Single-Crystal Elastic Moduli and the hcp → bcc Transformation in Ti, Zr, and Hf

The variations with temperature of the elastic moduli in single crystals of hcp Ti, Zr, and Hf have been experimentally determined from 4\ifmmode^\circ\else\textdegree\fi{}K up to 1155\ifmmode^\circ\else\textdegree\fi{}K, using an ultrasonic wave interference technique. The ${c}_{11}$, ${c}_{33}$, and ${c}_{44}$ differ considerably among the three metals, whereas the ${c}_{66}$ shear moduli for Ti and Zr are similar to within 1% between 4 and 300\ifmmode^\circ\else\textdegree\fi{}K and less than 10% at 1100\ifmmode^\circ\else\textdegree\fi{}K. There is very marked anisotropy in the temperature dependence of the shear moduli in Ti and Zr, ${c}_{66}$ decreasing by 75% between 4 and 1100\ifmmode^\circ\else\textdegree\fi{}K whereas ${c}_{44}$ decreases by 40%. A very pronounced deviation from a linear temperature dependence of ${c}_{66}$ occurs at a temperature which coincides with the anomalous curvatures in the published electrical resistivity data. The relatively small ${c}_{66}$ shear moduli at the hcp \ensuremath{\rightarrow} bcc phase transformation temperatures are as expected from the transformation shear mechanisms suggested by Burger. A survey of existing elastic modulus and thermal expansion data in hcp metals leads to the conclusion that a relatively high temperature dependence in ${c}_{66}$ and the occurrence of hcp \ensuremath{\rightarrow} bcc transformations are characteristic of metals with $\frac{c}{a}<1.60$. It is suggested that these characteristics due to the transverse thermal vibrations in the basal planes, and that the transformation results from anharmonic contributions to the free energy which are indicated by specific heat data. Contrary to comparison in other hcp metals, the Debye $\ensuremath{\theta}$ from 4\ifmmode^\circ\else\textdegree\fi{}K elastic moduli in Ti, Zr, and Hf are in excellent agreement with published specific heat data. A slight anomaly at 18\ifmmode^\circ\else\textdegree\fi{}K in Ti is indicated which may be associated with published resistivity minima in Ti alloys.