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The density and diffusivity in $\sigma =n{q}^{2}D/(k_{B}T)$ must refer to the same carrier species and correlation convention. Oxygen vacancy diffusivity belongs with vacancy density; tracer diffusion of oxygen atoms is not automatically the vacancy diffusivity. A transport inversion may determine an effective density without uniquely determining a chemical defect concentration.

Electrical conductivity arising from motion of ions. In a dilute, uncorrelated vacancy model, a mobile oxygen vacancy has effective charge $+2e$ and moves opposite to the oxygen ions that fill it.

For an oxide ion, or an oxygen vacancy measured relative to the ideal lattice, the charge magnitude is $|q|=2e$. The independent-carrier Nernst-Einstein inversion at $1250\mathrm{K}$ givesHere $D$ and $n_{\mathrm{eff}}$ must describe the same carrier convention. If the quoted diffusivity is a mobile-vacancy diffusivity, this is the vacancy density. If it is literally an oxygen-ion tracer diffusivity, it is an effective oxygen-ion density under an uncorrelated model. Vacancy density cannot be substituted for oxygen-ion density while retaining the same diffusivity. This distinction is consequential in the last part.

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. Thus $11.5\%$ is a conditional effective-vacancy answer, rather than a convention-free inference. TODO: establish the intended carrier/diffusivity convention if an authoritative clarification becomes available.

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.

There are four cation sites per conventional unit cell, hence $0.4$ Gd substitutions per cell on average. Two substitutions of Gd${}^{3+}$ for Ce${}^{4+}$ require one oxygen vacancy for charge neutrality, so the cell contains $0.2$ vacancies. Equivalently, the composition is $\mathrm{C}{\mathrm{e}}_{0.9}\mathrm{G}{\mathrm{d}}_{0.1}{\mathrm{O}}_{1.95}$. ThereforeThis uses the supplied lattice parameter literally and treats it as the conventional fluorite-cell edge.

Each missing oxide ion leaves two units of excess positive effective charge, compensated by reducing two Ce${}^{4+}$ ions to Ce${}^{3+}$. For $0.2$ oxygen vacancies per conventional unit cell, $0.4$ of its four cerium ions must be trivalent: 10% Ce${}^{3+}$. This assumes the same number of cation sites per unit volume as in the doped calculation and that oxygen loss is compensated entirely by this cerium-valence change. If reduction changes the cell volume appreciably, matching the absolute vacancy density requires correcting the percentage for that change.