Proximity effect of pair correlation in the inner crust of neutron stars

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 superfluid, 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 superfluid is affected by the presence of the nuclear cluster. It is found that the proximity effect is characterized by the coherence length of neutron superfluid 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 $5 \times 10^{-4}$ fm$^{-3} < \rho_b < 2\times 10^{-2}$ 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 $\rho_b < 2 \times 10^{-4}$ fm$^{-3}$, and in deep layers with $\rho_b > 5 \times 10^{-2}$ fm$^{-3}$.