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The equation of state (EOS) of symmetric nuclear and pure neutron matter has been investigated extensively by adopting the non-relativistic Brueckner-Hartree-Fock (BHF). For more comparison, the extended BHF approaches using the self-consistent Green’s function approach or by including a three-body force will be done. The EOS will be studied for different approaches at zero temperature. We can calculate the total mass and radius of neutron stars using various equations of state. A comparison with relativistic BHF calculations will be done. Relativistic effects are known to be important at high densities, giving an increased repulsion. This leads to a stiffer EOS compared to the EOS derived with a non-relativistic approach.

The properties of neutron stars depend on the equation of state (EOS) at densities up to an order of magnitude higher than those observed in ordinary nuclei. The EOS determines properties such as the mass range, the mass-radius relationship, the crust thickness and the cooling rate [_{סּ} to 1.9 M_{סּ}, where M_{סּ} is the solar mass [

Several theoretical approaches to calculations of the EOS have been considered. The hypothesis that strange quark matter may be the absolute ground state of the strong interaction [

The scope of this work is to derive the EOS from the underlying many-body theory, derived from realistic nucleon-nucleon (NN) interactions such as CD-Bonn potential [

In the self-consistent Green’s function approach, the binding energy as well as all single-particle observables in the nuclear matter are calculated from the exact in-medium single-particle propagator. The latter is obtained from the Dyson equation, where medium effects are taken into account by the irreducible self-energy that is obtained from an expansion in terms of the effective interaction obtained from the sum of all ladder diagrams. One important feature of the SCGF approach is that particles and holes are treated on an equal footing, whereas in BHF only intermediate particle

The paper is organized as follows. In the next section we will describe the formalisms within the non-relati- vistic approaches we have employed. Results for the EOS and pressure of symmetric and pure neutron matter using suggested models, together with neutron star matter observables will be presented in Section 3. A short summary and some conclusions will be given in Section 4.

Starting from realistic nucleon-nucleon (NN) interaction, we have to use more advanced many-body approximations like the BHF which have the capability to account for the effects of correlations, which are due to the strong tensor and short-range components of such realistic NN interaction. The single-particle energy of a particle in the BHF approximation corresponds to the Hartree-Fock expression using the G-matrix for the effective interaction. This means that the self-energy of a nucleon in nuclear matter with momentum k, isospin

with the occupation probability of a free Fermi gas of protons

This means for asymmetric nuclear matter with a total density

With Fermi momenta for protons

The matrix elements in Equation (1) denote antisymmetrized matrix elements of the Brueckner G-matrix that are determined by solving the Bethe-Goldstone equation for a given realistic NN interaction V

The single-particle energies

with a starting energy parameter

The Pauli operator _{1}, p_{2}, which are above the corresponding Fermi momentum. However, the single-particle spectrum is often parameterized in the form of an effective mass

so that a so-called angle-averaged propagator can be defined, which reduces the Bethe-Goldstone equation to an integral equation in one dimension. The exact Pauli operator has been treated in [

One of the drawbacks of the BHF approximation is the fact that it does not provide results for the equation of state, which are consistent from the point of view of thermodynamics. As an example we mention that BHF results do not fulfill, e.g., the Hugenholtz-Van Hove theorem [

The quasi-particle energy for the extended self-energy can be defined as

The spectral functions for hole and particle strength, ^{ }

where the plus and minus sign on the left-hand side of this equation refers to the case of hole

Integrating the spectral distribution of the hole states yields the occupation probability

In the case of two body interactions, the hole spectral function gives access, through the Koltun sum-rule [

with deg denoting the degeneracy of the single-particle level, which is 4 for nuclear matter. The kinetic energy per nucleon,

and the potential energy per particle has the form

One can express the total binding energy per nucleon as

In Brueckner-Hartree-Fock,

Neutron stars are very interesting physical systems and their properties, such as masses and radii as function of the central density, can be derived from the equation of state (EOS) of the β-stable matter contained in them. The EOS is microscopically calculated from the sections a and b. After that, briefly we outline the derivation of neutron-star properties from its EOS. One starts from the Tolman-Oppenheimer-Volkov (TOV) equations for the total pressure P and the enclosed mass m [

where

From the EOS pure neutron matter, we calculate the nuclear contribution

Then the total pressure and the total mass density

Being _{N} is the nucleon mass and c is the speed of light in vacuum. Starting with a central mass density

For the outer part of the neutron star we have used the equations of state by Baym et al. [^{−3} < ρ < 0.08 fm^{−}^{3}) we have used the results of Negele et al. [^{−}^{3}), the present EOS will be used.

All results of calculations, which refer to realistic NN interactions, have been obtained using the CD-Bonn interaction [

In _{0} = 0.16 fm^{−}^{3}; E_{0} = −16 MeV). In the low-density limit, the BHF approach and the SCGF approach coincide. As the density increases, the phase space for hole-hole propagation is no longer negligible, and this leads to an important repulsive effect on the total energy. Since particle-particle and hole-hole ladders are treated in a completely symmetrical way in the SCGF approach, the Green’s function scheme is also appropriated for calculations at higher densities [

As we see, non-relativistic calculations, based on purely two-body interaction, fail to reproduce the correct saturation point for symmetric nuclear matter. This well-known deficiency is commonly corrected by introducing three-body force (3BF). Relevant progress has been made in the theory of nucleon 3BF, but a complete theory is not yet available. A realistic model for nuclear 3BF has been introduced by the Urbana group [

There is another method used to achieve saturation properties in nuclear matter. One has to supplement the effective interaction or the self-energy of BHF and SCGF calculations by a simple contact interaction (CT), which we have chosen following the notation of the Skyrme interaction to be of the form

where _{0}, t_{3} and _{0} and t_{3} represent the zero range and 3-body strength while the exponent

Now the nuclear many-body approach plus the above contact term is fit to reproduce the saturation point ρ_{0} = 0.16 fm^{−}^{3} and E_{0} = −16.0 MeV at fixed _{3} term. Comparing the suggested approaches with the relativistic Brueckner-Hartree-Fock (RBHF) approach [_{0}, but the difference increases at high densities, which are a little bit softer than the BHF + CT results.

We will extend the analysis to pure neutron matter EOS, which is more suitable for neutron star studies, at densities up to about five times the saturation one. Moreover, we consider the calculations for two nucleon- nucleon potential, i.e., CD-Bonn interaction likes symmetric nuclear matter EOS. The results for the EOS of the pure neutron matter, obtained by including only two-body force, is reported (solid line for BHF and dashed line for SCGF approaches) in

The pressure of nuclear matter is defined in terms of the energy per nucleon as in Equation (16). In

Parameters | BHF | SCGF |
---|---|---|

t_{0} (MeV fm^{3}) | −221 | −327 |

t_{3} (MeV fm^{3+3α}) | 3750 | 3917 |

Model ρ (fm^{−3}) | BHF | SCGF | BHF + CT E/A (MeV) | SCGF + CT |
---|---|---|---|---|

0.08 | −12.373 | −7.209 | −13.364 | −12.729 |

0.16 | −18.28 | −10.754 | −16.013 | −16.061 |

0.24 | −21.966 | −11.285 | −13.676 | −12.647 |

0.32 | −23.624 | −8.914 | −7.073 | −3.313 |

0.40 | −23.889 | −4.499 | 2.853 | 10.672 |

0.48 | −22.749 | 1.249 | 15.91 | 28.315 |

0.56 | −20.169 | 7.863 | 31.979 | 48.948 |

0.64 | −16.667 | 15.722 | 50.427 | 72.789 |

0.72 | −12.811 | 25.164 | 70.589 | 100.05 |

0.80 | −8.822 | 35.524 | 92.167 | 129.96 |

Model ρ (fm^{−}^{3}) | BHF | SCGF | BHF+CT E/A (MeV) | SCGF+CT |
---|---|---|---|---|

0.08 | 8.094 | 10.43 | 7.103 | 4.91 |

0.16 | 11.871 | 17.572 | 14.137 | 12.265 |

0.24 | 16.247 | 25.886 | 24.538 | 24.525 |

0.32 | 21.704 | 35.563 | 38.255 | 41.165 |

0.40 | 28.559 | 46.614 | 55.301 | 61.784 |

0.48 | 36.583 | 59.33 | 75.242 | 86.396 |

0.56 | 45.576 | 73.946 | 97.724 | 115.03 |

0.64 | 55.989 | 90.175 | 123.083 | 147.241 |

0.72 | 68.164 | 107.572 | 151.563 | 182.458 |

0.80 | 81.354 | 125.569 | 182.344 | 220.006 |

and

There is another important characteristic of the EOS and it enters in the discussion of a variety of phenomena such as supernovae explosions or heavy ion collisions, that it is the incompressibility K. It measures the stiffness of the EOS, usually defined as a slope of the pressure at saturation point:

The experimental value of the incompressibility of symmetric nuclear matter at its saturation density ρ_{0} has been determined to be 210 ± 30 MeV [_{0} as one sees from

Using the methods just described we obtain for each model the EOS for cold pure neutron matter. The TOV general relativistic equations for a spherically symmetric (non-rotating) neutron star are solved, then the gravitational mass of the star M_{G} is obtained as a function of both the stellar radius R and a central density_{max} ≈ 1.98 M_{סּ}, in the case of BHF + CT calculation, at a central density of ρ_{c} = 2.82 × 10^{15} gm/cm^{3} with

Model | ρ_{0} (fm^{−3}) | E_{0} (MeV) | K (MeV) |
---|---|---|---|

BHF + CT | 0.16 | −16.01 | 209 |

SCGF + CT | 0.16 | −16.06 | 266 |

RBHF [ | 0.157 | −16.0 | 200 |

a radius R ≈ 10.05 km. This value is consistent with the recent observation of a (1.97 ± 0.04) M_{סּ} neutron star (Demorest et al. [_{סּ}, was observed very recently by Antoniadis et al. [_{max} ≈ 1.98 M_{סּ} at a central density of ρ_{c} = 2.52 × 10^{15} gm/cm^{3} with a radius R ≈ 10.5 km for SCGF+CT model. The present non-relativistic calculations are compared with RBHF results [_{max} ≈ 2.5 M_{סּ} at a central density of ρ_{c} = 1.995 × 10^{15} gm/cm^{3} with a radius R ≈ 11.58 km.

One can see that the results from our non-relativistic equation of state may look more reasonable than those from the relativistic one. The present results are more compatible with recent microscopic calculations of neutron matter based on nuclear interactions derived from chiral effective field theory [

Comparing the non-relativistic approaches from realistic interaction among themselves, we find that the effect of the hole-hole interaction has been found to be important; it significantly accounts for more repulsion at large densities. Also, it enhances the nuclear matter E/A and at the same time reduces the saturation density ρ_{0}, as compared with the BHF results. If we do not consider the contact term these characteristics lead to better values for the saturation point. In general, microscopic BHF or SCGF calculations cannot simultaneously reproduce nuclear matter E/A and ρ_{0}. To improve the situation, one may either extend to three-body force to the effective interaction or include relativistic corrections RBHF, then the EOS of pure neutron matter can be used to determine the structure of neutron stars.

Using the present EOS’s of pure neutron matter at absolute zero, the TOV equation of general relativity, from which the gross properties of neutron stars follow, e.g., mass and radius can be solved numerically. It turns out that both maximum stable masses and radii depend crucially on the stiffness of the suggested EOS. Neutron stars including realistic EOS give the following general results: star model calculated with a stiff EOS have a lower central density, a larger radius than do stars of the same mass computed from a soft EOS. There is another important consequence; NS parameters such as the total mass and radius, are sensitive to microscopic model calculations. Also, the results from non-relativistic equation of state may look more reasonable than those from the relativistic one.

The present research support from Taif University, Kingdom of Saudi Arabia under Project No. 1-434-2667 is acknowledged.