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Unusual Relationship between Magnetism and Superconductivity in FeTe$_{0.5}$Se$_{0.5}$

We use neutron scattering, to study magnetic excitations in crystals near the ideal superconducting composition of FeTe$_{0.5}$Se$_{0.5}$. Two types of excitations are found, a resonance at (0.5, 0.5, 0) and incommensurate fluctuations on either side of this position. We show that the two sets of magnetic excitations behave differently with doping, with the resonance being fixed in position while the incommensurate excitations move as the doping is changed. These unusual results show that a common behavior of the low energy magnetic excitations is not necessary for pairing in these materials.