1. Transport phenomena in heavy-ion reactions Lijun Shi NSCL MSU and Physics Department, McGill University Catania, Italy, Jan. 23, 2004

2. Page 2 Transport theory Boltzmann equation: Single particle energy Optical potential is: EOS: through total energy E optical potential U opt Transport theory

3. Page 3 Isospin diffusion coefficient D I : In the limit of weak nonequilibrium and small isospin gradient, isospin flow will be proportional to the isospin gradient Particle Flow: Isospin diffusion coef D I : v i : average velocity of particle i mean velocity: v = (  1 v 1 +  2 v 2 )/  Isospin asymmetry:  =(n 1 -n 2 )/(n 1 +n 2 ) Isospin Flow: Diffusion

4. Page 4 Numerical results: (diffusion coefficient for free Fermi gas) Diffusion coefficient Mean field enhances isospin diffusion: R = D I (with IEOS) / D I (free gas) Free gas

5. Page 5 Isospin diffusion in HIC: Basic ideas: Peripheral reactions 124 Sn+ 124 Sn, 112 Sn+ 112 Sn -- no diffusion 124 Sn+ 112 Sn, 112 Sn+ 124 Sn -- diffusion Relative change between the two systems is due to diffusion effect Measure isospin in the projectile-like region Isospin-diffusion Isospin changed Isospin Diffusion

6. Page 6 Isospin dependent Mean Field IEOS ~ diffusion coefficient (  =  /  0 ) Isospin-diffusion

7. Page 7 R i changes as a function of time (simulation) R i is a stable signal Non-diffusion effects: cancelled out R i ~ IEOS ->Diffusion effect Isospin-diffusion M. B. Tsang, et al. Projectile isospin asymmetry  from simulation  = (N-Z)/(N+Z),

8. Page 8 Compare with experiment data Exp. Data extracted from isoscaling parameter R i (exp)  0, incomplete isospin diffusion Exp. favors iso-SH type IEOS Iso-stiff type IEOS is favored, especially iso-SH NS and SKM: iso-soft type IEOS is not favored Isospin-diffusion See also discussion by M. B. Tsang

9. Page 9 Summary: Optical potential for Transport theory and simulation Asymmetric matter: symmetry energy, symmetry potential Isospin diffusion coefficient derived mean field enhances isospin diffusion Simulating isospin diffusion in HIC – compared with data, – favors iso-SH type

10. Page 10 Isospin change in the projectile-like region Basic ideas: Peripheral reactions 124 Sn+ 112 Sn, 112 Sn+ 124 Sn -- diffusion 124 Sn+ 124 Sn, 112 Sn+ 112 Sn -- no diffusion Relative change between the two system is the diffusion effect Measure the projectile- like region Isospin-diffusion app 1

11. Page 11 Isospin equilibration time scale: Consider case where D I ~ 0.41 fm. c 1) 1/t H ~ D I / (s * r), where s is the size of the spectator, r is the distance between two spectator, s~4fm, r~4fm, ==> t H ~ 39 fm/c 2) Another way is from diffusion equation with some assumption about the initial isospin profile, ==> t H ~ 35-44 fm/c BUU simulation does suggest a comparable time scale, for the system 96 Ru+ 96 Zr at 100MeV/u, b=5fm, ==> t ~ 40 fm/c. Diffusion coefficient app 2

12. Page 12 Calculate Isospin diffusion coefficient D I : 1) Start from Boltzmann equations, 2) Variation of distribution: 3) Self-consistency equation: 4) Resulting equation for D I : Diffusion coefficient  f = ( ) app 3

13. Page 13 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 ^^ app 4

14. Page 14 Mean-free-path estimate Classical Two component system model l 1 is the mean free path of particle 1 in the medium of particle 2, C 1 is the thermal velocity Estimate: T=60MeV,  =60mb, n=0.16fm -3, effective mass m=429MeV, ==> D I = 0.29fm.c app 5