12Cにおける3α共鳴状態と散乱への寄与

KEK原子核研究会
「現代の原子核物理-多様化する原子核の描像」
多体共鳴状態の境界条件によって解析した
3α共鳴状態の構造
C. Kurokawa1 and K. Kato2
Meme Media Laboratory, Hokkaido Univ., Japan1
Div. of Phys., Grad. Sch. of Sci., Hokkaido Univ., Japan2
KEK原子核研究会8１-8/3
Theoretical studies of 12C
D.M.Brink in Proceedings of the Fifteen Solvay Conference on Physics (19070)
○Microscopic 3α model (RGM・GCM・OCM)
Y.Fukushima and M.Kamimura in Proceedings of the International Conference on Nuclear Structure (1977)
M.Kamimura, Nucl. Phys. A351(1981),456
Y.Fujiwara, H.Horiuchi, K.Ikeda, M.Kamimura, K.Katō, Y.Suzuki and E.Uegaki, Prog Theor. Phys. Suppl.
68 (1980)60.
E.Uegaki, S.Okabe, Y.Abe and H.Tanaka, Prog. Theor. Phys. 57(1977)1262; 59(1978)1031; 62(1979)1621.
α
H.Horiuchi, Prog. Theor. Phys. 51(1974)1266; 53(1975)447.
K.Fukatsu, K.Katō and H.Tanaka, Prog. Theor. Phys.81(1988)738.
α
○3α+p3./2Closed shell
Γ=34keV
N.Takigawa, A.Arima, Nucl. Phys. A168(1971)593.
N.Itagaki Ph.D thesis of Hokkaido University (1999)
3
Γ=8.7eV
α
－
31
02 +
Y.Kanada-En’yo, Phys. Rev. Lett. 24(1998)5291.
α
α
α
○Deformation （Mean-Field）
G.Leander and S.E.Larsson, Nucl. Phys.A239(1975)93.
○Faddeev
Y.Fujiwara and R.Tamagaki Prog. Theor. Phys. 56(1976)1503.
H.Kamada and S.Oryu, Prog. Theor. Phys 76(1986)1260.
01 +
Excited states of cluster states?
KEK原子核研究会8１-8/3
Situation around Ex= 10 MeV
Energy level of 12C
02
+:


l=0
L=0

Alpha-condensed state
0+, 2+
A.Tohsaki et al., PRL87(2001)192501
0+ : Er=2.7+0.3 MeV, G= 2.7+0.3 MeV
2+ : Er=2.6+0.3 MeV, G= 1.0+0.3 MeV
[Ref.]: M.Itoh et al., NPA 738(2004)268
Can 3αModel reproduce both of
the 22+ and the 03+ states ?
What kind of structure dose the
03+ state have ?
Why 03+ has such a large width ?
[Ref.] E.Uegaki et al.,PTP57(1979)1262
Boundary condition for three-body resonances
Analysis of decay widths
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Our strategy
In order to taking into account the boundary condition for three-body
resonances, we adopted the methods to 3 Model;

Complex Scaling Method (CSM)
[Ref.] J.Aguilar and J.M.Combes, Commun. Math. Phys., 22(1971),269
E.Balslev and J.M.Combes, Commun. Math. Phys., 22(1971),280

Analytic Continuation in the Coupling Constant
[Ref.] V.I.Kukulin, V.M.Krasnopol’sky, J.Phys. A10(1977),
combined with the CSM (ACCC+CSM)
[Ref.] S.Aoyama PRC68(2003),034313
Both enables us to obtain not only resonance energy
but also total decay width
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Model : 3  Orthogonality Condition Model (OCM)
folding for Nucleon-Nucleon
interaction(Nuclear+Coulomb)
[Ref.]: E. W. Schmid and K. Wildermuth, Nucl. Phys. 26 (1961) 463
, -parity )
μ=0.15 fm-2
: OCM [Ref.]: S.Saito, PTP Supple. 62(1977),11
Phase shifts and Energies of 8Be, and Ground band states of 12C
22
11
c=3
c=2
c=1
1
2

132
3
33
,
[Ref.]: M.Kamimura, Phys. Rev. A38(1988),621
KEK原子核研究会8１-8/3
Methods for treatment of three-body resonant states

CSM
It is sometimes difficult for CSM
to solve states with quite large
decay widths due to the limitation
of the scaling angle  and finite
basis states.
2θ
Exp. Broad
state
In order to search for the broad 0+ state, we employed …

ACCC+CSM
: Atractive potential with
k
Im(k)
<0
δ→0
Re(k)
Resonance
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Energy levels obtained by CSM and ACCC+CSM
G= 0.375+0.040 MeV
Γ=0.12 MeV
(2+)
0+ : Er=2.7+0.3 MeV, G= 2.7+0.3 MeV
03+: Er=1.66 MeV, Γ=1.48 MeV
2+ : Er=2.6+0.3 MeV, G= 1.0+0.3 MeV
22+: Er=2.28 MeV, Γ=1.1 MeV
[Ref.]: M.Itoh et al., NPA 738(2004)268
ACCC+CSM
3α Model reproduce 22+ and
03+ in the same energy region by
E.Uegaki
et al.,PTP(1979)
taking into account the correct boundary
condition
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Structures of 0+ states through Amplitudes
Wave function of 0+ states
Y (12C) Jp=0+ = l=0,L=0 j0,0 + l=2,L=2 j2,2 + l=4,L=4 j4,4
8Be
l

jl,L= [ 8Be (l) x L ]

L
l,L2 : Channel Amplitudes

Channel Amplitudes of 01+, 02+ and 04+
E [MeV]
0,02
Rr.m.s. [fm]
2,22
4,42
Er
G
Re.
Im.
Re.
Im.
Re.
Im.
Re.
Im.
01+
-7.29
0
2.36
0
0.364
0
0.382
0
0.254
0
02+
0.76
2.4x10-3
4.29
0.29
0.775
0.033
0.149
-0.019
0.076
-0.014
04+
4.58
1.1
3.26
0.97
0.499
0.170
0.307
-0.017
0.194
-0.153
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Feature of the broad 3rd 0+ state
Channel amplitudes as a function of d
8Be


l=0
2
2
2
L=0
Dominated

Similar property to 02+
（ Rr.m.s= 4.29 fm）
Re(Rr.m.s) (d= -140): 5.44 fm
Large component of 0,02 makes such the large width.
Wave function of 03+ shows similar properties to 02+.
03+ is considered as an excited state of 02+. Higher nodal state of 02+ ?
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Summary of obtained 0+ states
04+
03+
I=0
L=0
but higher
nodal ?
02+
I=0
L=0
r.m.s.=4.29 fm
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Structure of the 04+ state
4th 0+ state ;
Large component of high angular momentum compared with 2nd 0+
0,02 =0.499, 2,22 =0.307, 4,42 =0.194
Total decay width is sharp: Er=4.58 MeV, G=1.1 MeV

3αOCM with SU(3) base : K.Kato, H.Kazama, H.Tanaka, PTP 77(1986),185.
Component of linear-chain configuration: 56%

AMD: Y.Kanada-En’yo, nutl-th/0605047.
FMD: T.Neff, H.Feldmeier, NPA 738(2004), 357.
Linear chain like structure is found
α α
α
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Probability Density of 1st 0+ and 4th 0+ states
(Preliminary)
Probability Density of ’s
r1
12
r2
r1 = r2 = r
r [fm]
01+
04+
12
12
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Summary and Future work





We solve states above 3αthresold energy taking into account the
boundary condition for three-body resonant states.
Obtained resonance parameters of many Jp states reproduce
experimental data well.
We obtained broad 3rd 0+ state near the 2nd 2+ state. The state has
similar structure to the 2nd 0+ state. It is thus expected to be an
excited state of 2nd 0+.
The 4th 0+ state has large component of high angular momentum
channel, [8Be (2+) x L=2], and has a sharp decay width.
These features reflect the linear-chain like structure of 3αclusters.
Members of rotational band built upon the 4th 0+ state ?
How do these states contribute to the real energy ? To investigate it
we calculate the Continuum Level Density in the CSM and partial
decay widths to 8Be(0+, 2+, 4+)+α in feature.
[Ref.] A.T. Kruppa and K. Arai, PLB 431(1998)237
R. Suzuki, T. Myo, and K. Kato, PTP 113 (2005) 1273
KEK原子核研究会8１-8/3
Probability Density of 0+ states
04+
0２+
12
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Contributions from resonant states to real energy
Continuum Level Density (CLD) Δ(E)
( E  0) =  ( E )   0 ( E ),
[Ref.] S.Shomo, NPA 539 (1992) 17.
( E ) =
1 dd l
p dE
δl: phase shift

 0 ( E ) =  d ( E  Ei )

 ( E ) =  d ( E  Ei )
i
i
  1 
=  Im Tr 

p   E  H  
1
=
  1 
Im Tr 


p   E  T 
1
Discretization with a finite number N of basis functions
( E )  N ( E )
[Ref.] A.T. Kruppa and K. Arai, PLB 431(1998)237.
N
N
i =1
i =1
=  d ( E  Ei )  d ( E  Ei0 )
Smoothing technique is needed,
but results depend on smoothing parameter.
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CLD in the Complex Scaling Method
[Ref.] R. Suzuki, T. Myo, and K. Kato, PTP 113 (2005) 1273
Bound state
NB
N ( E ) =  d ( E  EB ) 
B
Continuum
Resonance
1
p
N R
Im 
R
１
1
 Im
E  ER p
N  N B  N R

C
１
E   C ( )
ER, εc(θ) have complex eigenvalues in CSM
CLD in CSM:
N ( E  0) = N ( E )  0 N ( E )
=
+
N R
Gr / 2

p R ( E  Er ) 2 + Gr2 / 4
1
1
p
N  N B  N R

c
 cI 2
 c0 I 2
1 N
 
R 2
I2
(E   c ) +  c
p c ( E   c0 R ) 2 +  c0 I 2
Smoothing technique is not needed
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Application to ３α system
CLD of 3αsystem
( E ) =  3 B ( E )   30B ( E )
  1
1 
=  ImTr 


p   E  H 3 B E  H 30B  
1
1
α１
α２
2
3
3
3
i =1
3
i =1
3
H 3 B =  ti  TG +  VN+Cl +OCM ( i ) + V3 (1 ,  2 ,  3 )
H 30B =  ti  TG +  VCl( point) ( i )
i =1
i =1
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Continuum Level Density: 0+ states
8Be(0+)
+α
8Be(2+)
E [MeV]
+α
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Subtraction of contribution from 8Be+α
(
' ( E ) = 3 B ( E )  30B ( E )   2 B ( E )   20B ( E )
)


 
1
1
1
  1



=  ImTr 





p   E  H 3 B E  H 30B  E  H 2 B E  H 20B  
1
3
8Be
α１
1
α２
)
H 2 B =  ti  TG + VN+Cl +OCM (1 ) + V8ClBe(point
(1 )

i =1
• α1- α2: resonance + continuum
• (α1α2)- α3: continuum
1
α３
3
H
0
2B
)
=  ti  TG + VCl( point) (1 ) + V8ClBe(point
(1 )

i =1
• α1- α2: continuum
• (α1α2)- α3: continuum
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Contributions from 8Be+α are subtracted
‘
02+
04+
03+
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Subtraction of contribution from 8Be+α
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Search for broad 0+ state with
05+
04+
δ= －200 MeV
δ= －150 MeV
03+
δ= －110 MeV
δ= －50 MeV
03+
04+
04+
05+
δ= －250 MeV
05+
04+
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Trajectories of the broad 03+ state
Complex-Energy plane
Complex-Momentum plane
Obtained resonance parameter
Present calc.
Exp. data
Er (MeV)
1.66
2.73 + 0.3
Γ (MeV)
1.48
2.7 + 0.3
KEK原子核研究会8１-8/3
Methods for treatment of three-body resonant states
Complex Scaling Method (CSM)
It is sometimes difficult for CSM to solve state with a quite large decay
width due to the limitation of the scaling angle .
In order to search for the broad 0+ state, we employed …
 Analytic Continuation in the Coupling Constant combined with the CSM
(ACCC+CSM)

CSM U ( ) :    =  exp( i )
Branch cut
k
Im(k)
Bound state

Antibound
state
Re(k)
Resonance
ACCC+CSM
k
Im(k)
δ→0
Re(k)
Resonance

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