Figure 1

Minimum energy pathways of hcp Ti within the basal plane (⊥ $c$ axis) and normal to the basal plane (// $c$ axis) predicted by first-principles CI-NEB method using five images.Reuse & Permissions

Figure 2

Atomic positions for vacancy migration within a basal plane of hcp Ti from first-principles CI-NEB calculations (view along the $c$ axis); the hexagonal structure with one vacancy (i.e. Image 0) is indicated by the red lines. Positions 0–6 denote the migrating Ti atom in images 0–6 (see Fig. 1), respectively. The filled symbols represent atoms in the basal plane, the open symbols atoms in a neighboring basal plane. For images 1, 4, 5, and 6, only the migrating Ti atom (filled diamond) is shown. The dashed line from 0–7 denotes the direction of relative movement between adjacent basal planes according to Burgers shear deformation for a bcc-hcp phase transition (movement from point 0 up to line 9–10). Atom 8 is the nearest neighbor (shown for images 0, 2, and 3 only) with respect to the migrating atoms 0–6, e.g. the bond lengths are 2.80 Å between 0–8, 2.60 Å between 2–8, and 2.56 Å between 3–8.Reuse & Permissions

Figure 3

Predicted stretching force constants around the migrating atom within the first nearest neighbor region (<∼3 Å, see text about the definition) for images 0 ($\mathit{IS}$), 2, and 3 (see Fig. 1) and hcp Ti ($\mathit{PS}$). Note that $\mathit{TS}$ indicates the transition state, 11 atoms exist for Image 0 (both ⊥ and // $c$ axis), four for images 2 (// $c$ axis) and 3 (// $c$ axis), six for Image 2 (⊥ $c$ axis), and eight for Image 3 (⊥ $c$ axis).Reuse & Permissions

Figure 4

Arrhenius diagram of self-diffusion coefficients perpendicular (D${}_{\perp }$) and parallel (D${}_{//}$) to the $c$ axis of hcp Ti. The filled symbols are experimental data from ultrapure samples, the open symbols from less pure samples. Open triangles are the work of Herzig et al.,[19] other symbols the work of Köppers et al.,[20] and the vertical dotted line indicates the phase transition temperature of 1155 K between hcp α-Ti and bcc β-Ti.Reuse & Permissions