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 Incoming links: Moiré fringe spacing formula

Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2023 / Paper 1 / Section A(i) / 1 / ii / Solution 2026-10-20

Let the relevant projected reciprocal lattice vectors have magnitudes

g_{1} = 1/ d_{1}

and

g_{2} = 1/ d_{2}

, with angle

θ

between them. Subtracting their phases gives the slow spatial modulation

2 π (g_{1} - g_{2}) \cdot r

. The moiré fringe spacing formula is therefore

D = \frac{1}{∣ g _{1} - g _{2} ∣} = \frac{d _{1} d _{2}}{d _{1}^{2} + d _{2}^{2} - 2 d _{1} d _{2} cos θ} .

For parallel planes,

θ = 0

and

D = d_{1} d_{2} /∣ d_{1} - d_{2} ∣

. For equal spacings,

D = d / [2∣ sin (θ /2) ∣] ≃ d /∣ θ ∣

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.

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