structure of resonance and continuum states hokkaido university unbound nuclei workshop pisa, nov....
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Structure of Resonance and
Continuum States
Hokkaido University
Unbound Nuclei Workshop
Pisa, Nov. 3-5, 2008
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1. Resolution of Identity in Complex Scaling Method
Continuum st.
Spectrum of
Hamiltonian
Bound st.
Resonant st.
|~|~|1 1kkRnn
bn
dkuu
rr
e
rr
e
RR.G. Newton, J. Math. Phys. 1 (1960), 319
Completeness Relation (Resolution of Identity)
Non-Resonant st.
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Among the continuum states, resonant states are considered as an extension of bound states because they result from correlations and interactions.
From this point of view, Berggren said
“In the present paper,*) we investigate the properties**) of resonant states and find them in many ways quite analogous to those of the ordinary bound states.”
*) NPA 109 (1968), 265. **) orthogonality and completeness
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Separation of resonant states from continuum states
|~|~|~|1 1)(
kkL
LN
rnrrnn
bn
dkuuuur
Deformation of the contour
Resonant states
ˆ~ lim ˆ~2
*1
021
2
uOuedruOu r
R
Ya.B. Zel’dovich, Sov. Phys. JETP 12, 542 (1961).
N. Hokkyo, Prog. Theor. Phys. 33, 1116 (1965).
Convergence Factor Method
Matrix elements of resonant states
T. Berggren, Nucl. Phys. A 109, 265 (1968)
Deformed continuum states
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Complex scaling method
irer
ikek
coordinate:
momentum:
r
)()(ˆ)(~ )(
ˆ~ lim ˆ~
2*
1
2*
10
21
2
ii
R
i
r
R
reuOreured
uOuedruOu
|~||~||~|1 1
kkLnn
N
rnnn
bnk
r
dkuuuu
B. Gyarmati and T. Vertse, Nucl. Phys. A160, 523 (1971).
reiθ
T. Myo, A. Ohnishi and K. Kato. Prog. Theor. Phys. 99(1998)801]
|~|~|~|1 1
kkL
N
rnnnnn
bnk
r
dkuuuu
inclination of the semi-circle
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k k E E
Single Channel system
b1b2b3 r1r2 r3
Coupled Channel system Three-body system
E| E|
B.Giraud, K.Kato and A. Ohnishi, J. of Phys. A37 (2004),11575
B.Giraud and K.Kato, Ann.of Phys. 308 (2003), 115.
Resolution of Identity in Complex Scaling Method
0 0
0
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(Complex scaled)
Structures of three-body continuum states
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0+1 -
Physical Importance
of Resonant States
M. Homma, T. Myo and K. Kato, Prog. Theor. Phys. 97 (1997), 561.
red: 0+
blue: 1-
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B.S.
R.S.
Contributions from B.S. and R.S. to the Sum rule value
Sexc=1.5e2fm2MeV
• Kiyoshi Kato
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(A) Cluster Orbital Shell Model (COSM)
• Y. Suzuki and K. Ikeda, Phys. Rev. C38 (1988), 410
Core+Xn system
The total Hamiltonian:
X
jiji
cij
X
iii
X
iC mA
vUm
HH ppp)1(
1
2
1
1
2
1
where
HC : the Hamiltonian of the core cluster AC
Ui : the interaction between the core and
the valence neutron (Folding pot.)
vij : the interaction between the valence
neutrons (Minnesota force, Av8, …)
1
2
i
X X
2. Complex Scaled COSM
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which has a peak in a region :
The two-neutron distance :
(B) Extended Cluster Model ー T-type coordinate system ー
),(cos)][:2( 20 Pn Ji
1
The di-neutron like correlation between valence neutrons moving in the spatially wide region
Y. Tosaka, Y Suzuki and K. Ikeda; Prog. Theor. Phys. 83 (1990), 1140.K. Ikeda; Nucl. Phys. A538 (1992), 355c.
When R~5-7fm, to describe the short range correlation accurately up to 0.5 fm, the maximum -value is 10~14.
1
Rd
θ
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(C) Hybrid-TV Model
S. Aoyama, S. Mukai, K. Kato and K. Ikeda, Prog. Theor. Phys. 94, 343-352 (1995)
+
(p1/2)2(p3/2)2
2)14(
Rapid convergence!!(p,sd)+T-base
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Two-neutron density distribution of 6HeH
ybrid
-TV
model (C
OSM
9ch
+ E
CM
1ch
)H
arm
onic
osc
illato
r (0
p3/2 o
nly
)
S=0
S=1
Total
(0p3/2) 2 Hybrid-TV
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H.Masui, K. Kato and K.Ikeda, PRC75 (2007), 034316.
18O
6He
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Excitation of two-neutron halo nuclei (Borromean nuclei)
Soft-dipole mode
Structure of three-body continuum
Three-body resonant states
Complex scaling method
Resonant state Bound state
(divergent) (no-divergent)
S. Aoyama, T. Myo, K, Kato and K. Ikeda; Prog. Theor. Phys. 116, (2006) 1.
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Y. Aoyama ; Phys. Rev. C68 (2003) 034313.
1- ( Soft Dipole Resonance) pole in 4He+n+n (CSM+ACCC)
1- resonant state??It is difficult to observe as an isolated resonant state!!
Er~3 MeV
Γ~32 MeV
)()()1( rVrV
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7He: 4He+n+n+n COSM
T. Myo, K. Kato and K. Ikeda, PRC76 (2007), 054309
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Coulomb breakup reaction
3. Coulomb breakup reactions of Borromean systems
Structures of three-body continuum state
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Strength Functions of Coulomb Breakup Reaction
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|~|"|~|'|~|~|~|1 ""1
''11
"'
kkLkkLkkL
N
rnnnnn
bnkkk
r
dkdkdkuuuu
9Li+n+n 10Li(1+)+n 10Li(2+)+n Resonances
T. Myo, A. Ohnishi and K. Kato, Prog. Theor. Phys. 99 (1998), 801.
in CSMin CSM
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T. Myo, K. Kato, S. Aoyama and K. Ikeda, PRC63(2001), 054313
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coupled channel [9Li+n+n]
+
[9Li*+n+n]
PRL 96, 252502 (2006)
T. Myo
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)()( iEEE
A.T.Kruppa, Phys. Lett. B 431 (1998), 237-241
A.T. Kruppa and K. Arai, Phys. Rev. A59 (1999), 2556
K. Arai and A.T. Kruppa, Phys. Rev. C 60 (1999) 064315
Definition of LD:
iHETrE
1Im
1)(
iii EH
4. Unified Description of Bound and Unbound States
Continuum Level Density
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B
B
B
R RB
N
n
N
nL C
CRn
Bn EE
dEEEEE
iHETrE
111Im
1
1Im
1)(
1
Resonance:
Rotated Continuum:2
R
RR
nRn
Rn iE
IRC iE
LIR
ICN
n nRn
nN
n
Bn E
dEE
EER
R RR
RB
B
B 2222 )(
1
4/)(
2/1)(
Descretization
RI in complex
scaling
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2θ
2θ
E E
εI εI
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Continuum Level Density: )()()( 0 EEE
)()(Im1
11Im
1)(
0
0
EGEGTr
iHEiHETrE
Basis function method: n
N
nnc
1
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Phase shift calculation in the complex scaled basis function method
)()(
2
1)( ES
dE
dESTr
iE
In a single channel case, )}(2exp{)( EiES
dE
EdE
)(1)(
)'(')(0
EdEEE
S.Shlomo, Nucl. Phys. A539 (1992), 17.
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Phase shift of 8Be=+calculated with discretized app.
Base+CSM: 30 Gaussian basis and =20 deg.
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Description of unbound states in the Complex Scaling Method
EVH )( 0
H0=T+VC V ; Short Range Interaction
000 EH
Solutions of Lippmann-Schwinger Equation
00
1
V
iHE
( Ψ0; regular at origin )
Outgoing waves
Complex Scaling
00)(
)(
1
V
HE
A. Kruppa, R. Suzuki and K. Kato, phys. Rev.C75 (2007), 044602
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T-matrix
Tl(k) =
Second term is approximated as
where
Tl(k)
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●Lines : Runge-Kutta method●Circles : CSM+Base
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Complex-scaled Lippmann-Schwinger Eq.
Direct breakup
Final state interaction (FSI)
• CSLM solution
• B(E1) Strength
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Dalitz distribution of 6He
• Decay process– Di-neutron-like decay is not seen clearly.
coskkmA
EEEc
22
2
21
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6 . Summary and conclusion• It is shown that resonant states play an important role in
the continuum phenomena.
• The resolution of identity in the complex scaling method is presented to treat the three-body resonant states in the same way as bound states.
• The complex scaling method is shown to describe not only resonant states but also non-resonant continuum states on the rotated branch cuts.
• We presented several applications of the extended resolution of identity in the complex scaling method; sum rule, break-up strength function and continuum level density.
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Collaboration:
S. Aoyama(Niigata Univ.), H. Masui(Kitami I. T.),
T. Myo (Osaka Tech. Univ.), R. Suzuki(Hokkaido Univ.),
C. Kurokawa(Juntendo Univ.), K. Ikeda(RIKEN)
Y. Kikuchi(Hokkaido Univ.)