1. Sub-Nuclear Matter in Neutron Stars and Supernovae Nuclear Pasta and Complex Fluids
W.G.Newton1, J.R.Stone1,2
1University of Oxford, UK
2Physics Division, ORNL, Oak Ridge, TN, USA

2. Outline Overview of NS, SN Matter The Transition to Uniform Matter
Anatomy of Supernovae (SNe) and Neutron Stars (NSs)
Superfluidity in NSs
The Transition to Uniform Matter
Astrophysical consequences
Frustration and Complex Fluids
Nuclear Pasta
Self-consistent models
QMD
Hartree-Fock
A new Hartree-Fock study of nuclear pasta
Computational Method
Preliminary Results
Conclusions

3. Anatomy of Core Collapse SN
Collapse proceeds until core reaches few times nuclear saturation density (≈2.4×1014g cm-3 or 0.16 baryons fm-3)
Neutrinos initially trapped above densities of (≈1012g cm-3), temperatures reach up to ≈100MeV and the proton fraction is roughly constant at ≈0.3

4. Anatomy of a Neutron Star
<1011 g cm-3 Nuclear physics relatively well known (heavy nuclei)
>4×1011 g cm-3 (neutron drip): nuclear models begin to diverge
>5×1014 g cm-3 physics is extremely uncertain (Hyperons? Meson condensates? Quarks? When does description in terms of nucleonic degrees of freedom become unphysical?

5. Superfluids in Neutron Stars
At temperatures below a critical temperature of ≈ 109K K, neutrons in the inner crust and core are expected to become superfluid (and, in the core, protons superconducting).
Superfluids have zero viscosity, and so cannot support bulk rotation.
If a fluid, rotating with period P(s), is cooled below the critical temperature, it arranges itself into quantized vortices of spin, density 104 /P cm-2

6. Superfluid Vortices
Quantized vortices in a sodium gas cooled into a Bose-Einstein condensate and set into rotation (Onofrio et al, Phys Rev Lett 85, 2228, 2001)

7. Transition to Uniform Matter
The density regime 1013 < ρ < 2×1014 g/cm3 is important
It marks the transition from the outer crystalline crust of a NS, or the gas of nuclei, neutrons and leptons in a core collapse, to the liquid, homogeneous phase above nuclear saturation density

8. The Transition to Uniform Matter: Astrophysical Consequences
Neutrino opacities and emission mechanisms
during core collapse
neutron star cooling
Pulsar Glitches
star-quakes
superfluid vortex dynamics
Pinned vortices?
Change in crustal composition and reheating during accretion
NS Oscillation
GWs

9. Frustration and Complexity
If a system contains energetically favourable (attractive) and unfavourable (repulsive) interactions operating over the same range, matter will be frustrated.
Prototypical frustrated system: Ising anti-ferromagnet on triangular lattice.
Impossible to minimize energy with respect to all interactions simultaneously
Large number of low energy configurations result
At densities just below nuclear saturation (1013 – 1014 g cm-3) the
distances between Coulomb repelling nuclei becomes comparable
with the range of the attractive nuclear interaction that binds nuclei.
Complex structures thus develop – nuclear pasta.

10. Nuclear 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 (Ravenhall et al Phys. Rev. Lett. 50, 2066, 1983 and Hashimoto et al, Progress of Th. Physics, 71, 2, 320, 1984).
The basic sequence is
(a) spherical (meatball/gnocchi) → (b) rod (spaghetti) → (c) slab (lasagna) → (d) tube (penne) → (e) bubble (swiss cheese?) → uniform matter

11. Nuclear Pasta vs Complex Fluids
A wide range of mechanical properties are exhibited (liquid crystal,
sponge, rubber…)
Pethick, C.J. and Potekhin, A.Y. – Liquid Crystals in the Mantles of
Neutron Stars – Phys. Lett. B, 427, 7, 1998

12. Self-consistent Modeling: QMD
Quantum Molecular Dynamics (QMD): semi-classical dynamical simulations with nucleonic degrees of freedom (Watanabe and Sonoda, nucl-th/ ).
Pasta shapes emerge without pre-conditioning.
Pasta formation from compression and cooling demonstrated.
0.1ρ ρ ρ ρ ρ0

13. Self-consistent Modeling: Mean field
Magierski and Heenen PRC (2001): 3D HF calculation of nuclear shapes at bottom of neutron star crust at zero T
When examined self-consistently in three dimensions, many more configurations emerge
- has effect of smoothing EoS
An important new phenomenon emerges: the fermionic Casimir effect. Scattering of unbound nucleons off nuclear structures leads to an effective interaction between those structures of order the energy difference between configurations

14. Computational Method: Skyrme HF
Choose phenomenological nuclear interaction (Skyrme)
Assume one can identify (local) unit rectangular cells of matter at a given density and temperature, calculate one unit cell containing A nucleons (A up to 5000)
Hartree-Fock approximation: system can be represented by a single Slater determinant.
Minimize energy w.r.t. single particle wavefunctions: Schrödinger equation for A nucleons → A Schrödinger equations (A up to 5000)
Periodic boundary conditions φ(x,y,z) = φ(x+L,y+L,z+L) (More generally Bloch boundary conditions φ(x,y,z) = eikr 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 cell
Additional free parameters: A, (proton fraction yp), proton and neutron quadrupole moments Qp,20, Qp,22

15. Unconstrained calculation at 8 densities between 0. 01fm-3 and 0
Unconstrained calculation at 8 densities between 0.01fm-3 and 0.12fm-3, T=0MeV, yp=0.03: Self-consistent dissolution of nuclear structure

16. Integrated Densities at nb = 0.0195fm-3

17. Integrated Densities at nb = 0.0312fm-3

18. Integrated Densities at nb = 0.0390fm-3

19. Integrated Densities at nb = 0.0507fm-3

20. Integrated Densities at nb = 0.0585fm-3

21. Integrated Densities at nb = 0.0702fm-3

22. Integrated Densities at nb = 0.0780fm-3

23. Integrated Densities at nb = 0.0976fm-3

24. Minimization with respect to A
T = 2.5MeV, nb=0.04fm-3
Minimization with respect to quadrupole moments is obtained in a similar way

25. T=5MeV nb=0.12fm-3 Boundary Conditions and Shell Effects

26. Pasta phase superimposes
artificial and real oscillations,
and real minima, on the curve

27. Conclusions and Future
The properties of matter in the density region 1013 < ρ < 2×1014 g/cm3 are an important ingredient in NS and SN models
Thorough microphysical description of transition to uniform matter – the nuclear pasta phases – is underway
Generalize boundary conditions to the Bloch form: φ(x,y,z) = eikr φ(x+L,y+L,z+L)
Calculate entrainment coefficient
Examine response of matter to perturbation
neutrino interactions
mechanical properties
Investigate effects of BCS pairing
The Future(?)
Hydrodynamical modeling of pasta phases
Mesoscopic structures