Equation of State in a Generalized Relativistic Density Functional Approach
aa r X i v : . [ nu c l - t h ] A p r Compact Stars in the QCD Phase Diagram IV (CSQCD IV)September 26-30, 2014, Prerow, Germany
Equation of State in a Generalized RelativisticDensity Functional Approach
Stefan TypelGSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH, Theorie, Planckstraße 1,64291 Darmstadt, Germany
In many simulations of astrophysical objects and phenomena, the equation of state(EoS) of dense matter is an essential ingredient. It determines, e.g., the dynamicalevolution of core-collapse supernovae [1, 2] and neutron star mergers [3], and thestructure of compact stars [4]. The application of an EoS is reasonable if the timescalesof reactions are much smaller than those of the system evolution and thermodynamicequilibrium can be assumed to hold. In general, a global EoS is required that coversa wide range in temperature, density and isospin asymmetry. These conditions affectthe chemical composition of matter and the nucleosynthesis.A critical examination of existing global EoS models [5] suggests that the devel-opment of an improved EoS is worthwhile. The set of constituent particles should beenlarged considerably including not only nucleons, charged leptons and photons butalso a “complete” table of nuclei, mesons, hyperons or even quarks as degrees of free-dom at high densities and temperatures. The model parameters have to be contrainedbetter taking, e.g., properties of nuclei, results of heavy-ion collisions or compact starobservations into account. Correlations should be considered more seriously, e.g.,at low-densities where the virial equation of state (VEoS), which is determined bynucleon-nucleon correlations, is a model-independent benchmark [6, 7]. For compos-ite particles such as nuclei the dissolution in the medium (Mott effect) has to bedescribed properly [8, 9]. Electromagnetic correlations are essential in order to modelthe solidification/melting at low temperatures. Phase transitions and the appear-ance of ’non-congruent’ features have to be treated correctly [10] with non-negligibledifferences between nuclear matter and stellar matter. Obviously, it is a tremendouschallenge to cover the full range of thermodynamic variables in a single unified model.Information on correlations are encoded in spectral functions, which have a com-plicated structure in general. Often, a quasiparticles (QP) approach is employed as anapproximation. The QP properties change inside the medium and the size of residualcorrelations is reduced. The QP concept is very successful in nuclear physics, e.g., inphenomenological mean-field models (Skyrme, Gogny, relativistic) or the treatment ofpairing correlations using a Bogoliubov transformation [11]. In the ultimate limit, an1xact diagonalisation of the Hamiltonian of the interacting many-body system leadsto a system of independent QP that can be many-body states. At low densities, clus-ters appear as new degrees of freedom as described in the VEoS. In order to considerthese features, a generalized relativistic density functional (gRDF) was developed. Ittakes the correct limits and explicit cluster degrees of freedom into account.
The gRDF model [12, 13, 14, 15] is based on a grand canonical approach. It isan extension of a conventional relativistic mean-field model with density dependentcouplings [16]. All thermodynamic quantities are derived from a grand canonicalpotential density ω ( T, { µ i } ), which depends on the temperature T and the set ofchemical potentials µ i of all particles. The present set of particle species comprisesbaryons (nucleons and hyperons), nuclei, charged leptons and photons. Besides lightnuclei ( H, H, He, He) a full table of heavy nuclei ( A Z with A > N, Z ≤ S i ) and vector ( V i ) poten-tials. The effective interaction is modeled by an exchange of mesons ( σ , ω , ρ ) withdensity-dependent couplings to the nucleons, both free and bound in nuclei, using thewell constrained DD2 parametrization [12]. It gives very reasonable nuclear matterparameters at a saturation density of n sat = 0 .
149 fm − , such as a binding energy pernucleon E/A = 16 .
02 MeV, a compressibility K = 242 . J = 31 .
67 MeV and a slope parameter L = 55 .
04 MeV. The neutron matter EoS lieswithin the error bounds of recent chiral effective field theoretical calculations [18, 19].Both potentials S i and V i receive contributions from the meson fields. For compositeparticles, the scalar potential contains an additional mass shift ∆ m i that dependson all particle densities and temperature. It mainly takes the blocking of statesby the Pauli exclusion principle into account and serves to describe the dissolutionof clusters by reducing the particle binding energy. This microscopically motivatedapproach replaces the traditional, purely geometric concept of the excluded-volumemechanism [20]. The vector potential V i includes a “rearrangment” contribution dueto the density dependence of the meson-nucleon couplings, which is required for thethermodynamic consistency of the model, and an electromagnetic correction to ac-count for electron sceening effects in stellar matter.2 Symmetry energy and neutron skins of nuclei
The isospin dependence of the effective interaction in the gRDF model determinesthe density dependence of the symmetry energy. It is crucial for a proper descriptionof the structure of neutron stars, see, e.g., the topical issue on the symmetry energy[21]. A strong correlation of the neutron skin thickness ∆ r np of heavy nuclei withthe slope of the neutron matter equation of state [22, 23] or the slope parameter L of the symmetry energy is observed when the predictions of a large number ofmean-field calculations, both relativistic and non-relativistic, are compared, see, e.g.,[24]. In recent years, many attempts were made to determine the symmetry energy atsaturation J and the parameter L from experiments, e.g., by measuring the neutronskin thickness of Pb and using the ∆ r np vs. L correlation. Since the calculations ofneutron skin thicknesses are based on mean-field models, the question arises whetherfew-nucleon correlations can effect the results.The gRDF approach can be employed to describe the formation of nuclei insidematter at finite temperatures by using an extended Thomas-Fermi approximation inspherical Wigner-Seitz cells [14]. In a calculation with nucleons and light clustersas degrees of freedom, it is observed that the probability of finding light clusters isenhanced at the surface of the heavy nucleus as compared to the surrounding low-density gas. The gRDF model can be extended to the description of heavy nucleiin vacuum at zero temperature to study cluster correlations. In this case, only the α -particle remains as the relevant light cluster. Its density distribution is obtainedfrom the α -particle ground state wave function that is calculated self-consistentlyin the WKB approximation. For the chain of Sn nuclei, a distinct reduction of theneutron skin thickness is observed when α -particle correlations are considered [25].However, the effect vanishes for very neutron-rich nuclei or for nuclei with roughlythe same neutron and proton numbers without a neutron skin. A variation of theisovector dependent part of the effective interaction allows to study the ∆ r np vs. L correlation, e.g., for a Pb nucleus. A systematic shift is observed that might affectthe determination of the slope parameter L from measurements of the neutron skinthickness, at least as a systematic error. It is envisaged to investigate experimentallythe predicted formation of α -particles at the surface of Sn nuclei in quasi-elastic (p,p α )reactions at RCNP, Osaka [26]. The present version of the gRDF model includes only hadronic and leptonic degreesof freedom where nuclei are described as clusters composed of nucleons. At high den-sities or temperatures a phase transition to quark matter is expected. Hence, quarkdegrees of freedom should be incorporated into the approach. On the other hand, at3ow densities and temperatures, quarks should be confined in nucleons. In a prelim-inary extension of the gRDF model with quarks, a phenomenological description ofconfinement will be implemented. The idea is to apply an “inverse” excluded-volumeapproach that permits the quarks to propagate freely only above a certain (scalar)density of the system. For this purpose, the classical excluded-volume mechanism isgeneralized by allowing more general dependencies of the “available volume fraction”.The correct quantum statistics and a relativistic description are considered, too. Therelevant theoretical formulation to guarantee the thermodynamic consistency of theapproach has been developed and exploratory calculations have been preformed.Another extension of the gRDF model concerns the introduction of more generalmeson-nucleon couplings in the Lagrangian density. In conventional RMF approacheswith density-dependent couplings, the nucleon self-energies only depend on densities.As known from Dirac-Brueckner calculations of nuclear matter, they should also de-pend on the nucleon momentum or energy. This dependence can be mapped tomodified effective density dependent meson-nucleon couplings [27], but the full de-pendence should be kept in order to comply with the optical potential constraint athigh nucleon energies. This can be achieved in a RMF model with density-dependentand non-linear derivative meson-nucleon couplings of general functional form [28].Preliminary studies indicate a softening of the EoS at high densities, however, for areliable fit of the model parameters, the approach has to be applied to the descriptionof finite nuclei. Work in this direction is in progress.
Acknowledgement
The author thanks Sofija Anti´c, David Blaschke, Jaroslava Hrt´ankov´a, Thomas Kl¨ahn,Gevorg Poghosyan, Gerd R¨opke, Maria Voskresenskaya and Hermann Wolter for thecollaboration, discussions and encouragement during various stages in developmentof the gRDF model and extensions. This work was supported by the Helmholtz As-sociation (HGF) through the Nuclear Astrophysics Virtual Institute (VH-VI-417).The participation of the author at the CSQCD IV workshop was made possible byNewCompStar, COST Action MP1304.
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