A class of Fe--Mn--Si-based alloys exhibit a reversible martensitic transformation between the $gamma$ phase with a face-centered cubic~($fcc$) and an $epsilon$ phase with a hexagonal close-packed ($hcp$) structure. During the deformation-induced $gamma$--$epsilon$ transformation, we identified a new phase that is different from the $epsilon$ phase. In this phase, the electron diffraction spots are located at the 1/3 positions corresponding to the ${$0002$}$ plane of the $epsilon$ ($hcp$) phase with 2H structure, which suggests long-period stacking order (LPSO). To understand the stacking pattern and explore the possible existence of an LPSO phase as an intermediate between the $gamma$ and $epsilon$ phases, we examined the phase stability of various structural polytypes of iron using first-principles calculations with a spin-polarized form of the generalized gradient approximation in density functional theory. We found that an antiferromagnetic ordered 6H$_2$ structure is the most stable among the candidate LPSO structures and is energetically close to the $epsilon$ phase, suggesting that the observed LPSO-like phase adopts the 6H$_2$ structure. Furthermore, we determined that the phase stability can be attributed to the valleys depth in the density of states~close to the Fermi level.
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