Transcript A statistical model for hot hadronic matter
A statistical model for hot hadronic matter
Matthias Hempel, and Jürgen Schaffner-Bielich Institut für Theoretische Physik J. W. Goethe Universität, Frankfurt 44th Karpacz Winter School of Theoretical Physics 27.02.2008
A statistical model for hot hadronic matter
Outline
Motivation Description of the model Results for n -free matter Results for trapped n ’s Summary & outlook
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
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...285..294B ]
Hot Hadronic Matter – Assumptions
nuclear statistical equilibrium (T ≥ 0.5 MeV) full grand-canonical ensemble n
-free
charge neutrality: n e = n p b -equilibrium: m e = m B - m p matter described by (T, n B )
trapped
n
’s
charge neutrality: n e = n p no b -equilibrium / finite n chemical potential: m e - m n = m B - m p described by (T, n B , Y p ) Matthias Hempel Ladek Zdroj, February 27, 2008
Hot Hadronic Matter – Ingredients
T, n B , Y p nuclei (A ≥ 2) A 3 , Z 3 a A 1 , Z 1 A 2 , Z 2 Matthias Hempel Ladek Zdroj, February 27, 2008
Hot Hadronic Matter – Ingredients
T, n B , Y p nuclei (A ≥ 2) p A 3 , Z 3 nucleons n a n A 1 , Z 1 A 2 , Z 2 n Matthias Hempel Ladek Zdroj, February 27, 2008
Hot Hadronic Matter – Ingredients
T, n B , Y p nuclei (A ≥ 2) p A 3 , Z 3 nucleons electrons & positrons n a n e e + A 1 , Z 1 A 2 , Z 2 n Matthias Hempel Ladek Zdroj, February 27, 2008
Hot Hadronic Matter – Ingredients
T, n B , Y p nuclei (A ≥ 2) p A 3 , Z 3 nucleons electrons & positrons g n a photons n e e + A 1 , Z 1 A 2 , Z 2 n Matthias Hempel Ladek Zdroj, February 27, 2008
Hot Hadronic Matter – Ingredients
nuclei (A ≥ 2) nucleons electrons & positrons photons Matthias Hempel Ladek Zdroj, February 27, 2008
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 T, m B A 1 , Z 1 a A 3 , Z 3 A 2 , Z 2 Matthias Hempel Ladek Zdroj, February 27, 2008
Nuclei
experimentally unknown nuclei: mass table generated with theoretical nuclear model T, m B A 1 , Z 1 a A 3 , Z 3 A 2 , Z 2 Matthias Hempel Ladek Zdroj, February 27, 2008
Nuclei – Theoretical Nuclear Model
standard relativistic mean-field description parameter-set TMA with mass number-dependent coupling constants BCS d -force pairing axial deformations s rms (AW)~2.1 MeV but: neglect of temperature and medium effects T, m B A 1 , Z 1 a A 3 , Z 3 A 2 , Z 2 Matthias Hempel Ladek Zdroj, February 27, 2008
[Geng, L.; Toki, H.; Meng, J.; 2005PThPh.113..785G]
Nuclei – Thermodynamics
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 Matthias Hempel Ladek Zdroj, February 27, 2008 T, m B A 1 , Z 1 a A 3 , Z 3 A 2 , Z 2
[Fai, G.; Randrup, J.; 1982NuclPhysA.381..557]
Nuclei – Coulomb Energies
Wigner-Seitz approximation included as corrections to the nuclear masses: only valid if G>>1 : but if G<<1 ideal gas limit achieved Matthias Hempel Ladek Zdroj, February 27, 2008 A i , Z i R i e e + R WS T, m B e e + p A 1 , Z 1 A 3 , Z 3 a e e + e e + A 2 , Z 2
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: Matthias Hempel Ladek Zdroj, February 27, 2008 T, m B p n n n
[Gong, Z. et al.; 2001CoPhC.136..294G ]
Thermodynamics
finite size of baryons excluded volume principle e, P, s corrected in the same manner thermodynamic inconsistent due to neglect of derivative terms Matthias Hempel Ladek Zdroj, February 27, 2008 T, m B n p A 1 , Z 1 A 3 , Z 3 a e e + n A 2 , Z 2 n
[Kouno, H.; Takagi, F.; 1989ZPhysC.45..43]
Results –
n
-free – Composition
mass fractions n B (ND) = 2x10 -4 fm ³ ~ n B 0 (ND) = 2.7x10
-4 fm ³ Matthias Hempel Ladek Zdroj, February 27, 2008 neutron drip
Results –
n
-free – Composition
average mass number and standard deviation s full T=0 calculations with explicit lattice energy reproduced (smoothed) unexpected decreasing at large density (limited mass table) spread at transition points Matthias Hempel Ladek Zdroj, February 27, 2008
[ Rüster, S. B.; H. M.; Schaffner-Bielich, J.; 2006PhRvC..73c5804R ]
Results –
n
-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 –
n
-free – Composition
nuclide distribution temperature effects decrease neutrons begin to appear Matthias Hempel Ladek Zdroj, February 27, 2008
Results –
n
-free – Composition
mass fractions Matthias Hempel Ladek Zdroj, February 27, 2008
Results –
n
-free – Composition
mass fractions nuclei dissolve into a , p & n at low density Matthias Hempel Ladek Zdroj, February 27, 2008
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
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
Results –
n
-free – EoS
T=0 case reproduced important benchmark up to n B ~ 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
Results – trapped
n
’s – EoS
good agreement 1st order phase transition; due to limited mass table (?) Matthias Hempel Ladek Zdroj, February 27, 2008
[Lattimer, J.; Swesty, F.; 1991NuclPhysA.535..331]
Results – trapped
n
’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.637..435S ]
Results – trapped
n
’s – Composition
average mass number strong shell effects huge differences at large densities Matthias Hempel Ladek Zdroj, February 27, 2008
Results – trapped
n
’s – Composition
mass fractions nuclei and a ’s only at largest densities Matthias Hempel Ladek Zdroj, February 27, 2008
Results – trapped
n
’s – Composition
average neutron number
Results – trapped
n
’s – Composition
average of squared neutron number
Results – trapped
n
’s – Composition
nuclide distribution Matthias Hempel Ladek Zdroj, February 27, 2008
Results – trapped
n
’s – Composition
nuclide distribution almost all nuclei of the nuclear chart populated Matthias Hempel Ladek Zdroj, February 27, 2008
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
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) simple quadratic behaviour (?) extension of the Bethe Weizsäcker mass formula preliminary 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 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