Takuma Matsumoto(Kyushu Univ.)

K. Minomo, K. Ogataa, M. Yahiro, and K. Katob

(Kyushu Univ, aRCNP, bHokkaido Univ)

Description for Breakup Reactions of Three-body Projectiles

1

2

4

3

1

2

3

Introduction

The Continuum-Discretized Coupled-Channels method (CDCC)Developed by Kyushu group about 20 years ago

M. Kamimura, M.Yahiro, Y. Iseri, Y. Sakuragi, H. Kameyama, and M. Kawai, Prog. Theor. Phys. Suppl. 89, 1 (1986)

Three-body breakup reactions

Four-body breakup reactions

Four-body CDCC method

6He

Target

4He

n

n

The CDCC equation of four-body systems is the same as that of three-body systems.

6He projectile : n + n + 4He (three-body model) S2n ~ 1 MeV

Three-body bound and discretized continuum states

Four-Body Breakup Reactions

Vnn: D. Gogny, et al., PLB32, 591 (1970), Vna: KKNN interaction

Channel 1 Channel 2 Channel 3

n n nn

nn

4He 4He 4He

Gaussian Expansion Method : E. Hiyama et al., Prog. Part. Nucl. Phys. 51, 223 An accurate method of solving few-body problems. A variational method with Gaussian basis functions Take all the sets of Jacobi coordinates

I=0+ I=1- I=2+

Exc

itat

ion

en

ergy

of

6 He

[MeV

]

Ground and breakup states of 6He

6He+12C scattering at 18 MeV 6He+209Bi scattering at 22.5 MeV

Nuclear Breakup

Nuclear & Coulomb Breakup

For elastic scattering, CDCC well reproduces the experimental data.

T.M. Hiyama, Ogata, Iseri, Kamimura, Chiba, and Yahiro, Phys. Rev. C70, 061601 (2004).

T.M. Egami, Ogata, Iseri, Kamimura, and Yahiro,Phys. Rev. C73, 051602 (2006).

Elastic Cross Section

In CDCC breakup cross sections are discrete.

E* (MeV)

(m

b)

6He+12C scattering at 240 MeV/nucl.

How to calculate the continuum breakup cross section

EXP. PRC59, 1252(1999), T. Aumann et al.

CDCC EXP

Breakup Cross Section

Continuous breakup T-matrix element

Smoothing factor :

Discrete T-matrix element

Smoothing factor for 3-body system

Smoothing factor for 4-body system

Smoothing factor :

Three-body continuum wave function Difficult to solveNot good convergency

M.Rodriguez-Gallardo, J. M. Arias, J. Gomez-Camacho,A. M. Moro, I. J. Thompson, and J. A. Tostevin,PRC80, 051601(R) (2009).

T. Egami, T.M., K. Ogata, M.Yahiro, PTP121, 780(2009) T.M., T. Egami, K. Ogata, M.Yahiro, PTP121, 885(2009)

New description of continuum breakup cross section with Complex-scaling method (CSM).

T.M., K. Kato, and M. Yahiro, PRC82, 051602 (2010).

Complex-scaling operator:

Coordinate:

Momentum:

k Im[k]

Re[k]

}Bound states

Resonance

ContinuumUseful for searching many-body resonances

Green’s function with Complex-Scaling Method (CDCS Green’s function)

S. Aoyama, T. Myo, K. Kato, and K. Ikeda,Prog. Theor. Phys. 116, 1 (2006)

Complex-Scaling Method

New Smoothing Procedure with CSM

Final state of the projectileResponse function

T-matrix calculated by CDCC

Green’s function with Complex-Scaling Method (CDCS Green’s function)

T.M., K. Kato, and M. Yahiro, PRC82, 051602 (2010).

New description of differential breakup cross section

Differential Breakup Cross Section

ConvergenceSystem : 6He + 12C scattering @ 40 MeV/A

1. Convergence of T-matrix elements calculated by CDCC

2. Convergence of Green’s function in calculating continuum cross sections.

The number of bases Gaussian range max

Set I 10 10 fm

Set II 15 20 fm

Set III 20 50 fm

We should confirm the convergence with extending the model space

Convergence of T-matrix (2+)

2+ (set II)2+ (set I)

The T-matrix calculated with set I gives good convergence

Convergence of Green’s Function

1-

0+

2+

Dashed : set ISolid : set IIMark : set III

Dashed : set ISolid : set IIMark : set III

The result with set II gives good convergence for Green’s function

6He + 12C and 208Pb scattering at 240 MeV/A T. Aumann et al, PRC59, 1252(1999).

Microscopic optical potential (Double folding model with Melbourne g-matrix)

n - 12C and 4He -12C potentials n – 208Pb and 4He – 208Pb potentials

VnA

VcA

VnAn

n

4He

12C, 208Pb

6He+12C scattering @ 240 MeV/nucl.

Exp. data from PRC59, 1252 (1999), T. Aumann et al.

Underestimation → Inelastic breakup effect ~ 20%

Nuclear Breakup is dominant

Breakup to 3- continuum is negligible

6He+208Pb scattering @ 240 MeV/nucl.

Exp. data from PRC59, 1252 (1999), T. Aumann et al.

Underestimation → Inelastic breakup effect

Overestimation ???

Coulomb Breakup is dominant

In order to obtain continuous breakup cross sections for four-body breakup, we propose a new smoothing method with the complex scaling method.

The convergence of breakup cross sections is confirmed with extending the model space.

The new smoothing method is applied to analyses for 6He breakup reactions on 12C and 208Pb at 240 MeV/A.

In a future work, we will analyse a four-body breakup reaction of 6He, 11Li, 14Be with the new smoothing method.

Summary