The elastic constants of the ${\mathrm{Ti}}_{1-x}{X}_x$ ($X=\mathrm{V}$, Nb, Ta, Mo, and W) and ${\mathrm{Zr}}_{1-x}{X}_x$ ($X=\mathrm{Nb}$ and Mo) binary alloys were calculated for $x=0.0$, 0.25, 0.5, 0.75, and 1.0 by the ultrasoft pseudopotential method within the generalized gradient approximation to density functional theory to clarify the mechanisms by which the low elastic moduli of the Ti binary alloys are realized. The Young’s moduli of the polycrystals for these Ti or Zr binary alloys were calculated from the calculated elastic constants of the single crystal by using the Voigt-Reuss-Hill averaging scheme. The results show that the Young’s moduli of the $\mathrm{Ti}\text{−}X$ or $\mathrm{Zr}\text{−}X$ binary alloys have the minimum values in the vicinity of $x=0.25$. From the calculation results, we have found that $C_{11}-C_{12}$ is correlated with the valence electron number per atom and the value of $C_{11}-C_{12}$ becomes nearly zero with the valence electron number of around 4.20–4.24. $C_{11}-C_{12}$ also represents the stability of the bcc structure in these alloys and we thus emphasize that controlling the valence electron number at around 4.20–4.24 is important to realize a low-Young’s-modulus material in the Ti or Zr binary alloys having bcc structure.

• Received 25 November 2003

DOI:https://doi.org/10.1103/PhysRevB.70.174113