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# Symmetry energy of dilute warm nuclear matter 

J. B. Natowitz, 1 G. R¨ opke, 2 S. Typel, 3, 4 D. Blaschke, 5, 6 A. Bonasera, 1,7 K. Hagel, 1 

T. Kl¨ ahn, 5, 8 S. Kowalski, 1 L. Qin, 1 S. Shlomo, 1 R. Wada, 1 and H. H. Wolter 9 

$^{1}$Cyclotron Institute, Texas A & M University, College Station, Texas 77843-3366, USA 

$^{2}$Institut f¨ ur Physik, Universit¨ at Rostock, Universit¨ atsplatz 3, D-18055 Rostock, Germany 

$^{3}$Excellence Cluster Universe, Technische Universit¨ at M¨ unchen, Boltzmannstraße 2, D-85748 Garching, Germany 

$^{4}$GSI Helmholtzzentrum f¨ ur Schwerionenforschung GmbH, Theorie, Planckstraße 1, D-64291 Darmstadt, Germany 

$^{5}$Instytut Fizyki Teoretycznej, Uniwersytet WrocGLYPH<suppress>lawski, pl. M. Borna 9, 50-204 WrocGLYPH<suppress>law, Poland 

$^{6}$Bogoliubov Laboratory for Theoretical Physics, JINR Dubna, Joliot-Curie str. 6, 141980 Dubna, Russia 

$^{7}$Laboratori Nazionali del Sud-INFN, v. S. Sofia 64, 95123 Catania, Italy. 

$^{8}$Theory Group, Physics Division, Building 203, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL 60439, USA 

$^{9}$Fakult¨ at f¨ ur Physik, Universit¨ at M¨ unchen, Am Coulombwall 1, D-85748 Garching, Germany 

(Dated: November 8, 2018) 

The symmetry energy of nuclear matter is a fundamental ingredient in the investigation of exotic nuclei, heavy-ion collisions and astrophysical phenomena. New data from heavy-ion collisions can be used to extract the free symmetry energy and the internal symmetry energy at subsaturation densities and temperatures below 10 MeV. Conventional theoretical calculations of the symmetry energy based on mean-field approaches fail to give the correct low-temperature, low-density limit that is governed by correlations, in particular by the appearance of bound states. A recently devel- oped quantum statistical (QS) approach that takes the formation of clusters into account predicts symmetry energies that are in very good agreement with the experimental data. A consistent de- scription of the symmetry energy is given that joins the correct low-density limit with quasiparticle approaches valid near the saturation density. 

The symmetry energy in the nuclear equation of state governs phenomena from the structure of exotic nuclei to astrophysical processes. The structure and the com- position of neutron stars depend crucially on the density dependence of the symmetry energy [1]. The symmetry energy characterizes the dependence of the nuclear bind- ing energy on the asymmetry δ = ( N − Z ) /A with Z and N the proton and neutron numbers, and A = N + Z . As a general representation of the symmetry energy coeffi- cient we use the definition 

<img src='content_image/405757.jpg'>

where E ( n, δ, T ) is the energy per nucleon of nuclear mat- ter with density n , asymmetry δ , and temperature T . At low density the symmetry energy changes mainly because additional binding is gained in symmetric matter due to formation of clusters and pasta structures [2]. 

Our empirical knowledge of the symmetry energy near the saturation density, n$_{0}$ , is based primarily on the bind- ing energies of nuclei. The Bethe-Weizs¨ acker mass for- mula leads to values of about E$_{sym}$ ( n$_{0}$, 0) = 28 − 34 MeV for the symmetry energy at zero temperature and satura- tion density n$_{0}$ ≈ 0 . 16 fm $^{$_{−}$3}$, if surface asymmetry effects are properly taken into account [3]. 

In contrast to the value of E$_{sym}$ ( n$_{0}$, 0), the variation of the symmetry energy with density and temperature is 

PACS numbers: 21.65.Ef, 05.70.Ce, 25.70.-q, 26.60.Kp, 26.50.+x 

intensely debated. Many theoretical investigations have been performed to estimate the behavior of the symme- try energy as a function of n and T . A recent review is given by Li et al. [4], see also [5, 6]. Typically, quasipar- ticle approaches such as the Skyrme Hartree-Fock and relativistic mean field (RMF) models or Dirac-Brueckner Hartree-Fock (DBHF) calculations are used. In such calculations the symmetry energy tends to zero in the low-density limit for uniform matter. However, in accor- dance with the mass action law, cluster formation dom- inates the structure of low-density symmetric matter at low temperatures. Therefore, the symmetry energy in this low-temperature limit has to be equal to the binding energy per nucleon associated with the strong interac- tion of the most bound nuclear cluster. A single-nucleon quasiparticle approach cannot account for such struc- tures. The correct low-density limit can be recovered only if the formation of clusters is properly taken into account, as has previously been shown in Ref. [7] in the context of a virial expansion valid at very low densities, and in Ref. [8]. 

In this letter we employ a quantum statistical (QS) ap- proach which includes cluster correlations in the medium. It interpolates between the exact low-density limit and the very successful RMF approaches near the saturation density. We show that this picture is in agreement with recent experimental findings on E$_{sym}$ at very low densi- 

