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A statistical model for hot hadronic matter

A statistical model for hot hadronic matter. A statistical model for hot hadronic matter. Motivation. EoS and composition at finite T is of interest for Supernovae, cooling or accreting NS, collisions between compact stars, (heavy ion collisions) …

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A statistical model for hot hadronic matter

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  1. A statistical model for hot hadronic matter

  2. A statistical model for hot hadronic matter

  3. Motivation • EoS and composition at finite T is of interest for Supernovae, cooling or accreting NS, collisions between compact stars, (heavy ion collisions) … • at present only two models available (Shen & Lattimer Swesty) • focus on matter below saturation density (crust) and construct a model that describes the liquid-gas phase transition with a grand-canonical statistical ensemble • sub-saturated matter important for e.g.: • - SN dynamics (stall of the shock front) • - cooling of NS • directly accessible by heavy ion collisions in form of multifragmentation Matthias Hempel Ladek Zdroj, February 27, 2008

  4. Motivation • present models describe the system by one representative nucleus / the ground state of the simulated cell •  no thermal or chemical ensemble • “single nucleus approximation” has little influence on the EoS; but significant effect on the composition possible • composition & form of matter (one component plasma ↔ statistical ensemble) influences e.g.: • - neutrino scattering • - thermal conductivity [Burrows, A.; Lattimer, J. M.; 1984ApJ...285..294B] Matthias Hempel Ladek Zdroj, February 27, 2008

  5. Hot Hadronic Matter – Assumptions • nuclear statistical equilibrium (T ≥ 0.5 MeV) • full grand-canonical ensemble • n-free • charge neutrality: ne = np • b-equilibrium: me= mB - mp • matter described by (T, nB) • trapped n’s • charge neutrality: ne = np • no b-equilibrium / finite n chemical potential: me-mn= mB - mp • described by (T, nB, Yp) Matthias Hempel Ladek Zdroj, February 27, 2008

  6. Hot Hadronic Matter– Ingredients T, nB, Yp • nuclei (A ≥ 2) A3, Z3 a A1, Z1 A2, Z2 Matthias Hempel Ladek Zdroj, February 27, 2008

  7. Hot Hadronic Matter– Ingredients T, nB, Yp • nuclei (A ≥ 2) • nucleons A3, Z3 p a n n A1, Z1 A2, Z2 n Matthias Hempel Ladek Zdroj, February 27, 2008

  8. Hot Hadronic Matter– Ingredients T, nB, Yp • nuclei (A ≥ 2) • nucleons • electrons & positrons A3, Z3 p a n n e- e+ A1, Z1 A2, Z2 n Matthias Hempel Ladek Zdroj, February 27, 2008

  9. Hot Hadronic Matter– Ingredients T, nB, Yp • nuclei (A ≥ 2) • nucleons • electrons & positrons • photons A3, Z3 p g a n n e- e+ A1, Z1 A2, Z2 n Matthias Hempel Ladek Zdroj, February 27, 2008

  10. Hot Hadronic Matter– Ingredients • nuclei (A ≥ 2) • nucleons • electrons & positrons • photons Matthias Hempel Ladek Zdroj, February 27, 2008

  11. Nuclei • if available experimental data of Audi, Wapstra and Thibault (2003): binding energies of over 2000 precisely measured nuclei  direct use of experimental data for the construction of the EoS Matthias Hempel Ladek Zdroj, February 27, 2008

  12. Nuclei • experimentally unknown nuclei: mass table generated with theoretical nuclear model Matthias Hempel Ladek Zdroj, February 27, 2008

  13. Nuclei – Theoretical Nuclear Model • standard relativistic mean-field description • parameter-set TMA with mass number-dependent coupling constants • BCS d-force pairing • axial deformations • srms(AW)~2.1 MeV • but: neglect of temperature and medium effects [Geng, L.; Toki, H.; Meng, J.; 2005PThPh.113..785G] Matthias Hempel Ladek Zdroj, February 27, 2008

  14. Nuclei – Thermodynamics • Maxwell-Boltzmann gas for every nucleus (Ai,Zi) • classical, non-relativistic Boltzmann description always adequate • chemical potential: • number density: • empirical formula for level density [Fai, G.; Randrup, J.; 1982NuclPhysA.381..557] Matthias Hempel Ladek Zdroj, February 27, 2008

  15. Nuclei – Coulomb Energies • Wigner-Seitz approximation • included as corrections to the nuclear masses: Ri • only valid if G>>1: • but if G<<1  ideal gas limit • achieved Ai, Zi e- e+ RWS Matthias Hempel Ladek Zdroj, February 27, 2008

  16. Nucleons • free Fermi-gas at finite T (high accurate Fermi-Dirac integration routine) • same relativistic mean-field description as for nuclei (at finite T) • nuclear matter properties: [Gong, Z. et al.; 2001CoPhC.136..294G] Matthias Hempel Ladek Zdroj, February 27, 2008

  17. Thermodynamics • finite size of baryons  excluded volume principle • e, P, s corrected in the same manner • thermodynamic inconsistent due to neglect of derivative terms [Kouno, H.; Takagi, F.; 1989ZPhysC.45..43] Matthias Hempel Ladek Zdroj, February 27, 2008

  18. Results – n-free – Composition • mass fractions • nB(ND) = 2x10-4 fm-³ • ~ nB0(ND) = 2.7x10-4 fm-³ neutron drip Matthias Hempel Ladek Zdroj, February 27, 2008

  19. Results – n-free – Composition • average mass number <A> and standard deviation s • full T=0 calculations with explicit lattice energy reproduced (smoothed) • unexpected decreasing <A> at large density (limited mass table) • spread at transition points [Rüster, S. B.; H. M.; Schaffner-Bielich, J.; 2006PhRvC..73c5804R] Matthias Hempel Ladek Zdroj, February 27, 2008

  20. Results – n-free – Composition • nuclide distribution (mass fractions) • smeared out transition from nucleus 66Ni to 86Kr • can not be reproduced by one representative nucleus Matthias Hempel Ladek Zdroj, February 27, 2008

  21. Results – n-free – Composition • nuclide distribution • temperature effects decrease • neutrons begin to appear Matthias Hempel Ladek Zdroj, February 27, 2008

  22. Results – n-free – Composition • mass fractions Matthias Hempel Ladek Zdroj, February 27, 2008

  23. Results – n-free – Composition • mass fractions • nuclei dissolve into a, p & n at low density Matthias Hempel Ladek Zdroj, February 27, 2008

  24. Results – n-free – Composition • nuclide distribution • T=0 path still observable • thermal energy larger than differences in the chemical potentials of different nuclei  broad distribution Matthias Hempel Ladek Zdroj, February 27, 2008

  25. Results – n-free – Composition • nuclide distribution • transition from neutron magic number 50 to 82 •  broad distribution with two maxima Matthias Hempel Ladek Zdroj, February 27, 2008

  26. Results – n-free – EoS • T=0 case reproduced •  important benchmark up to nB ~ 10-4 fm-3 • softening above ND due to free n • P and r at small densities and large T generated by the electron positron plasma Matthias Hempel Ladek Zdroj, February 27, 2008

  27. Results – trapped n’s – EoS • good agreement • 1st order phase transition; due to limited mass table (?) [Lattimer, J.; Swesty, F.; 1991NuclPhysA.535..331] Matthias Hempel Ladek Zdroj, February 27, 2008

  28. Results – trapped n’s – EoS • good agreement for low T, but bumps from shell effects • differences at large T [Shen, H. et al.; 1998NuPhA.637..435S ] Matthias Hempel Ladek Zdroj, February 27, 2008

  29. Results – trapped n’s – Composition • average mass number <A> • strong shell effects • huge differences at large densities Matthias Hempel Ladek Zdroj, February 27, 2008

  30. Results – trapped n’s – Composition • mass fractions • nuclei and a’s only at largest densities Matthias Hempel Ladek Zdroj, February 27, 2008

  31. Results – trapped n’s – Composition • average neutron number <N> • Neutrino cross-sections /<N²> Matthias Hempel Ladek Zdroj, February 27, 2008

  32. Results – trapped n’s – Composition • average of squared neutron number <N²> • Neutrino cross-sections /<N²> • big effect coming only from the distribution Matthias Hempel Ladek Zdroj, February 27, 2008

  33. Results – trapped n’s – Composition • nuclide distribution Matthias Hempel Ladek Zdroj, February 27, 2008

  34. Results – trapped n’s – Composition • nuclide distribution • almost all nuclei of the nuclear chart populated Matthias Hempel Ladek Zdroj, February 27, 2008

  35. Results – trapped n’s – Composition • nuclide distribution • almost all nuclei of the nuclear chart populated • importance of statistical treatment Matthias Hempel Ladek Zdroj, February 27, 2008

  36. Summary • Statistical model for the EoS and composition at finite T: • grand canonical ensemble consisting of an ideal gas of nuclei (vacuum masses at T=0) and nucleons (RMF) • empirical formula for level densities • Coulomb energies included in Wigner-Seitz approximation as effective masses • excluded volume corrections for baryons • Results: • T=0 results reproduced • consistent with existing EoSs, 1st order phase transition • big differences in the composition, shell effects Matthias Hempel Ladek Zdroj, February 27, 2008

  37. Outlook • extension of nuclear mass table • investigate nuclear level density / temperature dependence of BE • investigate role of the excluded volume corrections • investigate Coulomb energies • inclusion of medium effects on the nuclear binding energies Matthias Hempel Ladek Zdroj, February 27, 2008

  38. Outlook – Density Dependence of BE • full RMF calculation with fixed external neutron density by Thomas Bürvenich (Frankfurt, FIAS) • simple quadratic behaviour (?) • extension of the Bethe-Weizsäcker mass formula preliminary Matthias Hempel Ladek Zdroj, February 27, 2008

  39. Outlook • extension of nuclear mass table • investigate nuclear level density / temperature dependence of BE • investigate role of the excluded volume corrections • investigate Coulomb energies • inclusion of medium effects on the nuclear binding energies • study different theoretical nuclear models (other parameter sets & mass tables, Skyrme-HF) • use more realistic low density homogenous nuclear matter EoS •  generate a full (nB, Yp,T) EoS table Matthias Hempel Ladek Zdroj, February 27, 2008

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