Friedel's law
Friedel's law, named after Georges Friedel, is a property of Fourier transforms of real functions.[1] Given a real function [math]\displaystyle{ f(x) }[/math], its Fourier transform
- [math]\displaystyle{ F(k)=\int^{+\infty}_{-\infty}f(x)e^{i k \cdot x }dx }[/math]
has the following properties.
- [math]\displaystyle{ F(k)=F^*(-k) \, }[/math]
where [math]\displaystyle{ F^* }[/math] is the complex conjugate of [math]\displaystyle{ F }[/math].
Centrosymmetric points [math]\displaystyle{ (k,-k) }[/math] are called Friedel's pairs.
The squared amplitude ([math]\displaystyle{ |F|^2 }[/math]) is centrosymmetric:
- [math]\displaystyle{ |F(k)|^2=|F(-k)|^2 \, }[/math]
The phase [math]\displaystyle{ \phi }[/math] of [math]\displaystyle{ F }[/math] is antisymmetric:
- [math]\displaystyle{ \phi(k) = -\phi(-k) \, }[/math].
Friedel's law is used in X-ray diffraction, crystallography and scattering from real potential within the Born approximation. Note that a twin operation (a.k.a. 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. http://hal.archives-ouvertes.fr/docs/00/13/05/48/PDF/AppliedGeminography_gc5001_.pdf.
- ↑ 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.
 |  |
