Takuma Matsumoto(Kyushu Univ.)

Analyses of breakup, one- and two-neutron removal reactions for three-body projectile

Contents

• Introduction

• Breakup of Borromean Nuclei

• Breakup of Samba Nuclei

• Summary

Introduction

Nuclear and Coulomb

Target

Unstable Nuclei

The unstable nuclear structure can be efficiently

investigated via the breakup reactions. Inelastic cross section

Breakup cross section

Momentum distribution of emitted particles

An accurate method of treating breakup processes

is needed.

Structure information

n + n + Core System Borromean nuclei

No bound sub-system (n + core) Two-neutron halo or skin structure 6He, 11Li, 14Be Breakup

n + n + core three-body channel

Samba nuclei Existence of bound sub-system (n + core) One-neutron halo structure (n + core sub-system) 12Be, 16C, 20C Breakup

n + n + core three-boy channel n + n-core two-body channel

CDCC

Breakup = Coupling to Continuumnuclear or/and Coulomb

Four-Body Breakup Reaction

CDCC method• Developed by Kyushu Univ. group• Fully quantum method• Treats breakup states explicitly

Essence of CDCC• Breakup continuum states are described by a finite number of discretized continuum states

• A set of eigenstates forms a complete set within a finite model space that is important for breakup processes

M. Yahiro, K. Ogata, T.M., K. Minomo, PTEP 2012(1), 01A206.

Channel 1 Channel 2 Channel 3

n n n nn

n

4He 4He 4He

Ground and Breakup States of 6He 6He : n + n + 4He (three-body model) 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

Ip=0+ Ip=1- Ip=2+

Exc

itatio

n en

ergy

of 6 H

e [M

eV]

ELASTIC SCATTERING OF 6HE

4He

n

n

6He12C

6He (n+n+4He) + 12C elastic scattering

initial final

0+ 2+1-

Double-folding model

vNN:DDM3Y → no imaginary part

Transition density

R

srTrp

ground state density

Coupling Potential (Double Folding)

Elastic Cross Section of 6He + 12C T.M. Hiyama, Ogata, Iseri, Kamimura, Chiba, and Yahiro, Phys. Rev. C70, 061601 (2004).

4He

n

n

6He209Bi

6He (n+n+4He) + 209Bi scattering

4He

n

n

initial final

Cluster folding model

VnBiVnBi

VaBi

Optical potentials are folded with the transition densities of 6He.

No free parameter

VaBi includes the Coulomb interaction

Coupling Potential (Cluster Folding)

Other calculationsM. Rodríguez-Gallardo, J. M. Arias, J. Gómez-Camacho, R. C. Johnson, A. M. Moro, J. Thompson, and J. A. Tostevin, PRC 77, 064609 (2008)

Elastic Cross Section 6He + 209Bi T.M. Egami, Ogata, Iseri, Kamimura, and Yahiro, Phys. Rev. C73, 051602 (2006).

BREAKUP CROSS SECTION OF 6HE

Breakup Cross SectionBreakup cross sections calculated by CDCC are discrete in the internal energy of the projectile.

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

E* (MeV)

s (m

b)

6He+12C scattering at 240 MeV/nucl.4-body CDCC calc.

How to calculate the continuum breakup cross section

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

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

The Cauchy integral contour for the completeness relation

Complex-scaling operator:

Coordinate:

Momentum:

k Im[k]

Re[k]

}Bound states

Resonance

Continuum

Useful for searching many-body resonances

Complex Scaling Method (CSM)

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

Asymptotic form of the outgoing wave

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

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

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

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

ONE- AND TWO-NEUTRON REMOVAL REACTION OF 6HE

Breakdown for Coulomb breakup!

Reaction theories

Glauber model

Eikonal approximation + adiabatic approximation

A new theory to treat the inclusive reactions accurately was proposed.

Eikonal reaction theory (ERT)→ 　 one- and two-neutron stripping and removal cross

sections

○Exclusive reaction ○Inclusive reaction

M. Yahiro, K. Ogata, K. Minomo, PTP126, 167 (2011).

M. Yahiro, K. Ogata, T.M., K. Minomo, PTEP 2012(1), 01A206.

The method of Continuum-Discretized Coupled Channels (CDCC)

○Exclusive reaction ×Inclusive reaction

Two-neutron removal from 6He

Good agreement!

2n removalT. Aumann, et al. PRC59, 1252 (1999).

SAMBA NUCLEI

Samba nuclei

• Sub-bound state• core + n (bound)• n-n (unbound)

• Breakup channel• Core + n + n (Three-body breakup channel)• (Core + n)bound + n (Two-body breakup channel)

n

n

10Be

12Be

Because of existence of two breakup channels, we have to distinguish two-body breakup from three-body breakup channels.

Breakup of 12Be

target

(10Be + n + n)

(11Be + n)

Complete BU

Incomplete BU

11Be

10Be

n

n

10Be

12Be

In principle, this breakup process can be analyzed by four-body CDCC framework.

Discretized continuum of 12BeModel Hamiltonian of n + n + 10Be n-10Be interaciton

Phys. Rev. C62 034305, Aoyama n-n interaction

Phys. Lett. B32 591, Gogny et al. Phenomenological 3-body force and OCM

n + n + 10Be

n + 11Be

Discretized states include two components, complete BU and incomplete BU states.

BU cross section calculated by CDCC includes two components.

How to distinguish incomplete BU from complete BU

Probability of 11Be in 12BeWave functions of 12Be

Probability of 11Be ground state component in 12Be

Ground state Pi ～ 0.89 (89 %) Breakup cross sections for n + 11Be channel and n + n + 10Be

σ 𝐵𝑈 (𝑖𝑛𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑒 )=∑𝑖𝑃 𝑖𝜎 𝑖 ,𝜎𝐵𝑈 (𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑒 )=∑

𝑖(1−𝑃 𝑖 )𝜎 𝑖

12Be + Si @ 50 MeV/A

Pi=0.61 (61%)

Pi=0.57 (57%)

Pi=0.05 (5%)

Pi=0.03(3%)

Pi=0.89 (61%)

Calculation (preliminary)

BU(complete) : 54 mbBU(incomplete) : 20 mb

𝜎 (𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑒 )>𝜎 (𝑖𝑛𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑒)

Phys. Rev. C64, 044601Experiment

𝜎 ¿140 mb 100 mb

11Be is a halo nucleus

Continuum Breakup Cross Section

n + n + 10Be Breakup

?

n + 11Be Breakup

?

Summary We propose a new smoothing method with the complex scaling method to obtain continuous breakup cross sections.

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

Inclusive cross sections of three-body projectile are calculated by Eikonal reaction theory.

For breakup of Samba nuclei, we have to consider two breakup channels, complete and incomplete breakup.

Prog. in part. and nucl. Phys. 67, 9390994 (2012)