1. Equation-of-State (EOS) of Nuclear Matter with Light Cluster Correlations Equation-of-State (EOS) of Nuclear Matter with Light Cluster Correlations XLVIII Internat. Winter Meeting in Nuclear Physics, Bormio, Italy, 25-29.01.2010, In Memoriam of Ileana Iori Hermann Wolter Ludwig-Maximilians-Universität München

2. Construct an EOS for use in astrophysics: neutron stars and supernovae wide parameter range: proton fraction Large charge asymmetry: thus investigation of symmetry energy 1.include light cluster correlations at low density taking into account medium dependence 2.Consistent with experimental results from heavy ion collisions 3.constitutes a unified approach for the EOS for a very wide range of densities and temperatures Main points: in collaboration with: Stefan Typel, GSI,Darmstadt and Techn. Univ. Munich, Germany Gerd Röpke, Univ. Rostock, Germany David Blaschke, Th. Klähn, Univ. Wroclaw, Poland J.B. Natowitz, A. Bonasera, S. Kowalsky, S. Shlomo, et al., Texas A&M, College Station, Tx, USA

3. The Equation of State of Nuclear Matter in diff. theor. Models Symmetric matter: High density behavior constrained by heavy ion collisions, in particular Kaon production reason for differences in the models: Short range tensor force (B-A.Li) Zero density limit? Neutron matter: no correlations Symmetric matter: most bound symmetric nuclear cluster  alpha particle  e~7 MeV  56Fe  e~12 MeV (without Coul) Neutron matter: large variations in different models Symmetry enery is finite at low densities! Not contained in mean field approaches C. Fuchs, H.H. Wolter, EPJA 30(2006)5

4. Astrophysics: Neutron stars and Supernovae E sym      MeV)     1 230 Asy-stiff Asy-soft Asy-superstiff Symmetry Energy: Importance at different density ranges Finite symmetry energy at zero density due to clust- ering effects p, n High density: HIC at relativistic energies: Differences in proton/neutron or light isobaric cluster flow Ratio of isospin partners of produced particles:  ; K + /K 0  Wolfgang Trautmann‘s talk heavy ion collisions in the Fermi energy regime Fragmentation phenomena: Isospin fractionation and migration

5. I.7 Importance of Symmetry energy: Supernovae and neutron stars  Supernova evolution: range of densities and temperatures and asymmetries: Typical neutron stars

6. EOS for astrophysical processes: wide range of conditions: global approach needed! Low densities: 2-, 3-,..many body correlations important. Bound states as new particle species. Change of composition and thermodyn- properties High densities (around saturation): homogeneous nuclear matter, mean field dominates In between: Liquid-gas phase transition at low temperatures, Inhomogeneous phases (lattice structures)  Approach necessary, that interpolates reliably commonly used EOS‘s: Lattimer-Swesty, NPA 535 (1991): Skyrme-type model, Liquid Drop modelling of finite nuclei embedded in nucleon gas Shen, Toki et al., Prog. Theor. Phys. 100 (1998): RMF model (TM1),  -particle with excluded volume procedure Horowitz, Schwenk, NPA776 (2006): virial expansion, n,p,  ‘s, using experimental information on bound states and phase shifts: exact limit for low density improvements: ( S.Typel, G. Röpke, et al., PRC 2010, arxiv 0908:2344) -medium effects on light clusters, quantum statistical approach -realistic description of high density matter (DD-RMF)

7. Nuclear Statistical Equilibrium (NSE): Mixture of ideal gases for each species (zero density limit) Virial expansion, expansion of in powers of fugacities Beth-Uhlenbeck, Physica 3 (1936) interactions included Energies and phase shifts from experiment: Horowitz-Schwenk, model independent, but only valid at very low densities, no medium dependence Thermodynamical GF approach (M. Schmidt, G. Röpke, H. Schulz, Ann. Phys. 202 (1990) self energy shifts, blocking effects (melting at Mott density), proper statistics, E k (P,T,  ), and generalized phase shifts Parametrization in density and temperature Needs quasiparticle energies  generalized mean field model Light clusters : Theoretical approaches:

8. Generalized Relativistic Mean Field Model with Light Clusters Unified approach for nucleon and light cluster degrees of freedom degrees of freedom: fermions bosons mesons Lagrangian DD : density dependent RMF, S. Typel, PRC71 (2005) mass shifts of the clustersnucleon self energies with „rearrangement terms“ coupling of nucleons to clusters from density dependent coupling

9. Summary of theoretical approach: Quantumstatistical model (QS) -Includes medium modification of clusters (Mott transition) -Includes correlations in the continuum (phase shifts) -needs good model for quasi-particle energies in the mean field -In principle also possible for heavier clusters Generalized Rel. Mean Field model (RMF) -Good description of higher density phase, i.e. quasiparticle energies -Includes cluster degrees of freedom with parametrized density and temperature dependent binding energies -no correlation in the continuum -Heavier clusters treated in Wigner-Seitz cell approximation (single nucleus approximation) Global approach from very low to high densities S.Typel, G. Röpke, et al., PRC 2010, arxiv 0908:2344

10. Particle Fractions very low density: p,n Increasing density: clusters arise: deuteron first, but then  dominates Mott density: clusters melt, homogeneous p,n matter; here heavier nuclei (embedded into a gas) become important, not yet fully implemented Dependence on temperature ( T=0(2)20 MeV ) symmetric matter  0) Thin lines: NSE, i.e. without medium modifications of clusters (melting at finite densities) proton deuteron 3 He tritonalpha S.Typel, G. Röpke, et al., PRC 2010, arxiv 0908:2344

11. Comparison to other approaches: alpha particle fractions Schwenk-Horowitz (black dash-dot; virial expansion with experimental BE and phase shifts for nucleons and alpha): exact for n  0, but no disappearance of clusters for higher densities Nuclear Statistical Equilibrium (NSE) (green, dotted): decrease at higher densities because of heavier nuclei, but no medium modifications (melting) Shen-Toki (blue, dashed; RMF for p,n, heavy nuclei in Wigner-Seitz approximation in a gas of p,n, , excluded volume method): medium modifications empirical,  survive very long Generalized RMF (red, solid: coupled RMF approach for p,n,d,t,h,  medium dep. BE from QS approach, no heavy nuclei): strong deuteron correlations suppress alpha at higher densities Quantum Statistical (orange, dashed; medium modified clusters consistently in cluding scattering contributions): increases  fraction

12. Heavier clusters (nuclei) in the medium: Our approach: QS+RMF Hempel, Schaffner-Bielich (arXiv 0911:4073): NSE, with excluded volume with procedure Calculation in RMF of heavy cluster in Wigner-Seitz cell in beta-equilibrium T=5 MeV, b=0.3

13. Equation of state: pressure vs. density RMFQS  NSE (thin lines) low density limit but breaks down already at small densities  Differences between approaches: too strong cluster effects in RMF, additional minima in QS  Regions of instability: phase transitions between clusterized und homogeneous phases

14. Equation of State (RMF): With (solid) and without (dashed) clusters): reduction at low and increase at intermediate densities, increase of critical temperature Maxwell construction for phase coexistence Coexistence region (left) and phase transition line(right) RMF w/o clusters (blue) RMF with clusters (red) QS (green)

15. Thermodynamical properties: symmetric nuclear matter, RMF approach, NSE (thin lines), temperature dependence (increasing from black to purple) Free energyInternal energyentropy

16. Symmetry energy: with (solid) and without (dashed) clusters Usually: but E A not quadratic for low temperatures with clusters. Thus use:

17. et al., Can this be measured?? 64 Zn+( 92 Mo, 197 Au) at 35 AMeV Central collisions, reconstruction of fireball Determination of thermodyn. conditions as fct of v surf =v emission -v coul ~time of emission with specified conditions of density and temperature:  temperature: isotope temperatures, double ratios H-He   densities  p,  n, from yield ratios and bound clusters  Isoscaling analysis (B.Tsang, et al., )  Free symmetry energy Isoscaling coefficients  and 

18. J. Natowitz, G, Röoke, et al., nucl-th/0908.2344, subm. PRL Comparision of low-density symmetry energy to experiment: F sym E sym Parametrization of nuclear symmetry energy of different stiffness (momentum dependent Skyrme-type) (B.A. Li) Quantum Statistical model, T=1,4,8 MeV) Single nucleus approx. (Wigner- Seitz), RMF

19. Equation-of-State with  medium dependent light clusters  consistent description of matter, light clusters (and heavy clusters)  generalized RMF and QS models combined  for use in supernova calculations (tables)  finite value of symmetry energy at low density Outlook  better inclusion of heavier clusters  two-dimensional phase transition construction for asymmetric nuclear matter Equation-of-State with  medium dependent light clusters  consistent description of matter, light clusters (and heavy clusters)  generalized RMF and QS models combined  for use in supernova calculations (tables)  finite value of symmetry energy at low density Outlook  better inclusion of heavier clusters  two-dimensional phase transition construction for asymmetric nuclear matter Summary: Thank you!

20. backup

21. Nuclear statistical Equilibrium (NSE) mixture of ideal gases, no interaction, particle species do not disappear at higher density Virial expansion: Expansion of in powers of fugacities e.g. for p,n,  (intersction terms in 2nd virial coefficients) Beth_Uhlenbeck: Quantum generalization in terms of binding energies and phase shifts Horowitz, Schwenk, taken from experiment: model independent at zero density Approaches for low density matter :

22. Light clusters : 2-, 3-, 4-…body correlations: Theoretical approaches: 1. Virial expansion, expansion of in powers of fugacities Beth-Uhlenbeck, Physica 3 (1936) Horowitz-Schwenk: neutrons, protons, alpha Energies and phase shifts from experiment,  model independent, but only valid at very low densities (cluster fractions increase monotonously); but benchmark for other approaches at low density

23. Light clusters : 2-, 3-, 4-…body correlations: Theoretical approaches: 2. Quantum statistical approach, medium effects included in thermodynamical GF approach (M. Schmidt, G. Röpke, H. Schulz, Ann. Phys. 202 (1990) Nucleon densities (  =p,n), including bound states with continuum correction Medium modification of bound states: Features: self energy shifts, blocking effects (melting at Mott density), proper statistics Result: Cluster binding energies as fct. of density and temperature Limit: Nuclear Statistical Equilibrium (NSE): use free cluster BE for ound states

24. Isoscaling Analysis B. Tsang, et al., PRL 86 (2001) 5023 -experimentally well obeyed in statistically dominated processes -apply here to results of BNV calculation for Sn+Sn collisions (central) For isotope yields before (primary fragments) and after (secondary fragments) evaporation grand canonical ensemble Isoscaling coefficients  and 

26. Equation-of-State (EOS) of Nuclear Matter with Light Cluster Correlations Equation-of-State (EOS) of Nuclear Matter with Light Cluster Correlations 28th Internat. Workshop on Nuclear Theory, Rila Mountains, Bulgaria, June 22-27, 2009 in collaboration with: Stefan Typel, GSI,Darmstadt and Techn. Univ. Munich, Germany Gerd Röpke, Univ. Rostock, Germany David Blaschke, Th. Klähn, Univ. Wroclaw, Poland Hermann Wolter Ludwig-Maximilians-Universität München