Friedel's law

Friedel's law, named after Georges Friedel, is a property of Fourier transforms of real functions. ^[1]

Given a real function ${\displaystyle f(x)}$, its Fourier transform

${\displaystyle F(k)=\int _{-\infty }^{+\infty }f(x)e^{ik\cdot x}dx}$

has the following properties.

• ${\displaystyle F(k)=F^{*}(-k)\,}$

where ${\displaystyle F^{*}}$ is the complex conjugate of ${\displaystyle F}$.

Centrosymmetric points ${\displaystyle (k,-k)}$ are called Friedel's pairs.

The squared amplitude (${\displaystyle |F|^{2}}$) is centrosymmetric:

• ${\displaystyle |F(k)|^{2}=|F(-k)|^{2}\,}$

The phase ${\displaystyle \phi }$ of ${\displaystyle F}$ is antisymmetric:

• ${\displaystyle \phi (k)=-\phi (-k)\,}$.

Friedel's law is used in X-ray diffraction, crystallography and scattering from real potentials 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 individual reflections are equal under Friedel's law. ^{[2] [3] [4]}

References

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