EOS and Collision Dynamics Energy of nuclear matter E(ρ, δ)/a = E(ρ, )/A + E sym (ρ)δ 2 δ = (ρ n ρ p )/ρ 1 6 E(ρ, ) (Symmetric matter ρ n = ρ p ) E sy

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1 Nuclear collision dynamics and the equation of state We want to measure EOS. Measure T, P and ρ of matter... Prepare matter in the state we want to measure HI collisions What are taking place in collisions? High density Low density 28/9/26 p.1/18

EOS and Collision Dynamics Energy of nuclear matter E(ρ,

) (Symmetric matter ρ n = ρ p ) E sym (ρ): Symmetry

energy LG phase transition (two components) Effective

DD ρδ DD TW var AV18+ δv+3 BF BHF(AV18+3BF) FSU gold x= 1

2 EOS and Collision Dynamics Energy of nuclear matter E(ρ, δ)/a = E(ρ, )/A + E sym (ρ)δ 2 δ = (ρ n ρ p )/ρ 1 6 E(ρ, ) (Symmetric matter ρ n = ρ p ) E sym (ρ): Symmetry energy Depends on temperature T free energy rather than energy LG phase transition (two components) Effective masses m n(ρ, δ), m p(ρ, δ) E sym [MeV] 4 2 NL3 ChPT DBHF DD ρδ DD TW var AV18+ δv+3 BF BHF(AV18+3BF) FSU gold x= 1 x= ρ / ρ NN cross sections σ NN (ρ, δ) 28/9/26 p.2/18

3 Isospin Effects in High Density Region 197 Au Au at 15 MeV/u, b < 1 fm, t = 3 fm/c fm/c 6 3 fm/c ρ(r) [fm -3 ] Gogny Au + Au E/A = 15 MeV b < 1 fm Gogny t = 3 fm/c p n ρ(r) [fm -3 ] Gogny-AS Au + Au E/A = 15 MeV b < 1 fm Gogny[C(ρ )] t = 3 fm/c p n Symmetry Energy [MeV] Gogny Gogny-AS 6 fm/c r [fm] r [fm] ρ [fm -3 ] fm/c r 2 ρ(r) [fm -1 ] Au + Au E/A = 15 MeV b < 1 fm Gogny t = 3 fm/c p n r 2 ρ(r) [fm -1 ] Au + Au E/A = 15 MeV b < 1 fm Gogny[C(ρ )] t = 3 fm/c p n (N Z)/(N+Z) 1.5 Au + Au t = 3 fm/c E/A = 15 MeV < b < 1 fm Gogny Gogny-AS dn/dρ [fm 3 ] 12 fm/c r [fm] r [fm] ρ [fm -3 ] 28/9/26 p.3/18

4 Probes of High Density Matter Compressed state ρ n p Observables π /π + ratio π /π + ( ρn ρ p ) 2, R = (π /π + )( 124 Sn Sn) (π /π + )( 112 Sn Sn) OR ρ r n p r Yong et al., PRC73(26)3463. Difference of neutron flow and proton flow 28/9/26 p.4/18

5 Neutron and Proton Flows n p Double ratio of neutron-proton spectra R = (Y n/y p )( 124,132 Sn Sn) (Y n /Y p )( 112 Sn Sn) OR ρ n p r E/A = 5 MeV E/A = 4 MeV r Li et al., PLB(26) /9/26 p.5/18

6 Experiments Lowdensity EOS Shetty et al., PRC 76 (27) 2466 Observables: Isoscaling, Isospin diffusion, Neotron/proton emission ratio, Giant resonances, Binding energy and neutron skin, Neutron star calc.,... 28/9/26 p.6/18

7 Approach to measure EOS t = fm/c t ns EOS ρ n /ρ p... ρ n /ρ p EOS π /π + ρ n /ρ p 28/9/26 p.7/18

8 Clusters are important Many experimental observables (to probe high and low densities) are related to clusters and fragments. (t/ 3 He, isoscaling etc) Clusters and fragments are the main part of the total system. Li B Be p α d 3 He t 25 MeV/u For example, four nucleons in the gas at T = 1 MeV. Uncorrelated: E = 3 T 4 = 6 MeV 2 α cluster: E = 28.3 MeV + 3 T = 13.3 MeV 2 Can we satisfy with coalescence? 28/9/26 p.8/18

9 VUU Equation VUU Equation BUU Equation, BNV Equation) f t = h r f p h p f r + I coll Collision term dp 2 I coll = (2π ) 3 ( ) { dσ dω v f (r, p 3, t) f (r, p 4, t) [ 1 f (r, p, t) ][ 1 f (r, p 2, t) ] dω v f (r, p, t) f (r, p 2, t) [ 1 f (r, p 3, t) ][ 1 f (r, p 4, t) ]} Gain term p p 2 Loss term p 3 p 4 p 3 p 4 p p 2 p, p 2, Ω p 3, p 4 (Energy and momentum conservation) v = p p 2 /M 28/9/26 p.9/18

10 Antisymmetrized Molecular Dynamics (AMD) Initial State AMD wave function [ { ( Φ(Z) = det exp ν rj Z ) i 2 } χαi (j) ] ij ν iĥt Branching Z i = νd i + i 2 ν K i d Antisymmetrization s ν : Width parameter = (2.5 fm) 2 x c + c + c χ αi : Spin-isospin states = p, p, n, n p Stochastic equation of motion for the wave packet centroids Z: d dt Z i = {Z i, H} PB + (NN collisions) + Z i (t) One-body motion in the mean field Two-nucleon collisions 28/9/26 p.1/18

11 Clusters in Collision Dynamics Extension of AMD to respect cluster correlations Cluster formation Propagation Breakup t d d h α α d α d t Low density EOS Horowitz and Schwenk, NPA776 (26) 55. x α virial LS Shen T=2 MeV 4 8 S E [MeV] T=2 MeV T=4 MeV T=8 MeV n b [fm -3 ] α-particle fraction n b [fm -3 ] Symmetry energy 28/9/26 p.11/18

12 Time evolution of number of clusters Number of nucleons in correlated clusters Sn Sn E/A = 5 MeV < b < 2 fm Nucleon Number 15 1 Gogny Non-clustered (2N) 5 (3N) (4N) Time [fm/c] 28/9/26 p.12/18

13 Effects of cluster correlations 4 Ca + 4 Ca, E/A = 35 MeV, filtered violent collisions w/o cluster correlations with cluster correlations experiment S- H S- H S-K H Na-P C-Ne Li-B He Na-P C-Ne Li-B He Na-P C-Ne Li,Be,B He p 6.7 d 1.5 t.3 3 He.3 α 2.7 p 4.4 d 1.8 t.5 3 He.6 α 5. 28/9/26 p.13/18

14 Low density matter (Liquid-gas phase transition) E 8A MeV W(E) W(E) e 2 ae ν 1 MeV MeV 36 Ar V = 4 3 π(9 fm)3 (?) 28/9/26 p.14/18

15 Equilibrium ensembles and caloric curves Microcanonical ensemble Simply solve the time evolution for a long time Total energy: E Volume: V = 4 3 πr3 (reflections at the wall of container) Neutron and proton numbers: N = 18, Z = 18 Temperature T(E, V) and Pressure P(E, V) V/V T [MeV] 1 5 T 2 / E * /A [MeV] P=.4 E * /A=T 2 /8 E/A=(3/2)T E * /A [MeV] 25, fm/c 13 combinations of (E, V) 3 CPU hours 28/9/26 p.15/18

16 Comparison of reaction and equilibrium 4 Ca + 4 Ca, E/A = 35 MeV, b = T. Furuta, Doctor Thesis, Tohoku University, 27. { States at the reaction time t } =? = = Equilibrium ensemble(e, V, A) M(Z) t = 1 fm/c 14 fm/c 18 fm/c 3 fm/c Multiplicity E*/A=6.5MeV, V=3.9V t=1fm/c E*/A=6.1MeV, V=6.2V t=14fm/c E*/A=5.7MeV, V=6.6V t=18fm/c E*/A=5.3MeV, V=13.2V t=3fm/c Z Z Z Z 1 E /A <E*/A> [MeV] E*/A=6.5MeV, V=3.9V t=1fm/c E*/A=6.1MeV, V=6.2V t=14fm/c E*/A=5.7MeV, V=6.6V t=18fm/c E*/A=5.3MeV, V=13.2V t=3fm/c A A A A 28/9/26 p.16/18

17 Summary AMD AMD 28/9/26 p.17/18

18 High Density EOS and Flow Transverse Flow Eliptic Flow Danielewicz et al., Science 298(22)1592. tan φ = p x /p y cos 2φ : 28/9/26 p.18/18

global global mass region (matter ) & (I) M3Y semi-microscopic int. Ref.: H. N., P. R. C68, ( 03) N. P. A722, 117c ( 03) Proc. of NENS03 (to be

Gogny hard core spin-isospin property @ RCNP (Mar. 22 24, 2004) Collaborator: M. Sato (Chiba U, ) ( ) global global mass region (matter ) & (I) M3Y semi-microscopic int. Ref.: H. N., P. R. C68, 014316