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)

General properties of a vortex line

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.

{

{

The Astrophysical Journal, 328; 680-690, 1988

R.I. Epstein and G. Baym,

(inside the cell)

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 vortex

Nucleus

Vortex in uniform matter

Single particle phase shifts

Dependence on the effective interaction

Pinned Vortex2=vIn this case Cooper pairs are made of single particle levels of the same parity

2=v

1=v

Vortex radii

P.M. Pizzochero and P. Donati, Nucl. Phys. A742,363(2004) Semiclassical model with spherical nuclei.

* *

*

**

Coupling the phonons of the nucleus with the phonons of the neutron sea

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