The growth of thin $MgF_2$ films on Si(111) was studied with in situ LEED and ex situ XRD. When $MgF_2$ was deposited onto the Si(111)-surface and annealed from room temperature, a series of LEED patterns (1×1→3×1→Si(111)-7×7) was observed. The 3×1 phase was formed upon annealing the film at 700~800℃ and was stable over a wide temperature range. At such high temperature, $MgF_2$ is likely to dissociate, with evaporation of fluorines and formation of a 3×1 structure. If the annealing temprature was slowly raised, complex LEED patterns were observed between 1×1 and 3×1 phase. These patterns are attributed to the partially disordered surface layer in the course of forming the 3×1 structure. One of them is a third-order splitting pattern. To identify this structure, we calculated diffraction patterns within kinematic approximation for a number of antiphase models and could obtain a similar pattern. The corresponding structure has a dominant 3×1 surface structure but a true surface exists with a large unit cell because of the regular antiphase arrangement. From XRD studies we found out that the deposited $MgF_2$ film was oriented in (110) direction.