Symmetry Energy Effects in Heavy Ion Collisions

1. Symmetry Energy Effects in Heavy Ion Collisions Che-Ming Ko Texas A&M University • Nuclear symmetry energy • Isospin-dependent transport model (IBUU) • Nucleon emission source • Two-nucleon correlation functions • Light clusters production Collaborators: Lie-Wen Chen, Vincenzo Greco, Bao-An Li (Arkansas State University) PRL 90, 162701 (2003); nucl-th/0302068; 0305036; 0306032

2. Nuclear symmetry energy EOS of asymmetric nuclear matter Symmetry energy Symmetry energy coefficient theoretical values -50 to 200 MeV Slope theoretical values -700 to 466 MeV Curvature experimental values

3. Symmetry energy Pressure Potential

4. Isospin-dependent transport model (IBUU) NN Cross sections Isospin-independent potential In-medium cross sections from Dirac- Brueckner approach based on Bonn potential (Li, PRC 48, 1702 (93))

5. using free cross sections, soft symmetry potential • 10,000 events with 100 test particles for a physical nucleon Formation of ring structure in transverse plane

6. Quadrupole moment of momentum distribution Maximum density More sensitive to incompressibility of isospin-independent part of EOS than stiffness of symmetry energy Thermal equilibrium after about 40 fm/c

7. Nucleon emission times • Longer emission duration for lower momentum nucleons • Earlier emissions for stiffer symmetry energy • Larger separation in neutron and proton emission times for softer symmetry energy • High momentum nucleons emitted earlier than low momentum ones

8. Size of nucleon emission source • Broader emission source size distribution for lower momentum nucleons • Source size larger in transverse direction than in longitudinal direction • Larger emission source size for softer symmetry energy

9. Momentum distributions of emitted nucleons • Peak momentum lower for stiffer symmetry energy • Symmetry energy effect larger for low momentum protons than neutrons

10. Effects of isocalar and Coulomb potentials, and NN cross sections • Nucleon emission rates insensitive to incompressibility of isospin- independent part of EOS • In-medium cross sections enhance slightly nucleon emission rate at later stage of collisions • Coulomb potential shortens slightly proton emission time

11. Two-nucleon correlation functions : emission function, i.e., probability for emitting a nucleon with momentum p from the space-time point x=(r,t) : relative wave function of two nucleons Correlation After Burner: including final-state nuclear and Coulomb interactions (Scott Pratt, NPA 566, 103 (1994))

12. Two-nucleon correlation functions in central collisions of 52Ca+48Ca at 80 AMeV • Correlations of low momentum or very energetic pairs insensitive to symmetry energy • Symmetry energy effects on high momentum pairs about 20-30% with stiffer one giving stronger correlations

13. Time evolution of two-nucleon correlation functions Effects of isoscalar potential and NN cross sections Symmetry energy effect appears at ~50 fm/c when density is slightly below normal density Effects less than 10% for both isoscalar potential and NN cross sections

14. Impact parameter and incident energy dependence Strength of correlation functions increases with incident energy, and symmetry energy effect remain similar Symmetry energy effect decreases with impact parameter with stiffer one reduced more

15. Two nucleon correlation functions in central collisions of 132Sn+124Sn at 80 AMeV • Symmetry energy effect oncorrelation functions of high momentum pairs of pp and pn are reduced by ~2 compared to 52Ca+48Ca collisions due to larger Coulomb effect • Effect on nn correlation function similar to 52Ca+48Ca collisions, i.e., ~20%

16. Light clusters production Coalescence model : nucleon phase-space distribution function : Wigner phase-space distribution function for clusters G: statistical factor; 3/8 for deuteron, 1/3 for triton and 3He

17. Wigner phase-space distribution function for deuteron Hulthen wave function

18. Wigner phase-space distribution function for triton and 3He Gaussian wave function Jacobi coordinates b=1.61 fm for triton and 1.74 for 3He correct radii

19. Light clusters production from collisions of symmetric nuclei • Deuteron energy spectra reproduced • Low energy tritons slightly underestimated • Inverse slope parameter of 3He underestimated; probably due toneglect of • larger binding effect • stronger Coulomb effect

20. Yields and energy spectra of light clusters • Symmetry energy effects are about 51%, 73%, and 100% on deuteron, triton and 3He yields with stiffer one producing more • Symmetry energy effects stronger on lower energy light clusters • Effects of isoscalar potential and NN cross sections small

21. Isobaric yield ratio of t/3He • Stiffer symmetry energy gives smaller t/3He ratio • With increasing kinetic energy, t/3He ratio increases for stiff symmetry energy but slightly decreases for soft symmetry energy

22. Impact parameter and incident energy dependence Symmetry energy effects decrease with increasing incident energy but only slightly with increasing impact parameter

23. Light clusters production in central collisions of 132Sn+124Sn at 80 AMeV Symmetry energy effects similar to 52Ca+48Ca collisions: ~ 53%, 74% and 120% for d, t, and 3He

24. Emission times of light clusters Average emission time earlier for heavier ones

25. Two-deuteron correlationfunctions Include final-state repulsive nuclear s-wave and Coulomb interactions Anticorrelations of pairs of high total momentum are affected by symmetry energy with stiffer one giving a larger strength, about 20% at q=40 MeV/c

26. Two-triton or two-3He correlation functions Include only final-state repulsive Coulomb interaction Stiffer symmetry energy gives stronger anticorrelation but the effect is smaller than in two-nucleon and two-deuteron correlation functions

27. Summary Density dependence of nuclear symmetry energy affects dynamics of heavy ion collisions induced by neutron-rich nuclei at intermediate energies with stiffer density dependence giving • Earlier emissions of nucleons and light clusters • Stronger two-nucleon correlation functions • Larger light clusters production • Larger ratio of high energy tritons and 3He • Larger light clusters correlation functions Light clusters production and two-particle correlation functions are useful probes of nuclear symmetry energy Effects of momentum-dependence? in progress