Experiment (PREX) at the Jefferson Laboratory [@prex] is currently running to measure the $^{208}$Pb neutron radius with an accuracy of less than 0.05 fm, using polarized electron scattering.

In the present work, we use two different hadronic models that incorporate density dependence in different ways. The first one, to which we refer next as the TW model is a density dependent hadronic model with the meson-to-nucleon couplings explicitly dependent of the density [@original; @tw]. In the following it is used to calculate the neutron skin thickness of $^{208}$Pb, which is a neutron-rich heavy nucleus. This model was chosen because it is based on a microscopic calculation, fits well many nuclei properties and, as stated above, has shown to provide results which are different from the usual NL3 [@nl3] and TM1 [@tm1] parametrizations for the non-linear Walecka model (NLWM), having a richer density dependence of the symmetry energy than most of the relativistic nuclear models. The original motivation for the development of this density dependent hadronic model [@flw; @lf] was to reproduce results obtained with the relativistic Dirac-Brueckner Hartree-Fock (DBHF) theory [@DB]. Later the DBHF calculations for nuclear matter were taken only as a guide for a suitable parametrization of the 