  In this letter we consider the superradiant phase transition of a
two-component Fermi gas in a cavity across a Feshbach resonance. It is known
that quantum statistics plays a crucial role for the superradiant phase
transition in atomic gases; in contrast to bosons, in a Fermi gas this
transition exhibits strong density dependence. We show that across a Feshbach
resonance, while the two-component Fermi gas passes through the BEC-BCS
crossover, the superradiant phase transition undergoes a corresponding
crossover from a fermionic behavior on the weakly interacting BCS side, to a
bosonic behavior on the molecular BEC side. This intricate statistics crossover
makes the superradiance maximally enhanced either in the unitary regime for low
densities, in the BCS regime for moderate densities close to Fermi surface
nesting, or in the BEC regime for high densities.
