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 / 42 / f / Solution

The fluorine sites form a simple cubic sublattice of spacing $\ell =a/2=3.10\mathring{\mathrm{A}}$. Adopt uncorrelated nearest-neighbour hopping, identify the field-correlation time with the total jump waiting time, and use homonuclear dipolar relaxation: $f_D=c=1$ and $x_{\ast }=0.616$. ThenIf the simplified single-frequency condition ${\omega }_0{\tau }_c=1$ is used instead, the corresponding estimates are $1.01\times 10^{-12}{\mathrm{m}}^2{\mathrm{s}}^{-1}$ and $1.9\times 10^{-4}{\mathrm{m}}^2{\mathrm{s}}^{-1}$. These are explicit model-dependent estimates, not a uniquely measured tracer diffusivity: a different correlation-time-to-jump-time conversion or defect correlation factor multiplies them by $f_{D}c$. The lattice constant supplies a jump distance, not proof that all possible neighbouring sites are vacant.