Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2023 / Paper 1 / Section A(i) / 1

Moiré fringes arise when two overlapping periodic structures contribute closely spaced reciprocal lattice vectors to the transmission electron microscopy image. Their relative phase varies slowly across the image, producing a coarse modulation with a period much larger than either atomic spacing. The structures may have different lattice spacings, a small relative rotation, or both. The modulation is an interference or beat phenomenon; its spacing is not itself a new atomic-plane spacing. Its orientation is perpendicular to the difference vector of the two contributing projected reciprocal-lattice vectors.

Let the relevant projected reciprocal lattice vectors have magnitudes $g_1=1/d_1$ and $g_2=1/d_2$, with angle $\theta$ between them. Subtracting their phases gives the slow spatial modulation $2\pi ({\mathbf{g}}_1-{\mathbf{g}}_2)\cdot \mathbf{r}$. The moiré fringe spacing formula is thereforeFor parallel planes, $\theta =0$ and $D=d_1{d}_2/|d_1-d_2|$. For equal spacings, $D=d/[2|\sin (\theta /2)|]\simeq d/|\theta |$ at a small angle measured in radians. The apparent divergence as both mismatch and rotation vanish means that there is no finite beat period; it does not predict an infinite number of visible fringes.

For each spacing pair, the largest geometrical period occurs with parallel projected reciprocal vectors. The closest pair is $d_{\mathrm{Cu}}=0.208\mathrm{nm}$ and $d_{\mathrm{substrate}}=0.212\mathrm{nm}$, givingHowever, the crystal is only $10\mathrm{nm}$ across. It contains at most $10/11.024\simeq 0.91$ of this period, so the largest formal period cannot appear as a repeated fringe train inside one crystal. It can instead give a broad variation of contrast. A period of order the particle diameter is the largest directly sampled scale, and even that is poorly constrained by one particle.

The distinction matters if “see” means a complete repeat. Of the exactly parallel spacing pairs, the largest other period is approximately $2.09\mathrm{nm}$ from the $0.208$ and $0.231\mathrm{nm}$ pair. If relative rotation is continuously variable, the closest pair can instead be rotated slightly to shorten its period below $10\mathrm{nm}$, so $2.09\mathrm{nm}$ is not a universal maximum visible period for arbitrary orientations.

A coarse fringe in a high-resolution transmission electron microscopy image need not directly identify the lattice spacing of a new phase. Check whether overlapping crystals, a particle and its substrate, or differently oriented domains can produce the corresponding moiré fringe. Compare the underlying reciprocal-space peaks and their difference, use the particle boundary to constrain how many periods are actually sampled, and examine neighbouring or differently oriented regions. For the closest pair here, fitting a periodicity from less than one complete period is particularly unreliable. Distinguishing lattice contrast from a beat envelope prevents misidentifying a phase or over-interpreting local strain.