  We study multiple period states of a two-component unpolarized superfluid
Fermi gas in an optical lattice along the Bardeen-Cooper-Schrieffer (BCS) to
Bose-Einstein condensate (BEC) crossover. The existence of states whose period
is a multiple of the lattice spacing is a direct consequence of the non-linear
behavior of the gas, which is due to the presence of the order parameter
associated with superfluidity. By solving Bogoliubov-de Gennes equations for a
superfluid flow with finite quasimomentum, we find that, in the BCS side of the
crossover, the multiple period states can be energetically favorable compared
to the normal Bloch states and their survival time against dynamical
instability drastically increases, suggesting that these states can be
accessible in current experiments, in sharp contrast to the situation in BECs.
