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^{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[edit]

  1. ^ Friedel G (1913). "Sur les symétries cristallines que peut révéler la diffraction des rayons Röntgen". Comptes Rendus 157: 1533–1536. 
  2. ^ Nespolo M, 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. 
  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.