Effect of heterovalent impurities codiffusing with monovalent tracers in ionic crystals

Tracer-diffusion measurements have been made for ${}_{}{}^{45}{}_{}{}^{}\mathrm{Ca}$ and ${}_{}{}^{22}{}_{}{}^{}\mathrm{Na}$ codiffusing into pure NaCl as a function of temperature, initial calcium concentration, and diffusion anneal time. The measured penetration profiles deviate markedly from the Gaussian dependence expected for diffusion from a thin source. The curvature is caused by each divalent Ca ion introducing an extra "extrinsic" vacancy into the diffusion zone in order to maintain charge neutrality. In such cases, the limiting slope has been shown not to be a valid measure of the tracer diffusivity. Computer fits have been generated by a model which corrects for the effect and is found to give excellent agreement with the data except for the most extreme cases. The model can find the initial concentration of the impurity and the diffusivity of the impurity within a factor of 2 from the penetration profile for the monovalent tracer. The effect enters to first order into the measurement of the isotope effect in ionic crystals and can result in major errors in the measured isotope effect for even small deviations from a Gaussian profile. The diffusion coefficients for ${}_{}{}^{22}{}_{}{}^{}\mathrm{Na}$ diffusing in NaCl over the temperature range of (575-775)°C are given by $D_{\mathrm{Na}}=73\mathrm{}\exp (-2.04\mathrm{}\pm 0.02 \frac{\mathrm{eV}}{\mathrm{kT}})$ ${\mathrm{cm}}^2$/sec, and for ${}_{}{}^{45}{}_{}{}^{}\mathrm{Ca}$, $D_{\mathrm{Ca}}=0.23\mathrm{}\exp (-1.61\mathrm{}\pm 0.03 \frac{\mathrm{eV}}{\mathrm{kT}})$ ${\mathrm{cm}}^2$/sec.