A deep understanding of the character of superconductivity in the recently discovered Fe-based oxypnictides ReFeAsO1-xFx (Re = rare-earth) necessarily requires the determination of the number of the gaps and their symmetry in k space, which are fundamental ingredients of any model for the pairing mechanism in these new superconductors. In the present paper, we show that point-contact Andreev-reflection experiments performed on LaFeAsO1-xFx (La-1111) polycrystals with Tc ~ 27 K and SmFeAsO0.8F0.2 (Sm-1111) ones with Tc ~ 53 K gave differential conductance curves exhibiting two peaks at low bias and two additional structures (peaks or shoulders) at higher bias, an experimental situation quite similar to that observed by the same technique in pure and doped MgB2. The single-band Blonder-Tinkham-Klapwijk model is totally unable to properly fit the conductance curves, while the two-gap one accounts remarkably well for the shape of the whole experimental dI/dV vs. V curves. These results give direct evidence of two nodeless gaps in the superconducting state of ReFeAsO1-xFx (Re = La, Sm): a small gap, Delta1, smaller than the BCS value (2Delta1/kBTc ~ 2.2 - 3.2) and a much larger gap Delta2 which gives a ratio 2Delta2/kBTc ~ 6.5 - 9. In Sm-1111 both gaps close at the same temperature, very similar to the bulk Tc, and follow a BCS-like behaviour, while in La-1111 the situation is more complex, the temperature dependence of the gaps showing remarkable deviations from the BCS behaviour at T close to Tc. The normal-state conductance reproducibly shows an unusual, but different, shape in La-1111 and Sm-1111 with a depression or a hump at zero bias, respectively. These structures survive up to T* ~ 140 K, close to the temperatures at which structural and magnetic transitions occur in the parent, undoped compound.
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