Elastic Properties of Vanadium. I. Temperature Dependence of the Elastic Constants and the Thermal Expansion

The thermal expansion and adiabatic elastic constants of several single crystals of vanadium were investigated over the temperature range 4.2-300K. Although certain anomalies in the thermal expansion and in the elastic constants were observed at specific temperatures, in this paper emphasis is placed on the over-all elastic properties of vanadium, and attention is focused on those specimens in which the anomalies were less striking. The anomalies and their origin are the subject of a second paper which follows. The 4.2-K values of the elastic constants of a specimen exhibiting essentially no anomalous behavior were measured to be $C_{11}=23.\mathrm{}70$, $C_{44}=4.\mathrm{}708$, $C^{\prime }=5.\mathrm{}793$, $C_{12}=12.\mathrm{}20$, and the bulk modulus $B=15.\mathrm{}98$, all in units of ${10}^{11}$ dyn/${\mathrm{cm}}^2$. Except for $C_{44}\mathrm{}\left(T\right)\mathrm{}$, the over-all temperature dependence of the elastic constants agrees with that predicted by current theoretical models. $C_{44}\mathrm{}\left(T\right)\mathrm{}$, however, exhibited an upward concavity over much of the temperature range, and was characterized by an athermal value lower than the measured 4.2-K value. Above 150 K the coefficient of thermal expansion was approximately constant for those specimens showing no anomalies, with a value $\alpha \cong \frac{8.5\times {10}^{-6\mathrm{}}}{\mathrm{K}}$. Below 150 K, $\alpha$ decreased monotonically to 0 at ∼ 10 K. The value of the Debye temperature extrapolated to 0 K was calculated to be ${\Theta }_D \left(\mathrm{elastic}\right)=403\mathrm{}$ K.