1. Modeling Nuclear Pasta and the Transition to Uniform Nuclear Matter with the 3D Hartree-Fock Method W.G.Newton 1,2, Bao-An Li 1, J.R.Stone 2,3 1 Texas A&M University - Commerce 2 University of Oxford, UK 3 Physics Division, ORNL, Oak Ridge, TN, USA Compstar 2009 Workshop, Coimbra

2. Contents Motivation Computational Method + Tests Results – SN Matter – NS Matter – Transition to Uniform Matter Future Developments and Conclusions

3. Supernova (Finite Temperature) Neutron Star Motivation: Why a New 3d-HF Study? Self-consistently develop the EoS from lower densities (1D-HF) to higher densities (uniform matter) through the crust-core transition regime which likely includes the pasta phases, where the spherical Wigner-Seitz approximation

4. Motivation: Why a New 3d-HF Study? (cf. – Magierski and Heenen PRC65 045804 (2001): 3D-HF calculation of nuclear shapes at bottom of neutron star crust at zero T – Gogelein and Muther, PRC76 024312 (2007): RMF approach, finite-T) A careful examination of the effects of the numerical procedure on the results is needed To self-consistently explore the energies of various nuclear shapes, a constraint on both independent nucleon density quadrupole moments is required To study supernova matter and properties such as the specific heat of the NS inner crust, finite temperature calculations are required Transport properties of matter such as conductivities and entrainment require a calculation of the band structure of matter Previously, 3D-HF calculations have covered only a limited number of densities, temperatures and proton fractions Self-consistent determination of density range of pasta and transition density; dependence on nuclear matter properties

5. Computational Method I 3D Hartree-Fock calculations with Skyrme energy-density functional Assume one can identify (local) unit cubic cells of matter at a given density and temperature, calculate one unit cell containing A nucleons (A up to 3000) Periodic boundary conditions enforced by using FTs to take derivatives and obtain Coulomb potentialPeriodic boundary conditions enforced by using FTs to take derivatives and obtain Coulomb potential φ(x,y,z) = φ(x+L,y+L,z+L) φ(x,y,z) = φ(x+L,y+L,z+L) In progress: general Bloch boundary conditions In progress: general Bloch boundary conditions (relevant in NS crusts) φ(x,y,z) = e ikr φ(x+L,y+L,z+L) φ(x,y,z) = e ikr φ(x+L,y+L,z+L) Impose parity conservation in the three dimensions: tri-axial shapes allowed, but not asymmetric ones. Solution only in one octant of cellImpose parity conservation in the three dimensions: tri-axial shapes allowed, but not asymmetric ones. Solution only in one octant of cell Currently spin-orbit is omitted to speed up computationCurrently spin-orbit is omitted to speed up computation BCS pairing (Constant gap)BCS pairing (Constant gap)

6. Computational Method II Quadrupole Constraint placed on neutron density > self consistently explore deformation space Parameterized by β,γ; β is the magnitude of the deformation; γ is the direction of the deformation Free parameters at a given density and temperature – A/cell size, – (proton fraction y p ) – neutron quadrupole moments β,γ Minimize energy density w.r.t. free parameters

7. Effects of Boundary Conditions? Pt I

8. T=5MeV n b =0.12fm -3 Spurious shell effects from discretization of neutron continuum Effects of Boundary Conditions? Pt II

9. SN Matter: Energy Surfaces in Deformation Space

10. SN Matter Results: Equation of State: SkM*, y p =0.3, T=2.5 MeV

11. NS Matter Results

12. Properties of Skyrmes Used:

14. Finding Beta Equilibrium Minima: Contour plot of Energy density vs A,Z; Sly4, n b = 0.06 fm -3

15. Minima w.r.t. A in beta-eqb.

16. Energy-Deformation Surfaces

18. Phase Transition to Uniform Matter

19. EoS Non-uniform vs Uniform Matter Free Energy

20. SN Matter EoS Non-uniform vs Uniform Matter Pressure Phase Transition: 1 st or 2 nd Order?

21. EoS Non-uniform vs Uniform Matter Entropy Phase Transition: 1 st or 2 nd Order?

22. T = 0.0 MeV, A = 500 n b =0.06–0.10fm -3 Transition to uniform matter with increasing density Z = 10Z = 20Z = 30

23. Sly4 Phase Transition 0.080 < n t < 0.083 fm -3 Pressure discontinuity indicative of first-order phase transition

24. Current Developments I: Transition density Detailed search over densities to find the transition point to uniform matter – 1 st or 2 nd order? – Dependence on nuclear matter properties (symmetry energy...)

25. Current Developments II: Subtraction of Spurious Shell Energy Semiclassical method (WKB) approximation: leading order term in the fluctuating part of the level density for a Fermi gas in a rectangular box:

26. Current Developements III: Addition of Bloch Boundary Conditions > (Carter, Chamel and Haensel, arXiv:nucl-th/0402057)

27. Conclusions and Future The properties of matter in the density region 10 13 < ρ < 2×10 14 g/cm 3 are an important ingredient in NS and SN models 3D HF method applied to pasta phases – Inclusion of microscopic (shell) effects – Band structure can be calculated > transport properties – Finite T > SN matter, specific heat – Effects of computational procedure well accounted for Calculation of the transition density to uniform matter and density (and temperature) region of pasta has begun; how does it depend on the properties of the nuclear force used (symmetry energy) Implications for crust phenomenology: – Pasta phases unlikely to be solid; extent of pasta phases, and hence solid region of crust, depends on the EoS of nuclear matter at sub-saturation densities – Pasta phases likely to be disordered; does an ordering agent exist?