  We study the effects of an external periodic potential on the critical
velocity of a superfluid Fermi gas in the crossover between the
Bardeen-Cooper-Schrieffer (BCS) phase and Bose-Einstein condensation (BEC). We
numerically solve the Bogoliubov-de Gennes equations to model a
three-dimensional (3D) gas of ultracold atoms in the superfluid phase flowing
through a 1D optical lattice. We find that when the recoil energy is comparable
to the Fermi energy, the presence of the periodic potential reduces the effect
of pair-breaking excitations. This behavior is a consequence of the peculiar
band structure of the quasiparticle energy spectrum in the lattice. When the
lattice height is much larger than the Fermi energy, the periodic potential
makes pairs of atoms to be strongly bound even in the BCS regime and
pair-breaking excitations are further suppressed. We have also found that when
the recoil energy is comparable to or larger than the Fermi energy, the
critical velocity due to long-wavelength phonon excitations shows a
non-monotonic behavior along the BCS-BEC crossover.
