Elastic Constants of Single Crystals of the bcc Transition Elements V, Nb, and Ta

Values of the elastic constants of single crystals of the bcc transition elements V, Nb, and Ta, are reported at T=27°C. They are, in units of 10¹¹ d/cm²: $$\begin{array}{cccc}& c_{11}& c_{12}& c_{44}\\ \mathrm{V}& \text{22.8}& \text{11.9}& \text{4.26}\\ \mathrm{Nb}& \text{24.6}& \text{13.4}& \text{2.87}\\ \mathrm{Ta}& \text{26.7}& \text{16.1}& \text{8.25}\end{array}$$ Comparisons are made of the values obtained for two crystals each of V and Ta. A high‐frequency cw resonance technique was used in these measurements. The shear anisotropies A=2c₄₄/(c₁₁–c₁₂) are anomalously small for these elements as compared to other cubic system elements. An analysis of the shear anisotropy, based on Fuchs' model, is given. It is found essential in this analysis to take into account next‐nearest as well as nearest‐neighbor ion‐ion interactions.