ1/2)2

2α+dineutron

2α+dineutron

Calculated results of 10Be

Structures on the β-γ plane

unit: fm

unit: e2fm4

Energy levels in 10Be

Our results reproduce the experimental results well.

Calculated results of 10Be

(1)

(2) (3)

(1) 2α+2n(π3/2+px)2

(2) K=2+ side band (3) α+6He or 2α+2n(σ1/2)2

(1,2) (3)

Calculated results of 10Be Structures of 10Be

The present results is consistent with the earlier works.

The triaxial basis has an essential role in the description of excited states.

Importance of triaxiality Comparison with the axial symmetric GCM

Summary

・β-γ constraint AMD+GCM This method is effective for describing various cluster and shell model structures. ・Structures in 12C and 10Be To describe excited states, the triaxial basis play an important role.

・Future work Structures of other nuclei 14C, T.S. and Y. kanada-En’yo, PRC 82, 044301 (2010). 11B, in preparation