2. The ralativistic mean-field theory
In the RMF theory, the strong interaction is described by the exchange of mesons between nucleons through the Yukawa couplings. But in the model of Walecka (1974) only isoscalar-scalar and isoscalar-vector mesons are included. In order to describe actual nuclear systems, it is necessary to introduce proton-neutron asymmetry effects. This is done by adding the isovector-vector meson contribution. We follow the notation of Walecka (1974). The Lagrangian density of the system is given by
Here , , and denote the fields of baryons, attractive isoscalar-scalar mesons, repulsive isoscalar-vector mesons, and isovector-vector mesons with masses of , and , respectively. The constants , and are coupling constants for interactions between mesons and nucleons. The is the effective nucleon mass. The Lagrangian density (1) with Eq. (4) is referred to as the ZM model.
In this work, we only investigate nuclear matter without strangeness, and we do not consider pion condensation, kaon condensation and fields of strange hadrons. Using the BB model, several authors (Glendenning 1985; Ellis et al. 1991) have studied neutron-star matter in the RMF theory with strangeness degrees of freedom and discussed contamination of strange hadrons in neutron stars. Using the ZM model, Prakash, et al. (1995) have studied quark-hadron phase transitions in protoneutron stars. The possibility of strange quark stars has been also discussed (e.g., Witten 1984; Alcock,et al. 1986; Haensel, et al. 1986). Pion condensation and kaon condensation in the RMF theory and their applications to neutron stars have been studied (Glendenning et al. 1983; Thorsson, et al. 1994; Dai & Cheng 1997).
Starting with Eq. (1), we derive a set of the Euler-Lagrange equations. The Dirac equation for the nucleon field is given by
and the Klein-Gordon euqations for the meson fields are written as
We consider static infinite matter so that we can obtain simplified equations, where the derivative terms vanish automatically, due to the translational invariance of infinite matter. Setting , , and , we have the meson fields expressed by the expectation values of the ground state,
Next, according to the standard procedure of Walecka (1974), we derive the energy density of the system,
and the pressure,
In Eqs. (12) and (14), and are the proton and neutron Fermi energies, and and are the proton and neutron number densities. The effective nucleon mass is calculated through the following equations:
In order to investigate the properties of supernova matter, we must give some conditions. In a supernova core, neutrinos are trapped and form an ideal Fermi-Dirac gas. Thus, the weak process is . The chemical equilibrium requires
with the chemical potentials of proton and neutron being
where is the total baryon number density. The second condition is charge neutrality,
where and are the electron and muon number densities respectively. The third condition is to fix leptonic concentration,
with being the neutrino number density.
From Centelles et al. (1992), we choose the parameter set in Table 1. Please note that there is a numerical factor of in front of the -meson coupling constant in the Lagrangian density of Centelles et al. (1992) and Cheng et al. (1996). But, this factor has been absorbed into the -meson coupling constant in Eq. (1), so in the present paper the value of this parameter has decreased by a factor of
Table 1. The parameters of the ZM model
two as compared to the quoted value of Cheng et al. (1996). We have used this parameter set to calculate the binding energies, radii and diffusenesses of 40 Ca and 208 Pb. The results are well consistent with those of Centelles et al. (1992) (Cheng et al. 1996; for details see Yao 1996). As shown next section, this parameter set also yields very satisfactory saturation properties (incompressibility, saturation density and binding energy per nucleon) of symmetric nuclear matter.
© European Southern Observatory (ESO) 1998
Online publication: January 16, 1998