Transcript 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. Kurokawa
1
and K. Kato
2
Meme Media Laboratory, Hokkaido Univ., Japan 1 Div. of Phys., Grad. Sch. of Sci., Hokkaido Univ., Japan 2
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Theoretical studies of
12
C
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.
H.Horiuchi, Prog. Theor. Phys. 51(1974)1266; 53(1975)447.
α K.Fukatsu, K.Katō and H.Tanaka, Prog. Theor. Phys.81(1988)738.
α α ○ 3α+p 3./2 Closed shell N.Takigawa, A.Arima, Nucl. Phys. A168(1971)593.
N.Itagaki Ph.D thesis of Hokkaido University (1999) Y.Kanada En’yo, Phys. Rev. Lett. 24(1998)5291.
3
Γ
=34keV
Γ
=8.7eV
3 1 － 0 2 + 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.
0 1 + Excited states of cluster states?
H.Kamada and S.Oryu, Prog. Theor. Phys 76(1986)1260.
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Situation around E
x
= 10 MeV
Energy level of 12 C
l=0
0 2 + :
L=0
0 + , 2 + Alpha-condensed state A.Tohsaki et al., PRL87(2001)192501 0 + : E r =2.7+0.3 MeV, G = 2.7+0.3 MeV 2 + : E r =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 2 2 + and the 0 3 + states ? What kind of structure dose the 0 3 + state have ?
Why 0 3 + 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 8 Be, and Ground band states of 12 C
2 2
1 1
, 3
3 3
2 1 [Ref.]: M.Kamimura, Phys. Rev. A38(1988),621
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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 Im(k) : Atractive potential with < 0 k
δ→0
Re(k)
Resonance
Energy levels obtained by CSM and ACCC+CSM
KEK 原子核研究会 8 １ -8/3 (2+) G = 0.375+0.040 MeV Γ=0.12 MeV 0 + : E r =2.7+0.3 MeV, G = 2.7+0.3 MeV 2 + : E r =2.6+0.3 MeV, G = 1.0+0.3 MeV [Ref.]: M.Itoh et al., NPA 738(2004)268 0 3 + : E r =1.66 MeV, Γ=1.48 MeV 2 2 + : E r =2.28 MeV, Γ=1.1 MeV 3α Model reproduce 2 2 + and 0 3 + in the same energy region by taking into account the correct boundary condition
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Structures of 0
+
states through Amplitudes
Wave function of 0 + states Y ( 12 C ) J p=0+ =
l
=0,
L
=0 j 0,0 +
l
=2,
L
=2 j 2,2 +
l
=4,
L
=4 j 4,4
8 Be
l L
j
l
,
L
= [ 8 Be (
l
) x
L
]
l
,
L
2 : Channel Amplitudes Channel Amplitudes of 0 1 + , 0 2 + and 0 4 + 0 1 + 0 2 + 0 4 + Er -7.29
0.76
4.58
E [MeV] G 0 2.4
x 10 -3 1.1
Rr.m.s. [fm] Re.
2.36
4.29
3.26
Im.
0 0.29
0.97
Re.
0.364
0.775
0.499
0,0 2 Im.
0 0.033
0.170
Re.
0.382
0.149
0.307
2,2 2 Im.
0 -0.019
-0.017
Re.
0.254
0.076
0.194
4,4 2 Im.
0 -0.014
-0.153
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Feature of the broad 3
rd
0
+
state
Channel amplitudes as a function of d 2 2 2
8 Be
l=0
Dominated
L=0
Similar property to 0 2 + （ R r.m.s
= 4.29 fm ） Re(R r.m.s
) ( d = -140): 5.44 fm Large component of
0
,
0
2 makes such th e large width. Wave function of 0 3 + shows similar properties to 0 2 + .
0 3 + is considered as an excited state of 0 2 + . Higher nodal state of 0 2 + ?
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Summary of obtained 0
+
states
0 4 +
I=0
0 3 +
L=0
but higher nodal ?
I=0
0 2 +
L=0
r.m.s.=4.29 fm
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Structure of the 0
4 +
state
4 th 0 + state ; Large component of high angular momentum compared with 2nd 0 + 0,0 2 =0.499
, 2,2 2 =0.307, 4,4 2 =0.194
Total decay width is sharp: E r =4.58 MeV, G =1.1 MeV 3 α OCM with SU(3) base : K.Kato, H.Kazama, H.Tanaka, PTP 77(1986),185.
Component of linear-chain configuration: 56% α α α AMD: Y.Kanada-En’yo, nutl-th/0605047.
FMD: T.Neff, H.Feldmeier, NPA 738(2004), 357.
Linear chain like structure is found
Probability Density of 1
st
(Preliminary) 0
+
and 4
th
0
+
states
KEK 原子核研究会 8 １ -8/3 Probability Density of ’s r 1 12 r 2 r 1 = r 2 = r 0 1 + 0 4 + 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 J p states reproduce experimental data well. We obtained broad 3 rd 0 + similar structure to the 2 nd state near the 2 nd 0 + 2 + state. The state has state. It is thus expected to be an excited state of 2 nd 0 + . The 4 th 0 + state has large component of high angular momentum channel, [ 8 Be (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 4 th 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 8 Be(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
Probability Density of 0
+
states
KEK 原子核研究会 8 １ -8/3 0 ２ + 12 0 4 +
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Contributions from resonant states to real energy
Continuum Level Density (CLD) (
E
0 ) (
E
) = = (
E
) 0 (
E
),
i
d (
E
E i
)
Δ
(E) [Ref.] S.Shomo, NPA 539 (1992) 17.
(
E
) = 1 p
d
d
dE l
δ l : phase shift
0 (
E
) =
i
d (
E
E i
) = 1 p Im Tr
E
1
H
= 1 p Im Tr
E
1
T
Discretization with a finite number
N
of basis functions (
E
)
N
(
E
) [Ref.] A.T. Kruppa and K. Arai, PLB 431(1998)237.
=
i N
= 1 d (
E
E i
)
N
= 1
i
d (
E
E i
0 ) 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
N
(
E
) =
N
B B
d (
E
E B
) 1 p Resonance Im
N
R R E
１
E R
1 p Continuum Im
N
N B
C N
R E
１
C
( )
E
R ,
ε
c (θ) have complex eigenvalues in CSM CLD in CSM:
N
(
E
0 ) =
N
(
E
) 0
N
(
E
) = 1 p
N
R
R
(
E
G
r E r
) 2 / 2 + G
r
2 / 4 + 1 p
N
N B
c N
R
(
E
R c c I
) 2 2 +
c I
2 1 p
N
c
(
E
c
c
0
I
2 0
R
) 2 +
c
0
I
2 Smoothing technique is not needed
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Application to ３ α system
CLD of 3αsystem
α
１ 3 1
α
２ 2 (
E
) = 3
B
(
E
) 3 0
B
(
E
) = 1 p Im Tr
E
1
H
3
B
E
1
H
0 3
B
H
3
B
0
H
3
B
= =
i
3 = 1 3
i
= 1
t i t i
T G T G
+ +
i
3 = 1 3
V
N
+
Cl
i
= 1 +
OCM V
Cl
( point ) ( (
i i
) ) +
V
3 ( 1 , 2 , 3 )
Continuum Level Density: 0
+
states
8 Be(0 + ) +α 8 Be(2 + ) +α E [MeV] KEK 原子核研究会 8 １ -8/3
KEK 原子核研究会 8 １ -8/3
Subtraction of contribution from
8
Be+α
' (
E
) = 3
B
(
E
) 0 3
B
(
E
) ( 2
B
(
E
) 2 0
B
(
E
) ) = 1 p Im Tr
E
1
H
3
B
E
1 0
H
3
B
E
1
H
2
B
E
1 0
H
2
B
α
１ 8 Be 1
α
２
α
３ 1
H
2
B
=
i
3 = 1
t i
T G
+
V
N
+
Cl
+
OCM
( 1 ) +
V
8
Cl Be
( point ) ( 1 ) • α 1 α 2 : resonance + continuum • (α 1 α 2 ) α 3 : continuum 0
H
2
B
=
i
3 = 1
t i
T G
+
V
Cl
( point ) ( 1 ) +
V
8
Cl Be
( point ) ( 1 ) • α 1 α 2 : continuum • (α 1 α 2 ) α 3 : continuum
Contributions from 8 Be+α are subtracted KEK 原子核研究会 8 １ -8/3
0 2 + 0 3 + 0 4 +
Subtraction of contribution from
8
Be+α
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Search for broad 0
+
state with
δ= － 50 MeV 0 4 + 0 5 + δ= － 110 MeV 0 4 + 0 5 + δ= － 150 MeV 0 3 + 0 4 + δ= － 200 MeV 0 3 + 0 4 + 0 5 + δ= － 250 MeV
KEK 原子核研究会 8 １ -8/3 Trajectories of the broad 0 3 + state Complex-Energy plane Complex-Momentum plane Obtained resonance parameter E
Γ
r (MeV) (MeV) Present calc.
1.66
1.48
Exp. data 2.73 + 0.3
2.7 + 0.3
KEK 原子核研究会 8 １ -8/3
Methods for treatment of three-body resonant states
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 … Complex Scaling Method (CSM) Analytic Continuation in the Coupling Constant combined with the CSM (ACCC+CSM) CSM
U
( ) : = exp(
i
) Branch cut Im(k) k
Bound state Anti bound state
Re(k)
Resonance
ACCC+CSM Im(k) k
δ→0
Re(k)
Resonance