Fig. 1 a Projection of atoms in a bcc structure along [10]bcc; b an atomic structure model of a primitive hexagonal ω crystal

Fig. 2 a Atomic structure model of a fcc crystal; b projection of the fcc structure along [001]fcc zone axis; c projection of the near-perfect ω unit cell, which is formed in the fcc lattice by an atomic shuffling or displacement (black spots and the open circles represent the atoms in different (001)fcc planes (or c plane))

Fig. 3 Two perpendicular ω variants outlined by blue lines in one fcc crystal along [001]fcc zone axis, where the parallel two ω unit cells are belonged to one ω variant, and these ω variants cannot schematically be repeated along its c axis

Fig. 4 Four ω variants along one basal direction of a fcc crystal: a two ω variants with a rotating angle of about 36.88°; b the other two ω variants

Fig. 5 A schematic formation mechanism of one bcc unit cell in a fcc crystal with the help of the ω structure: a two parallel ω unit cells meet together not in the same atomic plane; b another two parallel ω unit cells join in Fig. 5a; c one bcc unit cell formed

Fig. 6 A schematic illustration of a pair of one bcc unit cell and one ω unit cell in a fcc crystal along the [001]fcc direction

Fig. 7 Four pairs (A-D) of ω/bcc unit cells in one fcc crystal, where A-bcc and A-ω correspond to the bcc unit cell and the ω unit cell in the A-pair, respectively

Fig. 8 a A near-perfect bcc structure (red spots) in an ideal fcc crystal (black and blue spots and the red spots at B, O and B′ positions), where each green arrow represents a movement or displacement of the corresponding atom along the arrow direction which is paralleled to the B-O-B′ line with a distance of (/20)a fcc; b the near-perfect bcc lattice structure withdraws from a; the distance among the atoms and the angle among various planes have been calculated based on a fcc = 3.59Å

Fig. 9 Transmission electron microscope image taken along the [110] zone axis of ferrite from a furnace cooled Fe-0.7 wt% C plate sample with a thickness of about 2 mm after treated at 1100 °C for 1 h; the inset is the corresponding selected area electron diffraction pattern