(a) (b) (c) (d) (e) (f) x=0.4, SC (a) (b) (c) scan-a scan-b (d) scan-a x=0.4 x=0.4 x=0.45 x=0.4 3 FIG. 3: (Color online) Constant-energy scans along the scan- a direction above and below T$_{c}$ for energies (a) 2 meV; Here the transverse scans have three peaks in the normal state, centered at (0 . 5 − δ, 0 . 5 + δ ), (0 . 5 , 0 . 5), and (0 . 5 + δ, 0 . 5 − δ ), and these peaks are suppressed but not eliminated below T$_{c}$ . (b) 3 meV; (c) 4.5 meV; (d) 6 meV; (e) 7.5 meV; and (f) 9.5 meV. The solid lines are Gaussian fits to the data on linear sloped backgrounds. Superconductivity has opposite effect on spin excitations below and above E = 4 . 5 meV. FIG. 2: (Color online) (a) Constant energy scans at E = 1 meV along the scan- a direction below (solid squares) and above T$_{c}$ (open circles). The normal state incommensurate scattering is completely suppressed below T$_{c}$ . The solid line is a fit of the data using two Gaussians on a sloped linear background. The data at 1.5 K are featureless indicating the presence of a full spin gap at this energy. (b) Constant- energy scans along the scan- b direction below and above T$_{c}$ . The data confirm that the normal state magnetic scatter- ing is incommensurate and centered at (0 . 5 − δ, 0 . 5 + δ ) and (0 . 5 + δ, 0 . 5 − δ ) with δ = 0 . 132 ± 0 . 007 at E = 1 meV. The scattering again become featureless below T$_{c}$ . (c) The tem- perature dependence of the scattering at Q = (0 . 37 , 0 . 63) and E = 1 meV (left scale) decreases below T$_{c}$ , while the scatter- ing at the resonance energy [ Q = (0 . 5 , 0 . 5) and E = 6 meV] increases below T$_{c}$ (right scale). (d) Similar scans as (a) in the x = 0 . 45 NSC sample. in the x = 0 . 4 SC sample. Consistent with earlier work [24, 25], cooling through T$_{c}$ re-arranges the scattering profile by reducing the magnetic scattering below E ≈ 4 meV and creating a resonance at E = 6 . 5 meV (Fig. 4a). perature dependence for spin excitation energies below and above E = 4 . 5 meV. For energies between E = 2 and 4.5 meV, the scattering in the normal state is sup- pressed but not eliminated at 1.5 K (Figs. 3a-3c). The effect of superconductivity reduces the entire transverse scattering profile at E = 3 meV (Fig. 3b), but only suppresses the incommensurate magnetic scattering in- tensity at E = 4 . 5 meV (Fig. 3c). The E = 2 meV ex- citations in both the normal and superconducting state are difficult to fit, although the suppression of the signal is still clear. For energies of E = 6, 7.5, and 9.5 meV, the low-temperature scattering are strongly enhanced due to the resonance. All these data were fitted by two Gaussian functions with equal width as shown by the solid lines in Fig. 3a-3f, where the incommensurability δ is defined as half of the distance between two peaks. Based on these data and the results at E = 1 meV (Fig. 2), we plot in Fig. 1f the dispersions of spin excitations in the normal and SC states. It is immediately clear that spin excita- tions of FeSe$_{0}$$_{.}$$_{4}$Te$_{0}$$_{.}$$_{6}$ display an hourglass-like dispersion in the normal state. To determine the impact of superconductivity on the energy dependence of spin excitations, we show in Fig. 4a constant- Q scans at Q = (0 . 5 , 0 . 5) below and above T$_{c}$ In cuprates,the resonance and hourglass dispersion may arise from the d -wave nature of the superconducting gap [19, 20]. To see if the hourglass dispersion in Fig. 1f is directly connected with superconductivity, we especially prepared a nearly NSC x = 0 . 45 sample and carried out identical measurements as those for the SC x = 0 . 4 sam- ple. Although the NSC x = 0 . 45 exhibits an hourglass dispersion remarkably similar to that in the x = 0 . 4 SC sample (Figs. 1f and 1g), the incommensurate scattering at 1 meV show no temperature dependence (Fig. 2d). In addition, the system has no spin gap and shows only weak resonance above 5 meV (Fig. 4b). The transverse scans at 6 meV in Fig. 4c further demonstrate the weak resonance. Figure 4d shows that there is a large broad peak along K -direction at (0.5,0,0), which suggests that our x = 0 . 45 sample is indeed poorly SC [29]. The hourglass dispersion may be understood within a Fermi surface nesting picture similar to the case of pure chromium [30, 31], where the incommensurate spin density waves come from the interband nesting between the electron and hole bands. In the FeSe$_{x}$Te$_{1}$$_{−}$$_{x}$ system, if we consider two nesting wavevectors in a multi-band