Neutron Number N

1. Inclusion of surface terms in symmetry asym=30-42 MeV for infinite NM Proton Number Z Neutron Number N

2. Can we determine the density dependence of the symmetry energy? • E(, d) = E(, 0)+Ssym() d2 • Isospin asymmetry: d = (n- p)/ (n+ p) Isospin Mixing and Diffusion in Heavy Ion Collisions Betty Tsang Dresden, 8/2003 The National Superconducting Cyclotron Laboratory @Michigan State University

3. What is known about the EOS of symmetric matter Danielewicz, Lacey, Lynch (2002) • Prospects are good for improving constraints further. • Relevant for supernovae - what about neutron stars?

4. The density dependence of asymmetry term is largely unconstrained. • E(, d) = E(, 0)+Ssym() d2

5. Measured Isotopic yields

6. R21=Y2/ Y1 ^ ^ Isoscaling from Relative Isotope Ratios Factorization of yields into p & n densities Cancellation of effects from sequential feedings Robust observables to study isospin effects

7. R21(N,Z)=Y2 (N,Z)/ Y1 (N,Z) ^ ^ Origin of isoscaling` • Isoscaling disappears when the symmetry energy is set to zero • Provides an observable to study symmetry energy

8. A. Ono et al. (2003) Isoscaling in Antisymmetrized Molecular Dynamical model

9. Symmetry energy from AMD a(Gogny)>a(Gogny-AS) Csym(Gogny)> Csym(Gogny-AS) Multifragmentation occurs at low density

10. EES_fragment Density dependence of asymmetry energy Strong influence of symmetry term on isoscaling =0.36  =2/3 Consistent with many body calculations with nn interactions S()=23.4(/o) Results are model dependent

11. BUU SMM Y(N,Z) R21(N,Z) P N/Z F1,F3 T Freeze-out source Data g~2 g~0.5 Sensitivity to the isospin terms in the EOS Asy-stiff term agrees with data better PRC,C64, 051901R (2001).

12. Vary isospin driving forces by changing the isospin of projectile and target. Examine asymmetry by measuring the scaling parameter for projectile decay. symmetric system: no diffusion asymmetric systems strong diffusion weak diffusion neutron rich system proton rich system New observable: isospin diffusion in peripheral collisions

13. Isoscaling of mixed systems

14. Experimental results Y(N,Z) / Y112+112(N,Z)=[C·exp(aN+ bZ)] 3 nucleons exchange between 112 and 124 (equilibrium=6 nucleons)

15. Experimental results Y(N,Z) / Y112+112(N,Z)=[C·exp(aN+ bZ)] 3 nucleons exchange between 112 and 124 (equilibrium=6 nucleons)

16. The neck is consistent with complete mixing. Isospin flow from n-rich projectile (target) to n-deficient target (projectile) – dynamical flow Isospin flow is affected by the symmetry terms of the EoS – studied with BUU model

34. Isospin Diffusion from BUU

35. Isospin Transport Ratio Rami et al., PRL, 84, 1120 (2000) x=experimental or theoretical isospin observable x=x124+124Ri = 1. x=x112+112Ri = -1. Experimental: isoscaling fitting parameter a; Y21 exp(aN+bZ) Theoretical :

37. E(, d) = E(, 0)+Ssym() d2 Ssym()  ()  BUU predictions

38. E(, d) = E(, 0)+Ssym() d2 Ssym()  ()  BUU predictions Experimental results are in better agreement with predictions using hard symmetry terms

39. Summary • Challenge: Can we constrain the density dependence of symmetry term? • Density dependence of symmetry energy can be examined experimentally with heavy ion collisions. Existence of isoscaling relations • Isospin fractionation: Conclusions from multi fragmentation work are model dependent; sensitivity to S(). • Isospin diffusion from projectile fragmentation dataprovides another promising observable to study the symmetry energy with Ri

40. Acknowledgements Bill Friedman