SLIDE 1 Stability of pasta phases
- Rigorous treatment of pasta phases in liquid drop
model
- Stability of the spaghetti and lasagna phase
Sebastian Kubis Institute of Physics Cracow University of Technology
in cooperation with W.Wójcik
POLNS18, Warszawa
SLIDE 2 Neutron star structure
atmosphere (ionized light elements: C , O, ..) e – degeneracy, Coulomb crystal of nuclei neutron drip → exotic nuclei - proton clusters in quasi-free neutron liquid cluster deformation → pasta phases neutronization of nuclei
cartoon by W.G.Newton
SLIDE 3
Pasta appearance in the crust-core transition region
proton clusters in neutron gas = sharp boundary between two phases with opposite charge
d=3 - gnocchi d=2 - spaghetti d=1 - lasagna
contribution from surface and Coulomb energy
Ravenhall, Pethick 1983
SLIDE 4
Pasta appearance in the crust-core transition region
proton clusters in neutron gas = sharp boundary between two phases with opposite charge
d=3 - gnocchi d=2 - spaghetti d=1 - lasagna
contribution from surface and Coulomb energy
Ravenhall, Pethick 1983
isolated Wigner-Seitz cell
SLIDE 5
Development in other approaches to pasta phases Hartree-Fock
Time-Dependent HF
Schuetrumpf `13
variety of structures… not only rods and slabs
W.G. Newton '09 Temperature ??
Periodic boundary conditions
SLIDE 6 Development in other approaches to pasta phases Molecular Dynamics
- two body potential for pointlike particles
- 50 000 nucleons in a box
- temperature ~ 1 MeV
- periodic boundary
Horowitz '15 Schneider '13
SLIDE 7 Rayleigh-Plateau instability, 1878
perturbed cylinder volume preserving deformation
unstable if
L
SLIDE 8
- boundary of D, surface
- normal deformation
Surface area change up to the 2nd order
- mean curvature
- Gaussian curvature
surface variation under conserved volume
Volume change up to the 2nd order
SLIDE 9 mean curvature (external, depends on immersion) Gaussian curvature (internal, depends on the metric → Ricci scalar in GR)
s – arc length
SLIDE 10
instability – variation analysis
L
conserved volume surface with constant H, for cylinder H = -1/2R stable with respect to any deformation negative !
SLIDE 11
instability – variation analysis
L
conserved volume surface with constant H, for cylinder H = 1/2R stable with respect to any deformation first unstable mode Jacobi equation L – mode wavelength liquid cylinder is always unstable
SLIDE 12
CMC -surfaces
conserved volume Young-Laplace equation, 1805 Constant Mean Curvature surfaces surface of revolution – axial symmetry unduloid nodoid intersections – not interesting
unstable !
SLIDE 13
CMC -surfaces
conserved volume Young-Laplace equation, 1805 Constant Mean Curvature surfaces
unstable !
SLIDE 14 CMC -surfaces
conserved volume Young-Laplace equation, 1805 Constant Mean Curvature surfaces
stable CMC are: sphere
the only one compact CMC
OR
doubly periodic
genus = number of holes triply periodic of genus 3
SLIDE 15 shape variation of charged cluster
no CMC surface! mean curvature H prescribed by pressure difference ΔP and the potential Ф
Kubis '16
modified Young-Laplace equation modified beta-equilibrium
SLIDE 16
Kubis '16
stability condition for charged cluster
the shape change → changes the potential the cluster obeying the modified Y-L equation is it stable? it enters only into the 2nd order variation: negative contribution negative/positive ? solution of
SLIDE 17
at cell boundary
spaghetti phase
periodic solution not known Y-L - 2nd order nonlinear problem single charged rod in vacuum single charged rod in W-S cell boundary condition relevant ! cylinder: H=conts - ok cluster surface concides with equipotential surface
at infinity
SLIDE 18 spaghetti phase (in vacuum)
cylinder is not stable for any charge density
Rayleigh-Plateau instability
gyroid hourglass flattening surface energy dominates Coulomb energy dominates
SLIDE 19 spaghetti phase (in cylindrical W-S cell)
“Virial theorem” but on volume fraction and mode wavelength all modes are stable at all volume fraction cylindrical W-S cell too restrictive ! for gyroid mode:
highly positive!
SLIDE 20 Looking for nontrivial shapes - stability analysis
“Virial theorem” a - cell size b, c – mode wavelengths in y, z directions
- nly the geometry is valid !
symmetry energy does not matter ! lasagna – the only one periodic solution in the CLDM different modes for different faces x y z
SLIDE 21 Looking for unstable modes for the slab
i-th face, k,l - modes numbers for z and y directions for any mode wavelength and volume fraction ! the slab is locally stable → no “decay” channel ?
- modes for z and y directions directions decouple
- two class of normal modes
SLIDE 22 stable modes in lasagna – transport properties
acoustic “phonons” (surfons ?)
- ptical “phonons” (surfons ?)
more efficient energy transport
for
in agreement with Pethick&Potekhin '98 result
SLIDE 23 Conclusions
- modified Young-Laplace equation governs the pasta
shape – nonconstant mean curvature H
- stability of charged cluster – nontrivial question
- lasagna phase is locally stable for any volume fraction
how it changes to other shapes?
- spaghetti: stable/unstable ? periodic solution desirable !
Questions
- new kind of excitations – "surfons" ?
