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

The gyromagnetic ratio is the signed proportionality coefficient $\mathbf{\mu }=\gamma \mathbf{I}$ between a nuclear magnetic moment and its spin angular momentum. With the angular-frequency convention, ${\omega }_0=|\gamma |B_0$, and $\gamma$ is expressed in rad s${}^{-1}$ T${}^{-1}$.

For a positive gyromagnetic ratio the magnetic moment is parallel to the spin angular momentum; for a negative ratio it is antiparallel. The Hamiltonian ${\hat{H}}_Z=-\gamma B_0{\hat{I}}_z$ gives $E_m=-\gamma \hslash B_{0}m$. Thus $m=+1/2$ is lower for the proton, whereas $m=-1/2$ is lower for nitrogen-15. Their spin precession senses are opposite for a common positive field convention; measurable absorption frequencies and energy separations remain positive.

The energy separation is $\Delta E=\hslash |\gamma |B_0$. At 10 T this givesAt 300 K, $k_{B}T=4.142\times 10^{-21}\mathrm{J}=0.02585\mathrm{eV}$, so the splitting-to-thermal-energy ratios are $6.80\times 10^{-5}$ and $6.87\times 10^{-6}$. The two spin levels therefore have nearly equal populations. Their equilibrium population polarizations are approximately half these ratios, $3.40\times 10^{-5}$ and $3.44\times 10^{-6}$, explaining weak thermal NMR sensitivity despite well-defined resonant transition energies.

NMR probes the chosen isotope's local environments and motion in the bulk without electrical contacts. Line narrowing, relaxation and exchange measurements can expose hopping across a broad range of timescales, separate sites with resolved shifts, and provide microscopic activation energies. Pulsed-field-gradient measurements can sometimes measure longer-range displacement directly; local relaxation measurements usually infer it through a model.

Limitations include low sensitivity for dilute or low-abundance nuclei, overlap of inequivalent sites, quadrupolar broadening for spins above one-half, and contributions from paramagnetic impurities or other relaxation mechanisms. Heating can change defect populations or phase composition. Most importantly, local hopping need not produce long-range ionic conductivity: back-and-forth motion, correlated jumps and trapped ions can relax spins without carrying charge over macroscopic distances. Connecting $T_1$ to a diffusion coefficient requires an explicit jump model and comparison with a bulk transport measurement.

For an exponentially correlated local field, a single-frequency illustrative relaxation rate is $T_1^{-1}=C{\tau }_c/(1+{\omega }_0^2{\tau }_c^2)$. Its derivative with respect to ${\tau }_c$ changes sign at ${\omega }_0{\tau }_c=1$. Slow motion gives $T_1\propto {\tau }_c$, and fast motion gives $T_1\propto 1/{\tau }_c$; an Arrhenius plot consequently has a minimum of $T_1$ between opposite-slope limits.

For identical spin-one-half nuclei with homonuclear dipolar relaxation, include both transition frequencies:Writing $x={\omega }_0{\tau }_c$, the maximum of the rate satisfiesLet $\ell$ be jump length, ${\tau }_j$ the total waiting time, ${\tau }_c=c{\tau }_j$, and $f_D$ the diffusion jump correlation factor. In three dimensions $D=f_D{\ell }^2/(6{\tau }_j)=f_{D}c{\ell }^2/(6{\tau }_c)$. Combining this with $D=D_0\exp [-E_D/(k_{B}T)]$ givesThe numerical constant is one for the single-frequency illustration and 0.616 for the homonuclear dipolar model. Assumptions are a single activated process, exponential field correlations, a temperature-independent prefactor and coupling amplitude, and an established relation between local correlation and translational motion. Defect creation can enter the measured activation energy; it cannot automatically be identified with a bare migration barrier. The fluctuation-spectrum approach originates in the relaxation theory of Nicolaas Bloembergen, Edward Mills Purcell and Robert Pound.^[ref]

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.