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With the explicitly stated convention $H_R=D_{\mathrm{tr}}/D_{\sigma }$, the tracer and collective diffusion coefficients are related by $D_{\sigma }=D_{\mathrm{tr}}/H_R$. The Nernst-Einstein relation then gives $\sigma =n{q}^2{D}_{\mathrm{tr}}/(H_R{k}_{B}T)$. State the convention, since reciprocal definitions can otherwise cause confusion.

Assume one temperature-independent activation energy over this interval and an unchanged dopant-controlled carrier population. Write the Arrhenius equation as $D=D_0\exp [-Q/(k_{B}T)]$. The two measurements giveUsing $D(T)=10^{-13}\exp [(Q/k_B)(1/1000-1/T)]$ yieldsThe Nernst-Einstein relation includes the factor $1/T$: $\sigma =n{q}^{2}D/(k_{B}T)$. For fixed $n$ and a fixed correlation correction,Using conductivity proportional to $D$ alone would miss the temperature factor. Carrier association, a changing vacancy population or a changing diffusion correlation factor would invalidate this simple extrapolation.

The numerical vacancy fraction depends on what the supplied diffusivity describes. In the usual effective mobile-vacancy calculation, the $0.9236$ carriers per cell are oxygen vacancies, each of effective charge $+2e$. There are eight oxygen sites, so that convention givesThere is a real qualification: the printed description calls the measured particles oxygen carrier ions. If $D$ is their tracer diffusivity and every occupied oxygen site is counted as an uncorrelated carrier, the inversion instead gives $N_O=0.9236$, formally $f_v=1-N_O/8=0.8845$. Such an enormous oxygen deficiency is inconsistent with the ordinary lightly doped fluorite picture; it is not a credible deduction of $88.5\%$ vacancies in stabilized zirconia.

More generally the conductivity involves a collective diffusion coefficient, whereas tracer diffusion includes jump correlations. With the explicitly defined convention $H_R=D_{\mathrm{tr}}/D_{\sigma }$,Without a carrier definition and the relevant correlation information, the literal tracer data do not uniquely establish the vacancy fraction. The consistent vacancy-carrier interpretation gives the boxed $11.5\%$ estimate. Interpreting the same measured $D$ as an oxygen-ion tracer diffusivity requires the collective/tracer conversion above; ordinary Nernst-Einstein relation arithmetic cannot identify these two carrier populations. This states the model assumption needed for the numerical result without treating the physically implausible uncorrelated oxygen-ion inversion as an established material composition.

A larger barium cation expands the fluorite structure and can widen the constrictions through which fluoride ions move. A plausible explanation is therefore a smaller ionic migration barrier and greater fluoride mobility. In a simple dilute-carrier description, ${\sigma }_{\mathrm{ion}}=n|q|\mu$, or $n{q}^{2}D/(k_{B}T)$ when an uncorrelated Nernst-Einstein relation applies. fluoride vacancies and interstitial defects, including those generated as Frenkel pairs, provide possible carriers.

More open migration pathways can raise the conductivity, but ion size by itself is not a proof: defect formation free energies set $n$, migration free energies set mobility, and correlated hopping can alter the simple relation. A controlled comparison would use crystals with comparable impurity content, thermal history and electrode conditions, and measure conductivity over temperature to separate plausible activation behaviour from extrinsic defect effects.

An oxygen vacancy has effective carrier charge $q_v=+2e$. Combining its density with its own diffusion coefficient in the Nernst-Einstein relation givesat $T=873.15\mathrm{K}$. The oxygen ions move oppositely to the vacancies and carry the same conventional current. This estimate assumes independently migrating vacancies, negligible correlation corrections, and negligible electronic contribution to the reported ionic conductivity.