Coupled-Channel analyses of three-body and four-body breakup reactions Takuma Matsumoto

(RIKEN Nishina Center)

T. Egami1, K. Ogata1, Y. Iseri2, M. Yahiro1 and M. Kamimura1

(1Kyushu University, 2Chiba-Keizai College)

Unbound Nuclei Workshop 3-5 November 2008

Introduction

Nuclear and Coulomb

Target

Unstable Nuclei

The unstable nuclear structure can be efficiently

investigated via the breakup reactions. Elastic cross section

Breakup cross section

Momentum distribution of emitted particles

An accurate method of treating breakup processes

is highly desirable.

Structure information

Breakup ReactionsThree-body Breakup Reaction

Four-body Breakup Reaction

The projectile breaks up into 2 particles.

Projectile (2-body) + target (1-body) 3-body breakup reaction

The projectile breaks up into 3 particles.

Projectile (3-body) + target (1-body) 4-body breakup reaction

Ex.) d, 6Li, 11Be, 8B, 15C, etc..

Ex.) 6He, 11Li, 14Be, etc..

One-neutron halo

Two-neutron halo

1

2

3

2

1

4

3

Region of Interest In a simplified picture, light neutron rich nuclei can be described by a few-body model

n

p 2H 3H

3He 4He 6He 8He

7Li6Li 8Li 9Li

7Be 9Be 10Be

8B 10B 11B

10C9C 11C 12C 14C13C 15C 16C

13B12B 14B 15B

12Be11Be 14Be

11Li

17B 19B

18C17C 19C 20C

Neutron

drip line

n

p 2H 3H

3He 4He 6He 8He

7Li6Li 8Li 9Li

7Be 9Be 10Be

8B 10B 11B

10C9C 11C 12C 14C13C 15C 16C

13B12B 14B 15B

12Be11Be 14Be

11Li

17B 19B

18C17C 19C 20C

Neutron

drip line

core

n

n

core

n

Three-body System(Four-body reaction system)

Two-body System(Three-body reaction system)

The CDCC Method The Continuum-Discretized Coupled-Channels method (CDCC)

Developed by Kyushu group about 20 years ago M. Kamimura, Yahiro, Iseri, Sakuragi, Kameyama and Kawai, PTP Suppl.89, 1 (1986). Fully-quantum mechanical method Successful for nuclear and Coulomb breakup reactions Three-body and Four-body reaction systems

Essence of CDCC

Continuum

Bound

Breakup threshold

discretization Discretized states Breakup continuum states

are described by a finite number of discretized states

A set of eigenstates forms a complete set within a finite model space

Discretization of ContinuumMomentum-bin method

Pseudo-state method

Wave functions of discretized continuum states are obtained by diagonalizing the model Hamiltonian with basis functions

Cannot apply to four-body breakup reactions

Can apply to four-body breakup reactions

Three-Body model of 6He 6He is a typical example of a three-body projectile and many theoretical calculations have been done.

4He

n

n

1

2

3

4

1. T. M, , E. Hiyama, T. Egami, K. Ogata, Y. Iseri, M. Kamimura, and M. Yahiro, Phys. Rev. C70, 061601 (2004) and C73, 051602 (2006).

2. M. Rodríguez-Gallardo, J. M. Arias, J. Gómez-Camacho, R. C. Johnson, A. M. Moro, I. J. Thompson, and J. A. Tostevin, Phys. Rev. C77, 064609 (2008).

Four-body CDCC calculation of 6He breakup reactions

Gaussian Expansion Method

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

Channel 1 Channel 2 Channel 3

n n nn

nn

4He 4He 4He

Hamiltonian

Vnn : Bonn-A

Vn : KKNN int.

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

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

Exc

itat

ion

en

ergy

of

6 He

[MeV

]

Elastic Scattering of 6He

Nuclear Breakup of 6He E >> Coulomb barrier : Negligible of Coulomb breakup effects Elastic cross section

Coulomb Breakup of 6He E ~ Coulomb barrier : Coulomb breakup effects are to be significant Elastic cross section

Inelastic Scattering of 6He

Inelastic scattering to 2+

2+ resonance state

0+ 2+

4He

n

n

4He

n

n

ground state

2+ resonance state

p

6He+p@40 MeV/nucl.

2+ resonance state is shown as a discretized state.

A. Lagoyannis et al., Phys. Lett. B 518, 27 (2001)

Elastic and Inelastic of 6He

Elastic cross section Inelastic cross section

Exp data: A. Lagoyannis et al., Phys. Lett. B 518, 27 (2001)

6He+p@40MeV/A 6He+p@40MeV/A

6He(g.s.) -> 6He(2+)

For the inelastic cross section, the calculation underestimates the data.

without breakup effects

with breakup effects

Non-resonance Component

Add non resonance component

Breakup to continuum of 6He

Breakup Cross Section

g.s → 2+ cont.

g.s → 1- cont.

g.s → 0+ cont.

Nuclear and Coulomb BreakupNuclear Breakup6He+12C @ 229.8 MeV

resonance

nn[MeV] nn[MeV]

B

U [m

b]

B

U [m

b]

Discrete S matrix Continuum S matrix

Smoothing ProcedureDiscrete T matrix → Continuum T matrix

Discrete T matrix

Smoothing factor

Approximately complete set

Continuum T matrix

Triple Differential Cross Section

Three-Body Breakup

c

n

t

k

P

Triple differential cross section

Momentum distribution of c

Momentum Distribution18C + p → 17C + n + p

p18C

17C

n

Exp. Data: Kondo et al.

Differential Cross SectionFour-Body Breakup

c

n

t

n

kK

P

Quintuple differential cross section

Three-body smoothing function

Calculation of Smoothing FactorSchrodinger Eq.

Lippmann-Schwinger Eq.

Smoothing function :

T. Egami (Kyushu Univ.)

Complex-Scaled Solution of the Lippmann-Schwinger Eq.

E1 Transition Strength

ground state

I = 1- Discretized B(E1) strength

Smoothing procedure

T. Aumann et al., Phys. Rev. C59, 1252(99).

A. Cobis et al., Phys. Rev. Lett. 79, 2411(97).

D. V. Danilin et al., Nucl. Phys. A632, 383(98).

Calculated by Egami

B(E1) strength of 6He: 0+ → 1-

J. Wang et al., Phys. Rev. C65,034036(02).

Summary We propose a fully quantum mechanical method called four-body CDCC, which can describe four-body nuclear and Coulomb breakup reactions.

We applied four-body CDCC to analyses of 6He nuclear and Coulomb breakup reactions, and found that four-body CDCC can reproduce the experimental data.

Four-body CDCC is indispensable to analyse various four-body breakup reactions in which both nuclear and Coulomb breakup processes are to be significant

In the future work, we are developing a new method of calculation of energy distribution of breakup cross section and momentum distribution of emitted particle.

6He+p scattering @ 40 MeV/A

4He

n

n p

6He+p four-body system

p+4He scattering at 40 MeV

6He+p @ 25, 70 MeV/A

6He+p@25MeV/A 6He+p@70MeV/A

The Number of Coupling Channels

Nuclear BU: Emax = 25 MeV @ 230 MeV, 10 MeV @18MeV 0+ : 60 channels @ 230 MeV, 32 channels @18MeV 2+ : 85 channels @ 230 MeV, 42 channels @18MeV

Nuclear & Coulomb BU: Emax = 8 MeV 0+ : 43 channels 1- : 48 channels 2+ : 61 channels

Continuum-Discretized Coupled-ChannelsReview : M. Kamimura, M. Yahiro, Y. Iseri, Y. Sakuragi, H. Kameyama and M. Kawai, PTP Suppl. 89, 1 (1986)

Three-body breakup (Two-body projectile) Two-body continuum of the projectile can be calculated easily.

Four-body breakup (Three-body projectile) Three-body continuum of the projectile cannot be calculated easily.

Momentum-bin method

We proposed a new approach of the discretization method

The Pseudo-State methodWave functions of discretized continuum states are obtained by diagonalizing the model Hamiltonian with basis functions

Variational method of Rayleigh-Ritz

Eigen equation

Bound State

Pse

ud

o-S

tate

As the basis function, we propose to employ the Gaussian basis function. (Gaussian Expansion Method : E. Hiyama et al. Prog. Part. Nucl. Phys. 51, 223)