A statistical model for hot hadronic matter

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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

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A statistical model for hot hadronic matter

Outline
Motivation Description of the model Results for n-free matter Results for trapped ns Summary outlook


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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
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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
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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 ns
• 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
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Hot Hadronic Matter Ingredients

• nuclei (A 2)

T, nB, Yp
A3, Z3
a
A1, Z1
A2, Z2

Matthias Hempel Ladek Zdroj, February 27, 2008
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Hot Hadronic Matter Ingredients

• nuclei (A 2)
• nucleons

T, nB, Yp
A3, Z3
p
a
n
n
A1, Z1
A2, Z2
n

Matthias Hempel Ladek Zdroj, February 27, 2008
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Hot Hadronic Matter Ingredients

• nuclei (A 2)
• nucleons
• electrons positrons

T, nB, Yp
A3, Z3
p
a
n
n
e-
e
A1, Z1
A2, Z2
n

Matthias Hempel Ladek Zdroj, February 27, 2008
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Hot Hadronic Matter Ingredients

• nuclei (A 2)
• nucleons
• electrons positrons
• photons

T, nB, Yp
A3, Z3
p
g
a
n
n
e-
e
A1, Z1
A2, Z2
n

Matthias Hempel Ladek Zdroj, February 27, 2008
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Hot Hadronic Matter Ingredients

• nuclei (A 2)
• nucleons
• electrons positrons
• photons

Matthias Hempel Ladek Zdroj, February 27, 2008
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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
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Nuclei

• experimentally unknown nuclei mass table
  generated with theoretical nuclear model

Matthias Hempel Ladek Zdroj, February 27, 2008
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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
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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
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Nuclei Coulomb Energies

• Wigner-Seitz approximation
• included as corrections to the nuclear masses

Ri

• only valid if Ggtgt1
• but if Gltlt1 ? ideal gas limit
• achieved

Ai, Zi
e-
e
RWS

Matthias Hempel Ladek Zdroj, February 27, 2008
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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
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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
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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
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Results n-free Composition

• average mass number ltAgt and standard deviation s

• full T0 calculations with explicit lattice
  energy reproduced (smoothed)
• unexpected decreasing ltAgt 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
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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
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Results n-free Composition

• nuclide distribution

• temperature effects decrease
• neutrons begin to appear

Matthias Hempel Ladek Zdroj, February 27, 2008
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Results n-free Composition

• mass fractions

Matthias Hempel Ladek Zdroj, February 27, 2008
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Results n-free Composition

• mass fractions

• nuclei dissolve into a, p n at low density

Matthias Hempel Ladek Zdroj, February 27, 2008
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Results n-free Composition

• nuclide distribution

• T0 path still observable
• thermal energy larger than differences in the
  chemical potentials of different nuclei ? broad
  distribution

Matthias Hempel Ladek Zdroj, February 27, 2008
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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
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Results n-free EoS

• T0 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
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Results trapped ns 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
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Results trapped ns 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
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Results trapped ns Composition

• average mass number ltAgt

• strong shell effects
• huge differences at large densities

Matthias Hempel Ladek Zdroj, February 27, 2008
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Results trapped ns Composition

• mass fractions

• nuclei and as only at largest densities

Matthias Hempel Ladek Zdroj, February 27, 2008
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Results trapped ns Composition

• average neutron number ltNgt

• Neutrino cross-sections / ltN²gt

Matthias Hempel Ladek Zdroj, February 27, 2008
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Results trapped ns Composition

• average of squared neutron number ltN²gt

• Neutrino cross-sections / ltN²gt
• big effect coming only from the distribution

Matthias Hempel Ladek Zdroj, February 27, 2008
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Results trapped ns Composition

• nuclide distribution

Matthias Hempel Ladek Zdroj, February 27, 2008
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Results trapped ns Composition

• nuclide distribution

• almost all nuclei of the nuclear chart populated

Matthias Hempel Ladek Zdroj, February 27, 2008
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Results trapped ns Composition

• nuclide distribution

• almost all nuclei of the nuclear chart populated
• importance of statistical treatment

Matthias Hempel Ladek Zdroj, February 27, 2008
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Summary

• Statistical model for the EoS and composition at
  finite T
• grand canonical ensemble consisting of an ideal
  gas of nuclei (vacuum masses at T0) and nucleons
  (RMF)
• empirical formula for level densities
• Coulomb energies included in Wigner-Seitz
  approximation as effective masses
• excluded volume corrections for baryons
• Results
• T0 results reproduced
• consistent with existing EoSs, 1st order phase
  transition
• big differences in the composition, shell
  effects

Matthias Hempel Ladek Zdroj, February 27, 2008
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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
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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
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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