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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. 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

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

  24. Using a zero-range pairing interaction, only local quantities are needed  - The equations are solved self-consistently - SI Skyrme interaction (Brink-Vautherin) - Protons are constrained to have a spherical geometry - No spin-orbit interaction

  25. Single Particle Resonances in the nuclear potential obtained with the SII interaction. Positive Parity Negative Parity Angular momentum quantum number L= E [MeV] E [MeV]

  26. The Astrophysical Journal, 328; 680-690, 1988 Interstitial configuration The Pinning Energy was first defined by Epstein & Baym as the difference between the minimum of the free energy of the vortex-nucleus system and the free energy when the nucleus and vortex are distant. Vortex-nucleus distance [fm] Approach by Donati & Pizzochero: Pinned configuration In this approach the PinningEnergy is calculated as the difference between the energy of Pinned and Interstitial configurations.

  27. microscopic calculations Semiclassical calculations Vortex pinned Vortex pinned SII 7 MeV Cell 3 nucleus nucleus Vortex in uniform matter Vortex in uniform matter

  28. Green microscopic calculations SII 7 MeV Velocity field: Red: semiclassical calculations cell 3 Vortex pinned on a nucleus Vortex in uniform matter

  29. ++ - - Spin orbit effect: LS : + +, - -, + -, - + d + - - + Positive parity d + + - - d :+ +, - -, + -, - + + - - + Negative parity

  30. The vortex is expelled from the nuclear volume, because there are no levels available which can satisfy the parity and angular momentum conditions.

  31. Pairing gap of pinned vortex, ν = 2 In this case the vortex goes through the nuclear volume, essentially undisturbed, because the levels can satisfy the parity and angular momentum conditions.

  32. Argonne (bare and uniform case) Gogny (bare and uniform case) With the adopted interaction, screening suppresses the pairing gap very strongly for kF >0.7 fm-1 Screening + nucleus Screening, uniform case However, the presence of the nucleus increases the gap by about 50%

  33. Conclusions and perspectives • We have solved the HFB equations for a single vortex in the crust of neutron • stars, considering explicitly the presence of the nucleus, generalizing previous • studies in uniform matter. • We have found that (ν=1) vortex stay outside of the nuclear volume, • where the pairing goes to zero. • The nuclear surface acquires a slight prolate deformation • Preliminary numerical results at kF = 0.7fm-1 indicate that the pinning • energy is of the order of a few MeV. • Many open questions. Among them: • Include medium polarization effects • -3D calculations • -Vortex dynamics

  34. Going beyond mean field: medium polarization effects Self-energy Induced interaction (screening)

  35. The results of various calculations in neutorn matter can be understood in terms of Landau parameters

  36. Pairing gap in the Wigner cell (no vortex) Pairing gap for a vortex in uniform matter Pairing cap and velocity field for vortex in the Wigner cell z = 15 fm z = 12 fm z = 7 fm z = 2 fm

  37. Vortex pairing field z z r r Proximity effects in the presence of the nucleus Vortex in uniform neutron matter

  38. in order to minimize the energy of the system it develops an array of microscopic linear vortices of superfluid matter neutron matter of the inner crust superfluid in a rotating container Modificare!! Each vortex line carries one unit of angular momentum

  39. Glitches Taken from HartRAO observatory web page The changing rotation rate of the Vela pulsar over ten years is shown above. The glitches are the sudden decreases in the rotation rate. These occur every few years in this pulsar. These glitches are followed by a recovery which usually exhibits two or three exponential recovery rates, the fastest measured being of the order of one day, and the slowest, months.

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