Construct an EOS for use in astrophysics: neutron stars and supernovae

wide parameter range:

proton fraction

Large charge asymmetry: thus investigation of symmetry energy

1. include light cluster correlations at low density taking into account medium dependence

2. Consistent with experimental results from heavy ion collisions

3. constitutes a unified approach for the EOS for a very wide range of densities and temperatures

10/10 08 MeVT 1000

6.00 pY

Main points:

in collaboration with:Stefan Typel, GSI,Darmstadt and Techn. Univ. Munich, Germany

Gerd Röpke, Univ. Rostock, GermanyDavid Blaschke, Th. Klähn, Univ. Wroclaw, Poland

J.B. Natowitz, A. Bonasera, S. Kowalsky, S. Shlomo, et al., Texas A&M, College Station, Tx, USA

EOS for astrophysical processes:

wide range of conditions: global approach needed!10/10 08

MeVT 1000 pp YY 21;6.00

Low densities: 2-, 3-,..many body correlations important. Bound states as new particle species. Change of composition and thermodyn- properties

High densities (around saturation): homogeneous nuclear matter, mean field dominates

In between: Liquid-gas phase transition at low temperatures, Inhomogeneous phases (lattice structures) Approach necessary, that interpolates reliably

commonly used EOS‘s:

Lattimer-Swesty, NPA 535 (1991): Skyrme-type model, Liquid Drop modelling of finite nuclei embedded in nucleon gas

Shen, Toki et al., Prog. Theor. Phys. 100 (1998): RMF model (TM1), -particle with excluded volume procedure

Horowitz, Schwenk, NPA776 (2006): virial expansion, n,p, ‘s, using experimental information on bound states and phase shifts: exact limit for low density

improvements: (S.Typel, G. Röpke, et al., PRC 2010, arxiv 0908:2344)

-medium effects on light clusters, quantum statistical approach

-realistic description of high density matter (DD-RMF)

Nuclear Statistical Equilibrium (NSE):

Mixture of ideal gases for each species (zero density limit)

Virial expansion, expansion of in powers of fugacities

Beth-Uhlenbeck, Physica 3 (1936)

interactions included

Energies and phase shifts from experiment: Horowitz-Schwenk, model independent,

but only valid at very low densities, no medium dependence

Thermodynamical GF approach

(M. Schmidt, G. Röpke, H. Schulz, Ann. Phys. 202 (1990)

self energy shifts, blocking effects

(melting at Mott density), proper statistics ,

Ek(P,T,), and generalized phase shifts

Parametrization in density and temperature

Needs quasiparticle energies

generalized mean field model

)iT,Z(V, )T/exp(z ii

Light clusters : Theoretical approaches:

Generalized Relativistic Mean Field Model with Light ClustersUnified approach for nucleon and light cluster degrees of freedom

degrees of freedom: fermions

bosons

mesons ,,

tHenpii ,;,, 3 ,, dii

Lagrangian

DD : density dependent RMF, S. Typel, PRC71 (2005)

,,,);,(0 thdiTBBmNmZM iinipii mass shifts of the clusters

nucleon self energies with „rearrangement terms“

),(()(),()()(),(

,,,3300

TBTTTTT j

htdjsjs

Ri

npiRii ,;0300

coupling of nucleons to clusters

from density dependent coupling

Summary of theoretical approach:

Quantumstatistical model (QS)

-Includes medium modification of clusters (Mott transition)

-Includes correlations in the continuum (phase shifts)

-needs good model for quasi-particle energies in the mean field

-In principle also possible for heavier clusters

Generalized Rel. Mean Field model (RMF)

-Good description of higher density phase, i.e. quasiparticle energies

-Includes cluster degrees of freedom with parametrized density and temperature dependent binding energies

-no correlation in the continuum

-Heavier clusters treated in Wigner-Seitz cell approximation (single nucleus approximation)

Global approach from very low to high densities

S.Typel, G. Röpke, et al.,

PRC 2010, arxiv 0908:2344

Comparison to other approaches: alpha particle fractions

Schwenk-Horowitz (black dash-dot; virial expansion with experimental BE and phase shifts for nucleons and alpha): exact for n0, but no disappearance of clusters for higher densities

Nuclear Statistical Equilibrium (NSE) (green, dotted): decrease at higher densities because of heavier nuclei, but no medium modifications (melting)

Shen-Toki (blue, dashed; RMF for p,n, heavy nuclei in Wigner-Seitz approximation in a gas of p,n,, excluded volume method): medium modifications empirical, survive very long

Generalized RMF (red, solid: coupled RMF approach for p,n,d,t,h,medium dep. BE from QS approach, no heavy nuclei): strong deuteron correlations suppress alpha at higher densities

Quantum Statistical (orange, dashed; medium modified clusters consistently in cluding scattering contributions): increases fraction

Heavier clusters (nuclei) in the medium:

Our approach: QS+RMF

Hempel, Schaffner-Bielich

(arXiv 0911:4073):

NSE , with excluded volume with procedure

Calculation in RMF of heavy cluster in Wigner-Seitz cell in beta-equilibriumT=5 MeV,

b=0.3

Equation of state: pressure vs. density

RMF QS

NSE (thin lines) low density limit but breaks down already at small densities

Differences between approaches: too strong cluster effects in RMF, additional minima in QS

Regions of instability: phase transitions between clusterized und homogeneous phases

Symmetry energy:

with (solid) and without (dashed) clusters

Usually:

but EA not quadratic for low temperatures with clusters. Thus use:

et al.,

Can this be measured??

64Zn+(92Mo,197Au) at 35 AMeV

Central collisions, reconstruction of fireball

Determination of thermodyn. conditions as fct of vsurf=vemission-vcoul

~time of emission with specified conditions of density and temperature:

temperature: isotope temperatures, double ratios H-He

densitiesp, n, from yield ratios and bound clusters

Isoscaling analysis

(B.Tsang, et al., )

Free symmetry energy

ZN

Z

T

N

T econsteeZNY

ZNYR

pn

),(

),(

1

212

Isoscaling coefficients and

T

))()((F4 22Az2

Az

sym2

1

1 )NSE(

symsym STFE

J. Natowitz, G, Röoke, et al., nucl-th/0908.2344, subm. PRL

Comparision of low-density symmetry energy to experiment:

Fsym Esym

Parametrization of nuclear symmetry energy of different stiffness (momentum dependent Skyrme-type) (B.A. Li)

Quantum Statistical model , T=1,4,8 MeV)

Single nucleus approx. (Wigner-Seitz), RMF

Particle Fractions

very low density: p,n

Increasing density: clusters arise: deuteron first, but then dominates

Mott density: clusters melt, homogeneous p,n matter;

here heavier nuclei (embedded into a gas) become important, not yet fully implemented

Dependence on temperature

( T=0(2)20 MeV )

symmetric matter 0)

Thin lines: NSE, i.e. without medium modifications of clusters (melting at finite densities)

proton

deuteron 3He

triton alpha

S.Typel, G. Röpke, et al.,

PRC 2010, arxiv 0908:2344

Equation of State (RMF):

With (solid) and without (dashed) clusters): reduction at low and increase at intermediate densities, increase of critical temperature

Maxwell construction for phase coexistence

Coexistence region (left) and phase transition line(right)

RMF w/o clusters (blue)

RMF with clusters (red)

QS (green)