Active lines of development in microscopic studies of the inner crust (spherical nuclei, 1S0 pairing) Study the inhomogeneous structure of the Wigner-Seitz cell: Isotopic composition Mean field Collective excitations Superfluidity within BCS theory and beyond Specific heat Study the influence of the Coulomb lattice: - Band structure, Level density Entrainment, effective mass Transport properties Ion vibrations Specific heat

Superfluid gaps 1. What is the spatial dependence of the pairing gap? How important are the nuclear clusters? 2. Are the gaps affected by many-body processes ? By how much?

Commonly used approach: just use the value of the pairing gap at the Fermi energy calculated in neutron/neutron star matter Independence of the BCS pairing gap from the specific bare potential

Finite size effects on the pairing field (BCS with the bare force) Potential in the Wigner cell Pairing gap in uniform neutron matter eF=13.5MeV P.M. Pizzochero, F. Barranco, E. Vigezzi, R.A. Broglia,APJ 569(2003)381 N. Sandulescu, Phys. Rev. C70(2004)025801 F. Montani, C. May, H. Muther, PRC 69 (2004) 065801 M. Baldo, U. Lombardo, E.E. Saperstein, S.V. Tolokonnikov, Nucl. Phys. A750 (2005) 409

Spatial dependence of pairing densities and pairing gaps FINITE NUCLEI, FINITE RANGE FORCE HFB Equations are expanded on a basis

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)

it is not very relevant for global properties Even if the spatial dependence of the gap at the nuclear surface is strong, it is not very relevant for global properties (volume of the nucleus is much smaller than the volume of the Wigner-Seitz cell) Calculated gaps for unbound states in a cell with nucleus Calculated gaps for unbound states in a cell without nucleus origin of Pvc 5-10% difference EF

Neutron and electron specific heat going from the core to the surface of the star The presence of the nucleus increases Cv but the electronic contribution is dominant. But: effects beyond mean field can reduce the gap and change this picture…

Are the gaps affected by many-body processes ? By how much? Neutron Matter crust core N. Chamel, P. Haensel, Liv. Rev. Rel. 11 (2008)10

Schematic: global reduction of the pairing interaction strength C. Monrozeau, J. Margueron, N. Sandulescu, PRC 75 (2007) 065807

Going beyond mean field: medium polarization effects taking into account the inhomogeneous character of the crust Self-energy Induced interaction (screening)

Medium effects decrease the gap PAIRING GAP IN FINITE NUCLEI PAIRING GAP IN NEUTRON MATTER renorm. bare Exp. renorm. bare Medium effects increase the gap in 120Sn and bring it in agreement with experiment Medium effects decrease the gap F. Barranco et al., Eur. J. Phys. A21(2004) 57 C. Shen et al., PRC 67(2003) 061302

Why such a difference with neutron matter? Crucial: the surface nature of density modes. This assures an important overlap between the transition density and the single-particle wave-function at the Fermi energy. Volume nature of Spin-modes origin of Pvc

Interpolating between density functionals in nuclei and infinite matter Neutrons M. Baldo, U. Lombardo, E.E. Saperstein, S.V. Tolokonnikov, Nucl. Phys. A750 (2005) 409

Coupling quasiparticles to the vibrations of the nuclear cluster; requires an explicit calculation in the WS cell BCS Many body G. Gori et al., Nucl. Phys. A731 (2004) 401; S. Baroni et al., arXiv 2008

Much progress since 1995 …. C.J. Pethick, D.G.Ravenhall, Annu. Rev. Nucl. Sci. 45 (1995) 429

What about the coupling to lattice vibrations? P. Magierski, Int. Jou. Mod. Phys. E 2003

4. How good is the Wigner-Seitz approximation? First calculations of band structure beyond the Wigner-Seitz approximation N. Chamel, S. Naimi, E. Khan, J. Margueron, nucl-th/07_01851

Check: pairing gap in the box without the potential reproduces the infinite matter gap, independent of the boundary conditions for R = 30 fm R = 30 fm P. Avogadro, F, Barranco, R.A. Broglia, E. Vigezzi, Nucl. Phys. A811 (2008) 378

A few important questions about pairing correlations Does superfluidity affect the results found by Negele and Vautherin? 2. What is the spatial dependence of the pairing gap? How important are the nuclear clusters? 3. Are the gaps affected by many-body processes ? By how much? 4. How good is the Wigner-Seitz approximation? 5. Can observations prove that the crust is really superfluid?