Neutron Number N Proton Number Z a sym =30-42 MeV for infinite NM Inclusion of surface terms in symmetry

The National Superconducting Cyclotron State University Isospin Mixing and Diffusion in Heavy Ion Collisions Betty Tsang Dresden, 8/2003  S sym (    Isospin asymmetry  n  p  n  p  Can we determine the density dependence of the symmetry energy?

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

The density dependence of asymmetry term is largely unconstrained.  S sym (  

Measured Isotopic yields

Isoscaling from Relative Isotope Ratios Factorization of yields into p & n densities Cancellation of effects from sequential feedings Robust observables to study isospin effects R 21 =Y 2 / Y 1 ^^

Origin of isoscaling`  Isoscaling disappears when the symmetry energy is set to zero  Provides an observable to study symmetry energy R 21 (N,Z)=Y 2 (N,Z)/ Y 1 (N,Z)^^

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

Symmetry energy from AMD  (Gogny)>  (Gogny-AS) C sym (Gogny)> C sym (Gogny-AS) Multifragmentation occurs at low density

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

Sensitivity to the isospin terms in the EOS Data Y(N,Z) R 21 (N,Z) SMMBUU F1,F3 N/Z P T Freeze-out source Asy-stiff term agrees with data better PRC, C64, R (2001).  ~2  ~0.5

New observable: isospin diffusion in peripheral collisions 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 weak diffusion strong diffusion neutron rich system proton rich system asymmetric systems

Isoscaling of mixed systems

Experimental results 3 nucleons exchange between 112 and 124 (equilibrium=6 nucleons) Y(N,Z) / Y (N,Z)=[C·exp(  N +  Z)]

Experimental results 3 nucleons exchange between 112 and 124 (equilibrium=6 nucleons) Y(N,Z) / Y (N,Z)=[C·exp(  N +  Z)]

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

Isospin Diffusion from BUU

Isospin Transport Ratio x=experimental or theoretical isospin observable x=x  R i = 1. x=x  R i = -1. Experimental: isoscaling fitting parameter  ; Y 21  exp(  N+  Z) Theoretical : Rami et al., PRL, 84, 1120 (2000)

BUU predictions  S sym (   S sym (  (  

BUU predictions Experimental results are in better agreement with predictions using hard symmetry terms  S sym (   S sym (  (  

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 data provides another promising observable to study the symmetry energy with R i

Acknowledgements Bill Friedman