Fig. 1 Schematic illustration of ideal ω formation from bcc structure: a a bcc lattice structure; b an ideal ω phase was based upon the collapse model of one pair of (111)bcc planes. The No. 1 layer moves up 1/2 × 1/6[111], and the No. 2 layer moves down 1/2 × 1/6[111], then the bcc structure becomes an ideal ω structure; c a hexagonal ideal ω lattice structure

Fig. 2 a Dark-field TEM image revealing high density of 1–3 nm-sized ω particles in a water-quenched Ti–30Nb–3Pd (wt%) sample. Only one variant of ω particles can be shown in the dark-field image. The particles have a uniform size distribution and are lying on one of {112} along one of 〈111〉 direction in the plane. b The corresponding selected area electron diffraction (SAED) pattern showing the bcc matrix and two variants of the ω phase

Fig. 3 High-resolution TEM lattice image showing that the distribution of a high density of fine ω particles formed in a water-quenched Ti–30Nb–3Pd (wt%) sample. Several ω particles are outlined by dashed circles. The image was taken with the incident electron beam parallel to the [\( \bar 1 \)13]β zone axis

Fig. 4 a Tensile stress–strain curves and the corresponding fracture surfaces of the as-cast Ti–30Nb and Ti–30Nb–3Pd (wt%) alloys. b, c The dark-field TEM images obtained from the above samples, respectively. The dark-field micrographs imaged in 〈112〉β directions clearly reveal the density, size, and distribution of fine ω particles formed in those two cast alloys

Fig. 5 A dark-field TEM image revealing the interaction between twin tips and fine ω particles in the furnace-cooled Ti–30Nb–3Pd alloy. All the bright spots are fine ω particles. Two twin tips are outlined by white dashed lines. The twinning boundaries are indicated by arrows

Fig. 6 a A bright-field TEM image showing that a twin plate is completely embedded in one β matrix grain. The two white arrows indicate the twin edge sides or twin tips. b, c High-resolution TEM images obtained from the left and right twin tip regions outlined by dashed lines in a, respectively

Fig. 7 Schematic illustration of a {112}〈111〉 twin nucleating inside one ω particle in the bcc matrix: a a bcc lattice structure projected along [1–10], the unit cells of bcc and an ideal ω phase were outlined in blue and red lines, respectively; b an ideal ω particle formed via martensitic transformation in bcc matrix; c a reversed martensitic transformation (ω → β) happened inside the ω particle. Two possibilities for the atom movement were indicated in red and blue arrows; d if the atoms follow those red arrows, then ω transforms back to bcc matrix; the blue arrow movement results in a {112}〈111〉-type twinning structure

Fig. 8 TEM observation results from an oil-quenched spring steel: a a bright-field image with misty contrast; b the corresponding dark-field image revealing high density of nanoscale particles. A SAED pattern was inset in the middle, and an enlargement of a local region in the dark-field image was also inset

Fig. 9 a A SAED pattern of the [\( \bar 1 \)13]α zone axis from the oil-quenched spring steel and followed by 400 °C aging for 2 h. In this sample, the volume fraction of the second phase is much less than that in the oil-quenched state. b The [\( \bar 1 \)13]α zone axis diffraction pattern obtained from the region shown in Fig. 8. The second phase was indexed as the ω phase based on the results in β-type Ti alloys. The ω phase has a primitive hexagonal structure. c, d [011]α and [012]α zone axes diffraction patterns, respectively. It is clear that the (10\( \bar 1 \)0)ω//(\( \bar 1 \)2\( \bar 1 \))α

Fig. 10 A hexagonal atomic structure model of the ω phase. It is an AB2 type structure with a unit cell containing three atoms. At A layer, one atom position is 000. At B layer, there are two atoms: 2/3 1/3 1/2; 1/3 2/3 1/2. Here, both A and B layers are the same iron atoms

Fig. 11 a A bright-field TEM image of the [\( \bar 1 \)13]α zone axis from the oil-quenched spring steel and followed by 400 °C aging for 2 h. In this sample, the volume fraction of the ω phase is much less than that in the oil-quenched state. b The corresponding high-resolution TEM image clearly reveals the contrast and lattice fringes of the ω particle

Fig. 12 a A {112}〈111〉-type twinning structure morphology in an oil-quenched spring steel. b The {112}〈111〉-type twinning diffraction pattern with the ω phase