  We model the one-dimension (1D) to three-dimension (3D) crossover in a
cylindrically trapped Fermi gas with attractive interactions and
spin-imbalance. We calculate the mean-field phase diagram, and study the
relative stability of exotic superfluid phases as a function of interaction
strength and temperature. For weak interactions and low density, we find
1D-like behavior, which repeats as a function of the chemical potential as new
channels open. For strong interactions, mixing of single-particle levels gives
3D-like behavior at all densities. Furthermore, we map the system to an
effective 1D model, finding significant density dependence of the effective 1D
scattering length.
