powder samples indicated a gradual suppression of the concurrent structural and magnetic phase transitions with increasing K-doping. For the underdoped regime $0.2\leq x\leq 0.4$, commensurate AF order appears to microscopically coexist with superconductivity [@hchen09]. Subsequent neutron scattering and muon spin rotation ($\mu$SR) measurements on single crystals grown in Sn-flux suggested mesoscopic separation of the AF and superconducting phases [@jtpark09]. However, recent neutron [@savci], X-ray scattering, and $\mu$SR work [@ewiesenmayer11] on high-quality powder samples confirm the microscopic coexistence of the commensurate AF order with superconductivity in the underdoped region between $0.2\leq x\leq 0.3$ and the suppression of the orthorhombic phase below $T_c$ (Fig. 2a). Since currently there is no neutron diffraction work on high-quality single crystals of Ba$_{1-x}$K$_x$Fe$_2$As$_2$ grown using FeAs-flux, it is unclear if there are also short-range incommensurate AF order in Ba$_{1-x}$K$_x$Fe$_2$As$_2$ near optimal superconductivity.

The appearance of static incommensurate AF order along the transverse direction of the collinear AF ordering wave vector ${\bf Q}_{AF}=(\pm 1,0)$ in BaFe$_{2-x}T_x$As$_2$ suggests that such order arises from the electron doping effect of FS nesting [@dkpratt11; @hluo12]. Based on a five-orbital tight-binding model, fitted to the density functional theory (DFT) band structure for BaFe$_2$As$_2$ [@graser10], there should be five FS pockets with different 