moment of $\sim$$2\ \mu_B$/Fe in Fe$_{1+y}$Te [@webao08; @slli08] suggests that local moments may be important, spin waves are rather broad in energy and difficult to fit using a Heisenberg Hamiltonian with only NN and NNN exchange couplings [@lipscombe]. By including third-neighbor (NNNN) exchange couplings, a Heisenberg Hamiltonian can fit the spin-wave dispersion with an anisotropic ferromagnetic NN exchange couplings and strong AF NNN exchange coupling (Table I). In a separate INS experiment on Fe$_{1.1}$Te, the total integrated Fe magnetic moment was found to increase with increasing temperature from 10 K to 80 K [@zaliznyak]. These results suggest that in the temperature range relevant for superconductivity, there is a remarkable redistribution of the magnetism arising from both itinerant and localized electrons.

![The electronic phase diagrams and the evolution of FS’s, static AF order, and spin excitations upon electron or hole doping to BaFe$_2$As$_2$. (a) The AF and superconducting phase diagram for hole-doped Ba$_{1-x}$K$_x$Fe$_2$As$_2$. In the underdoped regime, there is a region of coexisting AF order and superconductivity [@ewiesenmayer11]. Incommensurate spin excitations appear for $x\ge 0.4$ [@castellan] and persist till $x=1$ at KFe$_2$As$_2$ [@chlee11]. (b) Phase diagram for electron-doped BaFe$_{2-x}$Ni$_x$As$_2$ [@hluo12]. The long commensurate AF order changes into short-range incommensurate AF order 