Transcript Modeling Nuclear Pasta and the Transition to Uniform

Modeling Nuclear Pasta and the Transition
to Uniform Nuclear Matter with the 3D
Hartree-Fock Method
W.G.Newton1,2, Bao-An Li1, J.R.Stone2,3
1Texas A&M University - Commerce
2University of Oxford, UK
3Physics Division, ORNL, Oak Ridge, TN, USA
Contents
•
•
•
•
•
Motivation
Computational Method + Tests
Results: SN matter
Results: NS matter
Future Developments and Conclusions
Structure of Supernovae and Neutron Stars
Supernova (Finite Temperature)
Neutron Star
General Motivation
Microphysics of (hot, >1010K ), dense matter
•Nuclear models/QCD
•Weak interactions
Bulk Properties of (hot, >1010K ) Matter:
•Thermal/electrical conductivity
•Elastic properties (Bulk, shear
modulus)
•Hydrodynamic properties
(entrainment)
•Equation of State P = P(ρ,T)
Macrophysical Stellar Models
•Inclusion of GR
Calculation of observables and
confrontation with observation
•SNe Energetics, neutrino signal
•Radio/X-ray Pulsars
•Bursts from NSs (XRBs/SGRs)
•NS cooling
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...
Specific Motivation: The Phase
Transition to Uniform Matter
Supernova (Finite Temperature)
Neutron Star
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.
Nuclear Pasta vs Complex Fluids
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
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 potential
φ(x,y,z) = φ(x+L,y+L,z+L)
•
In progress: general Bloch boundary conditions
(relevant in NS crusts)
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
Currently spin-orbit is omitted to speed up computation
BCS pairing (Constant gap)
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 yp)
– neutron quadrupole moments β,γ
• Minimize energy density w.r.t. free parameters
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)
Computational Method IV
• Initial Wavefunctions:
Gaussian x Polynomial (GP) or
Plane wave (FD)
– < 0.01% difference between choices of initial wavefunctions
• 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
Effects of Boundary
Conditions? Pt I
Effects of Boundary
Conditions? Pt II
T=5MeV
nb=0.12fm-3
Spurious shell
discretization
continuum
effects from
of
neutron
Results: SN Matter
• yp = 0.3
• Include only n,p,e
• SkM* (mainly) and Sly4
Constant
deformation
sequences
Energy Surfaces in Deformation Space:
Energy-density surfaces with
increasing density
Equation of State: T=2.5 MeV
Free Energy
EoS Non-uniform vs Uniform Matter
Pressure
EoS Non-uniform vs Uniform Matter
Phase Transition: 1st or 2nd Order?
Entropy
EoS Non-uniform vs Uniform Matter
Phase Transition: 1st or 2nd Order?
T = 0.0 – 7.5 MeV, yp=0.3, nb=0.10fm-3
Transition to
Uniform Matter
with Increasing
temperature
New Pasta!
Bicontinuous Cubic-P Phase
Results: NS Matter
• yp determined by beta equlibrium
• Include only n,p,e
• SkM* and Sly4
Contour Plot: Energy density vs A,Z; nb = 0.06 fm-3
β = 0.0
SLy4
SkM*
β = 0.12
SkM*
(A,Z) = (500,14)
(A,Z) = (900,30)
SLy4
(A,Z) = (500,10)
(A,Z) = (900,20)
Effect of Proton Fraction on Appearance of Pasta?
y
Z = 10
Transition to
uniform matter
with increasing
density
T = 0.0 MeV,
A = 500
nb=0.06–0.10fm-3
Z = 20
Z = 30
Current Developments I: Transition
density
• Detailed search over densities to find the
transition point to uniform matter
– 1st or 2nd order?
– Dependence on nuclear matter properties
(symmetry energy)
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:
Current Developements III: Addition of Bloch
Boundary Conditions
>
(Carter, Chamel and Haensel, arXiv:nucl-th/0402057)
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)
Speculation: Ordering a disordered
phase
• B = 1015G > EB=10 keV fm-3
• Energy differences between various minima in
deformation space = 1-10keV fm-3
• Possible ordering agent?
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
• 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?