Equilibrium concentration of vacancies in hexagonal metals

Abstract

Measurement of macroscopic ($\frac{\Delta L}{L_0}$) and microscopic ($\frac{\Delta d}{d_0}$) thermal expansion has been carried out for cadmium single crystals, between room temperature and the melting point. It was found that the difference $\delta =\frac{\Delta L}{L_0}-\frac{\Delta d}{d_0}$, in a given crystallographic direction, is a function of temperature which is sample dependent. The ratio $\frac{{\delta }_{\parallel }}{{\delta }_{\perp }}$, where $\parallel$ and $\perp$ refer to the directions parallel and perpendicular to the $c$ axis, respectively, was found to be sample and temperature dependent. The mole fraction of vacancies in equilibrium given by $C_{\nu }\mathrm{}\left(T\right)=2\mathrm{}{\delta }_{\perp }+{\delta }_{\parallel }$ was obtained from a sole single crystal with well-chosen orientation. At the melting point $C_{\nu }$ has the value 4.5×${10}^{-4\mathrm{}}$±0.5×${10}^{-4\mathrm{}}$ and $C_{\nu }\mathrm{}\left(T\right)\mathrm{}$ is consistent with an interpretation in terms of monovacancies only, with enthalpy and entropy of formation of 0.42±0.02 eV and $(0.5\mathrm{}\pm 0.1)k$, respectively.