Description for Breakup Reactions of Three-body Projectiles
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Description for Breakup Reactions of Three-body Projectiles. Takuma Matsumoto (Kyushu Univ.) K. Minomo, K. Ogata a , M. Yahiro, and K. Kato b (Kyushu Univ, a RCNP, b Hokkaido Univ). Introduction. The Continuum-Discretized Coupled-Channels method (CDCC)

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Takuma matsumoto kyushu univ k minomo k ogata a m yahiro and k kato b

Description for Breakup Reactions of Three-body Projectiles

Takuma Matsumoto

(Kyushu Univ.)

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

(Kyushu Univ, aRCNP, bHokkaido Univ)


Takuma matsumoto kyushu univ k minomo k ogata a m yahiro and k kato b

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

1

2

3

1

2

3

4


Takuma matsumoto kyushu univ k minomo k ogata a m yahiro and k kato b

Four-Body Breakup Reactions

  • 6He projectile : n + n + 4He (three-body model)

  • S2n ~ 1 MeV

n

n

6He

Three-body bound and discretized continuum states

4He

Target

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


Takuma matsumoto kyushu univ k minomo k ogata a m yahiro and k kato b

Ip=0+

Ip=1-

Ip=2+

n

n

n

n

n

n

Excitation energy of 6He [MeV]

4He

4He

4He

Channel 1

Channel 2

Channel 3

Ground and breakup states of 6He

  • 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

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


Takuma matsumoto kyushu univ k minomo k ogata a m yahiro and k kato b

Elastic Cross Section

6He+12C scattering at 18 MeV

6He+209Bi scattering at 22.5 MeV

Nuclear & Coulomb

Breakup

Nuclear Breakup

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).

For elastic scattering, CDCC well reproduces the experimental data.


Takuma matsumoto kyushu univ k minomo k ogata a m yahiro and k kato b

Breakup Cross Section

In CDCC breakup cross sections are discrete.

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

CDCC

EXP

s (mb)

E* (MeV)

How to calculate the continuum breakup cross section

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


Takuma matsumoto kyushu univ k minomo k ogata a m yahiro and k kato b

Smoothing factor for 3-body system

Continuous breakup T-matrix element

Discrete T-matrix element

Smoothing factor :


Takuma matsumoto kyushu univ k minomo k ogata a m yahiro and k kato b

Smoothing factor for 4-body system

Smoothing factor :

  • Three-body continuum wave function

    • Difficult to solve

    • Not 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).


Takuma matsumoto kyushu univ k minomo k ogata a m yahiro and k kato b

Complex-Scaling Method

S. Aoyama, T. Myo, K. Kato, and K. Ikeda,

Prog. Theor. Phys. 116, 1 (2006)

Im[k]

k

Complex-scaling operator:

Bound states

}

Coordinate:

Momentum:

Re[k]

Useful for searching many-body resonances

Continuum

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

Resonance


Takuma matsumoto kyushu univ k minomo k ogata a m yahiro and k kato b

New Smoothing Procedure with CSM

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

Response function

Final state of the projectile

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

T-matrix calculated by CDCC


Takuma matsumoto kyushu univ k minomo k ogata a m yahiro and k kato b

Differential Breakup Cross Section

New description of differential breakup cross section


Takuma matsumoto kyushu univ k minomo k ogata a m yahiro and k kato b

Convergence

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

  • Convergence of T-matrix elements calculated by CDCC

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

We should confirm the convergence with extending the model space


Takuma matsumoto kyushu univ k minomo k ogata a m yahiro and k kato b

Convergence of T-matrix (2+)

2+ (set I)

2+ (set II)

The T-matrix calculated with set I gives good convergence


Takuma matsumoto kyushu univ k minomo k ogata a m yahiro and k kato b

Convergence of Green’s Function

Dashed : set I

Solid : set II

Mark : set III

Dashed : set I

Solid : set II

Mark : set III

0+

1-

2+

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


Takuma matsumoto kyushu univ k minomo k ogata a m yahiro and k kato b

  • 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

n

n

VnA

VnA

VcA

4He

12C, 208Pb


Takuma matsumoto kyushu univ k minomo k ogata a m yahiro and k kato b

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

Nuclear Breakup is dominant

Underestimation

→ Inelastic breakup effect ~ 20%

Breakup to 3- continuum is negligible

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


Takuma matsumoto kyushu univ k minomo k ogata a m yahiro and k kato b

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

Coulomb Breakup is dominant

Underestimation

→ Inelastic breakup effect

Overestimation ???

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


Takuma matsumoto kyushu univ k minomo k ogata a m yahiro and k kato b

Summary

  • 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.