Equilibrium vacancy concentrations in copper investigated with the absolute technique.

Abstract

The molar fraction of thermally generated vacant lattice sites ${\mathit{N}}_{\mathit{v}}$ has been measured during thermal equilibrium up to temperatures near the melting point in 99.9998%-grade copper with an absolute technique. A high-resolution differential dilatometric experiment is performed. The thermal expansion of the lattice \ensuremath{\Delta}a(T)/${\mathit{a}}_{0}$ is determined by the Debye-Scherrer technique. The macroscopic expansion \ensuremath{\Delta}L(T)/${\mathit{L}}_{0}$ is measured with a special laser interferometer simultaneously as a function of temperature in the same specimen. The absolute concentration of equilibrium vacancies is obtained from ${\mathit{N}}_{\mathit{v}}$(T)=3[\ensuremath{\Delta}L(T)/${\mathit{L}}_{0}$-\ensuremath{\Delta}a(T)/${\mathit{a}}_{0}$]. The lattice expansion \ensuremath{\Delta}a(T)/${\mathit{a}}_{0}$ can be determined with a relative accuracy of 2\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}5}$. At high temperatures the error of the Laser interferometer is 2\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}6}$ for each point, respectively, equilvalent to an absolute length variation of 0.04 \ensuremath{\mu}m. An extrapolation of ${\mathit{N}}_{\mathit{v}}$ up to the melting point ${\mathit{T}}^{\mathit{m}}$ leads to ${\mathit{N}}_{\mathit{v}}$(${\mathit{T}}^{\mathit{m}}$)=(0.76\ifmmode\pm\else\textpm\fi{}0.03)\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}3}$. The usual Arrhenius evaluation gives a formation enthalpy ${\mathit{H}}_{\mathit{v}}^{\mathit{f}}$=(1.19\ifmmode\pm\else\textpm\fi{}0.03) eV and a formation entropy ${\mathit{S}}_{\mathit{v}}^{\mathit{f}}$=(3.0\ifmmode\pm\else\textpm\fi{}0.3)k.