Possible metastable rhombohedral states of the bcc transition metals

The energy $E(c/a)$ for a bcc element stretched along its [001] axis (the Bain path) has a minimum at c/a = 1, a maximum at c/a = $\sqrt2$, and an elastically unstable\footnote{M. J. Mehl, A. Aguayo, L. L. Boyer, and R. De Coss, {\em Phys. Rev. B} {\bf 70}, 014105 (2004).} local minimum at c/a $>~\sqrt2$. A rhombohedral strain is an alternative method of connecting the bcc and fcc structures. The primitive lattice keeps $R\overline3m$ symmetry, with the angle $\alpha$ changing from 109.4$^\circ$ (bcc), to 90$^\circ$ (simple cubic), to 60$^\circ$ (fcc). We studied this path for the non-magnetic bcc transition metals (V, Nb, Mo, Ta, and W) using both a full-potential LAPW and PAW VASP. Except for Ta, the energy $E(\alpha)$ has a local {\em maximum} at $\alpha=60^\circ$, with local minima near 55$^\circ$ and 70$^\circ$, the later having lower energy. We studied the elastic stability of the 70$^\circ$ minimum structure. Only W is elastically stable in this structure, with the smallest eigenvalue of the elastic tensor at 4~GPa, while the other three elements are unstable. We discuss the possibility that Tungsten is actually metastable in this structure. We also consider the possible epitaxial growth of this structure.