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

2. Contents Motivation Computational Method + Tests Results: SN matter Results: NS matter Future Developments and Conclusions

3. Structure of Supernovae and Neutron Stars Supernova (Finite Temperature)Neutron Star

4. General Motivation Microphysics of (hot, >10 10 K ), dense matter Nuclear models/QCD Weak interactions Bulk Properties of Matter: Bulk Properties of (hot, >10 10 K ) Matter: Thermal/electrical conductivityThermal/electrical conductivity Elastic properties (Bulk, shear modulus)Elastic properties (Bulk, shear modulus) Hydrodynamic properties (entrainment)Hydrodynamic properties (entrainment) Equation of State P = P(ρ,T)Equation of State P = P(ρ,T) Calculation of observables and confrontation with observation SNe Energetics, neutrino signal Radio/X-ray Pulsars Bursts from NSs (XRBs/SGRs) NS cooling Macrophysical Stellar Models Inclusion of GR

5. General Motivation: Consistency of NS/SN Models In order to derive real physics from observation: – Construct the EoS using the same underlying physical model and the same level of approximation over the whole range of densities and temperatures realised in SNe and NSs. – Calculate the EoS self-consistently across all relevant phase transitions and where multiple phases co-exist – Quantities that are specified by a given EoS (e.g. pressure, energy density) should be consistently extended to include, for example, specific heat, entrainment, shear moduli...

6. Specific Motivation: The Phase Transition to Uniform Matter Supernova (Finite Temperature) Neutron Star

7. Pasta Competition between surface tension and Coulomb repulsion of closely spaced heavy nuclei results in a series of shape transitions from the inner crust to the core Hashimoto, Seki and Yamada, Progress of Th. Physics, 71 no. 2, 320, 1984 Ravenhall, Pethick and Wilson Phys. Rev. Lett. 50, 2066, 1983 “… after all, the cooking of spaghetti, while it spoils the perfect straightness of the strands, does not destroy the characteristic short range order” Nuclear Pasta! (a) spherical (gnocchi) → (b) rod (spaghetti) → (c) slab (lasagna) → (d) tube (penne) → (e) bubble (swiss cheese?) → uniform matter Accounts for up to 20% mass of collapsing stellar core; up to 50% mass and radius of NS inner crust Unlikely to be solid at zero temperature; analogous to terrestrial condensed matter Pethick, C.J. and Potekhin, A.Y. – Liquid Crystals in the Mantles of Neutron Stars – Phys. Lett. B, 427, 7, 1998 3D structure demands a treatment beyond the spherical Wigner-Seitz approx.

8. Nuclear Pasta vs Complex Fluids

9. 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

10. Computational Method I 3D Hartree-Fock calculations with phenomenological Skyrme model for the nuclear force 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)

11. 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

12. Computational Method III Computer resources used – Jacquard (NERSC), Lawrence-Berkely (725 proc) – Jaguar (NCCS), Oak Ridge (11,000 proc) – Milipeia, Universidade de Coimbra (125 proc)

13. Computational Method IV Dependence on grid spacing: – Single particle energies differ by 0.01% when increasing grid spacing from 1fm to 1.1fm at T = 0MeV – Differences decrease with grid spacing (smaller spacing = smaller difference) – Differences increase with temperature (larger no. of wavefunctions required) – Optimal grid spacing: 1fm up to T = 5MeV Initial Wavefunctions: Gaussian x Polynomial (GP) or Plane wave (FD) – < 0.01% difference between choices of initial wavefunctions

14. Effects of Boundary Conditions? Pt I

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

16. Results: SN Matter y p = 0.3 Include only n,p,e SkM * (mainly) and Sly4

17. Constant deformation sequences

18. Energy Surfaces in Deformation Space:

19. Energy-density surfaces with increasing density

20. Equation of State: T=2.5 MeV

21. EoS Non-uniform vs Uniform Matter Free Energy

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

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

24. Transition to Uniform Matter with Increasing temperature T = 0.0 – 7.5 MeV, y p =0.3, n b =0.10fm -3

25. New Pasta! Bicontinuous Cubic-P Phase

26. Results: NS Matter y p determined by beta equlibrium Include only n,p,e SkM * and Sly4

27. SLy4 SkM* β = 0.0β = 0.12 Contour Plot: Energy density vs A,Z; n b = 0.06 fm -3

28. (A,Z) = (500,10) (A,Z) = (500,14) (A,Z) = (900,30) (A,Z) = (900,20) SLy4SkM*

29. Effect of Proton Fraction on Appearance of Pasta? y

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

31. 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)

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

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

34. Current Developements III: Addition of Bloch Boundary Conditions Kostas Glampedakis, Lars Samuelsson and Nils Andersson - A toy model for global magnetar oscillations with implications for quasi-periodic oscillations during flares MNRAS 371, Issue 1, L74 (2006)

35. Speculation: Ordering a disordered phase B = 10 15 G > E B =10 keV fm -3 Energy differences between various minima in deformation space = 1-10keV fm -3 Possible ordering agent?

36. 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 Limitations: long wavelength effects not included > complimentary to molecular dynamics simulations 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 – Pasta phases likely to be disordered; does an ordering agent exist?