1. THREE-BODY FORCE AND FRAGMENTATION IN NUCLEAR REACTIONS
Zhiqiang Chen
Institute of Modern Physics
Chinese Academy of Sciences
May 16, 2018
Fourth International Workshop on
“State Of the Art in Nuclear Cluster Physics”
May 13-18, 2018
Galveston, TX, USA

2. Collaborators Zhiqiang Chen, Weiping Lin, Guoyu Tian, Rui Han
Institute of Modern Physics (IMP), Chinese Academy of Sciences (CAS)
Gaolong Zhang, Weiwei Qu, I. Tanihata
School of Physics and Nuclear Energy Engineering, Beihang University, China
Research Center for Nuclear Physics, Osaka University, Japan
Roy Wada
Cyclotron Institute, Texas A&M University, USA
Akira Ono
Department of Physics, Tohoku University, Japan

3. Outline Motivation Experiment of 12C+12C scattering at 100 A MeV.
AMD model simulations of nuclear fragmentation reactions.
Summary and outlook

4. Motivation
Three-body forces(TBFs) are a frontier for understanding and predicting strongly interacting many-body systems.
TBFs are known to play an important role in the binding of nuclei and also in the equation of state(EOS) for nuclear matter.
For the binding of nuclei, ab initio type calculations that include the Fujita- Miyazawa interactions have demonstrated the importance of the attractive TBFs for understanding the structure of light nuclei.
For the EOS, TBFs are the important for reproducing the saturation properties and the compressibility at high density.
A high-density environment is produced by high-energy heavy-ion collisions so that sensitivity of the cross sections to repulsive TBFs is expected.
[1] Hans-Werner Hammer, et al., Rev.Mod.Phys.85,197(2013).
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[3] P.Navratil, et al., Phys.Rev.Lett.105,032501(2010).
[4]T.Otsuka, et al., Phys.Rev.Lett. 105,032501(2010).
[5]A.Deltuva and A.C. Fonseca, Phys.Rev.C75,014005(2007).
[6]J.Fujita and H.Miyazawa Prog.Theor.Phys. 17,360(1957).
[7]M.Baldo et al., Astron.Astrophys.328,274(1997).
[8]A. Lejeune, et al., Phys.Lett.B477, 45(2000).

5. TBF effects on EOS of symmetric nuclear matter
1. In order to describe reasonably the nuclear saturation properties within the nonrelativistic Brueckner-Hartree-Fock (BHF) framework, one has to take into account the TBF effect.
2. the saturation density and the saturation energy are ∼ 0.167fm−3 and ∼ −15.9 MeV respectively, in satisfactory agreement with the empirical values.
3. TBF contribution to the EOS is repulsive and leads to a stiffening of the EOS, especially at supra-saturation densities.
1. Wei Zuo Journal of Physics: Conference Series 420 (2013).
2. W. Zuo et al., Nucl.Phys.A 706(2002) 418.
3. Z.H.Li et al, Phys.Rev.C77(2008)

6. TBF effects on high-energy heavy-ion scattering
T. Furumoto et al. have proposed a theoretical model for constructing the complex optical potential for any composite projectiles through the double-folding-model (DFM) with a newly proposed complex G-matrix NN interaction called CEG07.
real part + imaginary part
T. Furumoto, Y. Sakuragi, Y. Yamamoto, PRC78(2008) ,
T. Furumoto, Y. Sakuragi, Y. Yamamoto, PRC79(2009) (R),
T. Furumoto, Y. Sakuragi, Y. Yamamoto, PRC80(2009)
T. Furumoto, Y. Sakuragi, Y. Yamamoto, PRC82 (2010) (E)
T. Furumoto, Y. Sakuragi, Y. Yamamoto, PRC82(2010) ( )

7. Complex G-matrix interaction (CEG07)
T.Furumoto, Y. Sakuragi and Y. Yamamoto, Phys.Rev.C 78 (2008)
derived from ESC04
“ESC04” : the latest version of Extended Soft-Core force
designed for NN, YN and YY systems
Th. Rijken, Y. Yamamoto, Phys.Rev.C 73 (2006)
1. Three-body attraction (TBA)
・ Fujita-Miyazawa diagram
・ important at low density region
2. Three-body repulsion (TBR)
・ originated the triple-meson correlation
・ important at high-density region
In the ESC04 model
density-dependent effective two-body force

8. Saturation curve in nuclear matter
with G-matrix interaction（CEG07） ＋
Three-body repulsive (TBR)
Three-body attractive (TBA)
Two body force only
ESC04 NN force (
Extended Soft-Core )
includes
Three body force
important Three-body force effect K
=260
MeV
=84

9. 12C +12C elastic scattering at E/A=100~400MeV
real potential : repulsive
around E/A = 300～400 MeV
CEG07b (with TBF)
CEG07a (without TBF)
Diffractive
oscillation
E/A=100 MeV, calculated cross section with CEG07a dominates over the cross sections with CEG07b.
E/A=200MeV, two kinds of cross sections show almost identical angular distributions.
E/A=400MeV, the situation becomes completely opposite to that of E/A=100MeV.
E/A=300MeV (CEG07b) and E/A=400MeV(CEG07a), the cross sections show a strong diffractive oscillation pattern.
T. Furumoto, Y. Sakuragi and Y. Yamamoto. Phys. Rev. C 82 (2010)

10. Experiment Setup RCNP Osaka University 100MeV/u 12C+12C@RCNP，
Experiment of 12C+12C scattering at 100 A MeV
Experiment Setup
100MeV/u
Osaka University
Beam line: WS course
Beam: 100 AMeV 12C
Beam intensity: pnA
Beam energy resolution: 500keV
Target: 1.181mg/cm2 natural C
and mg/cm2 CH2 target
Detector: VDC1 and VDC2,
PS1, PS2 and PS3
Measured angles: degrees,
Angular resolution: better than 0.1 degree
RCNP
Osaka University
W.W.Qu et al., Phys.Lett.B 751,1 (2015).
W.W.Qu et al., Phys.Rev.C 95,044616(2017).

11. Magnetic spectrometer
“GRAND RAIDEN”
Focal plane detectors
Two Vertical-type Drift Chambers(VDCs) +
Three Plastic scintillators(PSs)
(thickness: 3mm,10mm and 10 mm )
VDC1 and VDC2: particle trajectory
PS1,PS2 and PS3: identify particles

12. Particle identification and spectrum fitting
Particle identification during experiment.
Two-dimensional plot of excitation energy and laboratory angles for outgoing 12C particles for the spectrometer central angle of 2.0◦.
Excitation spectrum obtained at scattering angle of 1.5◦. The fitted spectrum for the 4.44 MeV excited state with the target and projectile excitation components is shown.

13. Theoretical analysis Interaction model Saturation curves
ESC: The two-body interaction ESC08 NN interaction model.
CEG07b: ESC04 NN interaction model and include TBF effect.
MPa: ESC08 NN interaction model and includes a three-body repulsive part expressed by the multi-Pomeron exchange potential (MPP).
Theoretical frame of microscopic coupled-channel(MCC) method
W.W.Qu et al., Phys.Rev.C 95,044616(2017).

14. Results: 1-ch calculations
Elastic-scattering differential cross
sections for 12C + 12C at 100 AMeV
Reaction cross sections for 12C + 12C
The NW value is fixed by the reaction cross section. Because the cross section is very sensitive to the strength of the imaginary potential.
TBF makes an important contribution to elastic scattering.

15. Results: Full-coupled-channel calculations
Reaction cross sections for 12C + 12C
Differential cross sections for 12C + 12C at 100 AMeV
ESC
CEG07b
MPa
1. ESC fails to reproduce the measured cross sections.
2. MPa model reproduces the data better than CEG07b.

16. Results: coupled-channel effect on elastic and inelastic cross sections
Elastic cross sections for 12C + 12C at 100 AMeV
Inelastic cross sections for 12C + 12C at 100 AMeV
The CC effect for the MPa interaction is seen in the elastic differential cross sections as a decrease of the cross section at large scattering angles.
The inelastic scattering is better reproduced with the inclusion of the CC effect.
For the inelastic cross sections, the TBF effect is also clearly seen to be important.

17. AMD model simulations of nuclear fragmentation reactions AMD model
A. Ono, H. Horiuchi, T.Maruyama, and A. Ohnishi, Prog. Theor. Phys. 87, 1185 (1992).
In AMD a reaction system with N nucleons is described by a Slater determinate of N Gaussian wave packets:
The centroid of Gaussian wave packets Zi is given as:
The equation of motion for Z is derived as:
H is the Hamiltonian and Ciσ,jτ is a Hermitian matrix defined by:
AMD treats the nucleon-nucleon collision process in the physical coordinate space. The physical coordinateW ≡ {Wi} for a given nucleon, i, is defined as
The Winger form of the ith nucleon at time t = t0 is represented as :

18. Fermi boost in AMD-FM
W. Lin, X. Liu, R. Wada, et al., Phys.Rev.C94, (2016).
In AMD-FM, the Fermi motion is taken into account in the two-body collision process. When two nucleons are within the collision distance , the momentum uncertainty increases.
In the actual calculations for given coordinate vectors r1 and r2 of two attempted colliding nucleons, the associated momenta P1 and P2 are given as:
where P0i is the centroid of the Gaussian momentum distribution for the particle i and ΔP'i is the Fermi momentum randomly given along the Gaussian distribution.
where G(1) is a random number generated along the Gaussian distribution with σ = 1. (ρi/ρ0)1/3 in Eq. (16) is used for taking into account the density dependence of the Fermi energy, ρi is the density at ri , and ρ0 is the normal nuclear density.

19. AMD-FM reproduces the experimental data well.
Proton energy spectra for 40Ar+51V at 44 MeV/nucleon
AMD
CoMD
AMD-FM
AMD-FM
AMD-FM reproduces the experimental data well.

20. AMD-FM reproduces the experimental data well.
Proton energy spectra for 36Ar+181Ta at 94 MeV/nucleon
Proton energy spectra of AMD-FM at θ ∼ 110° (open squares) and 4π solid angle (open circles) are compared to the experimental spectrum (solid squares) in the center-of-mass frame for the central collision events.
Proton energy spectra of AMD-FM at 75 ° (red solid histogram) and 105 °(green dashed histogram) are compared to those
of experiment (solid symbols) in the laboratory frame.
AMD-FM reproduces the experimental data well.

21. GEANT4 calculations for 12C +12C at 95 MeV/nucleon
GANIL Experimental data.
Energy distributions of 4He, 6Li, and 7Be fragments at 4◦ and 17◦. Black points are experimental data. Histograms are for simulations with QMD, BIC, and INCL models coupled to the FBU de-excitation model.
J. Dudouet, et al., Phys.Rev.C 89, (2014)
None of the toolkits provide good enough reproduction of the experimental data, especially for those from the intermediate velocity source.

22. CoMD,AMD,AMD-FM calculations for 12C +12C at 95 MeV/nucleon
proton
deuteron
triton
4He

23. AMD-FM calculations for 12C +12C at 95 MeV/nucleon
Comparing to the three transport model calculations, overall AMD-FM reproduces best the experimental data.

24. AMD-FM calculations for 12C +12C at 95 MeV/nucleon
Angular distribution for Li, Be, B and C isotopes.
AMD-FM failed to reproduce IMFs experimental data.

25. Cluster correlations in AMD (AMD-Cluster)
This extended version of AMD is developed mainly to improve the description of the IMF emission by taking into account the cluster correlation by Akira Ono.
Ono, J. Phys. Conf. Ser. 420(2013)
Ikeno, Ono et al., PRC 93 (2016)

26. AMD-Cluster calculations for 12C +12C at 95 MeV/nucleon
G.Tian Phys.Rev.C 97, (2018)
Angular distribution for Li, Be, B and C isotopes.
AMD-Cluster reproduced the IMFs experimental data well.

27. Three-nucleon interaction in heavy-ion collisions
Three-nucleon (3N) interaction is incorporated into
the AMD-FM model by Roy Wada.
R. Wada Phys.Rev.C 96, (R) (2017)
Proton energy distributions for the 40Ar+51V.
AMD-FM(circles), AMD-FM(3N)(histograms)
1. At 100 AMeV, the energy spectra are very similar at different angles.
2. When the incident energy increases to 200AMeV, the shapes of the energy spectra show a distinct difference. AMD-FM(3N) show much harder slopes.

28. Proton energy distributions for the 40Ar+40Ca and 40Ar+51V.
AMD-FM(dashed curves), AMD-FM(3N)(solid curves)
The experimental proton energy spectra at θ≥70°are reproduced reasonably well by AMD-FM(3N).
3N interaction is importance in intermediate heavy ion reactions.

29. 100- 400 AMeV 12C + 12C proposal experiment at HIRFL-CSR
HIRFL-CSR (Lanzhou, China) Layout

30. 200 AMeV 12C + 12C experiment at HIRFL-CSR (test run)

31. Summary and outlook
The angular distributions of differential cross sections of 12C +12C elastic and inelastic scattering are precisely measured at 100A MeV in RCNP Osaka University.
The present results provide clear evidence of the important roles of the repulsive TBF and the CC effect in high-energy heavy-ion collisions.
The experimental energy spectra and angular distributions of light charged particles are well reproduced by the AMD-FM calculations for 12C +12C at 95 MeV/nucleon.
The cluster correlation plays a crucial role for producing fragments in the intermediate-energy heavy ioncollisions.
AMD-FM (3N) simulations indicates for the first time the importance of the 3N interaction in intermediate heavy ion reactions.
In the future, AMeV 12C +12C experiments will be done in HIRFL-CSR (Lanzhou, China).

32. Thanks for your attention!