1. Probes of EOS in Heavy Ion Collisions : results from transport theories Maria Colonna INFN - Laboratori Nazionali del Sud (Catania) Nuclear Structure & Astrophysical Applications July 8-12, 2013 ECT*, TRENTO

2. Heavy Ion Collisions: a way to create Nuclear Matter in several conditions of ρ, T, N/Z… Nuclear effective interaction Equation of State (EOS)  Self-consistent Mean Field calculations (like HF) are a powerful framework to understand the structure of medium- heavy (exotic) nuclei. Source: F.Gulminelli  Widely employed in the astrophysical context (modelization of neutron stars and supernova explosion)

3. Dynamics of many-body systems Mean-fieldResidual interaction Average effect of the residual interaction one-body Fluctuations TDHF

4. Dynamics of many-body systems -- If statistical fluctuations larger than quantum ones Main ingredients: Residual interaction (2-body correlations and fluctuations) In-medium nucleon cross section Effective interaction (self consistent mean-field) Transition rate W Interpreted in terms of NN cross section see BHF

5. 1. Semi-classical approximation Transport equation for the one-body distribution function f Chomaz,Colonna, Randrup Phys. Rep. 389 (2004) Baran,Colonna,Greco, Di Toro Phys. Rep. 410, 335 (2005) Residual interaction: Correlations, Fluctuations k δkδk (1,2) (3,4) Two-body Collision Integral Fluctuations in collision integral Stochastic Mean-Field (SMF) model Fluctuations are projected in coordinate space (density profile) (semi-classical analog of Wigner function) Vlasov v rel 3 1 2 4 Boltzmann-Langevin (BL) equation

6. stochastic NN collisions Collision integral fully stochastic, but approx. description of mean-field effects… Molecular Dynamics approaches (AMD, ImQMD, QMD,…) 2. Zhang and Li, PRC74,014602(2006) J.Aichelin, Phys.Rep.202,233(1991) Colonna, Ono, Rizzo, Phys. Rev..C82,054613 (2010) A.Ono, Phys.Rev.C59,853(1999)

7. The nuclear interaction, contained in the Hamiltonian H, is represented by effective interactions (Skyrme, Gogny, …) E/A (ρ) = E s (ρ) + E sym (ρ) β² β=(ρ n -ρ p )/ρ The density dependence of E sym is rather controversial, since there exist effective interactions leading to a variety of shapes for E sym :  Asysoft,  Asystiff Investigate the sensitivity of the reaction dynamics to this ingredient Put some constraints on the effective interactions EOS Symmetry energy Asysoft Asystiff Neutron skin Isovector modes Pigmy resonances around ρ 0 Effective interactions and symmetry energy γ = 1 γ = 0.5 γ = L/(3S 0 )

8.  Low-energy reaction mechanisms: Fusion, Deep-Inelastic, Charge equilibration  Fragmentation mechanisms at Fermi energies  Study of the PDR with transport theories

9. Low energy reaction mechanisms Fusion vs Deep Inelastic

10. Competition between reaction mechanisms: fusion vs deep-inelastic Fusion probabilities may depend on the N/Z of the reaction partners: - A mechanism to test the isovector part of the nuclear interaction Important role of fluctuations 36 Ar + 96 Zr, E/A = 9 MeV Fusion or break-up ? C.Rizzo et al., PRC83, 014604 (2011)

11. 36 Ar + 96 Zr, E/A = 9 MeV, b = 6 fm Starting from t = 200-300 fm/c, solve the Langevin Equation (LE) for selected degrees of freedom: Q (quadrupole), β 3 (octupole), θ, and related velocities Competition between reaction mechanisms θ Examples of trajectories t = 0 t = 40 t = 100 t = 140 t = 200 fm/c z (fm) Break-up configurations β 2, β 3, E*~250 MeV, J ~70ћ l (ћ) soft stiff Shvedov, Colonna, Di Toro, PRC 81, 054605 (2010) Break-up times of the order of 500-1000 fm/c ! Extract the fusion cross section

12. Charge equilibration in fusion and D.I. collisions D(t) : bremss. dipole radiation CN: stat. GDRInitial Dipole A1A1 A2A2 Relative motion of neutron and proton centers of mass If N 1 /Z 1 ≠ N 2 /Z 2 132 Sn + 58 Ni, D 0 = 45 fm E/A = 10 MeV TDHF calculations C.Rizzo et al., PRC 83, 014604 (2011) 40 Ca + 100 Mo Simenel et al, PRC 76, 024609 (2007) SMF simulations

13. Bremsstrahlung: Quantitative estimation V.Baran, D.M.Brink, M.Colonna, M.Di Toro, PRL.87 (2001) C.Rizzo et al., PRC83, 014604 (2011) B.Martin et al., PLB 664 (2008) 47 36 Ar + 96 Zr 40 Ar + 92 Zr Dynamical dipole emission D.Pierroutsakou et al., PRC80, 024612 (2009) Experimental evidence of the extra-yield (LNS data) soft stiff soft Asysoft larger symmetry energy, larger effect (30 %) see also A.Corsi et al., PLB 679, 197 (2009), LNL experiments

14. -- Charge equilibration: from dipole oscillations to diffusion Overdamped dipole oscillation τ d E sym t D(t) Charge equilibration in D.I. collisions at ~ 30 MeV/A low energy Full N/Z equilibration not reached: Degree of eq. ruled by contact time and symmetry energy E diss / E cm Observable: (N/Z) CP = N/Z of light charge particles emitted by PLF as a function of Dissipated energy: (impact parameter, contact time) INDRA data, Galichet et al., PRC (2010) 1(PLF) 2(TLF)

15. B. Tsang et al., PRL 102 (2009) Mass(A) ~ Mass(B) ; N/Z(A) = N/Z(B) Isospin transport ratio R : A dominance mixing B dominance +1 0 -- AA and BB refer to two symmetric reactions between n-rich and n-poor nuclei AB to the mixed reaction -- X is an observable related to the N/Z of PLF (or TLF) A B “Isospin diffusion” in D.I. collisions stiff soft  More central collisions: larger contact time more dissipation, smaller R  Good sensitivity to Asy-EoS X = N/Z PLF SMF calculations, 124 Sn + 112 Sn, 50 AMeV J. Rizzo et al., NPA(2008) (PLF) (TLF) X = Y( 7 Li)/ Y( 7 Be) Comparison between ImQMD calculations and MSU data B. Tsang et al., PRL 102 (2009)

16. “Exotic” fragments in reactions at Fermi energies

17. TLF PLF Y.Zhang et al., PRC(2011) Fragmentation mechanisms at Fermi energies IMF 124 Sn + 64 Ni 35 AMeV: 4π CHIMERA detector  Fragment emission at mid-rapidity (neck emission)  neutron-enrichment of the neck region dynamical emission E. De Filippo et al., PRC(2012) Charge Z

18. Isospin migration in neck fragmentation  Transfer of asymmetry from PLF and TLF to the low density neck region: neutron enrichment of the neck region  Effect related to the derivative of the symmetry energy with respect to density PLF, TLF neck emitted nucleons ρ IMF < ρ PLF(TLF) Asymmetry flux Sn112 + Sn112 Sn124 + Sn124 b = 6 fm, 50 AMeV Density gradients derivative of E sym The asymmetry of the neck is larger than the asymmetry of PLF (TLF) in the stiff case J.Rizzo et al. NPA806 (2008) 79 asy-soft asy-stiff Larger derivative with asy-stiff larger isospin migration effects stiff - full

19. Comparison with Chimera data 124 Sn + 64 Ni 35 AMeV DE SE Properties of ‘dynamically emitted’ (DE) fragments Charge distribution Parallel velocity distribution E. De Filippo et al., PRC(2012) Good reproduction of overall dynamics The N/Z content of IMF’s in better reproduced by asystiff (L =75 MeV)

20. “Exotic” collective excitations in Nuclei

21. X neutrons – protons X c core neutrons-protons Y excess neutrons - core Neutron center of mass Proton center of mass Isovector dipole response PDR GDR The Isovector Dipole Response (DR) in neutron-rich nuclei X.Roca-Maza et al., PRC 85(2012) Pygmy dipole strength  Giant DR  Pygmy DR Klimkiewicz et al.

22. Pygmy-like initial conditions (Y) X Y X c Neutron skin and core are coupled The low- energy response is not sensitive to the asy-stiffness -- soft -- stiff -- superstiff Baran et al. 25.0 see M.Urban, PRC85, 034322 (2012) 132 Sn

23. GDR-like initial conditions (X) Fourier transform of D Strength of the Dipole Response PDR is isoscalar-like frequency not dependent on E sym GDR is isovector-like dependent on E sym The strength in the PDR region depends on the asy-stiffness (increases with L) Larger L larger strength, but also Larger L larger neutron skin 2.7 % soft 4.4 % stiff 4.5 % superstiff see A.Carbone et al., PRC 81 (2010) E (MeV)

24. The energy centroid of the PDR as a function of the mass for Sn isotopes, 48 Ca, 68 Ni, 86 Kr, 208 Pb ~ 41 A -1/3  PDR energy and strength (a systematic study) Neutron skin extension with soft, stiff and superstiff Correlation between the EWSR exhausted by the PDR and the neutron skin extension V.Baran et al., arXiv:1306.4969 soft superstiff stiff soft stiff superstiff

25. A.Carbone et al., PRC 81 (2010) “Summary” of E sym constraints V.Baran (NIPNE HH,Bucharest) M.Di Toro, C.Rizzo, J.Rizzo, (LNS, Catania) M.Zielinska-Pfabe (Smith College) H.H.Wolter (Munich) E.Galichet, P.Napolitani (IPN, Orsay) Galichet 2010 De Filippo 2012  Reduce experimental error bars  Perform more complete analyses  Improve theoretical description (overcome problems related to model dependence) Asysoft Asystiff Isospin diffusion

26. Heavy Ion Collisions (HIC) allow one to explore the behavior of nuclear matter under several conditions of density, temperature, spin, isospin, … HIC from low to Fermi energies (~10-60 MeV/A) are a way to probe the density domain just around and below normal density. The reaction dynamics is largely affected by surface effects, at the borderline with nuclear structure. Increasing the beam energy, high density regions become accessible. Varying the N/Z of the colliding nuclei (up to exotic systems), it becomes possible to test the isovector part of the nuclear interaction (symmetry energy) around and below normal density.

27. A.Carbone et al., PRC 81 (2010) M.B.Tsang et al., PRC (2012) “Summary” of E sym constraints V.Baran (NIPNE HH,Bucharest) M.Di Toro, C.Rizzo, J.Rizzo, (LNS, Catania) M.Zielinska-Pfabe (Smith College) H.H.Wolter (Munich) E.Galichet, P.Napolitani (IPN, Orsay)

28. Calculations: - N/Z increases with the centrality of collision for the two systems and energies (For Ni + Ni pre-equilibrium effects) - In Ni + Au systems more isospin diffusion for asy-soft (as expected) - (N/Z) CP linearly correlated to (N/Z) QP PLF CP PLF CP N/Z CP -- stiff (γ=1) + SIMON -- soft(γ=0.6) + SIMON SMF transport calculations: N/Z of the PLF (Quasi-Projectile) Squares: soft Stars: stiff After statistical decay : N = Σ i N i, Z = Σ i Z i Charged particles: Z=1-4 forward n-n c.m. forward PLF Comparison with data Data:  open points higher than full points (n-rich mid-rapidity particles)  Isospin equilibration reached for E diss /E cm = 0.7-0.8 ? (open and full dots converge)  Data fall between the two calculations

29.  Interpretation in terms of isovector-like and isoscalar-like modes in asymmetric systems Isoscalar-like Isovector-like soft stiff Energy-density variation Curvature matrix Normal modes α increases with L strength in the PDR region (isoscalar-like mode) increases with L α =45 0 for symmetric matter Zero temperature

30. stiff soft Fragment isotopic distribution and symmetry energy Fragmentation can be associated with mean-field instabilities (growth of unstable collective modes) Oscillations of the total (isoscalar-like density) fragment formation and average Z/A (see δρ n /δρ p ) Oscillations of the isovector density (ρ n - ρ p ) isotopic variance and distributions (Y 2 / Y 1, isoscaling) Study of isovector fluctuations, link with symmetry energy in fragmentation Isovector density ρ v = ρ n –ρ p λ = 2π/k nuclear matter in a box At equilibrium, according to the fluctuation-dissipation theorem What does really happen in fragmentation ? F v coincides with the free symmetry energy at the considered density

31. Full SMF simulations T = 3 MeV, Density: ρ 1 = 0.025 fm -3, 2ρ 1, 3ρ 1 I = 0.14 T ρ F’ follows the local equilibrium value ! The isospin distillation effect goes together with the isovector variance I high densitylow density ρ Nuclear matter in a box freeze-out F’ ~ T / σ stiff soft stiff soft M.Colonna, PRL,110(2013) Average Z/A and isovector fluctuations as a function of local density ρ

32. Dissipation and fragmentation in “MD” models ImQMD calculations, 112 Sn + 112 Sn, 50 AMeV  More ‘explosive’ dynamics: more fragments and light clusters emitted more ‘transparency’ What happens to charge equilibration ? Rather flat behavior with impact parameter b: - Weak dependence on b of reaction dynamics ? - Other dissipation sources (not nucleon exchange) ? fluctuations, cluster emission weak nucleon exchange Isospin transport ratio R Y.Zhang et al., PRC(2011) 124 Sn + 112 Sn, 50 AMeV

33. Comparison SMF-ImQMD 6 fm 8 fm γ = 0.5 SMF = dashed lines ImQMD = full lines  For semi-central impact parameters: Larger transparency in ImQMD (but not so a drastic effect) Other sources of dissipation (in addition to nucleon exchange) More cluster emission SMF ImQMD γ = 0.5 Different trends in ImQMD and SMF! What about fragment N/Z ? γ = 2  Isospin transport R around PLF rapidity : Good agreement in peripheral reactions Elsewhere the different dynamics (nucleon exchange less important in ImQMD) leads to less iso-equilibration

34. Liquid phase: ρ > 1/5 ρ 0 Neighbouring cells are connected (coalescence procedure) Extract random A nucleons among test particle distribution Coalescence procedure Check energy and momentum conservation A.Bonasera et al, PLB244, 169 (1990) Fragment excitation energy evaluated by subtracting Fermi motion (local density approx) from Kinetic energy Correlations are introduced in the time evolution of the one-body density: ρ ρ +δρ as corrections of the mean-field trajectory Correlated density domains appear due to the occurrence of mean-field (spinodal) instabilities at low density Fragmentation Mechanism: spinodal decomposition Is it possible to reconstruct fragments and calculate their properties only from f ?  Several aspects of multifragmentation in central and semi-peripheral collisions well reproduced by the model  Statistical analysis of the fragmentation path  Comparison with AMD results Chomaz,Colonna, Randrup Phys. Rep. 389 (2004) Baran,Colonna,Greco, Di Toro Phys. Rep. 410, 335 (2005) Tabacaru et al., NPA764, 371 (2006) A.H. Raduta, Colonna, Baran, Di Toro,., PRC 74,034604(2006) i PRC76, 024602 (2007) Rizzo, Colonna, Ono, PRC 76, 024611 (2007) Details of SMF model T ρ liquid gas Fragment Recognition