Matthias Hempel, and Jürgen Schaffner-Bielich Institut für Theoretische Physik J. W....

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

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...285..294B ]

Hot Hadronic Matter – Assumptions

nuclear statistical equilibrium (T ≥ 0.5 MeV)

full grand-canonical ensemble

-free

charge neutrality: ne = np

-equilibrium: e Bp

matter described by (T, nB)

trapped ’s

charge neutrality: ne = np

no -equilibrium / finite chemical potential: e Bp

described by (T, nB, Yp)

Matthias Hempel


nuclei (A ≥ 2)T, nB, Yp

A1, Z1

A3, Z3

A2, Z2

Matthias Hempel


Hot Hadronic Matter– Ingredients

p

n

n

n

a

A1, Z1

A3, Z3

A2, Z2

nuclei (A ≥ 2)

nucleons

Matthias Hempel



T, nB, Yp

nuclei (A ≥ 2)

nucleons

electrons & positrons

p

n

n

n

a

A1, Z1

A3, Z3

A2, Z2

e-

e+

Matthias Hempel



T, nB, Yp

nuclei (A ≥ 2)

nucleons


photons

p

n

n

n

a

A1, Z1

A3, Z3

A2, Z2

e-

e+

Matthias Hempel



T, nB, Yp

nuclei (A ≥ 2)

nucleons


photons

Matthias Hempel



Nuclei

if available experimental data of Audi, Wapstra and Thibault (2003): binding energies of over 2000 precisely measured nuclei

A1, Z1

A3, Z3

T, B

A2, Z2

Matthias Hempel


direct use of experimental data for the construction of the EoS

Nuclei

experimentally unknown nuclei: mass table generated with theoretical nuclear model

A1, Z1

A3, Z3

T, B

A2, Z2

Matthias Hempel


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.113..785G]

Nuclei – Theoretical Nuclear Model

A1, Z1

A3, Z3

T, B

A2, Z2

Matthias Hempel


Maxwell-Boltzmann gas for every nucleus (Ai,Zi)

classical, non-relativistic Boltzmann description always adequate

chemical potential:

number density:

empirical formula for level density

Nuclei – Thermodynamics

A1, Z1

A3, Z3

T, B

A2, Z2

Matthias Hempel

Ladek Zdroj, February 27, 2008[Fai, G.; Randrup, J.; 1982NuclPhysA.381..557]

Nuclei – Coulomb Energies

Wigner-Seitz approximation

included as corrections to the nuclear masses:

Ai, Zi

Ri

RWS

e-

e+

e-e+

A1, Z1

p

T, B

A3, Z3

A2, Z2

e-e+

e-

e+

Matthias Hempel


only valid if :

but if ideal gas limit

achieved

Nucleons

free Fermi-gas at finite T (high accurate Fermi-Dirac integration routine)

n

n

n

T, B

p

Matthias Hempel


same relativistic mean-field description as for nuclei (at finite T)

nuclear matter properties:

[Gong, Z. et al.; 2001CoPhC.136..294G ]

Thermodynamics

finite size of baryons excluded volume principle

P, s corrected in the same manner

thermodynamic inconsistent due to neglect of derivative terms

n

n

n

A1, Z1

A3, Z3

T, B

e-e+

A2, Z2

p

e-e+

[Kouno, H.; Takagi, F.; 1989ZPhysC.45..43]Matthias Hempel


Results – -free – Composition

neutron drip

nB(ND) = 2x10-4 fm-³

~ nB0(ND) = 2.7x10-4 fm-³

mass fractions

Matthias Hempel


full T=0 calculations with explicit lattice energy reproduced (smoothed)

unexpected decreasing <A> at large density (limited mass table)

spread at transition points

average mass number <A> and standard deviation


Matthias Hempel

Ladek Zdroj, February 27, 2008[Rüster, S. B.; H. M.; Schaffner-Bielich, J.; 2006PhRvC..73c5804R ]


nuclide distribution (mass fractions)

smeared out transition from nucleus 66Ni to 86Kr

can not be reproduced by one representative nucleus

Matthias Hempel



nuclide distribution

temperature effects decrease

neutrons begin to appear

Matthias Hempel



mass fractions

Matthias Hempel



mass fractions

nuclei dissolve into , p & n at low density

Matthias Hempel




T=0 path still observable

thermal energy larger than differences in the chemical potentials of different nuclei broad distribution

Matthias Hempel




Matthias Hempel


transition from neutron magic number 50 to 82

broad distribution with two maxima

Results – -free – EoS

T=0 case reproduced

important benchmark up to nB ~ 10-4 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


Results – trapped ’s – EoS

good agreement

1st order phase transition; due to limited mass table (?)

[Lattimer, J.; Swesty, F.; 1991NuclPhysA.535..331]Matthias Hempel


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.637..435S ]

Results – trapped ’s – Composition

average mass number <A>

strong shell effects

huge differences at large densities

Matthias Hempel


mass fractions

Matthias Hempel



nuclei and ’s only at largest densities

average neutron number <N>

Neutrino cross-sections /<N²>

Matthias Hempel



average of squared neutron number <N²>

Matthias Hempel


Neutrino cross-sections /<N²>

big effect coming only from the distribution



Matthias Hempel




almost all nuclei of the nuclear chart populated

Matthias Hempel




almost all nuclei of the nuclear chart populated

importance of statistical treatment

Matthias Hempel



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


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


Outlook – Density Dependence of BE

full RMF calculation with fixed external neutron density by Thomas Bürvenich (Frankfurt, FIAS)

Matthias Hempel


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 (nB, Yp, T) EoS table

Matthias Hempel


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Matthias Hempel, and Jürgen Schaffner-Bielich Institut für Theoretische Physik J. W....

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