1. Maria Colonna Laboratori Nazionali del Sud (Catania) Testing the behavior of n-rich systems away from normal density Eurorib’ 10 June 6-11, 2010 --- Lamoura

2.  Equation of State (EoS) of asymmetric nuclear matter the nuclear energy density functionals, effective interactions  Self-consistent MF calculations (and extensions) are a powerful framework to understand the structure of medium-heavy nuclei. Source: F.Gulminelli In this context relativistic non-relativistic …only a matter of functional Isoscalar, spin, isospin densities, currents …  Widely employed in the astrophysical context (modelization of neutron stars and supernova explosion)

3. The largest uncertainties concern the isovector part of the nuclear interaction : The symmetry energy E/A (ρ) = Es(ρ) + E sym (ρ) β² β=(N-Z)/A asy-stiff asy-soft zoom at low density asy-soft asy-stiff C. Fuchs, H.H. Wolter, EPJA 30(2006)5,(WCI book) E sym (ρ) = J γ = L/(3J) E sym = E/A (β=1) – E/A(β=0) Often used parametrization:  asy-soft,  asy-stiff

4. Focus on E sym at low density The crust-core transition density decreases with L Nuclear structure Nuclear astrophysics Correlation between n-skin and L Properties of n-rich nuclei depend on low-density Esym (because of surface effects !) Nuclei- neutron star connection ! M.Centelles et al, PRL(2009) I.Vidana et al., PRC80(2009)

5. Isospin effects in reaction mechanisms at Fermi energies  Symmetry energy parameterizations are implemented into transport codes (Stochastic Mean Field - SMF) and confronted to experimental data for specific reaction mechanisms and related observables Chomaz,Colonna, Randrup Phys. Rep. 389 (2004) Baran,Colonna,Greco, Di Toro Phys. Rep. 410, 335 (2005) asy-soft asy-stiff Parametrizations used in SMF simulations E sym pot = 18 r (2 – r) SKM*(soft) 18 r stiff 18 (2r 2 )/(1+r) stiff (superstiff) r = ρ/ρ 0 Transient states of nuclear matter in several conditions ! γ~0.6 γ~1

6. Reactions between systems with different N/Z Isospin diffusion (in the low density interface) is driven by the symmetry energy Information on E sym at low density INDRA data: Ni + Ni, Ni + Au @ 52, 74 MeV/A ISOSPIN TRANSPORT AT FERMI ENERGIES ISOSPIN TRANSPORT AT FERMI ENERGIES 1(PLF) 2(TLF) Reaction plane 1)If x = N/Z or f(N/Z) Isospin equilibration 2) Contact time measured by kinetic energy dissipation Symmetry energy x 1,2 (t) – x m = (x 1,2 – x m ) e -t/τ x m = (x 1 + x 2 )/2 t contact time τ dissipation time for observable x Path towards equilibrium of the observable x Galichet et al., Phys. Rev. C79, 064615 (2009) How to access the N/Z of the PLF ?  Isotopic content of light charged particle emission as a function of the dissipated energy Exchange of energy, mass, isospin between 1 and 2

7. 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

8. R 1,2 (t) = (x 1,2 (t) – x m ) / |x 1,2 – x m | R 1,2 = ±e -t/τ XmXm X2X2 X1X1 x 1,2 (t) – x m = (x 1,2 – x m ) e -t/τ x m = (x 1 + x 2 )/2 Path towards equilibrium of the observable x B. Tsang et al. PRL 92 (2004) 1(PLF) 2(TLF) Isospin transport ratio R N/Z of largest fragment y red Ni + Ni @ 15,40 AMeV P.Napolitani et al., PRC(2010) τ E sym

9. Focus on Esym below normal density Strength of PDR Mass formula Neutron skin thickness Li, Lombardo, Schulze, Zuo, PRC77, 034316 (2008) Microscopic BHF calculations Galichet et al.,(2009) A.Carbone et al., PRC(R) (2010) and ref.s therein Nuclear reactions: Isospin diffusion Tsang et al., PRL(2009) GMR (Li et al, PRL 2007) Pre-equilibrium dipole emission

10.  Need to enlarge the systematics of data (and calculations) to validate the current interpretation and the extraction of E sym (consensus on E sym ~(ρ/ρ 0 ) γ with γ~0.6-1 at low density) Still large uncertainties at high density (FAIR, NICA, RIKEN, …) 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) Conclusions

11. Ensemble average Langevin: random walk in phase-space Transport model: Semi-classical approach to the many-body problem Time evolution of the one-body distribution function BoltzmannLangevinVlasov BoltzmannLangevin Vlasov: mean field Boltzmann: average collision term Loss term D(p,p’,r) SMF model : fluctuations projected onto ordinary space density fluctuations δρ Fluctuation variance: σ 2 f = D(p,p’,r)w

12. Probes of the symmetry energy (at low density) Isospin diffusion J.Rizzo et al, NPA (2008) Pre-equilibrium dipole oscillation V.Baran et al, PRC79, 021603 (2009). Isospin distillation (liquid-gas) asy-stiff - - -asy-soft M.Colonna et al PRC78,064618(2008) Optical potentials (isospin & momentum dependence of forces) Li & Lombardo, PRC78,047603(2008)

13. Trippa, Colò, Vigezzi PRC77(2008)061304 P.Danielewicz J.Lee nucl-th/08073743 A.Klimkiewicz et al PRC76(2007)051603 B.Tsang et al PRL102(2009)122701 Pygmy dipole mass formula Isospin diffusion GDR L=3  dE sym /d  P 0 =  L/3 BHF Fragment N/Z, Central collisions GDR Constraints on E sym Li,Lombardo et al PRC77(2008)034316 Galichet,Colonna et al PRC79(2009)064615 M.Colonna et al PRC78,064618(2008) Symmetry energy at ρ 0 (normal density)

14. C. Fuchs, H.H. Wolter, EPJA 30(2006)5,(WCI book) E/A (ρ) = Es(ρ) + Esym(ρ) β² β=(N-Z)/A data Momentum dependence effective mass different for protons and neutrons m* n < m* p m* n > m* p Asy-soft Asy-stiff n p Often used parametrization:  asy-soft,  asy-stiff Symmetry energy and mass splitting asy-stiff asy-soft zoom at low density asy-soft asy-stiff Lane potential Symmetry potential

15. E diss 1(PLF) 2(TLF) The charge of the reconstructed PLF is in reasonnable agreement with the data The dissipated energy is well correlated to the impact parameter Sorting variable and PLF properties Galichet et al., Phys. Rev. C79, 064615 (2009)