Friedel's law explained

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 transformF(k) = \int ^{+inf ty} _{-inf ty} f(x)e ^{i} 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 (Opération de maclage) is equivalent to an inversion centre and the intensities from the individuals are equivalent under Friedel's law.^{[2] [3] [4]}

Notes and References

1. Friedel G . Sur les symétries cristallines que peut révéler la diffraction des rayons Röntgen . Comptes Rendus. 157 . 1533–1536 . 1913.
2. Nespolo M, Giovanni Ferraris G . Applied geminography - symmetry analysis of twinned crystals and definition of twinning by reticular polyholohedry . Acta Crystallogr A . 60 . 1 . 89–95 . 2004 . 10.1107/S0108767303025625.
3. 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.
4. Friedel G. (1923). Bull. Soc. Fr. Minéral. 46:79-95.