Breakup effects on 6 Li elastic scattering

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.

5/22

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.

21/22

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