We study proximity effect of pair correlation in the inner crust of neutron stars by means of the Skyrme-Hartree-Fock-Bogoliubov theory formulated in the coordinate space. We describe a system composed of a nuclear cluster immersed in neutron superuid, which is confined in a spherical box. Using a density-dependent effective pairing interaction which reproduces both the pair gap of neutron matter obtained in ab initio calculations and that of finite nuclei, we analyze how the pair condensate in neutron superuid is affected by the presence of the nuclear cluster. It is found that the proximity effect is characterized by the coherence length of neutron superuid measured from the edge position of the nuclear cluster. The calculation predicts that the proximity effect has a strong density dependence. In the middle layers of the inner crust with baryon density 5 × 10-4 fm-3 ≲ ρb ≲ 2 × 10-2 fm-3, the proximity effect is well limited in the vicinity of the nuclear cluster, i.e. in a sufficiently smaller area than the Wigner-Seitz cell. On the contrary, the proximity effect is predicted to extend to the whole volume of the Wigner-Seitz cell in shallow layers of the inner crust with ρb ≲ 2 × 10-4 fm-3, and in deep layers with ρb ≲ 5 × 10-2 fm-3.
We study proximity effect of neutron superfluid in the inner crust of neutron stars by solving the Hartree-Fock-Bogoliubov equation in a spherical Wigner-Seitz cell. Numerical analysis shows that the presence of nuclei affects the pair correlation of neutron superfluid in proximity region, which is characterized by the coherence length measured from the edge of the nuclei. Reflecting strong density-dependence of the coherence length, the proximity effect is small in lower layers of the inner crust while it becomes as large as the cell size in upper layers.
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