Probing the nuclear EOS with fragment production Maria
Probing the nuclear EOS with fragment production Maria Colonna Laboratori Nazionali del Sud (Catania)
ü 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 üA new method to incorporate full fluctuations into a transport treatment (Boltzmann-Langevin theory) üConclusions and perspectives
Semi-classical approach to the many-body problem Time evolution of the one-body distribution function Vlasov Boltzmann Langevin Vlasov: mean field Boltzmann: average collision term Vlasov Boltzmann Loss term Langevin: random walk in phase-space D(p, p’, r) Ensemble average Correlation function Focus on the variance σ2 f = <δfδf> Instantaneous equilibrium Langevin
Stochastic mean field (SMF) calculations (fluctuations projected on ordinary space) b = 4 fm b = 6 fm Central collisions Ni + Au, E/A = 45 Me. V/A Chomaz, Colonna, Randrup Phys. Rep. 389 (2004) Sn 124 + Sn 124, E/A = 50 Me. V/A
Isospin Transport and Chemical Potentials E/A (ρ) = Es(ρ) + Esym(ρ)I² currents I=(N-Z)/A drift asy-stiff diffusion asy-soft Diffusion Drift Direct Access to Value and Slope of the Symmetry Energy at ρ !
IMF properties in central collisions Sn 124 + Sn 124, E/A = 50 Me. V/A, b = 2 fm • bubble-like configuration • radial flow • correlations: Size, N/Z vs radial distance - velocity X. T. Liu et al, PRC 69(2004) V. Baran et al, NPA 703(2002) Primary fragment properties
§ Sn 112 + Sn 112 § Sn 124 + Sn 124 § Sn 132 + Sn 132 E/A = 50 Me. V, b=2 fm N/Z vs fragment energy (1) 1. 64 N = Σ i Ni , Z = Σ i Z i 3≤ Zi ≤ 10 1200 events for each reaction N/Z vs charge Δ Δ’ “gas” phase (pre-equilibrium) Double ratio = (N/Z)2/(N/Z)1 “liquid” asy-stiff asy-soft § 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 (2) Double ratio R = (N/Z)2/(N/Z)1 Δ Pre-equilibrium emission Δ’ (BUU calculations) Sn 124 – Sn 112 IMF emission Famiano et al. PRL 06 IMF emission asy-stiff asy-soft 3≤ Zi ≤ 10 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-Eo. S is observed in the double N/Zs ratio Large double ratio in the asy-stiff case ! (opposite to pre-equ. emission)
Fluctuations in phase space (Improve the treatment of fluctuations in p space) Our improved method • test particles i and j Original procedure by Bauer et al. cells I and J • • Cross section reduction • Spherical search around I and J 1 nucleon = NTEST nearest neighbours • “clouds” rotated to final states • Phase-space distance J’ • < pi > and < pj > ; Δp assigned to each “cloud” -Δp Δp p-space I J Still arbitrary: • r-space distance • Pauli blocking checked only for i and j t. p. I’ • Clouds translated to final states ( no rotation) Bauer, Bertsch, Das Gupta, PRL 58 (1987) 58 x Pauli violations: average trajectory altered x Shape of more similar to classical than to quantum case ü Procedure easily applicable to nuclear reactions ü Careful check of Pauli-blocking üTake into account possible nucleon deformations in p space
Illustrative results Check of the <f> profile r-space: periodic 3 -D box, l = 26 fm; ρ = 0. 16 fm-3 (2820 nucleons); 500 test particles p-space: Fermi-Dirac configuration , k. T = 5 Me. V t =0 t = 100 fm/c In this case: ‘nucleon’ volume Our result: Δp Set of coordinates t = 0 fm/c t = 100 fm/c p = 260 Me. V/c, Δp = 10 Me. V/c, Check of the fluctuation variance on the Fermi surface
Propagation of fluctuations by the unstable mean-field Box calculations : ρ = 0. 05 fm-3 , T = 3 Me. V Fourier analysis of the density variance <δρδρ> : rapid growth of density fluctuations Fragment multiplicity and charge distributions (300 nucleons)
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 3 D 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)
N/Z-IMF vs. Alignement Correlation in semi-peripheral co 124 Sn Experiment Histogram : no selection + 64 Ni 35 AMe. V ternary events Transp. Simulations (124/64) Asystiff Asysoft Asystiff: more isospin migration to the neck Chimera data: see E. De Filippo, P. Russotto NN 2006 Contr. , Rio E. De Filippo et al. , PRC 71(2005) V. Baran, Aug. 06
The variance of the distribution function spherical coordinates Ø fit the Fermi sphere Ø allow large volumes Best volume: p = 190 Me. V/c, θ = 20° p = 190 Me. V/c Clouds position Δθ = 30° Set of coordinates t = 0 fm/c p = 260 Me. V/c, Δp = 10 Me. V/c, t = 100 fm/c
ISOSPIN DIFFUSION AT FERMI ENERGIES b=8 fm b=9 fm b=10 fm 120 fm/c 100 fm/c 80 fm/c contact time 112 Sn at 50 AMe. V b=10 fm BNV - transport model + b=8 fm 124 Sn Imbalance ratios asysoft eos superasystiff eos asy-soft EOS – faster equilibration experimental data (B. Tsang et al. PRL 92 (2004) ) Baran, Colonna, Di Toro, Pfabe, Wolter, PRC 72(2005)
Competition between reaction mechanisms: fusion vs deep-inelastic a) soft b) stiff neutron-rich Elab = 30 Mev/A, b = 4 fm proton-rich M. Colonna et al. , PRC 57(1998)1410
Comparison with INDRA data -- stiff -- soft forward c. m. forward QP
INDRA data: Ni + Ni, Ni + Au @ 52, 74 Me. V/A: N/Z vs b H H if L H L L I = Iin + c(Esym) (Iav – Iin) RP = 1 – c ; RT = c - 1 soft stiff 20% difference in the slope between stiff and soft E. Galichet et al. , Nucl. Phys. A submitted IPN, Orsay b
asy-stiff Competition between deep-inelastic and neck emission More dissipative neck dynamics with asy-stiff ! 132 Sn + 64 Ni Elab = 10 Me. V/A b = 6, 7, 8 fm, t = 500 fm/c Octupole distribution asy-soft asy-stiff SPIRAL 2 proposal asy-soft V. Baran, Aug. 06
Isospin effects on dissipation INDRA data: Ni + Ni, Ni + Au @ 52, 74 Me. V/A soft stiff 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; r 1 - the same ratio for the pair TLF-IMF The IMF is weakly correlated with both PLF and TLF 124 Sn + 64 Ni 35 AMe. V Wilczynski-2 plot !
CM Vz-Vx CORRELATIONS v_par v_x (c) Sn 124 + Sn 124, E/A = 50 Me. V/A, b = 6 fm after secondary decay (SIMON) Distribution v_z (c)
58 Fe+58 Fe vs. 58 Ni+58 Ni b=4 fm 47 AMe. V: Freeze-out Asymmetry distributions Fe Ni Fe Fe: fast neutron emission Ni: fast proton emission Ni White circles: asy-stiff Black circles: asy-soft Asy-soft: small isospin migration
Angular distributions: alignment characteristics Out-of-plane angular distributions for the “dynamical” (gate 2) and “statistical” (gate 1) components: these last are more concentrated in the reaction plane is the angle, projected into the reaction plane, between the direction defined by the relative velocity of the CM of the system PLFIMF to TLF and the direction defined by the relative velocity of PLF to IMF
Dynamical Isoscaling primary 50 AMe. V Z=1 (central coll. ) Z=7 final not very sensitive to Esym ? 124 Sn T. X. Liu et al. PRC 2004 Carbon isotopes (primary) Asy-stiff Asy-soft A
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