Past exam of the condensed matter physics major option course of the University of Oxford / 2016 / 1 / b

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- Solution b

Solution

Use fractional coordinates and the structure factor convention

F (h) = \sum_{j} f_{j} e^{2 πi h \cdot r_{j}}, I (h) \propto ∣ F (h) ∣^{2} .

For real atomic scattering factors,

F (- h) = F (h)^{*}

, immediately proving Friedel's law. If an inversion centre is chosen as origin, each site at

r_{j}

has an identical partner at

- r_{j}

, with the same possibly complex atomic scattering factor. Each pair contributes

2 f_{j} cos (2 π h \cdot r_{j})

; sites fixed by inversion modulo a lattice translation also have even phase factors at reciprocal-lattice vectors. Hence

F (- h) = F (h), I (- h) = I (h)

even for scalar anomalous X-ray scattering in a centrosymmetric structure. This last equality is an equality of amplitudes, not generally complex conjugacy.

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