Measurements of Equilibrium Vacancy Concentrations in Aluminum

Abstract

Measurements of change in length and change in lattice parameter were made at identical temperatures on 99.995% aluminum in the temperature range 229 to 656\ifmmode^\circ\else\textdegree\fi{}C. Length changes, $\ensuremath{\Delta}L$, were measured on an unconstrained horizontal bar sample using a rigid pair of filar micrometer microscopes. X-ray lattice parameter changes, $\ensuremath{\Delta}a$, were observed using a high-angle, back-reflection, rotating-single-crystal technique. The measurements are compared to earlier work. The relative expansions $\frac{\ensuremath{\Delta}L}{L}$ and $\frac{\ensuremath{\Delta}a}{a}$ were equal within about 1:${10}^{5}$ from 229 to 415\ifmmode^\circ\else\textdegree\fi{}C. At higher temperatures additional atomic sites were found to be generated: the difference between the two expansions could be represented by $3(\frac{\ensuremath{\Delta}L}{L}\ensuremath{-}\frac{\ensuremath{\Delta}a}{a})=\mathrm{exp}(2.4)\mathrm{exp}(\ensuremath{-}0.76 \mathrm{ev}/kT)$. At the melting point (660\ifmmode^\circ\else\textdegree\fi{}C) the equilibrium concentration of additional sites is $3(\frac{\ensuremath{\Delta}L}{L}\ensuremath{-}\frac{\ensuremath{\Delta}a}{a})=9.4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$. This result is independent of the detailed nature of the defects, for example, the lattice relaxation or degree of association. The nature of the defects is considered and it is concluded that they are predominantly lattice vacancies; it is estimated that the divacancy contribution at the melting point may well be less than about 15%, corresponding to a divacancy binding energy \ensuremath{\leqslant} 0.25 ev. The observed formation energy agrees with the values obtained by quenching techniques and by interpretation of the high-temperature electrical resistivity of identical material by Simmons and Balluffi. The present work is the first direct measurement of formation entropy; the value is near that expected from theoretical considerations. The contribution of the thermally generated defects to other physical properties at high temperatures is considered briefly.