Nuclear Structure for the Crust of Neutron Stars

Transcription

1 Nuclear Structure for the Crust of Neutron Stars Peter Gögelein with Prof. H. Müther Institut for Theoretical Physics University of Tübingen, Germany September 11th, 2007

2 Outline Neutron Stars Pasta in a Wigner-Seitz Cell The Nuclear Many-Body Problem Relativistic Hartree-Fock Pairing Results Outlook P. Gögelein, IfTP Tübingen Nuclear Structure for the Crust of Neutron Stars 2/26

3 Creation of a Neutron Star Star burning ends, Supernova explosion, Final states of stars: white dwarf, neutron star, black hole. Properties of a typical neutron star: mass: 1.4 M, radius: 10 km, density: g/cm 3, temperature at creation: K (30 MeV), fast rotation up to 1000 Hz, frequency changes. P. Gögelein, IfTP Tübingen Nuclear Structure for the Crust of Neutron Stars 3/26

4 Main Processes in Neutron Stars The fermi energy of the neutrons causes a force against gravitation according to the Pauli principle. This fermi energy determines the abundance of electrons and protons via β equilibrium µ n = µ p + µ e. Superfluidity of matter influences the cooling rate and the rotational behaviour of the star. P. Gögelein, IfTP Tübingen Nuclear Structure for the Crust of Neutron Stars 4/26

5 The Structure of a Neutron Star Inner core: strange baryons? quark matter? Outer core: homogenous superfluid neutron and proton matter, magnetic flux tube, Crust: nuclei, spaghetti, lasagne, superfluid neutrons, Athmosphere: nuclei, magnetic field. P. Gögelein, IfTP Tübingen Nuclear Structure for the Crust of Neutron Stars 5/26

6 Pasta in a Wigner Seitz Cell Neutrons and protons in a Wigner Seitz Cell as representation of nuclear matter in the crust of neutron stars. A cartesian cell is considered so that the whole space can be covered by repeated cells. In addition non-spherical structures can be observed. P. Gögelein, IfTP Tübingen Nuclear Structure for the Crust of Neutron Stars 6/26

7 The Nuclear Many Body Problem a) The realistic approach: Construct NN potential, which reproduces phase shifts ( scattering amplitudes) of NN sattering, e.g. Bonn, Argonne, etc. Calculate nuclear structure using many body techniques, which resolve correlations, e.g. Brueckner Hartree Fock, effective low momentum interaction V lowk. b) The phenomenological approach: Construct approach for the many body Hamiltonian with sufficient free parameters (e.g. DFT: Skyrme force, effective meson exchange: RHF). Fit the free parameters to radii and binding energy of a set of nuclei and an equation of state. Capable of fast calculations or large systems. P. Gögelein, IfTP Tübingen Nuclear Structure for the Crust of Neutron Stars 7/26

8 Lagragian Density for Relativistic Hartree Fock consists of three parts: baryon, meson and interaction Lagrangian: L = L B + L M + L int, L B = Ψ( iγ µ µ M)Ψ, L M = 1 ( ) µ Φ i µ Φ i mi 2 Φ 2 i 2 i=σ,δ,π 1 2 κ=ω,ρ,γ ( 1 2 F (κ) µν m 2 κa (κ) µ A (κ)µ), L int = Ψg σ Φ σ Ψ Ψg δ τ Φ δ Ψ fπ m π Ψγ 5 γ µ τ [ µ Φ π ]Ψ Ψg ω γ µ A (ω)µ Ψ Ψg ρ γ µ τ A (ρ)µ Ψ + fρ 2M Ψσ µν τ [ ν A (ρ)µ ]Ψ Ψeγ µ 1 2 (1 + τ 3)A (γ)µ Ψ, with the field strength tensor: F µν (κ) = µ A (κ) ν ν A (κ) µ. P. Gögelein, IfTP Tübingen Nuclear Structure for the Crust of Neutron Stars 8/26

9 Model from Realistic NN Interaction Realistic approach: Dirac Brueckner Hartree Fock for nuclear matter (van Dalen, Fuchs, Faessler) Adjustment of density dependent coupling functions to reproduce self energy from DBHF results (LDA), and finite nuclei. Phenomenological approach: Density Dependent Relativistic Mean Field (DDRMF) for nuclear matter, finite nuclei, and the Wigner Seitz Cell. P. Gögelein, IfTP Tübingen Nuclear Structure for the Crust of Neutron Stars 9/26

10 Hartree Self Energy Density in the No-sea approximation ρ s = α η α ψ αγ 0 ψ α. Helmholtz equation with source term for the meson field ( + m 2 σ)φ σ = g σ ρ s. Self energy contributions to Σ H S, ΣH 0 or ΣH T Σ H σ = g σ Φ σ Σ H S. Rearrangement self energy contributes to Σ H 0 Σ (r),h σ = gσ ρ Φ σρ s Σ H 0. P. Gögelein, IfTP Tübingen Nuclear Structure for the Crust of Neutron Stars 10/26

11 Fock Self Energy The exchange integral over interacting nucleons Iβα(x) σ = d 3 y can be rewritten to a Helmholz equation [ ] g σ (y) ψ exp( mσ x y ) β (y)γ0 ψ α (y) x y ( + m 2 σ) I σ βα = g σ ψ β γ0 ψ α Fock self energy contributions are summarized in (Σ F ψ α ) (Σ F σψ α ) = g σ η β Iβαγ σ 0 ψ β (Σ F ψ α ). β Rearrangement self energy contributes to Σ H 0 Σ (r),f σ = gσ ρ η α η β ψ α Iβαγ σ 0 ψ β Σ H 0. α,β P. Gögelein, IfTP Tübingen Nuclear Structure for the Crust of Neutron Stars 11/26

12 Static Dirac HF Equation We obtain a Dirac equation for the baryons from the Dirac Hamiltonian in Hartree Fock approximation ε α ψ α = [ αp + iβασ H T + β(m + Σ H s ) + Σ H 0 + Σ F ] ψ α with the scalar, vector and tensor Hartree self energy ( Σ H S, ΣH 0 and ΣH T ), and the Fock self energy (Σ F ). Hartree self energy: σ, δ (scalar); ω, ρ (vector); Fock self energy: contains the π exchange in addition (pseudo vector). The rearrangement self energy contribution is contained in the time like Hartree vector self energy Σ H 0 due to variation of the baryon density. P. Gögelein, IfTP Tübingen Nuclear Structure for the Crust of Neutron Stars 12/26

13 Transforming Dirac s equation First Pauli spinor representation ist introduced ( ) ϕα ψ α =, then the eigenvalues are shifted χ α ε α ε α M, and finally we obtain the following equations: ε α ϕ α = σp χ χ α + U ϕ ϕ α + (Σ F ϕ α ) ε α χ α = σp ϕ ϕ α + U χ χ α + (Σ F χ α ). P. Gögelein, IfTP Tübingen Nuclear Structure for the Crust of Neutron Stars 13/26

14 Effective Schroedinger Equation Effective Schroedinger equation for upper Pauli spinor ε α ϕ α = σp χ B α ( σpϕ ϕ α + (Σ F χ α ) ) + U ϕ ϕ α + (Σ F ϕ α ), where the right hand side is called effective Hamiltonian H ϕ,α and contains an effective mass term : 1 B α = 2M + ε α + Σ H S. ΣH 0 The effective Schroedinger equation yields the spin orbit term: σp χ B α σp ϕ = p χ B α p ϕ + i(p χ B α ) (p ϕ σ). P. Gögelein, IfTP Tübingen Nuclear Structure for the Crust of Neutron Stars 14/26

15 Iterative Procedure The Helmholtz equations for the mesons are solved by the conjugate gradient method. The Dirac equation is solved by variation applying the imaginary time step. First the imaginary time step operates on the upper component ϕ (n+1) α then the lower component is calculated = exp( λh ϕ,α ) ϕ (n) α, χ (n+1) α = B α ( σp ϕ ϕ (n+1) α + (Σ F χ α ) (n)), and finally both components are orthonormalized together via the Gram Schmidt method considering the parities of the wave functions. P. Gögelein, IfTP Tübingen Nuclear Structure for the Crust of Neutron Stars 15/26

16 Pairing Pairs of nucleons with momentum k and k are considered. Gap equation with realistic potential V (k, k ) (k) = 2 π 0 dk k 2 V (k, k (k ) ) 2 (ɛ k ɛ F ) 2 + (k ). 2 Density dependent monopole pairing force ( ) κ ) ρ(r1 ) V (r 1, r 2 ) = V 0 (1 η δ(r 1 r 2 ). Parameters: V 0 = 481MeV fm 3, η = 0.7, κ = 0.45, and cut off energy ɛ c = 60 MeV (Garrido et al. Phys. Rev. C60 (1999) ). ρ 0 P. Gögelein, IfTP Tübingen Nuclear Structure for the Crust of Neutron Stars 16/26

17 Pairing and Finite Temperature Normal and anomalous occupation factors from FT-HFB: η α = (1 2n α )vα 2 + n α, ζ α = (1 2n α )u α v α. Anomalous density and local gap function χ(r) = 1 2 ζ α ψ α (r) 2, (r) = V (r) χ(r). α State dependend pairing gap α = d 3 r (r) ψ α (r) 2. (Montani, May, Müther, Phys. Rev. C69 (2004) ) P. Gögelein, IfTP Tübingen Nuclear Structure for the Crust of Neutron Stars 17/26

18 Shapes from Skyrme HF (SLy4) Quasi nuclear structures in a surrounding neutron sea are observed: droplet, rod, slab, inverse, and finally homogeneous matter. Smooth transition in a density range fm 3. P. Gögelein, IfTP Tübingen Nuclear Structure for the Crust of Neutron Stars 18/26

19 Shapes from Skyrme HF (SLy4) Rod structure Slab structure P. Gögelein, IfTP Tübingen Nuclear Structure for the Crust of Neutron Stars 19/26

20 Shapes from Relativistic Mean Field (DDRMF) T = 0 MeV T = 5 MeV Quasi nuclear structures less pronounced than from Skyrme HF: droplet, rod, and finally homogeneous matter. One structure less is observed at T = 5 MeV (right column). P. Gögelein, IfTP Tübingen Nuclear Structure for the Crust of Neutron Stars 20/26

21 Transition densities Transition densities at T = 0 MeV Skyrme DDRMF HF TF H TF droplet rod rod slab slab homogeneous Transition densities at T = 5 MeV Skyrme DDRMF HF TF H TF droplet rod rod slab slab homogeneous P. Gögelein, IfTP Tübingen Nuclear Structure for the Crust of Neutron Stars 21/26

22 Equation of State at T = 0 MeV Skyrme HF DDRMF Energy reduction and enhancement of proton abundance due to non homogenous phases and shell effects. Thomas Fermi Approximation needs isospin dependent surface energy constant F 0 to reproduce microscopic results. P. Gögelein, IfTP Tübingen Nuclear Structure for the Crust of Neutron Stars 22/26

23 Finite Temperature T = 5 MeV Skyrme HF DDRMF The free energy F = E TS shows less reduction than the nuclear energy E because of the entropy. Proton abundance slightly enhanced compared to T = 0 MeV. P. Gögelein, IfTP Tübingen Nuclear Structure for the Crust of Neutron Stars 23/26

24 Comparison of Pairing at ρ B = fm 3. Anomalous neutron density Local gap function Reduction of the pairing gap in the region of the quasi nucleus of about 25% (Skyrme) or 40% (RMF). Enhancement of the anomalous density χ(r) in the region of the quasi nucleus. = reduction due to density dependent pairing force. This reduction may attrack vortices and influence the neutrino opacity. P. Gögelein, IfTP Tübingen Nuclear Structure for the Crust of Neutron Stars 24/26

25 Summary Wigner Seitz cell Smooth transition from inhomogenous to homogenous matter is investigated. Shell effects lead to an increase of proton abundance. Non spherical structures are observed. Decrease of the local pairing gap in the region of the quasi nucleus. Finite temperature effects are explored. The method can be applied to equation of state calculations. (P.Gögelein and H. Müther, nucl.-th/ , Phys. Rev. C) Exotic nuclei Binding energy and radii can be reproduced. Neutron drip line not reproduced for Oxygen. Level crossings may influence excitations and decay modes. P. Gögelein, IfTP Tübingen Nuclear Structure for the Crust of Neutron Stars 25/26

26 Outlook Improvement of Thomas Fermi based EoSs for astrophysical application (e.g. Shen et al and 2002) possible. Application of realistic NN interactions: Implementation of low momentum interaction V low k in progress. The effective density dependent pairing force should be checked by realistic pairing interactions. Calculate other observables like response functions and the neutrino opacity. Further study of nuclei at the neutron drip line. P. Gögelein, IfTP Tübingen Nuclear Structure for the Crust of Neutron Stars 26/26