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ties. 

Suitable approaches to account for cluster forma- tion are the nuclear statistical equilibrium (NSE) model [9], cluster-virial expansions [7], and generalized Beth- Uhlenbeck approaches [10]. A thermal Green function approach that allows a generalization of the NSE model by introducing a quasiparticle description also for the bound states was already formulated some decades ago [11], but only recently analyzed with respect to the conse- quences for nuclear matter [12]. In this QS approach the cluster correlations are described in a generalized Beth- Uhlenbeck expansion. The advantage of this method is that the medium modifications of the clusters at finite density are taken into account. In Ref. [8] the thermo- dynamic properties of nuclear matter were derived using this approach. The formulation of Ref. [8] is valid in the density and temperature range where the formation of light clusters with A ≤ 4 dominates and heavier clus- ters are not yet important. The method requires a suf- ficiently accurate model for the quasiparticle properties, for which we employ a RMF model with density depen- dent couplings [13] which gives a good description both of nuclear matter around normal density and of ground state properties of nuclei across the nuclear chart. In order to extend the applicability of this RMF model to very low densities, it has been generalized in Ref. [8] to account also for cluster formation and dissolution. 

We note that at very low densities and temperatures below T ≃ 1 MeV new phases may occur. In fact, the for- mation of a solid phase using Overhauser orbitals includ- ing a triple point [14] or Bose-Einstein condensation [15] have been suggested. However, in this letter we are con- cerned with experimental data which probe nuclear mat- ter at considerably higher temperatures. The low- T be- havior is an interesting issue for future studies. 

In the following we focus on finite temperatures and on the sub-saturation region n < n$_{0}$ . Experimental informa- tion is derived from heavy-ion collisions of charge asym- metric nuclei, where transient states of different density can be reached, depending on the incident energy and the centrality of the collision. In the Fermi energy domain symmetry energy effects have been investigated using ju- 

diciously chosen observables [4, 16–19]. 

Recently, the experimental determination of the sym- metry energy at very low densities produced in heavy ion collisions of $^{64}$Zn on $^{92}$Mo and $^{197}$Au at 35 MeV per nu- cleon has been reported [20]. Results of this study are given in the first four columns of Tab. I. Note that as a result of an energy recalibration and reevaluation of the particle yields in different velocity bins these values are slightly different than those reported in Ref. [20]. The surface velocity v$_{surf}$ , i.e. the velocity before the fi- nal Coulomb acceleration, was used as a measure of the time when the particles leave the source under different conditions of density and temperature. Only values of v$_{surf}$ < 4 . 5 cm/ns are included here, since the system does not reach equilibrium for higher v$_{surf}$ , see Tab. I of Ref. [20]. The yields of the light clusters A ≤ 4 were de- termined as a function of v$_{surf}$ . Temperatures were deter- mined with the Albergo method [21] using a H-He ther- mometer based on the double yield ratio of deuterons, tritons, $^{3}$He and $^{4}$He, and are given in Tab. I as the average for the two reactions. 

The free neutron yield is obtained from the free proton yield and the yield ratio of $^{3}$H to $^{3}$He. To determine the asymmetry parameter of the sources the total proton and neutron yields including those bound in clusters are used. The proton chemical potential is derived from the yield ratio of $^{3}$H to $^{4}$He. The corresponding free proton and free neutron densities are calculated, and the total nucleon density is obtained by accounting also for the bound nucleons according to their respective yields [20]. The total nucleon densities are of the order of 1/100th to 1/20th of saturation density, as seen in Tab. I. 

An isoscaling analysis [22] has been employed (as a function of v$_{surf}$ ) to determine the free symmetry energy F$_{sym}$ via the expression α = 4 F$_{sym}$ Δ( Z/A ) $^{2}$/T . Here α is the isoscaling coefficient determined from yield ratios of Z = 1 ejectiles of the two reactions and Δ( Z/A ) 2 is the difference of the squared asymmetries of the sources in the two reactions. With Δ( Z/A ) 2 and the tempera- ture determined as above, the free symmetry energy is extracted. 

From the free symmetry energy derived in this way 

