Friedel's law

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^{i k \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.
↑ Nespolo M, Giovanni Ferraris G (2004). "Applied geminography - symmetry analysis of twinned crystals and definition of twinning by reticular polyholohedry". Acta Crystallogr 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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