Past exam of the materials science course of the University of Cambridge / 2016 / Paper 1 / 2

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.

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 conventional cubic cell has volumeConsequently the same effective carrier count isA noninteger mean is normal for a doped crystal: it describes an ensemble average, not a requirement that every cell contain the same number of defects. The fluorite structure has four cation sites and eight oxygen sites in this conventional cell.

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.