Friedel's law

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^{ik\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 (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" (PDF). Acta Crystallogr A. 60 (1): 89–95. Bibcode:2004AcCrA..60...89N. doi:10.1107/S0108767303025625. PMID 14691332.
^ 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.

Crystallography

Key concepts

Structure

Systems	Cubic Hexagonal Monoclinic Orthorhombic Tetragonal Triclinic

Phase
transition

Defects

Grain boundary	Disclination CSL Growth Abnormal growth

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