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Quantum calculation of vortices in the inner crust of neutron stars

Quantum calculation of vortices in the inner crust of neutron stars. P. Avogadro RIKEN. R.A. Broglia, E. Vigezzi Milano University and INFN. F. Barranco University of Seville. Outline of the talk. Introduction The model Results Comparison with other approaches

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Quantum calculation of vortices in the inner crust of neutron stars

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  1. Quantum calculation of vortices in the inner crust of neutron stars P. Avogadro RIKEN R.A. Broglia, E. Vigezzi Milano University and INFN F. Barranco University of Seville

  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 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 neutron gas density The thickness of the inner crust is about 1km. The density range is from to In the deeper layers of the inner crust nuclei start to deform.

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

  7. Spatial description of (non-local) pairing gap Essential for a consistent description of vortex pinning! 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 = 0.25 fm -1 k=kF(R) k=kF(R) K = 2.25 fm -1 K = 2.25 fm -1 R(fm) R(fm) The local-density approximation overstimates the decrease of the pairing gap in the interior of the nucleus. (PROXIMITY EFFECTS)

  8. PINNING ENERGY 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= - Contributions to the total energy: Kinetic, Potential, Pairing Same number of particles All the cells must have the same asymptotic neutron density. pinning energy<0 : vortex attracted pinning energy >0 :vortex repelled by nucleus

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

  11. (inside the cell)

  12. Pairing gap of pinned vortex

  13. Velocity of pinned vortex

  14. Sly4 5.8MeV vortex pinned on a nucleus nucleus Gap [MeV] Gap [MeV] 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.

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

  16. Vortex in uniform matter Nucleus Pinned vortex

  17. Single particle phase shifts

  18. Dependence on the effective interaction

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

  20. Vortex radii

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

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

  23. 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 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. - 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. For future work: -3D calculations -Include medium polarization effects - Vortex in pasta phase and in the core

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