Reconstructive Phase Transformations in Body-Centered Cubic Titanium

Herein, the symmetry of the experimentally observed soft phonons in the body-centered cubic β-phase (Im (Formula presented.) m) of titanium is analyzed. The harmonic phonon dispersion relations are calculated using the first-principles calculations. Using the group-theoretical methods, the symmetry of the calculated unstable phonons is determined. The symmetries of the unstable phonons observed at wave vectors (Formula presented.) (N) and (Formula presented.) ((Formula presented.)) are the same as the symmetries of the (Formula presented.) and (Formula presented.) irreducible representations, respectively. Transformations of the β-phase due to the atomic motion of unstable phonons and the subsequent structure relaxation are discussed. One possible way to explain the transformation of the β-phase to the hexagonal close-packed α-phase ((Formula presented.) /mmc) is through an orthorhombic structure (either Cmcm or Pnnm). The atomic motion of an unstable (Formula presented.) phonon results in the orthorhombic structure and following structure relaxation transforms the orthorhombic structure to the α-phase. Similarly, the transformation of the β-phase to another hexagonal close-packed ω-phase (P6/mmm) can be considered to be happening through a trigonal structure (either P (Formula presented.) m1 or P3m1). The atomic motion of an unstable (Formula presented.) phonon forms the trigonal structure and subsequent structure relaxation transforms the trigonal structure to the ω-phase. The space group of the intermediate phase is a common subgroup of the space groups of the initial β-phase and the final α/ω-phase. Therefore, the β–α/ω transformation can be described as an unstable phonon-induced reconstructive transformation of the second type. There is no activation energy barrier along each of the four energy-minimizing paths, and the transformation strains are accommodated.