  We consider finite-range effects when the scattering length goes to zero near
a magnetically controlled Feshbach resonance. The traditional effective-range
expansion is badly behaved at this point and we therefore introduce an
effective potential that reproduces the full T-matrix. To lowest order the
effective potential goes as momentum squared times a factor that is
well-defined as the scattering length goes to zero. The potential turns out to
be proportional to the background scattering length squared times the
background effective range for the resonance. We proceed to estimate the
applicability and relative importance of this potential for Bose-Einstein
condensates and for two-component Fermi gases where the attractive nature of
the effective potential can lead to collapse above a critical particle number
or induce instability toward pairing and superfluidity. For broad Feshbach
resonances the higher-order effect is completely negligible. However, for
narrow resonances in tightly confined samples signatures might be
experimentally accessible. This could be relevant for sub-optical wavelength
microstructured traps at the interface of cold atoms and solid-state surfaces.
