  The interaction between nuclei in the inner crust of neutron stars consists
of two contributions, the so-called "direct" interaction and an "induced" one
due to density changes in the neutron fluid. For large nuclear separations $r$
the contributions from nuclear forces to each of these terms are shown to be
nonzero. In the static limit they are equal in magnitude but have opposite
signs and they cancel exactly. We analyze earlier results on effective
interactions in the light of this finding. We consider the properties of
long-wavelength collective modes and, in particular, calculate the degree of
mixing between the lattice phonons and the phonons in the neutron superfluid.
Using microscopic theory, we calculate the net non-Coulombic contribution to
the nucleus--nucleus interaction and show that, for large $r$, the leading term
is due to exchange of two phonons and varies as $1/r^7$: it is an analog of the
Casimir--Polder interaction between neutral atoms.
