  The inner crust of a neutron star is a superfluid and inhomogeneous system,
consisting of a lattice of nuclei immersed in a sea of neutrons. We perform a
quantum calculation of the associated pairing gap and compare it to the results
one obtains in the Local Density Approximation (LDA). It is found that the LDA
overestimates the spatial dependence of the gap, and leads to a specific heat
of the system which is too large at low temperatures, as compared with the
quantal result. This is caused by the neglect of proximity effects and the
delocalized character of the single-particle wavefunctions close to the Fermi
energy. It is possible to introduce an alternative, simple semiclassical
approximation of the pairing gap which leads to a specific heat that is in good
agreement with the quantum calculation.
