Friedel's law

Friedel's law, named after Georges Friedel, is a property of Fourier transforms of real functions.cite journal |author=Friedel G |title=unknown title |journal=Comptes Rendus, Acad. Sci. (Paris) |volume=157 |pages=1533-1536 |year=1913]

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 [http://reference.iucr.org/dictionary/Twin_operation twin operation] (aka "Opération de maclage") is equivalent to an inversion centre and the intensities from the individuals are equivalent under Friedel's law.cite journal |author=Nespoloa N, Giovanni Ferraris G |title=Applied geminography - symmetry analysis of twinned crystals and definition of twinning by reticular polyholohedry |journal=Acta Crystal A |volume=60 |issue=1 |pages=89-95 |year=2004] 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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