Equilibrium Defect Concentration in Crystalline Sodium

Abstract

The thermal expansion of high-purity sodium crystals has been measured from -25\ifmmode^\circ\else\textdegree\fi{}C up to the melting point by precision interferometric and x-ray techniques. Above 15\ifmmode^\circ\else\textdegree\fi{}C the two expansion curves diverge in a manner indicating a predominance of vacancy-type defects. The difference at the melting point is equivalent to a net vacancy concentration of 7.5\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}4}$. Within experimental error, the divergence between the curves varies exponentially with reciprocal temperature and, if interpreted as due only to vacancies, yields a formation energy ${{E}_{1v}}^{f}$ of 0.42\ifmmode\pm\else\textpm\fi{}0.03 eV and a formation entropy: ${{S}_{1v}}^{f}=(5.8\ifmmode\pm\else\textpm\fi{}1.1)k$ ($k$ is Boltzmann's constant). This value of ${{E}_{1v}}^{f}$ accounts for almost all of the self-diffusion activation energy (0.45\ifmmode\pm\else\textpm\fi{}0.01 eV) and therefore leads to a very low value for the vacancy migration energy ${{E}_{1v}}^{m}$. Both the high value of ${{S}_{1v}}^{f}$ and the low value of ${{E}_{1v}}^{m}$ are consistent with the existence of a large lattice relaxation around a vacancy. Other possible interpretations of the results are also considered, involving the presence of divacancies or interstitials in addition to the monovacancies. However, the totality of evidence available from the present and other work favors the interpretation that monovacancies are the dominant defect in sodium both in the present measurement and in self-diffusion. Discrepancies between the present results and previous work have been attributed to the irreversibility in the macroscopic expansion caused by an oxide coating.