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The coexistence of cluster and shell model structures with AMD+GCM
Tadahiro Suhara (YITP) Yoshiko Kanada-En’yo (Kyoto U.)
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Introduction
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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
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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
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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.
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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)
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・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
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Framework
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],,,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
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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
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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
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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
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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
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Structures in 12C
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Calculated results of 12C
Shell-model
+ energy surface
0+ energy surface
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Calculated results of 12C
Shell-model
Structures on the β-γ plane
3α cluster structures
Equilateral triangle
Obtuse-angle triangle
Linear-chain
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Energy levels in 12C
unit: e2fm4
unit: fm
Calculated results of 12C
Our results reproduce the experimental values (B(E2) and rms radii) well.
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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
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The triaxial basis has an essential role in the description of excited states.
Importance of triaxiality Comparison with the axial symmetric GCM
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Structures in 10Be
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Calculated results of 10Be + energy surface
0+ energy surface
2α+2n(π3/2)2
2α+2n(π3/2+px)2
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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
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unit: fm
unit: e2fm4
Energy levels in 10Be
Our results reproduce the experimental results well.
Calculated results of 10Be
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(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.
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The triaxial basis has an essential role in the description of excited states.
Importance of triaxiality Comparison with the axial symmetric GCM
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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