Quantum calculation of vortices
in the inner crust of neutron stars
R.A. Broglia, E. Vigezzi
Milano University and INFN
F. Barranco
University of Seville
P. Avogadro
RIKEN
Outline of the talk
-Introduction
- The model
- Results
- Comparison with other approaches
- Conclusions and perspectives
Phys. Rev. C 75 (2007) 012085Nucl.Phys. A811 (2008)378
The inner crust of a neutron star
The density range is from 4x10 11 g /cm3to 1.6x1014 g /cm3
The thickness of the inner crust is about 1km.
In the deeper layers of the inner crust nuclei start to deform.
neutron gas density
Lattice of heavy nuclei surrounded by
a sea of superfluid neutrons.
Recently the work by Negele & Vautherin has been improved including the effects of pairing coorelations:
-the size of the cell and number of protons changes.
-the overall picture is mantained.
-the energy differences between different configurations
is very small.
M. Baldo, E.E. Saperstein, S.V. Tolokonnikov
Previous calculations of pinned vortices in Neutron Stars:
-R. Epstein and G. Baym, Astrophys. J. 328(1988)680 Analytic treatment based on the Ginzburg-Landau equation
-F. De Blasio and O. Elgaroy, Astr. Astroph. 370,939(2001)Numerical solution of De Gennes equations with a fixed nuclear mean field and imposing cylindrical symmetry (spaghetti phase)
-P.M. Pizzochero and P. Donati, Nucl. Phys. A742,363(2004)Semiclassical model with spherical nuclei.
HFB calculation of vortex in uniform neutron matter
Y. Yu and A. Bulgac, PRL 90, 161101 (2003)
Distances scale withDistances scale with FF
Distances scale with Distances scale with ξξ FF
Vortex in uniform matter: Y. Yu and A. Bulgac, PRL 90, 161101 (2003)
Spatial description of (non-local) pairing gap Essential for a consistent description of vortex pinning!
The local-density approximation overstimates the decrease of the pairing gap in the interior of the nucleus. (PROXIMITY EFFECTS)
R(fm) R(fm)
The range of the force is small compared to the coherence length, but not compared to the diffusivity of the nuclear potential
K = 0.25 fm -1
K = 2.25 fm -1
k=kF(R)
k=kF(R)
K = 0.25 fm -1
K = 2.25 fm -1
Pinning Energy= Energy cost to build a vortex on a nucleus - Energy cost to build
a vortex in uniform matter.
Energy cost to build a vortex on a nucleus=
Energy cost to build a vortex in uniform matter=
-
-
All the cells must have the same asymptotic neutron density.Same number of particles
pinning energy<0 : vortex attracted pinning energy >0 :vortex repelled by nucleus
PINNING ENERGY
Contributions to the total energy:
Kinetic, Potential, Pairing
Conventional wisdom
The pairing gap is smaller inside the nucleus than outside
The vortex destroys pairing close to its axis
Condensation energy will be saved if the vortex axis passes throughthe nuclear volume: therefore pinning should be energetically favoured.
{
{
Pairing of Pinned Vortex
-0,50
0,00
0,50
1,00
1,50
2,00
2,50
0,0 5,0 10,0 15,0 20,0
rho(fm)
Del
ta(M
eV) z=0fm
z=4fm
z=8fm
z=12fm
z=16fm
Nucleus without vortex,z=0fm
Pairing gap of pinned vortex
Velocity of Pinned Vortex
-0,10
0,00
0,10
0,20
0,30
0,40
0,50
0,60
0,0 5,0 10,0 15,0 20,0
rho(fm)
Ve
l(c
/30
)
z=0fm
z=4fm
z=8fm
z=12fm
z=16fm
Velocity of pinned vortex
nucleus vortex pinned on a nucleusSly4 5.8MeV
In this region
the gap is not
completely
suppressed
the gap of a nucleus
minus the gap of a
vortex on a nucleus:
on the surface there
is the highest
reduction of the gap.
density x 20 [ fm−3] density x 20 [ fm−3]
density x 10[ fm−3]
Gap [MeV] Gap [MeV]
The velocity field is suppressed in the nuclear region
The superfluid flow is destroyed in the nuclear volume.
Pinned Vortex2=vIn this case Cooper pairs are made of single particle levels of the same parity
2=v
1=v
P.M. Pizzochero and P. Donati, Nucl. Phys. A742,363(2004) Semiclassical model with spherical nuclei.
* *
*
**
Conclusions
-We have solved the HFB equations for a single vortex in the crust of neutronstars, considering explicitly the presence of a spherical nucleus, generalizing previousstudies in uniform matter.
- We find pinning to nuclei at low density, with pinning energies of the order of 1
MeV. With increasing density, antipinning is generally favoured, with associated
energies of a few MeV. At the highest density we calculate the results depend
on the force used to produce the mean field: low effective mass favours (weak)
pinning while high effective mass produces strong antipinning.
-We have found that finite size shell effects are important, (ν=1) at low
and medium density the vortex can form only around the nuclear surface,
thus surrounding the nucleus. This leads to a great loss of condensation
energy, contrary to semiclassical estimates.
For future work:
-3D calculations-Include medium polarization effects- Vortex in pasta phase and in the core