  A class of Fe-Mn-Si-based alloys exhibits a reversible martensitic
transformation between the $\gamma$ phase with a face-centered cubic~(fcc)
structure and an $\epsilon$ phase with a hexagonal close-packed (hcp)
structure. During the deformation-induced $\gamma$--$\epsilon$ transformation,
we identified a phase that is different from the $\epsilon$ phase. In this new
phase, the electron diffraction spots are located at the 1/3 positions that
correspond 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, the phase stability of
various structural polytypes of iron was examined 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, which suggests that the
observed LPSO-like phase adopts the 6H$_2$ structure. Furthermore, we
determined that the phase stability can be attributed to the valley depth in
the density of states, close to the Fermi level.
