Friedel's law

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Friedel's law, named after Georges Friedel, is a property of Fourier transforms of real functions.^[1]

Given a real function $f(x)$ , its Fourier transform

$F(k)=\int _{{-\infty }}^{{+\infty }}f(x)e^{{ik\cdot x}}dx$

has the following properties.

$F(k)=F^{*}(-k)\,$

where $F^{*}$ is the complex conjugate of $F$ .

Centrosymmetric points $(k,-k)$ are called Friedel's pairs.

The squared amplitude ( $|F|^{2}$ ) is centrosymmetric:

$|F(k)|^{2}=|F(-k)|^{2}\,$

The phase $\phi$ of $F$ is antisymmetric:

$\phi (k)=-\phi (-k)\,$ .

Friedel's law is used in X-ray diffraction, crystallography and scattering from real potential within the Born approximation. Note that a twin operation (aka Opération de maclage) is equivalent to an inversion centre and the intensities from the individuals are equivalent under Friedel's law.^[2]^[3]^[4]

References

↑ Friedel G (1913). "Sur les symétries cristallines que peut révéler la diffraction des rayons Röntgen". Comptes Rendus 157: 1533–1536.
↑ Nespoloa N, Giovanni Ferraris G (2004). "Applied geminography - symmetry analysis of twinned crystals and definition of twinning by reticular polyholohedry". Acta Crystal A 60 (1): 89–95. doi:10.1107/S0108767303025625.
↑ Friedel G (1904). "Étude sur les groupements cristallins". Extract from Bullettin de la Société de l'Industrie Minérale, Quatrième série, Tomes III et IV. Saint-Étienne: Societè de l'Imprimerie Thèolier J. Thomas et C.
↑ Friedel G. (1923). Bull. Soc. Fr. Minéral. 46:79-95.

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