cluster and density wave --- cluster structures in 28 si and 12 c---
DESCRIPTION
Cluster and Density wave --- cluster structures in 28 Si and 12 C---. Y. Kanada-En’yo (Kyoto Univ.) Y. Hidaka (RIKEN). Phys. Rev. C 84 , 014313 (2011) arXiv:1104.4140. a. a. a. a. a. a. a. a. a. a. a. a. Two- and four-body correlations in nuclear systems. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Cluster and Density wave --- cluster structures in 28 Si and 12 C---](https://reader036.vdocument.in/reader036/viewer/2022062517/568135c1550346895d9d23d1/html5/thumbnails/1.jpg)
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
![Page 2: Cluster and Density wave --- cluster structures in 28 Si and 12 C---](https://reader036.vdocument.in/reader036/viewer/2022062517/568135c1550346895d9d23d1/html5/thumbnails/2.jpg)
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
![Page 3: Cluster and Density wave --- cluster structures in 28 Si and 12 C---](https://reader036.vdocument.in/reader036/viewer/2022062517/568135c1550346895d9d23d1/html5/thumbnails/3.jpg)
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
![Page 4: Cluster and Density wave --- cluster structures in 28 Si and 12 C---](https://reader036.vdocument.in/reader036/viewer/2022062517/568135c1550346895d9d23d1/html5/thumbnails/4.jpg)
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
![Page 5: Cluster and Density wave --- cluster structures in 28 Si and 12 C---](https://reader036.vdocument.in/reader036/viewer/2022062517/568135c1550346895d9d23d1/html5/thumbnails/5.jpg)
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
![Page 6: Cluster and Density wave --- cluster structures in 28 Si and 12 C---](https://reader036.vdocument.in/reader036/viewer/2022062517/568135c1550346895d9d23d1/html5/thumbnails/6.jpg)
2. AMD method
![Page 7: Cluster and Density wave --- cluster structures in 28 Si and 12 C---](https://reader036.vdocument.in/reader036/viewer/2022062517/568135c1550346895d9d23d1/html5/thumbnails/7.jpg)
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.
![Page 8: Cluster and Density wave --- cluster structures in 28 Si and 12 C---](https://reader036.vdocument.in/reader036/viewer/2022062517/568135c1550346895d9d23d1/html5/thumbnails/8.jpg)
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
![Page 9: Cluster and Density wave --- cluster structures in 28 Si and 12 C---](https://reader036.vdocument.in/reader036/viewer/2022062517/568135c1550346895d9d23d1/html5/thumbnails/9.jpg)
3. AMD results( without assumption of existence of cluster cores )
![Page 10: Cluster and Density wave --- cluster structures in 28 Si and 12 C---](https://reader036.vdocument.in/reader036/viewer/2022062517/568135c1550346895d9d23d1/html5/thumbnails/10.jpg)
AMD results
AMD
Positive parity bandsoblate & prolate
Negative-parity bands
![Page 11: Cluster and Density wave --- cluster structures in 28 Si and 12 C---](https://reader036.vdocument.in/reader036/viewer/2022062517/568135c1550346895d9d23d1/html5/thumbnails/11.jpg)
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
![Page 12: Cluster and Density wave --- cluster structures in 28 Si and 12 C---](https://reader036.vdocument.in/reader036/viewer/2022062517/568135c1550346895d9d23d1/html5/thumbnails/12.jpg)
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
![Page 13: Cluster and Density wave --- cluster structures in 28 Si and 12 C---](https://reader036.vdocument.in/reader036/viewer/2022062517/568135c1550346895d9d23d1/html5/thumbnails/13.jpg)
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
![Page 14: Cluster and Density wave --- cluster structures in 28 Si and 12 C---](https://reader036.vdocument.in/reader036/viewer/2022062517/568135c1550346895d9d23d1/html5/thumbnails/14.jpg)
(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 !
![Page 15: Cluster and Density wave --- cluster structures in 28 Si and 12 C---](https://reader036.vdocument.in/reader036/viewer/2022062517/568135c1550346895d9d23d1/html5/thumbnails/15.jpg)
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 !
![Page 16: Cluster and Density wave --- cluster structures in 28 Si and 12 C---](https://reader036.vdocument.in/reader036/viewer/2022062517/568135c1550346895d9d23d1/html5/thumbnails/16.jpg)
4. Toy model of DW- Interpretation of cluster structure in terms of DW -
![Page 17: Cluster and Density wave --- cluster structures in 28 Si and 12 C---](https://reader036.vdocument.in/reader036/viewer/2022062517/568135c1550346895d9d23d1/html5/thumbnails/17.jpg)
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
![Page 18: Cluster and Density wave --- cluster structures in 28 Si and 12 C---](https://reader036.vdocument.in/reader036/viewer/2022062517/568135c1550346895d9d23d1/html5/thumbnails/18.jpg)
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,
![Page 19: Cluster and Density wave --- cluster structures in 28 Si and 12 C---](https://reader036.vdocument.in/reader036/viewer/2022062517/568135c1550346895d9d23d1/html5/thumbnails/19.jpg)
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.
![Page 20: Cluster and Density wave --- cluster structures in 28 Si and 12 C---](https://reader036.vdocument.in/reader036/viewer/2022062517/568135c1550346895d9d23d1/html5/thumbnails/20.jpg)
5. Summary
![Page 21: Cluster and Density wave --- cluster structures in 28 Si and 12 C---](https://reader036.vdocument.in/reader036/viewer/2022062517/568135c1550346895d9d23d1/html5/thumbnails/21.jpg)
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.