meV, and J ‖ 2 = 47 . 3 meV = J ⊥ 2 . This fit implies a ratio J$_{2}$/J$_{1}$ = 0 . 64 between the next- nearest-neighbor and the nearest-neighbor Heisenberg exchange constants deep inside the cSDW. Although the conventional J$_{1}$ - J$_{2}$ model over the square lattice can also account for the linear and anisotropic dispersion of the spinwaves that are observed in superconducting 15 BaFe$_{2}$$_{−}$$_{x}$Co$_{x}$As$_{2}$, it cannot account for the absence of long-range magnetic order. Both the semi-classical analysis about hidden magnetic order and the exact diagonalization studies on a 4 × 4 lattice described in the previous section indicate, on the other hand, that the moment associated with true cSDW order is extremely small at the quantum critical point into hidden magnetic order. (See Figs. 1 and 7.) In conclusion, a linear spin-wave analysis of the J$_{0}$ - J$_{1}$ - J$_{2}$ model Hamiltonian (1) that describes a square lattice of frustrated spin-1 iron atoms yields a quantum critical point as a function of Hund’s rule exchange coupling ( − J$_{0}$ ) that seperates a cSDW from hidden ferromagnetic or hidden N´ eel order. The observable spectral weight of the Goldstone modes associated with hidden order is predicted to be small in general (see Fig. 3). A strong variation between the diagonal and the off-diagonal exchange couplings of the multi-orbital Heisenberg model (1) across nearest-neighbors alone is sufficient for Hund’s rule to give way to hidden magnetic order. Recent DFT calculations obtain ferromagnetic direct exchange and antiferromagnetic superexchange across nearest-neighbor iron moments in the cSDW phase of ferro-pnictide materials. 6 Their superposition potentially can result in the required difference between J ‖ 1 and J ⊥ 1 . Fits of the critical linear spin-wave spectrum to recent experimental studies of magnetic excitations in ferro-pnictide materials 14 , 15 yield a Hund’s rule exchange coupling constant on the order of 100 meV. Figure 7 displays the dependence of the ordered magnetic moments on Hund’s rule coupling obtained from exact diagonalization of the model Hamiltonian (1) over a 4 × 4 lattice, with two spin-1/2 orbitals per site. The top graph implies a critical Hund’s rule coupling at the transition between hidden ferromagnetic order and cSDW order that is almost twice the classical prediction listed in Table I: − J$_{0}$$_{c}$ = 0 . 8 J ⊥ 1 . Correcting for this discrepancy brings the value of the Hund’s rule coupling closer to the estimate of 700 meV that is obtained for ferro-pnictides by dynamical mean field theory. 26 (Note that − J$_{0}$ is twice as big as the Hund’s rule exchange coupling that appears naturally in multi-orbital Hubbard models.) The success of the theoretical fits to the inelastic neutron scattering data displayed by Fig. 8 indicates that a local-moment picture for magnetism in ferro-pnictide materials is 19