Phonon instabilities in bcc Sc, Ti, La, and Hf

The lattice dynamics of the elements Sc, Ti, La, and Hf in the bcc structur e is studied using the density-functional linear-response theory. The eleme nts exhibit similar phonon instabilities which cover large parts of the Bri llouin zone. In particular, the entire T-[1 (1) over bar 1][xi xi 0] branch , where the zone-boundary phonon is responsible for the bcc-->hcp transitio n, and the L[2/3 2/3 2/3] mode (bcc-->omega) are unstable. However, the T[x i xi xi] branch is unstable for ail elements except Sc, and Ti and Sc exhib it distorted bcc energy minima not seen in the other elements.