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13Cにおける単極遷移強度と
クラスター構造
山田泰一、船木靖郎
Contents
• Monopole strength and cluster structureTypical case : 16O (12C)
• Alpha clustering and condensation in 13C
Monopole strengths in 1/2- states
Structures of 13C: 1/2-, (1/2+)
E0-strength of 16O: Exp. vs Cal.16O(α,α’) 16O(α,α’)
Lui et al., PRC64(2001)
RelativisticRPA cal.
Exp: Lui et al., PRC 64 (2001)Cal: Ma et al., PRC 55 (1997)
Multiplied by 0.25Shifted by 4.2 MeV
TEWSR=48±10%
0
10
5
15
20+
10+
30+40+
6.06
12.05 ( 1.5 keV)Γ =13.5
xE (MeV)
4α
12C α +
Exp.
Monopole matrix elements M(E0) in 16O and 12C
12C(01+)+α(S)
12C(21+)+α(D)
2( 0) 3.55 0.21 fm (3%)M E = ±
16O50+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 M eVα
12C
2( 0) 5.4 0.2 fm (16%)M E = ±
EWSR value
2( 0) 4.03 0.09 fm (8%)M E = ±
2( 0) 3.3 fm (6%)M E =
60+15.1
Monopole Strengths
2.6 (3.50*)4.03±0.090+1- 0+
3
Cal. [fm2]Exp. [fm2]
5.4±0.2
3.3±0.7
3.55±0.21
0+1- 0+
2
0+1- 0+
5
0+1- 0+
2
6.712C
3.0 (-*)
4.1 (3.98*)
16O
16O: 12C+α OCM*4α OCM
Suzuki, PTP56, 111 (1976)Funaki, Yamada et al., PRL 101 (2008)
12C: 3α RGM Kamimura, Nucl. Phys. A 351, 456 (1981)
No effective charge!
12Be: 0+1 – 0+
2 (shell-model structure), <r2>=0.83 fm2, S. Shimoura et al., PLB560 (2002)
Exotic characters of 3/2-3 of 11B
0.483
0.745
0.588Shell ModelExperiment
-
0.6160.782
< 0.0038.104 (3/2−)
0.487±0.0295.020 (3/2−)
0.398±0.0318.420 (5/2−)
0.453±0.0294.445 (5/2−)0.401±0.0322.125 (1/2−)
B(GT)Ex (MeV)
0.345±0.0080.000 (3/2−)
3/2−3 state has exotic characters.
・ Suppressed GT strength・ Large monopole strength・ Not predicted by the shell-model calculation・ 100-keV below the α-decay threshold.
0
2
4
6
8
10
12
1/2-1
3/2-1
3/2-2
3/2-3
3/2-4
5/2-1
5/2-2
5/2-3
5/2-4
7/2-1
Experiment
Exc
itatio
n E
nerg
y (M
eV)
1/2-1
1/2-2
3/2-1
3/2-2
3/2-3
3/2-4
5/2-1
5/2-2
5/2-3
7/2-1
Shell-Model
Exc
itatio
n E
nerg
y (M
eV)
T. Kawabata et al., Phys. Lett. B 646, 6 (2007).
α+α+t
7Li+α
AMD, 2α+t OCM: 3/2-3 state has α+α+t cluster structure
4( 0, ) 96 16 fm (13%)B E IS = ±
Enyo-Kanada Yamada, Funaki
Monopole strengths to cluster states: ~20% of EWSRHere, we have an interesting question.
Why cluster states are populated from the ground states with mean-field structures by the monopole transitions ?
Non trivial problem !
T. Yamada, Y. Funaki, H. Horiuchi,K. Ikeda, A. Tohsaki, to be published in Prog. Theor. Phys.
Monopole operator
16O: ground statemean-field structure
2nd ,3rd 0+ states: 12C+α structure
Monopole transition between 0+1 and 0+
2 states
0+2
Relative motion is excited.
12C(0+)+ α cluster structure
( ) ( ) ( )12 C
162 22 2
1
1 1 1 1 12 42 2 2 2 16i G
ii i
iC
i αα
∈∈=
×− = − + − +∑ ∑ ∑r r r r r r r
Monopole operator rel.
12C(0+)α
Monopole
(0s)4(0p)12
16O (g.s.)
=0+1
12C(0+)
α
40 ( ,3 )NR r ν
Bayman-Bohr theorem
Q=12
G.S. correlation Q>12: α clustering
activating the seed of α
10% comp.
having an seed of αin G.S.
SU(3)(00)90% comp.
Doubly closed shell-model w.f. 16O
(0s)4(0p)12
16O (g.s.)
=12C(0+)
α
40 ( ,3 )NR r ν
=12C(2+) α
42 ( ,3 )NR r ν
12C(0+,2+,4+) wf.: SU(3)(04) wfα cluster: intrinsic (0s)4
R4L(r,3vN): h.o.w.f. with Q=4
Bayman-Bohr theorem
16O g.s. can be excited throughcluster degree of freedom, namely, 12C+α relative motion, from R4L tohigher nodal states.
SU(3)(00)
[ ]4 1 12(01. . det ( )1
)6
(0!
) G Gg s s p φ −= × r
{ }120 0 40 0
12!4! ( C) ( ) ( )16!g L J
A R rN φ φ α= =⎡ ⎤= ⎣ ⎦
{ }122 2 42 0
12!4! ( C) ( ) ( )16!g L J
A R rN φ φ α= =⎡ ⎤= ⎣ ⎦
howf
So far, our discussion was qualitatively.Next, we study the monopole strengths in 16O quantitatively with use of the 12C+α OCM.
16O=12C+α OCM
12Cα
{ },
12
12!4! ' ( ,3 )16!
( C) ( ) ( )
J NJ NJi i
i N
NJ Ji i NL
Ji I L J
c
A h R r
h Y r
ν
φ φ α
Φ = Φ
Φ =
⎡ ⎤= ⎣ ⎦
∑
12( ) : SU(3)(0,4) w.f. ( 0, 2,4)( , ) : h.o. basis ( 2 4)
L
NL
C LR r N n Lφ
β=
= + ≥
Y. Suzuki, PTP55(1976)
r
SU(3) basis : (0,4)×(N,0) = (N-4,0) + (N-2,2) + (N,4)12C 12C-α
Relative motionexpanded by h.o. basis
N=4 : (0s)4(0p)12 closed shellLarger N : 12C-α clustering
N
1616O=O=1212C+C+α α cluster modelcluster model
Even-parity Odd-parity
EXP CAL EXP CAL
12C+α : molecular states
Y. Suzuki, PTP55 (1976), 1751
12C+α
4α
Am
plitu
de
(0,0)
(2,0)
(4,0)
(4,2)
(6,2)
(8,4)
Full model space
G.S. w.f. : SU(3) expansion
122 *
111 1
1(0 0 ) ( ) ( ) (0 0 0 )2
0k kk i Gi
M r r+ +
=
+ + + +− = − ∝ Φ Φ∑Product of amplitudes
Ground state correlation in 16O (12C+α OCM)
s4p12
G.S. correlation
12C+α OCM
11 %N>4
89 %N=4
comp.
16O g.s wf
Deviation from doubly closed shell w.f.
2
10( / 3 ) ( ) 0 : OCMNP βνβ +
+= Φ
Squared overlap on β/3νΝ
0.9580.847・・・
0.890・・・1/ 3 Nβ ν
P最大値になる
α clustering is activated !
Modified doubly closed shell w.f.
{ }120 400 0
12!4!( ) ( ) ( ) ( , ) ( )16! LN A C R rβ β βφ φ α+ =⎡ ⎤Φ = ⎣ ⎦
/ (3 ) 1Nβ ν = の時、doubly closed shell w.f: (0s)4(0p)12
/ (3 ) 1Nβ ν < α clustering is activated.
Nν : nucleon size parameter
12C40 ( , )R r β
α
Monopole Strengths & G.S. correlation
1 : G.S. wf within quanta model space ( =4,6,0 ; ,30)N N N+
122
1 1*
11
1(0 0 ) (0 ; 0 0) ( );2
0) (k i Gi
N k kN r NM r++ +
=
+ + +− = − ∝ Φ Φ∑
Study effect of the ground-state correlation
(12C-α clustering in g.s.)
1 1
(1 ) 0 ,
0 0 0
0
; 0 ;
Nkk k
k k k
N P
N N N
+
+ + + +
+ = −
⎡ ⎤= −⎣ ⎦
10 ; 0 0kN+ + =k=2,3
2 30 , 0 : obtained with full model space ( =30)N+ +
12C+α structures
12C+α OCM
Dep. of MN(0+1-0+
2,3) on model space of G.S. in 16O
01+ - 03
+
01+ - 02
+
Monopole strengths
Importance of G.S. correlation (12C-α clustering in g.s.)
(model space of ground state)
3.98 fm2 vs. 3.55±0.22 fm2
EXPCAL
3.50 fm2 vs. 4.03±0.09 fm2
Shell-model limit: G.S.=(0s)4(0p)12 T. Yamada et al.,to be published in Prog. Thor. Phys.
16O
12C+α OCM
Quanta N
24 ( 0) ( 0) 0 ( 0)
. : 44 6
G S NN
M E O EO E N
+=
=∝
=
=
Summary (I)Mechanism of M(E0) in light nuclei
(1) Structure of ground statedual aspects in g.s. : mean-field + cluster originally having a seed of α clustering (Bayman-Bohr theorem)
g.s. correlation enhanced α clustering or activating the seed
(2) Monopole operator:exciting relative motions between clusters by 2hw
(3) Cluster states are populated by E0 (about 20% of EWSR).
Monopole strengths are a good tool to explore cluster states.
Cluster structure and α condensation in 13C
1. Structures of 1/2- states and monopole strengths2. Structures of 1/2+ states
T. Yamada and Y. Funaki
Motivations
• 12C, 2nd 0+ (Hoyle); 3α condensate16O, 6th 0+; 4α condensate
• Addition of an extra neutron to Hoyle state (3α cond.)What happens ? Which state has the 3α+n gas-like (condensate) structure ?
: gateway to explore gas-like states composed of bosons and fermions
• 1/2- states excited by monopole transitions in 13C(α,α’): What kinds of structures they have ?
Monopole excitations are a good tool to explore cluster structures.
Sasamoto and Kawabata et al.Mod. Phys. Lett. 21 2393 (2006)Reanalayses: Mar. 2008
13C: monopole strengths
(5%)
(3%)
(4%)
EWSR
Shell modelM(E0) < 0.2 fm2
Monopole excitations in 13C
{ }
{ }
{ }
{ }
1/ 2
3 / 2
00
02
4, 23/ 2
4, 01
13 12
1/ 2
12
1/ 2
9
1/ 2
9
1// 2 2
12!( C) ( C) ( )13!
12! ( C) ( )13!9!4! ( Be) ( ) ( )13!9!4! ( Be) ( ) ( )13!
p
p
N L
N L
GS J
J
J
J
A u
A u
A u
A u
φ
φ
φ φ α
φ φ α
+ −
−
−
−
−+
−
= =
=
=
=
=
= =
Φ =
=
=
=
R
R
R
R
Bayman-Bohr theorem: GS of 13C SU(3)(31)×1/2
12C(3α)+n
9Be+α
13C
12C n
Monopole operator: O(E0:IS)
12C n13Cα
αα12C
n
9Beα
9Be α
9Be α
O(E0)
Monopole-excited cluster states
NeutronHalo (?)
12C(Hoyle)+n
9Be+α cluster
gs
gs
equivalent
13C=3α +n OCM with Gaussian basis
Approximately taken into account: Fully Solving 4-body problem
12C=3α OCM, 9Be=2α+n OCM : successful reproduction of 12C and 9Be
(OCM=Orthogonality Condition Model)
α3 3 3( , )ϕ νr
1 1 1( , )ϕ νr
2 2 2( , )ϕ νr
An
ααΨJ(13C) =
3 3
2 2 3 21 1
( ) ( ) ( )Coulij ij n in n Pauli
i j i
H T V r V r V r V V Vα α α α α +< = =
⎡ ⎤= + + + + + +⎣ ⎦∑ ∑
( )2( , ) ( ) exp ( )m mN r r Yϕ ν ν ν= −r rGaussian bais:
Angular momentum channels:
[ ] 144 84 2nI L Jl l l⎡ ⎤⎡ ⎤⎣ ⎦⎣ ⎦ 30 channels
Vαα: reproduction of α-α phase shiftsVαn: reproduction of α-n phase shifts (Kanada-Kaneko pot.)
K-H coordinates
Overlap amplitudes (reduced width amp.)
12C(gs)+n12C(2+)+n
12C(3-)+n12C(1-)+n9Be(gs)+α
12C(Hoyle)+n 12C(2+)+n9Be+α
13C(1/2-, gs)13C(1/2-, 2nd): M(E0,IS)=4.2 fm2
13C(1/2-, 3rd): M(E0,IS)=5.6 fm2 13C(1/2-, 4-th): M(E0,IS)=8.2 fm2
SU(3) character
12C (Hoyle) + n9Be + α
12C (2+) + n
9Be + α12C (3−) + n
12C (gs, 2+) + n
8.25.64.2
(fm2)M(E0;IS)
8.793.361/2−2
6.374.33
2.82(fm)R3α-n
2.971/2−3
3.191/2−4
2.391/2−1
(fm)RrmsJπ
3α+n OCM
M(E0:IS)
Rms=4.3 fm for Hoyle state
Cluster structure
Fujimura and Fujiwara et al., PRC69, 064327 (2004)
13C(3He,t)13N12C+p decay
1/22- 1/23
-
12C(g.s)+p decay
12C(2+)+p decay
12C(Hoyle)+p decay
13N
12C (Hoyle) + n9Be + α
12C (2+) + n
9Be + α12C (3−) + n
12C (gs, 2+) + nLarge decay branching to the Hoyle and 2+ states.
Large neutron transition strength
8.25.64.2
(fm2)M(E0;IS)
8.793.361/2−2
6.374.33
2.82(fm)R3α-n
2.971/2−3
3.191/2−4
2.391/2−1
(fm)RrmsJπ
3α+n OCM
M(E0:IS)
Rms=4.3 fm for Hoyle state
Summary (2)• Mechanism of M(E0) transition in light nuclei
good tool to explore cluster structures
• 13C(1/2-), structure and M(E0): 3α+n OCM2nd 1/2- ; 12C(0+,2+)+n 3rd 1/2- ; 9Be+α, 12C(3-,1-)+n4-th 1/2- ; 12C(Hoyle)+n, 9Be+α, 12C(2+)+n
Nuclear radii =3.0~3.4 fm : cluster structure
• 13C(1/2+)3nd 1/2+ ; strong candidate of dilute α condensation
R~5 fm, Occu. Prob. ~60%• Future : 3/2-, other states, LS-splitting.
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