Towards the Equation of State of Dense Asymmetric Nuclear Matter

1. Towards the Equation of State of Dense Asymmetric Nuclear Matter Outline • Present constraints on the EOS. • Relevance to dense astrophysical objects: • Probes of the asymmetry term of the EOS. • Current investigations: • Isoscaling—scaling of isotope ratios • Sensitivity to EOS • Probing high density matter. • Summary and Outlook.

2. Isospin Dependence of the Nuclear Equation of State Brown, Phys. Rev. Lett. 85, 5296 (2001) PAL: Prakash et al., PRL 61, (1988) 2518. Colonna et al., Phys. Rev. C57, (1998) 1410. E/A (,) = E/A (,0) + d2S() d = (n- p)/ (n+ p) = (N-Z)/A • The density dependence of asymmetry term is largely unconstrained. • Pressure, i.e. EOS is rather uncertain even at 0.

3. Constraints on S() from microscopic models Total Symmetry Energy 50 • Microscopic theory provides incomplete guidance regarding the extrapolation away from saturation density. 45 I.Bombaci (2000) 40 35 ) MeV 30 r BHF Varia- tional ParisBL S( 25 AV14 20 UV14 15 AV18 10 0 0.1 0.2 0.3 0.4 0.5 -3 r (fm )

4. Nucleus-nucleus collision provide the only terrestrial means to probe nuclear matter under controlled conditions. • What has been learned about the EOS of dense symmetric matter? • What is the relevance to dense objects? • How can one probe the asymmetry term?

5. Determination of EOS from nucleus-nucleus collisionsWhat is known about the symmetric matter EOS? • Measurements are constraining symmetric matter EOS at >20. • Observables: transverse, elliptical flow. Danielewicz et al., (2002) Danielewicz et al., (2002) • Prospects are good for improving constraints further. • Relevant for supernovae - what about neutron stars?

6. Uncertainty due to the density dependence of the asymmetry term is greater than that due to symmetric matter EOS. Macroscopic properties: Neutron star radii, moments of inertia and central densities. Maximum neutron star masses and rotation frequencies. Proton and electron fractions throughout the star. Cooling of proton-neutron star. Thickness of the inner crust. Frequency change accompanying star quakes. Role of Kaon condensates and mixed quark-hadron phases in the stellar interior. Extrapolation to neutron stars E/A (, ) = E/A (,0) + 2S()  = (n- p)/ (n+ p) = (N-Z)/A1 Danielewicz et al., (2002) Symmetry term influences:

7. “First order”: depend on isospin of detected particles. Prequilibrium particle emission. Asymmetry of bound vs. emitted nucleons. Transverse flow. Isospin dependencies of pion production. “Second order”: do not depend on isospin of detected particles. Probing the asymmetry term F2 F1 stiff F3 soft F1=2u2/(1+u) F2=u F3=u u = Observables: • Sign of mean field opposite for protons and neutrons. • Shape is influenced by incompressibility. • .

8. Studies of asymmetry term at 0. soft asymmetry term Bao-An Li (2000) stiff asymmetry term • Direct measurements of n vs. proton emission rates and transverse flows - Probes the pressure from asymmetry term at saturation density and below. • Predictions of transport theory: • Similar effects in t vs. 3He flow, etc? • Measurements will be performed at NSCL in July 2003.

9. Studies of asymmetry term at 0. multifragment final state bound matter pre-equilbrium nucleons • Complimentary measurements of the asymmetry of bound matter. • Experimental study: • Central 112Sn+112Sn, 124Sn+124Sn collisions at E/A=50 MeV. • Measure isotopic distributions of multifragmentation products.

10. Experimental investigations of isospin effects in fragmentation Liu et al., (2003) ycm 112,124Sn+112,124Sn, E/A =50 MeV Miniball with LASSA array Provides good coverage of central collisions:

11. Expected dependencies on asymmetry term asy-stiff (F1) asy-soft (F3) N/Zres=Ntot/Ztot more symmetric dense region neutron-rich emitted particles N/ZemNtot/Ztot For a neutron rich system at r<r0: BUU predictions forcentral 124Sn+ 124Sn (N0/Z0=1.48) collisions at E/A=50 MeV

12. Comparison of isotopic distributions exp. (N/Z=1.44) (N/Z=1.23) Asy-stiff F1 Asy-soft F3 final 124Sn+124Sn Central Collisions at E=50AMeV Tan (2001) • ISMM decay of BUU residue: b=1fm, E*/A=4MeV, /0=1/6 • There is a sensitivity to the density dependence of the asymmetry term of the EOS.

13. Scaling behavior of Isotopic Ratios R21=Y2/ Y1 Relative Isotope Ratios in SMM models • Compact parameterization of total isospin dependence. • Factorization of yields into p & n densities Xu PRL (2000) Tsang PRC (2001) • Slopes are insensitive to sequential decay calculations • Slopes are sensitive to masses, which are inaccurately calculated in many models.

14. What do the isoscaling parameters reveal about the asymmetry term?

15. Relative Neutron(Proton) Densities • and are not sensitive to secondary decays. • Sdfsdf increases more rapidly than (N/Z)0 fractionation. • Comparison favors the stiffer asymmetry term. • Similar conclusions obtained from comparisons of mirror nuclei. Tan et al. PRC (2001)

16. Model dependence of fragment isotopic distributions. EES model assumes that fragments are at normal density: EES model assume that residue is at low density: • Few precise calculations: • ISMM, EES, AMD and SMF. • Existing calculations break into two groups: • Bulk multifragmentation models predict that asy-soft EOS favors more symmetric fragments: • One important surface emission model (EES model), however, predicts that asy-soft EOS favors neutron rich fragments This occurs because such models assume that fragments are at similar low freezeout density:

17. Predictions of EES model Use scaling to simplify representation: • Theoretically: • Separation energies depend on density dependence asymmetry term: • Strong influence of symmetry term on fragment isotopic ratios. • trends are opposite to those of the equilibrium SMM approach. Tsang et al., PRL (2001)

18. Predictions of EES model Use scaling to simplify representation: • Theoretically: • Separation energies depend on density dependence asymmetry term: • Strong influence of symmetry term on fragment isotopic ratios. • trends are opposite to those of the equilibrium SMM approach. Tsang et al., PRL (2001) Data

19. Test: Measurements of isotopically resolved spectra • Energy spectra reflect the cooling dynamics. • high energies reflect higher temperatures, earlier times • Emission rates are temperature (and therefore time dependent). • More strongly bound particles are preferentially emitted at low temperature. (B=BEpar-BEdau-BEf) • Data display the expected trends, which were previously observed only for helium and beryllium isotopes. Eem (MeV) Liu et al., (2003) Data

20. Expectation: Kinetic energy reflects simply the thermal and collective contributions (neglecting recoil effects). Actual trends suggest cooling phenomena that can be reproduced by EES model calculations in which fragments are emitted from surface of expanding and cooling system. New results supporting the surface emission picture Data EES 112Sn+ 112Sn, E/A=50 MeV central collisions Liu, Friedman, Souza (2003) SMM carbon

21. Current Status: studies of asymmetry term for 0 • n vs. p emission rates: all models predict consistent qualitative trends. • Such measurements provide the least ambiguous information about the density dependence of the asymmetry term. (expect first results in early 2004) • Fragment isocaling parameters: there is a model dependence for the sensitivity to the asymmetry term. • Present constraints on the asymmetry term are model dependent. • The combination of n vs. p rates with isoscaling data will rule out at least one commonly employed multi-fragmentation model and thereby further our understanding of the liquid-gas phase transition.

22. Vary isospin driving forces by changing the isospin of projectile and target. measure the scaling parameter for projectile decay. isospin equilibrium is not achieved in peripheral collisions. Since, , renormalize to symmetric collisions: New observable: isospin diffusion in peripheral collisions Tsang et al., (2003) symmetric system no diffusion Measure scaling parameter asymmetric system weak diffusion asymmetric system strong diffusion neutron rich system proton rich system Rami et al., PRL, 84, 1120 (2000)

23. Vary isospin driving forces by changing the isospin of projectile and target. measure the scaling parameter for projectile decay. isospin equilibrium is not achieved in peripheral collisions. Since, , renormalized to symmetric collisions: New observable: isospin diffusion in peripheral collisions Tsang et al., (2003) symmetric system no diffusion Measure scaling parameter asymmetric system weak diffusion asymmetric system strong diffusion neutron rich system proton rich system Rami et al., PRL, 84, 1120 (2000)

24. Vary isospin driving forces by changing the isospin of projectile and target. measure the scaling parameter for projectile decay. isospin equilibrium is not achieved in peripheral collisions. Since, , renormalize to symmetric collisions: New observable: isospin diffusion in peripheral collisions Tsang et al., (2003) symmetric system no diffusion Measure scaling parameter asy- stiff asymmetric system weak diffusion asy- soft asymmetric system strong diffusion neutron rich system proton rich system Rami et al., PRL, 84, 1120 (2000)

25. Studies of asymmetry term at >0. • High density behavior of asymmetry terms can be group into two classes: i.e. increasing or decreasing with density. Bao-An Li NPA 2002 stronger density dependence (asy-stiff) weaker density dependence (asy-soft) typical of many Skyrme EOS’s or some variational calculations (UV14+UV11) Influences significantly the inner structure of neutron star

26. Observables: pion yields • Softer asymmetry term favors neutron rich dense regions and larger relative -/+ yield ratios: Bao-An Li NPA 2002 Bao-An Li NPA 2002 dashed: asy-soft neutron rich increasing -/+ solid: asy-stiff neutron deficient

27. Conclusion • Significant constraints on symmetric matter EOS have been obtained. • Density dependence of symmetry energy can be examined experimentally. • Observed isoscaling laws • Conclusions from fragmentation work are model dependent: • BUU-SMM favors 2 dependence of S(). • SMF favors 2 dependence but isotope distributions are poorly reproduced • EES favors 2/3 dependence of S(). • Signals have been identified for measurements of asymmetry term at higher densities 0-3.50 : • Isospin dependence of n vs. p flow, pion yields.

28. Acknowledgements

29. These equations of state differ only in their density dependent symmetry terms. Clear sensitivity to the density dependence of the symmetry terms Neutrino signal from collapse. Feasibility of URCA processes for proto-neutron star cooling if fp > 0.1. p+e- n+ n p+e-+ Some examples Neutron star radii: Cooling of proto-neutron stars:

30. Test: Measurements of isotopically resolved spectra • Energy spectra reflect the cooling dynamics. • high energies reflect higher temperatures, earlier times • Emission rates are temperature (and therefore time dependent). • More strongly bound particles are preferentially emitted at low temperature. (B=BEpar-BEdau-Bem) Eem (MeV) Data

31. Predictions of EES model • Theoretically: • Separation energies depend on density dependence asymmetry term: • Strong influence of symmetry term on fragment isotopic ratios. • trends are opposite to those of the equilibrium SMM approach. Use scaling to simplify representation: Tsang et al., PRL (2001) =0 Data =1

32. Predictions of EES model • Theoretically: • Separation energies depend on density dependence asymmetry term: • Strong influence of symmetry term on fragment isotopic ratios. • trends are opposite to those of the equilibrium SMM approach. Use scaling to simplify representation: Tsang et al., PRL (2001)

33. Isoscaling laws for isotopic distributions feeding correction • Basic trends from Grand Canonical ensemble: • Yields  relevant term in partition function. • Ratios to reduce sensitivity to secondary decays: • Scaling parameters C, ,

34. Pressure and collective flow dynamics • The blocking by the spectator matter provides a clock with which to measure the expansion rate. pressure contours density contours

35. n vs. p flow differences dashed: asy-soft • The exploration of the density dependent asymmetry will be an important objective at RIA. Bao-An Li NPA 2002 differences between neutron and proton flow values solid: asy-stiff

36. Figure shows the predicted ratio of neutron and proton center of mass spectra for two different asymmetry terms. There is an uncertainty regarding the expected magnitude of the effect. Open question whether these differences stem from differences in the models (QMD vs. BUU), the numerics of the simulations, or differences in the asymmetry terms. More theoretical work quantifying the signature is needed. Bao-al Li: spectra are in the lab frame and integrated over angle. Above spectra are in the C.M. frame, angular cut (?).... Magnitude of the effect? Zhuxia Li 2002 QMD calc.

37. Current Status: studies of asymmetry term for 0 • n vs. p emission rates: all models predict consistent qualitative trends. • Such measurements provide the least ambiguous information about the density dependence of the asymmetry term. (expect first results in early 2004) • Fragment isocaling parameters: there is a model dependence for the sensitivity to the asymmetry term. • Present constraints on the asymmetry term are model dependent. • The combination of n vs. p rates with isoscaling data will rule out at least one commonly employed multi-fragmentation model and thereby further our understanding of the liquid-gas phase transition.

38. Additional statistical mechanisms: Deep-inelastic Evaporation: Note that these simple scaling laws require the two systems to be at a common temperature. There are also generalized scaling laws systems at different temperatures. Isotopic scaling as a general phenomenon Tsang et al., PRL (2001)

39. Isoscaling laws provide a straightforward scaling of isospin dependence of isotope distributions : Other three reactions are scaled by applying the isoscaling parameters to the 124Sn+124Sn data. Parameters depend linearly on =(N-Z)/A Extrapolation of isotope distributions Y2 =R21Y1

40. Scaling behavior of Isotopic Ratios R21=Y2/ Y1 • Note: you can reduce the isotopic lines to a single line by removing the Z dependence as follows: • Could do the same thing for the isotonic dependence if desired. Xu PRL (2000)

41. Yield Ratios of Mirror Nuclei Ratios of Mirror nuclei excited after decay Tan et al., PRC (2001) • Comparison favors the stiffer asymmetry term.

42. Scaling behavior of Isotopic Ratios R21=Y2/ Y1 • Note: you can reduce the isotopic lines to a single line by removing the Z dependence as follows: • Could do the same thing for the isotonic dependence if desired. Xu PRL (2000)

43. Theoretically: Separation energies depend on density dependence asymmetry term: Strong influence of symmetry term on fragment isotopic ratios. trends are opposite to those of the equilibrium SMM approach. Predictions of EES model Use scaling to simplify representation: Tsang et al., PRL (2001) =0 Data =1

44. One Key observable: n.vs.p emission rates asy-stiff (F1) asy-soft (F3) For a neutron rich system: N/Zres=Ntot/Ztot more symmetric dense region neutron-rich emitted particles N/ZemNtot/Ztot

45. “First order”: depend on isospin of detected particles. Prequilibrium particle emission Asy_soft: <En> > <Ep> * Asy_stiff: <En>  <Ep>* Asymmetry of bound vs. emitted nucleons. Asy_soft: (N/Z)EM>N/Z)BND. Asy_stiff: (N/Z)EM(N/Z)BND. Transverse flow Asy_soft: Fp  Fn * Asy_stiff: Fp > Fn * “Second order”: do not depend on isospin of detected particles. Probing the asymmetry term F2 F1 stiff F3 soft F1=2u2/(1+u) F2=u F3=u u = Observables: (neutron rich systems) • Sign of mean field opposite for protons and neutrons. • Shape is influenced by incompressibility. • . *Different for neutron deficient systems

46. One Key observable: n.vs.p emission rates asy-stiff (F1) asy-soft (F3) For a neutron rich system: N/Zres=Ntot/Ztot more symmetric dense region neutron-rich emitted particles N/ZemNtot/Ztot

47. What may be wrong with the SMF calculations? • Neglect of impact parameter averaging? • Neglect of dynamical cluster emission? • What is needed in terms of the primary distributions? • Comparison with SMM calculations suggests that widths of SMF primary distributions need to be broader and the excitation energies lower.

48. Fragment emission rates dramatically increase for r/r0 < 0.4 where fragment separation energies sF become negative. Separation energies (and therefore) rates depend on the density dependence of asymmetry term. Expanding Emitting Source (EES) Model W.A. Friedman Phys. Rev. C 42 (1990) 667. Decay of Au Nucleus 10 MeV 11 MeV 12 MeV 13 MeV 14 MeV 15 MeV Weisskopf emission rates:

49. Problems with masses in standard SMM • Effects appear small but are sufficient to change scaling parameters for primary fragments by nearly a factor of two and secondary fragments by 35%. disagreement with known masses is large Tan et al., (2002) Tan et al., (2002)

50. Isospin Dependence of the Nuclear Equation of State Brown, Phys. Rev. Lett. 85, 5296 (2001) PAL: Prakash et al., PRL 61, (1988) 2518. Colonna et al., Phys. Rev. C57, (1998) 1410. E/A (,) = E/A (,0) + 2S()  = (n- p)/ (n+ p) = (N-Z)/A • The density dependence of asymmetry term is largely unconstrained.