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 Generally in scattering theory and in particular in quantum mechanics, the Born approximation consists of taking the incident field in place of the total field as the driving field at each point in the scatterer. The Born approximation is named a ...

. 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