Ionic Conductivity of Impure Polar Crystals

Experimental determinations of the ionic conductivity of impure or "doped" alkali halide crystals are often used to find the mobility of the cation vacancies. For this purpose it is important to know to what extent the vacancies and the impurity ions have associated together to form neutral "complexes," which do not contribute to the conductivity. Previous interpretations of the experimental data have relied largely on the law of mass action to give the degree of association, but have neglected the long-range Coulomb interactions between the unassociated impurity ions and vacancies. The effect of these interactions on the calculated degree of association and upon the dc conductivity is examined in this paper. The interactions are shown to be significant even in systems containing impurities in concentrations of no more than one part in ${10}^4$. A new analysis of the results obtained by Etzel and Maurer for the system NaCl+Cd${\mathrm{Cl}}_2$, leads to a binding energy of about 0.35 ev for the cadmium ion-vacancy complex. The mobility of the sodium ion vacancy is found to be about 30 percent larger, at all temperatures, than was inferred by Etzel and Maurer, using a simpler theory to interpret the data. The activation energy for the vacancy motion remains however at about 0.9 ev.