Friedel's law, named after

Georges Friedel Georges Friedel (19 July 1865 – 11 December 1933) was a French Mineralogy, mineralogist and Crystallography, crystallographer. Life Georges was the son of the chemist Charles Friedel. Georges' grandfather was Georges Louis Duvernoy, Louis Geor ...

, is a property of

Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...

s of real functions. Given a real function

f(x)

, its

F(k)=\int^_f(x)e^dx

has the following properties. *

F(k)=F^*(-k) \,

where

F^*

is the complex conjugate of

F

Centrosymmetric In crystallography, a centrosymmetric point group contains an inversion center as one of its symmetry elements. In such a point group, for every point (x, y, z) in the unit cell there is an indistinguishable point (-x, -y, -z). Such point groups ...

points

(k,-k)

are called Friedel's pairs. The squared amplitude (

, F, ^2

) is centrosymmetric: *

, F(k), ^2=, F(-k), ^2 \,

The phase

\phi

F

is antisymmetric: *

\phi(k) = -\phi(-k) \,

. Friedel's law is used in

X-ray diffraction X-ray crystallography is the experimental science determining the atomic and molecular structure of a crystal, in which the crystalline structure causes a beam of incident X-rays to diffract into many specific directions. By measuring the angles ...

crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wor ...

and scattering from real potential within the Born approximation. Note that
twin operation
( ''Opération de maclage'') is equivalent to an inversion centre and the intensities from the individuals are equivalent under Friedel's law.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.

References

{{Crystallography Fourier analysis Crystallography