1. Investigations of nuclear equation of state in nucleus- nucleus collisions and fission Martin Veselsky Institute of Physics, Slovak Academy of Sciences, Dubravska cesta 9, Bratislava, Slovakia G. A. Souliotis, P. Fountas, N. Vonta Department of Chemistry, University of Athens, Athens, Greece Yu-Gang Ma Shanghai Institute of Applied Physics of CASc, Shanghai, China

2. Nuclear Equation of State Definition : A relation that determines how nuclear matter (composing of neutrons & protons) behaves when subjected to different pressure, density & temperature In real gas, this relation is very well known, because we know exactly the force of interaction (Van der Waals) between the gas molecules The force of interaction between nucleons (neutrons & protons) in nuclear matter is not very well known molecular force of attraction

3. The “stiffer” (“softer”) the nuclear interaction the larger (“smaller”) is the predicted neutron star mass and radius J. Stone et al, Phys. Rev. C 68 (2003) 034324 The long-sought mass-radius relation of neutron star depends on the strength of interaction “Stiff” “Soft” Nucleon interactions too “Soft” Radius (km) Ruled Out Credit : S. Lee, CXC, NASA Mass (M Sol )

4. Neutron star cooling depends on the strength of symmetry energy “Stiffer” symmetry energy can lead to faster cooling of the neutron stars Nucleon interactions “Stiff” “Soft” Neutron stars are way too cooler than originally expected E sym =E sym (ρ 0 )(ρ/ρ 0 ) γ δ 2

5. Nucleus-nucleus collisions - Introduction Nucleus-nucleus collisions allow to study properties of the nuclear matter. Properties of the nuclear matter are formulated in terms of nuclear equation of state. Nuclear matter is a two-component system and the effect of changing neutron-to-proton ratio is typically treated in terms of symmetry energy and its density dependence. Various implementations of Boltzmann equation (Fokker-Planck equation, Boltzmann-Uehling-Uhlenbeck equation, Quantum Molecular Dynamics) are used for modeling of the nucleus-nucleus collisions and testing of various equations of state. Successful equations of state can be used for modeling of astrophysical objects (neutron stars) and processes (supernova explosions)

6. Boltzmann equation Boltzmann equation Various implementations of the collision term lead to different physics No collision term – Liouville or Vlasov equation, explains e.g. Boltzmann distribution, behavior of plasma in electromagnetic field Collision term – BGK form – describes simple transport phenomena – Ohm's law, thermodiffusion, thermo-electric phenomena, Fokker- Planck equation (deep-inelastic transfer) Collision term with pair collisions – Boltzmann-Uehling-Uhlenbeck, Vlasov-Uehling-Uhlenbeck, Landau-Vlasov – suitable for description of dynamical stage of nucleus-nucleus collisions. In-medium nucleon- nucleon cross sections ?

7. Boltzmann-Uehling-Uhlenbeck equation Boltzmann-Uehling-Uhlenbeck equation EoS

8. In-medium nucleon-nucleon cross section in the nuclear matter are important component in the nuclear implementations of the Boltzmann equations, nevertheless they are practically unknown. Typically, free nucleon-nucleon cross sections are used in simulations of nucleus-nucleus collisions, occasionally scaled down by some factor. Density-dependence of nucleon-nucleon cross sections is approximated only empirically. True in-medium nucleon-nucleon cross sections must depend on equation of state of the nuclear matter and thus the dependence of collision term on EoS needs to be implemented !!!

9. Calculation of in-medium nucleon-nucleon cross sections in the nuclear matter presents a considerable challenge to the nuclear theory. G-matrix theory was used to estimate in-medium nucleon-nucleon cross sections by Cassing et al. (W. Cassing, U. Mosel, Prog. Part. Nucl. Phys. 25, 235 (1990)). Density-dependence of in-medium nucleon-nucleon cross section was studied for symmetric nuclear matter (G. Q. Li, and R. Machleidt, Phys. Rev. C 48, 1702 (1993); Phys. Rev. C 49, 566 (1994); T. Alm et al., Phys. Rev. C 50, 31 (1994); Nucl. Phys. A 587, 815 (1995)), and significant influence of nuclear density on resulting in-medium cross sections was observed in their density, angular and energy dependencies. Using momentum-dependent interaction, ratios of in-medium to free nucleon- nucleon cross sections were evaluated at zero temperature via reduced effective nucleonic masses (B. A. Li, and L. W. Chen, Phys. Rev. C 72, 064611 (2005)) and used for transport simulations. Still, transport simulation are mostly performed using parametrizations of the free nucleon-nucleon cross sections, eventually scaling them down empirically or using simple prescriptions for density-dependence of the scaling factor (D. Klakow et al., Phys. Rev. C 48, 1982 (1993)).

10. Van der Waals-like equation of state By combining a long-range attractive and short-range repulsive interaction it allows to describe by a relatively simple model the main features of the liquid-gas phase transition in isospin- asymmetric nuclear matter, including the possible interplay of 1 st and 2 nd order phase transition with variation of the isospin asymmetry (M. Veselsky, IJMPE 17 (2008) 1883; arXiv.org:nucl- th/0703077) It provides a parameter, called “proper” or “excluded volume”, which describes the volume per constituent particle of the non-ideal gas. Can the proper volume estimate in-medium nucleon- nucleon cross section ?

11. Method: Formally transform EoS into Van der Waals form, extract the proper volume. M. Veselsky and Y.G. Ma, PRC 87 (2013) 034615 Starting from EoS:

12. Fermionic effects taken into account - Two effects which cancel out mutually

14. Global 1/ρ 2/3 dependence of nucleon-nucleon cross sections from EoS EoS-dependent free

15. It is apparent that while the isospin-dependent nucleon-nucleon cross sections essentially follow the 1/ ρ 2/3 -dependence, the nucleon-nucleon cross section parametrization of Cugnon et al. (J. Cugnon, T. Mizutani, and J. Vandermeulen, Nucl. Phys. A 352, 505 (1981)) leads to much larger spread, mostly due to its explicit energy dependence. Nevertheless, one observes that both parametrization cover essentially the same range of values of the nucleon-nucleon cross sections. Furthermore, from the comparison of the parametrization of Cugnon et al. to in-medium cross sections at saturation density, calculated using the G-matrix theory by Cassing et al. (W. Cassing, and U. Mosel, Prog. Part. Nucl. Phys. 25, 235 (1990)), it can be judged that the in-medium cross sections, obtained using the proper volume of the Van der Waals-like equation of state, are in better agreement with somewhat higher values of G-matrix in- medium cross sections of Cassing et al., which reflect properly the Fermionic nature of nucleons.

16. Systematics of proton directed flow in Au+Au, b=5-7fm N. Herrmann, J. P. Wessels, and T. Wienold, Ann. Rev. Nucl. Part. Sci. 49 (1999) 581 N. Herrmann, J. P. Wessels, and T. Wienold, Ann. Rev. Nucl. Part. Sci. 49 (1999) 581. Directed flow is a 1 st Fourier coefficient of the angular distribution in azimuthal (transverse) plane.

17. Systematics of proton elliptic flow in Au+Au, b=5-7fm A. Andronic, J. Lukasik, W. Reisdorf, and W. Trautmann, Eur. Phys. J. 30 (2006) 31 A. Andronic, J. Lukasik, W. Reisdorf, and W. Trautmann, Eur. Phys. J. 30 (2006) 31. Elliptic flow is a 2 nd Fourier coefficient of the angular distribution in azimuthal (transverse) plane.

18. ΔxΔp>2h Stiff EoS (K 0 =380 MeV) Semi-stiff Esym (γ=1.) Directed flow: symbols- calc. line - experiment Elliptic flow: Symbols - calc. line - experiment

19. EoS-dependent collision term (with in-medium cross sections) leads to correct (positive) directed flow, while free cross sections lead to incorrect (negative) directed flow !!!

20. Soft EoS (K 0 =200 MeV) Semi-stiff Esym (γ=1.) Directed flow: symbols- calc. line - experiment Elliptic flow: Symbols - calc. line - experiment

21. Starting from potential After applying standard conditions for saturation we arrive to solution And get compressibility (linear to ! )

22. Semi-stiff EoS (K 0 =270 MeV) Semi-stiff Esym (γ=1.) Directed flow: symbols- calc. line - experiment Elliptic flow: Symbols - calc. line - experiment

23. Simultaneous analysis of directed and elliptic flow in Au+Au semi-peripheral collisions (0.4-10 AGeV)

24. Production of n-rich nuclei at 15 AMeV Comparison of the model of deep-inelastic transfer (DIT) vs constrained molecular dynamics (CoMD) vs experimental data

25. Peripheral Collisions, Deep Inelastic Transfer (DIT)* DIT : Phenomenological model (Monte Carlo implementation of Fokker-Planck equation)  Formation of a di-nuclear configuration  Exchange of nucleons through a “window” formed by the superimposition of the nuclear potentials in the neck region  Dissipation of Kinetic energy into internal degrees of freedom *DIT : L. Tassan-Got, C. Stephan, Nucl. Phys. A 524, 121 (1991) DIT(modified): M. Veselsky, G.A. Souliotis, Nucl. Phys. A 765, 252 (2006) b: impact parameter θ: scattering angle Approaching phase: Target (Zt,At) θ b Projectile (Zp,Ap) Neutrons Protons excited projectile-like fragment (PLF) or quasi-projectile excited target-like fragment (TLF) or quasi-target Overlapping (interaction) stage Exchange of nucleons: Deep Inelastic Transfer (DIT) Model L. Tassan-Got and C. Stephan, Nucl. Phys. A 524, 121 (1991)

26. Modified DIT (Nucl.Phys. A765, 252 (2006)), phenomenological correction, effect of shell structure on nuclear periphery ( assuming validity of the R n -R p vs μ n -μ p correlation ) and thus on transfer probability estimated as: where  S... = S... exp - S... mac,  is a free parameter (  = 0.53 determined as optimal value ), s > 0 fm ( only non-overlapping configurations considered )

27. Microscopic Calculations: Constrained Molecular Dynamics (CoMD)* CoMD: Quantum Molecular Dynamics model (Semiclassical)  Nucleons are considered as Gaussian wavepackets  N-N effective interaction (Skyrme-type with K=200 MeV/fm3)  Several forms of N-N symmetry potential V sym (ρ)  Pauli principle imposed via a phase-space constraint  Fragment recognition algorithm (R min = 3.0 fm)  Monte Carlo implementation *M. Papa, A. Bonasera et al., Phys. Rev. C 64, 024612 (2001)

28. 86 Kr+ 58,64 Ni 15 AMeV Experimental data measured at MARS spectrometer at Texas A&M CoMD (red) and DITm (blue) reproduce data

29. 86 Kr+ 112,124 Sn 15 AMeV Experimental data measured at MARS spectrometer at Texas A&M CoMD (red) and DITm (blue) reproduce data

30. 92 Kr+ 64 Ni 15 AMeV Estimate of possible production of n- rich nuclei in reaction with secondary beam CoMD more optimistic than DITm, symmetry energy ?

31. Motivated also by results of our work (with GS) new RIB facility is built in Korea including secondary beams with energy within the Fermi energy domain. After 15 years finally someone pays attention !!!

32. p (30 MeV) + 235U, CoMD simulation of nuclear fission

33. Comparison between theoretical and experimental results: p (660 MeV) + 238U Grey dots: Exprimental data: [12] A. R. Balabekyan, et Al Phys. Atom. Nucl. 73, 1814- 1819 (2010) Red line: standard Vsym ~ ρ Blue line: standard Vsym ~ ρ strict selection in Zfiss = 93

34. Red line: standard Vsym ~ ρ Blue line: soft Vsym ~ ρ 1/2 Circles: fission of 235U Squares: fission of 238U Triangles: fission of 232Th Open symbols: experimental data [12] A. R. Balabekyan, et Al Phys. Atom. Nucl. 73, 1814-1819 (2010) [1] P. Demetriou et al Phys. Rev. C 82, 054606 (2010) [10] M. C. Duijvestijn, et Al Physical Review. C 64, 014607 (2001) Total fission cross section and fission cross section/residue cross section Sensitivity of the ratio fission cross section/residue cross section to the choice of the nucleon-nucleon symmetry potential

35. Total kinetic energy of the fission fragments Red line: standard Vsym ~ ρ Blue line: soft Vsym ~ ρ 1/2 Circles: fission of 235U Squares: fission of 238U Triangles: fission of 232Th Open sympbols: experimental data P. Demetriou et al Phys. Rev. C 82, 054606 (2010) M. C. Duijvestijn, et Al Physical Review. C 64, 014607 (2001) S.I. Mulgin et Al Nuclear Physics A, 824, 1–23, (2009)

36. Neutron multiplicity Red line: standard Vsym ~ ρ Blue line: soft Vsym ~ ρ 1/2 Circles: fission of 235U Squares: fission of 238U Triangles: fission of 232Th Open symbols: experimental data [1] P. Demetriou et al Phys. Rev. C 82, 054606 (2010) [10] M. C. Duijvestijn, et Al Physical Review. C 64, 014607 (2001)

37. Conclusions: - new version of BUU with N-N cross sections derived from the equation of state and applied in the collision term - systematics of directed and elliptic flow reproduced and constraints on incompressibility and symmetry energy found (K 0 =260-310 MeV and  =0.8-1.25) - consistent reproduction of experimental production cross sections of n- rich nuclei using both CoMD and DITm - collective dynamics in fission decribed using the CoMD - consistent constraints on symmetry energy from fission, low-energy and relativistic collisions ? - hard work ahead

38. Conference in Slovakia: ISTROS 2015 (2 nd edition) Častá-Papiernička May 1 – 6 Everybody interested to participate is welcome ! http://istros.sav.sk/