Measurement of the Equilibrium Concentration of Lattice Vacancies in Silver near the Melting Point

Relative changes in macroscopic length, $\frac{\Delta L}{L}$, and x-ray lattice parameter, $\frac{\Delta a}{a}$, of a 99.999% silver bar have been measured from 14 to 956°C, using a rigid parallel pair of filar micrometer microscopes and a rotating single-crystal x-ray method. The expansions agree within experimental precision (about 1:${10}^5$) at the lower temperatures. At the higher temperatures ($\frac{\Delta L}{L}-\frac{\Delta a}{a}$) progressively increases, showing that atomic sites are added, corresponding to the thermal generation of vacancy-type defects. The concentration of added sites, $\frac{\Delta N}{N}=3(\frac{\Delta L}{L}-\frac{\Delta a}{a})$, at the melting point (obtained by a 4° extrapolation) is (1.7±0.5) ×${10}^{-4\mathrm{}}$. This value is only $\frac{1}{5.5}$ of the corresponding value previously found for aluminum by the same technique. For reasonable values of the binding energies of vacancy aggregates, it is concluded that more than 90% of these extra sites are present as single vacancies. For an entropy of formation of $(1.5\mathrm{}\pm 0.5)k$ an energy of formation for monovacancies of 1.09±0.10 ev is obtained. This formation energy is slightly larger than half the activation energy for self-diffusion, as similar measurements have shown for aluminum, and as quenching experiments have indicated for gold.