  We explain why the experimental efficiency observed in the conversion of
ultracold Fermi gases of $^{40}$K and $^{6}$Li atoms into diatomic Bose gases
is limited to 0.5 when the Feshbach resonance sweep rate is sufficiently slow
to pass adiabatically through the Landau Zener transition but faster than ``the
collision rate'' in the gas, and increases beyond 0.5 when it is slower. The
0.5 efficiency limit is due to the preparation of a statistical mixture of two
spin-states, required to enable s-wave scattering. By constructing the
many-body state of the system we show that this preparation yields a mixture of
even and odd parity pair-states, where only even parity can produce molecules.
The odd parity spin-symmetric states must decorrelate before the constituent
atoms can further Feshbach scatter thereby increasing the conversion
efficiency; ``the collision rate'' is the pair decorrelation rate.
