2 shown the presence of the resonance at E = 6 . 5 meV and suppression of spin fluctuations below E ≈ 4 meV [24, 25, 27, 28]. We define the momentum transfer Q at ( q$_{x}$ , q$_{y}$ , q$_{z}$ ) as ( H , K , L )=( q$_{x}$a/ 2 π , q$_{y}$b/ 2 π , q$_{z}$c/ 2 π ) recip- rocal lattice units (rlu), where the lattice parameters of the tetragonal unit cell ( P 4 /nmm space group) are a = b = 3 . 786 ˚ A and c = 6 . 061 ˚ A (Figs. 1a and 1b). We co-aligned ∼ 15 grams of single crystals of x = 0 . 4 samples to within 1.5 ◦ (prepared using flux method sim- ilar to [24] with T$_{c}$ = 14 K) in the [ H, K, 0] scattering plane with c -axis vertical. All momentum transfers given as Q = ( H, K ) are read to be Q = ( H, K, 0). To fur- ther study the effect of superconductivity, we have also measured a poorly superconducting (NSC) FeSe$_{x}$Te$_{1}$$_{−}$$_{x}$ ( x = 0 . 45) sample ( ∼ 23 grams) with a T$_{c}$ of 10 K and a superconducting volume less than 30% in the same scat- tering zone. The samples were loaded inside either a variable temperature liquid He cryostat or a closed cycle cryostat. Since superconductivity and magnetic order in Fe$_{1+}$$_{δ}$Se$_{x}$Te$_{1}$$_{−}$$_{x}$ are extremely sensitive to the excess Fe content δ [5, 6, 32], we have measured the excess Fe in our samples using inductively coupled plasma atomic- emission spectroscopy analysis. We find that both sam- ples are essentially stoichiometric without excess Fe to within 2% of the measurement accuracy. This is con- sistent with earlier results that suggest Fe$_{1+}$$_{δ}$Se$_{x}$Te$_{1}$$_{−}$$_{x}$ samples in this Se-range have little excess Fe [4–6]. Since magnetic scattering of x = 0 . 4 sample near the resonance energy are highly two-dimensional and peak at (0 . 5 − δ, 0 . 5 + δ ) and (0 . 5 + δ, 0 . 5 − δ ) positions as shown in Fig. 1b [26–28], we focus our attention to the energy evolution of these two peaks by carrying out transverse scans along the [ H, 1 − H, 0] direction (scan- a direction in Fig. 1b). At 20 K ( T = T$_{c}$ + 6 K), the scatter- ing at E = 1 meV show two clear peaks centered at (0 . 5 − δ, 0 . 5 + δ ) and (0 . 5 + δ, 0 . 5 − δ ) positions respec- tively with δ = 0 . 132 ± 0 . 007 (Fig. 2a, open circles). Upon cooling to 1.5 K ( T = T$_{c}$ − 12 . 5 K), the incom- mensurate peaks disappear and the scattering becomes featureless indicating the opening of a spin gap (Fig. 2a, filled squares). To confirm that the magnetic scattering at E = 1 meV is indeed incommensurate as depicted in Fig. 1b, we carried out scans along the scan- b direction. The outcome of these scans (Fig. 2b) indeed shows a peak at the expected position ( H = 0 and δ = 0 . 13) in the normal state and it disappears below T$_{c}$ . The tem- perature dependence of the scattering at E = 1 meV and Q = (0 . 5 − δ, 0 . 5 + δ ), where δ = 0 . 13, in Fig. 2c shows a sudden reduction in intensity near T$_{c}$ . For compari- son, the intensity of the resonance at E = 6 meV and Q = (0 . 5 , 0 . 5) below T$_{c}$ (Fig. 2c) clearly increases below T$_{c}$ [24, 25]. Fig. 3 summarizes the energy dependence of the trans- verse scans at a series of energies below and above T$_{c}$ for the x = 0 . 4 sample. The scattering is incommensurate at all energies investigated but display quite different tem- E$_{r}$ E$_{g}$ Fully Gapped (f) H 0.5 0 0.5 K scan-a scan-b a b Fe (a) (b) x=0.4, SC Te x=0.45, NSC (g) !"#$% &"#$’ ( E E ) f * +, +- +. / (c) ) *$_{+-}$ +. / X Y (d) (e) FIG. 1: (Color online) (a) Schematic in-plane spin struc- ture of the nonsuperconducting FeTe, where the solid and hollow arrows represent two sublattices of spins which can be either parallel or antiparallel [6]. Upon substitution of Se for Te to form FeSe x Te 1 − x , the static long-range AF order is suppressed, and the system display strong spin excitations at incommensurate positions near Q = (0 . 5 , 0 . 5) as shown in (b). The incommensurate scattering only appear at posi- tions (0 . 5 − δ, 0 . 5 + δ ) and (0 . 5 + δ, 0 . 5 − δ ). Our transverse scans are along the scan direction a , and the scan along the incommensurate position that is perpendicular to scan- a is marked as scan- b . (c,d) Schematic diagram of Fermi surfaces near Γ and M points from results of recent photoemission experiments [33, 34]. (e) In a multiband itinerant picture, quasiparticle excitations from the α 2 band to the β band can give rise to the upper branch of the hourglass dispersion as shown in the solid red lines. The lower branch of the dis- persion is then a consequence of the excitations from the α 3 band to the β band. (f) Experimental determination of the spin excitation dispersions in the normal (open red circles) and SC (black filled squares) states of FeSe 0 . 4 Te 0 . 6 . A full spin gap opens below E ≈ 1 meV at 1.5 K. The magnitude of E$_{g}$ marks the energy below which intensity of spin excitations decrease below T$_{c}$ , whereas E$_{r}$ indicates the resonance energy. The dispersion curves are obtained by fitting two Gaussians on linear backgrounds through transverse scans in Figs. 2 and 3. We note that the incommensurability at 2 meV at 20 K is obtained by fitting the difference between the 20 K data and 1.5 K data. The horizontal error bars are the fitted errors of the incommensurability. (g) Hourglass dispersion in the NSC FeSe 0 . 45 Te 0 . 55 at 4 K and 15 K.