Fe$_{1+y}$Te actually has a bi-collinear spin structure (Fig. 1c) [@webao08; @slli08]. Here, the AF Bragg peaks occur at ${\bf Q}_{AF}=(\pm 1/2+m,\pm 1/2+n)$ (Fig. 1g) in the pseudo-tetragonal notation ($a\approx b\approx 5.41$ Å), suggesting that FS nesting cannot induce such AF order. Finally, the parent compounds of the alkaline iron selenide $A$Fe$_{1.6+x}$Se$_2$ superconductors are insulators [@jgguo; @mhfang] and form a $\sqrt{5}\times\sqrt{5}$ block AF structure as shown in Fig. 1d with a large ordered moment ($\sim$$3\ \mu_B$/Fe) along the $c$-axis and $T_N\approx 500$ K [@wbao1; @fye]. In reciprocal space, defined using the pseudo-tetragonal unit cell of iron pnictides ($a\approx b\approx 5.41$ Å), the block AF Bragg peaks appear at ${\bf Q}_{AF}=(\pm 0.2+m,\pm 0.6+n)$ and $(\pm 0.6+m,\pm 0.2+n)$ combining left and right chiralities (red circles in Fig. 1h).

Since the parent compounds of iron-based superconductors can have different AF spin structures and either metallic or insulating ground states [@johnston; @stewart; @jgguo; @mhfang], the microscopic origin of the AF order cannot be induced by a simple FS nesting. If magnetism is relevant for high-$T_c$ superconductivity, then it would be important to determine magnetic exchange couplings for different classes of Fe-based superconductors and compare the results with those of the copper-oxides [@coldea]. For pnictides, INS 