Cluster and Density wave--- cluster structures in 28Si and 12C---

Y. Kanada-En’yo (Kyoto Univ.)Y. Hidaka (RIKEN)

Phys. Rev. C 84, 014313 (2011)arXiv:1104.4140

Tohsaki et al., Yamada et al.,Funaki et al. K-E.,

12C*

Dilute 3Dilute 3gas gas

Itagaki et al.,Von Oertzen et al.

14C*(3-2)

matter

T. Suhara and Y. K-E.

14,15,16C* linear chain

triangle

-gas

BEC-BCS

dineutron

Roepche et al.

matsuo et al.

Two- and four-body correlations in nuclear systems

Cluster structures in finite nucleigas or geometric configurations of gcluster cores

-clystal 　 ?

BCS

pn pairing

Shape coexistence and cluster structures

in 28Si

1) Shape coexistence and cluster structure in 28Si What is density wave2) Results of AMD for 28Si structure

3) Interpretation with density wave

oblate, prolate, exotic shapes

Exc

itatio

n en

ergy

0+1

0+3

oblateoblateproblateproblate K=3 －K=5 －

3 －5 －

28Si

Experimental suggestions(1980)

Shape coexistence in 28Si

7-cluster model (1981) AMD (2005)

D5h symmetryof the pentagon shape

δ

Energy surface

6 MeV

0+1

0+3

oblate prolate

K quantaK

J

body-fixed axis

O C

Mg α

Molecularresonance-cluster

DW on the edge of the oblate statePentagon in 28Si due to 7-cluster

SSB from axial symmetric oblate shape to axial asymmetric shape

D5h symmetryconstructs K=0+, K=5- bands

DW in nuclear matter is a SSB(spontaneous symmetry breaking)for translational invariance i.e. transition from uniform matter to non-uniform matter

What is density wave(DW) ? Why DW in 28Si ?

Origin of DW: Instability of Fermi surface due to correlation

k

kk aa

Correlation between particle (k) and hole (-k)

has non-zero expectation valuewave number 2k periodicity (non-uniform)

kk aa kk aa

Other kinds of two-body correlation(condensation)are translational invariant

BCS exciton

2. AMD method

　 　　 　 　 　　　

　　

　　　

　

nnpp

ccc

A

or

2

2AMD

AMDAMDAMD

,,

)(exp)(

,,,

''''''

i

ijiΖ

iiΖi

Zrr

　

A

Wave　function

Complex parameter Z={ }AA ,,,,,, 121 　ZZZ

spatial

Slater det.

Gaussian

det

det

Cluster structure

Shell-model-like states

Formulation of AMD

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.

Energy　Variation

)( 0Z

model space(Z plane)

Energy surface

frictional cooling method 　 　

Z

Z E

ii

dt

d

1

)(

0)()(

)()(

EH

ZZ

ZZ

Simple AMD

VAP

Variation after parity projection before spin pro. (VBP)

Variation after spin-parity projection

Constraint AMD & superposition AMD + GCM~~

Energy variation and spin-parity projection

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

AMD results

AMD

Positive parity bandsoblate & prolate

Negative-parity bands

Intrinsic structure

K=0+, K=5- K=3-

K=3-K=0+

28Si: pentagon constructs K=0+, K=5- bands

12C: triangle does K=0+, K=3- bands

Features of single-particle orbits in pentagon

Consider the pentagon 28Si as ideal 7-cluster state with pentagon configuration

(s) π2(p) π

6(sd) π6

(s) ν2(p) ν

6(sd) ν6

d+f 　 mixing resultsin a pentagon orbit

(s) π2(p) π

6(sd)π2(d+f) π

4

(s) ν2(p) ν

6(sd) ν2(d+f) ν

4

axial symmetry Axial asymmetry-clusterdevelops

s-orbit

p-orbit

d

In d=0 limit

＋－

＋－

＋

＋

－－

＋

－

),(sinsin 3322 ii e e Y2+2 Y3-3

)5cos1(2)sin1(sin|),(| 22

det

oblate pentagon

single-particle orbits in AMD wave functions

＋－

＋－

＋

＋

－－

＋

－

3322 sinsin),( ii e e Y2+2 Y3-3

5 ～6%

Pentagonorbitsd+f mixing

Triangle orbitsp+d mixing

(s) π2(p) π

6(sd) π6

(s) ν2(p) ν

6(sd) ν6

d+f 　 mixing resultsin a pentagon orbit

(s) π2(p) π

6(sd)π2(d+f) π

4

(s) ν2(p) ν

6(sd) ν2(d+f) ν

4

axial symmetry Axial asymmetry-clusterdevelops

＋－

＋－

＋

＋

－－

＋

－

),(sinsin 3322 ii e e Y2+2 Y3-3

)5cos1(2)sin1(sin|),(| 22 6%

The pentagon state can be Interpreted as DW on the edgeof the oblate stateSSB:

lz

sdfp

2 3 zz lala

SSB in particle-hole representation

HF0

assumed to be HF vacuum

HF

aaaa 0114

23

4

23

SSB state 6/d

d+f mixing pentagon orbits Wave number 5 periodicity !

What correlation ?

2,3,

zz ll aa nnpp ,,,in Z=N system (spin-isospin saturated)

1p-1h correlation 1p-3p correlation

DW alpha correlation (geometric, non uniform)

28Si 12C 20CZ=N=14 Z=N=6 Z=6,N=14

SSB

oblateNo SSB in N>Z nucleibecuase there isno proton-neutroncoherence.

DW is suppressed

lz

sd

fpSSB: single-particle energy loss < correlation energy gain

proton-neutron coherence is important !

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

Toy model ： DW hamiltonian

particle operator

hole operator

　 　 　　 nnpp or,,1. Truncation of active orbits

2. Assuming contact interaction (r) and adopting a part of ph terms （ omitting other two-body terms ）

lz

sd

fp

Approximated solution of DW hamiltonian

Energy minimum solution in an approximation: determination of u,v

where

　 　 　　 nnpp or,,

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.

non-zero uv indicates SSB

　 　 　　 nnpp or,,

p 　 　 　　 nnp or,

Less corrlation energy

For neutron-proton coherent DW (spin-isospin saturated Z=N nuclei)

For neutron-proton incoherent (ex. N>Z nuclei)

　 　 　　 nnpp or,,

Proton DW in neutron-rich nuclei:

SSB condition

SSB condition

Correlation energy overcomes1p-1h excitation energy cost

Since protons are deeply bound, energy cost for 1p-1h Increases. As a result, DW is further suppressed at least in ground states.

5. Summary

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.