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

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.

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.