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

1
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

2
Introduction

• The unstable nuclear structure can be
  efficiently
• investigated via the breakup reactions.

Nuclear and Coulomb

• Elastic cross section
• Breakup cross section
• Momentum distribution
• of emitted particles

Unstable Nuclei
Structure information
Target

• An accurate method of treating breakup processes
  
• is highly desirable.

3
Breakup Reactions
Three-body Breakup Reaction
The projectile breaks up into 2
particles. Projectile (2-body) target
(1-body) 3-body
breakup reaction
1
2
Ex.) d, 6Li, 11Be, 8B, 15C, etc..
3
One-neutron halo
Four-body Breakup Reaction
The projectile breaks up into 3
particles. Projectile (3-body) target
(1-body) 4-body
breakup reaction
1
2
3
Ex.) 6He, 11Li, 14Be, etc..
Two-neutron halo
4
4
Region of Interest

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

n
n
core
n
Three-body System (Four-body reaction system)
core
Two-body System (Three-body reaction system)
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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

Discretized states
Continuum
discretization

• 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

Breakup threshold
Bound
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Discretization of Continuum
Momentum-bin method
Cannot apply to four-body breakup reactions
Pseudo-state method
Wave functions of discretized continuum states
are obtained by diagonalizing the model
Hamiltonian with basis functions
Can apply to four-body breakup reactions
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Three-Body model of 6He
6He is a typical example of a three-body
projectile and many theoretical calculations have
been done.
1
2
n
3
n
4He
4
Four-body CDCC calculation of 6He breakup
reactions
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).
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Gaussian Expansion Method

• 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

• 6He n n 4He (three-body model)

n
n
n
n
n
n

• Hamiltonian
• Vnn Bonn-A
• Van KKNN int.

4He
4He
4He
Channel 1
Channel 2
Channel 3
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Elastic Scattering of 6He
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Nuclear Breakup of 6He

• E gtgt Coulomb barrier
• Negligible of
  Coulomb breakup effects
• Elastic cross section

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Coulomb Breakup of 6He

• E Coulomb barrier
• Coulomb breakup
  effects are to be significant
• Elastic cross section

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Inelastic Scattering of 6He
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Inelastic scattering to 2
2 resonance state
0
2
n
n
4He
p
ground state
6Hep_at_40 MeV/nucl.
A. Lagoyannis et al., Phys. Lett. B 518, 27 (2001)
2 resonance state
2 resonance state is shown as a discretized
state.
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Elastic and Inelastic of 6He
Exp data A. Lagoyannis et al., Phys. Lett. B
518, 27 (2001)
Elastic cross section
Inelastic cross section
6He(g.s.) -gt 6He(2)
without breakup effects
with breakup effects
6Hep_at_40MeV/A
6Hep_at_40MeV/A
For the inelastic cross section, the calculation
underestimates the data.
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Non-resonance Component
Add non resonance component
16
Breakup to continuum of 6He
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Breakup Cross Section
Nuclear and Coulomb Breakup
Nuclear Breakup 6He12C _at_ 229.8 MeV
g.s ? 0 cont.
g.s ? 1- cont.
resonance
s BU mb
s BU mb
g.s ? 2 cont.
enna MeV
enna MeV
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Smoothing Procedure
Discrete T matrix ? Continuum T matrix
Continuum T matrix
Discrete T matrix
Smoothing factor
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Triple Differential Cross Section
Three-Body Breakup
k
n
P
c
t
Triple differential cross section
Momentum distribution of c
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Momentum Distribution
Exp. Data Kondo et al.
18C p ? 17C n p
n
p
17C
18C
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Differential Cross Section
Four-Body Breakup
k
K
n
n
c
P
t
Quintuple differential cross section
Three-body smoothing function
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Calculation of Smoothing Factor
Schrodinger Eq.
Lippmann-Schwinger Eq.
Smoothing function
T. Egami (Kyushu Univ.)
Complex-Scaled Solution of the Lippmann-Schwinger
Eq.
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E1 Transition Strength

• Discretized B(E1) strength

• Smoothing procedure

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B(E1) strength of 6He 0 ? 1-
Calculated by Egami
A. Cobis et al., Phys. Rev. Lett. 79, 2411(97).
D. V. Danilin et al., Nucl. Phys. A632, 383(98).
J. Wang et al., Phys. Rev. C65,034036(02).
T. Aumann et al., Phys. Rev. C59, 1252(99).
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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.

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6Hep scattering _at_ 40 MeV/A
6Hep four-body system
n
n
p
p4He scattering at 40 MeV
4He
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6Hep _at_ 25, 70 MeV/A
6Hep_at_25MeV/A
6Hep_at_70MeV/A
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The Number of Coupling Channels
Nuclear BU Emax 25 MeV _at_ 230 MeV, 10 MeV
_at_18MeV 0 60 channels _at_ 230 MeV, 32 channels
_at_18MeV 2 85 channels _at_ 230 MeV, 42 channels
_at_18MeV
Nuclear Coulomb BU Emax 8 MeV 0 43
channels 1- 48 channels 2 61 channels
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Continuum-Discretized Coupled-Channels
Review 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.

Momentum-bin method

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

We proposed a new approach of the discretization
method
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The Pseudo-State method
Wave functions of discretized continuum states
are obtained by diagonalizing the model
Hamiltonian with basis functions

• Variational method of Rayleigh-Ritz

• Eigen equation

Pseudo-State
Bound State
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)
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