Matthias Hempel, and Jürgen Schaffner-Bielich Institut für Theoretische Physik J. W. Goethe-Universität, Frankfurt 44th Karpacz Winter School of Theoretical Physics A statistical model for hot hadronic matter

Motivation Description of the model Results for -free matter Results for trapped ’s Summary & outlook Outline A statistical model for hot hadronic matter

Motivation Matthias Hempel Ladek Zdroj, February 27, 2008 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

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 Matthias Hempel Ladek Zdroj, February 27, 2008 [Burrows, A.; Lattimer, J. M.; 1984ApJ B ]

Hot Hadronic Matter – Assumptions nuclear statistical equilibrium (T ≥ 0.5 MeV) full grand-canonical ensemble -free charge neutrality: n e = n p  -equilibrium:  e   B  p matter described by (T, n B ) trapped ’s charge neutrality: n e = n p no  -equilibrium / finite chemical potential:  e     B  p described by (T, n B, Y p ) Matthias Hempel Ladek Zdroj, February 27, 2008

nuclei (A ≥ 2)T, n B, Y p  A 1, Z 1 A 3, Z 3 A 2, Z 2 Matthias Hempel Ladek Zdroj, February 27, 2008 Hot Hadronic Matter– Ingredients

p n n n a A 1, Z 1 A 3, Z 3 A 2, Z 2 nuclei (A ≥ 2) nucleons Matthias Hempel Ladek Zdroj, February 27, 2008 Hot Hadronic Matter– Ingredients T, n B, Y p

nuclei (A ≥ 2) nucleons electrons & positrons p n n n a A 1, Z 1 A 3, Z 3 A 2, Z 2 e-e- e+e+ Matthias Hempel Ladek Zdroj, February 27, 2008 Hot Hadronic Matter– Ingredients T, n B, Y p

nuclei (A ≥ 2) nucleons electrons & positrons photons p n n n a A 1, Z 1 A 3, Z 3 A 2, Z 2 e-e- e+e+ Matthias Hempel Ladek Zdroj, February 27, 2008 Hot Hadronic Matter– Ingredients  T, n B, Y p

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

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

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

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

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

Nuclei – Coulomb Energies Wigner-Seitz approximation included as corrections to the nuclear masses: A i, Z i RiRi R WS e-e- e+e+ Matthias Hempel Ladek Zdroj, February 27, 2008 only valid if  : but if  ideal gas limit achieved

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

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

Results – -free – Composition neutron drip n B (ND) = 2x10 -4 fm - ³ ~ n B 0 (ND) = 2.7x10 -4 fm - ³ mass fractions Matthias Hempel Ladek Zdroj, February 27, 2008

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

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

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

Results – -free – Composition mass fractions Matthias Hempel Ladek Zdroj, February 27, 2008

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

Results – -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

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

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

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

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

Results – trapped ’s – Composition average mass number strong shell effects huge differences at large densities Matthias Hempel Ladek Zdroj, February 27, 2008

mass fractions Matthias Hempel Ladek Zdroj, February 27, 2008 Results – trapped ’s – Composition nuclei and  ’s only at largest densities

average neutron number Neutrino cross-sections /  Matthias Hempel Ladek Zdroj, February 27, 2008 Results – trapped ’s – Composition

average of squared neutron number Matthias Hempel Ladek Zdroj, February 27, 2008 Neutrino cross-sections /  big effect coming only from the distribution Results – trapped ’s – Composition

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

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

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

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

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

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

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 (n B, Y p, T) EoS table Matthias Hempel Ladek Zdroj, February 27, 2008