A Statistical Model for a Complete Supernova Equation of State

1. A Statistical Model for a Complete Supernova Equation of State Matthias Hempel and Jürgen Schaffner-Bielich Institute for Theoretical Physics HIC for FAIR Workshop, Prerow October 13, 2009

2. Introduction • conditions in supernovae: • T = 0 – 100 MeV • r = 104 – 1015 g/cm³ • Yp=0 – 0.6 • so far only two EoS tables, based on Single Nucleus Approximation (SNA) • main difficulty below saturation density: • non-uniform nuclear matter • distribution of nuclear clusters •  Nuclear Statistical Equilibrium model which can describe the transition to uniform nuclear matter

3. Model • Nucleons: relativistic mean-field model, parameter set TMA • Nuclei: • experimental data of AWT (2003) • mass table generated with the same relativistic mean-field description TMA • intrinsic partition function of hot nuclei for excited states • Coulomb energies in Wigner-Seitz approximation [Audi et al.; 2003NPA729] [Geng et al.; 2005PTP113] [Gong et al.; 2001CoPhC136]

4. Model – Thermodynamics • model constructed from simple geometrical considerations • total canonical partition function • every baryon fills the same volume • nucleons must not be inside of nuclei V V` VA,Z  no nuclei: unmodified RMF model

5. Model – Thermodynamics • nuclei (non-relativistic, classical) must not overlap V • van-der-Waals like hard-core repulsion at short distances  no nuclei above saturation density VA,Z

6. Model – Thermodynamics • free energy density • ideal gas expressions, Coulomb contribution, filling factor for nucleons, direct excluded volume contribution • for if nuclei are present  uniform nuclear matter above saturation density always favored / nuclei have to disappear  no nuclei: unmodified RMF model

7. Model – Thermodynamics • all other thermodynamic variables also modified • e.g. resulting nucleus chemical potential •  thermodynamic consistent description of excluded volume effects • self-consistent numerical solution

8. Results – Composition • general isothermal density dependence: nucleons  light  (heavies)  uniform nuclear matter • stepwise change for T=1 MeV: narrow distributions, almost discrete change of nucleus • continuous change for larger T

9. Results – Nuclear Distributions • T=0.1: only one nucleus • T=1 MeV: neutron shell effects leading to narrow distributions with multiple peaks • T ≥ 5 MeV: exponential  power-law  u-shape as manifestation of the onset of the liquid-gas phase transition

10. Results – Light Clusters • T=1 MeV: light clusters well represented by as • T ≥ 5 MeV: large Deuteron contribution at low nB nB~ 10-2 fm-3: whole distribution of light clusters important, differences for a  behaviour not representable by as alone

11. Results – Equation of State • excellent agreement at low T across all densities • T=1 MeV: heavy nuclei lead to binding • T=5 MeV: interactions set in above 10-4 fm-3 • T=10 MeV: interactions above 10-3 fm-3 • exponential and power-law distributions (with light clusters) lead to later onset of heavies and additional binding around nB~10-2 fm-3, not reproducible by SNA • deviations appear where baryonic contribution is important

12. Conclusions • thermodynamic consistent model for an ensemble of nuclei in equilibrium with an interacting nucleon gas • based on experimental masses  very precise description at low densities and temperatures (PNS crust) Composition: • significant differences compared to SNA: neutron shell effects, light clusters not well represented by as EoS: • consistent with existing SNA models • deviations at the onset of the liquid-gas phase-transition (SNA only applicable for Gaussian distributions) • complete EoS table including distributions of nuclei available •  further studies to analyze implications in astrophysical scenarios like supernovae or cooling proto-neutron stars

13. Phase Transitions in Supernova Matter Matthias Hempel and Jürgen Schaffner-Bielich Institute for Theoretical Physics HIC for FAIR Workshop, Prerow October 13, 2009

14. Model – Nuclei • semi-empirical formula for intrinsic partition function of hot nuclei • Coulomb energies in Wigner-Seitz approximation [Fai, Randrup; 1982NPA381]

15. Results – Light Clusters • Yp=0.5 • red: T=4 MeV • orange: T=10 MeV • generalized RMF model [Typel et al.; arXiv:0908.2344]

16. Results – Light Clusters • Yp=0.5 • red: T=4 MeV • orange: T=10 MeV • generalized RMF model [Typel et al.; arXiv:0908.2344]

17. Results – Light Clusters • Yp=0.5 • red: T=4 MeV • orange: T=10 MeV • quantum statistical approach [Typel et al.; arXiv:0908.2344]

18. Results – Light Clusters • Yp=0.5 • red: T=4 MeV • orange: T=10 MeV • quantum statistical approach [Typel et al.; arXiv:0908.2344]

19. Results – Light Clusters • Yp=0.5 • red: T=4 MeV • orange: T=10 MeV • generalized RMF model [Typel et al.; arXiv:0908.2344]

20. Results – Light Clusters Conclusion: • T=10 MeV: qualitatively similar, light clusters up to slightly larger densities and larger fractions • T=4 MeV: present model very similar to ideal NSE, but: XA>0.5 for nB>5x10-3 fm-3  similar behaviour for detailed medium effects expected  further comparison (low Yp) necessary [Typel et al.; arXiv:0908.2344]

21. Results – Equation of State without Light Clusters • light clusters with A<20 besides as taken out from calculation • decreased fB not visible any more, increased fB before uniform nuclear matter still present • smaller differences in sB and eB remain

22. Results – Equation of State T=5 MeV Yp=0.5 • excellent agreement at low T across all densities • T ≥ 5 MeV: exponential and power-law distributions (with light clusters) lead to additional binding around nB~10-2 fm-3, not reproducible by SNA • at nB>10-2 fm-3: restricted mass table, strong excluded volume: increased free energy • kinks in fB from the Maxwell construction

23. Results – Phase Diagram • almost no clusters for Yp< 0.1 • low T, Yp>0.1: heavy nuclei dominant • light clusters favored for more symmetric matter • heavy nuclei disappear for T>10 MeV • large fraction of heavy (low T) or light (high T) clusters before phase transition to uniform nuclear matter

24. Results – Phase Diagram • XA: heavy nuclei (Z ≥ 6) • Xa: light nuclei (Z < 6) • analogy to liquid-gas PD • dissolution of clusters at low densities and large T • fraction of nuclei increases with Yp • transition to uniform nuclear matter at ~½ n0

25. Results – Heavy Nuclei • stepwise change for T≤1 MeV • largest nuclei for Yp~0.3 • T=5 MeV: sudden increase of A,Z when fraction XA increases continuous change

26. Results – Heavy Nuclei • comparison to T=0 calculation in b-equilibrium with explicit lattice energy • discrete changes • up to 10-4 fm-3 based only on experimental data  excellent description at low temperatures and densities b-equilibrium T=1 MeV Yp=0.3 [Rüster, MH, Schaffner-Bielich; 2006PRC73]

27. Introduction – Matter in Supernovae • first order liquid-gas phase transition below Tc~15 MeV • importance of EoS below saturation with mixed phase (formation of nuclei) • at r ~1011 – 1013 g/cm³: neutrino spheres • stall of the shock front EoS calls during SN simulation saturation density n0 mixed phase [Liebendörfer, Fischer et al.]

28. Introduction – Commonly Used Models • existing EoS describe the system by one representative nucleus “single nucleus approximation” (SNA) • only a-particles for light clusters • no shell effects • advanced models only focus on certain aspects of the EoS [Souza et al.; 2008PRC78] [Burrows, Lattimer; 1984ApJ285] [Shen et al.; 1998NPA637] [Lattimer, Swesty; 1991NPA535]

29. Introduction – Advanced Models • successful application of nuclear statistical equilibrium models (heavy clusters) to multifragmentation experiments and astrophysics • detailed studies of medium modifications of light clusters • important impact of light clusters & nuclear distributions on dynamical processes in supernovae (e.g. neutrino and electron emission, absorption & scattering) [Arcones et al.; 2008PRC78 ] [Caballero, Horowitz et al.; 2006PRC74 ] [Röpke; 2009PRC79] [Typel et al.; arXiv:0908.2344] [Botvina, Mishustin; 2004PLB233]

30. Model – Thermodynamics • energy density  besides geometrical effect for nucleons (filling factor) no excluded volume modifications • total pressure • enhanced pressure of nuclei

31. Results – Nuclear Distributions • peaks caused by neutron shell effects • neutron magic numbers (40), 50, 82, 126, 184

32. Results – Nuclear Distributions • comparison to SMM at T=5 MeV, Yp=0.4, nB=10-3 n0 • shell effects lead to peaks on top of SMM • very good agreement, also for isotope distributions • minor deviations for heavy nuclei • larger deviations for lower Yp (not shown here) [Botvina, Mishustin; 2004PLB233]

33. Model – Thermodynamics • total canonical partition function • every nucleus fills a volume • nucleons must not be inside of nuclei V V` VA,Z • filling factor of the nucleons  no nuclei: unmodified RMF model

34. Model – Thermodynamics • nuclei (non-relativistic, classical) must not overlap V • free volume fraction VA,Z • van-der-Waals like hard-core repulsion at short distances  no nuclei above saturation density

35. Model – Phase Transition to Uniform Nuclear Matter • Maxwell construction for simplicity • constant pressure mixed phase • based on assumption of local charge neutrality and YpI= YpII (artificial) • pressure equilibrium & chemical equilibrium condition: equality of mB=Yp(mp+me)+(1-Yp)mn • interpretation of phase II as infinitely large nuclear cluster [Bugaev, et al.; 2001PLB498] phase II: uniform nuclear matter phase I: NSE [MH, Pagliara, Schaffner-Bielich; arXiv:0907.2680]

36. Results – Composition

37. Results – Composition

38. Results – Free Nucleons • number density outside of nuclei • n’nuc>0.01 n0 only for Yp<0.1 or T=5 MeV • larger T: larger densities of free nucleons but heavy nuclei not important

39. Results – Light Clusters • general behavior at large densities: low Yp: very asymmetric and large light clusters high Yp: mainly as • T = 5 MeV: deuterons dominant up to ~10-3 fm-3 • T = 10 MeV deuterons dominant up to larger densities • Z>2 not important

40. Model – Thermodynamics • number density of nuclei • k depending on nA,Z, nn and np •  numerical solution of the implicit set of equations  thermodynamic consistent description of excluded volume effects • mn0, mp0 contain RMF interactions of nuclei • compared to Typel09: counteracting in-medium self energy and Pauli-blocking shifts of the clusters do not appear • Mott effect mimicked by excluded volume corrections

41. Model – Thermodynamics • entropy density • volume excluded by baryons reduces available states • leading to negative entropy contribution • for

42. Model – Thermodynamics

43. Results – Equation of State • increased entropy causes lower free energy even though energy is also increased

44. Results – Equation of State • baryon pressure negative, total pressure always positive • total pressure constant across the Maxwell transition

45. Results – Equation of State • previous features and deviations from ideal gas clearly visible

46. Results – Equation of State • T=1 MeV: stepwise change of nuclei affects chemical potential • strong kinks, because chemical equilibration wrt Yp is suppressed in Maxwell

47. Results – Equation of State

48. Introduction – Phase Diagram of Nuclear Matter • first order liquid-gas phase transition below Tc~15 MeV • mixed phase region with phase separation • dilute neutron-rich phase (neutron gas) • dense symmetric phase (clusters / nuclei) • low nB / high T: formation of light clusters liquid gas mixed phase

49. Model – Nucleons • meson fields: • nucleons: