Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Paper 3 / Section D / 43

The conventional lattice-defect description is vacancy-mediated diffusion: an ion moves into a neighbouring vacancy and leaves a vacancy behind. sodium chloride is another familiar example. In a real alpha-alumina specimen the dominant species and transport path depend on oxygen chemical potential, dopants and grain boundaries, so this label should not be read as a universal microscopic assignment for every aluminium and oxygen diffusion experiment. A vacancy mechanism is different from an interstitial ion moving between otherwise empty interstitial sites.

Here $\mathbf{j}$ is electric current density, $n$ the number density of independently mobile ions, $Q$ their charge, $T$ absolute temperature and $k_B$ the Boltzmann constant. The sum runs over all nearest-neighbour destinations $j$ of site $i$, and ${\mathbf{r}}_{ij}$ is the displacement from $i$ to $j$. The applied field is $\mathcal{E}$, ${\omega }_0$ is an attempt frequency, and $\Delta E_{ij}$ is the zero-field reaction energy barrier for the indicated jump. With the normalization of the supplied equation,It has units s${}^{-1}$; it is not an equilibrium occupation probability. The current formula is the linear-response result of a small field-induced asymmetry in forward and reverse rates, requiring $|Q{\mathbf{r}}_{ij}\cdot \mathcal{E}|\ll k_{B}T$. Use its explicit factor $1/2$ consistently rather than absorbing it into a redefined jump rate.

Expanding the scalar product gives $j_{\alpha }=[n{Q}^2/(2k_{B}T)]{\sum }_{j}f{r}_{ij,\alpha }{\sum }_{\beta }{r}_{ij,\beta }{\mathcal{E}}_{\beta }$. Comparing coefficients of the independently variable field components yieldsThe tensor is symmetric and positive semidefinite because its contraction with any vector twice is a nonnegative sum of squares.

Set $F={\omega }_0{e}^{-\Delta E/(k_{B}T)}$ and $K=n{Q}^{2}F/(2k_{B}T)$. The six simple cubic neighbour vectors are $\pm a\hat{x}$, $\pm a\hat{y}$, $\pm a\hat{z}$. Each diagonal direction receives $2a^2$, and every off-diagonal product vanishes. ThereforeThe result uses $a$ as the nearest-neighbour distance specified here, not automatically as the conventional-cell edge for the other lattices.

The eight body-centred cubic neighbour vectors are $(a/\sqrt{3})(\pm 1,\pm 1,\pm 1)$ with independent signs. Their diagonal sums are $8a^2/3$ and the sign-paired off-diagonal sums cancel. HenceThe conventional cubic cell edge is $2a/\sqrt{3}$ in this comparison; using that edge as the hopping distance would spoil the stipulated equal-distance comparison.

The twelve face-centred cubic neighbours are $(a/\sqrt{2})(\pm 1,\pm 1,0)$ and permutations of the zero coordinate. Eight vectors have a nonzero given component, each contributing $a^2/2$, while off-diagonal terms cancel between signs. ThusFor a cubic neighbour shell this can also be checked from ${\sum }_j{r}_{j,\alpha }{r}_{j,\beta }=(z{a}^2/3){\delta }_{\alpha \beta }$ with coordination number $z=12$.

Yes, for all three ideal lattices under the stated assumptions. Cubic symmetry and equal rates make $\mathbf{\sigma }=\sigma I$ with $\sigma >0$, so $\mathbf{j}=\sigma \mathcal{E}$. Positive ions drift along the field. The current coefficient contains $Q^2$, so negative charge carriers would also produce a conventional current along the field, although their particle drift would reverse. Anisotropic barriers or a field strong enough to invalidate linear response would require a new calculation.

At fixed number density, attempt frequency, nearest-neighbour distance and barrier, the conductivities have ratio ${\sigma }_{\mathrm{sc}}:{\sigma }_{\mathrm{bcc}}:{\sigma }_{\mathrm{fcc}}=1:4/3:2$. Thus the face-centred cubic array is largest. Numerically,The small value is consistent with the low supplied carrier number density and strongly activated hopping. This comparison holds $n$ fixed rather than comparing fully occupied crystal structures, whose site densities at the same neighbour distance differ.