Effect of Surface Impurities on Tracer Diffusion in Insulators

In self-diffusion experiments performed on insulators in which a radioactive tracer is deposited as a thin source, heterovalent impurities, initially present on the sample surface or in the tracer solution, may diffuse into the sample along with the tracer. This effect may be important, even for very small amounts of impurity, since each heterovalent ion entering the lattice automatically introduces one or more extrinsic vacancies to maintain charge neutrality. If the extrinsic-vacancy concentration is comparable to the thermal-equilibrium concentration, this effect can cause a marked deviation from the expected Gaussian behavior. A model is developed for iteratively solving the nonlinear diffusion equation for the case of an unknown divalent impurity co-diffusing with a monovalent tracer in an alkali-halide lattice. The solutions show a strong curvature in the tracer profiles, the amount of curvature depending on the amount and relative diffusivity of the impurity, and on the diffusion time. The model permits fairly precise determination of the "true" tracer diffusion coefficient and approximate values for the concentration and diffusivity of the impurity. Experimental results are presented which show good agreement between the predicted time dependence of the model and experiment for "carrier-free" Na²² diffusing in NaCl.