クラスターガス状態と モノポール励起 - osaka...
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クラスターガス状態と モノポール励起
関東学院大・工 山田泰一
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1. α-particle condensation in 4n nuclei 2. Hoyle-analog states in A≠4n nuclei 3. IS monopole transitions to search for cluster states “Dual nature of ground states” IS monopole strength function of 16O (typical case) : α-cluster states at low energy mean-field-type states at higher energy 4. Summary
Contents of my talk
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Introduction • Cluster picture as well as mean-field picture is important
viewpoint to understand structure of light nuclei. • Structure of light nuclei Cluster states + Shell-model-like states Microscopic cluster models, AMD,.... 8Be=2α, 12C=3α , 16O=12C+α / 4α, … • α-condensate states in 4n nuclei: α-gas-like state described by a product state of α-particles, all with their c.o.m. in (0S) orbit. Typical states: 12C: Hoyle state (2nd 0+)
Cluster gas α cluster
0 03 5ρ ρ~
R ≈ 3.8 fm Tohsaki, Horiuchi, Schuck, Roepke, Phy. Rev. Lett.87 (2001) Funaki et al., PRC (2003) Yamada et al., EPJA (2005), Matsumura et al.,NPA(2004)
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Cluster gas α cluster
0 03 5ρ ρ~
R ≈ 3.8 fm
shell-model –like structure
Condensed into the lowest orbit
3(0 )S α
70%
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+2Brink wf exact 0 state
Uegaki et al, PTP57(1977): 3α GCM The 0+
2 state has a distinct clustering and has no definite spatial configuration.
Chernykh, Feldmeir et al., PRL98(2007)
UCOM + FMD, 3α RGM
About 55 components of the Brink-type wave functions are needed to reproduce the full RGM solution for the Hoyle state.
Uegaki et al, PTP57(1977)
Overlap 3α GCM
Tohsaki,Horiuchi,Schuck,Roepke, PRL87(2001)
THSR wave function: α-condensate-type cluster wf 1 base THSR :
2
3 (3 RGM) 99%THSRα αΦ ≈+
2exact 0 state
Kamimura et al., (1977): 3α RGM
Funaki et al., PRC67, (2003)
α-gas-like nature of Hoyle state
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α
α α G
X3
X1 X2
3 2
3 21
2( ) exp ( )i ii
B XBα φ α
=
Φ = − ∏
THSR wave function Tohsaki, Horiuchi, Schuck, Roepke, PRL (2001)
3(0 )S α
Condensed into the lowest orbit
3(0 )S α
B → ∞3 2
3 21
2( ) exp ( )i ii B
B Xα φ α=
Φ → − ∏
B b→ 4 83 ( ) (0 ) (0 )B s pαΦ →
b: nucleon size parameter
B: parameter
2THSR3 ( ) (3 RGM) 0.9Bα αΦ ≈+
2exact 0 state3α RGM: Kamimura & Fukushima (1978)
Deformation (Bx,By,Bz) 0.999 R ≈ 3.8 fm: alpha-gas-like structure
Funaki et al., PRC (2003)
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Occupation probabilities of α-orbits in 12C
12C(g.s) Hoyle state 70%
(λµ)=(04)-nature 0S-orbit
Yamada & Schuck EPJA26 (2005) with 3α OCM wf Matsumura & Suzuki, NPA739 (2004) Funaki et al., PRC (2010) with 3α THSR wf
Single cluster density matrix:
' ( , ') ( ') ( ),ad α α αρ ϕ λ ϕ=∫ r r r r r ( )ϕ r : single-cluster orbital w.f. λ : occupation probability
ρ(r,r')
SU(3) nature: confirmed by no-core shell model , Dytrych et al., PRL98 (2007)
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α-density distribution and α-momentum distribution in 12C
Density × r2 Momentum distribution
01+ (g.s)
02+
02+
01+ (g.s)
Compact (01+) vs. Dilute (02
+) 02+ state: δ function-like
Similar to dilute atomic cond. Yamada & Schuck EPJA26 (2005) with 3α OCM wf
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Single α-orbits in 12C(0+)
Large oscillation : strong Pauli blocking effect Compact structure SU(3) model: [f](λµ)=[444](04)-like structure
Small oscillation: weak Pauli blocking effect Long tail: dilute structure Radial behavior: Gaussian form with a= 0.04 fm-2
01+ (g.s.)
D-orbit (Nα=35%)
G-orbit (Nα=27%)
S-orbit (Nα=35%)
02+
S-orbit (Nα=70%)
20ref.: ( ) exp( )r N a ar× −
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Cluster gas α cluster
0 03 5ρ ρ~
Condensed into the lowest orbit R ≈ 3.8 fm
3(0 )S α
shell-model –like structure
70%
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P. Navr’atil et al., PRC62 (2000) 12C
Family of the Hoyle state No-core shell model calculation
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Exp. for 2nd 2+ state
Itoh et al., NPA738, 268 (2004)
Freer et. al. PRC80, (2009)
M. Gai et. al, (2011)
Kurokawa, Kato, PRC71 (2005)
Funaki et al., EPJA24 (2005) Yamada et al., EPJA26 (2005)
Family of the Hoyle state
12C
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12C
3−
1−
Precursor of a 3α condensate state: 3-, 1-
(0P)α3 : superfluid?
vortex?
Condensed into the lowest orbit
3(0 )S α
70%
3a OCM: Occup. Prob. ~ 40% precursor of 3α condensate
Yamada et al., EJPA26, 185 (2005)
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Structure of 16O
1. 4α condensate ? 2. 4α linear chain state ? 3. Monopole excitation (later)
Funaki’s talk
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(0Sα)3 : 70%
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Cluster-model analyses of 16O
• α+12C OCM
• α+12C GCM
• 4α THSR wf
• 4α OCM
Y. Suzuki, PTP55 (1976), 1751
M. Libert-Heinemann, D. Bay et al., NPA339 (1980)
Tohsaki, Horiuchi, Schuck, Roepke, PRL87 (2001) Funaki, Yamada et al., PRC82(2010)
Funaki, Yamada et al., PRL101 (2008)
Not include α+ 12C configuration.
4α-gas, α+ 12C, shell-model-like configurations
Reproduction of lowest six 0+ states up to 4α threshold (15MeV)
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16O=12C+α cluster model Even-parity Odd-parity
EXP CAL EXP CAL
12C+α : molecular states
Y. Suzuki, PTP55 (1976), 1751
12C+α
4α
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Funaki et al., PRL101 (2008)
Suzuki, PTP55 (1976)
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Experimental data 4α OCM
Ex [MeV]
R [fm]
M(E0) [fm2]
Γ [MeV]
R [fm]
M(E0) [fm2]
Γ [MeV]
0+1 0.00 2.71 2.7
0+2 6.05 3.55 3.0 3.9
0+3 12.1 4.03 3.1 2.4
0+4 13.6 no data 0.6 4.0 2.4 0.60
0+5 14.0 3.3 0.185 3.1 2.6 0.20
0+6 15.1 no data 0.166 5.6 1.0 0.14
20% of total EWSR
over 15% of total EWSR
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4α cond. state
12C(01+)+α(S)
12C(21+)+α(D)
12C(01+)+α(S): higher nodal
12C(11-)+α(P)
shell-model-like: (0s)4(0p)12
Funaki, Yamada et al., PRL101 (2008)
4α OCM calculation
strong candidate 4(0 )S α 60%
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α-particle condensates in heavier 4n nuclei than 16O
1. THSR approach 2. Gross-Pitaevskii approach
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THSR approach Energy curve of HW equation
Tohsaki, Horiuchi, Schuck, Roepke, NPA738, 259 (2004)
nα condensate
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( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )
22
2 2 22
1
3
1, 12
1 21
2
n
ii
U rm n
n nU r n d d v
n
v d
α
φ εφα
φ φ φ
φ=
− − ∇ + =
− −′ ′ ′ ′′ ′ ′′ ′ ′′ ′= − +
Φ =
∫ ∫
∏ r r
r r r , r r r r r r , r , r
r
Single-α potential given via Gross-Pitaevskii approach
・Coulomb barrier → quasi-stable states
・Weakly interacting “gas-like” states
・Pure bosons ・nα states are assumed
Yamada & Schuck, PRC 69, 024309 (2004)
α condensates : up to about 10α (40Ca)
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Hoyle-analog states in A ≠ 4n nuclei (11B and 13C)
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Search for Hoyle-analogue states in A≠4n nuclei
Definition of Hoyle-analogue state: A cluster-gas-like state described mainly by a product state of clusters, all with their c.o.m. in respective 0S orbits
• A=4n nuclei 12C=3α : (0Sα)3 70% 16O=4α : (0Sα)4 60% • A≠4n nuclei, for example, 11B=2α+t : (0Sα)2(0St) 13C=3α+n : (0Sα)3(0Sn)
Condensed into lowest orbit 3(0 )S α
exit or not?
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Purposes of present study • Cluster structures in A≠4N nuclei: 11B (13C)
• Hoyle-analogue states? Conditions of appearance? 11B=2α+t : (0Sα)2(0St) 13C=3α+n : (0Sα)3(0Sn) • Which states of 11B (13C) correspond to the Hoyle-analogue? • What happens in 11B (13C) when an extra triton (neutron) is added into 8Be=2α (Hoyle=3α) state?
R.B. Wiringa et al., PRC(2000)
8Be density α
α
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11B
8Be(2α) + p-orbit
s-orbit reduced:
not so changed:
2α part in 11B triton Jπ (11B)
1/2+
1/2-,3/2-
Thus, one can expect cluster-gas-like states in 1/2+.
α-t potentials: parity-dependent odd waves : attractive enough to make resonances/bound states even waves: weakly attractive
αα α S-orbit
α
12C 2nd 0+ (Hoyle)
α α t S1/2-orbit
α t
11B 1/2+ : Hoyle-analogue exists or not ?
S-orbit
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α-n phase shifts α-triton phase shifts
α-n potential (1) P-wave: attractive
(2) S-wave: weakly attractive
α-triton potential
(1) P-, F-waves: attractive
(2) S-, D-waves: weakly attractive
Parity-dependent potentials
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states in 11B 3 ,− +2 1 2
Results of OCM calculations
Yamada and Funaki, PRC82 (2010)
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Structure study of 11B
• α+α+t OCM with H.O. basis
Nishioka et al., PTP62 (1979)
• AMD calculation
Enyo-Kanada, PRC75 (2007)
• No-core shell model
Navratil et al., JPG36 (2009)
Not well understood for even-odd states of 11B
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No-core shell model Navratil et al., JPG36(2009)
7Li+α
11B energy levels (T=1/2)
Experimentally, three 3/2- states
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No-core shell model Navratil et al., JPG36(2009) Not studied well for even-parity states
11B energy levels (T=1/2)
α+α+t
7Li+α
Even-parity states
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11 (12,3)
, ,
(23,1) (31,2)
, ,
(23,1) (31,2) (23,1) (31,2)
( , )
( B) ( , ) ( , )
( , ) ( , ) ,
( , ) ( , ) ( , ),
( , ) ( , ) ( , ) , ( , ) ( ) exp
L c cc
c cc
c c c
ij kc ij k mL
A
B
N r
ν µ
ν µ
λ
ν µ ν µ
ν µ ν µ
ν µ ν µ ν µ
ν µ ϕ ν ϕ µ ϕ ν ν ν
+
+
Φ = Φ
+ Φ
Φ = Φ + Φ
Φ = = −
∑
∑
r r r ( )2 ( ),mr Y r
Total wave function: 2α +t OCM
Hamiltonian
( ) ( )
2 12 2 12 2( ) (23),(31)
2(2) (2) 2
12 12
( ) ( ) ( ) ( ) ( ) ,
2( ) exp , ( ) erf
lim ( ) ( ) ( ) (
Coul c LS Coult ij t ij ij t ij t Pauli
ij
Coulx n n x
n
Pauli n n n ij n ij
H T V r V r V r V r V r V V
xeV r V r V r arr
V u r u r u r u r
α α α α α α
α α
λ
β
λ
+ + +=
+ +
→∞
= + + + + ⋅ + + +
= − =
= +
∑
∑
l s
2 4 2 3 (23),(31)
) ,n n ij+ < + < =
∑ ∑ ∑
[ ] 0=Φ−Φ HEδEquation of motion
: Pauli blocking operator
α
t
α α
t
α α
t
α
⊕ ⊕
α+α , α+t phase shifts 7Li:3/2-,, 1/2-, 7/2-, 1/2-
2, ( 7,8)
3( ) : 3( ) :tL Q
V SU L Q SU L Qπ
π πα η λµ λµ
=
= ∑ : effective 2α+t force
OCM+GEM (Gaussian expansion method)
![Page 34: クラスターガス状態と モノポール励起 - Osaka Universitytakahiro/acon11/pdfs/yamada.pdfall with their c.o.m. in (0S) orbit. Typical states: 12C: Hoyle state (2 nd 0+)](https://reader036.vdocument.in/reader036/viewer/2022071404/60f80a3fa9682210ff42d838/html5/thumbnails/34.jpg)
Single-cluster motions in 11B(α+α+t)
' ( , ') ( ') ( ),ad α α αρ ϕ λ ϕ=∫ r r r r r
α
α
triton ( )
1Gr
( )2
Gr
( )3
Gr G
( )ϕ r : single-cluster orbital w.f. λ : occupation probability
1α
λ =∑
11( B)JΦ : α+α+t OCM w.f.
(1 ) ( )1 11
1
21( B) ( ') ( ) ( B)2
( , ') ,G GnJ n J
nα δ δρ
=Φ − − Φ= ∑ r rr rr r
( )11 11( )3 3( B) ( ') ( ) ( B)( , ') ,t J J
G Gδ δρ Φ − − Φ= r rr rr r
alpha:
triton:
' ( , ') ( ') ( ),t t t td ρ ϕ λϕ=∫ r r r r r 1t
λ =∑
11B=α+α+t
Single-cluster density matrix:
Suzuki et al., PRC65 (2002); Matsumura et al., NPA739 (2004) Yamada et al., EPJA 26 (2005); Funaki, Yamada et al., PRL101 (2008) Yamada et al., PRA (2008), PRC(2009)
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Energy levels of 11B
3/2- and 1/2+ states Even-parity states
7Li+α
α + α + t OCM α + α + t OCM
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B(E0,IS)=96±16 fm4 B(E0,IS)=92 fm4
R=2.43 fm R=2.22 fm
R=3.0 fm
E0 E0
vs. Hoyle state: B(E0,IS)=120±9 fm4
11B = α +α + t OCM
T. Kawabata et al., PRC70 (2004)
shell-model-like state
shell-model-like state
cluster state α+7Li(3/2-) with S-wave
But 7Li part has a distorted α+t structure
-3 2
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B(E0,IS)=96±16 fm4 B(E0,IS)=92 fm4
R=2.43 fm R=2.22 fm
R=3.0 fm
E0 E0
vs. Hoyle state: B(E0,IS)=120±9 fm4
Overlap amplitude with α+ 7Li(g.s) channel
1st 1/2+
3rd 3/2-
1st 3/2-
3/2-3: α+7Li(3/2-) with S-wave
main configuration
But 7Li part has a distorted α+t structure
Yamada & Funaki, PRC82(2010)
11B = α +α + t OCM -3 2
T. Kawabata et al., PRC70 (2004)
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B(E0,IS)=96±16 fm4 B(E0,IS)=92 fm4
R=2.43 fm R=2.22 fm
R=3.0 fm
E0 E0
11B: 3rd 3/2-
α triton
No concentration on a single orbit !
NOT cluster condensate
vs. Hoyle state: B(E0,IS)=120±9 fm4
12C: Hoyle state
70%
S-orbit
3(0 )S α
11B = α +α + t OCM Occupation probabilities
T. Kawabata et al., PRC70 (2004)
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B(E0,IS)=96±16 fm4 B(E0,IS)=92 fm4
R=2.43 fm R=2.22 fm
R=3.0 fm
E0 E0
11B: 3rd 3/2-
α triton
No concentration on a single orbit !
NOT cluster condensate
vs. Hoyle state: B(E0,IS)=120±9 fm4
12C: Hoyle state
70%
S-orbit
3(0 )S α
11B = α +α + t OCM Occupation probabilities
T. Kawabata et al., PRC70 (2004)
3 MeV
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states in 11B +1 2
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Structure of 1/2+ states
α α t
α t
exists or not ?
7Li(3/2-) + α : P-wave Parity doublet partner of 3rd 3/2-
Rrms = 3.14[fm] vs. 3.00 fm for 3rd 3/2-
11B 1/2+ : Hoyle-analogue
S-orbit S1/2-orbit
0
-5
5
2α+t
11 2+
7Li+α (1 2 )+
1 2 ( 1 2)T+ =
EXP
11 2+
Candidate ? CAL
21 2+
E2α+t [MeV] 2α+t OCM
Overlap amplitudes with α+ 7Li(g.s) channel
1st 1/2+
3rd 3/2-
1st 3/2-
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Structure of 1/2+ states
α α t
α t
exists or not ? 11B 1/2+ : Hoyle-analogue
S-orbit S1/2-orbit
0
-5
5
2α+t
11 2+
7Li+α (1 2 )+
1 2 ( 1 2)T+ =
EXP
11 2+
Candidate ? CAL
21 2+
E2α+t [MeV] 2α+t OCM
1/2+: Ex=12.6 MeV (E2a+t=1.5 MeV)
Soic et al. NPA742 2004 7Li(9Be,α7Li)5He T=1/2
10Be(p,γ), 9Be(3He,p) T=3/2 Goosman et al. PRC1 1970 Zwiegliski NPA389 1982
Curis et al. PRC72 2006
7Li(7Li,11B*)t T=1/2
⇔Ajenberg-Selove
Rrms = 5.98[fm] : dilute !
7Li(3/2-) + α : P-wave Parity doublet partner of 3rd 3/2-
Rrms = 3.14[fm] vs. 3.00 fm for 3rd 3/2-
11B* → α+7Li
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Complex-scaling method for 1/2+ with α+α+t OCM
Bound state
Resonant state
α+α+t 7Li(1/2-)+α
7Li(3/2-)+α
1st 1/2+
2nd 1/2+
2nd 1/2+ exists as a resonant state.
Ex(cal)= 11.9 MeV, Γ=190 keV Ex(exp)=12.56 MeV, Γ=210±20 keV
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Occupation probabilities of cluster orbits
1st 1/2+
Concentration on S-wave orbits !
2nd 1/2+
α α triton triton
12C(g.s) Hoyle state 70%
52% 96%
2nd 1/2+: Hoyle-analogue state
T. Yamada et al., EPJA 2005
P-orbit S-orbit S-orbit
structure7 Li + α
S-orbit
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Occupation probabilities of cluster orbits
1st 1/2+
Concentration on S-wave orbits !
2nd 1/2+
α α triton triton
12C(g.s)
52% 96%
2nd 1/2+: Hoyle-analogue state
P-orbit S-orbit S-orbit
structure7 Li + α
2nd 1/2+: 65% (0Sα)2(0St)
65% = (2×52%+96%)/3
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(0Sα)2(0St): 65%
(0Sα)3 : 70% (0Sα)4 : 60%
Hoyle-analogue states
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states in 13C ,− +1 2 1 2
• What happens in 12C when an extra neutron (n) is added into the Hoyle state (3α) ? • Hoyle-analogue states?
• 3α+n OCM with GEM
α α n
α n
αS-orbit S1/2-orbit (0Sα)3(Sn) π +J = 1 / 2
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13C 12C(Hoyle) +
p-orbit
s-orbit reduced:
not so changed:
3α part in 13C neutron
11B
8Be(2α) + p-orbit
s-orbit reduced:
not so changed:
2α part in 11B triton
Jπ (13C) 1/2-,3/2-
1/2+
Jπ (11B)
1/2+
1/2-,3/2-
Thus, one can expect cluster-gas-like states in 1/2+.
α-n and α-t potentials: parity-dependent odd waves : attractive enough to make resonances/bound states even waves: weakly attractive
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α-n phase shifts α-triton phase shifts
α-n potential (1) P-wave: attractive
(2) S-wave: weakly attractive
α-triton potential
(1) P-, F-waves: attractive
(2) S-, D-waves: weakly attractive
Parity-dependent potentials
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3α + n OCM for 13C
J=1/2-, 1/2+ : Yamada et al., Int.J.Mod.Phys. E17 (2008) reproduction of M(E0), Hoyle-analogue state
energy spectra of 13C 12C (Hoyle) + n
9Be(1/2+) + α
9Be(3/2-,1/2-) + α [larger than 12C (Hoyle) + n]
12C (g.s) + n neutron-halo-like
R=2.68fm
1/2+ states Probability
Sα-orbit
20%
28%
55%
R=4.0fm
R=5.4fm
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Hoyle-analogue states
(0Sα)3 : 70% (0Sα)4 : 60%
(0Sα)2(0St): 65%
(0Sα)3(Sn): 55%
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IS monopole transition in light nuclei
1. Useful to search for cluster states 2. Dual nature of ground states 3. typical case: 16O IS monopole strength function: α-cluster states at low energy mean-field-type states at higher energy
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0
10
5
15
30+
20+
10+
30+40+
6.06
12.05 ( 1.5 keV)Γ =
[11.26 ( =2.6 MeV)]Γ
13.5
xE (MeV)
14.4 V4 4 Meα
12 7.16 eC M Vα +
Exp.
Monopole strengths M(E0) in 16O and 12C
12C(01+)+α (S)
12C(21+)+α (D)
2( 0) 3.55 0.21 fm (3%)M E = ±
16O
50+14.0
EWSR values
EWSR values
0
10
5
15
10+
14+
30+
20+
12+4.44
7.66
14.1
10.322+
xE (MeV)
3 7.272 MeVα
12C
2( 0) 5.4 0.2 fm (16%)M E = ±
EWSR value
Single particle M(E0)~5.4 fm2
2( 0) 4.03 0.09 fm (8%)M E = ±
2( 0) 3.3 fm (6%)M E =
60+15.1
12C(1-)+α (P)
Strong candidate (4α condensate)
higher nodal
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Isoscalar Monopole Excitations in 16O
α cluster states at low energy and mean-field states at higher energy
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• Isoscalar Monopole Excitation ↔ density fluctuation Typical example IS-GMR (heavy, medium-heavy nuclei)
208Pb(α,α’) IS-GQR
IS-GQR IS-GMR
breathing mode Ex=13.7 MeV Γ=3.0 MeV
Harakeh et al., PRL(1977)
exhaust EWSR ~ 100%
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IS-GMR is well reproduced by RPA cal.
Blaizot, Gogny, Grammticos, NPA(1976)
208Pb: RPA
Ex=18.0 MeV collective motion with coherent 1p-1h states
N=10 h.o.quanta
N=12
N=14
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• Light Nuclei (p-, sd-shell,,,) Isoscalar monopole strengths are fragmented. • For example, 16O, (12C,11B,13C,24Mg,…) IS-monopole response fun. of 16O(α,α’) (i) discrete peaks in Ex ≤ 15 MeV ~20% of EWSR large M(E0) states ⇔ cluster states (ii) three-bump structure : E=18, 23, 30 MeV
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IS Monopole Strength Function of 16O Exp. vs Cal.
Exp: histogram Lui et al., PRC 64 (2001) discrete peaks in Ex≤15 MeV three bumps in 18, 23, 30 MeV Cal: real line Relativistic RPA Ma et al., PRC 55 (1997) Multiplied by 0.25 Shifted by 4.2 MeV
16O(α,α’)
Relativistic RPA cal.
Not well reproduced by RRPA cal.
Exp. condition: Ex > 10 MeV
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Non-rela. Mean-field calculations of IS-monopole strengths for 16O
(1) RPA calculation: 1p-1h excitations Blaizot et al., NPA265 (1976) (2) Second-order RPA (SRPA) calculations: : 1p1h + 2p2h i) Drozdz et al., PR197(1980): D1-force ii) Papakonstantino et al., PLB671(2009): UCOM iii) Gambacurta et al., PRC81(2010) : full SRPA calculation with Skyrme force
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Blaizot et al., NPA(1976)
RPA calculation Isoscalar
16O(α,α’)
Relativistic RPA cal.
Experiment
Exp. condition: Ex > 10 MeV
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RPA(1p1h)
SRPA(1p1h+2p2h)
SRPA RPA
Gambacurta et al., PRC(2010)
61 Exp. condition: Ex > 10 MeV
Exp: histogram Lui et al., PRC 64 (2001)
Exp. condition: Ex > 10 MeV
10 20 30 40
10 20 30 40 50 0 10 20 30 40 0
Papakonstantino et al., PLB(2009)
Ex [MeV] Ex [MeV]
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Experiment 4α OCM Ex
[MeV] R
[fm] M(E0) [fm2]
Γ [MeV]
R [fm]
M(E0) [fm2]
Γ [MeV]
0+1 0.00 2.71 2.7
0+2 6.05 3.55 3.0 3.9
0+3 12.1 4.03 3.1 2.4
0+4 13.6 no data 0.6 4.0 2.4 0.60
0+5 14.0 3.3 0.185 3.1 2.6 0.20
0+6 15.1 no data 0.166 5.6 1.0 0.14
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RPA(1p1h)
SRPA(1p1h+2p2h)
SRPA RPA
Papakonstantino et al., PLB(2009)
Gambacurta et al., PRC(2010)
20+
No sharp peak !
20+
No sharp peak !
Exp. condition: Ex > 10 MeV
Exp: histogram Lui et al., PRC 64 (2001)
Exp. condition: Ex > 10 MeV
10 20 30 40
10 20 30 40 50 0 10 20 30 40 0
Ex [MeV] Ex [MeV]
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RPA calculations: (1) Reproduction of 3-bump structure, but the energy positions are by about 3-5 MeV higher than the data. (2) No reproduction of discrete peaks at Ex≤15 MeV
SRPA calculations: (1) Reproduction of 3-bump structure
(2) Discrete peaks at Ex≤15 MeV are not reproduced well.
In particular, M(E0: g.s – 2nd 0+) is not seen in SRPA cal. Ex(2nd 0+) = 6.1 MeV This peak should exit sharply at Ex=6.1 MeV because Γ ≤ 1 eV.
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Purposes of my talk
• What kind of states contribute to the discrete peaks? • Recently, 4α OCM succeeded in describing the structure of the lowest six 0+ states up to 4α threshold (Ex≈15 MeV). • The discrete peaks correspond to cluster states ? • We study the strength function of IS M(E0) with the 4α OCM framework.
Funaki’s talk
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4α cond. state
12C(01+)+α(S)
12C(21+)+α(D)
12C(01+)+α(S): higher nodal
12C(11-)+α(P)
shell-model-like: (0s)4(0p)12
Funaki, Yamada et al., PRL101 (2008)
4α OCM calculation
strong candidate 4(0 )S α 60%
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16O(α,α’)
Relativistic RPA cal.
Exp: histogram Lui et al., PRC 64 (2001) Cal: real line Ma et al., PRC 55 (1997) Multiplied by 0.25 Shifted by 4.2 MeV
60+
50+40+
30+
Exp. condition: Ex > 10 MeV
10 20 30 40
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IS monopole strength function of 16O within 4α OCM
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Monopole Strength Function
2
12 2
1( ) Im[ ( )]
/ 21 (0 0 )( ) ( / 2)
nn
n n n
S E R E
ME E
π
π+ +
= −
Γ= −
− + Γ∑
IS-monopole m.e.:
Energy weighted sum rule (EWSR): 22 2
1 12( ) (0 0 ) 16 ,n n
nE E M R
m+ +− − = × ×∑ 16
21 cm 1
1
1 0 ( ) 016 i
iR + +
=
= −∑ r R
+n0 : resonance state with En - iΓn/2
rms radius of 16O
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Energy weighted sum rule within 4α OCM (OCM-EWSR)
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4α OCM and OCM-EWSR
Total w.f. : internal w.f.s of α clusters × relative motion
[ ]1 1 1 2 2 2 3 3 312( ) ( , ) (( ) , ) ( , )c L J
f SJν
π ν ϕ ν ϕ ν ϕ νΨ × = ∑ ξ ξ ξ
( ) 0F Ju π =Ψ : Orthogonality condition to Paul forbidden states
4 4( ) (Coul)
OCM 2 21
4
3 4
( , ) ( , )
( , , ) (1, 2,3, 4)
Ni cm
i i j
i j k
H T T V i j V i j
V i j k V
α α
α α
= <
< <
= − + +
+ +
∑ ∑
∑
Hamiltonian for ( )J πΨ : 4-body problem with orthogonality condition
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4α OCM and OCM-EWSR
Total w.f. : internal w.f.s of α clusters × relative motion
IS monopole matrix element
where we used
rms radius of α
α α
α α
16 nucleons
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Interesting characters of IS monopole operator
4 42
4(11 1
21)
4
cm4( ( ) ) i k kk ki
kαα+ −= = =
−= − +∑ ∑∑ R Rr R
internal part of each α-cluster relative parts acting on relative motions of 4α’s with respect to c.o.m. of 16O
4 162 2
1 5
212 12C C( ) ( ) 3( )i i
i iαα
= =
− += − −+∑ ∑ R Rr R r R
internal part of α internal part of 12C relative part acting on relative motion of α and 12C
Decomposition into Internal part and relative part plays an important role in understanding monopole excitations of 16O.
α 12C
α α
α α
16 nucleons
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4α OCM and OCM-EWSR
Total w.f. : internal w.f.s of α clusters × relative motion
EWSR of IS monopole transition in 4α OCM: OCM-EWSR
where 4
2OCM cm
14( ) ,
kk
O α=
= −∑ R R16
21 cm 1
1
1 0 ( ) 016 i
iR + +
=
= −∑ r R
: r.m.s radius of α-particle
OCM-EWSR
: r.m.s radius of 16O
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4α OCM and OCM-EWSR
Ratio of OCM-EWSR to total EWSR:
4α OCM shares over 60% of the total EWSR value. ( c.f. α+12C OCM : 31%) This is one of important reasons that 4α OCM works rather well in reproducing IS monopole transitions in low-energy region of 16O.
22 21 1
2( ) (0 0 ) 16n nn
E E M Rm
+ +− − = × ×∑
Yamada et al.
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IS monopole strength function S(E) within 4α OCM framework
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Monopole Strength Function
2
12 2
1( ) Im[ ( )]
/ 21 (0 0 )( ) ( / 2)
nn
n n n
S E R E
ME E
π
π+ +
= −
Γ= −
− + Γ∑
+n0 : resonance state with En - iΓn/2
: calculated by 4α OCM 2
n2
n= Γ (OCM) + (exp. resolution)Γ
nE : experimental energy of n-th 0+ state
50 keV
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Experiment 4α OCM Ex
[MeV] R
[fm] M(E0) [fm2]
Γ [MeV]
R [fm]
M(E0) [fm2]
Γ [MeV]
0+1 0.00 2.71 2.7
0+2 6.05 3.55 3.0 3.9
0+3 12.1 4.03 3.1 2.4
0+4 13.6 no data 0.6 4.0 2.4 0.60
0+5 14.0 3.3 0.185 3.1 2.6 0.20
0+6 15.1 no data 0.166 5.6 1.0 0.14
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16O(α,α’)
Relativistic RPA cal.
IS monopole S(E) with 4α OCM
30+
40+
50+
60+
20+
Exp. vs. Cal.
Exp. condition: Ex > 10 MeV
It is likely to exist discrete peaks on a small bump in Ex < 15MeV This small bump may come from the contribution from continuum states of α+12C
10 20 30 40
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16O(α,α’)
Relativistic RPA cal.
IS monopole S(E) with 4α OCM
30+
40+
50+
60+
20+
Exp. vs. Cal.
Excitation to cluster states (α-cluster type)
Monopole excitation of mean-field type (RPA)
Exp. condition: Ex > 10 MeV 10 20 30 40
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IS monopole S(E) with 4α OCM
30+
40+
50+
60+
20+
Exp. vs. Cal.
Excitation to cluster states (α-cluster type)
Monopole excitation of mean-field type (RPA)
Two features of monopole excitations They come from a dual nature of the ground state of 16O
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IS monopole S(E) with 4α OCM
30+
40+
50+
60+
20+
Exp. vs. Cal.
Excitation to cluster states (α-cluster type)
Monopole excitation of mean-field type (RPA)
Two features of monopole excitations They come from a dual nature of the ground state of 16O
next discuss!
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Dual nature of ground state of 16O
mean-field character and α-clustering character
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Ground state of 16O
Dominance of doubly-closed-shell structure: (0s)4(0p)12 Cluster-model calculations: 4α OCM, 4α THSR, α+12C OCM,... Mean-field calculations : RPA, QRPA, RRPA,......
Supported by no-core shell model calculations: Dytrych et al., PRL98 (2007)
Bayman-Bohr theorem :
Doubly-closed-shell w.f, (0s)4(0p)12, is mathematically equivalent to a single α-cluster w.f.
This means that the ground state w.f. of 16O originally has an α-clustering degree of freedom together with single-particle degree of free dom.
We call it duality of α-clustering and mean-filed characters in g.s.
= SU(3)(0,0)
SU(3)[f](λµ) = cluster-model wf
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[ ]
{ }
{ }
14 12cm cm
120 40 3 0
122 42 23 0
0
1 det (0 ) (0 ) ( )16!
12!4! ( ,3 ) ( C) ( )16!
12!4! ( ,3 ) ( C) ( )16!
J
J
L
L
s p
N A u
N A u
φ
ξ ν φ φ α
ξ ν φ φ α
=
−
=
==
×
=
=
R3/4
2cm cm cm
32( ) exp( 16 )νφ νπ
= −
R R
Bayman-Bohr theorem
c.o.m. w.f. of 16O
G.S. has mean-field-type and α-cluster degrees of freedom.
Nucl. Phys. 9, 596 (1958/1959)
Excitation of mean-field-type degree of freedom in g.s 1p1h states (3-bump structure)
Excitation of α-cluster degree of freedom in g.s α+12C cluster states: 2nd 0+, 3rd 0+
4 162 2
1 5
212 12C C( ) ( ) 3( )i i
i iαα
= =
− += − −+∑ ∑ R Rr R r R
relative part internal parts
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4α cond. state
12C(01+)+α(S)
12C(21+)+α(D)
12C(01+)+α(S): higher nodal
12C(11-)+α(P)
shell-model-like: (0s)4(0p)12
Funaki, Yamada et al., PRL101 (2008)
4α OCM calculation
strong candidate 4(0 )S α 60%
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Monopole excitation to 0+2,3 and 0+
6 state
0+2,3 states: 12C(01
+,21+)+α(S) main configuration
Bayman-Bohr theorem: (0s)4(0p)12 has an α + 12C(01
+,21+) degree of freedom
relative motion (α-12C) is excited by IS monopole operator
0+6 state: 4α-gas like structure
Bayman-Bohr theorem: (0s)4(0p)12 has a 4α degree of freedom
relative motion among 4α is excited by IS monopole operator
main configuration
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(0s)4(0p)12
16O(g.s.) α
SU(3)(00) wf
= α α (N2α,L2α)=(4,0)
(N2α-α,L2α-α)=(4,0)
0+1
α (N3α-α,L3α-α)=(4,0)
Bayman-Bohr theorem
Bayman-Bohr theorem
G.S. has a 4α-cluster degree of freedom.
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(0s)4(0p)12
16O(g.s.) α
SU(3)(00) wf
= α α (N2α,L2α)=(4,0)
(N2α-α,L2α-α)=(4,0)
0+1
α (N3α-α,L3α-α)=(4,0)
α α
α α
4α gas-like or α+12C(Hoyle) ≈ 0+6 state
4 42
4( 1)1 1
42
1cm( ) ( 4 ) i k kk
k i kαα+ −
= = =
−+−= ∑∑ ∑ R Rr RMonopole transition
relative part
coherent excitation
Bayman-Bohr theorem
internal part
4α -gas-like M(E0)=1.0 fm2 by 4α OCM
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Dominance of SU(3)(04) : no-core shell model by Dytrych et al., PRL98 (2007)
α
α α
(0s)4(0p)8
α
SU(3)(04) wf
= α α (N44,L44)=(4,0)
(N84,L84)=(4,0)
12C(g.s.)
0+1
3α gas-like = 0+2 (Hoyle) state
3 42
4(
1
1 1
2
1)
3
cm4(
( )
)
i k kk
k
i
kα
α+ −= =
=
−
= −
+
∑
∑
∑
R R
r R
Monopole transition relative part
coherent excitation
internal part 2( 0) 5.4 0.2 fm (16%)M E = ±
portion of EWSR
Yamada et al., PTP120 (2008)
12C
Bayman-Bohr theorem
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Bayman-Bohr theorem
12 4 120 internal
0 40 2 40 10
( C) (0 ) (0 ) ;( , ) (0, 4), 0
4!4!4! 8 ( , ) ( , 2 ) ( ) ( ) ( )12! 3
J
J
s p J
N A u u
πφ λ µ
ν ν φ α φ α φ α
+=
=
= = =
= ξ ξ
Dominance of SU(3)(04) : confirmed by no-core shell model Dytrych et al., PRL98 (2007)
This means that the ground state w.f. of 12C originally has an α-clustering degree of freedom together with single-particle degree of free dom.
12C
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Monopole excitation to 0+5 and 0+
4 state
0+5 state: 12C(11
-)+α(P) main configuration
Bayman-Bohr theorem: (0s)4(0p)12 has no configuration of 12C(1-)+α
Why this state is excited?
Coupling with 12C(0+,2+)+α and 12C(Hoyle)+α configuration
Coherent contribution from these configurations
0+4 state: 12C(01
+)+α(S): higher nodal
Coherent contributions from 12C(0+,2+)+α and 12C(Hoyle)+α configurations
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Components of α+12C(Lπ) channels in 0+ states of 16O
SU(3)-nature (0s)4(0p)12 12C(01
+)+α(S) 12C(21+)+α(D)
12C(01+)+α(S)
higher nodal 12C(11
-)+α(P) 12C(02+)+α(S): 4α gas-like
R=2.7fm
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Components of α+12C(Lπ) channels in 0+ states of 16O
SU(3)-nature (0s)4(0p)12 12C(01
+)+α(S) 12C(21+)+α(D)
12C(01+)+α(S)
higher nodal 12C(11
-)+α(P) 12C(02+)+α(S): 4α gas-like
R=2.7fm
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Two features of IS monopole excitations in 16O
IS monopole S(E): 4α OCM
30+
40+
50+
60+
20+
Excitation to cluster states (α-cluster type) Monopole excitation
of mean-field type (RPA)
Fine structure Ex < 16 MeV
3-bump structure 16 < Ex < 40 MeV
4α OCM OK difficult
Mean-field difficult OK
Origin: dual nature of G.S. of 16O (1) α-clustering degree of freedom (2) mean-field-type one (0s)4(0p)12 : SU(3)(00)= 12C+α : Bayman-Bohr theorem
Two features in IS monpole exci.
Huge model space is needed to reproduce S(E) in the whole energy region.
cluster basis + collective basis
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Dual nature of the ground states in 12C and 16O : common to all N=Z=even light nuclei
20Ne
Excitation of mean-field degree of freedom
Excitation of α-cluster degree of freedom
α+16O cluster states of Kπ = 0+4, 0
- bands
α+16O comp. = 80% for low spins: Q=10 quanta
Almost pure α+16O structures for low spins: Q=9 quanta
20Ne, 24Mg, 32S, ….., 44Ti,….
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M. Kimura, PRC69, 044319 (2004)
AMD+GCM 20Ne
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Two features of IS monopole excitation
Cluster states at low energy, Mean-field-type states at higher energy
These features will persist in other light nuclei.
Increasing mass number, effect of spin-orbit forces becomes stronger: some SU(3) symmetries are mixed. Goodness of nuclear SU(3) symmetry: gradually disappearing. Two features of IS monopole excitation will be vanishing. Eventually only 1p1h-type collective motions are strongly excited.
Hope: Systematic experiments!!
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Summary • α-condensation in 12C, 16O, heavier 4n nuclei. • Hoyle-analog states in 11B, 13C • IS monopole tran. : useful to search for cluster states • IS monopole excitations have two features: 16O (typical) (i) α-cluster type: discrete peaks in Ex≤15 MeV (ii) mean-field type: 3-bump structure (18,23,30 MeV)
• The origin: dual nature of the ground state of 16O. (0s)4(0p)12 : SU(3)(00) = 12C+α : proved by Bayman-Bohr theorem Dual nature is common to all N=Z=even light nuclei • クラスターガス状態: 新しい物質形態 拡がりと深さの探求: 通常核(4n核), 中性子過剰核