Breakup effects on
6Li elastic scattering
11/Mar./2013YIPQS International Molecule on
Coexistence of weak and strong binding in unstable nuclei and its dynamicsYITP, Kyoto University
1S. Watanabe, 1T. Matsumoto, 1K. Minomo,2K. Ogata, and 1M. Yahiro
1Kyushu University, 2RCNP, Osaka University
Ⅰ. Introduction
Ⅱ. Formulation
Ⅲ. Results and Discussion
Ⅳ. Summary and Future work
Table of Contents
Background, Previous studies and Purpose
CDCC and Model Hamiltonian
Elastic cross sections for 6Li scattering
Ⅰ. Introduction
Ⅱ. Formulation
Ⅲ. Results and Discussion
Ⅳ. Summary and Future work
Table of Contents
Background, Previous studies and Purpose
CDCC and Model Hamiltonian
Elastic cross sections for 6Li scattering
Background: Why 6Li ?
t production reaction
fusion reactiond + t → 4He + n + 17MeV
n + 6Li → t + 4He
use as fuel
6Li: Key nucleus in fusion reactors
• Significance of energy alternative to nuclear power plant
• Difficulties of correcting t
• n has no charge
Necessity of precise theoretical prediction
Determination of the reaction rate is difi f cult.
We can utilize t produced by Li-isotopes.
d + t fusion reaction is realistic
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6Li & Breakup reaction
4-body breakup reaction
our interest2/22
Halo
n nweakly bound
6Li
6Li breaks up into 3 constituents (α, n, p).
Breakup reaction & CDCCWhat is BreakupTransition from bound state
to continuum states
g.s.
ε
-3.7 MeV
α, n, p-B.U.threshold
∞
0 MeV Importance of multistep transition
energy spectrum of 6Li
6Li goes back to and from these states.
Even for elastic scattering,breakup effect is significant.
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CDCC : Continuum Discretized Coupled Channels Fully quantum-mechanical method to treat B.U. Success of application for different types of B.U. reactions
Previous studies on 6He
T. Matsumoto et al., PRC 73 (2006), 051602(R).
3-body CDCC does not reproduce.
4-body CDCC solved this problem.
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N. Keeley et al., PRC 68 (2003), 054601.
underestimation
3-body CDCC4-body CDCC
experimental dataE. F. Aguilera et al., Phys. Rev. Lett. 84, 5058 (2000).E. F. Aguilera et al., Phys. Rev. C 63, 061603 (2001).
6He + 209Bi elastic scattering was analyzed with both 3-body and 4-body CDCC.
3-body CDCC cannot reproduce the experimental data.
6Li scattering should be treated with 4-body CDCC.
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N. Keeley et al., PRC 68 (2003), 054601.
Experimental dataE. F. Aguilera et al., Phys. Rev. Lett. 84, 5058 (2000).E. F. Aguilera et al., Phys. Rev. C 63, 061603 (2001).
3-body CDCC
underestimation
3-bodyCDCC
6Li + 209Bi elastic scattering was analyzed with 3-body CDCC.
Previous study on 6Li
Purpose 6/22
3-body CDCC 4-body CDCC
We find out why 3-body CDCC does not work well.
Application of 4-body CDCC to 6Li + 209Bi scattering
We treat d-breakup explicitly.
Ⅰ. Introduction
Ⅱ. Formulation
Ⅲ. Results and Discussion
Ⅳ. Summary and Future work
Table of Contents
Background, Previous study and Purpose
CDCC and Model Hamiltonian
Elastic cross sections for 6Li scattering
p
α
209BiUn
Uα
Up
Rn
6Li
CDCC describes 6Li breakup processesas a transition to continuum states.
(𝐻 4b−𝐸 )Ψ (𝑹 ,𝝃 )=0
𝐻4 b=𝐾𝑅+𝑈𝑛+𝑈𝑝+𝑈𝛼+𝑒2𝑍Li𝑍 Bi
𝑅 +h𝜉
(h𝜉−𝜀 )𝜙𝜀 (𝝃 )=0 : 6Li internal w. f.
4-body Hamiltonian 4-body Schrödinger eq.
7/22
CDCC wave function
CDCC wave functionΨ (𝑹 ,𝝃 )=𝜙0(𝝃 )𝜒 0(𝑹)+∫
0
∞
𝜙𝜀(𝝃 )𝜒 𝜀(𝑹)𝑑𝜀
truncated discretized
g.s.
ε𝜀max
ε
𝒊=𝟎𝒊=𝟏
𝒊=𝒊𝐦𝐚𝐱・・
・
ε
-3.7 MeV
bound state continuum state
∞
(𝐻 4b−𝐸 )Ψ (𝑹 ,𝝃 )=0
𝐻4 b=𝐾𝑅+𝑈𝑛+𝑈𝑝+𝑈𝛼+𝑒2𝑍Li𝑍 Bi
𝑅 +h𝜉
(h𝜉−𝜀 )𝜙𝜀 (𝝃 )=0
p
α
209BiUn
Uα
Up
Rn
6Li
ΨCDCC=∑𝑖=0
𝑖max�̂�𝑖(𝝃 )𝜒 𝑖(𝑹)
4-body Schrödinger eq.
8/22
: 6Li internal w. f.
: 6Li internal w. f.(h𝜉−𝜀 )𝜙𝜀 (𝝃 )=0
(𝐻 4b−𝐸 )Ψ (𝑹 ,𝝃 )=0
𝐻4 b=𝐾𝑅+𝑈𝑛+𝑈𝑝+𝑈𝛼+𝑒2𝑍Li𝑍 Bi
𝑅 +h𝜉
Coupled-Channels equation
substituting
Coupled-Channel equation for
[− ℏ22𝜇
𝑑2
𝑑𝑅2 +ℏ22𝜇
𝐿 (𝐿+1 )𝑅2 +𝑈𝛾𝛾 (𝑅 )+
𝑒2𝑍Li𝑍 Bi
𝑅 − (𝐸−𝜀𝛾 )] 𝜒𝛾 (𝑅 )=−∑𝛾 ′≠ 𝛾
𝑈𝛾𝛾 ′ (𝑅 ) �̂�𝛾 ′ (𝑅)
�̂�𝛾(𝑅)→¿
𝑈𝛾𝛾 ′ (𝑅 )= ⟨𝒴𝛾|𝑈𝑛+𝑈𝑝+𝑈𝛼|𝒴𝛾′ ⟩𝝃 , �̂�Boundary condition
Coupling potential
p
α
209BiUn
Uα
Up
Rn
6Li
CDCC wave functionΨ (𝑹 ,𝝃 )=𝜙0(𝝃 )𝜒 0(𝑹)+∫
0
∞
𝜙𝜀(𝝃 )𝜒 𝜀(𝑹)𝑑𝜀
4-body Schrödinger eq.
≡∑𝒴𝑖(𝝃 , �̂�)𝜒 𝑖(𝑅)
ΨCDCC=∑𝑖=0
𝑖max�̂�𝑖(𝝃 )𝜒 𝑖(𝑹)
9/22
bound state continuum state
Model Hamiltonian
A. J. Koning et al., NPA 713 (2003), 231-310.
A. R. Barnett et al., PRC 9 (1974), 2010.
Optical potential Optical potential
(𝐻 4b−𝐸 )Ψ (𝑹 ,𝝃 )=0
𝐻4 b=𝐾𝑅+𝑈𝑛+𝑈𝑝+𝑈𝛼+𝑒2𝑍Li𝑍 Bi
𝑅 +h𝜉
p
α
209BiUn
Uα
Upn
6Li
4-body Schrödinger eq. for the scattering of 6Li at 5MeV/nucleon
10/22
5 MeV
20 MeV(5 MeV/nucleon)
Model Hamiltonian
A. J. Koning et al., NPA 713 (2003), 231-310.
A. R. Barnett et al., PRC 9 (1974), 2010.
Optical potential Optical potential
(𝐻 4b−𝐸 )Ψ (𝑹 ,𝝃 )=0
𝐻4 b=𝐾𝑅+𝑈𝑛+𝑈𝑝+𝑈𝛼+𝑒2𝑍Li𝑍 Bi
𝑅 +h𝜉
p
α
209BiUn
Uα
Upn
6Li
Un Uα
4-body Schrödinger eq.
10/22
Internal Hamiltonian hξ
H. Kanada et al., Theor. Phys. 61, 1327 (1979).
h𝜉=𝑇𝒓 𝑐+𝑇 𝒚 𝑐
+𝑉 𝑛𝛼+𝑉 𝑝𝛼+𝑉 𝑛𝑝+𝑉 OCM
𝑉 OCM= lim𝜆→∞
𝜆∑|𝜙FS ⟩ ⟨𝜙FS|
(h𝜉−𝜀 )𝜙𝜀 (𝝃 )=0p
α
n
6Li
Vnα Vpα
R. Machleidt,Adv. Nucl. Phys. 19, 189 (1989).
Forbidden State
: 6Li internal w.f. Bonn-A interaction
KKNN interaction
Internal Hamiltonian of 6Li
Exp: B. Hoop et al., Nucl. Phys. 83, 65 (1966). Exp: P. Schwandt et al., Nucl. Phys. A 163, 432 (1972).
11/22
Gaussian Expansion Method
¿�̂�𝑖 (𝝃 )=∑
𝑛=0
𝑁
𝐶𝑛(𝑖 )𝜑𝑛 (𝝃)
𝒊=𝟎𝒊=𝟏
𝒊=𝑵・・
・
ε
Internal Hamiltonian hξ
h𝜉=𝑇𝒓 𝑐+𝑇 𝒚 𝑐
+𝑉 𝑛𝛼+𝑉 𝑝𝛼+𝑉 𝑛𝑝+𝑉 OCM 𝑉 OCM= lim𝜆→∞
𝜆∑|𝜙FS ⟩ ⟨𝜙FS| (h𝜉−𝜀 )𝜙𝜀 (𝝃 )=0
Gaussian Expansion Method (GEM)
The are obtained with the GEM.
Gaussian basis
• Bound state• Discretized continuum states (Pseudo states)
12/22
: 6Li internal w.f.
Results for 6Li
I π ε0 [MeV] Rrms [fm]
Calc. 1+ -3.68 2.34Exp. 1+ -3.6989 2.44±0.07
1+ 2+ 3+
Comparison between theory and experimental data for 6Li g.s.
g.s.
D. R. Tilley et al., Nucl. Phys. A 708, 3 (2002).A. V. Dobrovolsky et al., Nucl. Phys. A 766, 1 (2006).Exp.
13/22
eigenenergies of 6Li
We have no adjustable parameterfrom now on.
Introduction of effective three-body force
Ⅰ. Introduction
Ⅱ. Formulation
Ⅲ. Results and Discussion
Ⅳ. Summary and Future work
Table of Contents
Background, Previous study and Purpose
CDCC and Model Hamiltonian
Elastic cross sections for 6Li scattering
4-body CDCC
3-body CDCC
We analyzed 6Li + 209Bi scattering with 4-body CDCC.
Results
Experimental dataE. F. Aguilera et al., Phys. Rev. Lett. 84, 5058 (2000).E. F. Aguilera et al., Phys. Rev. C 63, 061603 (2001).
3-body CDCC
To begin with, how does 3-body CDCC treat d-breakup?
209Bi
3-body CDCC cannot reproduce the data.4-body CDCC reproduces.
14/22
Ud = Ud : d-optical potentialOP
209Bi
Ud
Uαdetermined from d + 209Bi scattering data
𝐻3b=𝐾𝑅+𝑈𝑑+𝑈𝛼+𝑒2𝑍 Li𝑍Bi
𝑅 +h𝜉′
experimental data & potentialA. Budzanowski et al., Nucl. Phys. 49, 144 (1963).
3-body CDCC
Ud SF
(without d*)
Strong interference ⇒ Importance of d*
(with d* ) Ud OP
How to treat d-breakup 3-body Schrödinger eq.
(𝐻 3b−𝐸 )Ψ (𝑹 , 𝝃)=0 With d*
d
Ud = Ud : single folding potentialSF
obtained by folding Un and Up with the deuteron g. s. w. f.Ud = 〈 φd |Un + Up|φd 〉SF
(gs) (gs)
d
inert
d*
6Li
15/22
d-breakup effects on d + 209Bi scattering
Ud SF Ud OP
d-breakup is significant for d + 209Bi scattering
Ud : d-optical potential(with d-breakup)
OP
Ud : Single folding potential(without d-breakup)
SF
Definition of Ud
experimental dataA. Budzanowski et al., Nuclear Physics 49, 144 (1963).
d-209Bi scattering
We can check d-breakup effects directly with 3-body CDCC.
3-body CDCC with d-B.U.
(3-body CDCCwithout d-B.U.)
16/22
Ud = Ud : d-optical potentialOP
Ud = Ud : single folding potentialSF
209BiUd
Uα
𝐻4 b=𝐾𝑅+𝑈𝑛+𝑈𝑝+𝑈𝛼+𝑒2𝑍Li𝑍 Bi
𝑅 +h𝜉
𝐻3b=𝐾𝑅+𝑈𝑑+𝑈𝛼+𝑒2𝑍 Li𝑍Bi
𝑅 +h𝜉′?
We reconsider 3-body CDCC.
experimental data & potentialA. Budzanowski et al., Nucl. Phys. 49, 144 (1963).
Ud = 〈 φd |Un + Up|φd 〉SF(gs) (gs)
3-body CDCC
d
d
inert
6Li
(with d*) Ud OP
Ud SF
(without d*) inert
Why does not 3-body CDCC work?
(with d-breakup)
(without d-breakup)
17/22
Ud : d-optical potential(with d-breakup)
OP
Ud : single folding potential(without d-breakup)
SF
d hardly breaks up in 6Li + 209Bi scattering
3-body CDCC analysis
negligible
Definition of Ud
experimental dataE. F. Aguilera et al., Phys. Rev. Lett. 84, 5058 (2000).E. F. Aguilera et al., Phys. Rev. C 63, 061603 (2001).
Ud SFUdOP
dominant
Ud has d-breakup
effect implicitly.OP
Deuteron breakup in 6Li
3-body (Ud ) reproduces experimental dataSF
(with d-B.U.) (without d-B.U.)
18/22
Ud and Ud
Real part Imaginary part
SFOP
Direct comparison between Ud and UdSFOP
Ud is much more absorptive as a result of d-breakup effect.
OP
19/22
Convergence
7 MeV
15 MeV
20 MeV1+ 2+ 3+
20/22
Goodconvergence
energy spectrum of 6Liconvergence with respect to increasing εmax
Ⅰ. Introduction
Ⅱ. Formulation
Ⅲ. Results and Discussion
Ⅳ. Summary and Future work
Table of Contents
Background, Previous study and Purpose
CDCC and Model Hamiltonian
Elastic cross sections for 6Li scattering
We will investigate whether d-breakup in 6Li scattering is negligible also for other targets or other incident
energies.
We have applied 4-body CDCC to 6Li + 209Bi scattering.
We have investigated d-breakup effect on 6Li- and d-scattering.
Future work
We should use the single folding potential as Ud for 6Li scattering.
4-body CDCC reproduces experimental data with no free parameter.
Summary and future work
In the 6Li + 209Bi scattering, d-breakup is negligible. ⇔ d-breakup is significant for d + 209Bi scattering.
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nn
halo Application to unstable nuclei very near to drip line• 「 Core + n + n 」 ⇒ Analysis similar to 6Li
Future work for unstable nuclei
Island of Inversion
We will apply 4-body CDCC to neutron rich nuclei, and figure out the reaction mechanisms.
I am researching the properties of neutron-rich unstable nuclei.
Nuclei very near drip line
22/22