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
DESCRIPTION
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. Theoretical studies of 12 C. D.M.Brink in Proceedings of the Fifteen Solvay Conference on Physics (19070) - PowerPoint PPT PresentationTRANSCRIPT
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
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Theoretical studies of 12CTheoretical 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.
H.Horiuchi, Prog. Theor. Phys. 51(1974)1266; 53(1975)447.
K.Fukatsu, K.Katō and H.Tanaka, Prog. Theor. Phys.81(1988)738.
○3α+p3./2Closed 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.
○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.
α
01+
02+
31-
α
α
α
α α
Γ=8.7eV
Γ=34keV
Excited states of cluster states?
3
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Situation around Ex= 10 MeVSituation around Ex= 10 MeV
[Ref.] E.Uegaki et al.,PTP57(1979)1262
0+, 2+A.Tohsaki et al., PRL87(2001)192501
Alpha-condensed state
02+ :
l=0
L=0
0+ : Er=2.7+0.3 MeV, = 2.7+0.3 MeV
2+ : Er=2.6+0.3 MeV, = 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 ?Boundary condition for three-body resonances
Analysis of decay widths
Boundary condition for three-body resonances
Analysis of decay widths
Energy level of 12CEnergy level of 12C
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Our strategyOur 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)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
: OCM [Ref.]: S.Saito, PTP Supple. 62(1977),11
Phase shifts and Energies of 8Be, and Ground band states of 12C
,1
2
3
1
1c=1
1
2
3c=2
221
2
3c=3 3
3
[Ref.]: M.Kamimura, Phys. Rev. A38(1988),621
μ=0.15 fm-2
, -parity )
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Methods for treatment of three-body resonant statesMethods 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.
In order to search for the broad 0+ state, we employed … ACCC+CSM
Exp. Broad state
2θ
Resonance
Im(k)
Re(k)
δ→0
k
: Atractive potential with < 0
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Energy levels obtained by CSM and ACCC+CSMEnergy levels obtained by CSM and ACCC+CSM
E.Uegaki et al.,PTP(1979)
03+: Er=1.66 MeV, Γ=1.48 MeV
22+: Er=2.28 MeV, Γ=1.1 MeV
0+ : Er=2.7+0.3 MeV, = 2.7+0.3 MeV
2+ : Er=2.6+0.3 MeV, = 1.0+0.3 MeV
[Ref.]: M.Itoh et al., NPA 738(2004)268
(2+)
= 0.375+0.040 MeVΓ=0.12 MeV
ACCC+CSM3α Model reproduce 22+ and 03
+ in the same energy region by taking into account the correct boundary condition
3α Model reproduce 22+ and 03
+ in the same energy region by taking into account the correct boundary condition
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Structures of 0+ states through AmplitudesStructures of 0+ states through Amplitudes
12CJl=0,L=0 0,0l=2,L=2 2,2l=4,L=4
4,4
l
Ll,L
2 : Channel Amplitudes
8Be l,L= [ 8Be (l) x L ]
Wave function of 0+ statesWave function of 0+ states
E [MeV] Rr.m.s. [fm] 0,02 2,2
2 4,42
Er 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 0.30
7 -0.017 0.194 -0.153
Channel Amplitudes of 01+, 02
+ and 04
+
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Feature of the broad 3rd 0+ stateFeature of the broad 3rd 0+ state
2
2
2
l=0
L=0
8Be
Dominated
Re(Rr.m.s) (= -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+ ?
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+ ?
Similar property to 02+
( Rr.m.s= 4.29 fm )
Channel amplitudes as a function of
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Summary of obtained 0+ statesSummary of obtained 0+ states
I=0
L=0
L=0 but higher nodal ?
I=0
r.m.s.=4.29 fm
03+03+
02+02+
04+04+
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Structure of the 04+ stateStructure of the 04+ state
4th 0+ state ; Large component of high angular momentum compared with 2nd 0+
0,02 =0.499 2,2
2 =0.307, 4,42 =0.194
Total decay width is sharp: Er=4.58 MeV, =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 1st 0+ and 4th 0+ states (Preliminary)
04+04+
r1 r2 r1 = r2 = r
01+01+
r [fm
]
Probability Density of ’s
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Summary and Future workSummary 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 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
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0 2+
0 2+
Probability Density of 0+ states
04+04+
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Contributions from resonant states to real energy
Contributions from resonant states to real energy
Continuum Level Density (CLD) Δ(E) [Ref.] S.Shomo, NPA 539 (1992) 17.
dE
dEEEE l
1
)(),()()0( 0
N
ii
N
ii
N
EEEE
EE
1
0
1
)()(
)()(
HE
EEEi
i
1TrIm
1
)()(
TE
EEEi
i
1TrIm
1
)()(0
Discretization with a finite number N of basis functions
Smoothing technique is needed,
but results depend on smoothing parameter.
Smoothing technique is needed,
but results depend on smoothing parameter.
[Ref.] A.T. Kruppa and K. Arai, PLB 431(1998)237.
δl: phase shift
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CLD in the Complex Scaling Method[Ref.] R. Suzuki, T. Myo, and K. Kato, PTP 113 (2005) 1273CLD in the Complex Scaling Method[Ref.] R. Suzuki, T. Myo, and K. Kato, PTP 113 (2005) 1273
ER, εc(θ) have complex eigenvalues in CSM
RB
R
NNN
c
N
cI
cR
c
Ic
Ic
Rc
Ic
N
R rr
r
NNN
EE
EE
EEE
2020
20
22
2
22
0
)(
1
)(
1
4/)(
2/1
)()()0(
Smoothing technique is not needed
CLD in CSM:
R RBB N
R
NNN
C CR
N
BB
N
EEEEEE
)(Im
1Im
1)()(
11
Bound state Resonance Continuum
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Application to 3 α systemApplication to 3 α system
),,()( 3213
3
1
3
13 VVTtH i
i
OCMClNG
iiB
)(3
1
)point(3
1
03 i
i
ClG
iiB VTtH
α 1
α 21
2
3
033
033
11TrIm
1
)()()(
BB
BB
HEHE
EEE
CLD of 3αsystem
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Continuum Level Density: 0+ statesContinuum Level Density: 0+ states
8Be(0+) +α8Be(0+) +α
8Be(2+) +α8Be(2+) +α
E [MeV]
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Subtraction of contribution from 8Be+αSubtraction of contribution from 8Be+α
022
033
022
033
'
1111TrIm
1
)()()()()(
BBBB
BBBB
HEHEHEHE
EEEEE
)()( 1)point(
81
3
12
ClBe
OCMClNG
iiB VVTtH
)()( 1)point(
81)point(
3
1
02
ClBe
ClG
iiB VVTtH
α 1
α 2
α 3
1
1
8Be• α1- α2: resonance + continuum
• (α1α2)- α3: continuum
• α1- α2: continuum
• (α1α2)- α3: continuum
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Contributions from 8Be+α are subtractedContributions from 8Be+α are subtracted
02+
04+
03+
‘
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Subtraction of contribution from 8Be+αSubtraction of contribution from 8Be+α
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Search for broad 0+ state withSearch for broad 0+ state with
δ= - 50 MeV δ= - 110 MeV δ= - 150 MeV
δ= - 200 MeV δ= - 250 MeV
04+ 04
+
03+
04+
04+
03+
05+
05+
05+
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Trajectories of the broad 03+ stateTrajectories of the broad 03+ state
Obtained resonance parameter
Present calc. Exp. data
Er (MeV) 1.66 2.73 + 0.3
Γ (MeV) 1.48 2.7 + 0.3
Complex-Energy plane Complex-Momentum plane
KEK 原子核研究会 8 1 -8/3
Methods for treatment of three-body resonant statesMethods 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)
Resonance
ACCC+CSMACCC+CSMCSMCSM
Im(k)
Re(k)
Im(k)
Re(k)
δ→0Bound state
Anti-bound state
Branch cut
:)(U )exp( i
k k
Resonance