1. Cluster and Density wave --- cluster structures in 28Si and 12C---
Y. Kanada-En’yo (Kyoto Univ.)
Y. Hidaka (RIKEN)
Today I will report our activities on nuclear structure studies.
Phys. Rev. C 84, (2011)
arXiv:

2. Two- and four-body correlations in nuclear systems
Cluster structures in finite nuclei
gas or geometric configurations of a gcluster cores
matter
pn pairing
Dilute 3a gas
a-gas a a
12C*
Tohsaki et al., Yamada et al.,
Funaki et al. K-E.,
a-clystal　?
Recently, further novel cluster structure has been proposed. A gas of weakly interaction alpha particle and cluster crystalization
in the excited states of 12C by Tohsaki et al. and Funaki et al. and 14C by Itagaki et al, respectively.
Then, one can usually speculate a alpha cluster gas or cluster crystalization in phases of nuclear matter.
Roepche et al.
14C*(3-2) a
triangle
Itagaki et al.,
Von Oertzen et al.
dineutron
BCS
14,15,16C*
linear chain a
T. Suhara and Y. K-E.
BEC-BCS
matsuo et al.

3. and cluster structures
Shape coexistence
and cluster structures
in 28Si
Shape coexistence and cluster structure in 28Si
What is density wave
Results of AMD for 28Si structure
Interpretation with density wave

4. Shape coexistence in 28Si
7a-cluster model (1981)
AMD (2005)
28Si
Molecular
resonance O C Mg
a-cluster α
D5h symmetry
of the pentagon shape
Excitation energy 5－
0+3 3－ δ
Energy surface
6 MeV
0+1
0+3
oblate
prolate
0+1
K=3－
K=5－
problate
oblate
Experimental suggestions(1980)
oblate, prolate, exotic shapes J
body-fixed axis
K quanta K

5. What is density wave(DW) ? Why DW in 28Si ?
DW in nuclear matter
is a SSB(spontaneous symmetry breaking)
for translational invariance
i.e. transition from uniform matter
to non-uniform matter
DW on the edge of the oblate state
Pentagon in 28Si due to 7a-cluster
SSB from axial symmetric oblate shape
to axial asymmetric shape
D5h symmetry
constructs
K=0+, K=5- bands
Origin of DW: Instability of Fermi surface due to correlation
Correlation between particle (k) and hole (-k)
has non-zero expectation value
wave number 2k periodicity (non-uniform)
Other kinds of two-body correlation(condensation)
are translational invariant k
BCS
exciton

6. 2. AMD method

7. Formulation of AMD det Slater det. Gaussian det Wave function
Cluster structure
Wave function
det
Slater det.
Gaussian
spatial
Shell-model-like states
det
Complex parameter Z={ }
Existence of any clusters is not apriori assumed. But if a system favors a cluster structure, such the structure automatically obtained in the energy variation.

8. Energy variation and spin-parity projection
Energy surface
frictional cooling method
model space
(Z plane)
Simple AMD
Variation after parity projection before spin pro. (VBP)
VAP
Variation after spin-parity projection
Constraint AMD & superposition ~ ~
AMD + GCM

9. （without assumption of existence of cluster cores）
3. AMD results
（without assumption of existence of cluster cores）

10. AMD results Negative-parity bands Positive parity bands
oblate & prolate
AMD

11. Intrinsic structure K=0+, K=5- K=3- K=0+ K=3-
28Si: pentagon constructs K=0+, K=5- bands
12C: triangle does K=0+, K=3- bands

12. Features of single-particle orbits in pentagon
s-orbit
Consider the pentagon 28Si as
ideal 7a-cluster state
with pentagon configuration
det
p-orbit d
In d=0 limit
Axial asymmetry
axial symmetry
a-cluster
develops
(s) π2(p) π6(sd) π6
(s) π2(p) π6(sd)π2(d+f) π4
d+f　mixing results
in a pentagon orbit
(s) ν2(p) ν6(sd) ν6
(s) ν2(p) ν6(sd) ν2(d+f) ν4
oblate
pentagon
Y Y3-3 ＋ ＋ － － － － ＋ ＋ ＋ －

13. single-particle orbits in AMD wave functions
Pentagon
orbits
d+f mixing
Triangle orbits
p+d mixing
5～6%
Y Y3-3 ＋ ＋ － － － － ＋ ＋ ＋ －

14. SSB in particle-hole representation
axial symmetry
Axial asymmetry
a-cluster
develops
(s) π2(p) π6(sd) π6
(s) π2(p) π6(sd)π2(d+f) π4
d+f　mixing results
in a pentagon orbit
(s) ν2(p) ν6(sd) ν6
(s) ν2(p) ν6(sd) ν2(d+f) ν4
assumed to be HF vacuum
SSB state fp sd lz
d+f mixing
pentagon orbits
Wave number 5 periodicity !
Y Y3-3
The pentagon state can be
Interpreted as DW on the edge
of the oblate state
SSB: ＋ ＋ － － － － ＋ ＋ ＋ － 6%

15. What correlation ? in Z=N system (spin-isospin saturated) 28Si 12C 20C
1p-1h correlation p-3p correlation DW
alpha correlation (geometric, non uniform) fp
SSB: single-particle energy loss
< correlation energy gain
proton-neutron coherence is important ! sd lz
28Si
Z=N=14
12C
Z=N=6
20C
Z=6,N=14
oblate
No SSB in N>Z nuclei
becuase there is
no proton-neutron
coherence.
DW is suppressed
SSB

16. - Interpretation of cluster structure in terms of DW -
4. Toy model of DW
- Interpretation of cluster structure in terms of DW -

17. Toy model：DW hamiltonian
1. Truncation of active orbits
particle operator
hole operator
2. Assuming contact interaction d(r) and
adopting a part of ph terms （omitting other two-body terms） fp sd lz

18. Approximated solution of DW hamiltonian
Energy minimum solution in an approximation: determination of u,v
non-zero uv indicates SSB
where
Coupling with condensations of other species of particles:
For , three-species condensation for
couple resulting in the factor 3. A kind of alpha(4-body) correlation.

19. For neutron-proton coherent DW (spin-isospin saturated Z=N nuclei)
SSB condition
Correlation energy overcomes
1p-1h excitation energy cost
For neutron-proton incoherent (ex. N>Z nuclei)
SSB condition
Less corrlation energy
Proton DW in neutron-rich nuclei:
Since protons are deeply bound, energy cost for 1p-1h
Increases. As a result, DW is further suppressed at least in ground states.

20. 5. Summary

21. Cluster structures in 28Si (and 12C)
K=0+ and K=5- bands suggest a pentagon shape because of 7alpha clusters.
The clusterization can be interpreted as
DW on the edge of an oblate state, .i.e., SSB of oblate state.
1p-1h correlation of DW in Z=N nuclei is equivalent to
1p-3p (alpha) correlation.
n-p coherence is important in DW-type SSB.
Future:
Other-type of cluster understood by DW.
Ex) Tetrahedron 4 alpha cluster : Y32-type DW.