1. 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

2. Outline of the talk -Introduction - The model - Results - Comparison with other approaches - Conclusions and perspectives Phys. Rev. C 75 (2007) 012085 Nucl.Phys. A811 (2008)378

3. The inner crust of a neutron star The density range is from to 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

4. 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)

5. General properties of a vortex line

6. Distances scale with F Distances scale with ξ F Vortex in uniform matter: Y. Yu and A. Bulgac, PRL 90, 161101 (2003)

7. 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) 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=k F (R) K = 0.25 fm -1 K = 2.25 fm -1

8. 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 repelled by nucleus PINNING ENERGY Contributions to the total energy: Kinetic, Potential, Pairing

9. 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 through the nuclear volume: therefore pinning should be energetically favoured. { {

10. The Astrophysical Journal, 328; 680-690, 1988 R.I. Epstein and G. Baym,

13. (inside the cell)

14. Pairing gap of pinned vortex

15. Velocity of pinned vortex

16. nucleus vortex pinned on a nucleus Sly4 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. Gap [MeV]

17. The velocity field is suppressed in the nuclear region The superfluid flow is destroyed in the nuclear volume.

18. Pinned vortex Nucleus Vortex in uniform matter

19. Single particle phase shifts

20. Dependence on the effective interaction

21. Pinned Vortex In this case Cooper pairs are made of single particle levels of the same parity

22. Vortex radii

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

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

26. Conclusions -We have solved the HFB equations for a single vortex in the crust of neutron stars, considering explicitly the presence of a spherical nucleus, generalizing previous studies 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