Martensitic transformations of β-phase in zirconium

Martensitic transformations in the body-centered cubic β-phase (Im 3 ¯m) of zirconium are studied using first-principles calculations, group-theoretical methods, and symmetry analysis. Phonon dispersion relations in the β-phase calculated within the harmonic approximation predicted an unstable phonon at wave vector 2 π a [ 1 2 , 1 2 , 0 ](N) and a soft phonon at wave vector 2 π a [ 2 3 , 2 3 , 2 3 ]( Λ). The symmetry of the unstable phonon is the same as the symmetry of the N 4 − irreducible representation, and the symmetry of the soft phonon is the same as the symmetry of the Λ 1 irreducible representation. Martensitic transformations are simulated considering two steps. Frozen phonon calculations are used to determine the first step, i.e., the transformation of the β-phase to an intermediate phase due to phonon motion. Structure relaxation is used to determine the second step, i.e., the transformation of the intermediate phase to the final phase. The unstable N 4 − phonon transforms the β-phase into an intermediate orthorhombic phase (Cmcm), which further transforms to a hexagonal close packed α-phase ( P 6 3/mmc) after structure relaxation. The soft Λ 1 phonon transforms the β-phase into an intermediate trigonal phase (P 3 ¯m1), which further transforms to a hexagonal close packed ω-phase (P6/mmm) after structure relaxation. The intermediate phase space group (Cmcm/P 3 ¯m1) is a common subgroup of the parent phase ( β) space group and the final phase ( α / ω) space group. Therefore, the martensitic transformations in zirconium are reconstructive transformations of the second kind. Symmetry characterization of the martensitic transformations is also presented.