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

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

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 twin operation (aka 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

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". Acta Crystallogr A 60 (1): 89–95. doi:10.1107/S0108767303025625.
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.