Probing the nuclear EOS with fragment production Maria Colonna Laboratori Nazionali del Sud (Catania)

Fragmentation events: IMF properties in central and semi-peripheral collisions (neutron-rich systems) Kinematical properties Size and asymmetry (N/Z) Insight into the reaction mechanism responsible for fragment emission and isospin transport: density vs N/Z concentration gradients Dependence of the results on the asy – EOS A new method to incorporate full fluctuations into a transport treatment (Boltzmann-Langevin theory) Conclusions and perspectives

Instantaneous equilibrium Ensemble average Langevin: random walk in phase-space 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 Correlation function Focus on the variance σ 2 f = D(p,p’,r)

Stochastic mean field (SMF) calculations b = 4 fmb = 6 fm Sn124 + Sn124, E/A = 50 MeV/A Chomaz,Colonna, Randrup Phys. Rep. 389 (2004) Central collisions Ni + Au, E/A = 45 MeV/A (fluctuations projected on ordinary space)

Isospin Transport and Chemical Potentials currents diffusion DiffusionDrift drift Direct Access to Value and Slope of the Symmetry Energy at ρ ! asy-soft asy-stiff E/A (ρ) = E s (ρ) + E sym (ρ)I ² I=(N-Z)/A

IMF properties in central collisions Sn124 + Sn124, E/A = 50 MeV/A, b = 2 fm bubble-like configuration radial flow correlations: Size, N/Z vs radial distance - velocity Primary fragment properties X.T.Liu et al, PRC69(2004) V.Baran et al, NPA703(2002)

N/Z vs charge 1200 events for each reaction  Proton/neutron repulsion: larger negative slope in the stiff case (lower symmetry energy)  n-rich clusters emitted at larger energy in n-rich systems (Δ’>Δ) N/Z vs fragment energy (1)  Sn112 + Sn112  Sn124 + Sn124  Sn132 + Sn132 E/A = 50 MeV, b=2 fm “gas” phase (pre-equilibrium) asy-stiff asy-soft “liquid” asy-stiff asy-soft N = Σ i N i, Z = Σ i Z i 1.64 Double ratio = (N/Z) 2 /(N/Z) 1 Δ Δ’Δ’ 3≤ Zi ≤ 10

N/Z vs fragment energy (2) To combine the two effects: Different slope vs. n-rich cluster emission “shifted” N/Z: N/Zs = N/Z – N/Z(E=0) Larger sensitivity to the asy-EoS is observed in the double N/Zs ratio IMF emission Double ratio R = (N/Z) 2 /(N/Z) 1 Large double ratio in the asy-stiff case ! (opposite to pre-equ. emission) Famiano et al. PRL 06 Pre-equilibrium emission IMF emission asy-stiff asy-soft 3≤ Zi ≤ 10 Sn124 – Sn112 (BUU calculations) Δ Δ’Δ’

Still arbitrary: r-space distance Procedure easily applicable to nuclear reactions Careful check of Pauli-blocking Take into account possible nucleon deformations in p space I J J’ I’ p-space Our improved method test particles i and j cells I and J Spherical search around I and J “clouds” rotated to final states x Pauli violations: average trajectory altered x Shape of more similar to classical than to quantum case Fluctuations in phase space Original procedure by Bauer et al. Cross section reduction 1 nucleon = N TEST nearest neighbours Phase-space distance and ; Δp assigned to each “cloud” Clouds translated to final states ( no rotation) Pauli blocking checked only for i and j t.p. -Δp-Δp ΔpΔp Bauer, Bertsch, Das Gupta, PRL58 (1987) 58 (Improve the treatment of fluctuations in p space)

r-space: periodic 3-D box, l = 26 fm; ρ = 0.16 fm -3 (2820 nucleons); 500 test particles p-space: Fermi-Dirac configuration, kT = 5 MeV Illustrative results t =0 t = 100 fm/c Set of coordinates t = 0 fm/c t = 100 fm/c p = 260 MeV/c, Δp = 10 MeV/c, Check of the profile Check of the fluctuation variance on the Fermi surface Our result: In this case: ‘nucleon’ volume ΔpΔp

Propagation of fluctuations by the unstable mean-field Box calculations : ρ = 0.05 fm, T = 3 MeV Fourier analysis of the density variance : rapid growth of density fluctuations Fragment multiplicity and charge distributions (300 nucleons) -3

CONCLUSIONS Study of IMF’s emitted in heavy ion collisions: Correlations between N/Z and velocity Fragments with smaller kinetic energy are more neutron rich (asy – stiff) Double ratios as sensitive observables to the asy-EOS Development of a full 3D treatment of the BL theory : Improvement of the treatment of fluctuations in p space (thermal fluctuations) V.Baran (NIPNE HH,Bucharest) M.Di Toro, J. Rizzo (LNS-Catania) Ph. Chomaz (GANIL, France) H.H. Wolter (Munich)

124 Sn + 64 Ni 35 AMeV ternary events N/Z-IMF vs. Alignement Correlation in semi-peripheral collisions Experiment Transp. Simulations (124/64) Chimera data: see E.De Filippo, P.Russotto NN2006 Contr., Rio Asystiff Asysoft V.Baran, Aug.06 Asystiff: more isospin migration to the neck fragments Histogram: no selection E.De Filippo et al., PRC71(2005)

Set of coordinates p = 260 MeV/c, Δp = 10 MeV/c, t = 0 fm/c t = 100 fm/c The variance of the distribution function p = 190 MeV/c Δθ = 30° spherical coordinates  fit the Fermi sphere  allow large volumes Clouds position Best volume: p = 190 MeV/c, θ = 20°

b=8fm b=10fm ISOSPIN DIFFUSION AT FERMI ENERGIES 124 Sn Sn at 50 AMeV BNV - transport model b=8fm b=9 fm b=10fm 120fm/c 100fm/c 80fm/c experimental data (B. Tsang et al. PRL 92 (2004) ) asysoft eos superasystiff eos contact time Baran, Colonna, Di Toro, Pfabe, Wolter, PRC72(2005) Imbalance ratios asy-soft EOS – faster equilibration

E lab = 30 Mev/A, b = 4 fm a) soft b) stiff Competition between reaction mechanisms: fusion vs deep-inelastic M.Colonna et al., PRC57(1998)1410 neutron-rich proton-rich

Comparison with INDRA data -- stiff -- soft forward c.m. forward QP

I = I in + c(E sym ) (I av – I in ) R P = 1 – c ; R T = c - 1 soft stiff b 20% difference in the slope between stiff and soft INDRA data: Ni + Ni, Ni + 52, 74 MeV/A: N/Z vs b E.Galichet et al., Nucl.Phys.A submitted IPN, Orsay if H LHLH HLL

asy-soft asy-stiff Sn + Ni E lab = 10 MeV/A b = 6,7,8 fm, t = 500 fm/c More dissipative neck dynamics with asy-stiff ! asy-stiff asy-soft Octupole distribution Competition between deep-inelastic and neck emission V.Baran, Aug.06 SPIRAL2 proposal

INDRA data: Ni + Ni, Ni + 52, 74 MeV/A Isospin effects on dissipation softstiff E.Galichet et al., Nucl.Phys.A submitted IPN, Orsay

DEVIATIONS FROM VIOLA SYSTEMATICS r - ratio of the observed PLF-IMF relative velocity to the corresponding Coulomb velocity; r1- the same ratio for the pair TLF-IMF The IMF is weakly correlated with both PLF and TLF Wilczynski-2 plot ! 124 Sn + 64 Ni 35 AMeV

v_z (c) v_x (c) Distribution after secondary decay (SIMON) Sn124 + Sn124, E/A = 50 MeV/A, b = 6 fm CM V z -V x CORRELATIONS v_par

58Fe+58Fe vs. 58Ni+58Ni b=4fm 47AMeV: Freeze-out Asymmetry distributions Fe Ni Fe Ni White circles: asy-stiff Black circles: asy-soft Asy-soft: small isospin migration Fe: fast neutron emission Ni: fast proton emission

Angular distributions: alignment characteristics  plane is the angle, projected into the reaction plane, between the direction defined by the relative velocity of the CM of the system PLF- IMF to TLF and the direction defined by the relative velocity of PLF to IMF Out-of-plane angular distributions for the “dynamical” (gate 2) and “statistical” (gate 1) components: these last are more concentrated in the reaction plane.

Dynamical Isoscaling Z=1 Z=7 primary final not very sensitive to E sym ? 124 Sn Carbon isotopes (primary) A Asy-soft Asy-stiff T.X.Liu et al. PRC AMeV (central coll.)