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Nuclear “pasta” phase in a triaxial relativistic mean field approach
Progress in Particle and Nuclear Physics 59 (2007) 206–208 www.elsevier.com/locate/ppnp
Review
Nuclear “pasta” phase in a triaxial relativistic mean field approach P. G¨ogelein Institut f¨ur Theoretische Physik, Universit¨at T¨ubingen, Germany
Abstract The crust of a neutron star is of special interest because of the phase transition from a gas of nuclei to homogeneous asymmetric nuclear matter. Several efforts have been made to explore the nuclear matter in this region of the neutron star. These calculations have been done mostly in spherical Wigner–Seitz-cells using Skyrme Hartree–Fock (HF) or relativistic mean field (RMF) calculations. We apply both approaches in a Cartesian Wigner–Seitz-cell and get a smooth transition from nuclei to homogeneous matter showing various shapes. In this article we pay special attention to the relativistic mean field. Pairing is included in the BCS approximation. c 2007 Elsevier B.V. All rights reserved.
Keywords: Neutron star; Neutron star crust; Relativistic mean field; Relativistic hartree fock; Equation of state
1. Relativistic mean field approach In nuclear theory there are non-relativistic and relativistic approaches. In contrast to nonrelativistic calculations the relativistic Brueckner Hartree–Fock approach is able to reproduce the nuclear saturation point from realistic NN potentials e.g. the Bonn potentials. Further it is possible to include a local density approximation from the Dirac Brueckner Hartree–Fock (DBHF) calculations [1] into the RMF calculations, which also provides a good description for finite nuclei. The Lagrangian is taken from Hofmann et al. [1] for the density dependent Hartree approximation (DDRH) or from Long et al. [2] for the density dependent Hartree–Fock (DDRHF) approximation. Density dependent meson–baryon vertices Γ (ρ) depending on the E-mail address: [email protected]. c 2007 Elsevier B.V. All rights reserved. 0146-6410/$ - see front matter doi:10.1016/j.ppnp.2006.12.001
P. G¨ogelein / Progress in Particle and Nuclear Physics 59 (2007) 206–208
207
Fig. 1. Results for L–HS: binding energy per nucleon, proton abundance and local gap function for droplet shape.
baryon density ρ = hΨ¯ γ0 Ψ i are introduced. The density dependence of the vertex functions is taken from Dirac Brueckner–Hartree–Fock calculations for asymmetric nuclear matter [3] by a local density approximation (LDA) for the Brueckner G-matrix. Performing the variation of the baryon field, rearrangement terms are obtained from the density dependent vertices. The Klein–Gordan and Proca equations have to be used to express the meson and photon fields in terms of nucleon fields Ψ only. Neglecting the retardation effects due to big meson masses m k 1 exp(−m k |x−y|) the meson propagators take on the Yukawa form Dk (x, y) = 4π and the nucleonic |x−y| Hamiltonian can be written as: Z Z 1 ¯ d3 xd3 y H = d3 x[ψ[−iγ · ∇ + M]ψ] + 2 X ¯ ψ(y)[Γ ¯ × ψ(x) (1) k (x, y)Dk (x, y)]ψ(y)ψ(x), k=σ,ω,ρ,π,A
with Γk the interaction vertices. This Hamiltonian is solved in the Hartree or Hartree–Fock approximation together with pairing. The latter is treated in the BCS approximation in a way similar to Montani et al. [5] with the parameters v0 = 481 MeV fm3 , η = 0.7, α = 0.45 and the cut-off energy c = 60 MeV. 2. First results All calculations are performed in β-equilibrium which is present in a cold neutron star. The electrons are treated as relativistic quantum gas. The different results in Fig. 1 are obtained in the Hartree approximation using a linear model from Horowitz and Serot (L–HS) described in [4]. In the first two graphs the line corresponds to the infinite matter calculation, the small square points to the homogenous matter calculation in the box and the circle points to the RMF calculations. The box size has been chosen from 20–32 fm in steps of 4 fm. A good agreement of the infinite matter calculation and the plane wave calculation in the box can be recognized, which means that the discretization is good. The energy in the non-homogeneous phase is reduced considerably and we get a smooth transition to the homogeneous phase which was not the case in spherical calculations [5]. Further the proton abundance is enhanced at low densities in the nonhomogeneous phase. The minimum of the proton abundance is lower in the RMF calculations than in the Skyrme HF calculations [5]. Pairing gaps are reduced in regions where protons are present as we see in the third graph in Fig. 1. This reduction may attract vortices and influence the neutrino opacity. As an example of a non-spherical shape the rod-like shape is displayed in Fig. 2. The shapes vary smoothly from droplet to rod-like over grid-like and finally to homogeneous matter.
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P. G¨ogelein / Progress in Particle and Nuclear Physics 59 (2007) 206–208
Fig. 2. Rod-like shape for RMF L–HS: proton density ρ p over the x = 0, y = 0 and z = 0 plane.
References [1] [2] [3] [4] [5]
F. Hofmann, C.M. Keil, H. Lenske, Phys. Rev. C 64 (2001) 034314. W.H. Long, N. Van Giai, J. Meng, nucl-th/0608009 v2 (preprint). E.N.E. van Dalen, C. Fuchs, A. Faessler, Nuclear Phys. A 744 (2004) 227. C.J. Horowitz, B.D. Serot, Nuclear Phys. A 368 (1981) 503. F. Montani, C. May, H. M¨uther, Phys. Rev. C 69 (2004) 065801.
Review
Nuclear “pasta” phase in a triaxial relativistic mean field approach P. G¨ogelein Institut f¨ur Theoretische Physik, Universit¨at T¨ubingen, Germany
Abstract The crust of a neutron star is of special interest because of the phase transition from a gas of nuclei to homogeneous asymmetric nuclear matter. Several efforts have been made to explore the nuclear matter in this region of the neutron star. These calculations have been done mostly in spherical Wigner–Seitz-cells using Skyrme Hartree–Fock (HF) or relativistic mean field (RMF) calculations. We apply both approaches in a Cartesian Wigner–Seitz-cell and get a smooth transition from nuclei to homogeneous matter showing various shapes. In this article we pay special attention to the relativistic mean field. Pairing is included in the BCS approximation. c 2007 Elsevier B.V. All rights reserved.
Keywords: Neutron star; Neutron star crust; Relativistic mean field; Relativistic hartree fock; Equation of state
1. Relativistic mean field approach In nuclear theory there are non-relativistic and relativistic approaches. In contrast to nonrelativistic calculations the relativistic Brueckner Hartree–Fock approach is able to reproduce the nuclear saturation point from realistic NN potentials e.g. the Bonn potentials. Further it is possible to include a local density approximation from the Dirac Brueckner Hartree–Fock (DBHF) calculations [1] into the RMF calculations, which also provides a good description for finite nuclei. The Lagrangian is taken from Hofmann et al. [1] for the density dependent Hartree approximation (DDRH) or from Long et al. [2] for the density dependent Hartree–Fock (DDRHF) approximation. Density dependent meson–baryon vertices Γ (ρ) depending on the E-mail address: [email protected]. c 2007 Elsevier B.V. All rights reserved. 0146-6410/$ - see front matter doi:10.1016/j.ppnp.2006.12.001
P. G¨ogelein / Progress in Particle and Nuclear Physics 59 (2007) 206–208
207
Fig. 1. Results for L–HS: binding energy per nucleon, proton abundance and local gap function for droplet shape.
baryon density ρ = hΨ¯ γ0 Ψ i are introduced. The density dependence of the vertex functions is taken from Dirac Brueckner–Hartree–Fock calculations for asymmetric nuclear matter [3] by a local density approximation (LDA) for the Brueckner G-matrix. Performing the variation of the baryon field, rearrangement terms are obtained from the density dependent vertices. The Klein–Gordan and Proca equations have to be used to express the meson and photon fields in terms of nucleon fields Ψ only. Neglecting the retardation effects due to big meson masses m k 1 exp(−m k |x−y|) the meson propagators take on the Yukawa form Dk (x, y) = 4π and the nucleonic |x−y| Hamiltonian can be written as: Z Z 1 ¯ d3 xd3 y H = d3 x[ψ[−iγ · ∇ + M]ψ] + 2 X ¯ ψ(y)[Γ ¯ × ψ(x) (1) k (x, y)Dk (x, y)]ψ(y)ψ(x), k=σ,ω,ρ,π,A
with Γk the interaction vertices. This Hamiltonian is solved in the Hartree or Hartree–Fock approximation together with pairing. The latter is treated in the BCS approximation in a way similar to Montani et al. [5] with the parameters v0 = 481 MeV fm3 , η = 0.7, α = 0.45 and the cut-off energy c = 60 MeV. 2. First results All calculations are performed in β-equilibrium which is present in a cold neutron star. The electrons are treated as relativistic quantum gas. The different results in Fig. 1 are obtained in the Hartree approximation using a linear model from Horowitz and Serot (L–HS) described in [4]. In the first two graphs the line corresponds to the infinite matter calculation, the small square points to the homogenous matter calculation in the box and the circle points to the RMF calculations. The box size has been chosen from 20–32 fm in steps of 4 fm. A good agreement of the infinite matter calculation and the plane wave calculation in the box can be recognized, which means that the discretization is good. The energy in the non-homogeneous phase is reduced considerably and we get a smooth transition to the homogeneous phase which was not the case in spherical calculations [5]. Further the proton abundance is enhanced at low densities in the nonhomogeneous phase. The minimum of the proton abundance is lower in the RMF calculations than in the Skyrme HF calculations [5]. Pairing gaps are reduced in regions where protons are present as we see in the third graph in Fig. 1. This reduction may attract vortices and influence the neutrino opacity. As an example of a non-spherical shape the rod-like shape is displayed in Fig. 2. The shapes vary smoothly from droplet to rod-like over grid-like and finally to homogeneous matter.
208
P. G¨ogelein / Progress in Particle and Nuclear Physics 59 (2007) 206–208
Fig. 2. Rod-like shape for RMF L–HS: proton density ρ p over the x = 0, y = 0 and z = 0 plane.
References [1] [2] [3] [4] [5]
F. Hofmann, C.M. Keil, H. Lenske, Phys. Rev. C 64 (2001) 034314. W.H. Long, N. Van Giai, J. Meng, nucl-th/0608009 v2 (preprint). E.N.E. van Dalen, C. Fuchs, A. Faessler, Nuclear Phys. A 744 (2004) 227. C.J. Horowitz, B.D. Serot, Nuclear Phys. A 368 (1981) 503. F. Montani, C. May, H. M¨uther, Phys. Rev. C 69 (2004) 065801.