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| '''Friedel's law''', named after [[Georges Friedel]], is a property of [[Fourier transform]]s of real functions.<ref name="Friedel1913">{{cite journal |author=Friedel G |title=Sur les symétries cristallines que peut révéler la diffraction des rayons Röntgen |journal=Comptes Rendus|volume=157 |pages=1533–1536 |year=1913}}</ref>
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| Given a real function <math>f(x)</math>, its [[Fourier transform]]
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| :<math>F(k)=\int^{+\infty}_{-\infty}f(x)e^{i k \cdot x }dx</math>
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| has the following properties.
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| *<math>F(k)=F^*(-k) \,</math>
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| where <math>F^*</math> is the complex conjugate of <math>F</math>. | |
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| [[Centrosymmetric]] points <math>(k,-k)</math> are called [[Friedel's pairs]].
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| The squared amplitude (<math>|F|^2</math>) is centrosymmetric:
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| * <math>|F(k)|^2=|F(-k)|^2 \,</math>
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| The phase <math>\phi</math> of <math>F</math> is [[antisymmetric]]:
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| * <math>\phi(k) = -\phi(-k) \,</math>.
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| 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.<ref name="Nespoloa2004">{{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 |doi=10.1107/S0108767303025625}}</ref><ref name="Friedel1904">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.</ref><ref name="Friedel1923">Friedel G. (1923). ''Bull. Soc. Fr. Minéral.'' '''46''':79-95.</ref>
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| ==References==
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| {{reflist}}
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| [[Category:Fourier analysis]]
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| [[Category:Crystallography]]
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Latest revision as of 17:03, 11 December 2014
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