Measurement of Equilibrium Concentrations of Vacancies in Copper

Abstract

The net added fraction of thermally-generated atomic sites, $\frac{\ensuremath{\Delta}N}{N}$, has been measured, during equilibrium heating to near the melting point, in a coarse-grained copper bar of 99.999% purity. The linear thermal expansion, $\frac{\ensuremath{\Delta}L}{L}$, and the x-ray lattice expansion, $\frac{\ensuremath{\Delta}a}{a}$, as measured simultaneously, agree within the measurement error of \ifmmode\pm\else\textpm\fi{}1\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}5}$ throughout the temperature interval between 56 and 850\ifmmode^\circ\else\textdegree\fi{}C. At 1000 and at 1075\ifmmode^\circ\else\textdegree\fi{}C, $\frac{\ensuremath{\Delta}N}{N}=3(\frac{\ensuremath{\Delta}L}{L}\ensuremath{-}\frac{\ensuremath{\Delta}a}{a})$ is (0.9\ifmmode\pm\else\textpm\fi{}0.5)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}4}$ and (1.9\ifmmode\pm\else\textpm\fi{}0.5)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}4}$, respectively. Using expected limits for the binding energies of vacancy clusters, it is concluded that $\frac{\ensuremath{\Delta}N}{N}$ consists almost entirely of monovacancies. An assumed value of $(1.5\ifmmode\pm\else\textpm\fi{}0.5)k$ for the formation entropy, combined with these concentrations, yields a monovacancy formation energy of (1.17\ifmmode\pm\else\textpm\fi{}0.11) eV. The ratio of this formation energy to the activation energy for self-diffusion is therefore about 0.57, near the ratios found for the other noble metals. The results imply a monovacancy migration energy of (0.88\ifmmode\pm\else\textpm\fi{}0.13) eV. The present results are compared with the results of a wide variety of other investigations of defects in copper which include: (1) theoretical calculations, (2) quenching and annealing, (3) thermal diffusion, (4) annealing after irradiation, and (5) annealing after cold work. In certain cases apparent disagreement is found. However, it is concluded that in no case is there a sufficiently firm body of experimental data available either to confirm or to contradict the present monovacancy data. The need for further definitive experiments in these areas is emphasized.