Elastic Moduli of Vanadium

Abstract

Using the technique of zone melting by electron bombardment, $\frac{3}{8}$-in. diameter single crystals of vanadium were grown in a high vacuum. Two crystals were prepared from seeds such that their lengths were parallel to the [100] and the [110] crystal directions. With these orientations, the elastic moduli ${C}_{11}$, ${C}_{44}$, and $\frac{1}{2}({C}_{11}\ensuremath{-}{C}_{12})$ were measured directly from 4.2\ifmmode^\circ\else\textdegree\fi{}K to 300\ifmmode^\circ\else\textdegree\fi{}K using the ultrasonic pulse-echo technique. The results are, in units of ${10}^{11}$ dyne ${\mathrm{cm}}^{\ensuremath{-}2}$, The moduli at 4.2\ifmmode^\circ\else\textdegree\fi{}K were used to calculate a Debye ${\ensuremath{\theta}}_{0}$ of 399.3\ifmmode^\circ\else\textdegree\fi{}K which disagrees with the ${\ensuremath{\theta}}_{0}$ of 338\ifmmode^\circ\else\textdegree\fi{}K obtained from calorimetry. The room temperature moduli agree very well with all other determinations. The moduli were used to calculate a phonon frequency distribution which is compared with the distribution measured by slow neutron scattering.